Keywords: Accelerators, Cryogenics, Radiofrequency, Cryomodule, Modelling, Automation

Transmutation shall be used to stabilize the long-lived high-level waste (LL-HLW) stocks produced by the thermal-neutron power plants worldwide. This process can be carried out in an Accelerator-Driven System (ADS): a neutron spallation source, featuring a proton accelerator, driving the criticality of a dedicated fast-neutron nuclear reactor.

In Europe, the MYRRHA project (Multi-purpose hYbrid Research Reactor for High-tech Applications) has matured through several R&D programs for about 25 years and aims at being a pre-industrial ADS in the 2040's horizon. Its neutron source requires a proton energy of 600 MeV and a beam current of 4 mA in continuous wave mode, that only the most powerful superconducting particle accelerators can provide. The reactor operation also sets stringent requirements to the accelerator beam dynamics, which in turn, implies a yet unreached reliability of its sub-systems, even more challenging in view of an energy-sober facility. The MYRRHA initial 100-MeV accelerating section, MINERVA, is under construction in Belgium by SCK•CEN for a first beam in 2027: on top of supplying 2 target facilities, the purpose of MINERVA is to demonstrate the MYRRHA reliability.

The heart of the superconducting accelerating section is the cryomodule; a series of 30 identical units will be produced and aligned, following a modular strategy.

Puis, chacune des personnes des équipes cryogénie et RF de l'IJClab : je me souviendrai en particulier d'Hervé Saugnac pour ses conseils avisés de cryogéniste, et pour les discussions passionnées sur les origines du hip-hop. Ensuite, de Didier Grolet, un bricoleur professionnel hors pair d'une bienveillance sans fond. Puis, de Matthieu Pierens, aussi calé en instrumentation d'un cryomodule supra que sur un carbu de Peugeot 103. Ensuite, de Frédéric Chatelet qui manie les blagues et les infra du 106 avec la même adresse. Ensuite encore, de Christophe Joly avec qui les soirées à raisonner sur les diagrammes de Smith n'ont pas semblé si longues. Ensuite encore, de Nicolas Gandolfo et Thierry Pepin-Donat pour la coanimation de la fête de la science style « C'est pas sorcier ». Enfin, de Sébastien Blivet avec qui monter/démonter un cryomodule devient aussi facile que pour un meuble IKEA. Sans oublier Guillaume Mavilla pour sa bonne humeur sans égal lors des longues après-midis à fouiller dans la documentation du PLC. Toujours à l'IJClab, du côté LAL, je remercie Walid Kaabi pour m'avoir rapidement intégré aux problématiques coupleurs, ainsi qu'à Alice Thiebault et Christopher Magueur, avec qui démêler les fils d'une cinquantaine de thermocouples ou éponger les fuites d'eau de la RF devient presque un plaisir. Enfin, du côté Physique Nucléaire, je voudrais remercier Marc Ernoult pour les discussions sur les réacteurs ainsi que pour sa relecture. Au LPSC, je remercie Pierre-Olivier Dumont et Frédéric Bouly pour m'avoir accueilli plusieurs fois et mis le pied à l'étrier, ainsi que Yolanda Gomez-Martinez pour m'avoir « coaché » sur le multipacting. Ensuite, j'adresse bien sûr mes remerciements aux équipes du SCK•CEN, en particulier Daniel Berkowitz, Frank Marhauser et Adrian Fabich pour leur accueil et leur conseils avisés. Je remercie aussi les équipes du GANIL, de FREIA, d'ESS, du CERN ainsi que du CEA/DSBT pour le temps accordé lors de visites personnalisées et pour leur expérience partagée sur l'opération et la modélisation des cryomodules. Evidemment, je les adresse à Patxi pour son encadrement infaillible, sa confiance et la relecture minutieuse de ce manuscrit ; et à Gaël Sattonnay pour sa bienveillance au cours de la thèse et pour avoir accepté d'en être le directeur.

Enfin, pas les moindres, à ma femme chérie qui m'a toujours soutenu, ainsi qu'à ma fille qui a attendu que son papa finisse (presque) son manuscrit avant de naître.

Résumé : La transmutation est un procédé permettant de stabiliser les stocks de déchets à haute activité et à vie longue (HA-VL), produits dans le monde par les réacteurs nucléaires à neutrons lents, qui peut notamment être mis en oeuvre dans des réacteurs pilotés par accélérateur : une source de spallation, utilisant un accélérateur à proton, contrôlant la criticité d'un coeur à neutrons rapides dédié.

En Europe, le projet de développement d'un tel concept, MYRRHA (Multi-purpose hYbrid Research Reactor for High-tech Applications), fort de nombreuses phases de R&D depuis 25 ans, prévoit la mise en marche d'un démonstrateur préindustriel à horizon 2040. Sa source de neutrons nécessite un faisceau continu de proton d'énergie de 600 MeV et de courant 4 mA, que seuls les plus puissants accélérateurs de particule supraconducteurs peuvent fournir. Le fonctionnement du réacteur impose de surcroît des contraintes aiguës de dynamique faisceau, qui en retour, implique une fiabilité jamais atteinte des sous-systèmes accélérateurs, un défi accentué par le besoin d'un système sobre en énergie. La première section accélératrice de MYRRHA allant jusqu'à 100 MeV, MINERVA, est en construction en Belgique par le SCK•CEN pour un premier faisceau en 2027 : en plus de fournir un faisceau à 2 stations de cibles, MINERVA doit démontrer la faisabilité de MYRRHA.

Le cryomodule constitue le coeur de la section accélératrice ; suivant une stratégie modulaire, 30 exemplaires identiques seront produits et alignés. Le cryomodule héberge les cavités accélératrices radiofréquence supraconductrices, et concentre toutes les fonctions nécessaires à leur fonctionnement optimal.

L'exigence stricte de fiabilité combinée à la contrainte d'efficacité énergétique globale exige une approche système, plutôt qu'une optimisation individuelle des fonctions.

Pour permettre cette approche, cette thèse propose de développer un modèle de process détaillé du cryomodule, qui intègre les couplages forts entre les fonctions cryogénie, radiofréquence et système d'accord mécanique, leur dynamique interne ainsi que leurs systèmes de contrôle respectifs.

Les efforts conséquents de développement d'un tel modèle se justifient par l'ensemble des applications pendant la phase de prototypage de MINERVA, et par la perspective d'une exploitation à long terme de ce modèle jusqu'à la mise en marche de l'accélérateur et audelà.

Suivant une démarche ascendante, les modèles de composants ont été développés à partir de l'analyse approfondie de leurs caractéristiques réelles, telles que construits. Le jumeau numérique ainsi créé est utilisé en support des tests du prototype, pour mettre au point ses systèmes de contrôle en amont des essais et pour en analyser ensuite les performances ; en retour, le modèle est validé sur des données expérimentales. Enfin, en vue de la série, plusieurs applications du modèle sont illustrées : la mise à jour du design des composants, l'étude de la robustesse du contrôle global, et des techniques de contrôle avancées couplées aux méthodes de Machine Learning.

The cryomodule hosts the accelerating superconducting radiofrequency cavities and concentrates all the necessary functions for their optimal operation. The strict reliability target combined with the overall accelerator efficiency objectives require a global system approach rather than individual optimizations of those functions.

To support this approach, the thesis proposes to develop a detailed cryomodule process model, that implements the strong couplings between the cryogenic, radiofrequency and mechanical tuning functions, their respective internal dynamic as well as their control systems. The substantial initial effort to develop such a model is motivated by the broad application panel within the current MINERVA prototyping phase, and the perspective of a long-term exploitation until the commissioning and beyond.

Following a bottom-up model creation, individual component models based on thorough investigations of their actual as-manufactured design are instantiated. The created digital twin is used to support the prototype test phase, through control system virtual commissioning and modelbased performance analysis; it is in turn, validated against the experimental results. Finally, in view of the series, several practical applications are illustrated: component design updates, the study of the global control robustness and advanced controls coupled with machine-learning techniques.

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EXTENDED SUMMARY IN FRENCH

La transmutation est un procédé permettant de stabiliser les stocks de déchets à haute activité et à vie longue (HA-VL), produits dans le monde par les réacteurs nucléaires à neutrons lents, qui peut notamment être mis en oeuvre dans des réacteurs dédiés. Les systèmes classiques de contrôle de la réactivité ne pouvant pas fonctionner avec des déchets en forte concentration dans le coeur, la réactivité est ajustée continûment par une source externe et pilotable de neutrons, elle-même alimentée et contrôlée par un accélérateur à protons.

En Europe, le projet de développement d'un tel concept, MYRRHA (Multi-purpose hYbrid Research Reactor for High-tech Applications), fort de nombreuses phases de R&D depuis 25 ans, prévoit la mise en marche d'un démonstrateur préindustriel à horizon 2040. Sa source de neutrons nécessite un faisceau continu de proton d'énergie de 600 MeV et de courant 4 mA, que seuls les plus puissants accélérateurs de particule supraconducteurs peuvent fournir. Le fonctionnement du réacteur impose de surcroît des contraintes aiguës de dynamique faisceau, qui en retour, implique une fiabilité jamais atteinte des sous-systèmes accélérateurs, un défi accentué par le besoin d'un système sobre en énergie. La première section accélératrice de MYRRHA allant jusqu'à 100 MeV, MINERVA, est en construction en Belgique par le SCK•CEN pour un premier faisceau en 2027 : en plus de fournir un faisceau à 2 stations de cibles, MINERVA doit démontrer la faisabilité de MYRRHA.

Le cryomodule constitue le coeur de la section accélératrice ; suivant une stratégie modulaire, 30 exemplaires identiques seront produits et alignés. Le cryomodule héberge les cavités accélératrices radiofréquence supraconductrices, et concentre toutes les fonctions nécessaires à leur fonctionnement optimal. L'exigence stricte de fiabilité combinée à la contrainte d'efficacité énergétique globale exige une approche système, plutôt qu'une optimisation individuelle des fonctions.

Pour permettre cette approche, cette thèse propose de développer un modèle de process détaillé du cryomodule, qui intègre les couplages forts entre les fonctions cryogénie, radiofréquence et système d'accord mécanique, leur dynamique interne ainsi que leurs systèmes de contrôle respectifs. Les efforts conséquents de développement d'un tel modèle se justifient par l'ensemble des applications pendant la phase de prototypage de MINERVA, et par la perspective d'une exploitation à long terme de ce modèle jusqu'à la mise en marche de l'accélérateur et au-delà.

Le modèle de process cryogénique a été entièrement développé à partir de zéro ; il reprend néanmoins la démarche proposée dans d'autres thèses pour des usines cryogéniques : une approche ascendante du composant vers le système et un langage orienté objet. Le modèle de process RF met également en oeuvre cette approche mais réutilise des modèles existants développés pour le projet MYRRHA et mis à jour. Enfin, une attention particulière a été portée à la création des modèles de composants manquants au sein desquels les process cryogénique et RF sont fortement couplés.

Chaque composant est formé par un ensemble de blocs élémentaires dans lesquels sont implémentées les équations de la physique ; ces lois ont fait l'objet d'analyses critiques.

Ces blocs sont paramétrés sur la base de l'analyse approfondie de leurs caractéristiques réelles, telles que construits.

Le jumeau numérique est créé par l'assemblage des composants cryogéniques et RF entre eux. Il est utilisé en support des tests du prototype, pour mettre au point ses systèmes de contrôle en amont des essais et pour en analyser ensuite les performances ; en retour, le modèle est validé sur des données expérimentales. Les différences entre essais et modèles sont minimisées par des ajustements empiriques dont la vocation est d'orienter les futures études d'amélioration du modèle, et à terme, de devenir des modèles physiques.

Enfin, en vue de la série, plusieurs applications du modèle sont illustrées : la mise à jour du design des composants, l'étude de la robustesse du contrôle global, et des techniques de contrôle avancées couplées aux méthodes de Machine Learning. [START_REF] Bouly | R&D pour l'accélérateur du projet MYRRHA[END_REF] [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF], [145] AND [START_REF] Munoz | Spoke and other TEM-class superconducting cavities[END_REF] [START_REF] Kostin | UPDATE ON THE EUROPEAN XFEL RF POWER INPUT COUPLER[END_REF] AND ESS FROM [START_REF] Rampnoux | Design of 352.21 MHz RF Power Input Coupler and Window for the European Spallation Source Project (ESS)[END_REF] [START_REF] Chevret | Mesure de la réactivité de réacteurs sous-critiques pilotés par accélérateur par l'analyse d'expériences d'interruptions de faisceau programmées[END_REF] ESS [START_REF] Abderrahim | MYRRHA -A multipurpose fast spectrum research reactor[END_REF] AND LINAC4 [START_REF]La réversibilité Thème 1 -La place de Cigéo dans le dispositif de gestion des déchets[END_REF] ...................

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Nuclear waste classification

In the nuclear industry, the term waste refers to the process by-products that are not meant for reintroduction in the process cycle, and which cannot be discharged in the environment due to the residual radioactivity level. Radioactive wastes in fact refers to a whole family of diverse wastes: each differs by the type of radioactivity, the level and the half-life of the main radioactive isotopes [START_REF] Andra | Waste classification[END_REF].

The level of radioactivity, or activity, is measured by the number of radioactive disintegrations per second or becquerel per second (Bq/s). The radioactive dose absorbed by a material is measured in gray (Gy), which is equivalent to 1 J of radiation energy absorbed by 1 kg. Finally, the radiotoxicity measures the harmful effect caused to biological tissues by a radioactive element; it varies according to the radiation types, and is given in sievert (Sv) 1 . Radiotoxicity is purely related to radioactivity and has nothing to do with the chemical hazard of the element.

The different radiation types correspond to the decay process of an instable atom spontaneously returning to a lower-energy and more stable state:

• α-decay: loss of neutrons and protons in the form of a helium nucleus • β-decay: emission of an electron or positron by transforming a neutron in proton or vice-versa • γ-decay: emission of high-energy photons Additionally, in case of a nuclear fission (see next section), isolated neutrons can be emitted. Figure 1-1 shows the material penetrative power of the different radiation type. 1 As nuclear science is recent, less than one century old, the literature is full of former obsolete units, such as the curie for the activity (1 Ci = 3,7.1010 Bq), the rad for the dose (1 Gy = 100 rad) and the rem for the radiotoxicity (1 rem = 0,01 Sv) [START_REF] Irsn | Les unités de mesure[END_REF]. A sievert is a large unit; on average, our body annually receives about 2.4 mSv.

Figure 1-1: the different emission types and associated penetration powers from [3]

Finally, another parameter for nuclear waste classification is the half-life: it is the time for a radioactive element to lose half of its population by means of one of the radioactive decay types. This data indicates how long the substance is potentially harmful. One can note that after seven half-times, less than one percent of the initial radioactive population remains (and hence the radiation level).

Each country has its own classification, more or less equivalent; in France, the wastes are classified according to their activity, and to their radioactive half-life, as seen in Figure 12. The activity levels feature four thresholds: from Very Low-Level to High-Level Wastes (HLW). In France, 98% of the total radioactivity is concentrated in the High-Level wastes, although they represent a tiny fraction of the waste volume and mass. The main problem caused by HLW is the heat generation, making them unsuitable for compact disposal.

There is also a distinction between short half-lives (less than 31 years), considering the radioactivity is attenuated enough after ten half-lives (300 years) and longer half-lives or Long-Lived (LL), i.e. 100000 years or more, commensurate with geological times.

The interest of a classification is to adapt the storage solution according to the specificity of the radioactive element (activity and half-life). Obviously, the Long-Lived High-Level Wastes (LL-HLW) need to be specifically addressed, as discussed in part 0.

Radioactive wastes come from five economic sectors: electricity generation, research, defence, non-electronuclear industry and medical. The electricity generation sector generates the largest share of the waste, as seen in Figure 1-3. This share is even higher when considering solely the LL-HLW. Frédéric and his wife Irène Joliot-Curie (Marie Curie's daughter) announced in 1934 the successful fabrication of a radioactive atom that did not exist on its own in nature. Put in the present perspective, it means that radioactivity is a general material property: stars are the melting pots where all the complex atoms are formulated, as products of fusion reactions, from the primordial hydrogen and helium atoms. With some exceptions, all the atoms and their isotopes are created this way, the heaviest ones in the catastrophic end-of-life of massive stars (supernova); but the most unstable have disappeared today, through radioactive decay. What Irène and Frédéric Joliot-Curie discovered, is the possibility to recreate them.

The most stable elements belong to the so-called valley of stability, represented in Figure 1-4: for each element (categorized by Z, the proton number), there is only some stable isotopes. The isotope of an atom X is noted 𝑋 𝑍 𝐴 or simply 𝑋 𝐴 . the same specie (thus same Z), but with a different number of neutrons N; A is the mass number Z+N.

Figure 1-4: nuclides by half-life as a function of their atomic number Z and neutron number N; the blacks squares are the stable elements, from [START_REF]Valley of stability[END_REF] Right before World War II, Enrico Fermi in its laboratory in Roma, Otto Hahn and Lise Meitner in Berlin and finally the Joliot-Curie couple proceeded to the neutron bombardment of uranium, in view of obtaining transuranic 2 elements, uranium being the heaviest natural element. Otto Hahn, together with Fritz Strassman discovered that the bombarded uranium gives rise to two lighter elements. Frédéric Joliot-Curie brings in early 1939 the final evidence that the uranium splits in two products, a process called fission and illustrated in [START_REF]Laradioactivite[END_REF].

Together with Hans Halban and Lew Kowarski, Frédéric Joliot-Curie examined the possibility of a sustained chain reaction, given that 2-3 neutrons are emitted along with the fission products. In May 1939, they applied for two patents: on the one hand for a steady process to produce energy, on the other hand, for an explosive charge concept [START_REF]Laradioactivite[END_REF]. 2 Atoms which proton number Z is greater than the one of uranium In the US, the context of the war tremendously boosted the nuclear researches: the first nuclear reactor was started in Chicago by Enrico Fermi, then exiled by Mussolini, and the Manhattan project, directed by Robert Oppenheimer, nicknamed the "atomic bomb father", led to the complete destruction of Hiroshima and Nagasaki. Respectively "Little boy" and "fat man" were dropped on those cities for their terrible fate. The names refer to their radically different shape as a consequence of their specific fissile material: the former had a uranium load, and the latter, a plutonium one.

Indeed, whereas 235 U is the only fissile material in nature, two isotopes of plutonium ( 239 Pu and 241 Pu) are also fissile but need to be artificially created (all their isotopes have disappeared today) by neutron captures from 238 U. Likewise, another artificial fissile material 233 U can be obtained from thorium.

Building a sufficiently-concentrated fissile material core is the technological enabler for nuclear weapons, where the fissile material grade must reach 90%. In nature, the fissile 235 U isotope stands only for 0.7 % of the mass of the uranium ore, the rest being 238 U; the concentration is done via isotopic separation, that requires incommensurately energyconsuming centrifugal facilities. Synthesizing concentrated plutonium has a different trade-off: whereas the separation can be chemically done (at least partially), the hard work is the required breeding process in fast-reactors.

As an opponent to nuclear weapons like many of his pairs, Frédéric Joliot-Curie was banned by Charles de Gaulle from dealing with nuclear reactors in the early 50's. He could still conduct nuclear physics, yet with particle accelerators [START_REF] Joliot-Curie | [END_REF]. Irène and Frédéric built their laboratory in Orsay, a countryside they appreciated at the terminal station of the Paris-Sceaux line; the facility was granted in 2020 the name IJClab. Many family descendants and relatives (Joliot, Curie and Langevin) are still in activity in the laboratory and its neighbourhood.

In the early 70's, enabled by 20 years of intense researches in the development of the nuclear weapon, and in parallel, in the development of civil nuclear reactors, France made the choice of nuclear-only electricity production. In the frame of the Messmer plan [8], within 25 years, a massive effort led to the construction of 58 reactors. Worldwide, about 440 reactors are in operation, mainly in Europe, USA, Canada, Russia and China and stands for about 5 % of the total world energy mix, and 11 % of the sole electricity mix.

Compared to chemical energy (the physics of the electrons), the nuclear physics involves energies that are several orders of magnitude higher. As an illustration, a typical 1 GWe reactor requires a thermal power core of 3 GWth (with a Carnot efficiency of about 33 %) [START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF]. Each fission releasing about 200 MeV, the required number of fissions is thus ~3 10 27 , or about 1175 kg of fissioned material yearly. Over the last 20 years, France has yearly produced ~400 TWh nuclear electricity, hence an average power of 45.7 GWe, equivalent to 0.79 g of fissioned mass per capita per year: a tiny amount with respect to a single car tank refuel.

The extremely short time for a new branch activity to develop is remarkable: the neutron, the key ingredient in fission processes, was only discovered by James Chadwick in 1934. Unfortunately, as often in the history of industry (automotive, chemicals), the problem of the wastes was handed over to posterity.

Nuclear wastes from thermal reactors

As mentioned, most of the radioactive wastes come from the operation of nuclear fission power plants. In fission reactions, the nuclei split into two lighter elements, which nature follows a probability law dependent on the fissile element, as illustrated in Figure 123456.

Those fission products (FP) are still radioactive and constitute the majority of the Intermediate-Level and High-Level Wastes and most of the heat generation. Since their half-lives are generally short, they belong to the Short-Lived Waste category; the shortest are Strontium-90 and cesium-137 with radioactive periods of 30 years. Among the fission products, some rare species like Technetium ( 99 Tc), Iodine ( 129 I) and zirconium ( 93 Zr) happen to be Long-Lived (~10 6 years) but are much less radiotoxic compared to some Short-Lived fission products [START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF].

Figure 1-6: mass distribution of fission products, from [8]

Nearly all the electrical nuclear power plants worldwide today make use of the rare 235 U as a fissile material. Their core contains low-enriched 235 U fuels, between 0.7 and 4.5 %.

In such cores, the neutron produced by the fission can either interact with the scarce 235 U, or be captured by the overwhelmingly numerous 238 U with probabilities quantized as equivalent cross-sections, as seen in Figure 1234567.

The neutrons from fission are produced at an energy of around 1 MeV: at this energy, the cross-section ratio between 235 U fission and 238 U capture is about 10, not enough to counteract the 238 U/ 235 U quantity ratio (~ 50 to 150). In order to use a more favourable cross-section ratio and enable the chain reaction, the neutrons are slowed-down, or moderated, to energies lower than 1 eV: they become thermal neutrons. It is the role of the moderator, typically graphite or water, to reduce the neutron energies by consecutive collisions. In the latter case, the water is also the reactor coolant, which transfers the produced heat toward the steam generator for electricity generation in a turbine, as in any coal or gas power plant.

Today, most of the nuclear power plants worldwide run pressurized water reactors (PWR) or boiling water reactors (BWR): in the former, due to the elevated pressure, the water remains liquid in the core; in the latter, it turns into vapor. This is the case of all the reactors in the US and in France; they all work at about 3-4 % fuel enrichment. Canada has a special reactor, the CANDU, that was historically developed to work with natural uranium (no enrichment). In this case, due to the 235 U atom scarcity, and the tendency of light water to absorb neutrons, heavy water 2 H2O (deuterium oxide), must be used instead of light water H2O.

Figure 1234567: 235 U fission cross-section (black) and 238 U neutron capture cross-section (pink) as a function of the neutron energy [START_REF] Joliot-Curie | [END_REF] In all those reactors, a somewhat troublesome side nuclear process occurs: the dominant 238 U atoms capture neutrons and give birth, in the spent fuel, along with fission products, to isotopes of transuranic species from the actinide family: neptunium (Np), plutonium (Pu), americium (Am) and curium (Cm).

In France, unlike many other countries, plutonium is not considered as a waste since it is separated and reused to produce nuclear fuel: the mix of oxides (MOX). As a side note, the thermal reactors produce a mix of plutonium isotopes, unsuitable for military application without isotopic separation.

Being produced in smaller quantities, Np, Am and Cm are called minor actinides; they are all highly radiotoxic and Long-Lived, and hence, belong to the HL-LLW category. In Figure 1-8, it is clear that the actinides dominate the total spent fuel radiotoxicity over the fission products after 300 years.

To illustrate, a rough waste inventory is given from [START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF] and [START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF]; a standard 3%-enriched, 1-GW thermal spectrum reactor starts with:

• 29 tons 238 U • 1 ton 235 U After 1 year of operation, the spent fuel has roughly the following composition at the reactor discharge: instant, CANDU reactors, due to the higher 238 U content produce more minor actinides.

On top of the fission products, the Pu and the minor actinides, operating thermal reactors generate the by-product of the enrichment, about 200 ton of depleted uranium to produce 30 ton of enriched uranium. An illustration of the material inventory generated by a thermal reactor is given in Figure 123456789. [START_REF] Hargraves | Liquid Fluoride Thorium Reactors[END_REF] To conclude, although produced in minor quantities, the thermal reactor operations accumulate substantial minor actinide stocks over years, whose high activity and long radioactive half-lives require specific disposal methods. On the other side of the cycle, the resources, given the fissile material scarcity, are also a worrying topic, for fission thermal reactors, with a potential of 100-300 years at the present consumption pace.

Figure 1-9: thermal reactor waste inventory schematics adapted from

THE ADS AS AN ANSWER TO THE MINOR ACTINIDES ACCUMULATION

Fast reactors to close the cycle

The problems of wastes and resources raise the need for another nuclear branch to close the fuel cycle: it requires to recycle a large part of the spent fuel, much more than in the case of the MOX, and several times (multi-recycling). That way, it may be possible to stabilize the waste stocks, Pu for instance [START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF]. Theoretically, the Pu produced in thermal reactors from 238 U could be indefinitely recycled and totally replace the fissile 235 U, making the ratio 238 U/Pu converge to a constant value, as illustrated in Figure 1-10. The problem being that the equilibrium core composition 238 U/Pu is not critical in the thermal spectrum (not enough fissile species); hence, an external fissile material 235 U is necessary but would spoil the closed-cycle approach. Another reactor type is thus needed: the fast-neutron spectrum. In such reactors, the multi-recycling equilibrium 238 U/Pu ratio is critical: non-moderated neutrons fission the plutonium, which is constantly regenerated by the 238 U neutron captures. If the system is provided with an initial close-to-equilibrium composition, it becomes self-sustainable.

Such an approach was initially planned in France, back in the 60/70s by Electricité de France (EDF) and the Commissariat à l'Energie Atomique (CEA): the thermal reactors would produce the necessary plutonium to gradually start the fast-reactor branch. The timescale of such a plan was meant to spread over decades, as about 20-30 tons of plutonium is needed per fast-reactor, whereas the whole French thermal-spectrum reactor fleet only produced 10 ton a year [START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF].

Fast-reactors have been developed alongside with their thermal counterpart. In France, the Phenix demonstration reactor has been working from 1974 to 2009. Its successor, Superphenix, a pre-series fast reactor was stopped for political reasons in 1996 [8]. Finally, the construction of the ASTRID3 reactor, a 4 th generation reactor in the GENIV point of view 4 , was stopped in 2019 for political reasons too, and postponed to the second half of the century.

The reasons for such decisions are many, but two groups are worth stressing:

• The public nuclear acceptance and the loss of expertise • Technological breakthroughs and the safety topic For the first one, although being again brought to light in the context of the energetic independence debate (in France) that followed the Russian attack in Ukraine in February 2022, the French nuclear industry has regularly been targeted by ecological movements; not the least on the topic of the waste management and their humanly unconceivable time scales. Bad press also came from the official occult communication after the Chernobyl accident in 1986, and continues nowadays with the tremendous delays and cost overruns in the generation 3 EPR5 constructions in France, England and Finland. Indeed, after 2 decades without reactor construction, the nuclear industry faces expected issues in manpower and expertise.

For the second group, the technological breakthroughs of new reactor types are numerous, especially to comply with the safety requirements, which became even more stringent after the 2012 Fukushima accident. As an example, in a fast-reactor, water cannot be used as a coolant because of its neutron-moderating property: a molten salt like sodium (Na) or a liquid metal like lead-bismuth eutectic (Pb-Bi) must be used, both solutions being part of the GENIV scenarios. To rapidly sketch-up the challenges, on the one hand, sodium is highly reactive with air and water, on the other hand, lead is extremely corrosive. Making the fast reactor reach the same safety levels as thermal reactors hence still requires years of R&D and validation.

Finally, the blurry nuclear energy future and the apparent short-terms abundance of 235 U did not urge the construction of a fast-reactor fleet, as planned; but as said, nuclear energy is on a tipping point and things might rapidly change.

Worldwide, US also stopped nearly all fast-reactor developments: their early concept, the IFR (Integral Fast Reactor) aimed at closing the fuel cycle with an advanced fully robotized fuel reprocessing method, where the hazardous species would never leave the containment area. The concept was stopped in 1994 by the Democratic party, fearing public opinion and without technical reasons [START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF]. By contrast, in Russia, the BN-800 fast reactor (sodium) has worked since 2016 and Russia has projects for yet another more powerful reactor, the BN-1200. China has operated an experimental sodium fast-reactor of 20 MWe since 2010 and had initially planned to even build two Russian BN-800 reactors to be in service in 2020. Instead, China is programming its own fast-reactor version, The CFR-600, a 600 MWe prototype and in the longer term, a 1200 MWe variant.

To conclude, fast reactors are the key concepts to solve the problems of the fuel resources and of the plutonium accumulation, and also to minimize the minor actinide generation (fast-reactors do also produce minor actinides). The minor actinide stocks could also be stabilized thanks to fast-reactor, yet, with a dedicated treatment.

Transmutation as a complement to geological repository

In France, the 1991 Bataille's law sets the plans for the construction of a laboratory in the town of Bure in the French Meuse, to study the reversible deep geological storage of the Long-Lived nuclear wastes. In 2006, a nuclear waste management law is enforced, that defines the geological repository as the reference solution for the HL-LLW management and gave the green light for the CIGEO project [START_REF]Centre d'informations et de ressources sur CIGEO[END_REF]: the construction of a demonstration storage site in the vicinity of the laboratory.

The main argument in favour of this storage is the fact that minor actinides are not mobile in some very specific geological formations. A famous evidence is often put forward: the fact that a natural nuclear reactor has been in operation 2 billion years ago in Gabon [START_REF]Réacteur nucléaire naturel d'Oklo[END_REF]. Indeed, about 500 ton of 235 U, back then still around 3 % concentrated, fissioned thanks to water infiltrations and other favourable circumstances. The produced minor actinides would have moved only by a few centimetres. CIGEO is meant to reproduce those conditions, supported by the undeniable progresses in the vitrification processes.

Yet, even with this solid proof and another body of evidence, projects like CIGEO may not be acceptable to public opinion, due to the humanly incommensurable times scales [START_REF] Poncelet | Myrrha: un (autre) regard sur l'énergie nucléaire[END_REF].

Hence, as a complement of the reversible storage, the reduction of the minor actinide burden is a necessity. Beyond the obvious need to minimize the storage number and footprint for the sake of the coming generations, the nuclear industry should incontrovertibly prove it can handle its wastes on historical rather than geological time scales which, in turn, would likely increase the public acceptance and clear the political horizon in the long term.

As for the plutonium, reducing the minor actinide inventory is possible via recycling and fission. Doing so, the minor actinides are transformed into lighter -still radioactive but manageable -fission products. This is the principle of the process called transmutation, which has to take over some important achievements.

A first (immense) challenge is the required minor actinide separation process (partitioning) from the wastes. Presently, several countries (France, China, UK, Russia, Japan) already isolate and extract Pu and U from the spent fuel, i.e. 95 % of the mass [START_REF] Poncelet | Myrrha: un (autre) regard sur l'énergie nucléaire[END_REF]. The breakthrough consists in separating the minor actinides from the fission products, then the minor actinides (Np, Am, Cm) between them. One difficulty is caused by the Am and Cm chemical behaviour being close to that of some 50 times more abundant fission products. In France, CEA has developed variants of the PUREX process at the laboratory scale [START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF], but substantial R&D efforts are still needed.

A second technical challenge is the minor actinide manipulation for fuel assembly fabrication at an industrial scale: the spent fuel for such a composition would be much more radioactive than any conventional unirradiated uranium fuel assemblies. This would get even more complex since several fuel cycles are required (multi-recycling), as the burn-up is limited by the reactivity change and radiation damages during operation. A significant effort is needed in this field [START_REF]Accelerator-driven Systems (ADS) and Fast Reactors (FR) in Advanced Nuclear Fuel Cycles: A Comparative Study[END_REF].

The process of actinide reprocessing and dedicated fuel assembly fabrication for transmutation purpose is called partitioning and transmutation (P&T). Many European and national P&T projects are on the way, even included in GENIV roadmaps [START_REF] Junghans | Transmutation of high-level radioactive waste -Perspectives[END_REF]. Yet, in France at least, opinions are divided on the interest of transmutation as a complement to the deep geological repository. According to French major actors like IRSN 6 and ASN 7 , on a technical point of view, there are fewer difficulties in permanently storing the minor actinides (especially the americium) rather than doing a multi-pass separation and fuel assembly, as the once-through is not possible [START_REF]Bilan du débat public, Projet de centre de stockage réversible profond de déchets radioactifs en Meuse / Haute-Marne (Cigéo)[END_REF]. For IRSN, the reversibility of the storage is necessary for return of experience and analysis, but in the long term, the storage shall be irreversibly closed [START_REF]La réversibilité Thème 1 -La place de Cigéo dans le dispositif de gestion des déchets[END_REF]. EDF has not planned to tackle with this challenge in the near future either 8 .

Transmutation has long been opposed to geological storage; both are in fact complementary in the perspective of a better public acceptance, especially as storages like CIGEO are only sized for the current nuclear wastes. In practice, how transmutation should be carried out is the object of another debate, as exposed in the following section. 6 Institut de Radioprotection et de Sûreté Nucléaire Due to the poor neutronic properties of the minor actinides (MA), especially americium and curium, the MA fuel closure method requires specific attention. The possible methods are more or less tightly connected to the Pu cycle closure management, previously discussed, yet always involving fast reactors (minimizing the parasitic captures from the minor actinides); three modes are possible [START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF], as illustrated in Figure 1-11:

• The homogeneous mode dilutes the minor actinides in the fast-reactor fleet:

keeping them in low concentration mitigates the differences in neutronic properties; but it has the drawback that it introducew minor actinides in all fuel assembly fabrication steps • The heterogeneous mode solves the latter problem by placing them in concentrated areas in the core periphery, thus decoupling the minor actinide cycle from the U-Pu one, with some remaining trade-off, such as more complex reactor design which could delay the deployment of fast reactors • The double-strata concentrates the minor actinides even more, by involving parallel dedicated fast-reactor, in burner mode, hence suppressing all the couplings with the U-Pu cycle reactors, which can optimally run for electricitygeneration CEA favours the heterogenous mode, that would enable a 10 % minor actinide concentration and that would globally bring a 20 % cheaper kWh compared to the double-strata mode. CEA also stresses that the technological breakthroughs are still important for the operation of dedicated MA burners. As a matter of fact, the aforementioned 2006 law, in addition to the geological storage, had planned for a transmutation roadmap, involving the ASTRID reactor: beyond the infinite plutonium recycling, a fuel cycle involving homogeneous and heterogeneous Am transmutation was under topic [START_REF] Junghans | Transmutation of high-level radioactive waste -Perspectives[END_REF]. Beyond CEA, at least on a technical standpoint, the opinions of the major nuclear actors are more or less aligned [START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF].

Yet, on a broader scope, opinions are more divided about the dedicated MA burners, even the favourable ones 9 . One of the main troubles of the other approaches (homogeneous and heterogeneous), is that they depend on the political decisions to go for the Pu multi-recycling, i.e. the development of a whole fleet of fast-reactors, which as exposed earlier, is not to be planned before 2050. The issue being, the more time goes, the least efficient transmutation is. The advantage of the dedicated MA burner solution, by contrast, is the complete decorrelation from the electricity generation, even compatible with a full nuclear phase-off, as decided for instance by Belgium and Germany.

Furthermore, on a more technical standpoint, the advocates of the dedicated burners stress that, with MA in the same core as U-Pu, the poor proportion of delayed neutrons (cf. next section), the degraded fuel temperature reactivity coefficient 10 and void coefficient 11 impede the safety and reduce the operational margin of the fast reactor [START_REF] Poncelet | Myrrha: un (autre) regard sur l'énergie nucléaire[END_REF]. According to them, within about 80 years, a fleet of twenty 400-MWth dedicated burners could process all the wastes from the European countries which still produce nuclear wastes, and from the countries which abandoned nuclear for electricity production.

In conclusion, opinions are torn about pure technical trade-offs. On a political standpoint, the single-component solutions might suffer from the unknowns around which plutonium fuel-cycle will ultimately be chosen, if any. By contrast, dedicated burners are compatible with any option, and probably better satisfies the public acceptance, and, as a consequence, the political decisions.

ADS to control the criticality in minor actinide-rich cores

Controlling a reactor is inherently linked to the problem of safety: the fission shall be controlled, without any risk of overrun or stall. Controlling a core consists in continuously keeping the neutron population constant, i.e. make the criticality 𝑘 𝑒𝑓𝑓 12 be exactly equal to 1. Standard electricity-generating fission cores are built around a criticality of 1; the control of the minor criticality fluctuations is done by adjusting the coolant flow and the neutron absorber position, in cadmium or boron. Those absorbers are put on mechanically-actuated rods (or bars) which penetrate more or less in the core. Very simple in appearance, this principle works for a single reason: a small proportion of the emitted neutrons (0.65 % for 235 U) are in fact delayed. They come from so-called precursors, created by the fission product decay chains, that can lose a neutron instead of a gamma photon to get de-excited. Depending on the involved nuclear reaction chains, the time delay spans from 0.1 to 10 seconds or more. This gift from nature gives engineers the necessary time to mechanically (by use of flow valves and motors) compensate for 𝑘 𝑒𝑓𝑓 variations and prevent divergence; it would be impossible in the presence of pure prompt neutrons 13 [START_REF]Laradioactivite[END_REF].

In the case of plutonium, the delayed neutron fraction is only 0.21 %, giving less time to react, but which is still compatible with conventional controls. Minor actinides, by contrast, have a too small delayed neutron ratio. Far from being a detail, this in fact totally prevents the concentration of actinides in conventionally-controlled cores (without forgetting the degraded reactivity coefficients, as said earlier). To obtain minor actinide concentrations as high as 40 % for the dedicated burners, another neutron-population control system is necessary, such as in an ADS.

In an ADS, the MA-rich fast core is built sub-critical, i.e., 𝑘 𝑒𝑓𝑓 < 1. To sustain the chain reaction, the missing neutrons are provided by a source, a spallation target, placed within the reactor core and supplied by an external proton accelerator. Since the core is subcritical, each source neutron generates a finite number of descendants in the fission chain, given by the expression 𝑘 𝑠 1-𝑘 𝑠 [START_REF] Chevret | Mesure de la réactivité de réacteurs sous-critiques pilotés par accélérateur par l'analyse d'expériences d'interruptions de faisceau programmées[END_REF]. In this expression, 𝑘 𝑠 is the equivalent core criticality that considers also the descendant neutrons induced by the external source, which energy and space characteristics differ from the core steady-state fission distribution represented by 𝑘 𝑒𝑓𝑓 . The production rate 𝑁 of fission-inducing neutrons is thus [START_REF] Chevret | Mesure de la réactivité de réacteurs sous-critiques pilotés par accélérateur par l'analyse d'expériences d'interruptions de faisceau programmées[END_REF]:

In this expression, 𝑆 is the neutron flux from the spallation source; S is continuously adjustable, and is related to the thermal reactor power by: 13 The notion of prompt/delayed neutron shall not be mixed up with the fast/thermal neutron antagonism

𝑁 = 𝑆 * 𝑘 𝑠 1 -𝑘 𝑠 ( 1-1 ) 𝑃 𝑡ℎ = 𝑁 * 𝐸 𝑓 𝜈 = 𝑆 * 𝑘 𝑠 1 -𝑘 𝑠 * 𝐸 𝑓 𝜈 ( 1-2 )
In the latter expression, 𝐸 𝑓 is the average fission energy and 𝜈 the average number of neutrons produced by one fission. This relation tells that the power of the ADS is proportional to the power of the adjustable source and independent from the fuel ratio of delayed neutrons. From the point of view of inherent nuclear safety, this control type is also advantageous since it prevents criticality overruns, by design: if the source is turned off, the reactor stops, even though, the problem of the remnant heat evacuation persists (origin of the Fukushima accident).

This latter relation also illustrates the coupling between the accelerator and the reactor, in particular, why a powerful ADS calls for a high-power accelerator; there is furthermore a trade-off between the required accelerator power versus the required reactor safety features: a criticality closer to 1 makes

𝑘 𝑠 1-𝑘 𝑠
become large, thus needs a weaker source power at iso 𝑃 𝑡ℎ ; the drawback is the substantial additional safety features and associated costs required.

A last important fact about the coupling between the accelerator and the reactor, unseen in ( 1-2 ) but highlighted in [START_REF] Chevret | Mesure de la réactivité de réacteurs sous-critiques pilotés par accélérateur par l'analyse d'expériences d'interruptions de faisceau programmées[END_REF] by measurements on the ADS mock-up GUINEVERE (see later), is the very fast -a few µs -neutron population decay after a proton beam stop. A constant heat production within the core assembly -necessary to avoid thermal stressthus requires a stable and continuous accelerator proton beam. Those two important ADS accelerator requirements, the high-power and the stability, have strong practical implications which are further detailed in the next two sections.

ADS DEMONSTRATOR SPECIFICATIONS

The ADS principle was suggested nearly 30 years ago at Los Alamos and at CERN for respectively thermal and fast spectrums [START_REF] Junghans | Transmutation of high-level radioactive waste -Perspectives[END_REF]. Nowadays, in Europe, the MYRRHA project centralizes the efforts and endeavours in this direction.

MYRRHA today

In 1998, in reaction to its aging reactor BR2, SCK•CEN14 (Mol, Belgium) initiated the MYRRHA (Multi-purpose hYbrid Research Reactor for High-tech Applications) project.

MYRRHA was, and still is, an ADS relevant-scale demonstrator aiming at studying the modality of the minor actinide transmutation. Significant R&D efforts were identified on the three components: the reactor (MA-concentrated fuel assemblies, sub-critical core reactivity control …), the spallation source and the accelerator, as well as their coupling.

Over years, several R&D projects, summarized in Table 1-1, some being supported by the European Commission (EC) framework programs, enabled to gradually freeze the specifications, as well as broadened its applications. Indeed, considering the international and European needs for transmutation and beyond, MYRRHA is designed as a flexible fast-spectrum irradiation facility for:

• GENIV innovative fast-spectrum fuel developments • Structural material irradiation resistance for fast-spectrum fission or even fusion To achieve these various goals, the reactor is able to work in both sub-critical (accelerator driven) and critical mode [START_REF] Abderrahim | MYRRHA -A multipurpose fast spectrum research reactor[END_REF].

Table 1-1: European Commission co-funded MYRRHA R&D programs

In parallel, but not connected to the reactor side, a small portion of the MYRRHA accelerator beam is diverted to dedicated test stations: a proton target facility for Radioactive Ion Beam (RIB) applications (ISOL@MYRRHA 15 project) and a fusion-oriented material irradiation target facility. A conceptual view is shown in Figure 1-12 (the accelerator side-target facilities are not shown). 15 Some of the main steps that led to the current MYRRHA design and specifications are worth highlighting, in particular those about the ADS specification. This is the topic of the two following sections.

The pre-2010 period: the genesis

The ADONIS project (Accelerator Driven Optimized nuclear irradiation system) was conducted by SCK•CEN from 1995 to 1997, involving a coupling between a 150 MeV beam energy accelerator and a 1.5 MWth thermal spectrum reactor. Its original goal (the radio-isotope production for medicine applications) was extended to demonstration of the ADS principle and, not the least, to fuel research [START_REF] Abderrahim | MYRRHA -A multipurpose fast spectrum research reactor[END_REF]. Fuel research is one of the pillars of minor actinide transmutation, especially via ADS (the MA are concentrated): as already mentioned, the neutronic flux and heat dissipation are without comparison much larger than with standard spent fuel: all steps of the fuel re-fabrication, among them the transport in ventilated containers, and the fuel pellet fabrication by sintering are extremely challenging [START_REF] Poncelet | Myrrha: un (autre) regard sur l'énergie nucléaire[END_REF].

It gave birth to the first MYRRHA project (1998)(1999)(2000)(2001)(2002)(2003)(2004), also carried by SCK•CEN. The initial intention was already to be a large facility that, beyond the ADS demonstration, would replace the 1962 BR2 reactor and its applications. A preliminary reactor design with a nominal power of 30 MWth coupled to a 350 MeV cyclotron was completed in mid-2002 ("Draft 1"), and ultimately evolved to a "draft 2" version, featuring a 50 MWth reactor and a 350 MeV linear accelerator.

In parallel, the MUSE experiments (1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004), an international effort led by CEA and CNRS/IN2P3, aimed at studying one of the transmutation pillars: the control of a sub-critical core by an external neutron source. More precisely, it focused on the neutronic multiplication kinetics (of prompt and delayed neutrons) in several sub-critical configurations, with known neutron sources (as spallation sources bring additional uncertainties) [START_REF] Vollaire | L'expérience MUSE-4: Mesure des paramètres cinétiques d'un système sous-critique[END_REF]. The MUSE experiments went through several phases on the CEA-Cadarache MASURCA reactor (fast-sodium core), until MUSE-4, where the GENEPI (GEnérateur de Neutrons Pulsé Intense) accelerator built by LPSC 16 served as a neutron source.

In the early 2000's, the EC-funded PDS-XADS program aimed at demonstrating the technological feasibility of an ADS concept. On the accelerator side, the initial beam profile, energy and stability were set, as well as the premises of the overall accelerator beam availability (the time spent delivering the expected beam) and reliability targets: the central concept of modular de-rated components was already put forward. Lastly, the linear accelerator (linac) was confirmed as being the reference. On the reactor side, several concepts were explored and the MYRRHA "Draft 2" design was proposed to be the starting base for the next EC-funded project, EUROTRANS, as illustrated in Figure 12345678910111213. FP5 (1998FP5 ( -2002) ) and FP6 (2002FP6 ( -2006) ) are the EC framework programs, image from [START_REF] Bruyn | The MYRRHA project: Current design status & Evolution ADS to FR system[END_REF] The main goal of the EUROTRANS project was to pave the way to the construction of an 16 LPSC: Laboratoire de Physique Subatomique et de Cosmologie, IN2P3/CNRS, Grenoble, France experimental facility, under the concept called XT-ADS, which, as said, MYRRHA "Draft 2" would be the baseline. The design of all the demonstrator sub-components was studied, including investigations on suitable structural materials. This project ultimately froze the initial specifications for a demonstration and for a serial minor actinide burner, as seen in Table 1-2 17 .

Table 1-2: demonstrator and serial minor actinide burner specification following EUROTRANS

From the MYRRHA earliest phases, the sub-system reliability was identified as a major requirement: on the accelerator side, the EUROTRANS project led to the assessment of acceptable beam trips based on a first accelerator reliability modelling. The experiments focused on advanced sub-critical core control items: the continuous operation (enabled by the updated GENEPI3C), the neutron population decrease in beam interruption, operational procedures (start-up, shut-down) as well as the core reactivity 18 monitoring. The layout of the facility is shown in Figure 1-14. 17 The spallation target, although using the same medium as the coolant (liquid heavy metal), is not in direct contact with it. Indeed, the very high neutron activated metal is confined in a separate loop. 18 Briefly, the derivative of the criticality The VENUS reactor could also work in critical configuration, in order to accommodate the new requirements from the CDT/FASTEF project. Started in 2009, this project upgraded the XT-ADS concept from EUROTRANS so that it better complies with some of the applications and requirements of MYRRHA, especially, the demonstration of the Lead Fast Reactor technology. The possibility to run in core critical configuration was one of the requirements (a major one, given the additional necessary safety features and control devices), a second one was the increase in the reactor power to 100 MWth.

The main features of the MYRRHA/FASTEF version are shown in Figure 1-15. In subcritical configuration (ADS), the core central position is occupied by the spallation target assembly; it is removed in the critical mode, whereas the control and shutdown rods are installed.

Figure 1-15: the MYRRHA/FASTEF core and its main sub-components

In connection with the parallel CDT/FASTEF and FREYA projects, the other EC-funded MAX project brought an updated consolidated design of the MYRRHA accelerator, through advanced beam dynamics simulations and R&D activities. During MAX (2011-2014), the accelerator design and test started: on the one hand on the injector, on the other hand on the main linac [START_REF] Bouly | DEVELOPMENTS AND TESTS OF A 700 MHz PROTOTYPICAL CRYO-MODULE FOR THE MYRRHA ADS PROTON LINEAR ACCELERATOR[END_REF], [START_REF] Saugnac | THE MYRRHA SPOKE CRYOMODULE[END_REF]. Much of the current prototype cryomodule design actually comes from the MAX project. On the reliability aspects, a major objective of MAX was to develop a far more accurate reliability model of the MYRRHA accelerator compared to EUROTRANS' one.

Finally, the last EU-funded project, MYRTE, continued the experiments on the VENUS reactor to validate neutron calculation tools and nuclear data libraries, to refine criticality measurements procedures and other core-related topics. Still related to the reactor, the lead-bismuth coolant thermal hydraulics was also investigated, as well as its chemical interaction with the fission products. Lastly, on the fuel side, studies on americiumbearing oxide were carried out.

In MYRTE, the largest work-package was devoted to the construction and commissioning of the accelerator injector in Louvain-La-Neuve, with a strong emphasis on the reliability aspects [START_REF] Gatera | MYRRHA-MINERVA Injector Status and Commissioning[END_REF]; many LLRF (see section 4.4) current prototype aspects are inherited from these developments.

Still on the accelerator topic, a work package coordinated by CNRS/IN2P3 19 included an extensive superconducting Spoke cavity (see 4.1.1) R&D program for the first section of the LINAC, leading to the test of the first two pre-series cavities [START_REF] Longuevergne | Performances of the Two First Single Spoke Prototypes for the MYRRHA Project[END_REF]. Last but not least, in this work-package, a major accelerator design update was performed, following the conceptual design of the previous MAX project, the 1 st international design review and the MAX beam dynamics simulations.

To summarize, the MYRRHA project has been for nearly 30 years undergoing many successive design and experimental phases. The scope was enlarged to comply with all the ADS pillar needs, as well as GENIV fission nuclear reactor, fusion reactors and other industrial needs. In particular, the accelerator specifications have gradually been frozen, as exposed in the next section.

MYRRHA ACCELERATOR HIGH-LEVEL SPECIFICATIONS

A high-power linear accelerator

The MYRRHA accelerator is designed to produce neutrons for the reactor criticality control, thanks to a spallation source. The requirements for the ADS demonstration and other irradiation applications require a 4-mA beam at 600 MeV energy, thus a total beam power of 2.4 MW.

To avoid thermal stress on the spallation source components as well as on the fuel assembly and reactor structure, the beam must be as continuous as possible: beam interruptions shall not be larger than 100 ms. This aspect, although trivial in appearance has in fact large consequences in normal operation, as seen in section 1.4.3, and in abnormal cases, as detailed in section 1.4.5.

In reality, the operation is not strictly continuous: a pattern at 250 Hz repetition rate (thus every 4 ms) contains a main 3.8 ms pulse for the spallation source, and a narrow 190-µs pulse, diverted upstream of the reactor to feed secondary target facilities. The narrow pulse is also used for the on-line reactor criticality monitoring, a topic studied during the MYRRHA experimental phases, as aforementioned. The beam time structure, inherited from MYRTE, is illustrated in Figure 12345678910111213141516. In order to drive the reactor power, the beam power needs to be controlled from 20 to 100 %. The preferred method is to keep the beam energy and peak beam current at their rated value, but to trim the main pulse length between 0 and 3.8 ms. This is to avoid retuning the control parameters (for the beam optics and the accelerating cavities) and to avoid affecting the secondary target facilities. This large range of beam power, in turn, requires the subsystems to be controllable over wide operating ranges, as detailed in 2.4.

As said, during the PDS-XADS project, the choice was to go for a linac. To give a little background, there are two families of machines: linear or circular.

In circular machines, one takes advantage that all the accelerated particles will not be used at the target point, and will thus circulate again many times in the same accelerating structures. This enables to reach very high beam energy with a relatively reduced number of cavities. This is used for instance in colliders for fundamental physics reaching TeVenergies, like LHC at CERN in Switzerland, or the former TeVatron at Fermilab in the US. Circular machines are also used in synchrotron light machines (DIAMOND in the UK, SOLEIL and ESRF in France): there is not any target, the electrons, slowed-down by the synchrotron emission, are continuously reaccelerated.

Linear machines on the other hand, are necessary when the whole beam is meant to be entirely lost on the target. The drawback being, as the particles only go through the accelerating devices once, that the capital costs are proportional to the specified maximum beam energy. This is the case for instance for cancer-treatment machines (with electrons or protons), or for material analysis as performed by the compact accelerator AGLAE (Accélérateur Grand Louvre d'Analyses Elémentaires) of the Louvre museum in Paris [START_REF] Aglae | Accélérateur Grand Louvre d'Analyses Elémentaires[END_REF] and, on a much larger scale, the ESS (Europeran Spallation Source) in Lund, Sweden [START_REF] Darve | The ESS Superconducting Linear Accelerator[END_REF]. This is also the case of the HIPA (high Intensity Proton Accelerator) at PSI in Switzerland [START_REF] Kiselev | Status and Future Projects of the PSI High Intensity Proton Accelerator[END_REF]: the cyclotron is also to some aspects a linear machine, as the particles never cross the same location from one turn to the other.

Obviously for MYRRHA, only a linear machine could be used; among them, both cyclotrons and linacs are suitable candidates. Although both machines can provide the required beam current and power, two reasons made the linac option more favourable:

• The required mean current and energy combination for the MYRRHA demonstrator beam is at the edge of what is currently feasible: although both cyclotrons and linacs would be relevant, it was believed that, for an industrial MA burner, the necessary power increase by a factor of 5 to 10 would be less difficult (very relatively) to achieve with a linac • The requirements such as the reliability, the beam availability, the beam stability and emittance growth were considered to be more easily fulfilled with a linac which, moreover, has a more flexible construction Figure 1-17 puts MYRRHA in the perspective of the current and future accelerators; MYRRHA is indeed at the limit of the current machines. In the range of MYRRHA's beam current and energy is the HIPA cyclotron from PSI which is, as discussed, not suitable; there is also ESS, a machine which operation should start in 2024-2025.

Figure 1-17: the landscape of high power proton beam accelerators, as of 2016 [START_REF] Wei | THE FRIB SUPERCONDUCTING LINAC: STATUS AND PLANS[END_REF] Also, contrary to MYRRHA, ESS is a pulsed machine, which has the advantage to significantly reduce the load on the cryogenics, as it will be seen in 1.4.3. The next part deals with the comparison between pulsed and continuous operation of a beam and RF systems.

Pulsed versus continuous operation

In modern accelerators, the particles are accelerated by the electrical component of an electromagnetic wave, trapped in a closed resonant structure called cavity (see part 4.1).

The particles are synchronized with the radio-frequency (RF) oscillating field 20 , and have the tendency to clump and slightly oscillate longitudinally around a given position perfectly synchronized with the RF wave, thus namely the synchronous particle, instead of evenly spreading along the accelerator (this is more detailed in 2.5). The RF period where a bunch of particles is locked this way, is called an RF bucket. The order of magnitude of the bunch length is about ~1000 times smaller than the RF period, thus of the order of the ps or hundred of fs.

One should keep in mind that not all the RF buckets are occupied in an accelerator: either when a minimum bunch spacing is desired (e.g. the bunch spacing at LHC is 25 ns; at XFEL it is 220 ns), or in linac high-energy section cavities, as they have additional buckets compared to the lower energy structures (since they operate at higher frequencies). For instance for MYRRHA, as the low-energy frequency is 176.1 MHz, a maximum of 1 bucket over 2 can be used in the 352.2 MHz structures, and 1 over 4 in the 704.4 MHz section.

The beam current is defined as the number of charges in the bunches per second, expressed in [µA] or [mA]. Actually, the beam current may not be constant over time, and instead may have a time structure: it could be pulsed or continuous. Likewise, the RF can be pulsed or continuous (the RF pulses are thus always a little larger than beam pulses), often refered as pulsed-wave (PW) or continuous-wave (CW).

The definitions and characteristics of RF and beam pulses, buckets and bunches are illustrated in Figure 123456789101112131415161718.

Figure 1-18: basic beam dynamic definitions, inspired from [START_REF] Alesini | Linear Accelerator Technology[END_REF] As said in the previous section, MYRRHA needs a nearly CW beam. This implies to make use of superconducting cavities; the next section explains the reason why.

Superconducting versus normal-conducting linacs

Not all linear accelerators need superconducting cavities. For instance, the nonsuperconducting LINAC4 at CERN inaugurated in 2017 features a 160 MeV and 40 mA proton beam; in this specification, 40 mA is the peak current of the 400 µs long beam pulse: it means that the mean current is much smaller, and so is the average power, of only 5.1 kW [START_REF]Taking a closer look at LHC -LINAC4[END_REF].

t (s)

Repetition rate = pulses / sec (Hz) [START_REF] Chevret | Mesure de la réactivité de réacteurs sous-critiques pilotés par accélérateur par l'analyse d'expériences d'interruptions de faisceau programmées[END_REF] ESS [START_REF] Abderrahim | MYRRHA -A multipurpose fast spectrum research reactor[END_REF] and LINAC4 [START_REF]La réversibilité Thème 1 -La place de Cigéo dans le dispositif de gestion des déchets[END_REF] What actually drives the need for superconducting sections, is the RF duty-cycle. A normal-conducting cavity, operating in CW at a relevant field of a few MV/m, would dissipate MW of Joule power. Beyond the technical feasibility to evacuate this heat, such inefficiency is not conceivable.

That is why normal-conducting cavities operate with short pulses: the RF is in fact, most of the time, off. The challenge though, to achieve reasonable mean currents, is to concentrate the RF power transmission to the beam on a narrow time window. Specific RF amplifier, klystrons and their modulators, are used for that matter.

But when the pulses need to be large, as required by the application, and in the extreme case that the beam (thus the RF) needs to be CW, superconducting cavities must be used. Superconductivity is detailed in section 0; but, to come to the point, once in superconducting state, the loads from the RF to the cavity walls (dynamic heat loads) only dissipate typically 5-10 W per cavity. Since superconducting cavities operate at cryogenic temperatures (see 3.1), heat is also transferred by conduction, convection and radiation from the surrounding environment, adding typically 2 to 5 W of so-called static heat loads.

One drawback though, is that the required cavity cryogenic process is highly inefficient and spoils the economics. Indeed, the best achievable thermodynamic performance is obtained with a Carnot cycle, which depends on the temperatures at stake, and is quantized by the 𝐶𝑂𝑃 (Coefficient Of Performance):

𝐶𝑂𝑃 𝑅,𝐶𝑎𝑟𝑛𝑜𝑡 = 𝐻𝑒𝑎𝑡 𝑝𝑢𝑚𝑝𝑒𝑑 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑐𝑜𝑙𝑑 𝑟𝑒𝑠𝑒𝑟𝑣𝑜𝑖𝑟 𝑆ℎ𝑎𝑓𝑡 𝑝𝑜𝑤𝑒𝑟 = 𝑇 𝑐𝑜𝑙𝑑 𝑇 ℎ𝑜𝑡 -𝑇 𝑐𝑜𝑙𝑑 ( 1-3 )
As the 𝐶𝑂𝑃 tends to be tiny when 𝑇 𝑐𝑜𝑙𝑑 drops down to cryogenics, one instead uses which is 66 when 𝑇 𝑐𝑜𝑙𝑑 = 4.5 𝐾 and 𝑇 ℎ𝑜𝑡 = 300 𝐾: to pump 1 W of heat from a cold reservoir at 4.5 K, rejecting it at room temperature, one ideally needs 66 W of mechanical or electrical power. On top, due to irreversible processes in pumps, compressors and turbines, real cryoplants only achieve about 26 to 32% of the ideal Carnot cycle [START_REF] Weisend | Energy Efficiency and Recovery at Large Scale Cryogenic Plants: A Survey[END_REF], which gives a ratio of 1 W of extracted heat per 250 W of electrical power for this cold temperature. For ESS, the design cryogenic heat loads are 9500 W (equivalent at 4.5 K), which requires nearly 2.37 MW of electrical power, about half of the beam power.

To illustrate the energy-consumption trade-off when going for superconducting technology, the total electrical power needed to rise a cavity field at 7 MV/m is calculated, on the one hand on a superconducting cavity, on the other hand, on a normalconducting one. First, one considers that the RF amplifier power is either transmitted to the beam (see the power formula ( 4-27 )) or dissipated as heat in the cavity walls, but without power reflection (a notion explicated later on). To calculate the heat dissipation as a function of the accelerating gradient, one uses the notion of quality factors (see section 4.2.2.1). Then, one considers the following assumptions:

• 50 % RF amplifier efficiency (internal heat dissipations)

• 5 W of static heat loads (heat leaks from conduction and radiation)

• 𝑄 0,𝑠𝑢𝑝𝑒𝑟 = 10 9 , the superconducting cavity quality factor • 𝑄 0,𝑛𝑜𝑟𝑚𝑎𝑙 = 10 4 , the normal-conducting cavity quality factor 21 to beam power with respect to the duty cycle. For duty-cycles larger than about 0.03 %, the superconducting technology is appealing. In practice, as a rule of thumb, superconducting technology is generally used above about ~1 % duty-cycle. Indeed, other considerations come into play, such as the cryogenic system capital cost and maintenance, the somewhat cumbersome cryogenic installation, the need for cryogenic expertise, the risk of helium shortage, etc.

• 1 𝐶𝑂𝑃 = 300

The electrical consumption problem

For the MYRRHA first 100 MeV accelerating section, the design figures is almost 3000 W of cryogenic heat loads [START_REF] Kochebina | Cryogenic Systems Studies for the MINERVA 100 MeV Proton SC LINAC Project[END_REF]. Compared to ESS, it means that for a twenty-time less powerful beam, still 30 % of the cryogenic capacity is needed. This is a consequence of the CW operation.

Accordingly, for the whole 600 MeV MYRRHA, and ultimately for the industrial burners, very large energy-consuming cryoplants will be installed, with the problem of the electricity price fluctuations, and, again, the public acceptance thus the political support.

It is true that the reactor of the industrial burner should be able to produce some 100 to 150 MW of electricity (about 30 % of the 400 MWth), thus covering the facility needs, with margin 22 .

But, in the first MYRRHA phases, that may last ten years, the accelerator will be operational without producing electricity from the reactor; given the current European energy crisis, the electrical consumption of the facility will likely be scrutinized.

The need to design an energy-efficient facility, in turn, sets strong requirements to the cryomodule cryogenic control: electrical heaters, which are commonly used to even out the RF dynamic heat load variations, may be banned in continuous operation and limited to short transient periods. Consequently, the cryogenic control will have to be robust over a large range of operating conditions; this point is all but trivial and will need to be properly studied, as explained in section 2.5.

The reliability requirements of the linac

In nominal operation, the accelerated beam must be as continuous as possible to limit thermo-mechanical stress and fatigue on the spallation target, fuel and reactor components. It implies that excessively long unwanted beam interruptions shall be avoided too.

A quantized requirement was established during EUROTRANS: there should not be more than 10 beam interruptions longer than 3 seconds, per 3-month missions. This figure of 3 seconds is the product of a negotiation between the accelerator and the reactor people, and is partially inherited from the Phenix reactor 23 . In addition, shorter trips are also limited to no more than 100 occurrences.

Beyond the thermo-mechanical aspects, this requirement of 3 seconds is also a guarantee of a high beam availability (the target is 80 %): in case of longer beam trips, the reactor should be shut down; the restart procedure (a nuclear facility needs clearances) typically involves 24 to 48 hours.

The equivalent required mean time before failure (MTBF) is about 250 hours, to be compared with a few hours at PSI, for instance. Again, this requirement puts MYRRHA, again, at the edge of what is feasible: in Figure 1-20, the requirement is compared in particular to the already-operating machine SNS (Spallation Neutron Source), in Oak Ridge in the US [START_REF] Sns | [END_REF]. Whereas the short trip duration meets the MYRRHA specification, there is a gap about the more-than-3-second beam trips 24 .

Figure 1-20: the MYRRHA reliability requirement compared to other machines, from [START_REF] Bouly | MYRRHA Superconducting linac & Fault Tolerance Concept[END_REF] This difficulty was identified soon: the PDS-XADS project already established that the reliability should be a design feature. This translates into robust conservative design, maintenance plans and above all, redundancy concepts.

Redundancy is fully standard for critical systems, as in space applications were repairing is almost never possible: to mention the most famous, the Hubble Space Telescope has for instance a spare set of giros for its alignment (although they were ultimately replaced during one of the Space Shuttle missions); the same goes for each of the 3 computer systems of the Voyager 1 & 2 space probes. The problem of redundancy is, of course, the additional costs.

In MYRRHA, the injector is made of a 30 keV proton source transported through the Low Energy Beam Transport line (LEBT) and matched to a 176 MHz RFQ for bunching and accelerating up to 1.5 MeV [START_REF] Debongnie | Modelization of an Injector With Machine Learning[END_REF]. CH-type cavities then increase the proton energy to 16.6 MeV that reach the main linac through the Medium Energy Beam Transport line (MEBT).

Whereas the injector components are all normal-conducting, the main linac is fully superconducting. It The injector was identified to have the greatest failure probability [START_REF] Saugnac | Spoke Cryomodule Prototyping for the MINERVA Project[END_REF]. The accelerator is thus equipped with a duplicated complete injector line; a fast-switching magnet placed at the beginning of the superconducting section is able to switch from one to the other, in case a failure is detected on one. This is a "parallel" redundancy approach: one injector is idling whereas the other one is operating; this sets strong constraints on the injector control system, as both units must rapidly switch over wide ranges of configurations [START_REF] Debongnie | Etude de la modélisation des accélérateurs de particules par des méthodes de «machine learning» pour optimiser et fiabiliser l'opération d'un linac de forte puissance[END_REF].

In the main linac, another approach was favoured, called the series redundancy: thanks to substantial operational margin (in nominal, the cavities work as de-rated), it is possible to distribute the needed energy gain not provided by a faulty cavity to the neighbouring ones, typically four ones. This operation margin was fixed to 30 % (the nominal accelerating gradient is 7 MV/m, the maximum one is 9.1 MV/m), leading to a cost overrun of 20 %, far less than a parallel main linac. Up to 30 to 35 failures are expected to need compensation simultaneously during the foreseen 3-month missions if no on-line repairs can be done [START_REF]Design of the MYRRHA 17-600 MeV Superconducting Linac[END_REF].

The process of distributing the faulty cavity function on the adjacent ones is called the fault-compensation and occurs in two parallel steps [START_REF] Biarotte | Beam operation aspects for the MYRRHA linear accelerator[END_REF]:

• The faulty cavity field is disabled and its resonance frequency is shifted from the excitation by 100 bandwidths to avoid beam-loading-induced parasitic gradients (see 4.5.2) • New setpoints (phase and voltage, see Figure 234567) are sent to the compensating cavity LLRF control system Before the compensation strategy is triggered, a diagnostic phase is necessary, to determine the origin of the fault. All this (diagnostic and cavity shutdown/compensation) should take place within three seconds.

This has strong implications on many control aspects. First, on actuators to be used to detune the faulty cavity, for instance the motor of the tuning system (presented in 4.5.2). Then, on the RF amplifier margin, as explained in [START_REF] Bouly | Fault Tolerance and Consequences in the MYRRHA Superconducting Linac[END_REF], and on the LLRF control algorithm reacting to new settings [START_REF] Bouly | Low Level Radio-frequency Developments toward a Fault-tolerant Linac Scheme for an Accelerator Driven System[END_REF]. Last but not least, on the optimal compensation strategy: faults can occur on any cavity and the linac can be in whatever configuration (with the potential presence of already compensated faults). Model-based approaches are thus necessary to, on-line, calculate the optimal beam-dynamics targets of the cavities used for the compensation [START_REF] Bouly | Method Development for Cavity Failure Compensation in a Superconducting Linac[END_REF].

Finally, considerable challenges are still to tackle, like the cavity behaviour when facing fast RF field changes (multipacting and quench, further defined in section 4.2.1), and last but not least, the interactions between the cryomodule control systems, as detailed in section 2.5.

MINERVA, THE MYRRHA PHASE 1 CONSTRUCTION

MYRRHA is an about 1.6 bn€ 25 project, not yet entirely funded: a non-profit legal entity was established to find industrial partners, and benefits from the Belgium government support 26 .

In 2018, Belgium committed to finance the first phase of the MYRRHA accelerator, called MINERVA, that reaches 100 MeV and 4 mA energy and current in CW, thus, a 400 kW 25 Currency value of year 2020 beam; for comparison, SPIRAL2 at GANIL has a target at 200 kW. MINERVA comprises 1 of the 2 injector lines, the MEBT and the first superconducting (Spoke) section, as shown in the global MYRRHA layout in Figure 1-21. The superconducting section comprises 60 identical cavities grouped in pairs within a total of 30 identical cryomodules; it spreads over a little more than 90 m. This is the reference scenario; an initial implementation and operation of a first batch of 24 cryomodules is also considered [START_REF] Zamora | Status of the MINERVA cryomodules and associated cryogenic system (MYRRHA phase 1)[END_REF]. The detailed layout of the current implementation is displayed in Figure 1-22. MINERVA has a budget of 558 m€: about a half is meant for the construction of the 100-MeV accelerator and for the two application buildings, with a first beam foreseen in 2027 [START_REF] Bruyn | MYRRHA, the Belgian Prototype that fascinates the World[END_REF]. A quarter of this amount is dedicated to the operation at 100 MeV in the 2027-2038 period. Another quarter is for the R&D and licensing studies in view of the MYRRHA phase 2 (the 600 MeV accelerator) which is planned for completion in 2033, and of phase 3 (the reactor), for a start of operation in 2036 [START_REF]About MYRRHA[END_REF]. Figure 1-23 shows the current frozen MINERVA implantation, in the context of phases 2 & 3: the place is chosen, and the building construction is about to start 27 . In the MINERVA phase operation, the PTF and FTS will make use of macro beam pulses of the beam (directed towards the stations by a septum kicker and dipoles); the other part of the beam being routed towards a 400 kW beam dump.

This beam dump makes it possible to test the beam maximum power in view of MYRRHA. Indeed, the primary objective of MINERVA, besides the two applications (ISOL experiments and irradiation of materials), is the demonstration of the feasibility of the ADS accelerator features, among which reliability is a major item. The strong incidence is that MINERVA should be already able to run fault-compensation schemes: the necessary accelerator sub-system and their respective controls must thus be operational for this quite short-term deadline.

To summarize, MYRRHA, the demonstrator of the ADS technology and test bed for future reactors, including fusion, is on the way to be built; it is among the most powerful accelerators worldwide, and has a yet never reached reliability target. Its short-term first phase, MINERVA, must demonstrate several of its features, among them:

• Considerable heat load, due to continuous wave operation

THE NEED FOR A MINERVA CRYOMODULE SYSTEM MODEL

The previous chapter presented the different contradictory objectives an ADS accelerator shall fulfil, and introduces the idea that the accelerator high-level constraints, in turn, requires high performances from the different sub-functions. To support the MYRRHA accelerator development and orient the necessary compromises, a specific modelling approach, centred on the active part of the accelerator, the cryomodule, seems necessary.

This chapter first comes back on the roles of the cryomodule and its functions; then, an industry-standard modelling approach for complex systems is reviewed and put next in the perspective of the MINERVA project development needs, to explain how and why such an investment would be profitable. Then, an emphasis is proposed on the model specific capabilities, given the expected problems to address; lastly, a practical model development methodology is detailed.

THE CRYOMODULE MODULAR APPROACH AND FUNCTIONS

In the superconducting accelerating section of most modern linear accelerators, like SPIRAL2, FRIB or ESS, an architecture based on a series of tens of cryomodules, is chosen [START_REF] Xu | FRIB Cryomodule Design and Production[END_REF], [START_REF] Ghribi | Spiral 2 Cryogenic System for The Superconducting LINAC[END_REF]. Each cryomodule contains only a limited number of superconducting RF cavities (SRF cavities), typically 1 to 8 units.

For MYRRHA, a similar approach is chosen: the superconducting sections are made of identically-repeated strings, featuring a cryomodule, focusing sections and beam diagnostics. This design complies with the previously-mentioned reliability specification, by offering [START_REF] Biarotte | Beam operation aspects for the MYRRHA linear accelerator[END_REF]:

• A large longitudinal acceptance • A wide RF operating margin • A high RF and cryogenic control flexibility

To make their construction, assembly and installation compatible with series industrial constraints, among other, the tooling investments, the cryomodules of the same accelerating section feature a modular design: they are all identical and concentrate all the necessary sub-functions for the acceleration.

To provide the maximum energy gain to the particles, each cavity should however be specific and have an optimal accelerating gap geometry, adapted to the reduced local particle speed β = 𝑣 𝑐 ⁄ (𝑣 the particle speed, 𝑐 the speed of light), that changes as the particles progress along the linac; this is especially true for protons in the GeV range, since β(proton) = 0.88 at 1 GeV. Specific cavities at each β would thus ensure the minimum number of cryomodules for a given desired total energy. As exposed, for industrial reasons, this is not practically feasible; the trade-off consists in splitting the accelerator into typically 2 to 6 different sections, each section having only one cavity type adapted to the particle average β over the section. MYRRHA and ESS for instance, feature three different SRF accelerating sections. Within one section, all cryomodules are almost identical in terms of hardware.

In terms of RF setpoint, by contrast, they completely differ from each other, as every cavity has its field amplitude and phase adapted to the beam dynamic requirements [START_REF]Design of the MYRRHA 17-600 MeV Superconducting Linac[END_REF]. There again, the series modular identical architecture is a drawback: as further detailed in 4.4.1, the RF process optimal power efficiency requires to use a specific RF power coupler antenna penetration, adapted to the cavity field (the power coupler is the component that couples the high-power RF line from the amplifier to the cavity). In practice, one can use, for instance, mobile couplers as for the European XFEL [START_REF] Kostin | UPDATE ON THE EUROPEAN XFEL RF POWER INPUT COUPLER[END_REF], or penetration-adjusted fixed couplers. The MINERVA project went rather for a third possibility, fixed-penetration couplers, resulting in an identical external quality factor for all the cavities in the section.

The role of the cryomodule and its processes is to provide the best environment for the cavity operation. In this sense, it is similar to a cavity test cryostat, like in [START_REF] Kern | Cryogenic Performance of the HNOSS Test Facility at Uppsala University[END_REF] or [START_REF] Suhane | ENGINEERING DESIGN OF VERTICAL TEST STAND CRYOSTAT[END_REF], but with additional environment constraints, as summarized by the functional diagram in Figure 2-1.

Figure 2-1: simplified functional diagram of a cryomodule

Cavities MF1: ensure the cavity alignment with respect to the beam, in the nominal operating conditions MF2: shield the cavities against the earth and ambient magnetic fields MF3: provide interfaces to the RF, cryogenic and vacuum networks (+ utilities) CF1: provide a controlable cryogenic environment for the cavities (cool-down, nominal, warm-up) CF2: provide a mechanical support to the cavities CF3: limit the parasitic heat loads by convection (insulation vacuum), conduction (appropriate supports) and radiation (active thermal shields) CF4: adapt the available space CF5: provide measurement signals of the on-going processes CF1 CF2

Beam

MF1

Environment

MF2

Main networks

• RF • Cryogenics • Vacuum MF3 Control system CF3 CF5 CF4

Cryomodule

Main Function (MF) Constraint Function (CF)

To address those functions, many cryomodule designs exist worldwide, with the same final objectives. Starting from the most zoomed perspective, the cavities are protected against the earth and surrounding device magnetic shields by a sheet of high magnetic permeability material, operating at cold or warm temperature. For MINERVA, they are cold, loosely fitted around the cavity tanks.

Then the cavities and their magnetic shields are protected against thermal radiation by a typically 10-layer MLI (multi-layer insulation) blanket. The cavities are carried by the supporting system; its role is to ensure the correct alignment, a robust and safe support for all the transport operations, while minimising the heat loads to the cavities. Those requirements being somehow contradictory, the design of the supporting system results from a trade-off. Different philosophies exist worldwide: hanged from a top plate like SPIRAL2, suspended to the cryogenic exhaust line like LCLS-II [START_REF] Peterson | LCLS-II 1.3 GHz Cryomodule Design -Modified TESLA-Style Cryomodule for CW Operation[END_REF], by side rods like ESS Spoke and Elliptical cryomodules [START_REF] Duthil | The ESS Spoke cryomodule and its test valve box[END_REF], [START_REF] Olry | Recent Progress of ESS Spoke and Elliptical Cryomodules[END_REF], or carried on a bottom plate like FRIB [START_REF] Xu | FRIB Cryomodule Design and Production[END_REF].

Every solution has doubtless its pros and cons; the full global picture is pretty opaque, as the return of experience is rarely fully communicated. For MINERVA, the choice was done to go the latter design, a bottom frame with supporting arms. The supporting system equipped with the cavity and their ancillaries is called the cavity string.

The next layer around the cavity string is the thermal shield (TS), an actively cooled shell that can be in copper (MINERVA prototype) or in aluminium (ESS SPOKE cryomodule), that intercepts both the radiation and the conduction from hotter parts. The TS is itself covered with a typical 30-layer MLI blanket. The whole assembly is then enclosed in a stainless-steel vacuum vessel, which is depressurized to provide the necessary thermal insulation. This vacuum vessel provides all the ports for the cryogenic circuits, the instrumentation and the insulation vacuum. The MINERVA prototype cryomodule breakdown is shown in Figure 2-2.

Figure 2-2: the MINERVA prototype cryomodule breakdown

To sum-up, a cryomodule is an accelerator building-block; as such it concentrates all the necessary processes for the proper beam dynamic operation, i.e. the beam vacuum, the insulating vacuum, and not the least important, the cryogenics and the RF ones. All the beam dynamics requirements, including the reliability, thus set all the constraints to this complex object, which is central in the accelerator project development.

A last constraint, overlooked so far but that becomes obvious by looking at Figure 23is the necessity for the cryomodule to be compact, to fit in the tunnel. It must also connect to all the cryogenic lines via the valve-box, to the RF via the coaxial waveguides and to the auxiliary networks, namely the pressurized air, the water and vacuum systems (insulation and beam).

SYSTEM MODEL OVERVIEW IN A PROJECT DEVELOPMENT CYCLE

Just making a model is no self-sufficient goal: a model depends on its final purposes, which drives its complexity. In a project, different kinds of model are used, with specific purposes whether used in the early sizing phase or during the commissioning or operating phases.

For instance, in the industry (automotive, space, aeronautics …), component models based on 2D or 3D-spatial discretization methods of different approaches (finite elements, finite volume or finite difference) are used to find an approximate solution of a differential equation set, at every mesh grid point. The equation domains can be mechanical, thermal, fluid, magnetic or any other physics, even mixing them (multiphysics). Many commercial software are available for this purpose, like COMSOL [START_REF]COMSOL[END_REF] and ANSYS [START_REF][END_REF]. Those models typically utilize meshes in the order of 10 6 mesh elements or (much) more. They are used primarily for detailed and local analysis, when no experiment is available yet, to support a component design. These "component design models" can also be used for test analysis, but limited to a narrow scope, as they run thousands of time slower than the real time, at least on a common computer hardware. As an illustration, over the MINERVA early studies, detailed 3D design models have been used for numerical studies of the critical parts of cryomodule and its valve box ( [START_REF] Dieudegard | TECHNICAL NOTE MECHANICAL AND THERMAL DIMENSIONING[END_REF], [START_REF] Junquera | Update on cryogenic plant and cryogenic distribution systems requirements for MINERVA 100 MeV[END_REF] and [START_REF] Saugnac | THE MYRRHA SPOKE CRYOMODULE[END_REF]). By contrast, in scientific fields where the sub-systems are especially numerous and interdependant, such as rocketry, combustion engine, batteries, etc, another class of models can additionally come into play. Those models are called "system models": they encompass the whole system physics, by modelling all the sub-system and their physical interactions. They are by construction, multi-physics and can even be real-time capable, if simplified enough. When sufficiently detailed, they can also capture the physical transient behaviours; they are then called "dynamic system models". As they physically react like the entire real process, they are sometimes also called "digital twins". The drawbacks compared to the "component design models", are the required simplifications to cope with large scopes while being able to run close to real-time; those simplifications result in a much lower accuracy.

A first important purpose of these models, is to provide a global design evaluation tool. In large projects, this is done in close connection with the requirement management tools like DOORS from IBM [START_REF] Doors | [END_REF]: any sub-system constraint or boundary condition update is made public to all sub-component responsible, which triggers new iterations until the global design gradually converges. When instantiated on a past project, a system model can be used to make very quick initial design investigations to orient the design according to the new project highest requirements.

A second main use is the control system design and optimization. Following widespread concurrent engineering methodology, a system control design should start as soon as possible, as it can be a show-stopper for some concepts. The digital twin can then be used to mimic the reaction of the physical process being designed, and perform series of virtual validations of the control system, years before the real equipment installation.

As said, the system model accuracy is not its strength; but, in practice, it does not matter, as its primary goal is the prediction of outputs, in reaction to design updates, boundary condition changes, control setpoint or even control algorithm changes. The problem of accuracy is addressed by using such a model relative to an absolute anchoring; this anchoring consists in a real-scale reference test, such as those provided by the consecutive prototype test phases. In the system model approach, hence, the link between the tests and the model must be very tight.

In practice, in large companies, the engineers in charge of the system modelling belong to a separate transverse team, close to the project management 28 .To perform accurate predictions, they must regularly interact with the component owners, to discuss simplifications and validate their results. They must also define a process to store the numerous model parameters in a transparent and update-friendly manner, over the whole project length. Such an internal company process, when well-established, is a great asset for the project development. However, starting such a process from scratch is a heavy task; the next section exposes why it would be relevant for the MYRRHA/MINERVA project.

PURPOSE OF A DYNAMIC SYSTEM MODEL FOR MINERVA

Back in 2016, the GANIL cryogenic team, and CEA/DSBT, started the implementation of a dynamic system model for SPIRAL2: a thesis [START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF] was dedicated to the creation of the models of the cryoplant, the cryogenic distribution and the cryomodules. This model brought major contributions during the many years of the commissioning [START_REF] Ghribi | Cryogenic Tests of the SPIRAL2 LINAC Systems[END_REF]: support for process analysis and dynamic heat loads measurement [START_REF] Vassal | UN FACTEUR QUALITÉ D'UNE CAVITÉ ACCÉLÉRATRICE D'UN ACCÉLÉRATEUR DE PARTICULES[END_REF], as well as control optimization [START_REF] Vassal | Dynamic modeling and control of the SPIRAL2 cryomodules[END_REF]. However, looking in particular at the cryomodule models, although successful, the implementation came at a very late project stage, and hence, did not benefit from the experience of the prototype tests. The model could not be used either for process optimization prior to the design freeze and construction. By contrast, this thesis covers the MINERVA prototyping period: first, compared to a series cryomodule, anchoring the model onto such a reference is facilitated by use of dedicated and extensive instrumentation and tests. During the prototype test phase itself, the dynamic system model is meant to play a major part: at the same time that the tests are used to validate the model, the model helps to prepare, run and analyse the tests.

Then, the commissioning of the series accelerator being planned for 2027, there is still some room for design changes, based on the predictions of the model, before the final freeze. In the longer term, in view of the site acceptance tests (SAT) and commissioning, a digital twin is doubtless necessary to conduct test analysis, failure investigations, and multi-level control optimization; the anchoring on the prototype tests will likely be very beneficial.

Figure 2-4: the intended dynamic system model main contributions along the MINERVA project cycle

Figure 2-4 summarizes the intended purposes of a dynamic system model, in the context of the MINERVA project development cycle. The blue-highlighted items are the one tackled in particular by this thesis, and respectively the object of parts 5 and 6.2.

The goals and motivations of the MINERVA dynamic system modelling have just been exposed; now the scope of the model, necessary to achieve those goals, is detailed.

TARGET SCOPE OF THE DYNAMIC SYSTEM MODEL

In the cryogenic domain, several famous example of dynamic process modelling exist, among others:

• The CERN cryoplant [START_REF] Bradu | PROCOS: a Real Time Process Simulator Coupled to the Control System[END_REF] • The ITER superconducting magnet system [START_REF] Savoldi Richard | The 4C code for the cryogenic circuit conductor and coil modeling in ITER[END_REF] • The SPIRAL2 cryomodule model [START_REF] Vassal | SPIRAL2 Cryomodules Models: a Gateway to Process Control and Machine Learning[END_REF] •

Commissioning support The CERN cryoplant model is a very detailed process model, encompassing the interactions between the cooling loops, as well as detailed models of the subcomponents. It is actively used for, control optimization, continuous process improvement [START_REF] Bradu | Liquid helium level regulation improvement in the LHC electrical distribution feedboxes[END_REF] and beam-induced heat load analysis [START_REF] Bradu | HOW DOES A CRYOGENIC SYSTEM COPE WITH E-CLOUD INDUCED HEAT LOAD ?[END_REF]. It is also used for operator training: the real process human-machine interface (HMI) is connected to the digital twin for practice, with no risk for the real machine 29 .

The ITER magnet system model is also very detailed, with an emphasis on the interactions between the components. It has been used to analyse transient operations, like cooldowns, and response to dynamic heat loads.

Conversely, the SPIRAL2 cryomodule model was mainly dedicated to control aspects; the cryomodule model was thus much simplified. Its synoptic is presented in Figure 2345. For instance, the different cryogenic process loops are not physically interacting. Another example is the low level of physics in the heat loading to the helium tank: it is split into static and dynamic loads models, but both are purely calibrated on measurements. This approach is nevertheless perfectly sufficient to cover the cryogenic control aspects.

For the MINERVA cryomodule dynamic system model, more physical details are necessary to achieve the purposes described in the previous section: on a cryogenic standpoint, to account not only for control aspects, but also for the cryogenic process detailed analysis. In practice for instance, physical static heat load models should be implemented, as well as transient features like thermal masses, fluid reservoirs, and all the internal cryogenic interactions.

The challenge is to implement enough details, while complying with real-time capabilities on a typical standalone calculation machine; the target simulation-to-run time ratio was set to 10-50, in order to enable quick simulation iterations.

A first added-value of this thesis is thus applying the same approaches as the previouslydescribed one, but focused on the cryomodule and with an unpreceded level of details. Plus, the fact that this is a different cryomodule process, with yet an additional subcooling step, as detailed in 3.1.1. 29 Seen during a visit at the CERN cryogenic operation centre The second originality is on the RF model standpoint. In all three previously described studies, the dynamic heat loads come into play and interact with the cryogenics. The particularity of those heat loads, is that they depend on RF (like the duty cycle) or even beam parameters (like the beam intensity). They can be transient and reach different level over a range that can be much wider than the static heat loads.

For instance, the CERN cryoplant and cryogenic distribution model was used to evaluate the reaction of the cryogenic system to the e-cloud induced heat loads. As for the SPIRAL2 cryomodule model, it was used to study the possibility of on-line dynamic heat load measurements.

Overall, there is a growing interest of the accelerator communities to deepen the understanding of the interactions between cryogenics and RF 30 . In particular, in France, this topic is very active within the M4CAST 31 collaboration, and is presently the object of several thesis and internships in the French accelerator community. The US accelerator modelling community is also promoting these transdisciplinary aspects [START_REF] Biedron | Snowmass21 Accelerator Modeling Community White Paper[END_REF]. The demand is driven in particular by the need for process optimization to reduce capital and operational costs in the context of the energy price increase.

In practice, instead of considering the dynamic heat loads as boundary conditions to the cryogenic system, as in the three mentioned studies, the need is to integrate the physics of the dynamic heat loads itself in a more global model, so that dynamic RF and cryogenic interactions can occur, as a function of the beam current and set cavity voltage.

Cavity RF process models have been extensively used over the past 20 years; the most recent studies include FPGA-based implementations of a cavity field and resonance models, for instance developed by LBNL for LCLS-II [START_REF] Serrano | Cryomodule-on-Chip Simulation Engine[END_REF], by HZB [START_REF] Echevarria | Superconducting radio-frequency virtual cavity for control algorithms debugging[END_REF] and by DESY, ISE WUT and DMCS TUL [START_REF] Rybaniec | Real-time Estimation of Superconducting Cavities Parameters[END_REF]. The main purpose is, as real cavities are not available yet, to use cavity models to perform real-time validations of RF control algorithms:

• The quench detection and mitigation • Control of the detuning during an RF pulse, due to the Lorentz forces • Rejection of the detuning due to microphonic perturbations This technique, widely used in the industry, is called hardware-in-the-loop (HIL), when the real control hardware that hosts the algorithm is used, or software-in-the-loop (SIL) 30 Discussions with the GANIL & ESS cryogenic and RF teams, during dedicated workshops 31 Multiphysics Modelling, Machine learning and Model-based Control in Accelerator Sciences and Technologies, a French national collaborative effort for AI if the control hardware behaviour is simulated too.

Those problematics have been extensively studied in the acceleration community. The quench is a well-known cavity failure, where a part or the whole cavity niobium lose its superconducting state. The Lorentz forces are a cavity detuning effect caused by the physical deformations induced by the RF field rise or fall. Finally, the microphonic noise is a random pressure perturbation, with a large frequency spectrum and some peaks at specified frequency, inducing a cavity detuning of a few Hz amplitude. All these notions are further detailed in part 4.5. Even more recent studies focus on model-based or Artificial Intelligence (AI) based fault-classification algorithm, more detailed in 0.

On MYRRHA specifically, those problematics have been studied for almost 15 years by the LPSC laboratory. In [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF], a model was set-up for the medium-β elliptical cavities (the double-Spoke section had not been decided yet), together with the RF controls. In [START_REF] Bouly | MYRRHA Superconducting linac & Fault Tolerance Concept[END_REF], the model was updated towards the single-spoke cavity section. Compared to the other previously-described RF studies, this model is implemented in MATLAB, rather than embedded in a fast computer chip; it is also meant for RF control virtual commissioning, yet not in real-time: for instance, the MINERVA prototype RF field control PID controller gains could be pre-calibrated. The scope of this model is also quite large, compared to the others: it encompasses the whole high-power RF process, i.e. the cavity, its power coupler and the high-power lines up to the high-power amplifier.

The truly original topic tackled by these studies is the feasibility of the control sequence associated to the fault-tolerance mode, in particular the fast-retuning procedures and the requirements it sets on the RF control loops and equipment. As further explained in 4.3, the model already incorporates all the necessary RF internal couplings. In addition, it computes the cavity dynamic heat loads. This RF model scope is thus a good starting point to satisfy the objectives detailed in the previous section, yet not sufficient.

First, on the cavity level, the physical coupling between the cavity wall and the helium bath that surrounds it must be implemented, as implicitly suggested in [START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF], to properly reproduce the dynamic response of the bath pressure to a dynamic heat load pulse.

Besides, a second RF part gives rise to dynamic heat loads: the power coupler. Although weaker than the cavity dynamic heat loads, the coupler dynamic heat loads are also visible for the cryogenics. Power coupler design for MYRRHA have been extensively studied [START_REF] Souli | Etude et développement d'un coupleur de puissance pour les cavités supraconductrices destinées aux accélérateurs de protons de haute intensité[END_REF], [START_REF] Geslin | Étude de coupleurs de puissance hyperfréquence pour accélérateurs supraconducteurs[END_REF] and [START_REF] Bondoux | Pré-étude thermique des coupleurs de puissance pour les cavités Spoke à 352 MHz[END_REF], by detailed 2D or 3D "component design models", but dynamic interactions with the cavities and the cryogenics have never been considered.

To sum-up, the cavity operation has been studied a lot in terms of field control, but very little in terms of process model coupled to the thermal aspects. On the contrary, the coupler thermal and RF designs have thoroughly been investigated, but not integrated in dynamic RF and cryogenic process models. This thesis proposes to fill the gap by developing a combined cryogenic and RF dynamic process model of the MINERVA prototype cryomodule. The desired scope of the model can be summarized by Figure 23456: the cryogenic and the high-power RF processes, coupled in the cryomodule, as well as their control systems.

Figure 2-6: the synoptic of the desired cryogenic and RF coupled models, and their respective control systems

Beyond the scientific research interest, combining RF and cryogenics in a common model is all the more relevant for MYRRHA: being a high-intensity high-power linac at the edge of what is currently feasible, the expected dynamic heat loads have a large impact on the cryogenic operation. Furthermore, although the reference strategy to control the large beam power variations implies to keep the RF fields constant, hence the cavity dynamic heat loads, the coupler heat loads are expected to vary as a function of the beam power. Finally, the fault-tolerance mode involves fast RF transients, that are doubtless not transparent for the cryogenic system; these procedures also require the cryogenic system to cope with at least two different RF setpoints (nominal and fault-tolerance).

These elements call for a representative modelling of the RF and cryogenic controls, on top of the respective detailed physical processes. The next section emphasizes this latter aspect by introducing the notion of mutual control perturbations.

Cryo PLC LLRF

PERSPECTIVE OF A CRYOMODULE CONTROL VALIDATION TOOL

In the previous part, an important topic, illustrated in Figure 2-6 was overlooked: the mutual perturbation of the cryomodule controls. Indeed, three control systems are simultaneously at work: the cryogenic one, monitoring the helium bath level and pressure, the Low-level RF (LLRF) one, controlling the cavity field amplitude and phase, and the cavity resonance frequency one (CTS for Cold Tuning System), acting by deforming the cavity.

As implicitly mentioned, in a cryomodule, the RF and cryogenic physics are mutually coupled, as highlighted in Figure 2-6 by the red circular arrows. The first coupling comes from the high-power RF wave heat load depositions along the coupler and on the cavity walls, that affects the helium bath pressure and level. The second coupling is due to the bath pressure variations that deform the cavity, thus shifting its resonance frequency and increasing the required amplifier power to maintain the accelerating gradient.

The problem being that the cryogenic controls on the one hand, the field and resonance controls on the other hand, are, by design and construction, independent. Practically, it means that almost no variable connects to one another: both control systems thus see the other process as non-measurable perturbation, and no anticipation is possible; hence, a tool that would enable to evaluate the cohabitation of the different control systems seems necessary.

True, the two RF processes (field and resonance) are also internally coupled: a change in the cavity RF field shifts the resonance frequency because of the Lorentz forces. However, since their respective control systems share the same hardware (see 4.4), anticipation strategies (like feed-forward) are possible.

Ultimately, the risk is to perturbate the cavity field regulation, which stability is the key to get a stable beam for the reactor. The beam dynamics study during MYRTE gave the stability requirement for the amplitude and phase of the accelerator field: respectively 0.3% rms and 0.3° rms [START_REF]MYRTE D2.3 Digital LLRF development[END_REF]. In practice, each cavity has specific 𝑉 𝑐𝑎𝑣 amplitude and phase 𝜙 𝑠32 targets relative to the bunches, as illustrated in Figure 2-7 (𝑉 𝑚𝑎𝑥 , 𝑉 𝑐𝑎𝑣 and 𝑉 𝑎𝑐𝑐 are defined in 0).

Figure 2-7: beam dynamic with respect to RF, adapted from [86]

To summarize, the MYRRHA ADS accelerator is much different from the already existing machines:

• Stringent beam stability and availability • A large number of required stable cryogenic operating points • A high energetic efficiency, banning the systematic use of heaters • Strong couplings implying a risk of mutual perturbations • Cryomodule-to-cryomodule hardware dispersions and drift For all these reasons, the series cryomodule cryogenic and RF control algorithm will likely feature major breakthroughs.

To conclude, an important additional objective of the developed system modelling in this thesis is to pave the way to a tool that can help evaluating the robustness of this complex and new future cryomodule control algorithm. The next section details the practical guidelines to achieve all the system modelling aforementioned goals.

GENERIC MODELLING GUIDELINES

As said, the high-level goals of the model are the support of the prototype test phase and the support of the series design and controls, addressed respectively in parts 5 and 6.2. In this purpose, the cryogenic and RF process models, respectively described in parts 3 and 4, were developed following those general guidelines:

• A low accuracy versus the tests (+/-10 %) is tolerated • The precision over the widest operating range is prioritized • Correct variable tendencies in transient state are also prioritized • A minimum simulation-to-run time ratio of 20 for the cryogenic process

• Re-use any available piece of model, even partial, instead of redoing • Gather a maximum of tests to compare to, even partial and low-quality

The first task consisted in identifying the set of differential equations that best describe the respective process physics, at the same time as finding the right level of simplification, to respect the minimum accuracy and simulation-to-run time requirements.

The second, the most-time consuming, consisted in investigating on the real parts that have been manufactured (geometry, materials) to get state-of-art assumptions for the model parameters. In some cases, simple process models were already available from former or present design teams, or even supplier tests or 3D calculations; on specific components, additional tests or 3D calculation were performed, to give an insight on which effects can be neglected or not.

A related requirement, is to make this quite complex model follow-up and updatefriendly as well as tightly connected to the project up-to-date assumptions. In short, it is planned, from the beginning, that this model can be integrated as a part of the MINERVA project requirement management and concurrent method tools.

THE CRYOGENIC PHYSICAL PROCESS AND CONTROL MODEL

THE MINERVA CRYOMODULE CRYOGENIC PROCESS

Basics of the subcooled cryogenic process

The fundamental role of the cryomodule (QM) and valve box (QVB) is to maintain the cavity niobium walls under the superconducting transition temperature of about 9.2 K, and to evacuate the heat produced by the cavities, called the dynamic heat loads and from unavoidable parasitic sources, called static heat loads.

All superconducting linacs worldwide use an isotope of helium, 4 He, as a heat transfer medium, in the form of a saturated bath surrounding the cavities, as its saturated liquid temperature ranges from about 4.2 K at 1 bar to 2 K at 30 mbar or even below.

Another isotope of helium 3 He is used, among others, in dilution cryocoolers and detectors, but in any case, not associated with cavities. In the following, the word helium will be simply used instead of 4 He.

Operating at lower temperature decreases the dynamic heat load at a given RF field, as explained in the cavity model, in part 4.3, but at the cost of a lower Carnot efficiency for the liquid helium production. For MINERVA, the optimum was found to be 2 K [START_REF] Junquera | Analysis of the optimal temperature of the MYRRHA superconducting Linac[END_REF].

Another advantage to use helium below 2.17 K, the temperature of the lambda point is that helium is in a specific state, that only helium exhibits, called "superfluid" or He II.

Normal helium, also referred as He I, has fully standard properties; for instance its dynamic viscosity, around 1.5 10 -5 Pa.s, and its thermal conductivity, about 3 10 -2 W/m.K are close to those of air at room temperature. By contrast, He II features unusual and counter-intuitive properties: a close-to-zero viscosity and a thermal conductivity about 1000 times larger than that of oxygen-free high thermal conductivity (OFHC) copper [START_REF] Van Sciver | Helium Cryogenics[END_REF]. Thanks to the latter, the heat deposited at whichever point of the cavity surface by the RF field is instantaneously transferred to the liquid/vapor interface, avoiding the emergence of bubbles, thus decreasing the microphonic noise, which is an asset, although no critical for MINERVA. Figure 3-1 shows the helium phase diagram, and points to some operating conditions of well-known SRF cavity and magnet projects.

One should keep in mind that, even extremely large (in the order of 10 kW/m 2 over a 1 meter space) compared to the typical dissipated cavity power, the He II thermal conductivity is not infinite, limiting the maximum of heat extracted by the superfluid helium in a saturated bath. This limit is set by the competition between the vertical temperature gradient, proportional to the heat flow and inversely proportional to the bath horizontal section, and the temperature margin provided by the hydrostatic pressure. To maximize the heat extraction, the horizontal section at every height should be as wide as possible (no bottleneck).

As a side note, niobium-based superconducting magnets are cooled by pressurized helium and thus do not experience such limitations. The reason for them to be cooled with pressurized helium is to avoid the poor dielectric strength of low-pressure helium vapor. On the other hand, the problem of very long distance heat transport remains problematic [START_REF] Quack | Cryogenic Engineering[END_REF].

Figure 3-1: helium 4 phase transition diagram and some operating points from [START_REF] Van Sciver | Helium Cryogenics[END_REF], [START_REF] Peterson | Normal Cryogenic Fluid Behavior[END_REF] In every accelerator facility that makes use of saturated He II for the cavity cooling process, such as ESS [START_REF] Arnold | ESS Cryogenic System Process Design[END_REF] or LCLS-II [START_REF] Soyars | Status of the LCLS-II Cryogenic Distribution System[END_REF], a cryoplant provides supercritical (strictly speaking still liquid) helium at about 3 bar and 4.5 K. The advantage of supercritical helium compared with the saturated liquid is that the helium remains monophasic when it undergoes heat loads over long-distance transport, from the cryoplant building to the accelerator tunnel, thus avoiding the formation and trapping of vapor.

The supercritical helium then undergoes a theoretically isenthalpic expansion, called the Joule-Thomson expansion from the supercritical pressure to the design sub-atmospheric pressure. Since this expansion occurs below the Joule-Thomson inversion temperature, this process decreases the helium temperature. A major drawback is that an important part (> 35%) of the helium turns into vapor. That is why a subcooling process, consisting in the pre-cooling of the supercritical helium by use of the exiting 2 K vapor is implemented. One utilizes a heat exchanger, that can be either common to many cryomodules, as in LCLS-II cryogenic process, or specific to each one, as it is the case for ESS or MINERVA. There are many project constraints that lead to the optimal position of the heat exchanger. Without listing them all, one constraint which is worst highlighting, is minimizing the 2 K vapor return heat load before the vapor enters the heat exchanger, for obvious thermodynamic efficiency reasons. This tends to set the position of the heat exchanger as close as possible to the 2 K cavity tanks.

As seen in the next section, the test conditions of the MINERVA prototype cryomodule are different from the final accelerator installation. Instead of supercritical helium the cryomodule is supplied with saturated helium, typically at 1.1 bar (thus at 4.31 K). But the superfluid helium production process is the same. As said, the role of the heat exchanger is to decrease the amount of vapour produced by the expansion down to 30 mbar. As seen in Figure 3-3, in the case of saturated helium at 1 bar: without the HX, about 37% of vapour is produced; only 8% if the liquid is subcooled down to 2,5 K before being expanded. In the case of supercritical helium, the amount of created vapour is substantially superior when not subcooled before expansion, as one can infer from the pressure and enthalpy isolines of Figure 3-3.

The MINERVA prototype cryogenic process

The design phase of the MINERVA prototype cryomodule ( [START_REF] Saugnac | THE MYRRHA SPOKE CRYOMODULE[END_REF], [START_REF] Saugnac | Spoke Cryomodule Prototyping for the MINERVA Project[END_REF]) started in end of 2017; then came a tendering phase and a manufacturing phase which, being delayed by the covid-19 pandemic, ended up in February 2020 when the components were delivered at IJClab.

The prototype setup consists in the cryomodule itself (protoQM or QM) and its valve box (testQVB or QVB) that makes the interface between the IJClab test infrastructure and the protoQM. The testQVB and the protoQM together form the prototype cryomodule cell (protoQCELL or QCELL). The goal of this prototype cryomodule was to:

• Validate the sub-components (functional performances and control)

• Validate the assembly procedure • Provide a test stand for the "fault-tolerance" procedures As a consequence, the prototype cryomodule was thought to be relatively close to the series. Only the cavity alignment procedure was not in the scope of the tests.

As in the series final configuration, the cryomodule contains a pair of SRF cavities which are maintained at 2 K in a pool of saturated liquid helium at about 30 mbar. The envelope

2 2 1' 1 Expansion X = 37% X = 8%
of this pool, made of titanium, is what forms the cavity tank. In order that the niobium cavities are always submerged and, as exposed earlier, to provide margin with respect to He II boiling, a buffer tank is placed above the cavity titanium tanks. The reading for the level control loop is taken in this tank. For convenience, in the following, the two cavity niobium tanks and the common buffer tank will be referred as "2 K tank".

The flow schemes and the protoQCELL main components are displayed in Figure 34all what is shown in this figure is in the scope of the model.

The level control of the 2 K tank is done via the control valves CV04 (heat exchanger bypass) in 4 K mode, i.e. when the cavities are operated with normal helium at atmospheric pressure, or via CV05 for superfluid operation, called the "2 K mode". A second difference between the 2 K and the 4 K modes is the flow path of helium through the infrastructures as seen later in this section. The CV03 control valve can also fill the 2 K tank, although from the bottom, by contrast with the CV04-CV05 common line that fills it from the top. The CV03 line is thus only used during the cool-down, in order to benefit from the vapour high enthalpy to cool first the bottom of the tanks. The filling from the top and from the bottom together form what is called the main process line.

Figure 3-4: the MINERVA valve box and cryomodule cryogenic process

To optimize the global thermal efficiency, two cryogenic lines intercept the incoming heat from the parts in contact with the 2 K tanks:

• A high-temperature (HT) line circulating in a serial pattern across the protoQM thermal shield (TS), the coupler and the cavity supporting system and controlled by CV651

To infra

Jumper QVB Jumper QM

• A low-temperature (LT) line circulating, also in a serial pattern, across the coupler and the cavity supporting system, and controlled by CV02

Those three process lines (the main one and the two intercepts) are supplied by the testQVB; the industrial philosophy chosen for MINERVA, was that the testQVB should contain all the process active parts, like the valves and the heat exchanger. As explained earlier, the position of the heat exchanger is a trade-off between industrial constraints and process efficiency: the further away the HX is from the 2 K tank, the larger the flash is because of the heat loads warming up the returning 2 K vapor.

Additionally, the testQVB has its own thermal shield, cooled by an independent process line and also fed with saturated liquid nitrogen, controlled by CV601.

Contrary to the protoQM, which main design was frozen during the MAX project, the testQVB was not designed for a validation of the serial design, the latter not being studied at the time of the prototype design. Instead, it was designed to be compatible with the IJClab cryogenic supply infrastructure constraints. The main difference versus the final accelerator conditions are:

• The HT intercept line is fed with saturated liquid nitrogen at about 80 K from a mobile Dewar instead of supercritical helium at about 50 K • The LT intercept line and the main process line are fed with saturated liquid helium at about 4.3 K from a mobile Dewar instead of supercritical helium at about 3 bar and 4.5 K • To get rid of the vapor from the Dewar transfer line, and to provide a buffer during helium Dewar rotation, the testQVB features a 70 L liquid helium tank

As a matter of fact, LT intercept line is called in the following the "10 K loop". This naming refers to the loop target exhaust temperature, after collecting the heat loads along its path. The mass flowrate is thus a resulting variable.

Although the testQVB design is far from the series, several functional items like the valve sizing and the heat loads could be accurately measured, providing valuable data for model validation, as well as a fine anchoring point in view of the series design study. This is precisely the role of a system model, anchored on a real test, to make predictions in view of the next project development phase.

The IJClab cryogenic helium recovery infrastructure

The previous section exposed the testQVB and protoQM, as well as the IJClab cryogenic supply infrastructure, comprising the saturated nitrogen and helium Dewar all in the scope of the model. The mobile Dewar are considered as the inlet boundary conditions, thus, the IJClab helium liquefier is not in the scope.

On the other side of the process, the IJClab recovery infrastructure is also in the scope of the model. Indeed, some actuators like the helium tank pressure control valves are outside of the testQVB and protoQM. The helium and nitrogen flowmeters, and their associated cryogenic lines, are also part of the helium recovery infrastructure. Most importantly, the warm compressors responsible for the sub-atmospheric helium pressure in 2 K mode, should be modelled for an optimal reproduction of the pressure evolution downstream of the cryomodule helium pressure-control valve.

The cryogenic recovery infrastructure is shown in Figure 345. Again, the whole schematic content is within the modelling scope.

Figure 3-5: the MINERVA cryogenic infrastructure process flow diagram at IJClab in Orsay

For all the helium and nitrogen lines, the first element entering the recovery system is the reheater (not shown in Figure 3-5 for the nitrogen): contrary to an accelerator facility that would use a refrigerator and keep all cryofluids as cold as possible for efficiency, the use of liquefier for testing the prototypes involves the recovery operation at ambient temperature.

The 80 K circuits of both the protoQM and the testQVB run through two consecutive flowmeters: a thermal mass flow meter for the data acquisitions and a volumetric (diaphragm) one for calibration and regular cross-checks. The nitrogen is then routed out of the test pit building to the atmosphere.

The 10 K loop is routed through a dedicated thermal flowmeter; for the prototype, a ΔPbudget analysis was done, considering a limited total pressure difference of 100 mbar (the available ΔP is limited), the need for a minimum valve ΔP (for flowrate controllability) and the ΔP of the lines (essentially the reheater low-density downstream lines). It was concluded that, to circulate 1 g/s for the cool-down sequence (as a comparison, a flow of about 200-300 mg/s is needed in nominal), the flowmeter pressure drop should remain below 10 mbar at 1 g/s, thus imposing a specific low-ΔP technology. As a matter of fact, this limited available pressure difference during the cryomodule test phase might become problematic for the coupler heat intercept concepts other than the MINERVA prototype (see 4.7.1.1), where the flow path is much longer and involves higher temperature thus lower densities, as explained in 6.1. The 10 K loop ends-up in the helium atmospheric recovery line, that leads to a helium reservoir at atmospheric pressure. The level of this reservoir is monitored and when close to full, the helium is pumped into a high-pressure 200 bar storage, for later re-use by the liquefier.

The testQVB phase-separator tank exhaust line, named "4 K" in Figure 3-5, meets two parallel pressure-control valves CV06 and CV07, which different flow coefficients enable to regulate a wide range of flow rates, from 0.01 to 0.6 g/s. This range was considered to encompass the cases when the mobile Dewar is disconnected (to be refilled) and when it is connected; in the former case, only the QVB internal loads contribute to the evaporation flowrate; in the latter, the loads of the filling line are dominating by far. The testQVB exhaust line encounters a flowmeter before being routed to the atmospheric helium recovery. The flowmeter is actually made of two volumetric-type units (rotary and diaphragm), to address the flowrate range.

Finally, the main process loop takes two different paths whether operating in 4 K or 2 K mode. In 4 K mode, the flow is routed to Site 3 flowmeters by the header line "2 K pumping line 3"; the path is thus similar to the QVB 4 K tank exhaust (Site 4). In 4 K mode, the pressure regulation "MKS valve" (from the MKS supplier) is not used: the 2 K tank pressure equals the recovery pressure, modulo the pressure drop. In 2 K mode, by contrast, the MKS valve regulates the pressure at 30 mbar, from a downstream suction pressure of about 1-10 mbar, depending on the flowrate and pumping speed curve of the warm compressor (hence the need to model them). In this mode, instead of Site 3, the flow is conveyed by the header line "2 K pumping line 1" or "2 K pumping line 2" (or both) towards one or both of the warm compressors. The warm compressors have a cumulated capacity of 120 m 3 /h helium, or roughly 120 W equivalent heat load to be extracted from the helium bath at 2 K 33 .

The different flowmeter characteristics are summarized in Annex 7.3. One major point to highlight, is the importance of the flowmeters to perform heat load measurement, as explained in 5.2.4. The IJClab infrastructure was hence deliberately equipped with the adequate flowmeters on each cryogenic loop. The same goes for the control valves (cryogenic and warm): they were sized for the specific operation of the cryomodule; the precise knowledge of their characteristics is crucial for the virtual commissioning of the control, as explained in 5.2.9.

MODELLING ENVIRONMENT

A common multiphysic simulation platform

The cryogenic process mixes several physical domains, such as the fluid dynamics and the heat transfer, whereas the RF processes involve the laws of electromagnetism. The common platform for cryogenic and RF processes must thus have multiphysic capabilities. It should be noted that separate simulation platforms for RF and cryogenics could be performed with co-simulation, but is not favoured in this study.

Since especially the cryomodule model is meant to be rather detailed, a bottom-up approach is favoured, where:

• The cryomodule and valve box are broken down into component models

• Each component model is made of generic elementary physical building blocks

In this approach, the physical building-blocks contain the basic physical equation (algebraic, differential or both) to solve, coded in a typical object-oriented language.

Those equations should contain time-dependant variables (like mass and heat storage) to enable transient behaviour reproductions. Clearly, these requirements rule out the possibility to use Excel as a main platform; however, Excel is used on the one hand, to store the fluid and material property database, see 3.2.4 and 3.2.5; on the other hand, to address the parameter allocation problem, as seen in 3.2.3.

By connecting those blocks -standing for pipes, rods, or valves for instance -one builds a model of a component, like a coupler, a heat exchanger or a supporting system. Generally, some tens or a hundred of blocks associated to 1-dimension (1D) discretization are enough to reach the desired accuracy.

This approach, a standard practice in the industry involving complex sub-system assemblies (batteries, engines, rocketry, bioscience …), enables a flexible component model construction, and the complexity can be tuned by adjusting the number and the type of blocks. The arrangement, the number and the internal building-block parameters are determined by thorough investigations of the components. Finally, the buildingblocks are designed to be common to the cryogenic process components and to the RF process, as much as possible, although specific RF blocks were also needed.

Cryogenic and RF process models have been developed over the past few years. On the cryogenic side, several libraries of cryogenic building blocks exist on different graphical interface and drag-and-drop platforms:

• Cryolib from the EcosimPro platform, originating from CERN [START_REF] Bradu | PROCOS: a Real Time Process Simulator Coupled to the Control System[END_REF], [START_REF] Bradu | Modélisation, simulation et contrôle des installations cryogéniques du Cern[END_REF] • Simcryogenics in MATLAB/Simulink (MathWorks), developed at CEA/DSBT [START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF], [START_REF] Bonne | Modélisation et contrôle des grands réfrigérateurs cryogéniques[END_REF] • 4C in Modelica, from Dipartimento di Energetica, Politecnico di Torino [START_REF] Zanino | Dynamic modeling of a supercritical helium closed loop with the 4C code[END_REF] • Dymola on Modelica from Dassault Systèmes [START_REF] Chao | Dynamic simulation of the target moderator cryoplant and cryogenic transfer line at the European Spallation Source[END_REF] • Aspen HYSYS from AspenTech [START_REF]Aspen HYSYS[END_REF] Those platforms are in fact generic multiphysic process modelling tools, which incorporate cryogenic fluid and material properties. None is fundamentally bad, but several reasons pointed toward the MATLAB/Simulink option 34 , and more particularly to the Simscape physical modelling library, since:

• Several RF models written in MATLAB already exist, as explained • The possibility to use the control optimization MATLAB toolboxes • The good compatibility with the Siemens software (prototype cryogenic controls) • MATLAB is widely spread in the industrial and academic sectors • The reasonable software costs, even including all the toolboxes • The support from MathWorks proved to be efficient and helpful Simscape is an object-oriented language that provides physical modelling building blocks to b used in the same graphical interface as Simulink blocks. There are several differences with Simulink. First, Simscape is a unit-commensurate language, meaning that every parameter and variable has an inherent declared physical unit. Another difference is the role of the links between the blocks. The connecting lines in Simscape transport a bunch of physical variables, which propagation direction respect the laws of physics: for instance, temperatures are transported in the flow direction, whereas pressure evolves in the counterflow direction. Simulink, by contrast, features arrows that transport only one variable, and which direction imposes the direction of the information (causality). This latter difference is of primary importance in order to get a compact and readable graphical code. In Figure 3456, Simscape blocks are compared to Simulink ones. The colours in the Simscape blocks has a purpose: each one is associated to a physical domain.

Figure 3-6: Simulink blocks (left) and Simscape (right)

Simscape and Simulink blocks can co-exist in the same model: variable extraction and conversion blocks exist for that matter. Indeed, some code are more relevant to be kept in Simulink, like the control algorithm, or even some physical processes which are simple enough.

Simscape natively offers a panel of more than ten physical domains, that spans from mechanics, to electronics and thermodynamics. A domain is defined by two sets of variables: "through" and "across". The "through" variables are the variable of flux type, which evolution is ruled by conservation laws, coded in the blocks. The "across" variables are variables of potential type, present at the boundary of each block. For instance, the "Thermal" domain (the orange colour in in Figure 3456) transports the heat flow as a "through" variable and puts the "across" variable temperature at the block boundaries. Likewise, the "Two-phase fluid" domain (the blue colour in in Figure 3456) transports the energy flowrate and the mass flowrate as "through" variables and sets the pressure and specific internal energy as the "across" variables.

Furthermore, Simscape provides the possibility to declare new physical domains: this was chosen for instance for the Simcryogenics library. In the frame of this study, it was indeed necessary to create a new "RF" domain.

For the cryogenic process, the two physical domains "Thermal" and "Two-phase fluid" were evaluated and proved to rightly incorporate the physics in order to support the development of new physical blocks, as explained in the next section.

Cryoscape, a new building-block library

Re-using the "Thermal" and "Two-phase fluid" domains and starting from the Simscape native component codes enabled to save a significant amount of time. Another advantage, is to benefit from the updates and improvements from MathWorks on their existing domains, as well as it eases the interaction with the MATLAB helpdesk. The main initial task was to incorporate the fluid and material properties, as explained in 3.2.4 and 3.2.5.

Although sharing the same domain as the existing native Simscape blocks, the newly created blocks were stored in a separate library, for easier model sharing; the new library is called Cryoscape, and showed in Figure 34567.

Figure 3-7: the Cryoscape library, as of May 2023

Over the course of the model development, several new blocks were created, following the approach exposed earlier, i.e. the steps of simplification, the identification of the equations and the investigation of the as-build properties. The building-blocks are detailed in part 3.3, and the component analysis in 0

Model parameter allocation

The detailed and modular modelling approach in this study can involves up to hundreds of building-blocks for a single component; to ease the model follow-up and update, a specific procedure was implemented and frozen from the very beginning. This robust parametrizing method relies first on an Excel layer: each component (for instance, a coupler or a heat exchanger) or a group of related components (for instance, the IJClab infrastructure) has a list of its building-block parameters written in several Excel tabs, each tab being a building-block type, such as a tube or a conductance. The second procedure feature is a MATLAB script that reads the Excel file and allocates the values in the right field of the right component in Simscape.

Hence, apart when debugging or for test purpose, changing a parameter value is always performed from Excel and, not from the Simscape block parameter interface. One advantage is to avoid errors when performing batch of changes (avoiding opening and closing hundreds of interfaces). Another advantage is the absence of formulae in the Simscape block parameter interface (the formulae are contained in the Excel file), making it much easier to read. The constraint, by contrast, is the necessity to be rigorous, e.g. in the block name choices and not to forget to report the block in the Excel file.

The process is illustrated in Figure 3-8. To conclude, this kind of procedure is also mandatory in order to document the model creation and ensure a proper potential takeover by other teams.

Figure 3-8: Simscape parameter allocation methodology

Excel table summary: x-axis = Simscape blocks, y-axis = list of parameters and their values MATLAB script to read the Simscape block parameter values and allocate them in the right block in Simscape

Geometrical measurement in CAD, info from datasheets …

The fluid properties

The fluid properties and the material properties are the key enabler for cryogenic simulations: the phase changes on the one hand, and the highly variable monophasic properties of the cryogenic fluids (especially helium) make the constant-property assumption highly inaccurate. The required fluid properties are:

• Density 𝜌 [kg/m 3 ] • Entropy 𝑠 [J/(kg.K)] • Temperature T [K] • Kinematic viscosity 𝜈 [m 2 /s] • Thermal conductivity 𝜆 [W/(m.K)] • Prandtl number 𝑃𝑟[-]
As a side note, the heat capacity is a combination of these parameters: 𝐶 𝑝 = 𝜆𝑃𝑟 𝜌𝜈 . Those properties are necessary to compute the evolution of the variables in all types of equation (conservation laws, behaviour law and macroscopic correlations), as detailed in 3.3.

The fluid properties are given by the equations of state, the simplest being the ideal gas law 𝑃 = 𝜌𝑟𝑇, with 𝑟 the specific ideal gas constant. For fluids within or close to the diphasic domain, this relation becomes inaccurate. The equations of state are given by the commercial library Hepak for helium [START_REF]HEPAK[END_REF], and by the open-source CoolProp library for the nitrogen [START_REF]Welcome to CoolProp -CoolProp 6.4.1 documentation[END_REF]. CoolProp actually also computes helium properties, but has some limitations at temperature below the lambda line. Hepak and CoolProp are Excel add-on functions: the idea is to generate permanent property tables which Simscape can locally interpolate as a function of 2 independent state variables.

Re-using the "two-phase fluid" domain from Simscape imposes a frame for this purpose: the state variables which are used to allocate this table are the normalized specific internal energy 𝑢 ̅ and the pressure 𝑃; 𝑢 ̅ is the specific internal energy 𝑢, normalized by its range boundaries: the minimum and maximum and the saturation boundaries.

𝑢 ̅ spans from -1 to 0 in the subcooled liquid side, from 0 to 1 in the 2-phase zone (thus equivalent to the vapor mass fraction x) and from 1 to 2 in the superheated gas phase. Figure 3456789shows the advantage of using 𝑢 ̅ versus 𝑢 : the liquid and vapor [START_REF] Peterson | Normal Cryogenic Fluid Behavior[END_REF] spaces are fully distinct and rectangular, which eases the table filling process. A dedicated MATLAB function calls the Excel Hepak and CoolProp functions and fills the table with about 70 ×140 values for 𝑃 × 𝑢 ̅, with mesh refinement in the areas of interest. To illustrate, Figure 3-10 displays the properties for helium.

Figure 3-10: the helium fluid properties as a function of P-u

Hepak actually features some property lacks at temperatures below 3-3.5 K. For instance, considering the thermal conductivity; the empty spaces were filled with values in order to avoid discontinuities that would bring about solver failures. As a general note on the helium superfluidity, the abrupt loss of (equivalent) conductivity was not computed in the fluid properties. It was implemented in the equations instead.

As a matter of fact, other independent combinations would be possible, such as 𝑃 and 𝜌 or 𝑃 and ℎ (the specific enthalpy); but not 𝑃 and 𝑇, since both are bounded in the 2phase zone.

To conclude, considering the specific helium and nitrogen variable properties is mandatory to perform accurate cryogenic simulations. To illustrate, Table 3-1 compares helium and nitrogen to water in terms of thermal properties: helium has a latent heat 10 times weaker than the one of nitrogen and 100 times less than water; in practice, it means that liquid helium evaporates extremely easily regarding a heat load. By contrast, helium has a significantly higher sensible heat than nitrogen, which means that cooling down or warming up with helium vapor is very effective.

Table 3-1: comparison of the fluid thermal properties of three substances

The material properties

Modelling a cryogenic process requires the thermal properties of the structural and functional materials: the thermal conductivity 𝜆 [W/m.K] for the evaluation of steady conduction, and the specific heat 𝐶 𝑝 [J/kg.K] for the transients (cool-down and heat-up). For the RF process, the electrical resistivity 𝜌 [Ω.m] of the layers run by currents is also necessary. This section sticks to the thermal properties, but the electrical property implementation follows the same approach.

Generally speaking, in the cryogenic domain, the material properties strongly depend on temperature and differ by orders of magnitude from one material to another. Just like for the fluids, specific material property tables are thus necessary in Simscape. The approach is similar: property tables are generated for Simscape from Excel reference functions, and made available for interpolation during the simulation.

For example, the thermal properties of metallic materials are functions of the temperature and, in the case of non-alloyed materials, to the material purity, assessed by the Ratio of Residual Resistivity (RRR). A material RRR is defined as the ratio of the electrical resistances at 273.15 K (the water melting point) over the one at 4.2 K (the helium boiling point at atmospheric pressure), although this definition can differ, in particular, from the considered reference temperatures [START_REF] Splett | A Comparison of Methods for Computing the Residual Resistivity Ratio of High-Purity Niobium[END_REF]. For a metal, the higher the RRR is, the purest the material is and the fewer crystallographic defects it contains, since at low temperature, in the absence of impurity and default-induced scattering mechanisms, the (residual) resistance converges towards 0.

The two main pure metals used in the MINERVA prototype cryomodule are the niobium for the cavities walls and ports, and the copper, where high thermal conductivity is favoured (thermal shields, thermal braid), or where high electrical conductivity is desired (as for the coupler cuff internal wall, seen in 4.7).

Besides, the MINERVA cryomodule features several alloyed materials for structural purposes: stainless-steel (AISI 304, 304L and 316) in the general case, titanium when a lower thermal conductivity is needed around 60-100 K, or Invar (iron and nickel) when a low thermal shrinkage coefficient is desired. For specific insulation purposes, like tube spacers, a glass fibre made of fibre glass and epoxy compound, like G10, is used.

The thermal conductivity of pure materials, niobium and copper, is compared to the ones of alloyed materials in Figure 3-11. As shown, the thermal conductivities at stake span over several orders of magnitude; hence, a careful attention is necessary. For the copper in particular, it is clear that the knowledge of the true RRR is of primary importance to assess the thermal conductivity, as seen in part 4.7.2.3, for the characterization of the cuff copper coating. For the niobium, the high thermal conductivity, at about 4 K (which is two orders of magnitude higher than the one of stainless-steel), is important to predict potential risks of superconductivity loss due to the heat loading of the cavity induced by the RF power couplers, as seen in part 5.2.4.

Figure 3-11: thermal conductivity comparison between the pure and the alloyed materials

By contrast, the material specific heat does not depend on the material purity, but strongly on temperature, as seen in Figure 3-12. At temperatures below 20 K, the ability of a material to store heat is extremely weak. This has a strong impact, in particular for the cavity heat transfer model, in part 4.6, and for the simulation of transient cryogenic sequence, like a cool-down. 3); the thermal conductivity and specific heat are also affected by this transition. In metals, the main thermal contributions come from the electrons; but as the electrons become superconducting, the effect of the lattice vibration (phonons) becomes of first order.

In [104], a study proposes an estimate of the global thermal conductivity model (𝐾 𝑠 ), as the sum of the normal and superconducting electron contribution (𝐾 𝑒𝑠 ), and the phonon one, (𝐾 𝑔𝑠 ); this resulting function, based on physical equations, is parametrized either using the BCS theory (see part 4.1.3) or using a tailored "Best fit" that gives even lower discrepancies versus their datasets. The respective contributions are illustrated in Figure 3-13.

From [104], the only two parameters needed to estimate the thermal conductivity of a niobium sample at a given temperature are its purity (RRR) and the phonon mean free path, which is roughly the average grain size (AGS), that results from the final heat treatment (hardening processes will make the grain smaller, while annealing will make them grow). Both parameters are available in the data sheet from the MINERVA prototype cavities: AGS = 50 µm and RRR = 300 on average. This model, and its related "Best fit", is implemented in a reference Excel function: it shows a discrepancy of +/-5 % with respect to IJClab measurements from [START_REF] Fouaidy | Low Temperature Thermal Conductivity of Niobium and Materials for SRF Cavities[END_REF]. As a last note, one has to be careful with the RRR definition in the case of niobium, since the resistivity at 4.2 K has no physical meaning; several methods, based for instance on extrapolation from 9.26 K are proposed in the literature [START_REF] Splett | A Comparison of Methods for Computing the Residual Resistivity Ratio of High-Purity Niobium[END_REF]; in this study, the niobium RRR is defined as

𝜌 293.15𝐾 𝜌 9.26𝐾
.

In the normal state, due to the lack of data in the literature for RRR > 50, the thermal conductivity was inferred from the Lorentz factor and the electrical resistivity 𝜌. The Lorentz factor is the product of the thermal conductivity times the electrical resistivity over the temperature 𝐿 = 𝜆𝜌 𝑇 . Several fits from [START_REF] Cannelli | Thermal properties of Nb from acoustic and electrical resistivity measurements in the temperature range 60-340 K[END_REF], [START_REF] Singer | RRR measurement techniques on high purity Niobium[END_REF], [START_REF] Webb | Low-Temperature Electrical Resistivity of Pure Niobium[END_REF] and [START_REF] Morton | The electrical resistivity of niobium and niobium-zirconium alloys[END_REF] enabled to build an electrical resistivity 𝜌 model, as a function of the RRR and temperature. Finally, in order to get a continuous law by sewing the normal conductivity to the superconducting one, the Lorentz factor was slightly linearly scaled to match the know values at 9.26 K and 300 K; this is reasonable regarding the Wiedemann-Franz law that assumes this factor as temperature independent. The overall niobium thermal conductivity model construction is summarized in Figure 3-14.

Figure 3-14: continuous niobium λ function between 1 K and 300 K built for the model developments

A continuous specific heat reference function over the niobium superconducting and normal state was also built: the fit for the specific heat function of the superconducting state is from papers [START_REF] Novotny | Single superconducting energy gap in pure niobium[END_REF], [START_REF] Leupold | Superconducting and normal specific heats of a single crystal of Niobium[END_REF], [START_REF] Van Der Hoeven | Specific Heat of Niobium between 0.4 and 4.2[END_REF] and [START_REF] Brown | The Superconducting and Normal Heat Capacities of Niobium[END_REF], whereas the normal state comes from the CryoComp library. As a side note, the sharp discontinuity around 9.26 K in the specific heat curve of Figure 3-12 is physical.

To conclude, the precise and wide fluid and material property knowledge is fundamental, given the planned broad operating range of the model; a last important point is to keep questioning and extending the bibliographic study. The next sections deal with the physical building blocks, the first layer of the component models.

THE PHYSICAL BUILDING BLOCKS

3.3.1 The generic duct model

The conservation equations

The duct model is used primarily to simulate the physical behaviour of internal pipes and external transfer lines, which are used to convey the fluids between the components. On top of being a connecting element itself, the duct model is also a building block to build other components (heat exchanger channel, thermal screen tubes). Contrary to a volume, a duct is characterized by the length L being large compared to the diameter D (L >> D), meaning that the flow has a preferential direction. Along this direction, the fluid properties evolve because of the energy transfers (pressure drops, heat transfer) or changes in diameter, according to physical laws.

In fluid mechanics, the thermodynamic variables are connected by a set of four equations that close the system and enable to solve it. Those equations are the state equations, as described in a previous section, and the three Navier-Stokes equations. As detailed in [START_REF] Lions | Mathematical topics in fluid mechanics[END_REF], those three equations result from the same conservation equation, applied to the mass m, to the momentum ρ V and to the total energy E. In this equation, the production rate and the transport at the speed V of a variable φ equal the creation or dissipation rate from either a source or sink S:

𝜕𝜑 𝜕𝑡 + ∇. (𝜑𝑉 ⃗ ) = 𝑆 ( 3-1 )
A standard approach to drastically save computational time is to simplify the Navier-Stokes equations by several assumptions into the Euler equations: one assumes that there is no internal viscous force, and no internal heat diffusion. Even in this form, solving those equations remains a solver intensive task, due to non-stationary effects like travelling pressure waves, requiring small time discretization steps. Indeed, although cryogenic processes are slow, the pipes may feature fast transient phenomena, due, for instance, to rapid valve closure (hammer effect) or due to thermally driven acoustic oscillations (Taconis). The choice was made to implement the inertia of the fluid in the momentum balance contained in the Euler equations, but let the user activate it or not when relevant in the simulation.

One point which should be further stressed, is that the axial conduction in the fluid is neglected over the convection, which can be problematic in the case of superfluid helium. Indeed, the superfluid helium forms some tens of cm downstream of the Joule-Thomson valve 35 after the expansion, and thus thermally connects the QM tank to the QVB circuitry and thus could bring about substantial additional transfer heat loads. However, no measurable additional transfer heat loads were observed in 2 K mode versus 4 K mode, validating our approach.

𝑉 * 𝑑𝜌 𝑑𝑡 = 𝑚̇𝐴 + 𝑚̇𝐵 ( 3-2 )
Where, ρ is the mean fluid density of the volume V of the discretized element, 𝑚̇𝐴 and 𝑚̇𝐵 respectively the mass flow rate through port A and B.

𝑑𝜌 𝑑𝑡 can be expressed as the sum of the partial derivatives in pressure P and specific internal energy u, so that one gets:

𝑉 * [𝑃 ̇𝐼 𝜕𝜌 𝜕𝑃 + 𝑢̇𝐼 𝜕𝜌 𝜕𝑢 ] = 𝑚̇𝐴 + 𝑚̇𝐵 ( 3-3 )
The partial derivatives of ρ are then simply interpolated as a function of the state variable P and 𝑢 ̅. The reason to do that instead of simply deriving 𝜌 is that the partial derivatives of ρ can be mapped as a function of P and 𝑢 ̅, and smoothed at will, across the sharp liquid-vapor transitions. The error generated by this smoothing is simply balanced by an additional closing equation. This approach, already present in the original Simscape blocks, is not compulsory as directly computing

𝑑𝜌 𝑑𝑡
works fine too in most cases. But it provides yet another lever to smooth the computationally speaking violent phase 35 Discussion with J-P Thermeau, APC, Paris

𝑃 𝐵 -𝑃 𝐴 = 1 2 𝜌 𝐼 [𝑣 𝐴 2 -𝑣 𝐵 2 ] + ∆𝑃 𝑓𝑟𝑖𝑐 + 𝜌 𝐼 𝑔(𝑧 𝐴 -𝑧 𝐵 ) ( 3-4 )
Where 𝑃 𝐴 and 𝑃 𝐵 are the static pressures, 𝑣 𝐴 and 𝑣 𝐵 the flow velocities and 𝑧 𝐴 and 𝑧 𝐵 the altitude respectively for nodes A and B. At the right-handed side, the first term expresses the total pressure change due to changes in velocity, the second term ∆𝑃 𝑓𝑟𝑖𝑐 is the dissipation by friction with the walls (parietal friction) and the third one is the change in altitude.

The dissipation of mechanical energy due to friction is given by the Darcy-Weisbach equation:

∆𝑃 𝑓𝑟𝑖𝑐 = 𝑓 * 𝐿 𝐷 ℎ * 𝜌 * 𝑣 2 2 ( 3-5 )
Where f is the Darcy friction factor 36 and 𝐷 ℎ the hydraulic diameter and L, the length of the fluid section; f is given by experimental laws, as seen in the next section.

Finally, the third and last conservation is the energy equation:

𝑑(𝑀 * 𝑢 𝐼 ) 𝑑𝑡 = 𝑚̇𝐴ℎ 𝐴 𝑡𝑜𝑡 + 𝑚̇𝐵ℎ 𝐵 𝑡𝑜𝑡 + 𝐻𝑇 ( 3-6 )
Where 𝑢 𝐼 is the specific internal energy in I, ℎ 𝐴 𝑡𝑜𝑡 and ℎ 𝐵 𝑡𝑜𝑡 the total (including the kinetic energy) specific enthalpy in A and B, and HT is the heat transfer from the fluid to the wall (parietal), given by Newton's law:

𝐻𝑇 = 𝑆 * ℎ * ∆𝑇 ( 3-7 )
Where h is the heat transfer coefficient averaged over the surface, S the heat transfer area and ΔT the temperature difference between the bulk fluid and the wall fluid-side temperature. The coefficient h is derived from the Nusselt number Nu:

ℎ = 𝜆 * 𝑁𝑢 𝐷 ℎ ( 3-8 )
Nu is given by experimental correlations rather than true boundary layer solution, as seen in the next section.

This new Simscape block, called the "generic duct" and which icon in shown in Figure 3-16, does not radically differ from the original Simscape duct block; the approach was actually to start from the latter (a major time saving versus starting from scratch) and gradually modify the code according to the needs and encountered computational issues. The major changes were a better management of the phase transition smoothing, and of the flow reversal. Minor changes also focus on the implementation of the change in fluid height in the momentum equation, as well as tuning of the parietal heat transfer and pressure drop correlations.

The parietal pressure drop and heat transfer correlations

For monophasic cryogenic fluids, studies report that classical heat transfer (HT) and pressure drop (ΔP) correlations for standard fluids like water at ambient temperature works fine for nitrogen and helium I at cryogenic temperatures [START_REF] Peterson | Normal Cryogenic Fluid Behavior[END_REF]; opinions are a less clear in the case of the diphasic state.

As often in fluid dynamics, the problem can be divided into the laminar flow and the turbulent flow regimes, as illustrated in the well-known Moody chart [START_REF] Moody | Friction factors for pipe flow[END_REF]. The shift between the two regimes occurs for internal flows at a Reynold (Re) number of about 2000. The formulation of the Darcy friction factor f for the ΔP correlation, and of Nu for the HT correlation differ according to the flow regime, and numerical blending is necessary to ensure a smooth transition.

For laminar flow, the wall rugosity has a negligible effect compared to the internal fluid friction. The resulting analytical expression is given by Poiseuille's law:

𝑓 = 𝑓 𝑆ℎ𝑎𝑝𝑒 𝑅𝑒 ( 3-9 )
Where Re is the Reynolds number and 𝑓 𝑆ℎ𝑎𝑝𝑒 is a factor which depends on the crosssection: it is equal to 64 for circular cross sections which are representative of the vast majority of the cryomodule pipes. For other cross-sections, this factor must be scaled. For instance, it becomes 96 for an infinitely large flat channel, such as the QVB plate heat In laminar flow, Nu is constant according to Poiseuille's law and is equal to 3.66 in a duct with circular section and uniform temperature [START_REF] Gh | General equations for fully developed fluid flow and heat transfer characteristics in complex geometries[END_REF], or 7.54 for an infinitely large flat channel.

In laminar flow, no distinction on the fluid monophasic or diphasic state was implemented, as the most critical item, the heat exchanger is indeed working in this laminar regime but exclusively in monophasic state.

In turbulent flow, the Moody chart gives a graphic expression of the Darcy friction factor of the implicit Colebrook-White relation for rough pipes, where 𝜀 𝑟 is the internal surface relative roughness:

1 √𝑓 = -2 log 10 ( 2.51 𝑅𝑒√𝑓 + 𝜀 𝑟 3.7 ) ( 3-10 )
Although solving this equation is trivial in Simscape, it is computationally lighter to use an approximated explicit relation. For Cryoscape, the explicit Haaland correlation [START_REF] Haaland | Simple and Explicit Formulas for the Friction Factor in Turbulent Pipe Flow[END_REF] was chosen:

𝑓 = 1 {-1.8 * 𝑙𝑜𝑔 [ 6.9 𝑅𝑒 + ( 𝜀 𝑟 3.7 ) 1.11 ]} 2 ( 3-11 )
Indeed, it is considered acceptable by [START_REF] Brkić | Review of explicit approximations to the Colebrook relation for flow friction[END_REF], which compare all the most common Colebrook approximations. This is also the approximation of the original Simscape duct. Another advantage also is the consistency with Simcryogenics library from CEA/DBST, which could later be an asset, in view of a larger model mixing the cryoplant and the cryomodule model.

Like in laminar regime, no specific correlation for parietal ΔP was implemented in 2-phase flow, although approaches do exist, assuming homogeneous or inhomogeneous diphasic flow. Yet, the additional computation time and the absence of experimental validation made this effort unnecessary.

Still in turbulent flow, on the heat transfer point of view, for a monophasic flow, Nu is given by the Gnielinski correlation [START_REF] Bergman | Fundamentals of heat and mass transfer[END_REF]:

𝑁𝑢 = ( 𝑓 8 ) (𝑅𝑒 -1000). 𝑃𝑟 1 + 12.7. ( 𝑓 8 ) 0,5 . (𝑃𝑟 2 3 -1) ( 3-12 )
Where Pr is the Prandtl number and f, again, the Darcy friction factor evaluated from the Haaland correlation : it is calculated only once for both the HT and ΔP correlations.

In 2-phase mixture specifically, contrary to the ΔP correlation, it was decided to use a specific correlation that depend on the vapor quality 𝑥 𝐼 at the internal node I. Nu is thus assessed from the Cavallini and Zecchin correlation [START_REF] Cavallini | A DIMENSIONLESS CORRELATION FOR HEAT TRANSFER IN FORCED CONVECTION CONDENSATION[END_REF]:

𝑁𝑢 = 0.05 [(1 -𝑥 𝐼 + 𝑥 𝐼 √ 𝜌 𝑆𝐿 𝜌 𝑆𝑉 ) * 𝑅𝑒 𝑆𝐿 ] 0.8 𝑃𝑟 𝑆𝐿 0.33 ( 3-13 )
Which is a classical Dittus-Boelter correlation, adapted to diphasic flow; SL and SV refer to saturated liquid and saturated vapor. Other correlations do exist, like Steiner-Taborek or Gungor-Winterton. It was decided to use the Cavallini and Zecchin correlation, as it was the one already implemented in Simscape.

Contrary to the ΔP case, the implementation of a diphasic relation to estimate the heat transfer coefficient is justified by the extensive prototype thermometric instrumentation that enables to see even subtle changes in the heat transfer coefficient. It was thus worth trying to refine the modelling. The experimental analysis of this approach is presented in section 5.2.4.

The discretized duct

The Simscape environment offers the possibility to combine blocks programmatically, which has several advantages:

• Less parametrization efforts since much fewer blocks and higher clarity • Flexible model discretization (adjusting the precision VS computing time)

• The combined blocks automatically inherit the individual blocks code updates Several combined blocks are implemented in Cryoscape; one of them is the segmented generic duct, a series repetition of the generic duct, as illustrated in Figure 3-17. This block can be used to model longitudinally-discretized heat exchangers or transfer lines.

Thermal conduction and thermal inertia

As seen, implementing thermal conduction and thermal inertia in solids requires to have a material specific heat and a thermal conductivity model. As those properties vary significantly with temperature, especially in the 1-50 K temperature range, the use of integral forms is necessary to avoid excessively refined space and time discretization. The native Simscape "Thermal" library does not include this option.

The heat flow φ leaving or entering a mass of material changing from temperatures 𝑇 1 to 𝑇 2 is the time derivative of its specific enthalpy ℎ, simply related to the specific heat by:

φ = 𝑚𝑎𝑠𝑠 * 𝑑 𝑑𝑡 (ℎ) = 𝑚𝑎𝑠𝑠 * 𝑑 𝑑𝑡 (∫ 𝐶 𝑃 (𝑇)𝑑𝑇 𝑇 2 𝑇 1 ) ( 3-14 )
Likewise, from Fourier's law, the conductive heat flow φ between two points at 𝑇 1 and 𝑇 2 , distant from L and connected by a cross section S, is expressed as:

φ = - 𝑆 𝐿 ∫ 𝜆(𝑇)𝑑𝑇 𝑇 2 𝑇 1 ( 3-15 )
The specific heat integrals and the conductivity integrals are the true tables generated with the reference Excel functions, and loaded at the start of the simulation for the interpolation by the dedicated Cryoscape blocks, displayed in Figure 3-18. One can note that all the Cryoscape blocks exhibit a small flake on their icon, to distinguish them from the Simscape native blocks.

Figure 3-18: the thermal conduction model (left) and the thermal inertia model (right) Cryoscape blocks

As for the generic duct, it is possible to make smart combinations of those blocks: Figure 3-19 shows the segmented "thermal link" block, which solves the transient heat equation without internal source. This block is especially adapted for modelling complex supporting systems (with rods and bars). This combined block can be used together with the segmented generic duct to model the conduction along a tube wall, coupled with convective heat exchange with a fluid circulating inside the duct.

Radiative heat transfer

The heat conductance between two solids which are not in contact and in vacuum is dominated by radiation. The radiative heat transfer is the balance between Stephan's law applied to each solid at respectively 𝑇 1 and 𝑇 2 :

𝐻𝑇 𝑟𝑎𝑑 = 𝜎 * 𝐸 * 𝐴 * (𝑇 1 4 -𝑇 2 4 ) ( 3-16 )
In this relation, 𝐴 is the surface, 𝜎 is the Boltzmann constant and E the emissivity factor E, obtained from the surfaces view factor and their respective emissivity. This equation is implemented in a native Simscape block from the thermal library, which is not modified; it is shown in Figure 3-20.

As seen in chapter 5.2, the lack of emissivity data and the complex 3D surfaces make it difficult to a priori evaluate the radiative heat transfer.

The fluid tank model

Development of a 1-zone model and limitations

The desired goal of the tank model is to reproduce the dynamics of the pressure and liquid level, as a response to inlet and outlet flow, and above all, to static and dynamic heat loads. Contrary to a duct, fluids in a tank are considered to have a net global velocity equalling zero. A common approach is the one described in [START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF]: the liquid and the vapor are considered as perfectly mixed in one unique zone; there is this no phase separation in practice, the diphasic fluid properties are read in the phase diagram as one unique point, in the liquid (L) and vapor (V) phase, as illustrated in Figure 3 This model is the same as the one implemented in the duct, except that: (i) since all port pressures are equal, the pressure conservation equation is much simplified; and (ii) the sorting out of the entering and exiting phases is specific:

• Only liquid can leave through the BL port • Only vapor can leave through the AV ports Nevertheless, the fact that heat exchange from the tank or cavity walls cannot be treated for the vapor and liquid phases separately has a very strong numerical impact in modelling the cool-down phase: since liquid cannot exist together with superheated vapor, the apparition of liquid is only possible at the exact point when vapor reaches the saturation, creating unphysical violent transients, prone to solver failures.

Finally, the tank model validation performed in [START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF] stresses the inability of the 1-zone model to reproduce the dynamics of the pressure in complex tanks, urging for an L-V phase decoupling, and more generally for a deeper modelling approach. The next section introduced general tank modelling approaches, while the following section develops the selected one.

Physics and time constants in tanks

Two fields, out of the accelerator domain, have particularly invested in tank modelling:

• Space industry: need to control the fuel tank flow (H2) for proper injection in the combustion chamber and avoid overpressure due to heat load, while saving the mass of pressuring gas (He), in references [START_REF] Grayson | CFD Modeling of Helium Pressurant Effects on Cryogenic Tank Pressure Rise Rates in Normal Gravity[END_REF], [START_REF] Ahuja | Computational Analyses of Pressurization in Cryogenic Tanks[END_REF], [START_REF] Demeure | Comportement thermodynamique de réservoirs d'ergols cryogéniques : étude expérimentale et théorique d'un système de contrôle pour des missions spatiales de longue durée[END_REF], [START_REF] Kassemi | Effect of interfacial turbulence and accommodation coefficient on CFD predictions of pressurization and pressure control in cryogenic storage tank[END_REF], [START_REF] Pooja | Cryogenic Tank Pressurization and Liquefaction System Modeling[END_REF], [START_REF] Wang | CFD investigation on helium pressurization behaviors in liquid hydrogen tank[END_REF], [START_REF] Hermsen | Pressurization system for a cryogenic propellant tank in a pressure-fed high-altitude rocket[END_REF], [START_REF] Imai | Study on break of thermal stratification in container targeted to thermodynamic vent system for future spacecraft[END_REF], [START_REF] Li | Simulation of mass and heat transfer in liquid hydrogen tanks during pressurizing[END_REF] • Liquefied natural gas (LNG) carriers: need to limit the evaporation due to parasitic heat loads in the ship tanks and, thus, save liquefaction power, in references [START_REF] De Schepper | Modeling the evaporation of a hydrocarbon feedstock in the convection section of a steam cracker[END_REF], [START_REF] Lee | Liquefied natural gas flow in the insulation wall of a cargo containment system and its evaporation[END_REF], [START_REF] Noba | Modélisation et simulation numérique du comportement de chargement de gaz naturel liquéfié dans une cuve de méthanier[END_REF], [START_REF] Ovidi | Numerical study of pressure buildup in vertical tanks for cryogenic flammables storage[END_REF], [START_REF] Lee | Practical Prediction of the Boil-Off Rate of Independent-Type Storage Tanks[END_REF] All those studies all highlight strongly coupled effects, listed below and illustrated in Beyond the couplings, the real difficulty lies in the space distribution of those effects throughout the tank: on the one hand, the heat loads deposited on specific wall areas (supports, connections), on the other hand, the natural convection and heat diffusion within and between the liquid and vapor bulks, that potentially give rise to temperature stratification.

The studies stress that these effects are in a large part responsible for the response time in large tanks: the changes in fluid properties (due to a heat or mass input) do not instantaneously propagate throughout the fluid mesh. This is compatible with the cryogenic routine experience when a 500 L helium Dewar is pressurized for before transferring liquid: the time to reach homogeneous temperature across the tank is of about several hours. This is the "energy transfer" time constant.

These effects can be solved with a 2D model, as shown in Figure 3-24 from [START_REF] Grayson | CFD Modeling of Helium Pressurant Effects on Cryogenic Tank Pressure Rise Rates in Normal Gravity[END_REF], where a phase-change formula was added to a commercial Computational Fluid Dynamics (CFD) code. The studies point out a second physical effect that could drive the time needed to reach the equilibrium: a "phase change" time constant. From a pure energetic standpoint, the coupling between the mass and the energy transfers at the location of the phase-change is expressed by the following relation, where L is the latent heat, 𝑚̇ the evaporated or condensed mass flowrate, 𝑄 ̇𝐿 and 𝑄 ̇𝑉 the cumulated conductive and convective heat respectively from the liquid and the vapor phases:

Yet, in reality, phase-change involves small-scale phenomenon, where mass diffusion across phases comes into play, with associated kinetics. According to the works from Knudsen (1934), the phase-change boundary temperature 𝑇 is dynamically related to the mass transfer flux 𝑚̇ due to evaporation (> 0) and condensation (< 0) by:

𝑚̇= 2𝜎 2 -𝜎 √ 𝑀 2𝜋𝑅𝑇 𝑠𝑎𝑡 * 𝐿 1 𝜌 𝑔 - 1 𝜌 𝑙 * 𝑇 𝑠𝑎𝑡 -𝑇 𝑇 𝑠𝑎𝑡 ( 3-18 )
In this relation, 𝑀 is the molar mass, 𝑅 is the ideal gas constant, 𝑇 𝑠𝑎𝑡 the saturation temperature, 𝜌 𝐿 and 𝜌 𝑉 respectively the liquid and vapor densities and finally 𝜎, the mass accommodation coefficient (unitless). From this equation, the increase or decrease of the phase-change boundary temperature 𝑇 bring about mass transfers, at a rate driven by "phase-change" time constant. This time constant depends on the fluid physical properties, and above all on 𝜎; the latter is an empirical coefficient to be tuned for each fluid type and tank shape configuration.

To conclude, most of the studies indicate that this "phase-change" time constant is in the order of the second or less, meaning the tank time evolution is rather dominated by the "energy transfer", and as a side effect, that the phase-change boundary temperature 𝑇 is close to the saturated temperature 𝑇 𝑠𝑎𝑡 . The problem then, is to incorporate enough of the spatially-localized "energy transfer" effects at stake, while keeping the model simple.

One solution is a 2-zone fluid model, as detailed in the next section.

Towards a 2-zone model for the tank

A simple dynamic tank model was developed by NASA [START_REF] Osipov | A Dynamical Model of Rocket Propellant Loading with Liquid Hydrogen[END_REF] to predict the transient evolution of the pressure and spatial distribution of the temperature in a diphasic cryogenic H2 tank, the "External tank" in Figure 3-25, submitted to transfer states (filling, emptying) and long storage states.

𝑚̇ * 𝐿 = 𝑄 ̇𝐿 -𝑄 ̇𝑉 ( 3-17 )

Figure 3-25: schematic of an H2 propellant loading from [START_REF] Osipov | A Dynamical Model of Rocket Propellant Loading with Liquid Hydrogen[END_REF] This model is well-known in the cryogenic community, all the more that it is open-source [START_REF] Daigle | LH2Sim[END_REF] and given the growing interest in H2 for extended application (process industries, mobility).

In this model, the liquid and vapor phases are 2 decoupled fluid control volumes: each phase evolves in its respective monophasic domain, when submitted to heat load or to mass flow input, as opposed to the 1-zone model in Figure 3-22. The return to equilibrium after any perturbation is ensured by energy transfer from the phases to the interface (a 3 rd control volume, massless and infinitely thin), that is translated into mass exchanges (evaporation -condensation) according to [START_REF] Chen | A Study on Properties of Novel Metallic Foam for Nuclear Applications[END_REF][START_REF] Edf | Les déchets radioactifs[END_REF][START_REF]Valley of stability[END_REF][START_REF]Laradioactivite[END_REF][START_REF] Joliot-Curie | [END_REF](8)[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF][START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF][START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF][START_REF] Hargraves | Liquid Fluoride Thorium Reactors[END_REF][START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF][START_REF]Centre d'informations et de ressources sur CIGEO[END_REF][START_REF]Réacteur nucléaire naturel d'Oklo[END_REF][START_REF] Poncelet | Myrrha: un (autre) regard sur l'énergie nucléaire[END_REF][START_REF]Accelerator-driven Systems (ADS) and Fast Reactors (FR) in Advanced Nuclear Fuel Cycles: A Comparative Study[END_REF]. The energy exchanges between the phases are computed at 2 different locations: through the L-V interface thin film, thanks to conduction and natural convection (through 0D correlations), and by convection (also 0D correlations) with the tank walls. As said previously, the "phasechange" time constant is neglected against the "energy transfer", and the L-V interface temperature is precisely at the saturated temperature. In the following, this model is called the "NASA lumped model".

In a more recent update, the authors added two specific features to improve the model representativity during storage phases (no transfer) [START_REF] Daigle | Temperature Stratification in a Cryogenic Fuel Tank[END_REF]:

• Natural convection correlations within liquid and vapor phases • Stratification layers in the liquid and vapor phases (no conduction between them)

• 2-zone wall discretization: in the vapor and in the liquid phase This second model is further called the "NASA stratified model".

A third reference model shall be presented: the Simscape Receiver Accumulator [138]. This block is proposed in the Simscape "Fluid" specialized commercial library and thus not open-source. This block is similar to the NASA lumped model, except 3 points:

• The L-V interface is not modelled, hence, the only heat exchange between the phases, for the return to equilibrium, occurs by convection with the walls • The phase-change can occur anywhere in the tank (not just at the L-V interface) • The phase-change is driven by a customizable time constant, similar to the one in [START_REF] Chen | A Study on Properties of Novel Metallic Foam for Nuclear Applications[END_REF][START_REF] Edf | Les déchets radioactifs[END_REF][START_REF]Valley of stability[END_REF][START_REF]Laradioactivite[END_REF][START_REF] Joliot-Curie | [END_REF](8)[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF][START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF][START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF][START_REF] Hargraves | Liquid Fluoride Thorium Reactors[END_REF][START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF][START_REF]Centre d'informations et de ressources sur CIGEO[END_REF][START_REF]Réacteur nucléaire naturel d'Oklo[END_REF][START_REF] Poncelet | Myrrha: un (autre) regard sur l'énergie nucléaire[END_REF][START_REF]Accelerator-driven Systems (ADS) and Fast Reactors (FR) in Advanced Nuclear Fuel Cycles: A Comparative Study[END_REF][START_REF] Junghans | Transmutation of high-level radioactive waste -Perspectives[END_REF] The NASA and Simscape models are all presented in Figure 3-26, together with the "tank model v2", which is what would be needed, ideally, for the MINERVA cryomodule tank model. The second point is important, since, in normal operation, the somewhat harsh dynamic heat loads are likely immediately followed by mass transfers (thus without the "energy transfer" time constant):

• In He I: due to its poor thermal conductivity, and to the weak subcooled margin provided by the hydrostatic pressure, the liquid helium might evaporate at the location of the heat load (nucleation) • In He II: the very high (yet not infinite) equivalent thermal conductivity immediately conveys the heat to the L-V interface, where it evaporates, without nucleation Furthermore, due to the high flowrates relative to the tank volume the forced convection is likely to prevail over diffusion and natural convection. By contrast, in stand-by or in some failure modes (like an electrical power cut), the pressurization of the tanks is driven by the diffusivity within and between the phases, thus calling for a "NASA lumped" model, or even a "NASA stratified" model.

2-zone model implementation

The previously aforementioned target tank model, called the tank model v2 in Figure 3-26, would enable to cover both the normal operation and the abnormal cases. It would incorporate features from both the NASA lumped model and from the Simscape Receiver Accumulator. Due to a task prioritization and the lack of data to validate a more complex model, only the tank model v1, which is equivalent to the Simscape Receiver Accumulator was implemented. Although more limited, this tank model already features some necessary concepts, such as the 2 decoupled fluid zones, more physical and numerically more stable, as said earlier. Furthermore, the phase-change can occur anywhere in the liquid and vapor zones.

2-P in

Liq out

Vap out Heat

Compared to the 1-zone model, dealing with 2 fluid zones requires first to duplicate the mass and energy conservation equations (one for each zones). Furthermore, it is important to pay attention, in the energy conservation equations, to the quasi-static compression or expansion power received by each control volumes from the other phase:

𝑊 ̇𝐿 𝑜𝑟 𝑉 = -𝑃 𝑑 𝑑𝑡 (𝑉 𝐿 𝑜𝑟 𝑉 ) ( 3-19 )
Second, the 4 total mass and energy conservation equations get a new term: the mass 𝑚̇ and the power generation 𝐽 (in W) from the phase-changes:

𝑚̇𝑣 𝑎𝑝𝑜 = 𝐽 𝑣𝑎𝑝𝑜 ℎ 𝑉,𝑠𝑎𝑡 = 1 𝜏 * 𝑚 𝐿 * ℎ 𝐿 -ℎ 𝐿,𝑠𝑎𝑡 ℎ 𝑉,𝑠𝑎𝑡 ( 3-20 )
𝑚̇𝑐 𝑜𝑛𝑑 = 𝐽 𝑐𝑜𝑛𝑑 ℎ 𝐿,𝑠𝑎𝑡 = 1 𝜏 * 𝑚 𝑉 * ℎ 𝑉 -ℎ 𝑉,𝑠𝑎𝑡 ℎ 𝐿,𝑠𝑎𝑡 ( 3-21 )
The parameter 𝜏 is the "phase-change" time constant; it is set to a tiny value, typically of 0.1 second according to the literature, but can be tuned to match experiments. The way the evaporation process is driven by the enthalpy difference between the saturated conditions and the present ones is similar to equation [START_REF] Chen | A Study on Properties of Novel Metallic Foam for Nuclear Applications[END_REF][START_REF] Edf | Les déchets radioactifs[END_REF][START_REF]Valley of stability[END_REF][START_REF]Laradioactivite[END_REF][START_REF] Joliot-Curie | [END_REF](8)[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF][START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF][START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF][START_REF] Hargraves | Liquid Fluoride Thorium Reactors[END_REF][START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF][START_REF]Centre d'informations et de ressources sur CIGEO[END_REF][START_REF]Réacteur nucléaire naturel d'Oklo[END_REF][START_REF] Poncelet | Myrrha: un (autre) regard sur l'énergie nucléaire[END_REF][START_REF]Accelerator-driven Systems (ADS) and Fast Reactors (FR) in Advanced Nuclear Fuel Cycles: A Comparative Study[END_REF][START_REF] Junghans | Transmutation of high-level radioactive waste -Perspectives[END_REF], and is drawn from the Simscape Receiver Accumulator block.

One final aspect to consider is the control of the localized static heat loads, listed in Figure 3-27. Despite the fact that the wall is not discretized (the wall is one unique point mass), one can ingenuously implement a rough sensitivity to the liquid level in the relations (Newton's law) connecting the wall temperature to the respective heat flows to the liquid and vapor phases:

𝑄 ̇𝑊𝑎𝑙𝑙,𝐿 = 𝐻𝑇𝐶 𝐿 * 𝑆 𝐿 * (𝑇 𝑊 -𝑇 𝐿 ) ( 3-22 ) 𝑄 ̇𝑊𝑎𝑙𝑙,𝑉 = 𝐻𝑇𝐶 𝑉 * (𝑆 𝑡𝑜𝑡 -𝑆 𝐿 ) * (𝑇 𝑊 -𝑇 𝑉 ) ( 3-23 )
In these equations, 𝐻𝑇𝐶 is the heat transfer coefficients between the fluid phases and the wall, 𝑇 𝑊 is the wall temperature; finally, 𝑆 𝐿 is the wall inner surface in contact with the liquid phase.

To summarize the time constants of the tank model v1:

• The "energy transfer" time constant is adjusted by the magnitude of the Heat Transfer Coefficient (HTC) from the phases to the wall. As explained, in the absence of heat transfer at the L-V interface, heat only circulates from one phase to another through the wall • The "phase-change" time constant is directly trimmed through 𝜏, in equations [START_REF] Chen | A Study on Properties of Novel Metallic Foam for Nuclear Applications[END_REF][START_REF] Edf | Les déchets radioactifs[END_REF][START_REF]Valley of stability[END_REF][START_REF]Laradioactivite[END_REF][START_REF] Joliot-Curie | [END_REF](8)[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF][START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF][START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF][START_REF] Hargraves | Liquid Fluoride Thorium Reactors[END_REF][START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF][START_REF]Centre d'informations et de ressources sur CIGEO[END_REF][START_REF]Réacteur nucléaire naturel d'Oklo[END_REF][START_REF] Poncelet | Myrrha: un (autre) regard sur l'énergie nucléaire[END_REF][START_REF]Accelerator-driven Systems (ADS) and Fast Reactors (FR) in Advanced Nuclear Fuel Cycles: A Comparative Study[END_REF][START_REF] Junghans | Transmutation of high-level radioactive waste -Perspectives[END_REF][START_REF]Bilan du débat public, Projet de centre de stockage réversible profond de déchets radioactifs en Meuse / Haute-Marne (Cigéo)[END_REF][START_REF]La réversibilité Thème 1 -La place de Cigéo dans le dispositif de gestion des déchets[END_REF] and ( 3-21 ) • These time constants are related to mass transfer • Then, there comes an overlooked time constant: the pressure build-up time constant. Indeed, the evolution of the pressure after liquid evaporation depends on the vapour phase compressibility and the volume of available compressible vapour This later effect is estimated to be the most important effect to model during the prototype cryomodule test phase. Even in its initial version, the 2-zone tank model is deemed suitable for this purpose. This model provides a good foundation for future extensions in view of the series predictions. For example, a hydrostatic pressure model could be introduced to consider limitations on the heat transfer in superfluid helium.

THE FUNCTIONAL SUB-COMPONENTS

The process pipe and transfer line models

The process pipes are the internal ducts of the QVB and QM, that carry the cryogenic fluids through the processes; a pipe encompasses both the fluid duct and its solid envelope, the wall. The transfer lines are the vacuum-insulated pipes which route the fluids between the QVB, QM and infrastructure. Although seemingly trivial, modelling pipes and transfer lines demands specific attention.

Firstly, it is necessary to obtain the precise "as-built" geometrical data. The final validated CAD files from the QVB and QM manufacturers provide the main source of information.

During the prototype integration and assembly phase, checks were conducted to detect any changes between CAD and the real components. The needed geometrical data includes:

• The cross-sectional area of the flow • The length and inner diameter • The wall thickness • The wall material properties Then, specifically for transfer lines, the estimate of the heat load from the ambient is crucial but limited: despite the large IJCLAB experience, the heat load assessment is impeded by unknowns, for instance on the vacuum quality, meaning the value of 1 W/m for a standard flexible vacuum-insulated line may in fact be 2 or 3 times more or worse, as seen in 5.2.1.3. Moreover, because of diphasic states, a direct temperature instrumentation to measure the heat load is useless.

Then, validating the pressure drops of the pipes or transfer line is not straightforward either, especially at cryogenic temperatures, both because of the cost of the instrumentation and because any error in the estimate of the fluid state (liquid, vapor, diphasic) results in large discrepancies. Pressure drop tests at room temperature on the other hand provide a reliable source of information and were performed for validation purposes for the heat exchanger (3.4.7) and for the valves (5.2.1); unfortunately, once assembled, the large valves pressure drops fully conceal pressure drops of the sole pipe. One thus sticks to the previously described pressure drop correlations.

The pipes primarily carry most of the thermal instrumentation, critical to measure the component performance. The principle relies on the measurement of the pipe external wall temperature, to get the running fluid temperature. Unfortunately, two physical effects compete at the location of the measure: the convection from the fluid (that one wishes to maximize), and the axial conduction from hotter parts along the pipe wall. For instance, the QVB thermal shield outlet sensors TT602 and TT603 are too close to the nitrogen reheater, thus do not fully reflect the fluid temperature. Coupled fluidic and thermal simulations are sometimes needed to model these competing effects, as for the heat exchanger (5.2.3) low-pressure side exhaust temperature analysis, or to spot the optimum location for the instrumentation, as used for the 10 K loop sensors TT081 and TT082.

To conclude, modelling pipes and transfer lines requires tailored models, fitted to the problem to address. For illustration, Figure 3-28 displays a 3 m long transfer line model: the thermal inertia is parametrised thanks to Cryoscape thermal mass blocks, the heat transfer from the ambient is modelled by Simscape native radiative heat transfer blocks and the fluid volume is modelled with Cryoscape generic ducts. Additionally, the axial conduction is not needed. The transfer line is discretized in typically 5-10 elements, seamlessly adjustable in the dedicated segmented Cryoscape block parameter. 𝐾 𝑣 is selected based on the maximum mass flow rate to be controlled in the process; the rangeability, in the order of 50-100, is the ratio of the maximum 𝐾𝑣 over the minimum controllable 𝐾 𝑣 . Typically, a large rangeability is favoured in order to address the cooldown situations (high flowrates) and the nominal steady-state ones (low flowrates). The MINERVA prototype valve functions and characteristics are summarized in Table 3-2.

Table 3-2: datasheet of the control valves of the MINERVA prototype

The flow model of the valve is broken-down as followed:

• • Linear (the slope is constant): the valve offers a constant sensitivity over the whole range but a limited permeability at full opening • Equal percentage (the slope is proportional to the % 𝑜𝑝𝑒𝑛𝑖𝑛𝑔): the valve offers a good sensitivity only below ~40 % but a large permeability at full opening In practice, the supplier adjusts the shape of the seal plug and seat, as seen in Figure 3-29.

Figure 3-29: typical valve characteristics

For all seven cryogenic control valves of the MINERVA prototype, the real characteristics 𝐾 𝑣 = 𝑓(% 𝑠𝑡𝑟𝑜𝑘𝑒) is known from the supplier "as manufactured" documentation, and is deemed reliable +/-3 %. This is also the case for the QVB 4 K tank warm pressure control valves, which were specially bought for the MINERVA test phase. On the other hand, it was not possible to get the 2 K tank pressure control valve 𝐾 𝑣 = 𝑓(% 𝑠𝑡𝑟𝑜𝑘𝑒) characteristics from the supplier; an estimate was done, based on the geometry of the seal (butterfly) and a pressure drop formula from [START_REF] Idelʹčik | Memento des pertes de charge: coefficients de pertes de charge singulières et de pertes de charge par frottement, Nouv. tirage[END_REF].

The flow model part 2 consists in singular pressure drop relations. The equation for a liquid (incompressible) is presented in , where 𝜌 𝑖𝑛 is the inlet density, 𝑃 𝑖𝑛 the inlet , where 𝑃 𝑜𝑢𝑡 is the outlet pressure.

𝑚̇𝑖 𝑛𝑐𝑜𝑚𝑝 = 0,1 * 𝐾 𝑣 3600 * √𝜌 𝑖𝑛 * 𝑃 𝑖𝑛 * 𝑋 ( 3-24 )
This equation is equivalent to the Darcy-Weisbach one ( 3-5 ): the Darcy friction factor 𝑓 and 𝐾 𝑣 are simply related by:

𝑓 * 𝐿 𝑑 = 2 ( 𝐴 * 36000 𝐾 𝑉 ) 2 ( 3-25 )
The equation for a compressible fluid is presented in , where 𝑋 𝐶 is the critical pressure ratio 𝑋, (approximately 0.5) and stands for the pressure ratio at which the flow becomes choked (sonic regime).

𝑚̇𝑐 𝑜𝑚𝑝 = 0,1
( 3-26 )
This equation is simply an approximation of the Barré-Saint-Venant's equation, which is the equivalent of the Bernoulli's equation applied to an ideal gas. Back to the flow model part 2, the physics behind the flow formula is a scaling of a mass flow rate at reference conditions towards new conditions. The formulae are meant for rough valve sizing, not for extensive and accurate physical reproductions 37 . A thorough validation was thus carried out, as presented in 5.2.1. One crucial point is that nearly all the valves are equipped with position measurement: the % 𝑜𝑝𝑒𝑛𝑖𝑛𝑔 from 0 to 100 % is thus not just the software command of the local control loop, but the true physical measure. This removes a large uncertainty, provided the position calibration is properly done.

Valve heat transfer model

The model of the valve is not limited to the flow model only: indeed, since the valve is in contact with the ambient and with the cryogenic fluid, conductive heat transfer occur. To limit the heat loads, the valve body is mounted at the tip of a long hollow rod, in the order of 800-900 mm, and the heat is intercepted by the 80 K loop, as shown in Figure 3-31.

A straightforward thermal model was developed to account for the conduction and for the thermal inertia of the valve. It consists in a unidirectional discretisation of the hollow rod and internal control stem: an upper "warm" part located between the ambient and the 80 K intercept, and a "cold" part located between the intercept and the valve seat. Each part is modelled by one Cryoscape "thermal link" block.

Using the actual geometries, the simulated heat flow rates to the 80 K and to the 4 K heat sinks do not deviate more than +/-20 % from the manufacturer datasheet; the valve hollow rod diameter is slightly trimmed to cancel out the discrepancy. The model predicts about 24 hours for the valve to fully thermalize, once the valve body is at 4 K, which is compatible with the common experience.

The full cryogenic valve model is displayed in Figure 3-32, comprising the flow model 37 Discussion with IJClab cryogenic experts

The 2 K pumps

To make accurate control predictions, like PID gains or a more advanced control law, it is essential to simulate the evolution of the pressure downstream of the MKS valve. This pressure evolves according to the mass flowrate reaching the pumping groups.

The pumping capacity in the B106 building at IJClab consists in two pumping units (a third line exists but is capped), as seen in 3.1.3. Unit 1 consists in a vane pump in series with a root pump, offering together a capacity of about 1.92 g/s at 10 mbar. Unit 2 is similar to group 1 one, plus a second root pump, providing a total capacity of 3.42 g/s at 10 mbar.

The pumping group curves are available from the supplier simulated data, as shown in Figure 3-33, and from commissioning experiments conducted in 2014, that confirmed the simulations. These functional data curves were implemented in Simulink, to drive a Simscape native mass flowrate source block.

Bottom communication tube

Top communication tube

Buffer heat loads coming from the different warmer sources (80 K, 300 K) are well separated by means of individual physical models, as detailed in the next part 3.4.5. Then, as explained in 3.3.3, the heat load from the wall mass is distributed to the liquid and vapor phases as a function of the respective heat transfer coefficient and heat transfer surfaces.

Instrumentation model

For the purposes of control commissioning and test analysis, the tank models must have inputs and outputs comparable to the experimental ones.

For both the 4 K and 2 K tanks, their pressure sensors are embedded in the warm area, situated approximately about 1-2 m above the liquid level. As previously stated, hydrostatic pressure is not modelled within the tank (nor in the liquid, nor in the gas phase): the pressure is thus considered homogeneous throughout the tank and equal to the measured pressure.

Both tanks are also equipped with electrical heaters: in the 4 K tank to maintain the pressure when the He Dewar is disconnected for refill, as the sole QVB heat loads are not enough; in the 2 K tank to simulate the dynamic heat loads. The precise knowledge of the injected heat is crucial and requires specific measurements and procedures, as detailed in 5.2.1.2.

Last but not least, the tanks have liquid level sensors to monitor the amount of liquid. The measurement principle is simple: a niobium wire lies in the tank; the immerged section is superconducting, whereas the section in the gas phase is normal-conducting. The measurement of the wire resistance yields the level. As seen in Figure 3-36, the QM level probes (they are redundant) do not reach the 2 K tank bottom, contrary to the 4 K tank. The probe measures in fact a 20 cm range centred around the nominal regulation level, at 10 cm (the middle of the buffer tank). It means that, during cool-down, the level must be inferred from other sensors, like dedicated thermometers. In addition, the probe is located in the turbulent flow of the exiting vapor, instead of ideally, in a quite area, to avoid parasitic transitions in the vapor phase. Yet, with a proper trimming of the current supply strategy, the probes proved to work fine; this was one point to validate during the prototype test campaign.

In the Cryoscape tank block, the level reading output is simulated from the volume variable. The as-built CAD data are used to build the non-linear function between the level and the tank volume which is shown in Figure 3-37.

Figure 3-37: level of liquid in the 2 K tank as a function of the volume of liquid

This curve shows that the dynamic of the level as a function of the filling and emptying flow rates varies considerably across the whole range. Therefore, the PI controller gains should be adjusted depending on the targeted level.

The cavity supporting system

As previously stated, physical models are needed for each of the static heat loads to the 2 K loop. The main static heat load sources to the 2 K tank are:

1. The couplers (conduction and radiation) 2. The beam tubes (conduction) 3. The cavity vertical supporting arms (conduction) 4. The invar horizontal clamping rods (conduction) 5. The thermal shield (radiation) Heat sources number 3 and 4 come from the cavity supporting system, the table, already presented in 2.1. Being able to model and confirm these heat loads with measurements is crucial to validate the choice of the table, over the other existing supporting system concepts.

More widely, the goal is to have a model that predicts the heat from all sources to all sinks, in steady-state and in transient. The cavity supporting system is one of the cryomodule sub-system where most of the heat transits. The supporting system is in contact with the following items (Figure 3-38):

• The "mobile" and "fixed" feet connected to the ambient environment • The 12 thermal braids connected to the QM thermal shield • The table horizontal frame This model considers the physical coupling between the cryogenic fluids and solid heat transfer: it enables to make simple steady-state comparison to experiments, as seen in part 5.2.4 ,up to the prediction of abnormal transient events such as unintentional shutdown of the cryogenic intercepts.

In addition to scrutinising the as-built geometry and instrumentation location, during the prototype integration, the proper tightening and the presence of thermal grease in the heat intercept clamps were checked, in view of the calibration of the contact resistances.

The parasitic thermal contacts were also tracked and suppressed.

3.4.6

The QM and QVB thermal shields (TS)

The thermal shields also have a significant interaction with several systems and are important elements to consider to simulate transient cryomodule sequences. A model like the one developed for the supporting system, even with only a few tens elements, enables to spot potential flaws like warm points, localized in space.

The thermal shields not only cut the radiation from the 300 K vacuum vessel, but also Each element consists in a thermal mass, connected to the others by a conductive block, which considers the conductive average distance and cross-section between them, as well as the contact resistance. Each element also receives thermal radiation from the ambient, proportionally to its area. Finally, each element is thermally connected to the appropriate other components (valves, safety chimney, tie rods).

As for the valves and the table, the interaction between the fluid and the thermal domain is considered: the relevant elements can exchange heat with the 80 K circuit; the heat exchange coefficient is computed in the fluid duct physical blocks. A complete evaluation of the models is performed in 5.2.6. 3.4.7 The subcooling heat exchanger

Technology overview and modelling strategy

Following the choices made by IJClab for ESS and for the new IJClab vertical test cryostat "CV 1250", a plate heat exchanger from DATE was chosen for MINERVA. This heat exchanger (HX) is in counter-flow configuration; the core technology is a stack of plates, with frames in between to form alternate channels. The side where the liquid enters is called the high-pressure side (HP), and the side where the returning vapor enters is called the low-pressure side (LP). The channels are stamped with dimples, which roles are preventing the collapse of the channel due to the pressure difference, and boosting the heat transfer. Both sides feature typically 30-60 channels.

DATE's heat exchanger (HX) design is illustrated in Figure 3-41 for the ESS Spoke prototype heat exchanger. The material is in stainless-steel, to reduce the longitudinal conduction and to allow for compact heat exchangers. Heat exchangers are complex objects to design, manufacture and simulate 38 . A capital thing, all the more relevant here, is the precise knowledge of the as-build characteristics. Cross-checks were conducted between the MINERVA heat exchanger data and the previously manufactured HX for the ESS and CV1250 projects, thanks to the documentation and to interactions with the supplier. Those two projects plus the experience from CERN on similar HX fulfilled a second important point, also highlighted several times: providing maximum test and simulation data for initial model comparison.

The interesting functional performance are the heat transfer (to maximize), and the pressure drop (to minimize).

Large variations of fluid properties occur in the axial direction. There are 2 common methods to solve this problem, either unidimensional (such as LMTD and NTU 39 ) as in [START_REF] Kumar | Experimental and numerical investigation of 2 K heat exchanger for superfluid helium cryogenic systems at KEK[END_REF] or by discretization, as in [START_REF] Gupta | Numerical modeling of a 2 K J-T heat exchanger used in Fermilab Vertical Test Stand VTS-1[END_REF]. Discretizing of the fluid domain was deemed the most appropriate and adaptable method; 10 elements are enough in most simulation cases (the mesh is converged). Given the high number of channels, one can reasonably assume that they all experience the same conditions (except the two end ones). Hence, the HX model can be reduced to only 1 channel with the total equivalent cross-section section and the same unitary hydraulic diameter, to implement as Cryoscape generic duct 38 Experience from TitanX (former Valeo subsidiary) in the 2017-2019 period The generic duct block parietal laws enable to calculate both the heat transfer and the pressure drop. An important point is that the influence of the dimples is not included, as discussed later. For the parasitic longitudinal wall conduction, the total mass and crosssection area is considered homogeneously distributed along the unique equivalent channel, and implemented as Cryoscape thermal mass and conduction blocks.

The model assembly is showed in Figure 3-42. For the purpose of illustration, the model in this figure is using the normal Cryoscape blocks and not the segmented ones.

Thermal performance model initial check

A first set of data for thermal performance comes from the supplier simulations, which conditions (IJClab specifications) are summarized in Table 3-3. For MINERVA, no simulation was performed, given that the "CV 1250" HX has precisely the same geometry. One should note that some cases are not representative of cryomodule operating conditions: for instance, in lines 2, 4 and 8, the HP side enters in vapor state and exits in liquid; moreover, in line 8, the LP side enters in liquid state instead of vapor. This nevertheless does not impede the comparison with the model.

HP in HP out

LP out LP in

Table 3-3: summary of the working conditions of the DATE simulations over all the different projects

A second set of data comes from the CERN heat exchangers, those cooling the superconducting magnets of the LHC, abbreviated "HXA", also manufactured by DATE; the two heat exchanger version, "long" and "short", were tested in cryogenic conditions [START_REF] Gilbert | Performance Assessment of 239 Series Helium Sub-Cooling Heat Exchangers for the Large Hadron Collider[END_REF]. DATE claim their own thermal model are in full agreement with these tests, within 3% on the whole power range. As an illustration of the impact of the longitudinal conduction due to the stainless-steel plates (about 0.6 mm thick), a parameter sweep is performed on one of the "ESS Spoke Series" working points:

• No plate (material conductivity set to 0 W/m.K) The results are presented in Figure 3-44. One first notices that there hardly is any longitudinal conduction with stainless-steel plates. Second, that copper plates bring about significant performance drops, even at low RRR. Those simple results illustrate that the model physics is compatible with the common experience: such compact-design HX are always made in low-conductivity material. Copper heat exchangers do exist, yet in a different core technology, for instance, coil-wounded.

Hydraulic performance model initial check

Two pressure drop (ΔP) reference datasets were available for the ESS and CV1250 projects:

• Simulations with Helium in cryogenic conditions (the same working points as the thermal model) • Characterisations with demineralized water at room temperature For the latter, the supplier measured the ΔP with water, at mass flowrates that match the Reynolds number met with helium at cryogenic temperature (to skip the costs of a cryogenic ΔP test bench). The Darcy friction factor f, which is needed to make ΔP predictions, actually only depends on this non-dimensional number, as seen in 3.3.1.2.

The Cryoscape model reproduced both of the datasets and the comparison is seen in Figure 3-46 for the LP side.

Figure 3-46: Simscape comparison to the cryogenic simulations (left), and to the DATE tests and simulations at room temperature (right), on the LP side

There is an obviously large discrepancy between the Simscape model versus the simulations and tests from the supplier. This outcome was fully expected, as the Simscape model does not include the negative impact of the dimples, which tend to increase the pressure drop. The deviation is even worse on the HP side (not shown), again fully expected as there are significantly more dimples on this side. On the thermal model, the effect of the dimples (positive, this time) is not implemented either; this means the HX likely performs better than the model; this is detailed in part 5.2.3.

To predict the ΔP due to the dimples, the supplier deforms the Darcy friction factor f following:

𝑓 𝑚𝑜𝑑 = 𝑎 * 𝑓 𝑟𝑎𝑤 + 𝑏 ( 3-28 )
In this relation, 𝑓 𝑟𝑎𝑤 is the laminar friction factor from Poiseuille's law; 𝑎 and 𝑏 are the deformation coefficient. 𝑏 gives a quadratic trend to the ΔP as a function of the mass flow, to simulate turbulent pressure losses, whereas 𝑎 influences the linear trend, representative of laminar pressure losses. Both are adjusted by the supplier, so that their model reproduces the room temperature tests, as seen in right-handed chart in Figure 3-46. This model deformation is then used to make predictions at cryogenic temperatures. The deformation coefficients are summarized in Table 3-4: they are pretty large on the HP side, and moderate on the LP side, as expected.

Table 3-4: deformation of the Darcy friction factor from DATE on the ΔP tests with water

However, it can be seen on Figure 3-46 that: (i) under cryogenic conditions, Simscape underestimates the pressure drop by a factor of 30 when compared with the supplier's model; (ii) in the case of the water test conditions, Simscape also underestimates the pressure drop compared to the Supplier's model, but only by a factor of 3 to 4. As Simscape does not make use of the modified friction factor, this indicates that the Supplier possibly used a set of coefficients (a;b) which differ between simulations of water tests and those of cryogenic ones.

As a clear explanation could not be obtained, a test campaign was conducted at IJClab to assess the performances at room temperature of a spare ESS Spoke series heat exchanger (due to technical constraints, neither the CV 1250 nor the MINERVA HX could be disassembled). A dedicated test bench was specified and build, as shown on the schematics of Figure 3-47 and on the pictures of Annex 7.4. Instead of water, it was estimated that nitrogen had more suitable fluid properties (essentially the viscosity and density) to reach the whole Reynolds range (1000 on the HP side, 2000 on the LP side at 4.5 g/s helium mass flowrate) while lying in the range of standard commercially available ΔP and mass flow sensors. As a side note, the supplier test with water stopped at Reynolds of respectively 100 and 1400 for the HP and LP sides.

On top of being inexpensive, nitrogen gas can be seamlessly provided in racks of several tanks, to also serve as a pressure source (hence, no compressor needed); a single rack of nine 50-liter tanks pressurized at 200 bar was enough for the whole characterization, with margin. A simple manual control valve was used to regulate the mass flowrate at different steps. A manual air compressor was also installed in order to sweep the static pressure of the HP channel while testing the LP channel; this was to see the influence of the plate deformation.

The LP side was characterized with 24 runs of about 5-6 mass flowrates:

• LP inlet pressure = 2 -3 -4 -5 bar • Static HP pressure = 1 -2 -3 -4 -5 -6 bar
And on the HP side, 3 runs:

• HP inlet pressure = 3 -4 -5 bar The status is the same as on the LP side; the optimal 𝑎 coefficient is set to 1 but is actually insignificant, as the pressure loss is fully dominated by the dimples, thus by the quadratic effect expressed in the b coefficient.

To conclude, the pressure drop measurements using gaseous nitrogen at room temperature proved that the supplier tests with water are perfectly reliable. The plausible reason why the supplier simulations overestimate the pressure drops at cryogenic temperature is a safety margin.

Those new tests enable the development of a modified model of the pressure drop that accommodates the whole Reynolds number range; this modified model was implemented into the Cryoscape generic duct. However, the accuracy of the model is not fully relevant since the pressure drop in the heat exchanger is minor and does not cause any measurable thermodynamic impact on the cryomodule operation. In the long term, when considering the production of the 30 series heat exchangers, such tests and model analysis raise relevant questions and prepare for the series production follow-up strategy.

THE CRYOGENIC PROCESS AND CONTROL MODEL

The component models presented so far enable to setup a full MINERVA cryomodule cryogenic process model; the Simscape model and its corresponding physical parts are shown in Figure 3-50. This model can capture all the physical process variable interactions; it is called "open-loop" since the commands (valve, heaters) are not connected to any control box in this representation. Therefore, a model of the control layer is thus also needed. The first purpose of the PLC is to communicate with the field equipment, i.e. the instrumentation and actuators, some needing a dedicated signal conditioner. The prototype PLC uses several communication protocols to adapt the wide equipment, from analogue I/O cards to the digital Profibus and Profinet networks. On an accelerator facility by contrast, the field equipment should be more standardized and, moreover, radiation-resistant; but the overall principle remains the same. The developer team programs process variables (PV) from those raw I/O signals, with the correct physical units, for use by the control algorithms. In the model, it is crucial to simulate the exact same variables and associated units.

The second role of the PLC is to execute the control algorithms, which are split in 2 layers:

• The sequences: a series of command execution typically in a grafcet code, that for instance tells how to cool-down to 80 K, or 4 K, or how to stand-by, by setting appropriate targets to the different control loops • The control laws: the method which a control loop moves an actuator, to make a measured process variable reach its target

The MINERVA prototype cryomodule features several control loops with two main types of control laws: the proportional integral (PI), as seen in Table 3-5 and the ON-OFF, summarized in Table 3- As seen in part 2.5, the performance of those control laws is crucial for the cryomodule operation in the accelerator facility, and already during the prototyping phase, calling for the need to model them too.

The control laws (and sequences too) are implemented in a Simulink sub-system, called the Simulink "control box". This Simulink code is like a copy of the PLC control code, both were actually developed in parallel for the prototype. The Simscape cryogenic process model and the Simulink control box is shown in Figure 3-52.

THE RF PHYSICAL PROCESS AND CONTROL MODEL

As exposed in part 2.5, considering the strong coupling between the cryogenic and the RF processes requires to be able to dynamically simulate the reaction of the RF systems in any RF setpoint, beam parameter and cryogenic conditions. First, the different heat loads generated by the two main RF parts, the cavity and power coupler (summarized in Figure 4-1) need to be simulated; second, the cavity detuning due to the RF field and to the helium tank pressure variations need to be considered.

To accomplish these goals, a dynamic process model of the RF and its control is needed, just like for the cryogenic aspects. The following list outlines the necessary models for the cavities: The four top models are developed based on an existing model, developed for MYRRHA in 2010 [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF] in preparation of the 700 MHz medium-beta elliptical prototype cryomodule tests at IJClab during MAX [START_REF] Bouly | DEVELOPMENTS AND TESTS OF A 700 MHz PROTOTYPICAL CRYO-MODULE FOR THE MYRRHA ADS PROTON LINEAR ACCELERATOR[END_REF], [START_REF] Bouly | Low Level Radio-frequency Developments toward a Fault-tolerant Linac Scheme for an Accelerator Driven System[END_REF]. The model was scaled on the 352.2 MHz Spoke cryomodule by the same author, now at the LPSC laboratory in Grenoble, and shared in the framework of the thesis. The thesis added-value on those cavity model sticks to the development of automated visualisation tools, which enable to perform initial model validations, as exposed in 4.3.4.

•
For the last two cavity models, the thesis work focused on the update of the mechanical models and on the creation of a simple heat transfer model. Additionally, techniques are developed to solve RF and cryogenic models, which have drastically different timesteps, in the same environment.

Most of the thesis added-value is on the coupler couplers models: Here also, as explained in the following sections, several model parts of the RF power coupler were inherited from previous studies of MYRRHA and MINERVA, and from the LPSC laboratory team, as exposed in [START_REF] Saugnac | Spoke Cryomodule Prototyping for the MINERVA Project[END_REF].

The next sections present the individual RF models in a specific order for better understanding.

SRF CAVITY MAIN CONCEPT

Brief cavity structure classification

The function of the cavity is to accelerate a bunch of particles, thanks to a longitudinal electrical field created on the beam axis. This differs from magnets which make use of the magnetic field to curve the particle trajectory, in focusing or bending applications. Moreover, in the cavities, the beam defocusing due to transverse electrical or longitudinal magnetic field should be minimized.

To generate large accelerating voltages, in the order of MV, the idea is to amplify a kVorder signal, supplied by a power amplifier, using a resonant cavity. There are two type of cavities: travelling wave (TW) cavities, where the particles "surf" on a co-propagating RF wave, and standing-wave (SW) cavities, where the beam is synchronous with a resonating field. The latter are the one used in superconducting proton linacs.

SW cavities are designed to operate in one useful resonant mode, where the E-field component complies with the previously mentioned requirements. This resonant mode is unfortunately not unique and other modes, such as high-order modes (HOM), can be excited by the particle wake field, causing perturbations if these parasitic modes are not damped enough. Several types of accelerating modes, and subsequent cavity geometries, are used. For the well-known pill-box cavity, the mode TM010 is used. This terminology is commonly used in the accelerator community to identify which resonant mode is operated: TM means that the magnetic field is transverse to the main direction, so no 𝐵 𝑧 component, thus only 𝐸 𝑧 . The first 0 means that 𝐸 𝑧 has no variation along the azimuthal direction, the second 1 denotes the number of 𝐸 𝑧 zeros in the radial direction and, ultimately, the last 0 specifies the number of 𝐸 𝑧 zeros in the z direction [START_REF] Uspas | Chapter 4 Accelerator Structures I Single-Cavity[END_REF].

The elliptical cavities also use this TM010 mode, in a π-resonant setup for the multicell ones. By contrast, half-wave (HWR), quarter-wave (QWR) and Spoke cavities utilize the TEM mode. This mode is the one used in coaxial transmission lines, as one can infer from the cavity inner shape from Figure 4-2. The fact that different cavity families (HWR, QWR, Spoke and Elliptical) are used comes from the optimum particle speed these cavities are designed for. Indeed, the synchronism condition, according to which the particles cross the gap while the accelerating field is positive, dictates the gap optimal length: it is proportional to the reduced particle speed β and to λ. Moreover, the accelerating gap should be the largest in order to deliver the maximum energy gain. Consequently, it is interesting to use low frequencies at low β and vice versa. The families of cavities are thus different types of resonators operating at precise frequencies and complies with the many other constraints such as the maximum magnetic fields, the manufacturability, the mechanical stiffness and the compactness.

In practice, elliptical cavities are used for high-β sections of linacs. They are operated at frequencies around 700-800 MHz and have typical dimensions of 0.5-1 m. Then, Spoke cavities also have typical dimensions of 0.5-1 m but operating at about 350 MHz [START_REF] Padamsee | RF Superconductivity[END_REF]. Finally, HWR and QWR are adapted for the lowest β sections: they are operated at lower frequencies, typically 100-200 MHz. Since the frequencies are low, the structure size is large, in the order of 1 m.

For medium β values, of about 0.5, there is no clear distinction between the domains of Elliptical and Spoke cavities. Within the MAX project, a β=0.47 5-cell elliptical cavity was tested. One of the key findings was the difficulty to maintain the resonance frequency stable, which is due to the high sensitivity to the Lorentz forces resulting from significant mechanical softness. This is because the optimum gap length decreases as β decreases. This causes the cell walls of the cavity to become flatter and more vertical, making them increasingly prone to mechanical deformation.

As previously mentioned, the energy gain per cavity is one parameter to optimize necessary for achieving a compact linac. Another parameter to optimize is the cavity heat loads: indeed, although in superconducting state, SRF cavities generate some watts of heat loads which significantly affects the electrical consumption of the cryogenic system.

The following sections introduce the concept of the quality factor, 𝑄 0 , which is central to characterize and predict the cavity heat loads.

Cavity RF voltage definitions

Before introducing the quality factor 𝑄 0 , it is worth defining some useful notions about cavity voltages and accelerating fields, already mentioned in 2.5. Figure 4-3 enables to introduce the maximum RF voltage 𝑉 𝑚𝑎𝑥 :

𝑉 𝑚𝑎𝑥 = ∫ 𝐸(0, 𝑧) 𝐿 2 - 𝐿 2 𝑑𝑧 ( 4-1 )
In this equation, 𝐸(0, 𝑧) is the amplitude of the RF field, pulsating at 𝑤𝑡, at radius r = 0.

Figure 4-3: cavity gap and field definition adapted from [START_REF] Wangler | RF Linear Accelerators[END_REF] This voltage is what would be experienced by a particle travelling in a DC field of amplitude 𝐸(0, 𝑧), along the cavity accelerating gap of length g, between positions -𝐿 2 and 𝐿 2 . On should keep in mind that 𝐸(0, 𝑧) has no reason to be square or sinusoidal as a function of z, as see in Figure 4-3 (right). Only in the case of the theoretical pill-box cavity can this field be perfectly square. This is no more possible when the beam tubes and the true cavity shape are considered.

Then, 𝑉 𝑐𝑎𝑣 is the voltage a particle travelling at the reduced speed β = 𝑣 𝑐 ⁄ would experience in a time-varying AC field:

𝑉 𝑐𝑎𝑣 = ∫ 𝐸(0, 𝑧) 𝐿 2 - 𝐿 2 cos (2𝜋 𝑧 𝛽𝜆 ) 𝑑𝑧 ( 4-2 )
This expression is correct if one considers 𝐸(0, 𝑧) as an even function close to the gap, and if one ignores the change in the particle speed 𝑣 (i.e. that the energy gain is much smaller than the particle energy), meaning 𝑤𝑡 ≈ = 𝑤𝑧 𝑣 ⁄ = 2𝜋𝑧 𝛽𝜆 ⁄ [START_REF] Wangler | RF Linear Accelerators[END_REF]. The ratio between 𝑉 𝑚𝑎𝑥 and 𝑉 𝑐𝑎𝑣 is the famous transit time factor:

𝑇 = 𝑉 𝑐𝑎𝑣 𝑉 𝑚𝑎𝑥 ( 4-3 )
Compared to 𝑉 𝑚𝑎𝑥 , 𝑉 𝑐𝑎𝑣 is lower because the RF field varies as the particle passes, contrary to an electrostatic field which remains constant. T quantifies the resultant loss of energy gain. For a given maximum voltage 𝑉 𝑚𝑎𝑥 , T thus characterizes how concentrated the field 𝐸(0, 𝑧) is around the cavity gap centre, which makes the acceleration process more effective because the cosine is maximum at z = 0. T also characterizes the cavity velocity acceptance, since a more concentrated field means the energy gain is be less dependent on β.

From 𝑉 𝑐𝑎𝑣 , it is common to define the accelerating field, which is normalized over a reference accelerating length 𝐿 𝑎𝑐𝑐 41 as:

𝐸 𝑎𝑐𝑐 = 𝑉 𝑐𝑎𝑣 𝐿 𝑎𝑐𝑐 ( 4-4 )
𝐸 𝑎𝑐𝑐 is the reference parameter considered for the cavity field control. However, the real voltage felt by a particle in the case of a non-zero synchronous phase 𝜙 𝑐𝑎𝑣 is:

𝑉 𝑎𝑐𝑐 = 𝑉 𝑐𝑎𝑣 cos(𝜙 𝑐𝑎𝑣 ) ( 4-5 )
It is worth noting that, even though the particle bunch has a non-zero length, it is typically 1000 times smaller than the RF wavelength. In the RF point of view, it is thus reasonable to consider the bunch as one particle. 41 By convention 𝐿 𝑎𝑐𝑐 = 𝑁𝛽𝜆 2

, with 𝑁 the number of cavity gaps

The particle energy gain, with q the particle charge, is then:

∆𝑊 = 𝑞𝑉 𝑎𝑐𝑐 = 𝑞𝑉 𝑐𝑎𝑣 cos(𝜙) = 𝑞𝑉 𝑚𝑎𝑥 𝑇 cos(𝜙 𝑐𝑎𝑣 ) ( 4-6 )
As a side note, one also gets 𝑊 = 𝑞𝐸 𝑎𝑐𝑐 𝐿 𝑎𝑐𝑐 cos(𝜙 𝑐𝑎𝑣 ). The energy gain is thus connected to both cavity control parameters 𝐸 𝑎𝑐𝑐 and 𝜙 𝑐𝑎𝑣 ; both are measured by the RF system. A consistency check can be done with the beam diagnosics, that also measure ∆𝑊.

The superconductivity

Many pure or alloyed materials, exhibit a sudden electrical conductivity fall to nearly zero, below a given critical temperature. This effect was first discovered on mercury in 1911 by H. K. Onnes, a Dutch physicist and Nobel Price, the same who had been able to liquify helium for the first time a few years earlier.

This observation was completed in 1933 with the discovery of the Meissner effect: thanks to permanent superconducting induced currents (supercurrents), a superconducting material features a perfect diamagnetism: it creates a field perfectly opposed to the applied one. Just like the one that induced Eddy currents create in some material, but never dissipated. This is how well-known public demonstrations can exhibit the levitation of a magnet over a superconductor as pictured in Figure 4-4.

In 1935, London's "liquid" theory of superconductivity gives a phenomenological explanation of both of those phenomena. This theory explains that below a certain critical temperature 𝑇 𝐶 a percentage of the charge carriers, the electrons, starts turning into superconducting ones, their share among the global electron population growing as the temperatures comes near 0 K, as illustrated in Figure 4 [START_REF] Xu | FRIB Cryomodule Design and Production[END_REF] moreover suggests that the penetration of the magnetic field in a superconducting material is very limited, explaining the Meissner effect. This elegant equation ( 4-7 ) combines the one of the four Maxwell laws of electromagnetism (the Maxwell-Faraday one), and the laws of movement from Newton, applied to the superconducting charge carriers [START_REF] Longuevergne | Etude et test d'un module accélérateur supraconducteur pour le projet Spiral2[END_REF]:

𝑚 𝑛 𝑆 𝑒 2 ∇ × 𝐽 𝑆 + 𝐵 = 0 ( 4-7 )
With 𝑛 𝑆 being the density of the superconducting charge carriers (temperaturedependent, as said), 𝐽 𝑆 the field-induced supercurrent density, m and e the mass and charge of the electron.

Knowing that 𝐽 𝑆 = 1 µ 0

∇ × B, one can infer:

𝜆 𝐿 2 (∇ × ∇ × B) + 𝐵 = 0 ( 4-8 )
With 𝜆 𝐿 = √ 𝑚 µ 0 𝑛 𝑆 𝑒 2 the so-called London penetration depth and µ 0 the vacuum magnetic permeability: both the magnetic field and the supercurrent density decay exponentially with this characteristic scale 𝜆 𝐿 , which is in the order of 40 nm for niobium, thus explaining the observed magnetic field expulsion from the material bulk. The supercurrents thus circulate only in a thin layer below the material surface. As remarked in [START_REF] Longuevergne | Etude et test d'un module accélérateur supraconducteur pour le projet Spiral2[END_REF], contrary to the skin depth 𝛿 = √ 2𝜌 2𝜋𝑓𝜇 0 𝜇 𝑟 which characterizes the current penetration (𝜌 the electrical conductivity) in a normal material as a function of the AC current frequency f, 𝜆 𝐿 has no dependency in f, explaining why the Meissner effect also appears with a DC field.

Yet, the London equation cannot explain why the superconducting effect not only is limited by the temperature, but also by the magnetic field, when strong enough. The phenomenological Ginzburg-Landau theory gives a macroscopic thermodynamic explanation to this phenomenon. The theory involves a parameter called the free energy of the normal phase on the one hand, and of the superconducting phase on the other hand. It turns out that below 𝑇 𝐶 , the superconducting phase can exist, as there are potential wells that appear when comparing the normal and superconducting free energies.

On the one hand, the applied magnetic field reduces the deepness of the well, on the other hand, that the deepness itself depends on the temperature. There is thus an interdependency between the maximum applicable field 𝐵 𝑐 and the temperature, that can be approached by the following formula:

𝐵 𝑐 (𝑇) = 𝐵 𝑐 (0) [1 -( 𝑇 𝑇 𝑐 ) 2
] and illustrated in Figure 4-6. This is strictly true for type-I superconductor. For type-II superconductor, like niobium, the red line in Figure 4-6 is actually not that sharp; instead, at each temperature, there is a first limit, 𝐵 𝑐1 beyond which the field partially penetrates, but the material remains superconducting. Beyond a second threshold 𝐵 𝑐2 , the material transits to the normal state. This intermediate state is made possible because normal zones appear among the superconducting ones, where the magnetic field can penetrate.

A few years after this theory came yet another one, that was the first to give a true theoretical background, i.e. to identify which quantum interactions are at stake. This is the Bardeen-Cooper-Schrieffer theory, or BCS theory. This theory explains how two electrons can arrange themselves into so-called Cooper pairs, which are no longer scattered by the lattice of the material [START_REF] Padamsee | RF Superconductivity[END_REF]. In short, under certain conditions, the phonons, which are the quantum form of the vibrations of the lattice (partly responsible for the heat transfer), can interact with the electrons. But a phonon created by a first electron can be absorbed by a second electron, according to the Feynman interaction given in [151]. The momentum is thus conserved and not dissipated.

SRF CAVITY HEAT LOAD MEASUREMENT AND PREDICTIONS

Although orders of magnitude lower than the normal one, the superconducting electrical resistance of a superconductor is however not zero. A first electrical resistance arises from the BCS theory, thus called 𝑅 𝐵𝐶𝑆 [START_REF] Gurevich | Theory of RF superconductivity for resonant cavities[END_REF]:

𝑅 𝐵𝐶𝑆 = 𝐴𝑓 2 𝑇 𝑒 - ∆ 𝑘 𝐵 𝑇 ( 4-9 )
Where A is a factor depending on the purity of the material, f is the field frequency and This relation shows an exponential decrease of the resistance as the material temperature drops due to the dependence on the superconducting electron density [START_REF] Longuevergne | Etude et test d'un module accélérateur supraconducteur pour le projet Spiral2[END_REF]. It also reveals a square dependency on the frequency: for DC fields, this BCR resistance is simply zero, but for AC fields applied to RF cavities, this term is for instance 1.16 nΩ at 2K and 50 nΩ at 4.3 K. This phenomenon can be seen as the effect of the inertia of the charge carriers.

A second and last feature of the superconductor is given by its residual resistance 𝑅 𝑠 . 𝑅 𝑠 is caused by two effects: the material defects (making it less predictable than 𝑅 𝐵𝐶𝑆 ) and the trapping of magnetic vortices generated by the RF field. The latter contribution is proportional to the magnetic field of the Earth and the surroundings with a sensitivity of about 5 • 10 -2 𝑛𝛺/𝑚𝐺 [START_REF] Longuevergne | MYRTE -D2.10 MYRRHA SRF spoke R&D nov[END_REF]. The Earth's magnetic field of 500 mG would lead to a large residual resistance of 25 nΩ, hence the need to shield the cavities, as seen in 2.1. The total electrical resistance 𝑅 𝑠 is then:

𝑅 𝑠 = 𝑅 𝐵𝐶𝑆 + 𝑅 𝑟𝑒𝑠 ( 4-11 )
The advantage of the superconductivity clearly appears when computing the electrical resistance 𝑅 𝑛𝑜𝑟𝑚𝑎𝑙 for normal materials. From the previously described normal skin depth 𝛿, one gets 𝑅 𝑛𝑜𝑟𝑚𝑎𝑙 = 𝜌(𝑇) 𝛿 , thus:

𝑅 𝑛𝑜𝑟𝑚𝑎𝑙 = √𝜌(𝑇)𝜋𝑓𝜇 0 𝜇 𝑟 ( 4-12 )
Applying the formula to high-purity copper (RRR = 100) at 2 K yields 𝑅 𝑛𝑜𝑟𝑚𝑎𝑙 = 0.5 𝑚𝛺 compared to typically 10 nΩ for 𝑅 𝑠 for superconducting niobium at 2 K, meaning a factor of 50000 in the dynamic heat loads.

Niobium is the current material of choice for SRF cavities: although its critical temperature is pretty low at 9.26 K compared to other superconductors, niobium has the highest lower critical field 𝐵 𝑐1 , about 180 mT [START_REF] Gurevich | Theory of RF superconductivity for resonant cavities[END_REF]. Its thermal conductivity, important to transport the heat from the cavity inner surface to the outer surface as seen in 4.6, is also 2-3 orders of magnitude higher than other common superconductors like Nb3Sn, or even normal metals like stainless steel. Finally, niobium has good mechanical properties, compatible with sheet metal forming and for in-operation cavity tuning.

Another extremely common way for the accelerator community to express the cavity surface resistance is to include it in the so-called quality factor 𝑄 0 . This factor is defined as the ratio between the total stored electromagnetic energy, 𝑊 = , where H is the magnetic field 42 [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF], [START_REF] Junginger | Investigations of the surface resistance of superconducting materials[END_REF] pulsating at the frequency 𝑓 0 :

𝑄 0 = 2𝜋𝑓 0 𝑊 𝑃 𝑐𝑎𝑣 = 2𝜋𝑓 0 ∭ µ 0 |𝐻| 2 𝑑𝑉 𝑉 ∬ 𝑅 𝑠 |𝐻| 2 𝑑𝑆 𝑆 ( 4-13 )
This ratio is without unit and can even simplify if one considers the geometrical factor G:

𝐺 = 2𝜋𝑓 0 µ 0 ∭ |𝐻| 2 𝑑𝑉 𝑉 ∬ |𝐻| 2 𝑑𝑆 𝑆 ( 4-
𝑄 0 = 𝐺 𝑅 𝑠 ( 4-15 )
As a side note, the stored electromagnetic energy W is actually pretty modest, some tens of J, compared to superconducting magnets, storing MJ of energy. The consequence is that a cavity quench and a magnet quench have completely different consequences on the cryogenic system.

This parameter is thus fundamental as it drives how much heat the RF field will dissipate in the helium bath, thus the power consumption of the whole accelerator facility. The next section deals with the techniques to optimize it.

The physics of the quality factor

Q0 degradation effects and mitigation

Equation [START_REF] Edf | Les déchets radioactifs[END_REF][START_REF]Valley of stability[END_REF][START_REF]Laradioactivite[END_REF][START_REF] Joliot-Curie | [END_REF](8)[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF][START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF][START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF] shows the evolution of 𝑅 𝑠 that drives the quality factor 𝑄 0 ; the variables that has an impact on it at a given RF frequency and ambient magnetic field, are the temperature and the material properties.

𝑅 𝑠 and thus 𝑄 0 can evolve even once installed in the cryostat, because of the material 42 From now on, the « magnetization field » H, units [A/m], will be used and called « magnetic field », instead of B, which will be referred as « magnetic induction », units [T] properties. Indeed, when kept at about 100 K during a sufficient time, typically some tens of hours or less, the trapped interstitial hydrogen in the niobium, coming from the ambient air and from the chemical etching bath during the cavity preparation, can precipitate into non-superconducting niobium hydrides [157]. This effect, commonly called the Q-disease affects the 𝑅 𝑟𝑒𝑠 part of the surface resistance 𝑅 𝑠 . On the Spoke cavities, this problem was addressed during the cavity R&D phase, MYRTE [START_REF] Longuevergne | MYRTE -D2.10 MYRRHA SRF spoke R&D nov[END_REF]: the solution is to degas hydrogen at 650°C during 10 hours under vacuum, in a dedicated furnace. Thanks to this solution, the MINERVA Spoke cavities should not be subject to this effect. However, as the return of experience is still limited, a good practice is to limit the time that niobium stands between 50 and 150 K, by optimizing the cryogenic cooldown, stand-by and warm-up procedures.

A second major effect on 𝑄 0 is the field emission, which this time, depends on another variable, the electrical field. Field emission is a natural emission of electrons in metals: the electric field enhances the tunnel effect, where a particle, even having a lower energy that the potential well where it is bound, can escape it with a given probability [START_REF] Fowler | Electron emission in intense electric fields[END_REF]. The electrical field lower the potential well, this increases the probability for the particle (the electron) to escape. On top, this effect is increased by surface pollution that created field emitter spots [START_REF] Junginger | Investigations of the surface resistance of superconducting materials[END_REF].

The field emission (FE) is responsible for the negative slope in the common plot of 𝑄 0 against 𝐸 𝑎𝑐𝑐 , as seen in the next section. The escaping electrons, being accelerated, hit the niobium wall and heat it up, entailing thus an increase in the cavity heat load, in turn a direct degradation of 𝑄 0 , roughly proportional to the electrical field and thus to 𝐸 𝑎𝑐𝑐 . Another indirect degradation comes from the rising niobium temperature, which in turn makes the BCS resistance rise.

This effect can thus be discriminated against the Q-disease, which rather features a 𝑄 0 constant offset, as function of 𝐸 𝑎𝑐𝑐 . Another way to detect field emission is the emergence of X-ray emissions, created by the emitted electron acceleration by the RF fields and then bremsstrahlung [START_REF] Geng | Field Emission in Superconducting Accelerators: Instrumented Measurements for Its Understanding and Mitigation[END_REF]. The X-ray emission is roughly proportional to the field emission. The main contributor to field emission is thus surface pollution, which can be overcome by the cavity preparation and by the cavity RF conditioning, as seen next.

A third major effect is the multi-impact electron amplification, or multipacting (MP).

Multipacting is a self-sustained electron emission, where the first impact of an incoming electron yields several secondary electrons. These, in turn, are accelerated by the RF field creating new secondary electrons. A resonant peak can thus appear which depends primarily on the geometry. In the 60's the cavities were limited at about 2 MV/m by this effect. The multipacting was then mitigated by rounding the edges as illustrated in Figure 4-8 from [START_REF] Junginger | Investigations of the surface resistance of superconducting materials[END_REF],

where the secondary electron trajectory ultimately ends up in the curvature, without enough energy to excite a chain-reaction. Nowadays, multipacting is rather easily predicted by codes like MUSIC3D from CNRS [START_REF] Hamelin | MUSIC3D: code pour la modélisation du multipacting[END_REF]. back MP, like FE, only occurs in vacuum; it is known and studied in other fields, like the development of RF antenna onboard satellites and space probes.

Surface quality is also crucial to reduce the multipacting: the SEY (secondary electron yield) measures the number of secondary electrons per collision of a primary election. The SEY can be improved by material coating, as it is done for fundamental couplers window with titanium nitride [START_REF] Kaabi | Titanium Nitride Coating as a Multipactor Suppressor on RF Coupler Ceramic Windows[END_REF]. The material SEY is studied on dedicated test benches, like the one at LPSC [START_REF] Gomez-Martinez | Multipacting at LPSC[END_REF]. Finally, an appropriate surface preparation that efficiently suppresses roughness peaks (which are electron sources) is highly beneficial.

Like field emission, because it involves accelerated electrons, MP also generates X-rays, and dissipations (thus a 𝑄 0 degradation). By contrast, the 𝑄 0 drop induced by multipacting is extremely sharp, creating a so-called multipacting barrier. This is because almost all the incoming power of the RF line couples to the MP. This effect is thus easy to recognize and can be conditioned in practice (as the field emission).

Finally, the high-field 𝑄 0 slope (HFQS) is a fourth 𝑄 0 degradation, featuring a sharp drop for instance at13 MV/m for the Spoke cavities within MYRTE [START_REF] Longuevergne | MYRTE -D2.10 MYRRHA SRF spoke R&D nov[END_REF]. This effect does not come with X-ray, thus enabling to isolate it. Its physics is still not fully understood, but recent studies points that the sudden drop could come again from hydrogen material inclusions, that become non-superconducting after a given magnetic field threshold [START_REF] Checchin | High-field Q-slope mitigation due to impurity profile in superconducting radio-frequency cavities[END_REF]. There again, the surface preparation by low-temperature baking is strongly beneficial.

To conclude, this section highlighted the role of the material choice, the cavity preparation, and the cavity design to maintain a high 𝑄 0 ahead of the cavity operation.

It must be highlighted that this optimization process remains highly empirical. Likewise, from a typical 𝑄 0 = 𝑓(𝐸 𝑎𝑐𝑐 ) curve as the one seen in 4.2.2.2, identifying all the previously described effects (excepted for the MP) is not straightforward.

Before the operation, the RF parts must undergo a preliminary step, the RF conditioning.

The role of the RF conditioning

RF conditioning is a compulsory step for all RF parts operating in vacuum, thus which are subject to field emission or multipacting or both. Indeed, even though the RF parts were thoroughly prepared and handled in iso 5 clean rooms, more stringent than most space applications, pollutions do remain on the surfaces. The RF field must therefore smoothly be increased from zero to a maximum field that covers the whole operating range. This rise in power occurs stepwise: for each power step, a procedure lasting typically for some tens of minutes makes the field rise and fall several times per minute.

Over the course of the conditioning procedure, that can take place once mounted in the cryomodule for the cavities, at cold temperature, or on dedicated test benches at room temperature for the couplers, the RF surfaces experience many controlled occurrences of MP and FE. This procedure gradually makes the surface degas the deposited pollutions, which are then sucked by the vacuum system, and soften the major rugosity peaks.

Other processing techniques exist, like the introduction of low-density helium gas to decrease the field emission [START_REF] Padamsee | RF Superconductivity[END_REF] or the cleaning of RF cavities by plasma-created UV rays, but are still experimental.

Extreme heat loads: the quenches

The critical RF field is the theoretical maximum field the niobium can sustain before transiting into normal state. As seen in Figure 4-6, the maximum field 𝐵 𝑐1 that can be achieved is limited by the maximum generated magnetic field, which itself depends on the niobium temperature. For superconducting cavities between 2 and 4.3 K, 𝐵 𝑐1 is typically near 0.2 T [START_REF] Wangler | RF Linear Accelerators[END_REF], [START_REF] Longuevergne | Etude et test d'un module accélérateur supraconducteur pour le projet Spiral2[END_REF]. This sudden transition from superconducting to normal state is commonly called "quench".

In practice though, the cavity accelerating gradient does not reach the 𝐵 𝑐1 limit: first, the non-superconducting defects or pollutions, create very high local heating, typically on sub-millimetre areas [START_REF] Junginger | Investigations of the surface resistance of superconducting materials[END_REF]. Such a hot spot can make the local niobium temperature be higher than the transition temperature; the spot then propagates by conduction and gradually leads to a cavity total quench, as said, below the maximum field 𝐵 𝑐1 . A second mechanism can lead to similar hot spots: trapped residual flux during the superconducting transition can oscillate under the RF field and create dissipations. As quenches are destructive for the cavities and for the cryogenic system, monitoring techniques are set-up, which principle is briefly explained in 4.2.1.3. These effects are all the more pronounced in high magnetic field areas, as the transition temperature is even lower.

A high material thermal conductivity is a first way to mitigate this effect: the higher it is, the lower hot spot temperature rises. On the cavity design, it is also crucial to decrease the peak magnetic and electrical fields over the whole cavity surface. The indicators , which quantify those peaks relative to the accelerating gradient, are optimized during the cavity design.

Lastly, as said, X-ray emissions come along with FE and MP. For personal protection matter, in a test or accelerator facility, those emissions are limited 43 . This maximum in Xray can thus limit the cavity field, even when there is a sufficient margin with regard to the quench.

Measurement of the quality factor Q0

Another way to express Q0

Equation [START_REF] Edf | Les déchets radioactifs[END_REF][START_REF]Valley of stability[END_REF][START_REF]Laradioactivite[END_REF][START_REF] Joliot-Curie | [END_REF](8)[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF][START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF][START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF][START_REF] Hargraves | Liquid Fluoride Thorium Reactors[END_REF][START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF][START_REF]Centre d'informations et de ressources sur CIGEO[END_REF][START_REF]Réacteur nucléaire naturel d'Oklo[END_REF] gives the relation between 𝑄 0 and 𝑅 𝑠 . A clever technique, exposed in [START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF] and patented [START_REF] Vassal | Procédé de détermination d'un facteur de qualité d'une cavité accélératrice d'un accélérateur de particules[END_REF], was set up to derive 𝑄 0 from 𝑃 𝑐𝑎𝑣 using this relation, in the context of a cryomodule installed in an accelerator facility. The advantage of this approach being that it enables to calculate 𝑄 0 independently from any RF information, like the accelerating gradient 𝐸 𝑎𝑐𝑐 , but requires a precise model of the surface resistance, in particular, the temperature dependency. This approach is fully adapted for a cavity 𝑄 0 monitoring by the cryogenic controls, in a long-term drift detection strategy.

In practice, when the accelerating gradient can be known, another approach is possible with fewer uncertainties. First, the cavity shunt impedance should be defined as the ratio between the accelerating voltage and the cavity heat loads:

𝑟 = 𝑉 𝑐𝑎𝑣 2 𝑃 𝑐𝑎𝑣 = (𝐸 𝑎𝑐𝑐 𝐿 𝑎𝑐𝑐 ) 2 𝑃 𝑐𝑎𝑣 ( 4-16 )
It is clear that the objective of the cavity designer is to maximize the shunt impedance. Another useful parameter is the ratio between the shunt impedance and the quality factor:

𝑟 𝑄 0 = 𝑉 𝑐𝑎𝑣 2 𝑃 𝑐𝑎𝑣 * 𝑃 𝑐𝑎𝑣 2𝜋𝑓 0 𝑊 = 𝑉 𝑐𝑎𝑣 2 2𝜋𝑓 0 𝑊 ( 4-17 )
This parameter, abbreviated 𝑟 𝑄 , is the second one, in addition to G, that is independent from the cavity wall material properties: it is fully determined by 3D RF codes. A point of attention, as 𝑉 𝑐𝑎𝑣 is involved, 𝑟 𝑄 depends on the particle β. Mixing ( 4-16 ) and [START_REF] Edf | Les déchets radioactifs[END_REF][START_REF]Valley of stability[END_REF][START_REF]Laradioactivite[END_REF][START_REF] Joliot-Curie | [END_REF](8)[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF][START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF][START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF][START_REF] Hargraves | Liquid Fluoride Thorium Reactors[END_REF][START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF][START_REF]Centre d'informations et de ressources sur CIGEO[END_REF][START_REF]Réacteur nucléaire naturel d'Oklo[END_REF][START_REF] Poncelet | Myrrha: un (autre) regard sur l'énergie nucléaire[END_REF][START_REF]Accelerator-driven Systems (ADS) and Fast Reactors (FR) in Advanced Nuclear Fuel Cycles: A Comparative Study[END_REF] results in a widely used relation:

𝑃 𝑐𝑎𝑣 = (𝐸 𝑎𝑐𝑐 𝐿 𝑎𝑐𝑐 ) 2 𝑟 𝑄 𝑄 0 ( 4-18 )
This relation has the great advantage to connect 𝑄 0 to 𝑃 𝑐𝑎𝑣 and 𝐸 𝑎𝑐𝑐 , which are measured by the RF system, with sometimes, the help of the cryogenic system, as seen in the next section.

Q0 measurement in practice

looking at Figure 4-9, it is theoretically always possible to make a power balance on the cavity, where 𝑃 𝑖 is the incident power from the amplifier (also called 𝑃 𝑔 for "generator"), 𝑃 𝑟 is the reflected power from the cavity and 𝑃 𝑡 is the transmitted power, the image of the field sensed by the pick-up antenna:

𝑃 𝑐𝑎𝑣 = 𝑃 𝑖 -𝑃 𝑟 -𝑃 𝑡 ( 4-19 )
This power balance is valid in steady-state: transient power occurs when the cavity field changes, since energy leaves or enters the cavity. In reality, 𝑃 𝑖 , 𝑃 𝑟 and 𝑃 𝑡 are measured thanks to powermeters, devices that converts and RF signal to power reading. As a side note, the phase of those RF signals, namely 𝜙 𝑖 , 𝜙 𝑟 and 𝜙 𝑡 can also be measured (not always done in operation and highly dependent on the calibration drift). Those 6 values are the only RF field signals normally acquired by the RF control.

From ( 4-19 ) it is clear that 𝑃 𝑐𝑎𝑣 is seamlessly obtained if 𝑃 𝑖 , 𝑃 𝑟 and 𝑃 𝑡 are measured with enough precision. By combining with equation ( 4-18 ), one gets the evolution of 𝑄 0 as a function of 𝐸 𝑎𝑐𝑐 .

In practice, in a cavity test cryostat, the antenna that brings the RF power (the inlet coupler, see Figure 456789) is deliberately chosen so that 𝑃 𝑟 is the smallest; 𝑃 𝑐𝑎𝑣 can then be derived with the highest precision from the RF system.

On a cryomodule, meant for the beam operation, by contrast, the inlet coupler is a power coupler; it is adapted to transfer power to the beam in the most efficient way. It results that 𝑃 𝑟 is very close to 𝑃 𝑖 (if the beam is OFF) and consequently that 𝑃 𝑐𝑎𝑣 is acquired with a very low precision by the RF measurements. This is a major obstacle for cryomodule test and ultimately for accelerator operation, as 𝑄 0 is then only accessible through 𝑃 𝑐𝑎𝑣 measured via the cryogenic system.

MINERVA prototype Q0 curves

The prototyping phase comprised the manufacturing of two new representative Spoke cavities, delivered by Zanon, namely BSC03 and BSC04. Those cavities are built upon the experience from the MYRTE project on the first two MYRRHA Spoke cavities, BSC01 (code name Amelia) and BSC02 (code name Virginia). Those had suffered from flaws on the tank welding, leading to an excessively low stiffness, thus an extreme sensitivity to the surrounding pressure [START_REF] Longuevergne | Performances of the Two First Single Spoke Prototypes for the MYRRHA Project[END_REF].

BSC03 and BSC04 were measured twice, as a problem was discovered after the first test, requiring new machining and welding steps, thus a new validation in cavity test cryostat.

The tests were done in the IJClab vertical cryostat CV800 and the results are shown in ( 4-9 ).

Both cavities were measured at different temperatures, which, as seen before, affects 𝑅 𝑠 mainly through 𝑅 𝐵𝐶𝑆 . The cold tuning system, abbreviated CTS, has an expected negative impact on the 𝑄 0 , likely due to its magnetic field that affects 𝑅 𝑠 , through 𝑅 𝑟𝑒𝑠 this time.

Up to 10 MV/m, both cavities have the same 𝑄 0 gentle drop, still above 10 10 , limiting the heat loads below 2 W at 10 MV/m. Beyond, BSC03 was limited to about 15 MV/m in field due to strong field emission, whereas BSC04 was validated up to 18 MV/m and could have even higher.

Figure 4-10: BSC03 (left) and BSC04 (right) Q0 and X-ray curves as a function of Eacc

What is at stake for the cryomodule test phase is to measure 𝑄 0 as precisely as possible, to compare against the reference vertical cryostat tests.

To conclude, so far, the important quality factor 𝑄 0 was defined, as well as its physics and the measurement methods. It was also seen that 𝐸 𝑎𝑐𝑐 was somehow necessary to derive 𝑄 0 , and finally, that the only measurable system states are 𝑃 𝑖 , 𝑃 𝑟 and 𝑃 𝑡 and their respective phases. Being able to predict those values and their interdependency is thus crucial; this is the object of the cavity field model, presented in the next part.

THE CAVITY "FIELD" MODEL

As said in introduction of this chapter, the cavity field model, the high-power and the LLRF models could be directly reused from [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF] and were updated by the same author.

Besides, the relations between the cavity variables using an equivalent RLC analogy are already well documented. The purpose of this part is thus not to demonstrate again some well-established results, but rather to complement this approach by some new relevant formula and additional interpretation useful to understand the connection with the cavity field control and, above all, with the coupler field model. Last but not least, some preliminary cavity field validations against measurements are exposed.

To start with, some well-known fundamental notions like the coupling factors in the next section still deserve a short overview.

Coupling and quality factor

The cavity field control studies presented in 2.4 all rely on a widely accepted notion that, in a frequency range (some MHz) around the used resonant mode (TEM or TM), the cavity can be assimilated to an RLC resonator. This analogy makes it possible to use the standard resonant circuit analysis tools to describe and analyse the relations between the cavity signals. The only task is to scale the RLC circuit model parameters to the cavity one.

The first parameter, 𝑟 𝑄 , is already known; the additional model parameters are the other quality factors, namely 𝑄 𝑖 and 𝑄 𝑡 . By analogy with 𝑃 𝑐𝑎𝑣 , those quality factors relate the importance of the power leaks respectively through the inlet coupler and by the pick-up antenna with respect to the stored energy. Finally, 𝑄 𝐿 qualifies the importance of all the combined cavity losses (except the beam), so that one gets the well-known relation [165]:

1 𝑄 𝐿 = 1 𝑄 0 + 1 𝑄 𝑖 + 1 𝑄 𝑡 ( 4-20 )
This relation can also be seen as:

𝑄 0 𝑄 𝐿 = 1 + 𝛽 𝑖 + 𝛽 𝑡 ≈ 1 + 𝛽 𝑖 ( 4-21 )
Here, 𝛽 𝑖 = 𝑄 0 𝑄 𝑖 and 𝛽 𝑡 = 𝑄 0 𝑄 𝑡 are the coupling coefficients. 𝛽 𝑖 in particular is important as it quantifies how the antenna coupling to the RF field is chosen.

The cavity is "undercoupled" when 𝛽 𝑖 ≪ 1, which occurs when the superconducting cavity is still at room temperature, meaning 𝑄 0 is very low (~10 4 ) with a typical 𝑄 𝑖 of ~10 6 for a power coupler in a cryomodule for beam acceleration.

With the same 𝑄 𝑖 , the coupler antenna becomes "overcoupled" (𝛽 𝑖 ≫ 1), typically ~1000 when the cavity is cooled at 2 K (𝑄 0 rise to ~10 10 ). For beam operation in a cryomodule, 𝑄 𝑖 is chosen so that most of 𝑃 𝑖 is transferred to the beam (critical coupling with the beam), and of course, not to the walls. When the beam is off with this same strong coupling, most of the power that got into the cavity generates a reflected wave that travels backward and leaks out by the coupler. One therefore sees that for the same field 𝐸 𝑎𝑐𝑐 in the cavity, thus the same heat load 𝑃 𝑐𝑎𝑣 , the coupler does not experience the same field with or without beam. To predict the coupler heat loads, it is then crucial to predict the coupler field too in any working conditions (beam ON or OFF), as developed in 4.7.

By contrast, in a cavity test stand (so, without beam), one favours 𝛽 𝑖 close to 1 (thus 𝑄 𝑖 = 𝑄 0 = 2𝑄 𝐿 ) meaning this time, that the incoming power ends-up almost fully in the cavity walls. The cavity is said to be critically coupled in this case. This ensures that 𝑃 𝑟 is the smallest, and that the measurement precision on 𝑃 𝑐𝑎𝑣 is the highest, as seen in 4.2.2.2.

In practice, this change of 𝛽 𝑖 configuration requires a change in the physical penetration of the inlet antenna in the cavity, as well as in the antenna size: in the case of a cavity loaded with a beam, the antenna must be large and penetrate substantially to couple to the cavity field and transmit the required power to the beam. That is why cavity test and cryomodule antenna are radically different. Part 4.4.1 further develops how to optimally chose 𝑄 𝑖 and highlights the trade-off with another notion, the bandwidth.

On the contrary, the pick-up antenna is chosen weakly coupled, to minimize the losses, as the antenna purpose is to make an image of the cavity voltage, thus implying 𝛽 𝑡 ≪ 1.

All the scaling parameters for the RLC circuit being described, the next section presents the main relation one can derive from this approach. The red square in Figure 4-11 stands for the total impedance of the transmission line and wall losses seen from the cavity, thus is the loaded impedance 𝑍 𝐿 that corresponds to 𝑄 𝐿 . Still seen from the cavity, the transient equation of the cavity complex voltage 𝑉 ̃𝑐𝑎𝑣 in reaction to the excitations from the generator current 𝐼 ̃𝑔 (the RF power amplifier source) and by the beam current 𝐼 ̃𝑏 is adapted from [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF] and gives:

Voltage VS current equation and power balance in the RLC analogy

𝑉 ̃ċ𝑎𝑣 + 𝜔 0 2𝑄 𝐿 [1 -𝑗 tan 𝜓]𝑉 ̃𝑐𝑎𝑣 = 𝜔 0 𝑟 𝑄 4 (2𝐼 ̃𝑔 + 𝐼 ̃𝑏) ( 4-22 )
On top of the couplings, this equation tells us that the cavity voltage also depends on the resonance frequency 𝑓 0 of the circuit, which is hidden in a parameter called 𝜓, which is related to 𝑓 0 by:

tan 𝜓 = 𝑄 𝐿 ( 𝑓 0 2 -𝑓 2 𝑓 0 * 𝑓 ) = 𝑄 𝐿 𝛿 ( 4-23 )
Physically, 𝜓 is the angle from all the complex currents to their respective voltages (in this direction) involved in the cavity. It is 0 at resonance frequency, as one would expect from all resonator (the currents and voltages are in phase). 𝑓 0 is dynamically actuated by the tuning system, so that the power is optimally transmitted to the beam.

To express the relations between the measured power and the couplings, it is interesting to perform a cavity transient power balance. To do so, first, the incoming voltage from the generator 𝑉 ̃𝑖𝑛𝑐 seen from the transmission line and the reflected voltage 𝑉 ̃𝑟𝑒𝑓 must be expressed: From left to right, one identifies respectively:

𝑉 ̃𝑖𝑛𝑐 =
• 𝑃 𝑡𝑟𝑎𝑛𝑠𝑖𝑡 : the transient power filling or leaving the cavity as the field rises or falls • 𝑃 𝑐𝑎𝑣 : the power dissipated in the walls (the dynamic heat loads)

• 𝑃 𝑡 : the power taken by the pick-up antenna • 𝑃 𝑔 or 𝑃 𝑖 : the amplifier incident power to the cavity

• -𝑃 𝑟 : the opposite of the reflected power • -𝑃 𝑏 : the opposite of the beam power One can then check that 𝑃 𝑖 = 𝑃 𝑡𝑟𝑎𝑛𝑠𝑖𝑡 + 𝑃 𝑐𝑎𝑣 + 𝑃 𝑡 + 𝑃 𝑟 + 𝑃 𝑏 , as expected.

Thanks to this power balance, it is now possible to relate 𝑃 𝑖 , 𝑃 𝑟 and 𝑃 𝑡 and to the couplings. One also should note that the incident power amplitude 𝑃 𝑔 (= 𝑃 𝑖 ) and the reflected power amplitude 𝑃 𝑟 are the first boundary conditions of the coupler field model.

Measurement of the couplings Qi, Qt and relation to Eacc

At resonance, meaning tan 𝜓 = 0, the ratios 𝑄 𝑖 and 𝑄 𝑡 which are constants whatever the field or the cavity temperature; they are thus connected to 𝑄 𝐿 by these relations [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF]. So, in addition to the power ratio measured, for instance, by a vector network analyser (VNA 44 ), one additionally needs 𝑄 𝐿 .

𝑄 𝐿 is straightforward to measure, either by measuring the bandwidth of the cavity with a VNA, which is related to 𝑄 𝐿 by equation or by measuring the time constant τ of the response to a 𝑃 𝑖 step [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF], [START_REF] Longuevergne | Etude et test d'un module accélérateur supraconducteur pour le projet Spiral2[END_REF].

𝐵𝑊 = 𝑓 0 𝑄 𝐿 ( 4-30 )
As a side note, the bandwidth is defined as the frequency range where the transmitted power 𝑃 𝑡 is above +/-half of its maximum value [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF].

The ± signs in ( 4-28 ) and ( 4-29 ) comes from the fact that the couplings can have 2 solutions in the equation, only one being physical. The "+" sign is used in overcoupled configuration, the "-" in undercoupled configuration [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF], [START_REF] Longuevergne | Etude et test d'un module accélérateur supraconducteur pour le projet Spiral2[END_REF].

As both of the antenna (power coupler and pick-up) were specific in the MINERVA cavity test cryostat described in 4. On the other hand, the undercoupled configuration seamlessly gives 𝑄 0 at room temperature, since this time, 𝑄 𝑖 ≫ 𝑄 0 and 𝑄 𝑡 ≫ 𝑄 0 . This measurement also enables to check the estimated G from CST, from the formula ( 4-15 ) since the electrical resistance of niobium is known at room temperature.

The summary of all cavity parameters measured and estimated is given in Annex 7.5.

To conclude this part on the couplings, as said the antenna pick-up purpose is to give an image for the accelerating field. From 𝑃 𝑡 expression in the power balance, and from ( 4-4 ), the relation between 𝐸 𝑎𝑐𝑐 and 𝑃 𝑡 becomes simple:

𝐸 𝑎𝑐𝑐 = √ 𝑟 𝑄 𝑄 𝑡 𝑃 𝑡 𝐿 𝑎𝑐𝑐 ( 4-31 )
This relation is used in practice by the RF control to compute 𝐸 𝑎𝑐𝑐 from the measured 𝑃 𝑡 signal.

Validation of the cavity S-parameters

In circuit analysis, in order to fully describe a resonant system, it is common and convenient to refer to s-parameter [166], which are another way to express the power signal ratios. The value of these parameters depends on the excitation frequency; they are used in this part to compare the prototype cavity field RLC-model to the cavity measurements.

A first validation can be performed, by looking at 𝑆 21 , rather its module |𝑆 21 |, defined as follows: A second validation can be performed on another commonly used s-parameter, called 𝑆 11 , and defined as follows:

𝑆
𝑆 11 = 𝑏 1 𝑎 1 ↔ 𝑆 11 = 𝑉 ̃𝑟𝑒𝑓 𝑉 ̃𝑖𝑛𝑐 ( 4-33 )
The complex value of 𝑆 11 depends on the location where the voltages are probed; the cavity RLC model predicts the voltages exactly at the accelerating gap. There is no such probe in the cryomodule, but it is possible to simulate it with 3D RF tools.

The expression of 𝑆 11 must be simplified using the expression of the generator current, from [START_REF] Edf | Les déchets radioactifs[END_REF][START_REF]Valley of stability[END_REF][START_REF]Laradioactivite[END_REF][START_REF] Joliot-Curie | [END_REF](8)[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF][START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF][START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF][START_REF] Hargraves | Liquid Fluoride Thorium Reactors[END_REF][START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF][START_REF]Centre d'informations et de ressources sur CIGEO[END_REF][START_REF]Réacteur nucléaire naturel d'Oklo[END_REF][START_REF] Poncelet | Myrrha: un (autre) regard sur l'énergie nucléaire[END_REF][START_REF]Accelerator-driven Systems (ADS) and Fast Reactors (FR) in Advanced Nuclear Fuel Cycles: A Comparative Study[END_REF][START_REF] Junghans | Transmutation of high-level radioactive waste -Perspectives[END_REF][START_REF]Bilan du débat public, Projet de centre de stockage réversible profond de déchets radioactifs en Meuse / Haute-Marne (Cigéo)[END_REF][START_REF]La réversibilité Thème 1 -La place de Cigéo dans le dispositif de gestion des déchets[END_REF][START_REF] Chevret | Mesure de la réactivité de réacteurs sous-critiques pilotés par accélérateur par l'analyse d'expériences d'interruptions de faisceau programmées[END_REF][START_REF] Abderrahim | MYRRHA -A multipurpose fast spectrum research reactor[END_REF] in steady-state and considering the case without beam:

𝐼 ̃𝑔 = 𝑉 ̃𝑐𝑎𝑣 𝑄 𝐿 𝑟 𝑄 [1 -𝑗 tan 𝜓] ( 4-34 )
Then, injecting this relation in the previously seen 𝑉 ̃𝑖𝑛𝑐 and 𝑉 ̃𝑟𝑒𝑓 expressions ( 4-24 ) and ( 4-25 ) yields:

𝑉 ̃𝑖𝑛𝑐 = 𝑄 𝑖 𝑉 ̃𝑐𝑎𝑣 2𝑄 𝐿 [1 -𝑗 𝑡𝑎𝑛 𝜓] ( 4-35 ) 𝑉 ̃𝑟𝑒𝑓 = 𝑉 ̃𝑐𝑎𝑣 [1 - 𝑄 𝑖 2𝑄 𝐿 [1 -𝑗 tan 𝜓]] ( 4-36 )
Putting both equations in ( 4-33 ) results in a simple expression for 𝑆 11 , again, only valid for the gap seen from the transmission line point of view (not from the cavity), and assuming a zero-length line:

𝑆 11 = 𝛽 𝑖 -𝛽 𝑡 -1 -𝑗𝑄 0 𝛿 1 + 𝛽 𝑡 + 𝛽 𝑖 + 𝑗𝑄 0 𝛿 ≈ 𝛽 𝑖 -1 -𝑗𝑄 0 𝛿 1 + 𝛽 𝑖 + 𝑗𝑄 0 𝛿 ( 4-37 )
This well-known expression for an RLC circuit [166] can be plotted in a Smith chart, another standard tool in circuit analysis, that enables to represent the impedance matching; it is shown in Figure 4-13. The background dark curves show the isolines of the reduced impedance 𝑧 = 𝑍 𝑐𝑎𝑣 𝑍 0 imaginary part (the surrounding values preceded by "j")

and its real part (the dark circles that cut the x-axis). This chart shows both 𝑧 and 𝑆 11 since:

𝑆 11 = 𝑧 -1 𝑧 + 1 ( 4-38 )

Figure 4-13: Smith chart of the cavity voltage at the accelerator gap

Each coloured circle represents an RLC model simulation, where the excitation frequency 𝑓 is swept at iso 𝑄 0 . The different circles are for different 𝑄 0 values (like when the cavity is RT or at 2 K), at iso 𝑄 𝑖 . The point where all the circles converge to the left-handed side → → 1 → of the x-axis, is where 𝑓 is far from the resonance 𝑓 0 . The other intersection point between each circle and the x-axis gives the value of the cavity gap reduced impedance at resonance.

For the red circle (undercoupled), the reduced impedance at resonance is near 0 (𝑆 11 = -1), meaning the cavity gap behaves like a short-circuit (SC), just like all coupling configurations far from the resonance. For the critical coupling case (dark blue), at resonance 𝑧 = 1 (𝑆 11 = 0) meaning the gap behaves like a matched source. Finally, the largest light blue circle is the overcoupled case; it crosses the x-axis at a high reduced impedance (𝑆 11 = 1), meaning the gap behaves like an open-circuit (OC).

Three particular points, (undercoupled off-resonance, matched and overcoupled at resonance) were simulated using CST Microwave Studio by the LPSC team, and perfectly match with the RLC model, as seen in Figure 4-14.

Figure 4-14: E field along the beam axis in both gaps predicted by CST (left) and RLC model smith chart (right) for the same 𝑃 𝑖

A third and last validation of the cavity field model comes from the literature [166]: the fact that the variation of 𝑆 11 when off-resonance versus at resonance is so different when undercoupled or overcoupled is a well-known result, and is even used to know in which coupling configuration a resonator is. This summarizes as follows:

• Resonator detuned + 𝛽 𝑖 > 1 or 𝛽 𝑖 < 1 → reflexion with phase change (𝑆 11 = -1) • Resonator tuned + 𝛽 𝑖 > 1 → reflexion without phase change (𝑆 11 = 1)

Here, the term "phase change" refers to the change between 𝜙 𝑖 and 𝜙 𝑟 , the respective phases of the complex voltages 𝑉 ̃𝑖𝑛𝑐 and 𝑉 ̃𝑟𝑒𝑓 .

To make it more visual, the latter 4 presented cases are plotted in the so-called cavity phaser (Figure 4-15), a tool widely used in [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF]: it shows the evolution of all the complex voltages (so far only 𝑉 ̃𝑖𝑛𝑐 and 𝑉 ̃𝑟𝑒𝑓 ) and their associated current in whatever coupling and resonance condition. The phaser clearly shows when 𝜙 𝑖 equals 𝜙 𝑟 or not.

Tuned Detuned

Over-coupled

Under-coupled

Figure 4-15 : the cavity phaser diagram for the 2 configurations under/overcoupled crossed with the 2 cases on/off resonance

As a conclusion for this part, a primary validation of the RLC model of cavity was conducted based on VNA measurements, 3D RF simulation and experiments. In section 5.4.1, the cavity field model is this time challenged against experiments with a highpower amplifier in a true cryomodule.

This model is crucial to the RF field control model which is presented in the next section. It also generates suitable boundary conditions for the coupler field model, as seen in 0.

THE MINERVA CAVITY RF FIELD AND RESONANCE CONTROL MODEL

The previous section highlighted the importance of the cavity field model for the global RF field and resonance control model. Before presenting the RF control model itself, the concept of the optimal coupling and optimal frequency tuning must be introduced.

Optimal Qi choice and optimal frequency concept

Given the amount of RF power involved, it is crucial that the highest share of the amplifier power is transmitted to the beam. As seen earlier, 2 levers enable to optimize this transmission:

• The coupling 𝑄 𝑖 , fixed for the whole operation by trimming the antenna penetration during the cryomodule assembly (unless the coupler is variable) • The resonance frequency 𝑓 0 , or the detuning ∆𝑓 = 𝑓 0 -𝑓, by dynamically slightly changing the shape of the cavity The optimal ∆𝑓 (independent of 𝑄 𝑖 ) is about +35 Hz, given by the equation . 45 . This value deserves a short historic: over the course of the project and its research phase, the 𝑄 𝑖 specification has varied from 3 10 6 (the energy optimum), to 1.5 10 6 in order to get a larger cavity bandwidth (doubled from about 117 Hz to 234 Hz), safer for the resonance control but requiring more installed RF power (+12%). It was then raised again for a better trade-off at a target of 2.3 10 6 and a bandwidth of 153 Hz. When the cavities were ordered, the specification was still at 1.5 10 6 . However, the 𝑄 𝑖 as manufactured for BSC03 and BSC04 is around 1 10 6 . This highlights the difficulty to predict (with 3D models) the required antenna penetration to reach the desired 𝑄 𝑖 , and also the central role of the prototypes to provide experimental data.

∆𝑓 𝑜𝑝𝑡 =
Back to the cavity phasers, each of the 3 cases displays a specific detuning ∆𝑓, that is also reported in the transmission diagram:

• Case #1: ∆𝑓 = ∆𝑓 𝑜𝑝𝑡 • Case #2: ∆𝑓 > ∆𝑓 𝑜𝑝𝑡 • Case #3: ∆𝑓 < ∆𝑓 𝑜𝑝𝑡
In the phaser plot, 2 yet unseen voltages appeared, the cavity voltage induced by the beam current 𝑉 ̃𝑏 (by convention 𝐼 ̃𝑏 is oriented in the minus direction), and the one induced by the generator 𝑉 ̃𝑔, that obviously respect:

𝑉 ̃𝑏 + 𝑉 ̃𝑔 = 𝑉 ̃𝑐𝑎𝑣 = 𝑉 ̃𝑖𝑛𝑐 + 𝑉 ̃𝑟𝑒𝑓 ( 4-41 )
The phaser clearly illustrates that the optimum detuning ∆𝑓 𝑜𝑝𝑡 is reached when 𝜓 makes 𝑉 ̃𝑖𝑛𝑐 aligned with 𝑉 ̃𝑐𝑎𝑣 , thus minimizing the length of the 𝑉 ̃𝑟𝑒𝑓 complex vector. The fact that ∆𝑓 𝑜𝑝𝑡 is non-zero comes from the fact that the synchronous phase 𝜙 𝑠 is non-zero: physically, the additional reactance due to the phase offset between the beam voltage and the cavity voltage needs to be compensated. How exactly ∆𝑓 is actuated is explained in part 4.4.3.

The phasers also show the energetically non-optimal 𝑄 𝑖 choice (2.3 10 6 instead of 3 10 6 ): even at ∆𝑓 𝑜𝑝𝑡 , one sees that 𝑉 ̃𝑟𝑒𝑓 is non-zero. With an optimal 𝑄 𝑖 , 𝑉 ̃𝑖𝑛𝑐 would perfectly be superimposed on 𝑉 ̃𝑐𝑎𝑣 .

Those 3 cases illustrate once more, that even at constant field 𝐸 𝑎𝑐𝑐 , 𝑉 ̃𝑖𝑛𝑐 and 𝑉 ̃𝑟𝑒𝑓 change in amplitude and phase during the operation, when the beam parameters are changed or when the tuning is changed, which have consequences on the coupler heat loads.

To give some orders of magnitude, the needed powers to maintain the cavity field and feed the beam (𝑃 𝑏 = 7.64 𝑘𝑊) are: This principle is not specific to MINERVA but is actually widely used, on similar machines like SPIRAL2 or ESS, or on fully different ones like the synchrotron light machine ESRF 46 .

The tuning control law on the other hand can take several forms, from a simple constantincrement like at GANIL, to a feed-forward approach like at ESS or X-FEL [START_REF] Johansson | LLRF System for the ESS Proton Accelerator[END_REF], [START_REF] Branlard | LLRF SYSTEM DESIGN AND PERFORMANCE FOR XFEL CRYOMODULES CONTINUOUS WAVE OPERATION[END_REF]. It depends on the physics of the detuning effects at stakes and if the machine is pulsed or not, as further explained in 4.5.

Going back to the field control, in reality, 𝑃 𝑖 and 𝜙 𝑖 are not even the true command sent by the LLRF control law. The true command is a low-level voltage 𝑉 ̃𝐿𝐿𝑅𝐹 that drives the amplifier. On the other side of the control law, the true measurements are in fact not 𝐸 𝑎𝑐𝑐 and its phase 𝜙 𝑐𝑎𝑣 but rather 𝑉 ̃𝑡. Likewise, the targets 𝐸 𝑎𝑐𝑐 𝑡𝑎𝑟𝑔𝑒𝑡 and 𝜙 𝑠 are translated into a target voltage 𝑉 ̃𝑡 𝑡𝑎𝑟𝑔𝑒𝑡 . Lastly, the control law in fact does not consider directly 𝑉 ̃𝑡 and 𝑉 ̃𝐿𝐿𝑅𝐹 but rather their vector modulated I/Q counterpart, that provides a full decoupling of the amplitude and phase [169], [START_REF] Mosnier | CONTROL OF SCRF CAVITIES IN HIGH POWER PROTON LINAC[END_REF].

In order to perform model-based virtual commissioning, it is then crucial to account for all of these signal processing in the model; this way, the control law parameters can seamlessly be transposed from the model to the real device, as highlighted in [START_REF] Bouly | Superconducting Cavity and RF Control Loop Model for the SPIRAL2 Linac[END_REF].

On top of this signal processing, it is important to account for the signal delay and attenuations, as detailed in the model implementation in part 4.4.3.1.

Abnormal operation: fast interlock

As a final step before looking at the implementation of the model, it is worth making a short detour into the RF control of abnormal events.

The RF signals for the field and resonance control discussed earlier are acquired, processed and actuated by a dedicated LLRF control system. As RF processes can lead to destructive events, like arching, multipacting or quench, there must be a fast interlock system to cut or mitigate the RF, also part of the LLRF system.

Arching, for example, can occur with excessive clearances in any high power RF part, such as the circulators or the couplers. They can cause severe and irreversible damage to the hardware. Arc detectors, consisting in a dedicated optical line connected to a photodiode onboard the so-called FMC board (the board of the RF control hardware dedicated to the interlock, as seen in the next part), monitor both the circulators and the coupler windows.

Multipacting can occur in high power RF parts under vacuum (mainly cavities and couplers) if the RF surfaces are not properly conditioned; multipacting is especially dangerous in a cryomodule with a high-power amplifier since all the available power can potentially couple to the multipactor. The couplers are equipped with a multipactor detector, consisting in an electron pick-up antenna connected to the same FMC board as the arc detectors.

Quenching are also a major problem for the operation of cryomodules: as exposed in 4.2, cavities, unlike magnets, store only a relatively small amount of energy, a few tens of J. The problem is rather that when the cavity goes from its superconducting state to its normal state, 𝑄 0 decreases and inevitably crosses the critical coupling value. At this point, if nothing is done, the several kW of power from 𝑃 𝑖 will end-up in the cavity walls, and ultimately in the helium bath. But the cryomodule cryogenic system is only sized for typically 100 to 150 W of heat load: the tank pressure, initially at 30 mbar can reach 1 bar in a few hundreds of a second, thus requiring a sufficiently fast response from the RF interlock. More details on quench detection are given in part 6.2.3.

Those three faults, arching, multipacting and quenches require the RF interlock to react within a few ms. However, some faults do not need such a fast response time; their diagnostic signals are sent by PLCs to the FMC board:

• The cryogenic PLC sends the "Cryogenic OK" signal, which is simply a bit indicating whether the cryogenic system is OK or not • The RF auxiliaries PLC which monitors the cavity vacuum, and the cooling water flow and temperature of the high power RF devices

Model implementation

The low-level hardware model

Cryogenic regulations need a time constant in the order the second and can be handled by a standard industry-grade PLC. By contrast, the field regulation on the one hand, the resonance control and the interlocks on the other hand, have characteristic times in the order of respectively 1 µs and 1 ms. Since the early 2000's, the LLRF control is partially digital: the heart of the signal processing and control law is implemented on an FPGA chip, programmed in the VHDL language. Compared to a pure analogue approach, it provides the flexibility to update the LLRF algorithm (PID gains, detection threshold) and the signal filtering strategies (against parasitic modes) without the need to change the hardware. This is at the cost of the latency, but which has made tremendous progress over the last 20 years, carried by the development of the telecommunication technologies, that also drastically cut the costs [169].

In this approach, the RF signals get in or out of the FPGA chip after an I/Q vector modulation or demodulation (demodulation is done onboard, numerically), and after an analogue-to-digital conversion (ADC) or a digital-to-analogue conversion (DAC). The DAC, ADC and FPGA are implemented on a digital board, itself onboard a so-called µTCA.4 crate, as illustrated for MINERVA in Figure 4-17. Each cavity has its own µTCA.4 crate. In the simplified view of Figure 4-17, the missing detail is the modular architecture of the µTCA.4 crate standard. Indeed, each function is carried by a specific board, as detailed for MINERVA in [START_REF] Joly | MTCA.4-Based LLRF System Prototype Status for MYRRHA[END_REF]. Without too much details, the main board, the IFC1420 contains the RF field control algorithms and the signal processing implemented on an FPGA chip, and the communication protocols via EPICS (IOC). This board also computes the image of the detuning 𝜙 𝑠 -𝜙 𝑔 , and sends it to the cold tuning system (CTS) control board via a dedicated high-speed protocol; this board, with dedicated FPGA, controls the slow tuner (motor) and the fast tuner (piezo), detailed in part 4.5. Finally, as previously exposed, a third board, the FMC, is specialized in collecting interlock signals and is connected to the IFC1420 board.

On top of these three control and data acquisition boards, three auxiliary boards complement the µTCA.4 architecture:

• The timing board, that synthetizes the reference frequency from the external master oscillator input, and distributes the timing to the other boards • The cooling system card, that ensures a homogeneous and constant temperature in the rack, to prevent component drift • The power supply rack For this modular architecture to be efficient, high-speed protocols between the boards are set-up; the advantage of separating the functions, in addition to the reliability, is that each board can be individually developed and optimized for its function, either taken off the shelve or tailor-made.

To summarize, this LLRF crate based on a µTCA.4 tightly joins the cavity field control and the resonance control in the same hardware. It stands to reason that this is a clear advantage, as both regulations are highly coupled and share the same input variables.

The cryogenics control hardware (a PLC) on the other hand, has nearly no connection with the LLRF crate, except, as seen, via the interlock. This justifies the cryogenic-side model-based control algorithms, for instance in [START_REF] Vassal | SPIRAL2 Cryomodules Models: a Gateway to Process Control and Machine Learning[END_REF] or in [START_REF] Bradu | HOW DOES A CRYOGENIC SYSTEM COPE WITH E-CLOUD INDUCED HEAT LOAD ?[END_REF], that consider the RF as non-measurable perturbations, and the efforts to measure RF-side parameters from the cryogenic controls.

In terms of LLRF modelling, as discussed in 4.4.2.1, it is necessary to simulate the correct physical signals. But to get the real dynamic of the regulation (correct transfer function), again, necessary for a successful virtual commissioning, the dynamic features of the LLRF itself are considered:

• Delays due to the LLRF sampling and computing • Band-stop filter against parasitic cavity modes • Noises in the data acquisition • Signal saturations inherent to the boards • Attenuations: cables and connectors (measurement and command signals)

The last RF item to consider are the high-power RF parts, described in the next section.

The high-power hardware models

For each cavity, the low-level command voltage is transposed into a high-power signal by the amplified, then fed into the cavity by a coaxial waveguide via the circulator and the power coupler.

The role of the circulator is to prevent the reflected signal from the cavity and coupler to end-up back in the amplifier, although the latter can tolerate some reflected power. The circulator routes the reflected power towards a water-cooled load.

The amplifier planned for MINERVA is a solid-state amplifier (SSA), delivering up to 20 kW power per cavity. The same technology is installed on the SPIRAL2 type A and B cryomodules. The principle is a stack of several typically 5-kW modules, themselves made of a stack of many solid-state amplifiers. This modular architecture makes it possible to change a faulty module within a few minutes. They are thus easily scalable, an advantage as each cryomodule has its own power needs. This technology has grown in the accelerator community over the last 15 years and prices have tremendously dropped due to mass-production. It is nowadays mature for typically up to 50 kW, CW applications and are even intended, in the future, for higher-power PW applications, instead of klystrons and their complex RF distribution systems [START_REF]RF Power System of ILC by all Solid-State Amplifiers[END_REF].

To transport the high power on long distances, ESS utilizes rectangular waveguides for its 352.2 MHz double-Spoke cryomodules. For MINERVA, the high-power is transported by coaxial waveguides: a series of trade-off led to this choice, among which the compactness and the absence of doorknob transitions, against higher costs and more heat dissipation.

The amplifier, circulator and RF load temperatures and cooling water flows are thoroughly monitored and connected to the interlock. Figure 4-18 shows the RF power line path and main devices.

Figure 4-18: the prototype cryomodule RF setup at the IJClab test pit

In terms of modelling, there is no specific need for dynamic transfer functions; the important things are the following static characteristics:

• Amplifier power curve as a function of the LLRF power • Amplifier phase shift as a function of the amplifier delivered power • Amplifier saturation • Attenuations: coaxial waveguides and circulators

The Simulink RF field and cavity resonance model

The direct implementation of the cavity model on the same FPGA that carries the LLRF and cavity control is possible and widespread, as seen in 2.3. But this is essentially for real-time execution that makes the HIL approach fully possible. Actually, real-time is not fully necessary to make a proper controller virtual commissioning, provided the controller (or the virtual controller in case of SIL) has a slow-motion mode.

The choice for the MINERVA RF model is anyway not to use a pure HIL or even SIL approach, but rather to perform off-line simulations. The control algorithm and the RF process model are thus both implemented in Simulink, just like the cryogenic process model. A screenshot of the model with a highlight of the main blocks is exposed in Figure 45678910111213141516171819.

Figure 4-19: Simulink model of the RF process and its control

As said earlier, this model originates from the MAX cryomodule test preparation, and was then updated for the MINERVA 352.2 MHz Spoke cryomodule by the same author. In parallel, it was also scaled on SPIRAL2 and benchmarked against real acquisitions [START_REF] Bouly | Superconducting Cavity and RF Control Loop Model for the SPIRAL2 Linac[END_REF].

The orange arrow materializes the detuning effects, which are showed in next section.

To conclude, this model contains the physics of the cavity field, in dynamic, from the transient cavity field equation [START_REF] Edf | Les déchets radioactifs[END_REF][START_REF]Valley of stability[END_REF][START_REF]Laradioactivite[END_REF][START_REF] Joliot-Curie | [END_REF](8)[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF][START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF][START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF][START_REF] Hargraves | Liquid Fluoride Thorium Reactors[END_REF][START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF][START_REF]Centre d'informations et de ressources sur CIGEO[END_REF][START_REF]Réacteur nucléaire naturel d'Oklo[END_REF][START_REF] Poncelet | Myrrha: un (autre) regard sur l'énergie nucléaire[END_REF][START_REF]Accelerator-driven Systems (ADS) and Fast Reactors (FR) in Advanced Nuclear Fuel Cycles: A Comparative Study[END_REF][START_REF] Junghans | Transmutation of high-level radioactive waste -Perspectives[END_REF][START_REF]Bilan du débat public, Projet de centre de stockage réversible profond de déchets radioactifs en Meuse / Haute-Marne (Cigéo)[END_REF][START_REF]La réversibilité Thème 1 -La place de Cigéo dans le dispositif de gestion des déchets[END_REF][START_REF] Chevret | Mesure de la réactivité de réacteurs sous-critiques pilotés par accélérateur par l'analyse d'expériences d'interruptions de faisceau programmées[END_REF][START_REF] Abderrahim | MYRRHA -A multipurpose fast spectrum research reactor[END_REF]. The dynamics of the LLRF are also considered, as well as the physics of the high-power parts. In Figure 4-19, the orange arrow stands for the detuning and tuning effects, which models are tackled in section 4.5.

Prior to them, the technique to solve RF and cryogenic models in the same Simulink environment is shortly exposed.

Solving a combined RF and cryogenic model

The target simulation-to-run time ratio, noted 𝑅, mentioned in part 2.4, is meaningful only without RF models: simulating 1-2 second of RF process takes typically 1 minute (𝑅 is about ) on a performant simulation laptop computer; so, real-time simulation is not possible. This is mainly because the RF timestep is in the order of the µs, rather than due to the amount of modelling details. The developed cryogenic process model, on the other hand, reach 𝑅 = 30 on average; it is much more detailed but updates typically only every 1 second of simulation time. These are just orders of magnitude, as 𝑅 strongly depends on the complexity of the simulated sequences (many transients or not). The problem being, when implementing the models together without special care, the cryogenic process model variables are updated simultaneously with the RF variables, leading to unmanageable run times.

The first simple trick is to start the combined RF and cryogenic simulations with the cryogenic block variables already initialized at their steady value. In practice, it means splitting the cryogenic simulation sequences (cool-down, 4K nominal, pump-down at 2K, warm-up). Each cryogenic sequence initially begins with the steady-state from the previous sequence. This is actually also common sense, as RF is never on unless the cryogenic operation is steady.

The second trick, as the RF models have a much smaller time step, in the order of 10 -6 second, is to update the cryogenic variables at a much lower rate to avoid useless calculations. In practice, it means separating the RF solver from the cryogenic solver. Simscape actually provides a tool to associate a specific solver to each physical network (a physical network is defined by a set of Simscape blocks connected together). This socalled "local solver" can indeed be used to solve for the cryogenic variables separately from the RF ones. This local solver is however less performant, and a validation against the standard one is sometimes necessary.

When using these 2 techniques in combined RF and cryogenic simulations, the Simscape blocks of the cryogenic process model do not even influence the global run time, which is fully driven by the RF Simulink blocks.

THE CAVITY DETUNING EFFECTS AND TUNING SYSTEM MODEL

The cavity detuning models

As mentioned earlier, superconducting cavities have a narrow bandwidth of about 100 Hz, 1 or 2 orders of magnitude narrower than warm cavities. As seen also, the shape of the cavity determines its resonance frequency. Anything modifying its shape can shift the resonance frequency off the excitation frequency, requiring thus more RF power to achieve the same field and same beam loading.

There are 3 categories of cavity detuning:

• Bath pressure • Lorentz forces • Microphonics
As already mentioned, microphonics are cavity periodic frequency perturbations that come from the vibrations close to the cryogenic environment and transmitted by beam tubes, cryomodule supports and cryogenic lines. For MINERVA, with enough attention to limit the transmission of mechanical vibrations to the cryomodule, and considering the superfluid operation (no bubbles), the microphonic detuning is believed to be within 1 Hz, thus not requiring any active retuning.

It is nevertheless interesting to introduce the notion of dynamic detuning, thanks to Figure 4-20: when the excitation is periodic like a vibration, the resonance frequency oscillates around its average with a pulsation 𝑓 𝑚𝑖𝑐 . This pulsation is usually in the 10 Hz to 10 kHz range, the acoustic ranges. If no tuning system counteracts this dynamic detuning, this periodic perturbation has to be compensated by the field control, that must actuate 𝑃 𝑔 and 𝜙 𝑔 , which causes:

• More RF power consumption • Field amplitude and phase regulation errors

Figure 4-20: oscillation of the cavity resonance frequency due to vibrations

The problem is even more obvious in the case of the dynamic Lorentz forces. Static Lorentz forces are induced by the electromagnetic fields inside the cavity, that deform the walls, as illustrated in Figure 4-21 from [START_REF] Ch | RF System Modeling and Controller Design for the European XFEL[END_REF].

Figure 4-21: geometrical deformations induced by the Lorentz forces, from [174]

The radiation pressure applied on the wall is function of the field [START_REF] Mosnier | CONTROL OF SCRF CAVITIES IN HIGH POWER PROTON LINAC[END_REF]:

𝑃 = (µ 0 𝐻 2 + 𝜀 0 𝐸 2 ) 4 ( 4-42 )
In practice, the generated force is proportional to the square of the accelerating gradient, and thus the static detuning too:

∆𝑓 = -𝐾 𝐿 * 𝐸 𝑎𝑐𝑐 2 ( 4-43 )
This is the how the Lorentz force detuning (LFD) is modelled in the RF process model; this parameter 𝐾 𝐿 was measured on each MINERVA BSC03 and BSC04 cavity at respectively 10.2 and 8.0 Hz/(MV/m) 2 , as summarized in 7.5. This value is in agreement with the 3D simulations carried out during the MYRTE R&D phase, at about -8 Hz/(MV/m) 2 [START_REF] Longuevergne | MYRTE -D2.10 MYRRHA SRF spoke R&D nov[END_REF]. As the deformation depends also on the mechanical boundary conditions (how the cavities are supported), a verification of this parameter is performed in part 5.4.2.1, in the cryomodule condition. As a side note, already in static, if the cavity is sensible enough, the LFD can be a show-stopper, as it was the case during the MAX 700 MHz cryomodule tests [START_REF] Bouly | DEVELOPMENTS AND TESTS OF A 700 MHz PROTOTYPICAL CRYO-MODULE FOR THE MYRRHA ADS PROTON LINEAR ACCELERATOR[END_REF]. The trouble is, when 𝛽 is low, as for this medium-𝛽 elliptical cavity, the cavity walls tend to be vertical, thus decreasing the mechanical stiffness [START_REF] Padamsee | RF Superconductivity[END_REF].

To cope with the static LFD, a slow tuner is enough, as it is the case for SPIRAL2: some cavities are equipped with a mobile plunger [START_REF] Longuevergne | Etude et test d'un module accélérateur supraconducteur pour le projet Spiral2[END_REF]. By changing the penetration, this device changes the cavity volume, thus the resonance frequency. The resonance can thus tune the cavities at whichever RF field to optimize the power transmission. This is mandatory, as the LFD can induce constant detuning that are in the same order of magnitude as the cavity bandwidth, or wider, especially on high-gradient and high-𝑄 𝐿 machines. This kind of tuner is said to be slow, as it is set in motion along with the slow RF field increase, at least on a CW machine.

Dynamic LFD on the other hand occur only on pulsed RF accelerators. Machines like XFEL and ESS are pulsed at respectively 10 and 14 Hz, meaning the RF field makes several pulses per second, and so does the LFD. However, due to the mechanical resonant response of the cavity, the dynamic LFD does strictly not follow equation in case of sudden Lorentz forces, but instead exhibit a dynamic response.

The resulting detuning oscillation affects the RF flat top, that lasts typically a few ms or less, and must be compensated to avoid RF field regulation errors, or anyway increased RF overhead. The oscillations have time to damp between 2 pulses, resulting in a repetitive and relatively predictable detuning. This detuning can be compensated during each RF pulse thanks to a piezo actuator, and using feed-forward control, more adapted than feedback [175] [176]. This piezo tuner equips most pulsed linacs and is called the fast tuner.

Each MINERVA Spoke cavity, although in CW, is nevertheless equipped with fast piezo tuners (two, for redundancy), on top of a slow tuner. Indeed, in the course of a cavity detuning-retuning procedure, during a fault-recovery sequence, a fast tuner is necessary to:

• Retune the compensating cavity to match the new field target • Help the slow tuner to rapidly detune the faulty cavity

The MINERVA RF process models the LFD via equation and by a 1 st order filter, as described in [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF] to account for the contraction and relaxation time. But there is no proper vibrational model that can provide a real impulse response of the LFD to a sudden 𝐸 𝑎𝑐𝑐 change. Such a model could be estimated during a measurement. This was not performed and could be a further improvement to the model 47 , following for instance the work from [START_REF] Devanz | Numerical Simulations of Dynamic Lorentz Detuning of SC Cavities[END_REF] that included a second-order resonant model for each eigen mode.

On the other hand, the dynamic transfer function of the piezo action on the frequency shift has constantly been updated along the MYRRHA project phases, as explained in 4.5.3. Indeed, just like the LFD or any other forces, the piezo excites the cavity mechanical modes.

In general operation for MINERVA, still for reliability reasons, the piezo shall be used whenever possible (if the tuning range is enough) instead of the slow tuner, which has a much lower lifetime. For instance, to compensate for the static LFD, and for another sort of detuning, the slow bath pressure variations.

Indeed, the variation of the pressure around the cavity walls (niobium and tank walls) can detune the cavity because of the induced mechanical deformations. If one neglects the ambient pressure variations, that create slight coupler displacements, the major contributor is the helium bath pressure detuning (BPD).

3D mechanical simulations were performed on the BSC03 and BSC04 MINERVA cavities, which, as seen in Figure 4-22, agree with the pressure difference tests 48 conducted in the cavity test cryostat (line 1) and at room temperature (lines 2 to 4). One can notice that, during MYRTE, the predicted 𝐾 𝑃 was much higher, around 20-30 Hz/mbar, and that the two prototype cavities (BSC01 and BSC02, or Amelia and Virginia) having tank welding flaws could not provide any experimental confirmation (𝐾 𝑃 was measured around 170 Hz/mbar). Considering the real cavity geometry after manufacturing (material thinning, surface curvature) enabled to refine the predictions and match the experiments.

The planned variation of the pressure downstream of the pressure control valve (caused by the cold compressor) is assumed to be within +/-2 mbar 49 , with slow oscillations of 1 mbar in the 1-mn range and 0.1 mbar oscillations in the 10-second range 50 . The pressure control valve will have no difficulty to even out those oscillations, even with a simple control law like a PI or PID.

By contrast, in case of sudden events, caused by beam loading or tuning change (the coupler heat load change), or by a change in the accelerating gradient during a retuning procedure (the cavity heat load changes) the error on the pressure may be large, even with an advanced control law, as studied in [START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF]. Those errors in turn, cause detuning and thus the need for corrective actions from the field and resonance regulations. Considering this impact is thus primordial to evaluate the global RF and cryogenic controllability, as detailed in 0.

The RF process model considers a static relation between the detuning and the pressure variation, as described in . Just like the LFD, a dynamic transfer function of the BPD could be considered for further improvements.

In general, for the dynamic LFD and BPD, as well as for the piezo dynamic transfer function, 3D model simulations can provide the location in the frequency domain of the dangerous frequencies; but the amplitude of the transfer function cannot be reliably predicted.

So far, the detuning effects were presented and the roles of the cavity slow and fast tuners were introduced. The next two parts deal in more details with the tuners and their models.

The cavity slow tuner model

The MINERVA slow and fast tuners are illustrated in Figure 4-23. The principle is a lever arm, that pulls the cavity beam tube to deform its volume (by contrast with the SPIRAL2 plunger that changes the volume by displacement). The slow tuner is a stepper motor, connected to a worm screw via reduction gearhead; the fast tuner, as said earlier, a piezo (two in parallel).

Both systems act in parallel, but have radically different ranges:

• The slow tuner works in the [0 158] kHz range [START_REF] Gandolfo | Development and Performances of Spoke Cavity Tuner for MYRRHA Linac Project[END_REF] 49 MINERVA project assumption as of July 2023

• The fast tuner works in the [0 3.5] kHz range [START_REF] Gandolfo | Spoke Tuner for the Minerva Project[END_REF] As a side note, the slow tuner range is limited by the maximum effort the cavity can tolerate before plastic deformations start. At cold temperature, the yield point is much higher than at RT, allowing about 0.87 mm maximum flange displacement, against about 0.1 mm. The generated force at cold temperature is about 13.1 kN.

Figure 4-23: MINERVA slow and fast tuner overview, adapted from [178]

In the previous part, the main role of the slow tuner was overlooked: due to the extremely narrow cavity bandwidth ~200 Hz as regard of the operating frequency ~350 MHz, it is impossible to manufacture a cavity which resonance frequency at cryogenic operation falls exactly at the RF excitation frequency. On top, there are unknowns due to resonance frequency shifts during the preparation processes (chemical etching, high-pressure rinsing, heat treatments) as well as during the cool-down and vacuum pumping. It is possible to anticipate most of those frequency shifts, but no better that a few 10 kHz, requiring thus a slow tuner in this range. What is more, the acceptable dispersion can only be in the negative detuning range (𝑓 0 𝑖𝑛𝑖𝑡 < 𝑓), as the tuning system can only increase the resonance frequency, from the beam tube relaxed position. The role of the slow tuner is thus to make this coarse adjustment of the resonance frequency.

To illustrate, the BSC03 and BSC04 initial resonance frequencies, once under helium As already mentioned, the slow tuner has a limited lifetime: due to cold operation, the moving parts (in the gearhead in particular) are coated with dry lubrification and corrosion protection. A shock or a prolongated use can wear off this coating and lead to failure. On top of the roles of the piezo during the fault-recovery procedure, as said in the previous section, on MINERVA, the piezo has the fundamental role to save slow tuner displacements for LFD and BPD compensation and increase its lifetime 51 .

A second major role for the slow tuner, taking advantage of its large tuning range, is the detuning of the faulty cavities, during a fault-recovery sequence. As said, the piezo ensure a fast retuning of the compensating cavity to balance the higher LFD, corresponding to the new field target.

On the faulty cavity side, the target is a detuning of 14 kHz in 1 second, to limit the perturbating accelerating voltage induced by the beam loading [START_REF] Dominiczak | Modeling of Superconducting Spoke Cavity with its Control Loops Systems for the MYRRHA Linac Project[END_REF], and make the cavity "transparent". The piezo tuner can act as a fast detuner, but is limited by its range; the stepper motor is thus required to perform the fast detuning, which sets constraints on the motor and gearhead choice. ESS double spoke cavity tuners are similar to MINERVA but do not have the fast-detuning requirement; the slow tuner thus features a 200:1 reduction gearhead ratio to provide the required torque from a 32-mm calibre motor.

For MINERVA, the gearhead reduction ratio is set down to 6.25:1 to ensure a fast worm screw rotation. As a consequence, to provide the required torque out of the gearhead, the motor calibre was increased to 57 mm. The fast detuning capacity of this setup was tested and confirmed during the BSC04 cavity cryostat test in 2021, summarized in [START_REF] Gandolfo | Spoke Tuner for the Minerva Project[END_REF]. The tuning sensitivity was also measured: 128 kHz/mm (kHz resonance frequency shift per mm of flange displacement), as shown in Figure 4-25. In this figure, the motor position is translated into equivalent flange linear displacement, 51 Discussion with the IJClab RF simulation expert [START_REF] Bonne | Modélisation et contrôle des grands réfrigérateurs cryogéniques[END_REF] from the kinetic model relations, assuming the tuning system stiffness is infinite compared to the cavity. The initial transition is due to all the mechanism clearances that gradually cancel, as the tuner engages and gets away from its fully relaxed position.

The motor kinematic parameters are summarized in Annex 7.6: given the lever arm 𝐷 𝑑 = 20, the gearhead reduction ratio and the worm screw pitch (2 mm / gearhead rotation), the tuning sensitivity is 2048 Hz per motor revolution. Given that the motor has 200 steps per turn (a step is the motor control unit, and smallest increment), the frequency sensitivity (and resolution) is 10.24 Hz/step.

As usual for digital twin applications, it is crucial to account for the real control and measurement units. This last sensitivity is the one implemented in the RF process model; the motor model is verified in cryomodule operation in part 5.4.1.

The cavity fast tuner dynamic model

As said in introduction of this chapter, the mechanical tuning and detuning models were already implemented in the model inherited from the MAX 700 MHz cryomodule; just like the slow tuner model, the fast tuner model needed to be updated. During the same BSC04 cavity test that provided slow tuner sensitivities, the fast piezo tuner was tested in dynamic.

What is needed is a transfer function in the form ∆𝑓 𝑝𝑖𝑒𝑧𝑜 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 (𝑓 𝑒 ) where 𝑓 𝑒 is the piezo voltage excitation frequency, typically between 1 and 2000 Hz. The measurement of this transfer function consists in sending a piezo excitation in the form of a white noise, and to carry out an FFT (Fast Fourier Transform) of the frequency shift signal. One thus needs a mean to measure the resonance frequency oscillations of the cavity; a VNA can measure the resonance frequency of a cavity, but not its fast variations. To do so, two methods exist whether the bandwidth of the cavity is large or not with respect to the input perturbations, as explained in details in [START_REF] Longuevergne | Etude et test d'un module accélérateur supraconducteur pour le projet Spiral2[END_REF] and [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF].

In short, if the bandwidth is large, like when coupled with a power coupler (𝑄 𝑖 dominates in 𝑄 𝐿 ) or at room temperature (𝑄 0 dominates in 𝑄 𝐿 ), the piezo perturbations induce ∆𝑓 oscillations that remains within the cavity bandwidth. Referring to equation ( 4-23 ) and assuming 𝑓 0 close to 𝑓 yields:

tan 𝜓 = 2𝑄 𝐿 ∆𝑓 𝑓 ( 4-45 )
It is then enough to measure the fast variations of 𝜓. It was previously shown that the optimal detuning was achieved when the phase of 𝑃 𝑖 equals the phase of the 𝑃 𝑡 , i.e. 𝜙 𝑖 = 𝜙 𝑡 . What is more, in the particular case without beam, as seen in Figure 4-26, the detuning angle is exactly 𝜓 = 𝜙 𝑖 -𝜙 𝑡 (or 𝜙 𝑖 -𝜙 𝑠 since 𝜙 𝑡 is regulated on 𝜙 𝑠 ).

The dynamic changes of 𝜙 𝑖 -𝜙 𝑡 can be tracked with a device called a phase comparator, available in the form of digital or analogue devices.

The MAX 700 MHz cavity piezo transfer function was measured in 2010 using this principle (not with a piezo blank noise but by incremental piezo excitation frequency) and reported in [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF]. During MYRTE, the Virginia cavity (BSC02) had been measured this way in 2018 at room temperature, as reported in [START_REF] Dominiczak | Modeling of Superconducting Spoke Cavity with its Control Loops Systems for the MYRRHA Linac Project[END_REF].

Still during MYRTE, the Amelia cavity (BSC01) was measured in 2019 at cryogenic temperature, as reported in [START_REF] Gandolfo | Development and Performances of Spoke Cavity Tuner for MYRRHA Linac Project[END_REF] and also in [START_REF] Dominiczak | Modeling of Superconducting Spoke Cavity with its Control Loops Systems for the MYRRHA Linac Project[END_REF]. Finally, BSC04 was measured at cryogenic temperature in 2021, as previously mentioned and summarized in [START_REF] Gandolfo | Spoke Tuner for the Minerva Project[END_REF]; in this last paper, the reader should be aware that the captions "early" and "new" are in fact inverted.

For those tests at cryogenic temperature, due to the chosen very high 𝑄 𝑖 (critical coupling with the walls) for the cavity tests, the previous method in "open loop" is not adapted since the perturbations are way larger than the bandwidth (some tenth of Hz). Instead, a self-excited loop (SEL) is used to follow the cavity resonance frequency, in "closed loop". Such analogue devices are widely used for cavity tests, when there is no need to excite at the accelerator reference frequency; instead, the cavity is excited at its resonance frequency. The principle is that 𝑃 𝑡 is fed back to drive 𝑃 𝑖 via an amplifier, hence the term "loop". When the correct loop phase shift is found, the system locks on the cavity resonance frequency. The frequency of the loop is compared to a fixed reference signal thanks to a device called a CRM (Cavity Resonance Monitor), which output voltage is the image of the instantaneous cavity resonance frequency shifts. For implementation in the RF process model, the MINERVA BSC04 measurement is fitted into a MATLAB transfer function model, using the System Identification toolbox. Compared to the method used in [START_REF] Bouly | Etude d'un module accélérateur supraconducteur et de ses systèmes de régulation pour le projet MYRRHA[END_REF], where the author manually tunes about 40 order-2 transfer functions, the toolbox carries out the fit within seconds. The resulting model function (purple) is compared to the test (black) in Figure 4-28.

Figure 4-28: BSC04 cold temperature piezo dynamic characterization model-to-test comparison

The automated fit is excellent as regard of the amplitude, but rather poor as for the phase. Some potential solutions were investigated, like piecewise fitting and manual phase steps; the results are promising. The goal is to make the method less userdependent and fully automated. What is at stake is to have a method that could be applied to the post-processing of the series cavity testing.

In the same figure, the step response of the model is calculated. Two items are crucial to verify here to validate the automated fit: first, the model should not diverge, second, the steady-state value should match the piezo voltage sweep tests, which is the case here. The model is implemented in the RF process model, in the form of a Simulink transfer function.

To summarize, considering a relevant (at cryogenic temperature) and up-to-date dynamic piezo transfer function is mandatory to predict potential control problems in a dynamic sequence, like during a compensating cavity retuning, in a fault-recovery sequence. As seen, the piezo excite the cavity mechanical modes, causing detuning oscillations that can make the amplifier saturate; the RF process model hence provides a test platform for oscillation-damping algorithms 52 .

CAVITY « HEAT TRANSFER » MODEL

In part 3.3.3, it was showed that the dynamic (level and pressure) of a tank submitted to a heat load step is driven by the phase-change time constant, and by the helium vapor compressibility (and thus by the available vapor volume), in a tank where stratification has not the time to develop.

In this former part, the heat load was considered as a pure step; however, the heat load, whether coming from RF or electrical heater, does not occur in the bath but goes through a series of heat resistances and capacities, as illustrated in Figure 4-29.

The heat path from the RF heat load location (cavity niobium wall inner side) to the helium bath is modelled as follow:

• The RF load is evenly distributed over the whole surface (≈1.13 m 2 ) • Conduction across the cavity niobium wall (≈ 3 mm) • Heat transfer from the wall to the helium, with 2 different laws whether dealing with normal (4 K mode) or superfluid helium (2 K): o For He I, fluid convection, using standard correlations for boiling heat flux o For He II, the Kapitza resistance, using the heat transfer coefficient ℎ 𝐾 = 𝛼𝑇 𝑛 from [START_REF] Van Sciver | Helium Cryogenics[END_REF] The mass of the wall (≈ 31 kg) is considered in this heat equation. It is assumed to be a unique mass point.

Some principle simulations at 4 K and 2 K showed that, due to the large surface area relative to the thickness, and also to the still high niobium thermal conductivity (≈5 W/m.K) as well as the very low specific heat, the filtering effect is negligible with respect to the bath pressure build-up constant.

The same analysis was done in the case of a heater attached to the cavity tank wall. The heat path is similar to the RF heat load, except the material properties of the wall and the fact that the heat is not evenly distributed but limited to the heater surface. The result is the same here: the wall provides not noticeable filtering to sudden heat loads. There should thus not be any difference in pressure reaction to a heat load step, whether heating via RF or electrical heater. The experience from RF cavity operation experts 53 yet pointed that RF heat loads are more "sudden" than an equivalent heater load. This might be due to many other factors; for instance, during the RF field rise, additional transient heat loads (multipacting for instance) may happen, or in the case of the heater, the casing and support masses, plus the additional contact resistance may indeed play a significant filtering role.

To conclude, a simple cavity wall heat transfer model was developed; as said, it does not play a major role in the dynamic of the helium pressure build-up. The wall mass model included in this approach is nevertheless necessary for the prediction of the cryomodule cool-down.

THE FUNDAMENTAL POWER COUPLER MODEL

4.7.1 The MINERVA prototype power coupler

Power coupler design highlights

The fundamental power coupler (FPC) is a particularly complex item to design and consequently to model. As seen earlier, the coupler tip is designed to couple with the cavity electrical field; this kind of coupling is capacitive, as opposed to inductive for other coupler types, that couple to the magnetic field thanks to a loop antenna design, as seen in Figure 4-30.

Figure 4-30: coaxial loop (left) or antenna (right) coupling, from [165]

In the pure RF point of view, the FPC is the continuity of the RF coaxial lines that lead the 53 During modelling workshops organized at IJClab

Coaxial Coaxial

power from the RF amplifier: the RF main requirement is that its global characteristic impedance should be the same as the upstream lines. However, as the final interface between the cavity and the ambient side, the coupler answers to many other requirements, fulfilled by specific features. For the MINERVA prototype test phase, 6 couplers were manufactured; the detail of their design choses is as follows (see Figure 4-31):

• Mechanical: compensation of the insulation vacuum suction forces and of the thermal shrinkage, while allowing some degrees of freedom for the cavities, in the plane normal to the couplers. This function is fulfilled by the barometric compensator and bellows on the inner and outer conductors. The term "barometric" refers to the fact that the insulating vacuum pressure itself drives the compensator counter force. • Water cooling loop integration and connections: the coupler antenna (in copper) is cooled from the inside by an annular water loop. The water inlet and outlet connectors exit from the inner conductor thanks to an RF "T". The length of the stub is sized so that this T is transparent for the RF, and as a secondary role, it compensates for the RF mismatch caused by the bellows. • Active heat load limitation the coupler "cuff" is the part of the outer conductor in contact with both the coupler warm parts at 300 K and the cavity at 2 K. Two intercept heat exchangers, a low temperature "LT" in series in the 10 K loop, and a high temperature "HT" in series with the QM thermal shield limit the conduction heat load to the 2 K helium bath • Trade-off electrical/thermal conductivity: the cuff is in stainless steel, which has a poor thermal and electrical conductivity at cryogenic temperature. To improve the electrical conduction while limiting the thermal conduction, the cuff inner surface is coated with a few µm-thick copper layer (just thicker than the skin depth plus margin). The precise knowledge of this layer property is fundamental, as detailed in part 4.7.2.3. • Vacuum separation: the coupler shares the cavity beam vacuum. The separation with the ambient pressure is done by a 7-mm thick ceramic window, almost transparent to RF and, of course, dielectric. Its fabrication and assembly (brazing) are critical manufacturing steps. On top, 3 out of 6 MINERVA windows were coated with a few nm-thick titanium nitride (TiN) layer, which roles are preventing charge accumulation and mitigating the multipacting occurrences (lower SEY) • Temperature control: the RF window should not be exposed to mechanical stresses caused by temperature differences. A heater is thus glued in its vicinity (not represented in Figure 4-31) and controlled by a simple PI law at about 20°C.

Likewise, the coupler environment should not be exposed to condensing water: a second heater controls the coupler main flange temperature (ON/OFF control law). Both heater electrical power is thoroughly monitored for power balance purposes, as exposed in 5.2.4.

Figure 4-31: MINERVA prototype coupler environment main features

The presented design is specific to the MINERVA requirements, and is much different on other accelerators, as seen in Figure 4-32. For instance, ESS chose a spring-based rather than barometric compensator, and a serpentine continuous heat exchanger rather than 2 discrete ones; the external conductor diameter is about 100 mm versus 60 mm, to cope with higher powers. XFEL chose a variable coupler and as a consequence, features a cylindrical ceramic window (more robust to axial stress) rather than a radial one.

Figure 4-32: XFEL (left) from [START_REF] Kostin | UPDATE ON THE EUROPEAN XFEL RF POWER INPUT COUPLER[END_REF] and ESS from [START_REF] Rampnoux | Design of 352.21 MHz RF Power Input Coupler and Window for the European Spallation Source Project (ESS)[END_REF] main coupler design features

Cuff

Coupler main flange

HT intercept

The MINERVA initial design choices are inherited from the EUROTRANS project, that led to the fabrication of 2 couplers, developed by IJClab for the MAX project [START_REF] Geslin | Étude de coupleurs de puissance hyperfréquence pour accélérateurs supraconducteurs[END_REF], and from the SPIRAL2 couplers developed by LPSC [START_REF] Cabanel | SPIRAL 2 coupler preparation and RF conditioning[END_REF]. The 6 MINERVA "pre-series" couplers were manufactured by PMB and their sub-contractors, according to LPSC and IJClab codesign. As seen in details in part 6.1, in parallel to the prototype tests, the 2 coupler intercept heat exchangers were changed into a continuous heat exchanger (like ESS) for the series. The coupler model developed for the prototype was used as baseline for the series predictions 54 .

To summarize, although simple in principle, the coupler is a rather complex part, to say the least. The heat load contribution of this element has in the past been limited to engineering requirement rather than true evaluations verified by the experience 55 .

One of the of the thesis added-value is hence to model the heat path accurately enough in the coupler "heat transfer" model, developed in the next chapter, and to confirm it by measurements, exposed in part 5.2.4. Also, as mentioned, in introduction of this chapter, a "field model" was developed to calculate the RF heat loads dynamically, exposed right after the "heat transfer" model.

Prior to exploring those models, a short overview of the coupler conditioning test bench and procedure is given.

MINERVA coupler conditioning test bench

As all RF parts, the coupler surfaces exposed to the vacuum need to be conditioned. To avoid pollutions in the cavities and to save cryogenic operation time, couplers are conditioned in dedicated room-temperature test benches. The principle is to feed the couplers with a gradually increased RF power field, until the multipacting stops occurring; such a procedure typically takes 1-2 weeks.

The MINERVA coupler conditioning test bench is inherited from the XFEL installation at IJClab. The conditioning cavity is taken back from EUROTRANS. The role of this cavity is to connect two couplers top-to-tail, so that they are conditioned 2-by-2 and under roughly the same RF power: the RF travelling wave (TW) is sent by the 80 kW RF amplifier, travels across coupler at post 1, then in the cavity and next through coupler at post 2, to end-up in an RF load. The main conditioning bench features are described in Figure 4-33.

Figure 4-33: RF coupler room temperature conditioning test bench at IJClab

The required 80 kW amplifier comes from the need to condition the surfaces at the maximum power a coupler will experience in operation. Considering some couplers may see 20 kW (the maximum installed power on the accelerator) forward power, and that this power may be fully reflected by the cavity (in case there is no beam), the resulting standing wave (SW) amplitude reaches 80 kW at the location of the maximum (the voltages are doubled, so the power is quadrupled). An 80 KW TW conditioning thus covers the 20 kW SW for all the possible locations of the amplitude peak. The notions of standing and travelling wave is further detailed in part 4.7.3.

As on the cryomodule, the incident, reflected and load RF power are measured by bidirectional couplers. The coupler and RF line outer surfaces are instrumented in temperature, primarily to monitor temperature excesses due to multipacting. To evacuate the heat, the antennas are cooled by circulating water, the same way as on the cryomodule. The cuff intercept heat exchangers are also fed with running water for the same reason, instead of cryogenic fluids. The cooling loops were all instrumented (flowmeter and inlet/outlet thermocouples) in order to measure the heat loads to the water. This instrumentation provided valuable data for the model-based analysis and model validations; the last instrumentation version (after several improvements steps) is described in Figure 4-34. Each of the 6 couplers were tested at least once, as summarized in Table 4-1. The couplers nicknamed "CPL4" and "CPL5" were chosen for respectively BSC03 and BSC04 since they had the lowest heat loads. But during the cavity train assembly in clean room, the CPL5 coupler antenna collided with BSC04 port. CPL5 was thus replaced by CPL2; no damage was observed on BSC04, although dust very likely polluted the cavity, as seen in part 5.4.2.2.

The results of the conditioning of CPL2 and CPL4 are used for comparison with the model, as exposed in parts 5.4.3. Some results are also exploited, as exposed in part 5.4.4, to analyse the abnormal multipacting occurrences on CPL1. Previous studies like [START_REF] Souli | Etude et développement d'un coupleur de puissance pour les cavités supraconductrices destinées aux accélérateurs de protons de haute intensité[END_REF] highlighted the importance of modelling the thermal coupling between a power coupler and a superconducting cavity, to avoid the danger of normalstate transition in case of too high heat loads. More generally, being able to predict the loads from the ambient side to the intercepts and 2 K helium bath, is necessary for the RF dynamic process model. For the MINERVA coupler, LPSC had developed in 2018 [START_REF] Bondoux | Pré-étude thermique des coupleurs de puissance pour les cavités Spoke à 352 MHz[END_REF] a finite element model, in MATLAB code. The objects of the study were:

• The static and dynamic heat loads as a function of the cuff copper clad RRR and thickness in cryomodule conditions • The evaluation of the risk of water freezing in the antenna, in case of pump failure • Room temperature heat loads for the conditioning This MATLAB model comprises 3 main elements: the antenna, the window and the external conductor (comprising the cuff), which are supposed axisymmetric and are discretized (≈ 100 elements) along the coupler length. The parts exchange heat between them via conduction, and with the ambient via convection with air or water. Radiation from the antenna to the cavity is computed in a separate Excel sheet. The MATLAB code schematic (displaying the thermal elements as coloured rectangles) is showed in Figure 4-35. This model was shared by the LPSC teams and served as a reliable baseline. The knowledge of the "as built" geometry of the coupler is thus crucial and was the subject of thorough investigations. In particular, as exposed in the next section, complementary investigations on the precise composition of the copper coating on the coupler cuff were carried out.

Characterization of the cuff copper coating

The coupler cuff is a standalone separate coupler part; it is fabricated from a machined stainless-steel tube brazed to the two tip flanges (the main one, and the cavity flange). The final step is the copper coating: the copper layer thickness and RRR was specified by LPSC at respectively 15 µm and 30<RRR<60. Due to business secrets, staff turnover and the sub-contractor cascade, the exact electroplating process was not disclosed, and only partial information was available about the requirement respect:

• Documents from the copper coating subcontractor mentioned 15 +/-3 µm copper thickness measured on the fabricated cuff, but without procedure details • Informal discussions with PMB staff pointed to a 0.5 µm-thick additional nickel layer ("flash") to improve the copper adherence on the stainless-steel • A destructive measurement on a first sample batch (non-representative coating process) in mid-2020 by IJClab showed a copper RRR of roughly 29

Given the importance of the material characteristics and the incomplete initial information, it was decided to conduct:

• A new material thickness characterisation, with statistics as a function of the position along the cuff

• A new copper RRR measurement
To do so, a second sample batch was used, which coating procedure and recipe was exactly identical to the 6 manufactured cuffs. The sample batch consists in 5 sticks cut in a sacrificial cuff, as seen in Figure 4-39, and resized in the form of four 45-mm samples, deliberately picked-up at different locations of the cuff.

Figure 4-39: original sacrificial cuff (left) cut into 5 sticks by PMB. Sample #5 copper layer was removed with acid. Samples # 1-2-3-4 were then cut in 45-mm samples for analysis

Given the necessary detail resolution (less than 0.1 µm), a particular thorough polishing method of the cross-sections was necessary. The polishing was developed and carried out by IJClab material analysis team. The samples are put in a resin in order to hold them during the polishing process, as pictured in Figure 4-40.

Figure 4-40: sample put in resin and installed on a Lam Plan grinding machine at IJClab

The sample copper and adherence layer thicknesses were analysed using 2 different devices:

• A scanning electron microscope (SEM) with an energy-dispersive X-ray (EDX) detector, in order to get the chemical composition (atom level, not molecule) • Laser confocal microscope to perform copper layer thickness measurements on large areas and get statistics An EDX detector can discriminate the atoms of a sample by measuring the characteristic X-ray energy leap of the de-exciting electrons. A continuous composition measurement all the way from the stainless-steel to the copper layer enabled to detect gold in addition to nickel in the adherence layer, as seen in Figure 4-41 (right). Due to the thickness of the nickel and gold layers (less than 1.5 µm in total), the statistical measurement was carried out with the SEM, that could produce an image with varying contrast for each layer, due to the different chemical composition. The adherence layers proved to be homogeneous over the whole cuff platted surface.

On the contrary, the copper layer variations, across all 45 mm long samples, were measured with the confocal microscope and automated routines. An example of such measurement is shown on Figure 4-41 (left). Some periodic undulations are visible, likely due to the machining process of the stainless-steel tube, but the thickness is rather constant at around 15.5 µm, with no noticeable variation. The table of measurements is presented in Annex 7.7.

The RRR was measured on a dedicated test bench at IJClab. The cryogenic insert housed six samples, as pictured on Figure 4-42: the five 45 mm samples (including sample #5 from which copper had been removed with acid), and one niobium control sample. The method is straightforward: conduct a voltage measurement whilst imposing a current, at both room temperature and 4.2 K (using AC to cancel out the thermocouple effect). The sample from which copper was removed served as a reference in order to evaluate the copper's contribution to the RRR value.

The samples were measured twice: one with the voltage needles anchored in the copper layer, another time in the steel layer, to check any impact of the current distribution. Both runs led to the same conclusions: the copper layer had a higher quality than expected, with an RRR of 120 +/-30 (complete result table in Annex 7.7).

The values were unexpectedly higher compared to the ones obtained in 2020 on the first batch (all samples were measured in the same test bench). But these values are consistent with the additional heat loads observed during the cryomodule tests, as explained in part 5.2.4.

To conclude, a thorough examination of the material properties of a procured cuff was conducted to feed the heat transfer model of the RF power coupler. This investigation enabled to identify the chemical composition and size of the layers the measurement of the copper RRR. To further update the model for the series components, gathering this set of information would be valuable. The coupler exhibits two types of heat loads due to the RF field: the first one is the Joule RF load deposit on the antenna and on the external conductor (including the cuff), according to the following formula [START_REF] Souli | Etude et développement d'un coupleur de puissance pour les cavités supraconductrices destinées aux accélérateurs de protons de haute intensité[END_REF]:

⟨𝑃 𝐽𝑜𝑢𝑙𝑒 ⟩ = ⟨𝑅 𝑆 ∬ 𝐻 𝜑 (𝑟, 𝑧, 𝑡) 2 𝑆 𝑑𝑆⟩ = 1 2 𝑅 𝑆 ∬ 2 * ⟨𝐻 𝜑 (𝑟, 𝑧, 𝑡) 2 ⟩ 𝑆 𝑑𝑆 ( 4-46 )
In this expression, ⟨ ⟩ is the time-averaged operator; the term 𝑅 𝑠 is the electrical resistance of the layer where the current flows, dependent on the skin depth (see 4.1.3) and resistivity at the local material temperature.

The second heat load type is the RF dielectric losses in the ceramic window, computed according to the formula [START_REF] Souli | Etude et développement d'un coupleur de puissance pour les cavités supraconductrices destinées aux accélérateurs de protons de haute intensité[END_REF]:

⟨𝑃 𝐷𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 ⟩ = ⟨2𝜋𝑓𝜀 0 𝜀 𝑟 𝑡𝑎𝑛𝛿 ∭ 𝐸 𝑟 (𝑟, 𝑧, 𝑡) 2 𝑉 ⟩ = 1 2 (2𝜋𝑓𝜀 0 𝜀 𝑟 𝑡𝑎𝑛𝛿) ∭ 2 * ⟨𝐸 𝑟 (𝑟, 𝑧, 𝑡) 2 ⟩ 𝑉 ( 4-47 )
Here, f is the frequency, 𝜀 0 is the vacuum permittivity, 𝜀 𝑟 the material relative permittivity, and 𝛿 the loss angle.

While 𝑅 𝑠 is easily accessible by combining the coupler heat transfer model with a property table look-up, the space integrals of the double time-averaged square of E an H are cavity RF-field dependent, and furthermore, depend on the location along the coupler.

To compute those integrals, the formerly described LPSC model used an interpolation of those integrals from a 3D HFSS simulation. Yet, the simulated HFSS field configuration stick to the pure TW mode, i.e. the perfectly matched case; hence, the interpolation was a pure function of 𝑃 𝑖 (linearly scaled from a reference 80 KW case), and of the location along the axis.

In order to compute those integrals in all beam and tuning configurations while updating them dynamically in the course of the simulation, as required for the RF dynamic process model, a simple analytical coupler field model is necessary.

E and H analytical field model development and validation

The coupler is considered as a lossless coaxial line, as seen in Figure 4-43. This is reasonable, as the losses are in the order of some tens of W, compared to the transmitted power of a few kW.

Doing so, E has a pure radial component and H has a pure azimuthal component. Their value in any point z of the coupler is given by the following expressions, in pure travelling wave mode:

𝐸 𝑟 (𝑟, 𝑧, 𝑡) = 1 𝑟 √ 𝑃 𝑖 ̅ * ξ 0 𝜋 ln ( 𝑏 𝑎 ) cos (𝜔𝑡 -𝛽𝑧) ( 4-48 ) 𝐻 𝜑 (𝑟, 𝑧, 𝑡) = 1 𝑟 √ 𝑃 𝑖 ̅ ξ 0 𝜋 ln ( 𝑏 𝑎 ) cos (𝜔𝑡 -𝛽𝑧) ( 4-49 )
In these expressions, 𝑃 𝑖 ̅ is the forward average power of a pure travelling wave, ξ 0 is the vacuum impedance and a and b respectively the inner and outer conductor radii.

The case of an incident and reflected wave superposition is now considered; for the electrical field E, the expressions of those fields are:

𝐸 𝑖𝑛𝑐 (𝑟, 𝑧, 𝑡) = 1 𝑟 √ 𝑃 𝑖𝑛𝑐 ̅̅̅̅̅ * ξ 0 𝜋 ln ( 𝑏 𝑎 ) cos(𝜔𝑡 -𝛽𝑧 + 𝜙 1 ) = 𝐸 𝑖𝑛𝑐 cos(𝜔𝑡 -𝛽𝑧 + 𝜙 1 ) ( 4-50 )
𝐸 𝑟𝑒𝑓 (𝑟, 𝑧, 𝑡) = 1 𝑟 √ 𝑃 𝑟𝑒𝑓 ̅̅̅̅̅̅ * ξ 0 𝜋 ln ( 𝑏 𝑎 ) cos(𝜔𝑡 + 𝛽𝑧 + 𝜙 2 ) = 𝐸 𝑟𝑒𝑓 cos(𝜔𝑡 + 𝛽𝑧 + 𝜙 2 ) ( 4-51 )
In these expressions, 𝜙 1 and 𝜙 2 are respectively the phases of the incident and reflected travelling waves. The superposition of those fields following 𝐸 𝑟 (𝑟, 𝑧, 𝑡) = 𝐸 𝑖𝑛𝑐 (𝑟, 𝑧, 𝑡) + 𝐸 𝑟𝑒𝑓 (𝑟, 𝑧, 𝑡) results in a standing wave in the coupler, which amplitude yields, with a few

𝐸 𝑟 (𝑟, 𝑧 ) 𝑚𝑖𝑛-𝑚𝑎𝑥 = ∓ √ (𝐸 𝑖𝑛𝑐 + 𝐸 𝑟𝑒𝑓 ) 2 -4𝐸 𝑟𝑒𝑓 𝐸 𝑖𝑛𝑐 sin (-𝛽𝑧 + 𝜙 1 -𝜙 2 2 ) 2 ( 4-52 )
Likewise, the amplitude of the standing wave H field can be expressed considering first the expression of the incident and reflected travelling waves:

𝐻 𝑖𝑛𝑐 (𝑟, 𝑧, 𝑡) = 1 𝑟 √ 𝑃 𝑖𝑛𝑐 ̅̅̅̅̅ 𝜋ξ 0 𝑙𝑛 ( 𝑏 𝑎 ) 𝑐𝑜𝑠(𝜔𝑡 -𝛽𝑧 + 𝜙 1 ) = 𝐻 𝑖𝑛𝑐 𝑐𝑜𝑠(𝜔𝑡 -𝛽𝑧 + 𝜙 1 ) ( 4-53 ) 𝐻 𝑟𝑒𝑓 (𝑟, 𝑧, 𝑡) = 1 𝑟 √ 𝑃 𝑟𝑒𝑓 ̅̅̅̅̅̅ 𝜋ξ 0 𝑙𝑛 ( 𝑏 𝑎 ) 𝑐𝑜𝑠(𝜔𝑡 + 𝛽𝑧 + 𝜙 2 ) = 𝐻 𝑟𝑒𝑓 𝑐𝑜𝑠(𝜔𝑡 + 𝛽𝑧 + 𝜙 2 ) ( 4-54 )
Then, considering the superimposed magnetic field 𝐻 𝜑 (𝑟, 𝑧, 𝑡) = 𝐻 𝑖𝑛𝑐 (𝑟, 𝑧, 𝑡) -𝐻 𝑟𝑒𝑓 (𝑟, 𝑧, 𝑡), where the "-" sign is mandatory this time to respect the electromagnetic wave orientation convention (Maxwell's Cork Screw), yields:

𝐻 𝜑 (𝑟, 𝑧 ) 𝑚𝑖𝑛-𝑚𝑎𝑥 = ∓ √ (𝐻 𝑖𝑛𝑐 -𝐻 𝑟𝑒𝑓 ) 2 + 4𝐻 𝑟𝑒𝑓 𝐻 𝑖𝑛𝑐 𝑠𝑖𝑛 (-𝛽𝑧 + 𝜙 1 -𝜙 2 2 ) 2 ( 4-55 )
To summarize, the E an H fields now have an analytical expression for their respective time envelope. Those formula were implemented first in MATLAB and verified against CST 3D simulations from IJClab RF team, in the pure travelling wave mode (Figure 4-44) and in the pure standing wave mode (Figure 4-45) for the E field only (same conclusions with H). For the comparison, a MATLAB 3D plot with the same colour scales as the CST calculation is used, the abscise 0 is where the antenna tip stands. The comparison showed a maximum of 7% difference on the electrical field amplitude between the models, which is enough for validation. A remark here is that the standing wave case in Figure 4-45 is obtained because a specific boundary condition at the coupling plane (roughly the antenna tip) is applied, resulting in an open-circuit behaviour. This boundary condition comes from the cavity field model, as explained in the next section.

Connection with the cavity field model

As seen, the analytical model of the coupler field needs the values of 𝑃 𝑖 , 𝑃 𝑟 , 𝜙 1 and 𝜙 2 .

The powers 𝑃 𝑖 and 𝑃 𝑟 are obtained directly from the cavity field model but it is more complex for 𝜙 1 and 𝜙 2 . The first intuition would be simply considering that 𝜙 1 = 𝜙 𝑖 and 𝜙 2 = 𝜙 𝑟 (the respective phases of the complex cavity voltages 𝑉 ̃𝑖𝑛𝑐 and 𝑉 ̃𝑟𝑒𝑓 ). It would mean that the equivalent impedance of the coupling plane seen from the transmission line is the same as the accelerating gap, the one described in part 4.3.4.

This intuition is wrong, as confirmed by the cryomodule tests (part 5.4.3) and by past experiences at CERN 56 . It is also confirmed by CST simulations from LPSC, in Figure 4-46, that shows the H field in the cavity and coupler: the coupling plane impedance is in fact the opposite of the accelerating gap impedance. It means off-resonance, the coupling plane is like an open-circuit (OC) and at resonance & overcoupled, the coupling plane is like a short-circuit (SC). In practice, the coupler field phases are related to the cavity phases by:

𝜙 1 = 𝜙 𝑖 - 𝜋 2 𝑎𝑛𝑑 𝜙 2 = 𝜙 𝑟 + 𝜋 2 ( 4-56 )
This phase shift is not clearly understood 57 but could intuitively be explained by the capacitive-type coupling, that like a capacitance, induces a 𝜋 2 phase shift. The Simscape implementation is detailed in a coming section, right after the very next section, the analytical expressions of the heat loads.

Heat load analytical expression and illustrations

So far, the E an H field amplitudes were expressed and compared to 3D simulation for validation. Yet, to compute 𝑃 𝐽𝑜𝑢𝑙𝑒 and 𝑃 𝐷𝑖𝑒𝑙𝑒𝑐𝑡𝑟𝑖𝑐 , the space integrals of the double timeaveraged square of E an H are needed, as seen in ( 4-46 ) and ( 4-47 ).

Starting with the E field, by expanding the total incident and reflected field 𝐸 𝑟 (𝑟, 𝑧, 𝑡) 2 from the compact expressions in ( 4-50 ) and ( 4-51 ), then using the trigonometric relations to separate the time-dependant from the z-axis-dependant terms, one gets:

𝐸 𝑟 (𝑟, 𝑧, 𝑡) 2 = 𝐸 𝑖𝑛𝑐 2 𝑐𝑜𝑠 2 (𝜔𝑡 -𝛽𝑧 + 𝜙 1 ) + 𝐸 𝑟𝑒𝑓 2 𝑐𝑜𝑠 2 (𝜔𝑡 + 𝛽𝑧 + 𝜙 2 ) + 2 * 𝐸 𝑖𝑛𝑐 𝐸 𝑟𝑒𝑓 𝑐𝑜𝑠 2 (𝜔𝑡 + 𝜙 1 + 𝜙 2 2 ) -2 * 𝐸 𝑖𝑛𝑐 𝐸 𝑟𝑒𝑓 𝑠𝑖𝑛 2 (-𝛽𝑧 + 𝜙 1 -𝜙 2 2 ) ( 4-57 )
Then, considering the time-average, the term under the integral becomes:

2 * ⟨𝐸 𝑟 (𝑟, 𝑧) 2 ⟩ = 𝐸 𝑖𝑛𝑐 2 + 𝐸 𝑟𝑒𝑓 2 + 2 * 𝐸 𝑖𝑛𝑐 𝐸 𝑟𝑒𝑓 -4 * 𝐸 𝑖𝑛𝑐 𝐸 𝑟𝑒𝑓 sin 2 (-𝛽𝑧 + 𝜙 1 -𝜙 2 2 ) ( 4-58 )
Finally, by injecting the developed expressions of 𝐸 𝑖𝑛𝑐 and 𝐸 𝑟𝑒𝑓 , using again trigonometric relations, and by doing the space integration, one gets, in [V 2 .m]:

∭ 2 * ⟨𝐸 𝑟 (𝑟, 𝑧, 𝑡) 2 ⟩ 𝑉 = 2 * ξ 0 (𝑃 𝑖𝑛𝑐 ̅̅̅̅̅ + 𝑃 𝑟𝑒𝑓 ̅̅̅̅̅ )∆𝑧 + 4 * ξ 0 √𝑃 𝑟𝑒𝑓 ̅̅̅̅̅ * 𝑃 𝑖𝑛𝑐 ̅̅̅̅̅ ( 1 𝛽 cos(𝛽(𝑧 2 + 𝑧 1 ) + 𝜙 2 -𝜙 1 )sin(𝛽∆𝑧)) ( 4-59 )
Likewise, from ( 4-53 ) and ( 4-54 ), one gets for the total incident and reflected squared H field:

𝐻 𝜑 (𝑟, 𝑧, 𝑡) 2 = 𝐻 𝑖𝑛𝑐 2 𝑐𝑜𝑠 2 (𝜔𝑡 -𝛽𝑧 + 𝜙 1 ) + 𝐻 𝑟𝑒𝑓 2 𝑐𝑜𝑠 2 (𝜔𝑡 + 𝛽𝑧 + 𝜙 2 ) -2 * 𝐻 𝑖𝑛𝑐 𝐻 𝑟𝑒𝑓 𝑐𝑜𝑠 2 (𝜔𝑡 + 𝜙 1 + 𝜙 2 2 ) + 2 * 𝐻 𝑖𝑛𝑐 𝐻 𝑟𝑒𝑓 𝑠𝑖𝑛 2 (-𝛽𝑧 + 𝜙 1 -𝜙 2 2 ) ( 4-60 )
Then, considering the time-average gives:

2 * ⟨𝐻 𝜑 (𝑟, 𝑧) 2 ⟩ = 𝐻 𝑖𝑛𝑐 2 + 𝐻 𝑟𝑒𝑓 2 -2 * 𝐻 𝑖𝑛𝑐 𝐻 𝑟𝑒𝑓 + 4 * 𝐻 𝑖𝑛𝑐 𝐻 𝑟𝑒𝑓 𝑠𝑖𝑛 2 (-𝛽𝑧 + 𝜙 1 -𝜙 2 2
) (

And finally, the expression of the space integral, in [A 2 ]:

∬ 2 *
Beyond the cavity boundary conditions, the squared E an H space integrals depend on the coordinates along the coupler, 𝑧 1 and 𝑧 2 (the positions of the transverse planes where the integral begins and finishes). They are computed locally by a dedicated Cryoscape block, as seen in the next section.

To conclude and illustrate this part, the influence of the cavity tuning and beam current on the external conductor heat loads is shown in Figure 4-47. In this figure, the amplitude of the H field as well as the resulting Joule heat loads are plotted for 3 different cases. To simplify, the copper electrical conductivity is constant in this example, plotted with MATLAB (in the Simscape model by contrast, the electrical conductivity is a function of the local temperature).

• Case #1 from part 4.4.1, with beam and cavity optimally tuned, is taken as a reference, and renamed Case A; the amplitude is a mix of a travelling wave and a standing wave, a consequence of the non-optimal 𝑄 𝑖 • For case B, the same forward power 𝑃 𝑔 = 7.75 𝑘𝑊 is considered, still optimally tuned but without beam • Case C is additionally detuned by -1kHz.

Figure 4-47: coupler H field amplitude (left) and lineic Joule heat loads (right)

As demonstrated, the Joule heat load distribution varies significantly. This also applies to the heat loads on the dielectric window, as further illustrated in section 4.8.

Implementation in Simscape

On Figure 4-37 depicting the heat transfer model, the coupler is discretized along its axial direction. At each thermal node, a Cryoscape block computes the heat loads based on the local integrals of the squares of E and H, which is in line with the size and position of the adjacent thermal element. Figure 4-48 shows the Cryoscape block and its parameter menu.

This block is the same for modelling the Joule and dielectric loads and the type of loads is chosen from a menu is still also possible for the user to choose between the analytical model and an interpolation of some 3D model simulations to perform comparisons. The four boundary conditions at the cavity interface are obtained from the cavity field model. To that purpose, a new Simscape "domain" was instantiated: it is a set of variables that is transported through the connecting wires between the ports of the blocks. The dark blue colour is this RF physical domain colour.

ILLUSTRATIONS OF CAVITY AND COUPLER DYNAMIC HEAT LOAD

The major advantage of the RF process model, is its capability to dynamically predict the heat load variations, as a function of the RF field, beam current or tuning. As an illustration of this capability, and as a summary of the previous parts, a dynamic use case is presented here, where all the previously described sub-models are working together, in the cryogenic environment.

The scenario consists in 5 sequences, as shown in the three charts of Figure 4-49.

1. The RF field is at 0 MV/m, no power is fed through the RF coupler 2. The field is raised to 7 MV/m. The motor is controlled to compensate exactly for the static LFD (just for simulation purpose) and applies a static detuning corresponding to the optimal detuning at the target 𝜙 𝑠 , 𝐸 𝑎𝑐𝑐 and beam current, hence about +28 Hz. The piezo control law cancels this detuning since, without beam, the optimal detuning is 0 Hz; it does it by minimizing the difference 𝜙 𝑔 -𝜙 𝑠 . The reflected power equals the incident one, since there is no beam to match the inlet power line. 3. The piezo control is forced to apply an additional -120 Hz detuning: the power to maintain a constant field increases since the detuning is not optimal 4. The piezo control is back to the optimum detuning given by 𝜙 𝑔 -𝜙 𝑠 = 0 5. The beam is set on, the piezo no longer needs to counteract the motor predetuning. The reflected power is not 0 since 𝑄 𝑖 is not optimal For each of these RF sequences, in particular for sequences 3-4-5, it is interesting to look at the field configuration in the coupler. Figure 4-50 shows that change in tuning and beam loading modifies the locations of the maxima and minima of the E and H fields. shows the variation of the RF and non-RF heat load sources in the coupler and in the cavity; the coupler heater controls (especially the window) react to the RF heat loads by decreasing the electrical power. In the same chart, to the righthanded side, the variation of the heat distribution to the heat sinks is shown. It appears that, the cavity RF loads 58 and the antenna loads, which have the highest amplitude almost immediately reach their respective heat sinks (helium bath and water). 58 A 𝑄 0 of 2.10 9 is considered, although, as previously seen, a much better 𝑄 0 (even as high field) was achieved on both BSC03 and BSC04 in vertical cryostat; the intention is to take the worst case.

Conversely, the coupler RF loads have less magnitude but take several minutes to reach the heat sinks (intercepts and helium bath) because of the heat diffusion time constant.

This simple sequence highlights the importance of the location of the RF window for evaluating the heat load. Specifically, from sequence 3 to 4, the window dielectric heat loads vanish because the minimum of the RF standing wave matches the window's location. To put it briefly, when there is no beam and near the resonance, the dielectric heat load is at its minimum. This will be exposed in further details in section 5.4.3 for validating the coupler field on the cryomodule.

Figure 4-51: RF heat load sources (left) and distribution on the heat sinks (right)

This sequence also illustrates how unsteady the RF heat loads are, even when the RF field is constant, and highlights the challenges of the MINERVA cryogenic and ambient (heaters and water loop) controls to cope with heat load variations of different time scales and magnitude. Those challenges are all the more difficult on accelerators with higher coupler RF power.

In conclusion, a complete dynamic process model was developed upon previous studies for the cavity and the coupler. Although the model's accuracy is within +/-10% it can produce forecast for all coupling, beam, field and tuning configurations requisite for control and model-based analysis purposes.

MODEL-TO-TEST COMPARISONS DURING THE CRYOMODULE TEST CAMPAIGN

The component model initial comparisons that were performed in the previous 2 chapters had a perimeter limited to the components themselves, isolated on a dedicated test bench or a 3D calculation. However, there are many reasons to suspect that the cryomodule will not behave like the simple sum of those component models. A first reason is the uncontrolled difference in boundary conditions, worsen by the very compact arrangement and potential component interactions in the cryomodule vessel; a second one is the potential damages or assembly flaws during the fastidious cryomodule integration procedure, and a last reason is simply the missed physics. For all these reasons, a comparison to representative cryomodule tests is mandatory; a quantification of the model/test discrepancy is going to be performed.

In parallel, to take the most of this precious and rare cryomodule test phase, the developed system model is used to perform preparation tasks ahead of the tests, as well as post-treatment analysis. A broader intention is to demonstrate, in view of the future SAT and accelerator commissioning, the interest of similar accelerator projects to create and sustain such a large model.

The plan follows the same pattern as previously: first, an evaluation of the cryogenic process model is proposed, complemented then by a specific section on the cryogenic controls; the last section addresses the RF process models. Prior to that, an overview of the cryomodule test campaign is first given.

TEST CAMPAIGN OVERVIEW

The MINERVA prototype cryomodule test period spread over nearly 3 years from the reception of the equipment. The prototype QM parts and the test QVB were received in February-March 2021. The QM was assembled at IJClab with mock-up cavities, in view of test phase 1, in late 2021: the role of this phase was to start commissioning the cryogenics, as the real cavity string was not ready yet, and to validate the integration, as the mock-up cavities featured representative packaging. The differences versus the real cavity train configuration, available from test phase 2 on, is illustrated in Figure 5-1.

Figure 5-1: configuration main differences between test phase 1 and tests phases 2

During this first assembly phase, the testQVB which had been assembled by the manufacturer, was partially disassembled for tube welding quality checks and inspection of the valve seat and valve tightness. Overall, this phase where both the QM and the QVB were disassembled, gave a precious opportunity to inspect the internal instrumentation (presence, location, test and debugging).

After test phase 1, the QM was disassembled and reassembled once with the real cavity string, to perform a series of piping adjustments, fitting checks and assembly procedure validations. The QM was then disassembled again and reassembled back, this time with the full cavity string instrumentation in view of test phase 2. Again, this second assembly step was crucial to take note about the real instrumentation position, check the cable thermalization and watch for thermal contacts.

The summary of the test phases and their highlights is given in Table 5-1; overall, most of the cryogenic model evaluations was done against test phase 2.1, and cross-checked with test phase 1. On the RF side, the main reference period is test phase 2.2.

Test phase 1 300K → 2K transitions No Yes

Cavity mock-up tanks Ø25 SS tube → 0.5 L Cavities = 2 x 35 L Couplers Mock-up (bracket and tubes)

Real couplers

In both cases, an 11 L guard tank

Test phase 2

uncertainty negligible. This method is used to break-down the conductive heat loads between the different source and sinks.

A second method, the "fluid sensible heat" methods gives the heat flow absorbed by a fluid running in a thermalization. It follows the equation:

𝑄 ̇= 𝑚̇ * [ℎ 2 (𝑃 2 , 𝑇 2 ) -ℎ 1 (𝑃 1 , 𝑇 1 )] ( 5-1 )
The mass flow rate 𝑚̇ is measured by a flowmeter located downstream of the cryogenic lines, on the warm side, after the reheater. The enthalpy before (ℎ 1 ) and after (ℎ 2 ) the thermalization is known from the temperature measurement and interpolation in the fluid table. The limitation though, is when the fluid is in diphasic state: the temperature is thus not enough to derive the enthalpy. To overcome it, a trick was used, in particular in the 10 K loop, as seen later: a heater (EH080 in Annex 7.2) is used to evaporate the helium at the cryomodule inlet, so that the helium is fully in vapor state at the intercepts. This method allows to cross-check measurements with the previously seen conduction method.

Lastly, the "tank fluid latent heat" method gives an estimate of the heat loads to a tank, by considering the total evaporation power:

𝑄 ̇= (𝑚̇𝑉 ,𝑒𝑣𝑎𝑝 𝑙𝑖𝑛𝑒 + 𝑚̇𝑉 ,e𝑣𝑎𝑝 𝑡𝑎𝑛𝑘 ) * (h 𝑉,𝑠𝑎𝑡 -h 𝐿 , sat)

( 5-2 )

Here, 𝑚̇𝑉 ,𝑒𝑣𝑎𝑝 𝑙𝑖𝑛𝑒 is the vapor mass flowrate coming from the filling line (jumper in the case of the 2 K tank), and 𝑚̇𝑉 ,e𝑣𝑎𝑝 𝑡𝑎𝑛𝑘 is the vapor produced by the tabk heat loads only; h stands for the mass enthalpy in saturated vapor or liquid, measurable by the pressure.

The key is then to link those evaporation flow rates to 𝑚̇𝑒 𝑥ℎ𝑎𝑢𝑠𝑡 , the sole measured flowrate, considering also the vapor production from the J-T valve flash 𝑚̇𝑉 , 𝑓𝑙𝑎𝑠h , as illustrated in Figure 5-2; a mass balance on the vapor in the tank yields:

𝑑𝑚 𝑉 𝑑𝑡 = 𝑚̇𝑉 ,𝑒𝑣𝑎𝑝 𝑙𝑖𝑛𝑒 + 𝑚̇𝑉 ,e𝑣𝑎𝑝 tank + 𝑚̇𝑉 , 𝑓𝑙𝑎𝑠h -𝑚̇𝑉 𝑎𝑝 ( 5-3 )

Then, if the pressure is considered constant in the tank:

-𝜌 𝑉𝑎𝑝 * 𝑉 𝑇𝑜𝑡 * 𝑥̇𝐿 𝑖𝑞 𝑣𝑜𝑙 = 𝑚̇𝑉 , 𝑒𝑣𝑎𝑝 𝑙𝑖𝑛𝑒 + 𝑚̇𝑉 ,e𝑣𝑎𝑝 𝑡𝑎𝑛𝑘 + 𝑚̇𝑉 , 𝑓𝑙𝑎𝑠h -𝑚̇𝑉 𝑎𝑝 ( 5-4 ) Here, 𝑥̇𝐿 𝑖𝑞 𝑣𝑜𝑙 is the liquid volumic fraction derivative in the tank (commensurable to the level derivative) and 𝑉 𝑇𝑜𝑡 is the tank volume. Then, 2 cases can simplify this equation. First, if the liquid level is kept constant by means of a control loop, it becomes:

𝑚̇𝑉 ,𝑒𝑣𝑎𝑝 𝑙𝑖𝑛𝑒 + 𝑚̇𝑉 ,e𝑣𝑎𝑝 tank = 𝑚̇𝑉 𝑎𝑝 -𝑚̇𝑉 ,𝑓𝑙𝑎𝑠h ( 5-5 )
The flash vapor mass flow rate 𝑚̇𝑉 , 𝑓𝑙𝑎𝑠ℎ is estimated thanks to the pressures measured upstream and downstream of the J-T valve, knowing the liquid mass flowrate equals the vapor one. In these particular conditions, one thus gets the jumper heat loads blended with the tank heat loads.

A second condition simplifies ( 5-4 ): when the inlet valve is fully closed. First, a mass balance on the liquid side gives, in the general case:

𝑑𝑚 𝐿 𝑑𝑡 = 𝑚̇𝐿 𝑖𝑞 -𝑚̇𝑉 , 𝑒𝑣𝑎𝑝 𝑡𝑎𝑛𝑘  𝜌̇𝐿 𝑖𝑞 * 𝑉 𝐿𝑖𝑞 + 𝜌 𝐿𝑖𝑞 * 𝑉 𝑇𝑜𝑡 * 𝑥̇𝐿 𝑖𝑞 𝑣𝑜𝑙 = 𝑚̇𝐿 𝑖𝑞 -𝑚̇𝑉 , 𝑒𝑣𝑎𝑝 𝑡𝑎𝑛𝑘 ( 5-6 )
If the inlet valve is closed, 𝑚̇𝐿 iq = 0, and if the pressure is considered constant in the tank, 𝜌̇𝐿 𝑖𝑞 = 0; hence yields:

𝜌 𝐿𝑖𝑞 * 𝑉 𝑇𝑜𝑡 * 𝑥̇𝐿 𝑖𝑞 𝑣𝑜𝑙 = -𝑚̇𝑉 , 𝑒𝑣𝑎𝑝 𝑡𝑎𝑛𝑘 ( 5-7 )
By combining ( 5-4 ) and ( 5-7 ), and considering 𝑚̇𝑉 , 𝑒𝑣𝑎𝑝 𝑙𝑖𝑛𝑒 = 𝑚̇𝑉 , 𝑓𝑙𝑎𝑠h = 0, one gets:

𝑚̇𝑉 , 𝑒𝑣𝑎𝑝 𝑡𝑎𝑛𝑘 = 𝑚̇𝑉 𝑎𝑝 * 𝜌 𝐿𝑖𝑞 𝜌 𝐿𝑖𝑞 -𝜌 𝑉𝑎𝑝 ( 5-8 )
In these second particular conditions, one gets the tank heat loads alone. By subtracting it to the first measurement, one isolates the jumper heat loads.

Those 2 particular conditions where the liquid mass fraction derivative is not needed are widely used and particularly convenient when the tank level reading is too noisy. It will be seen, later on, that a correction based on the derivative of the level is required.

Electrical heater power

Heaters are necessary to both calibrate the previously seen heat flow measurements, and get rid of diphasic state. In both cases, the precise estimate of the injected electrical power is required, in the order of +/-1 %. For instance, during test phase 1, as the real couplers were not installed, heaters were placed at their expected location along the 10 K loop, to compare to the "fluid sensible heat" method. Heaters are also used associated with the cavity tanks, to simulate dynamic heat loads.

Figure 5-3: cavity heater electrical supply scheme

The main problem, as seen in Figure 5-3 is the different non-negligible resistances (lines and connections) lying between the power supply and the heaters, that introduce unknows in the actually supplied heater power, especially since the internal line resistance varies with temperature. A proper 4-wire measurement would have solved the problem, but at the cost of extra PLC and QM feedthrough slots. The problem was overcome by considering that the heater resistance was the same at 300K and 2 K, and by measuring the total resistance (line + connections + heaters) at 300 K and 2 K.

The transfer line problem

One of the roles of the protoQVB helium tank is to get rid of the incoming vapor from the Dewar and external transfer line, so that the QM is supplied with pure liquid, which greatly facilitates the heat load calculation; the heat loads from the external helium transfer line was measured at about 7 W, which, for a 3-meter long flexible insulated line, is more than twice the commonly accepted value of 1 W/m. This illustrates the poor knowledge about those heat loads, which is not really problematic for the helium loops thanks to the phase separation. Conversely, there is no such phase separator for the nitrogen loop; there is thus a large uncertainty on the nitrogen state entering the 80 K loop, that furthermore, varies from one LN2 Dewar to the other.

The experimental reference working points

A few relevant experimental working points are going to be used in the following sections, for the comparisons with the model. The detail of these working points is deliberately presented later in section 5.2.8: those points are referred as "WPx", where x stands for the working point number. Two sets of working points are going to be used: the one from test phase 2.1, presented in Figure 5-29 from section 5.2.8 and the one from test phase 2.2 presented in Annex 7.9.

Measurement uncertainty evaluation

A good agreement between a test and a simulation is meaningful only if the measurement uncertainty is much smaller than the target model accuracy; it is thus important to evaluate the experimental uncertainties. First, the estimate of the uncertainty of the heat flow calculated by the "material conduction" method is performed, by using the GUM method [183]. According to equation [START_REF] Chen | A Study on Properties of Novel Metallic Foam for Nuclear Applications[END_REF][START_REF] Edf | Les déchets radioactifs[END_REF][START_REF]Valley of stability[END_REF][START_REF]Laradioactivite[END_REF][START_REF] Joliot-Curie | [END_REF](8)[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF][START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF][START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF][START_REF] Hargraves | Liquid Fluoride Thorium Reactors[END_REF][START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF][START_REF]Centre d'informations et de ressources sur CIGEO[END_REF][START_REF]Réacteur nucléaire naturel d'Oklo[END_REF], this measurement is the product of three terms; the first two are:

• The cross-section S, whose relative uncertainty 𝑢 𝑆 equals ~15 % • The length L, whose relative uncertainty 𝑢 𝐿 equals ~10 %

The uncertainties associated to these terms are of type B, i.e. not inferred from a statistical analysis, but based on the characteristics of the measurement instrument (a tape measure) and a qualitative evaluation of the measurement procedure (by hand). Then comes a third term, the conductivity integral, that brings two additional uncertainties:

• The material properties, whose relative uncertainty 𝑢 𝑀𝑃 is supposed at ~5 % • The process instability, whose relative uncertainty 𝑢 𝑖𝑛𝑠𝑡𝑎𝑏 is supposed at ~5 % The latter uncertainty, 𝑢 𝑖𝑛𝑠𝑡𝑎𝑏 , would converge toward 0 % if the measurements were averaged over several hours; but for practical reasons, the averaging only spans over typically 10-15 min. As for the temperature sensors, also involved in this heat flow measurement method, the vast majority are of CERNOX type; these sensors are all individually calibrated: there is thus no uncertainty caused by any unknown error. The temperature sensor precision brings no uncertainty either, since the resolution and the fidelity both are oversized in this application.

The four identified uncertainties are statistically independent; they are furthermore all assumed to follow the same type of probability distribution (a square one). It is thus possible to calculate the propagated value 𝑢 𝑡𝑜𝑡𝑎𝑙 of the relative uncertainties by the following law:

𝑢 𝑡𝑜𝑡𝑎𝑙 = √𝑢 𝑆 2 + 𝑢 𝐿 2 + 𝑢 𝑀𝑃 2 + 𝑢 𝑖𝑛𝑠𝑡𝑎𝑏 2 ( 5-9 )
This formula yields a total relative uncertainty 𝑢 𝑡𝑜𝑡𝑎𝑙 = 19 % for the "material conduction" method. One can then apply the same approach for the "fluid sensible heat" method. Equation ( 5-1 ) involves three variables and their uncertainties, also evaluated as type B:

• The fluid properties, whose relative uncertainty 𝑢 𝐹𝑃 is supposed at ~5 % • The fluid temperature, whose relative uncertainty 𝑢 𝐹𝑇 is supposed at ~10 % • The flowrate, whose relative uncertainty is no larger than 1 %

The reason why the fluid temperature has such a large uncertainty is because it is not measured directly (the fluid temperature sensors are attached around the flowing tube, not in the fluid stream side). As for the flowrate, its uncertainty is driven by the precision of the flowmeter (given in Annex 7.3). Additionally, the process instability gives rise to the uncertainty 𝑢 𝑖𝑛𝑠𝑡𝑎𝑏 , this time supposed at 10 % since both the flowrate and the temperatures are affected (for the "material conduction" method, only the temperatures were involved).

Assuming again that all the uncertainties follow the same probability distribution (a square one) and that they are all statistically independent, applying formula for the uncertainty propagation yields a combined uncertainty 𝑢 𝑡𝑜𝑡𝑎𝑙 = 15 % for the "fluid sensible heat" method.

To conclude, the two heat flow calculation methods happen to have similar interval of uncertainty, and that these intervals are larger than the target model accuracy with respect to the test (+/-10 %). Ideally, the combined uncertainties should be within a 5 % interval (then smaller than the target model target accuracy); a way to reduce the uncertainties would be to perform principle tests, e.g. by comparing a calibrated heat source for each of the heat flow measurement of interest. This was actually done, but only for the coupler 10 K heat loads, as stated in 5.2.1.2.

Valve flow model

As seen in 3.4.2.1, the valve flow model is a linear combination of a compressible and an incompressible flow equation. It is important to validate both formula and the weighting between them.

The QM 2 K tank upper filling valve CV04 was chosen as a representative example of the incompressible flow model evaluation, since the valve inlet state is pure liquid. The comparison is made around a steady-state 4 K sequence from test phase 2.1, where the incoming liquid precisely compensates for the different evaporation sources, so that the 2 K liquid level remains constant. The knowledge of the valve flowrate is tricky since not directly measured: during the steady-state sequence, between 15:30 and 16:30 in Figure 5-4, the level is constant, thus the valve liquid mass flowrate equals the measured vapor mass flowrate. Outside of this sequence, the derivative of the liquid level, and the volume/level relation from the tank model in 0 must be used to infer the liquid mass flowrate; this is the green curve in Figure 5-4.

A Simscape model of the CV04 line, comprising the valve and all the elements from the 4 K tank to the 2 K tank is used to reproduce this test, under the exact same boundary conditions; this is the pink curve in Figure 5-4. The use of such a large model is compulsory since there is not intermediate pressure sensor between the 4 K and the 2 K tank.

Figure 5-4: CV04 simulated mass flowrate, with a position offset of -7%, against the levelcorrected measured mass flowrate

The simulated mass flowrate agrees with the level-corrected measured mass flowrate by +/-10 % in average, with peaks at -25 % in very unsteady conditions. This result is satisfying since the valve opening range is almost complete; moreover, the global tendencies are correct. A valve opening offset of -7 % was however mandatory to obtain this result, that would otherwise only reach +/-15 % in average. This valve offset can likely be explained by an error in the true valve position; this explanation is compatible with the MINERVA cryomodule experience that, below 5 %, CV04 hardly let any flow circulate, and with the fact that the valve fully-closed position, at cold, had not been calibrated. Lastly, as sensitivity analysis was conducted on the jumper heat loads, to see the influence of the vapor-induced pressure drop; the conclusion is that there is nearly no impact, since the valve fully dominated over the line ΔP. On the other hand, the hydraulic pressure due to the liquid level difference between the 4 K and the 2 K tank (about 61 cm) is significant (about 8 mbar) as regard of the pressure difference of only 60 mbar and must be considered.

For the compressible flow model evaluation, the CV05 valve was this time chosen. In nominal operation however, CV05 regulates the level only in 2 K mode and no vapor exists upstream of the valve, since the HX makes the saturated liquid go even deeper in the subcooled zone. Instead, the trick consists in using the 4 K mode with CV05: the warm vapor at 4.2 K from the 2 K tank, heated by the jumper heat loads up to 4.8 K, partially evaporates the incoming liquid on the other side of the HX, upstream of CV05. In this configuration, the HX works the wrong way, heating the liquid instead of cooling it. Such a working point (WP) was carried out during test phase 2.2, called WP11 in the bar chart from Annex 7.9 (explained in details in 5.2.8); a Simscape model comprising the CV05 filling line, as well as the HX and the return line from the 2 K tank is used to compare against this WP, again with the exact same boundary conditions.

The reason to include the HX in the valve model comparison is for the correct calculation of the valve inlet vapor mass fraction (about 0.1); indeed, the heating power (about 2.5 W) on the HP-side is needed, but, as seen in the next section about the HX validation, the measured heat balance on the LP-side is not reliable. The combination of the valve and HX models thus solves this problem.

As for the CV04 comparison, although the liquid level is very stable +/-0.1 cm, a levelbased correction of the vapor mass flowrate must still be used to derive the liquid-side mass flow rate. This is the drawback of the large section of the 2 K buffer tank at the target level of 10 cm: a small level variation has a significant impact on the tank inletoutlet mass flow balance. The comparison of the model versus the test is shown in Figure 5-5, for the 2 versions of the compressible flow formula. It is clear that the ANSI compressible flow equation is the one performing the best: both the absolute level and the tendency is correctly reproduced. Moreover, this validates the choice of the vapor mass fraction as a weighting factor between the compressible and the incompressible formula. One should keep in mind that other indicators, such as the isotherm compressibility from [START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF] could likely work fine too.

This time, no valve opening offset was necessary, consistent with the experience on the field; the HX pressure drop model was included, assuming the dimple distribution was the same as the ESS heat exchangers (the only ones having measured ΔP data), but stand only for 0.1 mbar out of 40 mbar total pressure difference (dominated again by the valve) and is thus negligible. By contrast, the difference in liquid height had to be included, as for CV04.

Finally, a last test at room temperature was conducted, to challenge the compressible flow models in warm conditions: CV601 and CV651 were chosen for this matter. During test phase 2.1, a simple ΔP test was run with a gas nitrogen 200-bar tank, regulated at 1.5 bar at the inlet; the 2 valves were consecutively swept in opening position from 0 to 90 %.

A Simscape model, comprising both of the complete QVB and QM 80 K loop models was used for the comparison at iso boundary conditions; the resulting mass flowrates are shown in Figure 56. The predicted valve flows of the QVB and QM 80 K lines are underestimated by respectively 30 and 40 %. As the valve stands for 90 % of the total ΔP along the line, the errors likely come from the valve model itself, not from the other components along the lines. These errors are quite large compared to the target at +/-10 % set in 2.6, but are not a real big issue for the control-oriented tasks; indeed, the tendencies are globally respected (linearity of the mass flowrate versus the opening) as well as the similarity between the QVB and the QM 80 K loops. The discrepancy can simply be overcome, if more accuracy is needed, by a model adjustment, by trimming, for instance, the valve maximum Kv. Other components really need such experimental adjustments, as seen later on.

To conclude, the valve models show satisfying absolute performance and, above all, good tendencies over a large range of operating conditions, which is what is needed for a dynamic process model. A coupling between the component models was necessary to correctly reproduce the test conditions. Unfortunately, one of the most critical valves, the MKS valve which controls the 2 K tank pressure, could not be directly evaluated due to the lack of an operational pressure sensor downstream of the valve; this is all the more problematic that this valve model is very hypothetical, as explained in 3.4.2.

Test Sim

Heat exchanger model comparison

The HX thermal model is the first one reviewed in this section. The HX hydraulic model comparison, shorter, is reviewed in conclusion.

The performance of the thermal model is evaluated looking at the exchanged power output. On the testside, this output is calculated using the mass flowrate and the inlet/outlet temperatures, on both the HP and LP sides for consistency check. Two problems must be overcome:

• The HP-side mass flowrate is not measured, thus must be derived from the level-corrected vapor mass flowrate, or by the valve flow model • The LP-side outlet temperature, TT030 in This test in 2 K mode was performed while the level probe power supply was still not optimal, causing much parasitic noise (corrected during test phase 2.2); this made the HP-side mass flow rate calculation by the level derivative method unreliable, as seen in the green curve, in the left-handed chart, and in the blue curve in the centre chart. Rather, the mass flowrate from the CV05 valve model was used (the pink curve in the left chart, the red curve in the centre chart) as a reference against the fully modelled heat exchanged, the green curve.

Looking at the HP-side (centre chart), the agreement in exchanged power is better than 5 % up to 1 g/s and drops to 15 % at 3 g/s: the model becomes pessimistic versus the test; this was actually expected: the dimples that increase the heat exchange coefficient are not considered in the model. Now, looking at the right-handed chart, one can see a large deviation between the simulated power and the LP-side measured power; this is because, as mentioned, the LPside outlet temperature TT030 is heated by some source that virtually raises the measured power. A Simscape model of the HX LP-side downstream pipe was added to the combined HX and valve model, showed in Figure 56789, to investigate on the physics of this parasitic heat. It was first assumed that it was caused by the conduction from the QVB top cap at 300 K along the pipe wall; however, the simulation showed that the vapor heat transfer coefficient is large enough so that the incoming heat is evacuated and does not reach the sensor. Moreover, trimming the HTC to a low value at the location of the double wall improves the average agreement but yields in inaccurate tendencies over the whole WP20-23 range, as seen on the pink curve in Figure 56789, against the measured TT030 in red, and the raw simulation in green.

TT030

This is the black curve in Figure 5-9; this much better agreement proves that this sensor mismatch can be in fact be approximately explained, reinforcing both the test and the model consistency.

A second working point, WP11 from test phase 2.2 (same as the valve flow model validation, in Annex 7.9), was investigated: as said, the heat exchanger works the other way around by heating the vapor (the power is thus negative in the following charts). An additional problem though, is that the helium is diphasic on the HP side: the calculation of the exchanged power based on the temperatures is thus not physical, as seen in the centre chart of Figure 5-10; leaving aside the previous method to correct TT030, another method is needed to measure the exchanged power and compare it to the simulated average 2.5 W, seen in the green curves in the centre and right charts of A first clue is given by comparing WP11, where the 2 K tank level is regulated via CV05 to WP10, where it is regulated through CV04 (nominal 4 K mode), again in Annex 7.9: the 2 K tank sees an additional vapor flow in WP11, due to the vaporization in the HX, corresponding to precisely 2.6 W. A second clue comes from the fact that the LP side, in this working point, is the lowest enthalpy side: its outlet temperature equals the HP inlet temperature. In short, the vapor goes back to the temperature it had when leaving the 2 K tank by releasing all the heat it captured on its way to the heat exchanger; this captured heat is the jumper heat loads, which, as seen in 5.2.7, is 2.5 W in average, measured by an independent method.

This second working point illustrates again the correct thermal performance of the HX component model; this working point analysis is also an opportunity to perform a first global cross-check of the different measured heat loads on the prototype QM. Overall, this heat exchanger thermal analysis, featuring failing sensors, partial information, as well as unsteady and unbalanced flowrates, is a typical example where a detailed dynamic process model, combining physical component models, is a powerful diagnostic tool.

To finish, the HX hydraulic model was compared to the only available working point, a single control test carried out with the CV1250 HX on the LP side, at room temperature with water. Indeed, no extensive characterization with water, like on the ESS project, was performed, nor did the prototype cryomodule include any ΔP instrumentation. The measured ΔP at 14000 l/h (water) was 240 mbar, against 297 mbar for the Simscape model, with the assumption that the dimple distribution is the same as for ESS (thus using the same deformation model). This very good prediction is yet another hint that the physics of the heat exchanger is globally correctly captured by the model.

Overall, the HX physics seems correctly captured by the component model, without the need for experimental adjustments, except for the thermal performance at very high mass flow rates.

Table model comparison

Test consistency analysis

As for the previous model, experimental heat balances are performed where possible, in order to evaluate the test data consistency; for clarity, the full table instrumentation is shown in Figure 5-15 from the next sub-section. The first relevant location is around the table feet: as seen in Figure 5-11, the heat from the feet can be compared to the heat that diffuses in the table frame and in the cavity Invar rods. The heat balance, seen in Figure 5-12, is accurate within +/-30%, which, given the pretty limited instrumentation is good; it also contributes to confirm that the contribution of the feet to the global heat loads to the 80 K loop is low, as seen later. The heat from the table right-handed side can be thus directly compared to the heat reaching the 2 K tank on the right side, seen in Figure 5-14 (right-handed chart): the agreement is within +/-20 %.

By contrast, in the table left-handed side, the heat leaving the cavity frame, about 0.9 W, does not reach the cavity tank, as seen in Again, those heat balances, although oversimplified, give an insight on the test data consistency, which is only correct within +/-20 % in average; they tell also, that a meticulously accurate model on just one test output is irrelevant, and instead, that one must focus on a model that globally matches a maximum of outputs, furthermore over a wide operating range.

Simscape reproduction

This section reviews the reproduction of the table temperature map during WP5 by the Simscape model, put at iso conditions (80 K, 10 K and 2 K loop mass flowrates). As for the coupler thermal model, several contact resistances had to be calibrated, and proved to be consistent between them, and in line with the observed tightening torques. As seen in Figure 5-15, the global agreement is accurate +/-10 K; the major discrepancy is on the table frame sensor, on the right-handed side, as the contact resistance with the thermal shield was the sole being not calibrated. This deviation is actually consistent with the model comparison during the 80 K cool-down, as seen in the next section.

Figure 5-15: table support temperature instrumentation mapping and Simscape reproduction

The table is yet another component where much effort was allocated, both on the instrumentation and on the test protocols, and furthermore, where the involved physical laws can be parametrized almost exclusively based on the as-build geometrical and material characteristics, without the need for adjustments. By contrast, the next two sections illustrate components where this approach is more limited.

Coupler thermal model comparison

The previous section highlighted the importance to perform experimental cross-checks between heat flow measurement methods, prior to comparing versus the model. The coupler can be split into 4 areas of interest, where such heat balances can be performed:

• The air-side area: the sum of all the heat input on the air-side can be compared to the heat towards the 80K loop • The 80 K, 10 K and 2 K loops: the heat measured by the material conduction method can be verified against the fluid-side methods

In practice, it was not possible to check the 80 K loop balance, as the nitrogen is in

LEFT

RIGHT diphasic state at this location, and no vaporization heater was installed, by contrast with the 10 K loop. The air-side, the 10 K and the 2 K loop heat balances are respectively reviewed in the following sections.

Heat balance on the air-side

Figure 5-16 defines the 4 main heat flows around the coupler air side: 𝑄 ̇𝑊𝑖𝑛° ℎ𝑒𝑎𝑡𝑒𝑟 and 𝑄 ̇𝐹𝑙𝑎𝑛𝑔𝑒 ℎ𝑒𝑎𝑡𝑒𝑟 are the known injected electrical power in the window and main flange, and 𝑄 ̇𝑎𝑖𝑟 𝑐𝑜𝑛𝑣 is the convection with the ambient air, rather small as the air-side parts are regulated at ambient temperature, but is still non-zero and cannot be directly measured. The sum of these heat loads can be compared to 𝑄 ̇300→80𝐾 , the heat towards the 80 K loop, measured thanks to the material conduction method along the cuff; Figure 5-17 shows this balance, for both couplers, on a 4 K mode nominal steady-state working point from test phase 2.1, again WP5.

The difference between the combined heater power and the heat to the 80 K loop is about 0.5 W, and can be allocated to convection with the ambient air, as this figure is actually consistent +/-0.5 W over all the MINERVA test phase.

As a side note, at this location, the weight of the copper layer in the heat to the 80 K loop is rather limited, as seen in Figure 5-17, thus cannot help to confirm the copper RRR measurement. The next cross-check to be performed is the total coupler heat to the 10 K loop.

There is actually a 5 th important heat load 𝑄 ̇𝑤𝑎𝑡𝑒𝑟 𝑎𝑛𝑡𝑒𝑛𝑛𝑎 , the heat load from the antenna cooling water that transits across the window; however, over test phase 2.1, the antenna water loop was not running at all, 𝑄 ̇𝑤𝑎𝑡𝑒𝑟 𝑎𝑛𝑡𝑒𝑛𝑛𝑎 is thus 0. The reference heat from the coupler to the 10 K loop 𝑄 ̇10𝐾 is computed from the difference between 𝑄 ̇80→10𝐾 and 𝑄 ̇10→2𝐾 , both measured by the material conduction method, as defined in Figure 5-18. WP5 is also used to compare this flow to the one measured by the fluid sensible heat method.

In this sequence, the 10 K loop is regulated at its nominal mass flowrate until 17:48; the helium is partially liquid down to coupler intercept #1, meaning the heat loads from the intercept arms and a part of coupler #1 heat loads are hidden to the fluid sensible heat method, as seen in Figure 5-19. Hence, 2 heating steps with EH080 are performed: 1 W from 17:48 on, then 2 W at 18:00; this second step is enough to fully vaporize the liquid helium prior to the cavity support arm intercepts. From 18:00 to 18:04, all the component 10 K loop heat loads which are downstream of EH080 can thus be derived by the fluid sensible heat method. Furthermore, upstream of EH080, the heat load from the jumper inlet line can be derived from the balance between the total heat to the 10 K loop and the sum of the individual heat loads (including EH080). In short, this method provides all the component-bycomponent heat loads to the 10 K loop, by the fluid sensible heat method, for comparison with the material-side methods.

For the couplers in particular, the comparison is shown in Figure 5 As said, the fluid-side measurement for coupler #1 is only valid from 18:00 on (grey arrow), while coupler #2 can be compared over the whole chart. The fluid-side measurement agrees within 10 % with the material-side measurement that assumes RRR = 120 and only within 30% when assuming an RRR = 30, thus confirming the RRR characterization; furthermore, the heat load difference between coupler 1 and 2 due to the series configuration is respected (coupler #1 gets more heat since the 10 K loop is at a lower temperature).

RRR Cu = 30 RRR Cu = 120

Heat load to the 2 K loop cross-check

The reference heat to the 2 K loop 𝑄 ̇10→2𝐾 is calculated from the material conduction method, thanks to a dedicated sensor on the niobium flange (TT130 on cavity #1), seen in Figure 5-21 and considering at the other end, that the titanium tank is at the liquid helium temperature. However, due to the high niobium thermal conductivity relative to stainless-steel, the temperature difference is only about 0.5-1 K, and, above all, the distance between the temperature measurement is short. To evaluate the method reliability, 2 crosschecks are performed.

First, against the 2 K loop heat load, by comparing WP5 against WP8, another case from Figure 5-29, where the 10 K loop mass flow rate is set to 0. From WP5 to WP8, the total heat load to the 2 K loop increases from 14 to 18 W, a difference of 4 W; the reference material-side coupler heat load method gives at total of 1.5 W to 7.5 W, a difference of 6 W. This 30-% excess of the reference method is not fully understood, but is nevertheless pretty good, given the uncertainties in both measurement methods.

A second cross-check of 𝑄 ̇10→2𝐾 can be done on the other side of the 10 K loop intercept, against 𝑄 ̇80→10𝐾 , also measured by the material conduction method. Indeed, when the 10 K loop is not running at all and thus does not take away any heat (again, WP8 in abnormally high, meaning there is only a slight difference (15 %) in the copper layer participation at RRR = 120 versus RRR = 30, which does not allow to rigorously confirm the RRR = 120, this assumption giving only a slightly better agreement.

To conclude, the measured reference heat flow to the 2 K loop due to the coupler conduction is about 1.5 W in total for both couplers; as seen, this figure is robust versus 2 other measurement methods, although likely pessimistic. This total heat flow is in fact about 4 times the specification at the same conditions, as seen in the next section, fully explained by the too high copper RRR.

Simscape synthesis reproduction

This section focuses on the full reproduction of 2 series of tests by the coupler heat transfer model presented in 0, put at iso conditions (10 K and 80 K loop mass flowrates): on top of the material conduction, the convection with the cryogenic fluids is modelled.

The reproduction against the first series of tests, WP5 and WP8 from test phase 2.1 in 4 K mode, is summarized in Table 5-2: two RRR assumptions are taken, at 30 (specification) and at 120 (the measurement). For all 3 cryogenic loops, the RRR = 120 assumption matches the experimental heat flows within +/-20 %. The simulation with RRR = 30 has a larger mean discrepancy, especially on the 2 K loop where it fails to explain the 1.5 W heat flow from the reference method material conduction. On this specific heat flow, the model foresees rather 1.3 W in total, in line with the fact that the reference method is pessimistic. Two items are especially important to make correct 2 K loop heat flow reproduction: the HTC with the 10 K loop, and the contact resistance at the coupler port. The former is correctly reproduced by the Simscape model, confirming that the heat intercept, consisting in a simple 2.5-turn machined channel, fails to fully anchor the coupler temperature at the helium temperature. The latter, impossible to physically model, was measured at about 5 K/W thanks to the surrounding thermometers, and imposed by a Simscape block. Overall, the coupler port temperatures (TT130/230 in Figure 5-18) are correctly reproduced, at about 4.8 K, confirming, that, although the 10 K loop intercept is inefficient, the high niobium thermal conductivity on the one hand, and the port contact resistance on the other hand, drastically reduce the risk of a niobium normal-conducting transition at this location.

The second test to be reproduced with the Simscape coupler model is a window heater failure test from test phase 2.2. This enables in particular, to evaluate the model of the convection with the ambient air, and more generally the warm source heat distribution; this time, the coupler antenna water loop was set on. Table 5-3 shows that the total heat to the 80 K loop as well as the heat from the antenna water are correctly reproduced. By contrast, the distribution of the heat between the convection with air and the flange heater is inaccurate. This is very likely due to a historical conservative assumption of 5 W/m 2 .K for the HTC with air, relevant for free convection in confined area; in reality though, the prototype cryomodule features several pumps and cooling fans in the vicinity of the coupler; the needed HTC of 25 W/m 2 .K to explain the heat load distribution seems more relevant.

Table 5-3: Cryoscape reproduction of the coupler heat loads and comparison to test in the case of the window heater failure test

To conclude, the complex heat path along the coupler is, in average, accurately enough reproduced throughout a wide operating range; model parameters like contact resistance and free convection, difficult to physically model could be calibrated with physical and consistent values, giving confidence for the model prediction capability. Two critical aspects could unfortunately not be experimentally confirmed: the radiative heat load of the 300 K antenna to the 2 K loop, and the antenna water freezing risk in the occurrence of a water circulation breakdown. The latter case is a particularly critical safety topic for the utilities, so that, although the model predicts an antenna minimal temperature of 12 °C, an experimental confirmation is required. A prototype test protocol, consisting in a thermocouple inserted in the antenna cooling loop is ready and should be performed in future test phases.

As a general conclusion, just like the heat exchanger, the MINERVA prototype coupler is a particularly well understood component, thanks to the extensive instrumentation and test protocols to challenge the models, as well as the easily parametrizable involved

Thermal shield comparisons

Contrary to the previous components, the QVB and QM thermal shields get a large share of their heat load via radiative heat transfer, which physics is rather simple, as seen in 3.3.2.2, but relies on surface parameters that are complex to a priori evaluate. As an introduction to the notion of model adjustment, the QM and QVB thermal shield transient behaviours are reviewed against the model first; then, the QM thermal shield steady-state heat loads are analysed.

Thermal shield cool-down

A 80 K cool-down test sequence from test phase 1 is used to compare to the Simscape thermal shield models developed in part 3.4.6, at iso CV601 and CV651 valve positions; the comparison is done based on the thermal shield envelope temperature outputs (PT-100), which precise locations are indicated in 7.8. For these comparisons, the global cryomodule process model presented in 0 is used. Starting with the QM, the comparison is shown in Figure 5-23. Overall, the dynamics of the thermal instrumentation is correct, and the order between the sensors is globally respected. One improvement could be to further discretize the very first inlet tubes, which would enhance the reproduction of the first PT-100 to drop (TT680, TT670, TT074 and TT688). A second improvement, as already mentioned, would be to consider the poor thermal contact between the thermal shield and the table, on the cryomodule righthanded side: doing so, the thermal shield would further drop in temperature; this would also improve the table cool-down reproduction, which by contrast, drops too much in temperature.

Looking now at a similar sequence from test phase 1, but on the testQVB, the statement is the same, as seen in Figure 5-24. Again, the distribution of the first temperatures could be improved with a refined inlet tube discretization; moreover, the experimental sharp decrease of TT622, TT623 and TT624 at 14:50 could be more faithfully reproduced if the change in boiling regime in the vertical section of the LN2 serpentine was considered. To conclude, this thermal shield cool-down reproduction illustrates the fact that the physical process model, although detailed, needs adjustments in some specified complex locations. This was introduced earlier with the contact resistances, and now with the valve equivalent permeability. This notion of model adjustment is primordial: they stand for the difference between the physical model and the test anchoring; in short, they are the picture of the model versus test deviations. The next section pushes this notion even further.

QM thermal shield heat loads

In the case described in the previous section, the radiative heat loads to the thermal shields are secondary compared to the other physics at stake in transient; in steady-state by contrast, they theoretically account for about half of the heat loads for the QM, about 30 W considering a standard value of 5 W/m 2 . Since radiative heat loads cannot be measured by one of the methods in 5.2.1, they must be derived from the difference between the sum of the individual measurable contributions and the total heat to the 80 K loop, measured on the fluid-side, considering the latent and the sensible heat between the LN2 Dewar and the LN2 reheater. This latter value is illustrated by the height of the bars in Figure 5-27, and the "Rest", in blue, is the difference versus the sum of the measurable heat loads. Figure 5-27 is a picture of the heat load distribution over 12 steady-state working points during test phase 2.1. The first 3 points are "screenshots" some hours/days after the 80 K cool-down, while the cavity string is still rejecting heat, explaining the excess of total heat loads. The following points, WP4 and WP5 are taken during an 80 K warmup, hence the thermal shield enthalpy change large contribution. Finally, WP7 to WP12 are taken at different cold mass temperatures, close to 4 K working points. The blue "Rest" is in average about 120 W; as said, it accounts for all the non-measurable heat loads: the thermal radiation, the external transfer line, as well as the jumper inlet line (impossible to subtract, since diphasic). For the latter two, observations during all the test campaigns enable to confidently allocate about 30 W, meaning 90 W for the sole thermal radiation. This figure in turn means 15 W/m 2 or 3 times the standard value for copper thermal shields; a couple of evidence actually reinforce this somehow surprising statement:

• During WP4 and WP5, the 80 K loop is not running, the sole non-measurable heat loads is thus the radiation, and the blue "Rest" being at about 50 W; furthermore, the calculation of the material enthalpy change rate (the yellow bar) is rather rough and proved to underestimate the heat loads by 30 %, when compared to calibrated electrical heater. The blue "Rest" in those 2 points may hence actually be rather 80 W, close to what is looked for • This blue "Rest" is constant +/-10 W in several working points over several weeks, once the cold mass is in steady-state

• Last but not least, the blue "Rest" was also at this value during test phase 1, meaning it is robust to a cryomodule complete disassembly/assembly step To conclude, beyond the figure, the important lesson-learned of the thermal shield reproduction is again the need for experimental model adjustments to complement the physical models; this is especially true for the QM thermal shield, that features many holes for the instrumentation, beam tubes, and other fittings, and on top, that has an irregular shape. The same analysis on the testQVB thermal shield, much simpler in shape, actually concluded in a radiative heat load of 4.5 W/m 2 , close to the expected value.

Those adjustments could be further reinforced by additional parameter sweeps in future prototype test phases, such as the nitrogen mass flowrate, or, simply, by a deeper analysis of the many free warmups (without active heaters) that were performed.

QM chimney heat loads

The QM chimney is yet another location where the pure physical models reach their limits: the raw measured heat loads are indeed way higher (Table 5-4) than the physical model would theoretically predict (a few tenth of a W). In particular, the 2 K inlet heat loads are calculated by a specific method: they are derived from the difference in two latent heat measurements on the 2 K tank (with versus without transfer), following the principle described in 5.2.1.1.

Table 5-4: QM chimney heat loads to the 10 K and 2 K loop, split between the inlet (from the QVB) and outlet (from the QM) flow lines

This statement is due to the QM chimney being an especially complex object, as seen in Figure 5-28, that cumulates several QM functions:

• Inlet pit for the level probe • Helium routing to the burst disk • The connections to the QVB • The QM / QVB vacuum barrier This design strategy first prevents from optimizing each function, for instance, minimizing the heat loads to the 2 K outlet vapor flow, responsible for a less efficient subcooling process, thus additional vapor production by flash. In addition, as the chimney manufacturing is also complex, it is impossible to put MLI on the helium and nitrogen tubes, thus increasing the radiation from the ambient and thermal shields, that might

Inlet Outlet 10K 3,5 W 0,25 W 4K/2K 0,75 W 2,5 W
actually explain the high heat loads to the 2 K outlet line. Finally, the horizontal displacements of the QM tubes likely make them thermally connect their double-wall sleeve through the vacuum barrier, which is probably responsible for the high heat loads on the 10 K inlet (measured following the method in 5.2.5.2), also confirmed by direct vacuum barrier temperature measurements.

The QM chimney is clearly a component where the physics of the heat loads is not captured at all by physical models, even if they are detailed, and thus where experimental adjustments have a great importance. Further experiments, such as clean 10 K loop parameter sweeps and a continuous measurement of the vacuum barrier temperature would be beneficial to test the previous parasitic thermal contact assumption. Likewise, at the occasion of a disassembly-reassembly step, testing to install MLI, even partially, would confirm the radiation problem or not. The key at least, is to understand whether those adjustments are specific to the prototype device, or in which extend they will also exist in the series version.

Radiative heat loads to the 2 K tank

It was previously seen that the cavity support table heat loads on the one hand, and the coupler ones on the other hand, were well understood, and that their respective component models were accurate enough. Those two components, theoretically, accounts for nearly all the heat loads to the 2 K tank; a last contribution is the radiation from higher-temperature parts, for instance the thermal shield, which again is not possible to directly measure. Instead, as for the 80 K loop, a balance can be done between the sum of the material conduction measurements, and the total heat flow to the 2 K loop, measured by the fluid methods described in 5.2.1.1. This balance is the blue bar called "Rest" in Figure 5-29, which was introduced earlier for the naming of the working points over test phase 2.1; a similar chart for test phase 2.2 is presented in Annex 7.9.

As a short overview of this chart, WP1 to WP4 are cavity heater (EH100 + EH200) parameter sweeps in nominal 4 K mode, hence the large red bars. Then, WP5 to WP8 are 10 K loop mass flowrate sweeps, still in 4 K mode, as already seen in the coupler measurement cross-checks. Then, WP9 to WP15 are consecutive screenshots after 2 hours in 4 K mode steady-state, while the filling valve (CV04) is fully closed; the level is thus gradually dropping. Then WP16 to WP19 are cross-check of the previous The yellow bar is the pure radiative (theoretically) heat load to the 2 K buffer tank only. It was isolated during test phase 1, as the buffer tank and the mock-up cavity tanks could be isolated from each other, when the level was below 17 cm (the "communication tube" in Figure 3-36 only exists since test phase 2.1 on). This heat load proved to be linear with the level, reaching about 2 W at 10 cm; whereas the level-dependence makes full sense, the magnitude is about 10 times the theoretical radiative heat load, considering a reference textbook value of 0.3 W/m 2 for a fully MLI-covered tank and an ideal 80-K thermal shield. In reality, though, due to the complex buffer tank shape, the MLI cover is only partial; furthermore, much radiation likely comes from the chimney located nearby, as already reviewed. Although not fully understood, a model adjustment of the buffer tank radiative heat load makes relatively sense.

The blue bar, the "Rest", hence encompasses the radiation to the cavity tank, buffer tank excluded, as well as all other non-measurable heat loads, i.e. transients, and any unexplained source. From WP5 to WP8, the decrease in the "Rest" bar is explained by the excess of the coupler heat loads from the reference material method, as mentioned earlier. Then, from WP9 to WP15, the cold mass enthalpy change is not considered, which explains why the "Rest" becomes negative. Over WP20-22, the heat loads to the 2 K tank did not have enough time to reach the steady-state. Finally, the measured transient CTS heat loads is overestimated by 2-5 when compared to the CTS mass enthalpy decrease rate, which explains why the blue "Rest" is not constant, between WP1 and WP16 in test phase 2.1, or between WP1 and WP10 in test phase 2.2. Those blue "Rest" variations could be further reduced simply with more post-treatment work.

The blue "Rest" can thus be considered constant in 4 K or 2 K steady-state, at about 3 W +/-0.5 W, which again deviate from the radiation textbook values by a factor of 10. Nevertheless, this value is robust to large electrical heater variations (thus mass flowrate), over a 2-week span with regular 4 K / 2 K steady-states sequences, and to a complete cryomodule reheating/cool-down step. Furthermore, during test phase 1 where the 2 K tank surface was much smaller (mock-up cavities), the "Rest" was much lower too, at about 0.5 W; this indicates that a large share of this unexplained heat load could indeed be due to an excess of radiation. As for the 80 K loop, this could originate from the complex shapes, multiple holes and partial MLI cover versus a perfectly MLI-protected, spherical cold mass. This idea is also supported by a similar approach conducted on the testQVB 4 K tank, which shape is way simpler than the 2 K tank assembly, that concluded in a good agreement of the textbook radiative model. A model adjustment of the radiative heat load to the 2 K tank (below the buffer tank) is thus calibrated on the "Rest" mean value. Further prototype tests during the coming phases, like a thermal shield temperature parameter sweep thanks to heaters or mass flowrate variations could challenge this interpretation.

Conclusions

The analysis of the cryomodule test campaign revealed a general poor repeatability of the cryogenic test, +/-5 %, caused in particular by the many transients, as the cryomodule does not stay at 4 K overnight; the test instrumentation consistency checks also highlighted 15 % discrepancy in average between the different measurement methods. This stresses the role of the process model, to serve as a physical reference in these test diagnostics, and the importance of the agreement in global tendency and wide output range.

The valves, the heat exchanger, the coupler and the supporting table detailed models, in most cases, proved to satisfy the dynamic process model accuracy target at +/-10 % over a broad operating range; only minor adjustments were necessary in order to cancel out the deviations of the physical models versus the reference measurements.

On the other hand, the physical models turned out to be limited against a priori unknown or overlooked model parameters, like 3D diphasic flow separation, parasitic thermal contacts and thermal radiation in non-ideal conditions, requiring significant model parameter experimental adjustments. Those large adjustments are not surprising ; they would also be needed to explain, for instance, the excessive average ESS Spoke cryomodule static heat loads [START_REF] Maiano | Production, Test and Installation of the ESS Spoke, Medium and High Beta Cryomodules[END_REF]; the advantage though, is that this statement comes ahead of the MINERVA series production, leaving time to react.

An important lesson-learned from this test versus model comparison, is the central role of these adjustments, listed in Table 5-5: they are a picture of the model deviation versus the anchoring tests. What is primordial then, in order to make reliable predictions, is to ensure the adjustments are:

• Allocated to a clearly identified physical model parameter • That they are constant whatever the test conditions Finally, the adjustments put a clear flag on critical areas, ahead of the series part design freeze: the large deviations versus the physical models reflect a lack of control of the design, thus stress an increased risk for unexpectedly high heat loads, and significant cryomodule-to-cryomodule dispersion for the series. It would be thus worth redesigning those items into a simpler design, robust to manufacturing or assembly dispersion and errors.

CRYOGENIC CONTROL MODEL

The previous section highlighted the role of the cryogenic process model, in the preparation and post-treatment of the prototype experimental phases, essentially focused on the thermo-hydraulic performance. This section deals with the role of the model for the prototyping phase, this time in control-optimization tasks.

The first section deals with the creation and virtual commissioning of the cryogenic prototype control loops, exclusively based on PI (Proportional Integral) control laws. In the second section, a Linear Quadratic (LQ) controller is synthetized and tested on the prototype cryomodule, as a second option to control the 2 K tank pressure and level.

Validation of the PI controller pre-calibrations

Problematic and approach

The prototype cryomodule operation in 4 K or 2 K mode was limited both in terms of consecutive hours, typically 4 hours a day, and in terms of occurrences, between three and seven days for each phase. Hence, the need to drastically reduce the time allocated to control optimization, that was moreover planned to be substantial, since the PLC codes (equipment signal routing, sequence and control laws) were fully new.

This task could instead be performed ahead of test phase 1, in parallel of the first cryomodule assembly, by using the cryogenic process model as a digital twin to validate the PLC control laws. As a matter of fact, the implementation of the control laws in the PLC was started only a few weeks prior to the test start; the PLC code itself could not be tested directly. Instead, the control laws (and sequences too) were first developed as a pure Simulink code, the Simulink "control box", as presented in 0 to control the process model. Once validated, the control laws could be copied into the PLC CPU, in STEP7 code; the only true difficulty in this conversion step, is the strict consistency needed between the Simulink control box and the PLC code, in terms of I/O signal unit and sign convention in the control laws. This parallel virtual commissioning approach is illustrated in Figure 5-30. actual measurement. Errors mainly arise from RF dynamic heat load changes (real or simulated), or due to cryogenic transients, especially at the start of a new test phase, or after a new sequence; the controller is used primarily in disturbance rejection rather than setpoint tracking. As a side note, not mentioned in the previous figure, the generic controller has an anti-windup strategy, to avoid the integrator response build-up while the command is stuck on one of the saturation limits.

The goal of the synthesis and virtual commissioning step, is to find, for each controller:

• The nominal pre-command • The optimal proportional gain P and integral time Ti

Although MATLAB offers dedicated tools to automatically tune P and Ti, based on the system identification and classical system analysis, a simple try-and-fail approach was chosen, more convenient to get a feeling on this brand-new system, and anyway suited, given the control law simplicity. Those 3 controller parameters are called "precalibrations" in the following sections, that review their model-based construction and performance validation during the true experimental phases.

Coupler flange and window temperatures

As presented earlier, each coupler is temperature-controlled at 2 locations by heaters: at the ceramic window by EH143 (cavity #1) or EH243 (cavity #2) so as to limit the window temperature changes, thus the stress and fatigue, and at the main flange by EH144 or EH244 to avoid water condensation. The heater positions were illustrated in Figure 5-16.

The coupler temperature controls are highly critical and should be active whenever the QM 80 K loop is running, as both coupler high temperature intercepts are in series with the QM thermal screen.

Contrary to all of the controllers mentioned in this chapter, the flange controller follows an ON-OFF control law, that controls the temperature between 290 and 296 K. This control law was reproduced in the Simulink control box too. A few simulation iterations were needed to pre-calibrate the EHx43 controller PI gains ("x" refers to coupler 1 or 2); the order of magnitude of Ti is 3 minutes: it was designed purposely slow, not to induce any thermal stress.

In order to pre-validate the gains, a 10 hour-sequence was simulated, comprising 3 usecases, as seen in Figure 5-32: a change in TTx43 setpoint at 09:00 (+ 5°C), then a breakdown of the EHx43 controller at 11:00, finally, a breakdown of the EHx44 controller from 13:00 to 18:00. At the beginning of this test phase, both coupler temperatures were at 294 K; during the 80 K cool-down, the nitrogen temperature drops through the coupler intercepts spread over 4 hours, due to the high thermal shield mass; the window and flange controllers seamlessly prevent the temperatures from dropping more than 5 K. The difference of dynamic between both couplers is due to dispersions in temperature sensor positions.

Then, a series of 2 experimental fault simulations were carried out, following the same approach as with the Simscape model; first, on EH144 and EH244, as seen in Figure 5-34. The reaction of the window temperature controller is satisfactory, with an error limited to 2 K only; interestingly also, the drop in TT144/244 is close to the simulated one, in Figure 5-33, another hint that the coupler model works fine. On the other hand, the response of the window heater is only +2 W, versus +4 W for the simulated one; this is in line with a remark from 5.2.5.1, that the simulated heating power due to the air convection is underestimated, hence an overreaction of the window heater.

Lastly, a fault simulation was carried out experimentally on EH143/243, following again the same approach as in the simulation: at 15:00, the controller output voltage is reinitialized to 0. The window temperature drops only by 6 K before the controller catches-up, consistently with the simulation in Figure 5-33.

Figure 5-35: experimental fault simulation of the EH143/243 window temperature controllers

To conclude, the accurate enough coupler component model enabled to create correct pre-calibrations for the coupler temperature controllers on the first try, ahead of the tests; as matter of fact, the model had also been used to size the flange heater itself. This successful pre-calibration is actually not surprising, as those control loops are not demanding, in the absence of strong interactions with the other feedback loops. The true difficulty, overcome thanks to the digital twin, was the impossibility to train on the real equipment, given the risks at stake.

80 K and 10 K loops

The 80 K loop controllers (QM and QVB) act on the respective nitrogen valves (CV601 and CV651) in order to reach 90 K in fluid temperature at the thermal shield outlets. This target was chosen to be clearly above the nitrogen saturated temperature, for obvious controllability reasons, and not too high in order to minimize the thermal shield temperature differences. For these control loops, the stability is favoured versus the performance, since:

• The strong variations of mass flowrate in critical parts like the coupler induce mechanical fatigue • The other control loops are potentially disturbed by those 2 loops • The heat load characterization requires steady flows The model-based optimization turned out to be straightforward, reaching acceptable performance within a few iterations: the resulting optimal 80 K loop feedback loop are slow, Ti being in the order of the hour, consistent with the fact that the reaction of the thermal shield is long, due to the large thermal mass at stake. The pre-calibrations are summarized in conclusion of this part.

The pre-calibrations were flashed into the PLC CPU, and did not need further retuning during the whole prototype QCELL test phases; the performances are illustrated in Figure 5-36, during test phase 2.2 initial 80 K cool-down. What is satisfying, is, although not designed for this purpose, the correctors were able to actually make the whole cool-down from 300 K to 80 K, with only some minor manual valve corrections to accelerate the convergence. Furthermore, the controllers are able to compensate for the large nitrogen Dewar pressure deviations, as seen in the righthanded chart. The controllers were finally able to compensate for the transient heat loads, inherent to initial 80 K cool-downs, especially visible in the centre chart on the QM nitrogen mass flowrate; this makes it possible to get reliable measurements of the 80 K heat loads, at the same temperature conditions. The 10 K loop has the same requirements as the 80 K loop, i.e. favour the stability and disturbance rejection to performance and setpoint tracking; the tolerance on the regulated value, the 10 K loop mass flowrate, is +/-20 %. The tuning on the process model was also straightforward, and resulted in more aggressive gains than the 80 K loop, with a Ti in the order of the minute, in line with the fact that the controlled value is, this time, the mass flowrate, not the outlet temperature; these pre-calibrations are also summarized in conclusion of this part.

2, in 4 K mode

The pre-calibrations were, as for the 80 K loop, directly copied in the PLC code; the performance of the 10 K loop controller is displayed in Figure 5-37. Over this sequence, the 10 K loop is started from 0 mg/s to 100 mg/s, then increased to the nominal mass flowrate of 200 mg/s. The controller is able to follow the setpoint change within a few minutes, without overshooting; the controller is also able to stay within the tolerances, despite 2 K tank pressure variations at 12:00 and 12:45 (right-handed chart), caused by the infrastructure recovery compressors; on the other hand, it cannot efficiently reject the variations of the 4 K tank pressure around 10:30, but not critical since those are due to an exceptional failure of the helium Dewar pressure regulation.

To conclude, the good experimental performance of the pre-calibrations is again a hint that the physics of the dynamic process model is correct. One could rightly argue that, as for the coupler heater regulations, the absence of any strong coupling with the other control loops and the relatively low performance expectations make those control loops tolerant to a large range of gains. The main process control loop, on the other hand, is far more demanding, as seen in the next section.

Main process loop in 2 K mode

The QM 2 K operation consists in a series of 2 helium transfers: from the helium Dewar to the 4 K tank, and from the 4 K tank to the 2 K tank. The 2 K mode is used to illustrate the main process loop pre-calibration, since both the level and pressure are regulated (10 cm and 30 mbar respective targets), by contrast with the 4 K mode where the pressure is floating.

Upstream in the 4 K tank, both the level and pressure are regulated too (typically at 35 cm and 1080 mbar), to ensure enough buffer for the Dewar rotation as well as enough pressure difference to run the 10 K loop. Finally, upstream, the helium Dewar pressurization is controlled, by an ON/OFF heater at 1180 mbar +/-5 mbar. Leaving aside this latter controller, which is robust and does not require any care, the 2 K operation thus needs 4 controllers, 2 pressures and 2 levels, to work in parallel.

The complete Simscape model presented in 0 was used to perform trials and fails, ahead of test phase 1; one major problem, as highlighted in details in [START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF], is the interactions between the level and the pressure regulations: indeed the pressure-regulating valve also influences the level and the other way round too (the physics are coupled). The adopted strategy for the MINERVA prototype, is a time-based decoupling of the regulations: the pressure controllers were pre-calibrated with an aggressive Ti, in the order of the second, while the level controllers had Ti in the order of the minute. Again, the controller gains were copied into the PLC STEP7 code; a first commissioning step took place, in 4 K mode (thus without the 2 K tank pressure controller), shown in Figure 5-38 from test phase 1.

Figure 5-38: test phase 1 (4 K mode) experimental commissioning and fine-tuning of the 4 K tank pressure controller, together with the 4 K and 2 K tank level controls active

To be conservative against overshoots, the 4 K tank pressure controller had been precalibrated quite slow in Simscape, in the order of 30 second, still 4 times quicker than the 4 K tank level controller. At the first trial, from 10:15 to 11:15, the 2 K tank level controller turned out to interact with the 4 K tank pressure controller, causing major oscillations. The 4 K tank pressure controller Ti was reduced to 8 second from 11:15 on, which drastically cut the oscillations, as expected. This need to further reduce the integrator time was attributed to the unexpected 4 K tank external transfer line heat loads from the helium Dewar, causing much more vapor mass flowrate versus the model. In the same chart, one can also see that the 2 K tank level is still oscillating: this is caused by the level controller pre-calibrations being optimal at 10 cm, whereas, over test phase 1, the level had to be regulated at 17 cm instead, where the level reading is much more dynamic since the buffer tank section is narrow. Lastly, one can also see that the 4 K tank pressure controller valve (CV06 in test phase 1, CV07 in the following phases) opening signal is noisy; this is caused by the valve regulating too close to its fully closed position, as a consequence of a valve Kv sizing error, which was actually predicted by the Simscape model.

The complete commissioning of the protoQCELL controllers (including the 2 K tank pressure control) occurred in test phase 2.2, since, during the previous phases, the 2 K tank pressure was regulated by a commercial "black box" controller, with mitigate performance. As for the 4 K tank, the 2 K tank pressure and level controller were copied into the PLC STEP7 code; the resulting behaviour is shown in Figure 5-39, a sequence in 2 K mode; during all the sequence, the 4 K tank regulations are not working in nominal (they are put in a specific model in view of a mobile Dewar change), but this does not affect the 2 K tank.

Between 17:00 and 17:30, large 2 K tank pressure oscillations were observed, characteristic of a lack of gain aggressiveness: from 17:30 on, P is doubled, and Ti decreased by two; the oscillations fully disappear.

(K mode) experimental commissioning and fine-tuning of the K tank pressure controller

As a final illustration, a sequence in 2 K mode (again, from test phase 2.2) is shown in Figure 5-40, where the four pressure and level regulations are active and stable; what is especially satisfying, is that the whole QCELL converges on its own, within about 1.5 hour from the end of the 2 K pump-down sequence. Considering the coupler heaters, the 10 K and 80 K loops, at this moment, a total of nine PI controllers are in stable operation. The experimental robustness to dynamic heat loads of the four pressure and level controllers could however not be checked, as the cavities were not conditioned yet. Hence, the achieved RF field was too low to induce any visible heat: at 15:00 in Figure 5-40, the field is risen to 2 MV/m in BSC03, indeed too low to be noticed.

Conclusion

In this section, the dynamic process model was used to produce pre-calibrations for all the prototype QCELL controllers, ahead of the tests. The experimental return of experience highlighted that the physics of the model, especially the dynamics, was sufficient so that only little fine-tuning was needed; the need for fine-tuning was furthermore consistent with the mismatches observed in the physical models, in part 5.2. The final calibrations are shown in Table 5-6.

Table 5-6: the prototype QCELL PI calibrations after experimental fine-tuning

Beyond the second confirmation of the process model physical accuracy, and the role of these pre-calibrations in the prototype experimental time optimization, this step was also a real-scale demonstration, in view of the MINERVA series site acceptance tests, of one of the possible controller virtual commissioning methods. A further method, based on the direct virtual commissioning of the PLC code, without the Simulink step, is presented in 6.2.2.

Still in view of the MINERVA series test, those calibrations are an important baseline to save time during the commissioning of the test stand. Those calibrations were crossedchecked against the FREIA facility, where the ESS Spoke cryomodule were tested in the 2021-2022 period, and against the ESS Test Stand 2, where the elliptical cryomodule were tested in parallel; they actually match remarkably. In particular, the same time-based decoupling of the level and pressure regulations was chosen. This decoupling can nevertheless be undertaken differently, as in [START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF], where an LQ regulator was developed to control the SPIRAL2 cryomodule tank pressure and level, which works exclusively in 4 K mode. Although the PI controllers proved to be enough for the MINERVA prototype tests, it was estimated that developing an LQ for MINERVA could enable to:

• Evaluate if the LQ is also superior to the double PI controllers, in the context of MINERVA, a 2 K mode process • Demonstrate that the detailed cryogenic process model can seamlessly be used to synthetize an LQ The schematic of the pressure and level regulation of the 2 K tank, based on 2 independent PI is presented in Figure 5-41 (left): each controller monitors one output and act solely on one valve, without considering the mutual disturbances; they are blind to the process as a whole. On the other hand, the LQ (right-handed side) controls both output and both valves simultaneously; its algorithm, based on the physics of the model (linearized), can counteract the mutual couplings, and thus provide better disturbance rejection [START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF]. ( 5-11 )

Equation ( 5-10 ) is the state equation and tells how the state variables (or states) evolve according to the states themselves and to the inputs, where x is the state vector and u the input vector. The state vector is the minimum set of independent variables that can fully describe the system, just like a coordinate system.

Equation [START_REF]Valley of stability[END_REF][START_REF]Laradioactivite[END_REF][START_REF] Joliot-Curie | [END_REF](8)[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF][START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF][START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF] is the output equation and describes the connection between the outputs y and the states x and the inputs u; the outputs are indeed not necessarily state variables, but most of the time, linear combination of them.

This formalism can represent whichever system of differential equation of any order -for order 2, one just substitutes 𝑧̇= 𝑥̈ in ( 5-10 ) -provided it is linear. The A, B, C and D matrices are the parameters that contain the model physics. A, the state matrix is especially important, as it tells the relation between the states and their derivatives.

The power of this formalism is that, on the one hand, it fits MATLAB standards for solving linear systems and even for any continuous non-linear system. On the other hand, and not the least, this formalism is made to facilitate the synthetization of advanced control laws, like an LQ regulator, a Model Predictive Controller (MPC) or any state observer. In particular, all the tools from the MATLAB Control System Toolbox are based on it, taking the A, B, C and D matrices as direct arguments in the functions.

Model linearization and comparison to the non-linear process model

As said, the synthesis of an LQ controller requires a linear representation of the process model, stored in A, B, C and D matrices. The Simscape process model being highly nonlinear, a linear approximation must be performed, around the target operating point, i.e. a 2 K mode steady-state point. The "Linearize" function from the MATLAB Control System Toolbox was used for that matter: for simplification, the input vector u was limited to CV05 and the MKS valve openings, and the output vector y, to the 2 K tank level and pressure.

The resulting A, B and C raw matrices contained the dependency of the u and y vectors to all the ~5000 model states of the complete process model. A second simplification could be done, by identifying the negligible state dependencies, with the "balreal" function from MATLAB Control System toolbox, and consequently reducing the model order using the "modred" function. This model reduction is not mandatory but alleviates all the following calculations; it removes the unnecessary physics (the 80 K and 10 K loops) to focus only on the main process loop around the 2 K tank. In the end, only ~100 model states are necessary so that the linearized model is consistent with the original process model.

Figure 5-46: a controller based on an LQ combined with a state observer

As for the previously described K gain, the MATLAB LQR command was used to find the optimal observer gain: 𝐿 = 𝑙𝑞𝑟(𝐴 𝑇 , 𝐶 𝑇 , 𝑄, 𝑅) 𝑇 .

The performance in disturbance rejection of this LQ and observer was then evaluated versus the double PI strategy (with the final calibrations presented in the previous section), on the cryogenic process model. For this purpose, a new Simulink control box was developed, implementing the LQ and the observer, specifically to control the process 2 K tank pressure and level instead of the double PI controllers.

The use-case is as follow: the process model is first converged towards the steady-state, in 2 K mode, at 10 cm and 30 mbar respective 2 K tank level and pressure; then, 2 consecutive 2 K tank heater steps are imposed. The comparison is shown in Figure 5-47.

At first sight, the LQ seems way better than the double PI to reject the disturbances from the heater. It has to be noted that this performance was obtained with a very small R matrix, thus allowing a high actuator effort. The double PI are nevertheless very good too, with limited output errors and about 1 hour to fully converge back, consistent with the observations during the real prototype tests. A pure performance-based comparison between both methods is actually difficult, since strongly dependant on their respective degree of optimization. The evaluation has to be done on a larger scope, as detailed in conclusion of this part. One of the positive conclusions so far, is the successful use of the cryogenic process model to synthetize a model-based controller, and to serve, again, as a controller test platform. In the last sub-section, the "LQ + observer" Simulink control box is used to control, this time, the real prototype.

LQ test on the prototype

The test of the "LQ + observer" Simulink control box was performed during test phase 2.2. The control architecture is presented in Figure 5-48: the Simulink control box runs on a desktop computer, and drives the real prototype cryomodule 2 K tank pressure and level, using the OPC-UA protocol to read and write in the PLC CPU, as introduced in 0. This architecture, although not safe due to the risk of the desktop computer crash, is suited for supervised tests and has the advantage to preserve the PLC code from draft modifications. The reaction of the prototype cryomodule to the LQ + observer Simulink control box is displayed in Figure 5-49: after a steady-state phase using the double PI controllers, the Simulink control box is activated at 18:24; the Q and R weighting matrices are the same as presented in the previous section.The charts are self-explanatory: the LQ did not perform as expected; although not entirely understood, this is likely caused by the too aggressive Q and R matrices, as one could have foreseen from the previous section, combined with the heavy measured signal noise. Unfortunately, the very limited test time allocated to this work did not permit to test other Q and R parameters. The positive result from this test is nevertheless the successful demonstration of the flexible control model architecture, that can drive the real process or the process model interchangeably and without modification.

Double PI LQ

Conclusions

As seen, judging whether the LQ performs better than the PI controllers in the MINERVA context is not straightforward, due to the need to equally optimize both; a fair evaluation would furthermore need to be done in series conditions. A more relevant evaluation should thus encompass a broader scope: which method has the best compromise in performance, robustness to hardware dispersion, fine-tuning and deployment effort on a series of 30 QMs? This is a long-term question, which the process model can clearly contribute to answer.

In perspective too, the process model can help overcome the weakness of the LQ, which is being narrow-band, i.e. performing well only close to the design operating point. Studies are currently underway, for instance, in the frame of the French collaborative effort M4CAST, or at CERN, to improve the state observer bandwidth by using, among others, neural networks trained by physical models.

To go further on this topic of extending the cryogenic controller robust range, and thus avoiding the use of heaters for the reasons mentioned in 1.4.4, some techniques from the engine industry could be adapted; indeed, car engines operate in extremely wide operating range and ambient conditions; modelling, furthermore, has been integrated in the control system development and validation since the early 2000's. One simple but efficient idea for instance, is to discretize the cryomodule cryogenic operating range, and allocate specific PI controller calibrations (pre-commands and gains) to each zone; arrays of pre-calibrations could be seamlessly created by the process model, as demonstrated.

As a side note, on top of being helpful for the MINERVA prototype phase, these controloriented studies were a demonstration of the possible uses of the process and control models, in order to synthetize and test control laws; other roles are further developed in part 6.2.

RF PROCESS MODEL

The last section of the model versus prototype test comparison covers the RF aspects developed in part 4, that could be tested over test phases 2.2 and 2.3. The first 2 sections deal with RF cavity-oriented topics: the RF field model is reviewed for both cavities, in tuned and detuned conditions; then, the cavity detuning sensitivities as well as the cavity 𝑄 0 are cross-checked against the measurements in vertical cryostat. The last section tackles with the RF coupler heat loads, at room temperature and cryogenic conditions.

Cavity RF field validation

RF field in tuned condition

In part 4.3.4, the cavity field model S21 parameter was showed to perfectly fit the VNAmeasured real BSC04 S21, at 2 K; the model was equally good on BSC03, which has a slightly lower bandwidth, since higher 𝑄 𝑖 . The goal of this section is to check that the model works against another measurement: the one from the powermeter that measures 𝑃 𝑡 , and from which 𝐸 𝑎𝑐𝑐 is derived following ( 4-31 ).

The reproduced RF sequence takes place over test phase 2.2, during the very first field rise of BSC03 (in 2 K mode) with one of the 20-kW MINERVA amplifier; indeed, for safety reasons, the cold conditioning and initial field rises were performed using a well-know 500 W amplifier, instead of the newly commissioned amplifiers, which limits the field at about 2 MV/m. At this period, the resonance frequency regulation loop with the CTS motor was not ready yet; instead, an SEL was used to excite the cavity at its resonance frequency. The reproduction of the RF sequence is shown in Figure 5-50: the RF field model is parametrized with BSC03 coupling factors and set at the resonance frequency; the reproduction is done at iso 𝑃 𝑖 , and shows both 𝑃 𝑡 and 𝐸 𝑎𝑐𝑐 that almost reach the nominal cavity field of 7 MV/m. The reproduction of 𝑃 𝑡 and 𝐸 𝑎𝑐𝑐 is correct +/-5 %, in line with the precision of the

Abnormal measured P t values

coupling factor 𝑄 𝑖 measurement. This result is satisfactory, since it confirms that the physics is correctly captured, but not surprising, given the return of experience on this model approach, and the few parameters needed. On the other way around, this confirms that the measurements are correct versus this reference model; as a matter of fact, the 𝑃 𝑡 powermeter happens to sporadically display abnormally low values, as it occurs during this sequence (highlighted in red, in the centre chart). This introduces one of the potential roles of the model, to serve as a digital twin also for on-line diagnosis, i.e., with the model integrated for instance in the HMI. The next section then reviews the RF field model response, off the resonance frequency.

RF field in all detuned conditions

In the previous section, the transmitted power 𝑃 𝑡 was always at its maximum, thanks to the excitation frequency being at the cavity resonance one; furthermore, the angles 𝜙 𝑖 , 𝜙 𝑟 and 𝜙 𝑡 were all equal since the detuning angle 𝜓 is 0. A specific test was performed during test phase 2.3, on BSC03 in 2 K mode to further challenge the RF field model against the measurements.

In this test, the cavity is excited at the constant target frequency 𝑓 of 352.21 MHz, and the cavity resonance frequency 𝑓 0 is manually driven close to it, thanks to the CTS motor.

The detuning ∆𝑓 = 𝑓 0 -𝑓 is then slowly swept from 0 to -2000 Hz, and from -2000 to +2000 Hz. Meanwhile, the input power 𝑃 𝑖 is maintained constant, at about 400 W, to rise enough field for a good signal-to-noise ratio, while mitigating the Lorentz force detuning; the resulting 𝑃 𝑡 and 𝐸 𝑎𝑐𝑐 are exposed in Figure 5-51. Then, the measured relative changes in the 𝜙 𝑖 , 𝜙 𝑟 and 𝜙 𝑡 angles during the same test are compared to the model, as seen in Figure 5-52; by convention here, 𝜙 𝑖 is set as the reference to 0, consistent with the fact that the cavity is driven in open-loop. What strikes first when comparing the model to the experiment, is the inversion in the angle evolutions: this trouble was not fully solved back then, but was suspected to originate from a wrong convention in the LLRF I/Q modulation step. The second point is the non-symmetric experimental phase evolutions around 0, responsible for the phase offset in 𝜙 𝑟 at 18:37, but only due to the starting point not being exactly at resonance frequency. Beyond that, the evolutions of the phases are fully compliant.

On top of another confirmation of the RF field model relevance, this comparison also helps again analysing the MINERVA prototype LLRF acquisitions. A hidden good point also, is the indirect cross-check of the CTS motor detuning model; this simple model is based in particular on the sensitivity from Figure 4-25 measured in vertical cryostat, and enables to know the imposed detuning in Hz, as a function of the motor rotations. The observed global accurate tendency shows that this model is indeed correct.

Cross-checks against the vertical cryostat tests

In the previous RF model-versus-test comparison, the influence of the detuning effects was bypassed, in order to challenge the main cavity field equation; it was the same for the cavity quality factor 𝑄 0 . In this section, all those 3 inherent cavity parameters are verified in cryomodule conditions against the reference values obtained in cavity vertical test cryostat CV800 (Annex 7.5). The goal is to quantify, as much as possible, the transfer functions between the vertical test cryostat and the cryomodule.

Static Lorentz force and tank pressure sensitivities

The influence of the static Lorentz force detuning had been measured on each cavity in CV800. Since the supporting system is different, the sensitivity in cryomodule condition is expected to be slightly different too.

The Lorentz force detuning sensitivity 𝐾 𝐿 was measured in cryomodule condition on BSC03, over the same sequence previously shown for the cavity field validation, during test phase 2.2, in 2 K mode. The resonance frequency was free, and tracked by the SEL; it is thus possible to see its evolution, as the field increases. The BSC03 frequency shift is compared to the models in ( 4-43 ), with 𝐾 𝐿 fitted on BSC03 and BSC04 measured in CV800. The resulting BSC03 sensitivity in cryomodule is about -7.5 Hz/(MV/m) 2 , 25 % less than in CV800. Although not completely understood, this reasonable difference was expected, as said, due to the different supporting system, and also due to the difference in the CTS mechanical effort: in CV800, the CTS is relaxed, whereas it was pulling on the cavity flange during this cryomodule test. In subsequent test phases, the BSC04 Lorentz force detuning sensitivity shall also be measured, which was hindered in test phases 2.2 and 2.3 by strong field emission at low field.

The helium bath pressure detuning, expressed in equation ( 4-44 ) by the sensitivity 𝐾 𝑃 was also measured on each cavity in CV800. The cross-check against the cryomodule conditions was performed also during test phase 2.2 and on BSC03. The pressure sensitivity is known by experience to be constant over the whole QM helium operating pressure range, so that 2 points are enough. To minimize the error, the sensitivity was derived from a measurement at ambient pressure (QM 4 K mode) versus one at 30 mbar (QM 2 K mode); moreover, both measurements were repeated 3-4 times at different occasions over test phase 2.2, always with the CTS relaxed. The mean sensitivity 𝐾 𝑃 value is 4.5 Hz/mbar +/-5 %, consistent with the measurements in CV800. Due to the little allocated time, the bath pressure detuning sensitivity could not be reliably checked on BSC04.

Overall, those 2 sensitivity measurements, although partial, are consistent with the data from the CV800 cavity characterization; differences do exist, but small enough not to suspect any flaw in the MINERVA prototype supporting system or during the integration steps. Further tests shall be carried out, to enhance the picture in cryomodule condition; in particular, the influence of the CTS motor position, and the dynamic detuning transfer functions. 3D modelling investigation shall be performed too, to check the consistency of the differences in 𝐾 𝐿 and 𝐾 𝑃 in vertical cryostat versus cryomodule.

Cavity quality factor Q0

As highlighted in part 4.2.2, in highly overcoupled conditions as in the MINERVA cryomodule, the RF power balance from [START_REF] Edf | Les déchets radioactifs[END_REF][START_REF]Valley of stability[END_REF][START_REF]Laradioactivite[END_REF][START_REF] Joliot-Curie | [END_REF](8)[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF][START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF][START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF][START_REF] Hargraves | Liquid Fluoride Thorium Reactors[END_REF][START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF][START_REF]Centre d'informations et de ressources sur CIGEO[END_REF][START_REF]Réacteur nucléaire naturel d'Oklo[END_REF][START_REF] Poncelet | Myrrha: un (autre) regard sur l'énergie nucléaire[END_REF][START_REF]Accelerator-driven Systems (ADS) and Fast Reactors (FR) in Advanced Nuclear Fuel Cycles: A Comparative Study[END_REF][START_REF] Junghans | Transmutation of high-level radioactive waste -Perspectives[END_REF][START_REF]Bilan du débat public, Projet de centre de stockage réversible profond de déchets radioactifs en Meuse / Haute-Marne (Cigéo)[END_REF] does not give enough precision on the dynamic heat load 𝑃 𝑐𝑎𝑣 . The dynamic heat load shall thus be derived from cryogenic measurements. The tank latent heat method from part 5.2.1.1 gives the static and dynamic 2 K tank heat loads, blended; a method is hence needed to separate both. A simple offset is not enough, since the static heat loads and the vapor production by flash are not constant as a function of the 2 K loop mass flowrate. A calibration step is thus required, by a heater, as detailed in [START_REF] Longuevergne | Etude et test d'un module accélérateur supraconducteur pour le projet Spiral2[END_REF]. As a matter of fact, another global heat load measurement technique, overlooked so far, can be used: the pressure rise rate, which consists in closing all input and output valves and observing the pressure rise; this method must also be calibrated against a heater. The associated process stops and starts and slow tank refilling and stabilization time however make the first method more suited to time-limited tests. This is the standard method to derive 𝑃 𝑐𝑎𝑣 .

𝑃 𝑐𝑎𝑣 was measured on BSC03 during test phase 2.3, in 2 K mode. A crucial prerequisite is to complete the cavity cold conditioning step, so that 𝑃 𝑐𝑎𝑣 only reflects the RF losses in the walls, and not the multipacting or transient degradation effects. To derive 𝑄 0 from 𝑃 𝑐𝑎𝑣 , the relation ( 4-18 ) is used; this requires the knowledge of the accelerating gradient. The BSC03 𝑄 0 curve in cryomodule is shown in Figure 5-54, against the test in CV800. The errors bars are caused by the poor resolution and the sporadic fluctuations of the 2 K tank exhaust mass flowrate, that make it impossible to reliably measure 𝑄 0 at low field.

The chart quantifies what was obvious during the BSC03 tests: even after a substantial cold conditioning, the cavity exhibits a very low 𝑄 0 and very high X-ray emissions, at about half of the field it could reach in CV800. This is the signature of field emission caused by pollutions, originating either from the integration step, or more likely during the warm and cold conditionings. Indeed, BSC04 has an even worse behaviour, with a maximum achievable field hardly above 2 MV/m: this cavity had been hit on its coupler flange, during the first coupler assembly attempt in clean room; the coupler antenna had been damaged and swapped. One plausible scenario is that this cavity was heavily polluted and contaminated BSC03 during the conditioning steps. As a consequence, unless properly cleaned and conditioned again, none of the prototype cavities will be able to reach the fault-tolerance field of 9.1 MV/m. Hence the role of the detailed model to complement those tests.

Beyond the status on the MINERVA prototype cavities, this section also pinpoints the inherent limits of the standard cryogenic method to precisely measure very high 𝑄 0 , and calls for the need to improve it in view of the series cryomodule SAT. One perspective resides in the method already mentioned in 4.2.2.1 from [START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF], suited for the accelerator cryogenic distribution system, where individual cryomodule mass flowrates are not possible to acquire. This method relies on a special state observer, a Kalman filter, that combines the measurement of the valve positions, the tank pressure and level to reconstruct 𝑃 𝑐𝑎𝑣 , following the same principle as detailed in 5.3.2. Although also prone to strong errors, this pure model-based method enriched with the mass flowrate information has the potential to increase the precision at low field by one order of magnitude.

RF coupler dynamic heat load validation

In this section, the heat loads predicted by the coupler field model and diffused by the coupler wall model are reviewed against the test phase 2.2 experiments. Over this phase, both couplers were first conditioned at room temperature in the cryomodule, at highpower and off-resonance, the object of the first sub-section. A consistency check of the observed temperature rises is performed against the results from the room-temperature conditioning test bench.

The couplers were then conditioned at cryogenic temperature, still off-resonance and at high-power and finally, they were operated at the cavity resonance frequency; this is the topic of the second sub-section. Since the coupler dynamic heat load peak position varies with the cavity tuning and coupling configuration (over or undercoupled), the main purpose is to check if the tendencies are respected.

Heat loads at room temperature

Although both couplers had been conditioned on a dedicated bench, as seen in 4.7.1.2, the assembly step with the cavities called for a new, normally shorter, coupler conditioning phase, first at room temperature. The coupler warm conditioning in cryomodule spread over 2-3 days for each coupler; during this phase, the couplers have only one active cooling loop: the antenna water channel. Indeed, unlike on the warm conditioning test bench where water runs in the cuff cryogenic intercepts too, although not impossible, running water in the cryomodule cryogenic lines was deemed troublesome.

The developed coupler Simscape model is fitted in the same ambient boundary conditions and same inlet power 𝑃 𝑖 as the experiments; compared to the model presented in 4.7, only one modification was done: the niobium cavity flange, not in superconducting state, hence exhibits heat loads predicted by the coupler field model and its electrical resistance, just like in the rest of the cuff. Coupler CPL2 from BSC04 was the first to be conditioned, reaching up to 14 kW (CW) inlet power 𝑃 𝑖 without a hitch. Its measured temperature elevations during the final validation 𝑃 𝑖 power rises are displayed in The Simscape reproduction agrees with the test, both in amplitude and in the sensor hierarchy. The field configuration here is the undercoupled detuned one, the same as in case C in Figure 4-47: all the power is reflected and a standing-wave hence forms; the H field is maximum in the middle of the cuff, hence, the Joule heat loads too, consistently with the temperature profiles in the test and simulation. A deviation is seen in TT234, but not relevant to consider since this sensor actually fell off its support during the cryomodule assembly.

A relevant deviation by contrast, lies in the window temperature TT243: a parasitic heat load is at work in the test; the weak electron-pick-up signal indicates it is not multipacting, as expected, since the ceramic window of this coupler is coated with a titanium nitride layer. Instead, this heat load is thought to emanate from the TiN coating itself, which by construction, is a short-circuit, although very small, between the inner and outer conductors; it is further commented in conclusion of the RF section. This effect is boosted by the E field being maximum at the location of the window, in this field configuration. This calls for a first model adjustment (not applied in any of the comparisons), following the same approach as the cryogenic models.

As an illustration of the importance to consider the right field configuration, the coupler is intentionally put in a fictive overcoupled tuned cavity configuration, corresponding to case B in Figure 4-47; the H field is this time minimum in the middle of the cuff, giving rise to much lower temperature increases and totally mismatching temperature order versus the test, as seen in Figure 5-56.

Finally, the water antenna heat loads could be measured with a pretty low precision of +/-5 W, but enough for a basic consistency check. The comparison versus Simscape is shown in Figure 5-57. While the tendencies are correct, there is a 2-3 ratio in the measured heat loads versus the simulated ones. Although not fully understood, this difference is thought to originate on the one hand, from the missing Joule heat loads in the RF coaxial line internal conductor, upstream of the coupler, that diffuses by conduction towards the antenna water loop. On the other hand, on the Joule heat load on the antenna tip, also not considered in the model. One good point, as seen further on, is that this 2-3 ratio is constant whatever the test bench and test conditions. Here, a second model adjustment is necessary (not applied either in the models for the comparisons, in these sections).

The same analysis was performed on CPL4 (BSC03), conditioned one week later, also during test phase 2.2. The conditioning was almost completed, but persistent remnant multipacting, clearly identified by the electron pick-up and expected, as this ceramic window was not TiN-coated, was encountered in the vicinity of the ceramic window, where the E field is maximum. It impeded to rise 𝑃 𝑖 over 5 kW, upon risk of window damage. A Simscape reproduction of the temperature profile at iso 𝑃 𝑖 and ambient conditions is proposed in Figure 5-58. In this reproduction, a multipacting heat load model, function of the square of the local E field, was applied to the ceramic window Simscape RF blocks, and calibrated by a multiplier coefficient to match the average observed window temperature elevation (rather than the unique large peak). In short, the proposed multipacting model is in the form:

𝐻𝑒𝑎𝑡 𝑀𝑢𝑙𝑡𝑖𝑝𝑎𝑐𝑡𝑖𝑛𝑔 = 𝛼 * 𝐸 2 ( 5-13 )
Leaving aside the poor temperature initializations at the start of the reproduction, the resulting evolutions of the other temperatures agree very well, confirming the assumption of a local multipacting spot. This multipacting model is a third necessary adjustment (always applied on this coupler in the following comparisons); the dependency on the E field is consistent with the window-type multipacting physics, but the square exponent, that best matches the tests, should be the object of further theoretical investigations 59 . In the following comparisons, this model will be challenged against different coupler field configurations.

To illustrate the major contribution of the multipacting heat loads, in Figure 5-59, the same reproduction is displayed, but without multipacting model, resulting in totally wrong tendencies.

As for CPL2 on BSC04, the heat loads to the antenna water are compared to the Simscape reproduction, in Figure 5-60. The mentioned 2-3 ratio (adjustment #2) is still here, although lower in this case, likely due to the fact that the major contribution is the window multipacting, not the Joule heat loads, hence confirming with another independent measurement, the order of magnitude of the multipacting heat load model. 59 Discussed during the recurrent meetings with LPSC multipacting and IJClab RF experts As seen, the coupler wall and field models can faithfully reproduce the observed tendencies and orders of magnitude during the conditioning in cryomodule; 3 model adjustments are necessary however, to complement the physical models. The next section comes back on the conditioning test bench results, to cross-check these facts.

Consistency versus the conditioning test bench

The main goal is to see whether the observed deviations (the 3 adjustments) were caused by the assembly in cryomodule, or by inherent flaws on the couplers; in parallel, to challenge the coupler models in this other field configuration, a pure TW regime, corresponding to a perfectly matched case, close to case A in Figure 4-47.

All the 6 prototype couplers were conditioned, at least once, on the dedicated conditioning test bench at IJClab, as seen in Table 4-1. The full temperature instrumentation of this bench is detailed in 4.7.1.2 and zoomed on the coupler in Annex 7.10; overall, it is very close to the cryomodule one, especially the window PT-100 which is the same sensor, deliberately put at the exact same location. Compared to the cryomodule, a few boundary condition updates were performed, fitted to the conditioning test bench:

• Water is running in the cuff cryogenic intercepts • Convection with air takes place around the cuff exterior side • The measured conditioning cavity temperature (heating-up much) is imposed at the coupler tip First, CPL2 (BSC04) is reviewed, over a sequence occurring at the end of its conditioning phase, hence with all multipacting spots conditioned. The comparison against Simscape is shown in Figure 5-61. Over this sequence, the field is risen up to 35 kW (in TW); the oscillations observed in the input power P i chart are glitches due to the measurement system. The temperature reproduction shows some deviation, essentially due to the temperature initializations, and again, to the window that heats up more, likely because of the TiN layer (adjustment #1), consistently with the observations in cryomodule conditioning.

Overall, both the model and the test tell that there is no significant temperature rise. To complement, Figure 5-62 shows the water heat loads in the antenna, and in both intercepts; on the latter, the precision is as low as the measured value, +/-2 W, thus should only be considered as indications. What appears here, again, is a factor 3 between the experiment and the model, consistent with model adjustment #2 in warm cryomodule condition. Looking only at the experimental data, it is obvious that this CPL4 coupler does not behave as CPL2, but, just like in the cryomodule, exhibits large heat loads near the ceramic window. However, the multipacting model, calibrated to match the warm cryomodule tests, shows 2 discrepancies versus the experiment here:

• The multiplier coefficient is about 1.5 times too strong • There is a threshold effect at about 15-20 kW 𝑃 𝑖 These observations are not surprising, given the multipacting model simplicity with respect to the phenomenon real complexity, that is likely sensitive to the shape of the RF wave, TW versus SW. They can be complemented with the water heat load comparison, in This time, the model matches better, but likely due to 2 compensating errors, i.e., the same previously seen Joule heat load underprediction, and a too strong simulated multipacting power. The threshold effect is clearly visible on the HT intercept water heat loads, consistent with its close position to the window.

Overall, by confirming the need to consider multipacting (adjustment #3) at the location of the window, in roughly the same order of power magnitude, this comparison strengthens the consistency between the cryomodule in warm conditions, and the warm conditioning test bench, proving anyway that the cryomodule assembly step is not responsible for the CPL4 abnormal behaviour. The same goes for adjustment #2 (the underestimated water heat loads) and #1 (the TiN layer parasitic heating on CPL2), the latter being even clearer in the next part, that shows the coupler model comparison in the last field configuration: the tuned overcoupled case.

Heat loads in cryogenic operation

The overcoupled configuration is achieved when the cavities are cooled at 4 K or 2 K. First, the detuned overcoupled case is cross-checked against the detuned undercoupled case from the cryomodule warm conditioning, both having the same coupler field shape.

As in the previous 2 parts, CPL2 from BSC04 is first compared to the model, in a 4 K mode sequence, where the coupler is submitted to a 0 to 14 kW 𝑃 𝑖 rise, while the cavity is offresonance; this phase is the coupler cold conditioning. The thermal boundary conditions are much different than in the cryomodule warm conditioning:

• The cavity is at 4. By contrast, in the latter comparison, the model predicted almost no temperature rise at the coupler-cavity interface, but the tests showed clear temperature increases, proportional to the 𝑃 𝑖 power. This is yet another parasitic dynamic heat load, which origin is yet not clearly identified, but for sure, is located at the coupler-cavity interface gap, and is consistent on both couplers, as seen later; this is adjustment #4. The associated heat load minor compared to the coupler static heat loads, as it was neither spotted on the 10 K or on the 2 K process loops. On the other hand, it may also explain the model underprediction of TT230 and TT231 in Figure 5-55 from the warm cryomodule conditioning phase. This is further commented in conclusion of the RF section. Unfortunately, as seen in 5.4.2.2, BSC04 exhibited tremendous field emission at low field, impeding the rise of P i above ~500 W in tuned condition. The tuned-versus-detuned comparison was thus performed on CPL4 from BSC03 solely. In a 2-K mode sequence, 2 identical P i steps are performed up to 5 kW, while the cavity is tuned (the frequency is tracked by the SEL). A few minutes later, the same P i rise is performed, this time, as the cavity is detuned. In Simscape, the exact same multipacting model (adjustment #3) as before is again implemented on the ceramic window. The result of the comparison is shown in Figure 5-68, looking first at the window temperature. When the cavity is tuned, the coupler field becomes similar to case B in Figure 4-47: the E field is minimum at the window, the opposite of the detuned case, which theoretically would cause the E field-induced multipacting to vanish; this is precisely what is observed in tuned conditions. In detuned conditions, the multipacting model works as fine as in warm cryomodule conditioning, with only a minor overprediction. This is a new direct evidence that the coupler field model, although much simple, is good in tendency.

To complement the analysis, again, the antenna water heat loads are examined against Simscape, in Figure 5-69. Here too, the multipacting model (adjustment #3) seems a little too strong in detuned conditions. The overall tendency between tuned and detuned conditions is globally respected. To conclude, although the cavities and couplers have flaws that limit the available 𝑃 𝑖 range, it was possible to demonstrate that the coupler field and thermal models, complemented with 4 adjustments, perform well in all conditions, at least in tendency and orders of magnitude. In return, the identified 4 model adjustments enable to spot and quantify specific unexpected deviations of the real parts, to further look after during the next test phases, and to further investigate based on the already available data on all the couplers.

Conclusion

As for the cryogenic models, the comparisons showed that RF physical models, even based on component as-build properties and reference dedicated tests were not sufficient: several adjustments are needed to account for all the missing and a-priori unknow effects, like the coupler parasitic heat loads or the uncontrolled cavity 𝑄 0 degradations during the critical cryomodule integration phases.

One lesson-learned from these comparisons is, again, the role of the adjustments and the importance to give them a physical meaning, in a perspective of the process model representativity gradual improvement and real part investigation in view of their optimal series design and operation. Some aspects of the coupler parasitic heat loads are for instance worth further investigating: first, adjustment #1, assumed to be due to the TiN coating Joule heating. In fact, this unexpected heating could also be due to some wrong parametrization in the dielectric heat load expression ( 4-47 ), like the loss angle or the relative permittivity, which are actually generic values, not measured on the real ceramic parts. Here, it would be worth performing additional material analysis, build a physical adjustment model, and challenge it against the measured heat loads on each coupler. The same goes for adjustment #4, the coupler-cavity interface gap: it is so far unclear what, among the RF gap, the gasket low conductivity or the supposed copper layer discontinuity causes the heat release; here too, geometrical measurements and additional observations would enable to build a Simscape physical adjustment model, to compare to the acquisitions.

Finally, adjustment #3 (the window multipacting) reflects a problem which better understanding would particularly support the MINERVA series design choices: actually, not all the coupler window that lack a TiN layer experience multipacting; deeper modelbased investigations and experimental cross-checks have the potential to help identify if a TiN layer is truly required or not. Beyond MINERVA, one particular point would be interesting to the accelerator community: whereas 3D software like SPARK3D (CST modelling suite) can accurately predict the multipacting occurrence conditions, the released heat depends on the overly complicated prediction of the total involved electron population. Although needing further reworking to better respect the multipacting physics, the model adjustment #3 does quantify the multipacting heat loads. The same adjustment calibration should be done next on the other couplers, and connected to measured material properties, as well as measurable variables like the multipacting pickup current; in view, the potential to better monitor the risks of overheating when conditioning an RF part. More generally, this approach can give information on abnormal multipacting events, as it occurred on the MINERVA coupler CPL1 (Table 4-1): the multipacting spot had the particularity to be very energetic, in the order of 200 W, and to change location as a function of the inlet power 𝑃 𝑖 , according to investigations in the frame of this thesis [START_REF] Lhomme | An Approach for Component-Level Analysis of Cryogenic Process in Superconducting LINAC Cryomodules[END_REF].

Overall, the RF model reproductions in this chapter confirmed the representativity of the coupler and cavity field models, as well as the coupler wall model in all the MINERVA RF field and thermal configurations. In the perspective of the series, it means it is possible to rely on the Simscape process model to prepare, for instance, safe conditioning procedures, and also, to support the redesign of the RF series part, as further detailed in the next and last chapter.

CONCLUSION AND PERSPECTIVES

GENERAL CONCLUSION

The MINERVA accelerator, the first phase of the MYRRHA project, is planned to produce a first beam in 2027. In order to validate the design and further upgrade the series components, a full-scale prototype of the cryomodule of the Spoke section was assembled and extensively tested during the 2021-2023 period. This phase provided a unique opportunity to create and validate a detailed and comprehensive system model of the cryomodule processes. The correct level of details and the implementation of the couplings between the RF and cryogenic processes give the model the ability to replicate and predict transients throughout the entire range of the cryomodule operating conditions, while running faster than real-time. These are the two main added-values brought by this thesis.

In a first part (Chapter 0), the physical model of the cryogenic process was setup from scratch in an existing multiphysic simulation platform which was adapted to accommodate the cryogenic material and fluid properties. A library of physical building blocks was created and a critical review of the embedded physical equations was performed. These blocks were used to form the component models of the cryomodule.

In addition to a thorough analysis of the manufactured parts, each of these component models underwent validation through experimental or numerical characterizations. The resulting process model can be controlled either by the PLC of the real prototype cryomodule or by a twin code implemented into Simulink.

In a second part (Chapter 0), the physical model of the RF process was implemented into the code. Initially, a model of the cavity RF field and of the cavity mechanical resonance, as well as their respective controls, was inherited from former studies but improved with visualization tools and updated to reflect the latest experimental characterizations. Secondly, the code includes a model of the power coupler: the thermal part was taken over and enhanced to address fluid heat transfer and thermal transients. Moreover, it was complemented by a dedicated material characterization. Finally, a new model of the coupler RF field was created and validated in all cavity tuning configurations. As it is implemented in the same platform as the cryogenic process, the RF process model can dynamically interact with it.

In a last part (Chapter 5), the developed system model was compared against the prototype cryomodule tests on a full-scale. The cryomodule was equipped with an unprecedentedly degree of instrumentation, while dedicated test protocols provided output data in several operating conditions. On the one hand, this in-depth picture of the cryomodule outputs enabled to identify parts of the system model that met the goals outlined in Chapter 2, demonstrating correct trends within +/-10 %. On the other hand, it also raised significant deviations, such as in the 2 K tank static heat loads or in the coupler dynamic heat loads. These differences are not entirely unexpected, since this is one of the first time that such detailed comparisons are carried out. To reduce those discrepancies and attempt to provide them with a physical meaning, so-called model adjustments were made; those adjustments are preliminary model patches, intended to be replaced by physical models in future studies. To accomplish this goal, these studies shall include further post-processing, new tests with extended operating conditions, hardware changes and new theoretical investigations. They would therefore provide important feedbacks for the coming MINERVA series cryomodule, as well as for the next generation of cryomodules to be designed. This feedback is all the more important to preserve as the long development cycles relative to the team turnover time can cause a loss of knowledge.

In a broader view, the comparisons and model applications discussed in Chapter 5 confirmed the role of the process model in supporting a full-scale test phase. In particular, they demonstrated that the chosen modelling is suitable for addressing various problems, and laid the foundations of methodologies. Firstly, on the control aspects, the model architecture modularity enabled to perform off-line calibration, synthetization of control laws, and on-line tests of prototype controllers. Secondly, for RF and cryogenic process diagnostic purposes, the user-friendly parametrization enabled to run models in the same conditions as the tests, while the common model/test chart plot tool seamlessly compares both. Finally, the developed modelling offers the perspective of model-based sensors to increase the precision of the measurements that currently reach their limits, or even provide some measurements which are not accessible. As a matter of fact, the model-to-test comparisons also highlighted the interest to maximize the test data, even very rough, in all possible conditions; a relevant example is the coupler conditioning test bench, initially not meant for that, which was gradually extensively instrumented; the developed system model proved to be suited to make the most of low-precision, transient tests.

As stated in the introduction, a second major role of the cryomodule process model is to help updating the series design of the MINERVA cryomodules: in fact, the anchoring of the RF and cryogenic models and the validation of the general tendencies obtained from the prototype tests allow reliable predictions to be made in order to optimise the cryomodule design; this topic is addressed in the next section.

PERSPECTIVE OF SERIES OPTIMIZATIONS FOR MINERVA

The aim of this section is to demonstrate the role of the cryomodule system model in supporting series optimization tasks for MINERVA through three practical applications. First, the experience gained from the MINERVA cryomodule prototyping campaign and the recent cryomodule design evaluations conducted by SCK•CEN necessitated the redesign of certain cryomodule components prior to the tendering phase [START_REF] Zamora | Status of the MINERVA cryomodules and associated cryogenic system (MYRRHA phase 1)[END_REF]; the role of the system model in this phase is illustrated in the first section by means of a specific example: the helium heat exchanger of the power coupler. Then, the forthcoming cryogenic series control design and validation of the MINERVA project will require the experience and input from the prototype test phase; the second section therefore shows how the cryomodule system model can be optimally used as a virtual commissioning platform for PLC control. Lastly, in a longer-term perspective, the third section exposes how the developed system model can be used to build advanced fault detection tools.

Series design optimization: the coupler heat exchanger

The prototype tests pointed out that some components did not perform optimally and needed to be redesigned. The QM chimney, as previously discussed, is one such component, while the power coupler cuff cryogenic intercept system, which is the subject of this section, is another.

The details of the two cryogenic intercepts of the coupler cuff were thoroughly studied in order to limit the heat loads to the 2 K tank, as mentioned in 4.7.1. Additionally, the concept itself of two separate intercepts was purposely matching with the prototype test conditions: saturated nitrogen was used instead of circulating supercritical helium in the high-temperature intercept. Finally, the idea of a low-temperature intercept was developed to meet the requirement for a low-temperature line that would intercept the cavity supporting table too. Nevertheless, having two distinct intercept lines with fixed temperatures (hereafter referred to as "2 loop") presented certain limitations:

• A unique continuously-variable temperature intercept would be thermodynamically more efficient • The return line of the low-temperature intercept occupies precious space in the jumper, collects additional heat loads and needs adequate interfaces • The layout of the low-temperature intercept line, cooling the table and the couplers in series, is a potential limitation when independent cooling control of these objects is required For these reasons, SCK•CEN investigated the use of a continuous heat exchanger along the coupler cuff, referred in the following as "1 loop". This concept is also the one adopted for ESS, as mentioned in 4.7.1.1, and at SNS [START_REF] Campisi | The fundamental power coupler prototype for the Spallation Neutron Source (SNS) superconducting cavities[END_REF], whereas the "2 loop" concept was preferred at SPIRAL2 [START_REF] Cabanel | SPIRAL 2 coupler preparation and RF conditioning[END_REF], XFEL [START_REF] Kostin | UPDATE ON THE EUROPEAN XFEL RF POWER INPUT COUPLER[END_REF] and FRIB [START_REF] Popielarski | Development and Testing of Prototype Fundamental Power Couplers for FRIB Half Wave Resonators[END_REF], resulting from compromises on specific project constraints. An additional incentive for the "1 loop" concept was the chance to make use of the ESS lesson-learned, resulting in the simplification of certain design elements in order to limit the risk of manufacturing defects.

As detailed in section 5.2.4, the prototype coupler Simscape model accurately reproduced the static heat loads observed during the prototype cryomodule tests, requiring only a few experimental adjustments. This positive outcome suggested that, in particular, the Simscape coupler cuff "2 loop" model could confidently be updated into the "1 loop" concept to provide predictions for the series; the layout of the planned series coupler cuff is compared to the prototype one in Figure 6-1. To meet these goals, the "1 loop" circuit geometry and length were trimmed by simple parameter sweeps. Additionally, the instrumentation and auxiliaries (flowmeter, temperature sensor, reheater) could be sized too. The final design was significantly different from the one adopted for ESS, in part since the design target were not the same, and in part because the ESS coupler cuff could not be equally optimized.

To conclude, a part of the developed cryomodule system model could be already used to optimize the actual series cuff cryogenic intercept system; this was made possible since the fidelity of the prototype cuff model was high enough in a wide range of conditions, with almost no need for experimental adjustments. This basic design optimization provides a good small-scale demonstration of the potential results from a dynamic and detailed process model associated to reference tests.

However, the potential of the system model in design optimization is not limited to individual components; for instance, a global energetic optimization between the cryoplant and the cryomodules could be performed. Such an optimization is usually done but is limited by rigid requirements on both sides, which tends to make both systems fall in a local optimum; by combining both models, it would be possible to find a better global optimum. In particular, the optimal choice of the operating conditions (mass flowrate and inlet temperature) of the cryomodule and valve box thermal shields results from a trade-off between the efficiency of the cryoplant and the heat loads to the cryogenic loops. This trade-off could be seamlessly identified with a more global model combining the cryoplant, the cryodistribution system and the cryomodules. In practice, a first necessary step in this direction would of course be to update the prototype parts of the cryomodule model into their series production version. A limitation is, of course, the presence of the experimental adjustments and the unknowns about their behaviour when a part of the design is changed; but as said earlier, this shall be gradually overcome by dedicated parallel studies. A second more challenging issue is the necessary simplification of the cryomodule models in order to combine the 30 units of the MINERVA project; this could be addressed by creating surrogate models, based for example, on neural networks.

Implementation of a PLC virtual commissioning platform

In part 5.3, the flexibility of the dynamic system modelling architecture was used to perform several control-oriented tasks for the prototype MINERVA cryomodule; in particular, light virtual validations were carried out, for the calibration of the prototype PI controllers and for a real test of an LQ control law. The term "light" refers to the fact that this approach can be much further extended: in many industrial branches, system models are fully integrated in the control development and validation cycle, through the aforementioned software-in-the-loop (SIL) and hardware-in-the-loop (HIL) methods. These techniques are so widespread and popular that PLC manufacturers like Siemens even developed their own physical modelling platform, SIMIT [188]. The purpose is the same, i.e. starting the control development without waiting for the real equipment to be installed; but, another important component can be validated: the control hardware itself.

Putting the control hardware in the virtual commissioning loop on top of the control algorithms, enables, first, to anticipate on several additional tasks:

• Starting the PLC code dedicated to the communication with the field equipment • Developing and validating the Human-Machine Interface (HMI) and definitively establishing the connections to the PLC process variables

To sum-up, the whole control work which is above the field equipment layer can be fully finalized without it. Most importantly, the potential limitations of the PLC computer chip can be considered; this is crucial when the control algorithm is especially complex, as it will probably be the case for MINERVA, given the cryogenic control challenges.

For the MINERVA prototype, a complete SIL workflow was actually implemented in the frame of this thesis, illustrated in Figure 6-2 (valid for an HIL workflow too). In details, the PLCSIM software from Siemens [189] was used to emulate a PLC, the same as the one used for the MINERVA prototype; the prototype cryomodule PLC code was flashed into the CPU of this virtual PLC. The same HMI program as the real prototype was used to display the process variables and send commands; at the other end, the system model was communicating with the virtual PLC via the OPC-UA protocol, as explained in 0.

Practically, this SIL application was developed after test phase 1, and thus, was not used to commission the cryogenic sequences; additionally, some work is still required to connect all the Simscape variables to the OPC-UA server ones. Nevertheless, the main principles were tested: a few valves of the system model could be controlled from the real prototype HMI, which in turn, could read some of the simulated temperature sensors.

The SIL workflow was preferred over the HIL one for a practical reason: it is more convenient to connect to a virtual PLC hosted by the same computer as the one running MATLAB, than to the real PLC, which implies to physically connect to its network, and which furthermore is not available when coding work is on-going. Another advantage of the SIL over the HIL workflow, is the possibility to run the PLC at different virtual time paces: in slow motion if the system model is slower than the real time (like when the RF models are running), or at an accelerated pace (when only the cryogenic process model runs). This means that in practice it is possible to skip the step of creating and maintaining a Simulink control box, and instead, always use the PLC code to control the cryogenic process model.

In a more general perspective, the SIL and HIL approaches are usually widely used for operator training. True, for the MINERVA project, this might not be relevant in the short term; by contrast, in the near future, these approaches could be used to train the newcomers in the project design and control teams, and alert them on critical points; this is yet another way to capitalize on the experience of the prototype test phase.

Towards an online cavity fault classification tool

Over the past two decades, Artificial Intelligence (AI) techniques have progressed considerably and some industries have been incorporating AI in their mass-produced goods for over 15 years. This is particularly evident in modern car engines, even in more affordable models. Due to the increasing complexity of engine systems, automotive engineers have gradually introduced more modelling in engine controls, ultimately leading to the use of AI. As an example, neural networks are used in some common 2010's cars to predict the behaviour of the intake air system, in reaction of the excessively numerous control parameters. These neural networks, which have been trained on genuine or simulated data, function as surrogate models and replicate the output of the dataset they have been trained on. Being much lighter than physical models with the same accuracy, they can run in real-time on the engine's on-board computer [190].

Furthermore, recent studies have demonstrated the application of these artificial intelligence techniques to abnormal engine working conditions, contributing to the detection and classification of faults, thereby providing car owners with more comprehensive information [START_REF] Français | Le machine learning au secours du diagnostic moteur[END_REF].

This latter topic seems to have been also broadly studied in the accelerator communities all over the world, in particular around the SRF cavity quench detection and classification algorithms. Quench is a major problem in the operation of SRF cavity, as introduced in part 4.4.2.2, and require the detection and RF mitigation to occur within a few ms. The trouble being that quench-dedicated sensors do not exist, contrary to the other cavity faults such as arching and multipacting. As previously mentioned, in cryomodule conditions, there is no direct way to measure the cavity 𝑄 0 or the power dissipated in the wall 𝑃 𝑐𝑎𝑣 . In principle, the cryogenic control system could detect the drop in 𝑄 0 , yet would not be fast enough, at least, with a classical pressure sensor and PLC acquisition card. The current quench detection techniques thus all rely on the analysis of the three available RF signals 𝑉 ̃𝑖𝑛𝑐 , 𝑉 ̃𝑟𝑒𝑓 and 𝑉 ̃𝑐𝑎𝑣 . More precisely, they aim at detecting the changes in 𝑄 𝐿 as a consequence of the strong drop in 𝑄 0 .

A classical technique, that works only for RF short-pulse machines is as follows: at each 𝑃 𝑖 pulse falling edge, the measurement of the accelerating field response time enables to derive the instantaneous cavity loaded factor 𝑄 𝐿 , as discussed in 4.3.3. Once averaged over several pulses, this method can reliably detect drops in 𝑄 𝐿 and trigger adequate RF mitigation. This relatively simple method has been used at XFEL since the commissioning of the machine. Although working fine, this method cannot detect a quench during the pulse itself. Moreover, this detection method can be fooled by other RF phenomena that affect the cavity gradient during the decay phase [START_REF] Rybaniec | Real-time Estimation of Superconducting Cavities Parameters[END_REF].

Another method is being studied, at XFEL too [START_REF] Rybaniec | Real-time Estimation of Superconducting Cavities Parameters[END_REF]: it relies on a cavity equivalent electrical model, similar to the one discussed in part 4.3, implemented in an FPGA chip onboard the LLRF for real-time calculations. From the three RF measured signals, 𝑉 ̃𝑖𝑛𝑐 , 𝑉 ̃𝑟𝑒𝑓 and 𝑉 ̃𝑐𝑎𝑣 it is possible to compute the instantaneous cavity 𝑄 𝐿 (and the detuning too) even during the pulse itself. This method would be applicable to long-pulse and even CW machines, as investigated at the Helmholtz-Zentrum Berlin (HZB) [START_REF] Echevarria | Simulation of quench detection algorithms for Helmholtz Zentrum Berlin SRF cavities[END_REF]. However, it would still suffer from noise trigger by other RF faults or normal RF transients.

A similar method is presently in use at SPIRAL2, a CW machine, and is also the one planned for MINERVA: instead of computing the instantaneous loaded quality factor 𝑄 𝐿 by an embedded cavity model, the changes in 𝑄 𝐿 are indirectly detected by monitoring the To generalize, while the detection of the 𝑄 0 sudden drops seems to be mastered, the classification of the detected fault as a real or fake quench seems to be a challenge for the accelerator community. This topic is being actively worked on worldwide, e.g. by the XFEL RF team: the detection method described above is being enriched with a classification algorithm [START_REF] Branlard | MACHINE LEARNING ASSISTED CAVITY QUENCH IDENTIFICATION AT THE EUROPEAN XFEL[END_REF]. To do so, the virtual signal produced by the embedded cavity model is compared to the measured one: leveraging on years of acquisition and data labelling, the generated residual is analysed by a classification algorithm based on machine learning. In the US, the RF team of the Continuous Electron Beam Accelerator Facility (CEBAF, Jefferson Laboratory [196]) is also developing a pattern recognition algorithm, based on machine-learning: they gathered years of labelled RF post-mortem time-series, including height sorts of faults of which three quench types, and are using them to train a classification model [START_REF] Tennant | Superconducting radio-frequency cavity fault classification using machine learning at Jefferson Laboratory[END_REF].

𝑃
For MINERVA and other future accelerators, such fault classification techniques should undoubtedly be developed; in this way, a detailed dynamic cryomodule model would be a major asset. In practice, the model could generate datasets of signal traces corresponding to different types of quench and normal RF transients under different operating conditions; these data would then be used to train a machine learning-based classification tool, just like with actually measured traces. The prerequisite for this, is the integration of the physical modelling of quenches; a simple 1D model, such as the one proposed in [START_REF] Bousson | Etude des phénomènes thermiques dans les cavités accélératrices supraconductrices en niobium rigidifiées par projection thermique[END_REF] could be seamlessly integrated into Simscape. Although probably not optimal, many concepts from this quench classification tool could be virtually validated prior to installation and commissioning; similarly, the requirements for the RF acquisition system could be anticipated.

In conclusion, a physical process model, such as the one developed in this work, paves the way for advanced control methods, as also noted in [START_REF] Vassal | SPIRAL2 Cryomodules Models: a Gateway to Process Control and Machine Learning[END_REF]. As many scientific fields have been investing for a long time, a large panel of AI-based techniques is becoming available; with a relatively limited workload, the accelerator community could possibly catch-up with these methods by trying them on challenging topics. For example, this has recently been done at GANIL with the participation of CEA/DSBT, who investigated two methods to improve the helium tank level and pressure control in the cryomodule: by Deep Neural Network inference [199] and by a machine learning-based state observer60 .

CONTROL ROBUSTNESS STUDIES FOR MYRRHA AND BEYOND

In order to have a clear view of all the applications of the developed model presented so far, it is possible to illustrate them thanks to the mind-map 61 presented in Figure 6-3. In chapter 5, the practical model-based analysis of the prototype phase and a brief demonstration of the synthesis of advanced control laws were presented; those topics can be identified as those which are tagged #1, #2 and #5 in the mind-map. The applications presented so far in this conclusion can in turn be identified as follows:

• The design updates towards the series cryomodule matches with topic #3

• The PLC virtual commissioning platform with topic #4

• The AI-based fault detection tool with topic #6

Figure 6-3: mind-map of the combined RF and cryogenic combined modelling and their applications

The mind-map shows a last important role of the system model on its left side (topic #7), overlooked so far but introduced in chapter 2.5: the ability to evaluate the robustness of the cryomodule controls over a wide range of cryogenic and RF operating conditions.

It was indeed anticipated before the prototype test campaign that such an approach would be necessary to pre-validate the controls before the commissioning of the MINERVA series cryomodules, in particular by checking the cohabitation between the cryogenic, mechanical and RF field regulators. This idea was confirmed by the experience 61 Result of the regular workshops, brainstormings and private discussions with accelerator community actors, shared with the M4CAST collaboration gained on the prototype tests and by the construction of the system model itself. Firstly, because both the RF and cryogenic processes proved to be highly nonlinear, meaning that a single fixed control law would certainly not be able to cope with the numerous operating points. Therefore, a wide-band control architecture has to be developed and validated, as already pointed out in section 5.3. Secondly, due to the fast RF transients specifically required for the fault-recovery procedures: even slow RF transients proved to be extremely destabilizing to the cryogenic process during the prototype tests. This was likely caused by prototype cavities not being properly conditioned, but this could actually happen again on series cryomodules too. Finally, the fact that the fast RF transients involved in the fault-recovery procedure could not be tested during the prototype cryomodule test period calls for a numerical validation campaign in a relatively short term.

This validation campaign should include fast RF field changes (e.g., from 0 MV/m to the nominal and maximum field) over all the RF and cryogenic operating points, i.e. across all the helium pressure and levels and across all cavity detuning and beam currents. This numerical campaign should also include representative dispersion and drift in the RF parts (e.g., in 𝑄 0 or in the multipacting level) as well as dispersion and drift in the cryogenic parts (e.g., in the valve permeability and in the heat exchanger efficiency).

The proposed system modelling is suited for this kind of studies: based on first principles and experimental adjustments, it captures the physical mutual strong couplings between the cryomodule RF, cryogenic and mechanical tuning processes and also reacts to some of the beam parameters.

Beyond MYRRHA, this type of modelling seems to be even more attractive, for instance, for the design of future project like PERLE [200], where the beam dynamics will feature large transient variations, a wide operating range, as well as involve high RF power. It would also be relevant for the design of a project like FCC, whose couplers will supply a power of several hundreds of kW in CW to the cavities [START_REF] Brunner | RF system challenges for future e+e-circular colliders[END_REF]. More generally, in the context of ever-increasing system complexity and optimization, individual accelerator sub-system optimization is no longer possible: the RF and cryogenics systems should no longer be solely viewed as beam dynamics ancillaries, but as integral parts of a global system to optimize. At the other end of the development cycle, the system modelling is relevant in view of implementing on-line digital twins via AI/ML-based surrogate models.

For these reasons, it seems clear that accelerator projects have an interest in investing early in such transverse modelling tools, to support the coming development phases as well as secure the experience.

The system modelling is standard in many industries, but not yet in the accelerator community: it probably suffers from the specificity of each machine, and from the tremendous work needed to bring low Technology-Readiness-Level devices at the right maturity level. All this likely makes the I/O standardization and the cross-disciplinary coordination between the teams not a priority. Yet, the benefits are worth the costs.

As previously mentioned, there is a cost involved in setting up such a model, as well as abandoning it, as it must be adequately maintained and continuously enhanced. Therefore, it is worth noting some good modelling practices, which are also recommended by the US accelerator modelling community [START_REF] Biedron | Snowmass21 Accelerator Modeling Community White Paper[END_REF]:

• Sustainable and reliable code maintenance The test conditions are detailed in the legend: "2K" or "4K" means 2 K or 4 K mode, "TRF" means transfer through the jumper, "L" means 2 K tank level and "10K" means 10 K loop mass flow rate. Each bar is a test measurement averaged over 5 to 15 min.

  Supraconducteur du projet MYRRHA Mots clés : Accélérateurs, Cryogénie, Radiofréquence, Cryomodule, Modélisation, Automatique

FIGURE 1 - 1 :

 11 FIGURE 1-1: THE DIFFERENT EMISSION TYPES AND ASSOCIATED PENETRATION POWERS FROM [3] .................................... FIGURE 1-2: NUCLEAR WASTE CLASSIFICATION IN FRANCE, FROM [1]......................................................................... FIGURE 1-3: RADIOACTIVE WASTE VOLUME PER ECONOMIC SECTOR (2018), ADAPTED FROM [4] ................................... FIGURE 1-4: NUCLIDES BY HALF-LIFE AS A FUNCTION OF THEIR ATOMIC NUMBER Z AND NEUTRON NUMBER N; THE BLACKS SQUARES ARE THE STABLE ELEMENTS, FROM [5] ............................................................................................ FIGURE 1-5: 235 U FISSION PRINCIPLE ................................................................................................................... FIGURE 1-6: MASS DISTRIBUTION OF FISSION PRODUCTS, FROM [8]........................................................................... FIGURE 1-7: 235U FISSION CROSS-SECTION (BLACK) AND 238 U NEUTRON CAPTURE CROSS-SECTION (PINK) AS A FUNCTION OF THE NEUTRON ENERGY [7] ........................................................................................................................ FIGURE 1-8: RADIOTOXICITY OF THE FISSION PRODUCTS AND ACTINIDES RELATIVE TO THE ORIGINAL URANIUM ORE, AS A FUNCTION OF THE TIME AFTER DISCHARGE FROM [8] ..................................................................................... FIGURE 1-9: THERMAL REACTOR WASTE INVENTORY SCHEMATICS ADAPTED FROM [12] ................................................ FIGURE 1-10: EXAMPLE OF PU STOCK STABILIZATION IN A MULTIPLE-RECYCLING STRATEGY VERSUS MONO-RECYCLING AND NO RECYCLING, FROM [9] .............................................................................................................................. FIGURE 1-11: THE POSSIBLE MINOR ACTINIDE IRRADIATION METHODS IN THE CONTEXT OF THE PU FUEL CLOSURE CYCLE, ADAPTED FROM [13] ............................................................................................................................... FIGURE 1-12: MYRRHA THREE-COMPONENT COUPLING, FROM[23] ....................................................................... FIGURE 1-13: FROM THE CONCEPTS EVALUATED DURING PDS-XADS TO THE EUROTRANS BASELINE; FP5 (1998-2002) AND FP6 (2002-2006) ARE THE EC FRAMEWORK PROGRAMS, IMAGE FROM [25] ............................................. FIGURE 1-14: THE VENUS REACTOR COUPLED TO THE GENEPI3C AT SCK•CEN, FROM [27] ...................................... FIGURE 1-15: THE MYRRHA/FASTEF CORE AND ITS MAIN SUB-COMPONENTS .......................................................... FIGURE 1-16: MYRRHA BEAM STRUCTURE PROPOSED IN MYRTE ........................................................................... FIGURE 1-17: THE LANDSCAPE OF HIGH POWER PROTON BEAM ACCELERATORS, AS OF 2016 [35] .................................. FIGURE 1-18: BASIC BEAM DYNAMIC DEFINITIONS, INSPIRED FROM [36] .................................................................... FIGURE 1-19: ELECTRICAL CONSUMPTION TRADE-OFF SUPERCONDUCTING VERSUS NORMAL CAVITIES .............................. FIGURE 1-20: THE MYRRHA RELIABILITY REQUIREMENT COMPARED TO OTHER MACHINES, FROM [41] .......................... FIGURE 1-21: THE MYRRHA ACCELERATOR LAYOUT, FROM [43] ............................................................................. FIGURE 1-22: PLANNED MINERVA STAGED IMPLEMENTATION OF THE LINAC, AS OF NOVEMBER 2021, PERSONAL COMMUNICATION, BY COURTESY OF SCK•CEN ............................................................................................ FIGURE 1-23: THE MINERVA SITE AT SCK•CEN IN MOL, PERSONAL COMMUNICATION, BY COURTESY OF SCK•CEN ....... FIGURE 2-1: SIMPLIFIED FUNCTIONAL DIAGRAM OF A CRYOMODULE .......................................................................... FIGURE 2-2: THE MINERVA PROTOTYPE CRYOMODULE BREAKDOWN ....................................................................... FIGURE 2-3: CRYOMODULES AND AUXILIARY SYSTEMS IN THE TUNNEL, PERSONAL COMMUNICATION BY COURTESY OF SCK•CEN ............................................................................................................................................ FIGURE 2-4: THE INTENDED DYNAMIC SYSTEM MODEL MAIN CONTRIBUTIONS ALONG THE MINERVA PROJECT CYCLE ......... FIGURE 2-5: SYNOPTIC VIEW OF THE QM MODEL DEVELOPED FOR SPIRAL2 FROM [17] ............................................... FIGURE 2-6: THE SYNOPTIC OF THE DESIRED CRYOGENIC AND RF COUPLED MODELS, AND THEIR RESPECTIVE CONTROL SYSTEMS ........................................................................................................................................................... FIGURE 2-7: BEAM DYNAMIC WITH RESPECT TO RF, ADAPTED FROM [86] .................................................................. FIGURE 3-1: HELIUM 4 PHASE TRANSITION DIAGRAM AND SOME OPERATING POINTS FROM [88], [90] ............................ FIGURE 3-2: SUBCOOLED JT EXPANSION .............................................................................................................. FIGURE 3-3: T-S DIAGRAM OF THE HELIUM EXPANSION CYCLES, WITH OR WITHOUT SUBCOOLING STEP ............................. FIGURE 3-4: THE MINERVA VALVE BOX AND CRYOMODULE CRYOGENIC PROCESS ....................................................... FIGURE 3-5: THE MINERVA CRYOGENIC INFRASTRUCTURE PROCESS FLOW DIAGRAM AT IJCLAB IN ORSAY ....................... FIGURE 3-6: SIMULINK BLOCKS (LEFT) AND SIMSCAPE (RIGHT) ................................................................................. FIGURE 3-7: THE CRYOSCAPE LIBRARY, AS OF MAY 2023 ........................................................................................

  FIGURE 1-1: THE DIFFERENT EMISSION TYPES AND ASSOCIATED PENETRATION POWERS FROM [3] .................................... FIGURE 1-2: NUCLEAR WASTE CLASSIFICATION IN FRANCE, FROM [1]......................................................................... FIGURE 1-3: RADIOACTIVE WASTE VOLUME PER ECONOMIC SECTOR (2018), ADAPTED FROM [4] ................................... FIGURE 1-4: NUCLIDES BY HALF-LIFE AS A FUNCTION OF THEIR ATOMIC NUMBER Z AND NEUTRON NUMBER N; THE BLACKS SQUARES ARE THE STABLE ELEMENTS, FROM [5] ............................................................................................ FIGURE 1-5: 235 U FISSION PRINCIPLE ................................................................................................................... FIGURE 1-6: MASS DISTRIBUTION OF FISSION PRODUCTS, FROM [8]........................................................................... FIGURE 1-7: 235U FISSION CROSS-SECTION (BLACK) AND 238 U NEUTRON CAPTURE CROSS-SECTION (PINK) AS A FUNCTION OF THE NEUTRON ENERGY [7] ........................................................................................................................ FIGURE 1-8: RADIOTOXICITY OF THE FISSION PRODUCTS AND ACTINIDES RELATIVE TO THE ORIGINAL URANIUM ORE, AS A FUNCTION OF THE TIME AFTER DISCHARGE FROM [8] ..................................................................................... FIGURE 1-9: THERMAL REACTOR WASTE INVENTORY SCHEMATICS ADAPTED FROM [12] ................................................ FIGURE 1-10: EXAMPLE OF PU STOCK STABILIZATION IN A MULTIPLE-RECYCLING STRATEGY VERSUS MONO-RECYCLING AND NO RECYCLING, FROM [9] .............................................................................................................................. FIGURE 1-11: THE POSSIBLE MINOR ACTINIDE IRRADIATION METHODS IN THE CONTEXT OF THE PU FUEL CLOSURE CYCLE, ADAPTED FROM [13] ............................................................................................................................... FIGURE 1-12: MYRRHA THREE-COMPONENT COUPLING, FROM[23] ....................................................................... FIGURE 1-13: FROM THE CONCEPTS EVALUATED DURING PDS-XADS TO THE EUROTRANS BASELINE; FP5 (1998-2002) AND FP6 (2002-2006) ARE THE EC FRAMEWORK PROGRAMS, IMAGE FROM [25] ............................................. FIGURE 1-14: THE VENUS REACTOR COUPLED TO THE GENEPI3C AT SCK•CEN, FROM [27] ...................................... FIGURE 1-15: THE MYRRHA/FASTEF CORE AND ITS MAIN SUB-COMPONENTS .......................................................... FIGURE 1-16: MYRRHA BEAM STRUCTURE PROPOSED IN MYRTE ........................................................................... FIGURE 1-17: THE LANDSCAPE OF HIGH POWER PROTON BEAM ACCELERATORS, AS OF 2016 [35] .................................. FIGURE 1-18: BASIC BEAM DYNAMIC DEFINITIONS, INSPIRED FROM [36] .................................................................... FIGURE 1-19: ELECTRICAL CONSUMPTION TRADE-OFF SUPERCONDUCTING VERSUS NORMAL CAVITIES .............................. FIGURE 1-20: THE MYRRHA RELIABILITY REQUIREMENT COMPARED TO OTHER MACHINES, FROM [41] .......................... FIGURE 1-21: THE MYRRHA ACCELERATOR LAYOUT, FROM [43] ............................................................................. FIGURE 1-22: PLANNED MINERVA STAGED IMPLEMENTATION OF THE LINAC, AS OF NOVEMBER 2021, PERSONAL COMMUNICATION, BY COURTESY OF SCK•CEN ............................................................................................ FIGURE 1-23: THE MINERVA SITE AT SCK•CEN IN MOL, PERSONAL COMMUNICATION, BY COURTESY OF SCK•CEN ....... FIGURE 2-1: SIMPLIFIED FUNCTIONAL DIAGRAM OF A CRYOMODULE .......................................................................... FIGURE 2-2: THE MINERVA PROTOTYPE CRYOMODULE BREAKDOWN ....................................................................... FIGURE 2-3: CRYOMODULES AND AUXILIARY SYSTEMS IN THE TUNNEL, PERSONAL COMMUNICATION BY COURTESY OF SCK•CEN ............................................................................................................................................ FIGURE 2-4: THE INTENDED DYNAMIC SYSTEM MODEL MAIN CONTRIBUTIONS ALONG THE MINERVA PROJECT CYCLE ......... FIGURE 2-5: SYNOPTIC VIEW OF THE QM MODEL DEVELOPED FOR SPIRAL2 FROM [17] ............................................... FIGURE 2-6: THE SYNOPTIC OF THE DESIRED CRYOGENIC AND RF COUPLED MODELS, AND THEIR RESPECTIVE CONTROL SYSTEMS ........................................................................................................................................................... FIGURE 2-7: BEAM DYNAMIC WITH RESPECT TO RF, ADAPTED FROM [86] .................................................................. FIGURE 3-1: HELIUM 4 PHASE TRANSITION DIAGRAM AND SOME OPERATING POINTS FROM [88], [90] ............................ FIGURE 3-2: SUBCOOLED JT EXPANSION .............................................................................................................. FIGURE 3-3: T-S DIAGRAM OF THE HELIUM EXPANSION CYCLES, WITH OR WITHOUT SUBCOOLING STEP ............................. FIGURE 3-4: THE MINERVA VALVE BOX AND CRYOMODULE CRYOGENIC PROCESS ....................................................... FIGURE 3-5: THE MINERVA CRYOGENIC INFRASTRUCTURE PROCESS FLOW DIAGRAM AT IJCLAB IN ORSAY ....................... FIGURE 3-6: SIMULINK BLOCKS (LEFT) AND SIMSCAPE (RIGHT) ................................................................................. FIGURE 3-7: THE CRYOSCAPE LIBRARY, AS OF MAY 2023 ........................................................................................

FIGURE 3 -

 3 FIGURE 3-8: SIMSCAPE PARAMETER ALLOCATION METHODOLOGY ............................................................................. FIGURE 3-9: THE FLUID PROPERTY P-U AND P-𝑢 SPACES FROM [90] ......................................................................... FIGURE 3-10: THE HELIUM FLUID PROPERTIES AS A FUNCTION OF P-U ........................................................................ FIGURE 3-11: THERMAL CONDUCTIVITY COMPARISON BETWEEN THE PURE AND THE ALLOYED MATERIALS ......................... FIGURE 3-12: SPECIFIC HEAT COMPARISON........................................................................................................... FIGURE 3-13: PHONON (KGS) AND ELECTRON (KES)THERMAL CONDUCTIVITY CONTRIBUTIONS IN PURE NIOBIUM, RRR = 30, AGS = 50 µM, FIT "BCS THEORY" ............................................................................................................. FIGURE 3-14: CONTINUOUS NIOBIUM Λ FUNCTION BETWEEN 1 K AND 300 K BUILT FOR THE MODEL DEVELOPMENTS ......... FIGURE 3-15: FLUID VOLUME BETWEEN A AND B AND NODE I .................................................................................. FIGURE 3-16: THE GENERIC DUCT MODEL CRYOSCAPE ICON ..................................................................................... FIGURE 3-17: THE SEGMENTED GENERIC DUCT ...................................................................................................... FIGURE 3-18: THE THERMAL CONDUCTION MODEL (LEFT) AND THE THERMAL INERTIA MODEL (RIGHT) CRYOSCAPE BLOCKS ..FIGURE 3-19: SEGMENTED HEAT EQUATION COMBINED BLOCK, THE THERMAL LINK ...................................................... FIGURE 3-20: NATIVE SIMSCAPE RADIATIVE HEAT TRANSFER BLOCK ........................................................................... FIGURE 3-21: 1-ZONE PHASE SEPARATOR MODEL PRINCIPLE .................................................................................... FIGURE 3-22: 1-ZONE TANK MODEL ................................................................................................................... FIGURE 3-23: RELATIONSHIPS AMONG INDIVIDUAL PHYSICAL MODELS, FROM [129] ................................................... FIGURE 3-24: HYDROGEN TANK TEMPERATURE AND VELOCITY PROFILES FROM [121] ................................................. FIGURE 3-25: SCHEMATIC OF AN H2 PROPELLANT LOADING FROM [135] ................................................................. FIGURE 3-26: COMPARISON OF THE DIFFERENT 2-ZONE TANK MODELS, THE GREEN ARROWS STAND FOR THE MASS TRANSFERS, THE YELLOW ONES, FOR THE HEAT TRANSFERS ............................................................................................. FIGURE 3-27: QVB TANK HEAT LOAD SUMMARY ................................................................................................. FIGURE 3-28: CRYOSCAPE SEGMENTED FLUID DUCT WITH WALL MASS AND EXTERNAL RADIATION HEAT TRANSFER ........... FIGURE 3-29: TYPICAL VALVE CHARACTERISTICS ................................................................................................... FIGURE 3-30: THE VALVE FLOW MODEL ............................................................................................................. FIGURE 3-31: MINERVA CRYOGENIC VALVE MAIN FEATURES ................................................................................ FIGURE 3-32: COMBINED THERMAL AND FLOW MODEL OF A CRYOGENIC VALVE (LEFT) INTEGRATD IN A SUB-ASSEMBLY (RIGHT) ......................................................................................................................................................... FIGURE 3-33: SIMULATED DATA OF IJCLAB B106 GROUP PUMPING CAPACITY AS A FUNCTION OF THE PRESSURE .............. FIGURE 3-34: 2 K PUMPING GROUP SIMSCAPE MODEL (LEFT) AND THE CORRESPONDING CAD PICTURE (RIGHT) ............. FIGURE 3-35: THE VALVE BOX "4 K" TANK ......................................................................................................... FIGURE 3-36: QM PHASE SEPARATOR MAIN VIEW AND CROSS-SECTIONS ................................................................. FIGURE 3-37: LEVEL OF LIQUID IN THE 2 K TANK AS A FUNCTION OF THE VOLUME OF LIQUID ........................................ FIGURE 3-38: TABLE LAYOUT (LEFT) AND CORRESPONDING CRYOSCAPE MODEL (RIGHT) .............................................. FIGURE 3-39: THE TESTQVB THERMAL SHIELD DISCRETIZATION (CENTRE), THE 80 K CIRCUITRY (RIGHT) AND SIMSCAPE IMPLEMENTATION (LEFT) ........................................................................................................................ FIGURE 3-40: THE PROTOTYPE QM THERMAL SHIELD DISCRETIZATION (RIGHT), THE 80 K CIRCUIT IN YELLOW ARROWS AND THE SIMSCAPE MODEL (LEFT) .................................................................................................................. FIGURE 3-41: THE PROTOTYPE OF THE "2.5 G/S" HEAT EXCHANGER OF THE ESS SPOKE CRYOMODULE .......................... FIGURE 3-42: SIMSCAPE MODEL OF THE HX ....................................................................................................... FIGURE 3-43: SIMSCAPE AGAINST CERN TESTS OR DATE SIMULATIONS .................................................................. FIGURE 3-44: SIMULATION WITH SIMSCAPE OF DIFFERENT PLATE MATERIALS ............................................................ FIGURE 3-45: TEMPERATURE PROFILE ALONG THE HX .......................................................................................... FIGURE 3-46: SIMSCAPE COMPARISON TO THE CRYOGENIC SIMULATIONS (LEFT), AND TO THE DATE TESTS AND SIMULATIONS AT ROOM TEMPERATURE (RIGHT), ON THE LP SIDE ...................................................................................... FIGURE 3-47: SCHEMATICS OF THE IJCLAB ROOM TEMPERATURE ΔP TEST BENCH ...................................................... FIGURE 3-48: LP-SIDE DEFORMED CORE ΔP MODELS VERSUS THE EXPERIMENT, THE SOLID LINES ARE THE LINEAR REGRESSIONS OF THE DOTS ........................................................................................................................................ FIGURE 3-49: HP-SIDE DEFORMED CORE ΔP MODELS VERSUS THE EXPERIMENT, THE SOLID LINES ARE THE LINEAR REGRESSIONS OF THE DOTS .....................................................................................................................

FIGURE

  FIGURE 3-50: THE MINERVA OPEN-LOOP CRYOGENIC PROCESS MODEL .................................................................. FIGURE 3-51: MINERVA PROTOTYPE CRYOGENIC CONTROL ARCHITECTURE ............................................................. FIGURE 3-52: MINERVA CRYOGENIC PROCESS MODEL ARCHITECTURE, COMPLIANT WITH WHICHEVER CONTROL BOX ...... FIGURE 4-1: THE DIFFERENT HEAT LOADS GENERATED BY THE RF PARTS ................................................................... FIGURE 4-2: CAVITY RESONANT FIELD CLASS AS A FUNCTION OF THEIR FAMILY, ADAPTED FROM [80], [145] AND [146] -THE COLOUR BARS ARE THE E-FIELD INTENSITY .................................................................................................. FIGURE 4-3: CAVITY GAP AND FIELD DEFINITION ADAPTED FROM [85]...................................................................... FIGURE 4-4: THE MEISSNER EFFECT ILLUSTRATED IN A VIDEO FROM [54] .................................................................. FIGURE 4-5: LONDON TWO-FLUID CHARGE CARRIER MODEL, PICTURE FROM [54] ..................................................... FIGURE 4-6: ILLUSTRATION OF THE VARIATION OF THE CRITICAL MAGNETIC FIELD AS A FUNCTION OF T, FROM [54] .......... FIGURE 4-7: FEYNMAN DIAGRAM OF THE ELECTRON-PHONON INTERACTION PROCESS ................................................. FIGURE 4-8: SECONDARY ELECTRON TRAJECTORIES IN A MULTIPACTING EMISSION FROM [61] ...................................... FIGURE 4-9: CAVITY POWER BALANCE................................................................................................................ FIGURE 4-10: BSC03 (LEFT) AND BSC04 (RIGHT) Q0 AND X-RAY CURVES AS A FUNCTION OF EACC ................................ FIGURE 4-11: CAVITY EQUIVALENT RLC CIRCUIT LOADED AND EXTERNAL CIRCUIT AND BEAM ........................................ FIGURE 4-12: S21 PARAMETER, MEASURED ON BSC04 AT 2 K IN THE MINERVA PROTOTYPE CRYOMODULE, AND REPRODUCED WITH THE CAVITY FIELD MODEL, AT +/-2000 HZ AROUND THE RESONANCE FREQUENCY .................. FIGURE 4-13: SMITH CHART OF THE CAVITY VOLTAGE AT THE ACCELERATOR GAP ....................................................... FIGURE 4-14: E FIELD ALONG THE BEAM AXIS IN BOTH GAPS PREDICTED BY CST (LEFT) AND RLC MODEL SMITH CHART (RIGHT) FOR THE SAME 𝑃𝑖 ................................................................................................................................. FIGURE 4-15 : THE CAVITY PHASER DIAGRAM FOR THE 2 CONFIGURATIONS UNDER/OVERCOUPLED CROSSED WITH THE2 CASES ON/OFF RESONANCE .............................................................................................................................. FIGURE 4-16: PHASER DIAGRAM AND TRANSMISSION CHART FOR A FIELD REGULATED AT 7 MV/M, WITH A 4-MA BEAM AND SYNCHRONOUS PHASE -30° .................................................................................................................... FIGURE 4-17: FPGA-BASED DIGITAL CAVITY FIELD AND RESONANCE CONTROL MAIN PHILOSOPHY ................................. FIGURE 4-18: THE PROTOTYPE CRYOMODULE RF SETUP AT THE IJCLAB TEST PIT ........................................................ FIGURE 4-19: SIMULINK MODEL OF THE RF PROCESS AND ITS CONTROL ................................................................... FIGURE 4-20: OSCILLATION OF THE CAVITY RESONANCE FREQUENCY DUE TO VIBRATIONS ............................................. FIGURE 4-21: GEOMETRICAL DEFORMATIONS INDUCED BY THE LORENTZ FORCES, FROM [174] .................................... FIGURE 4-22: MINERVA SPOKE CAVITY SENSITIVITY TO PRESSURE. EACH OF THE 3 FIGURES IS THE AVERAGE PRESSURE (IN BAR) IN THE ZONE INDICATED ON THE LEFT-HANDED SIDE. FOR INSTANCE, "010" MEANS 1 BAR IN THE CAVITY TANK AND 0 BAR FOR THE BEAM AND INSULATION VACUUMS; THE BOLD DIGITS ARE WERE THE PRESSURE VARIATION IS APPLIED FIGURE 4-23: MINERVA SLOW AND FAST TUNER OVERVIEW, ADAPTED FROM [178] ................................................ FIGURE 4-24: MINERVA PROTOTYPE SPOKE CAVITY S21 SHIFT DURING A 4 K COOL-DOWN ....................................... FIGURE 4-25: RESONANCE FREQUENCY SHIFT VERSUS FLANGE DISPLACEMENT, AT CRYOGENIC TEMPERATURE ON BSC04, FROM [94] .......................................................................................................................................... FIGURE 4-26: PHASER OF AN OVERCOUPLED CAVITY, WITHOUT BEAM AND 45 HZ DETUNING ....................................... FIGURE 4-27: MEASURED PIEZO DYNAMIC TRANSFER FUNCTIONS OF BSC01, BSC02 AND BSC04 ............................... FIGURE 4-28: BSC04 COLD TEMPERATURE PIEZO DYNAMIC CHARACTERIZATION MODEL-TO-TEST COMPARISON .............. FIGURE 4-29: CAVITY "WALL MODEL" ............................................................................................................... FIGURE 4-30: COAXIAL LOOP (LEFT) OR ANTENNA (RIGHT) COUPLING, FROM [165] ................................................... FIGURE 4-31: MINERVA PROTOTYPE COUPLER ENVIRONMENT MAIN FEATURES ....................................................... FIGURE 4-32: XFEL (LEFT) FROM [56] AND ESS FROM [181] MAIN COUPLER DESIGN FEATURES .................................. FIGURE 4-33: RF COUPLER ROOM TEMPERATURE CONDITIONING TEST BENCH AT IJCLAB ............................................ FIGURE 4-34: TEMPERATURE INSTRUMENTATION OF THE COUPLERS IN THE CONDITIONING TEST BENCH. EACH WINDOW IS EQUIPPED WITH A PT100, THE OTHER NUMBERS REFER TO THERMOCOUPLES. THE ANTENNA AND CUFF HEAT EXCHANGER WATER FLOWMETERS ARE NOT SHOWN. ................................................................................... FIGURE 4-35: LPSC'S MATLAB MODEL SCHEMATIC (TOP) AND CAD CORRESPONDENCE (BOTTOM) ............................ FIGURE 4-36: LPSC'S MODEL STRICTLY TRANSLATED INTO SIMSCAPE ...................................................................... FIGURE 4-37: THE LATEST VERSION OF THE SIMSCAPE COUPLER MODEL ...................................................................

  FIGURE 3-50: THE MINERVA OPEN-LOOP CRYOGENIC PROCESS MODEL .................................................................. FIGURE 3-51: MINERVA PROTOTYPE CRYOGENIC CONTROL ARCHITECTURE ............................................................. FIGURE 3-52: MINERVA CRYOGENIC PROCESS MODEL ARCHITECTURE, COMPLIANT WITH WHICHEVER CONTROL BOX ...... FIGURE 4-1: THE DIFFERENT HEAT LOADS GENERATED BY THE RF PARTS ................................................................... FIGURE 4-2: CAVITY RESONANT FIELD CLASS AS A FUNCTION OF THEIR FAMILY, ADAPTED FROM [80], [145] AND [146] -THE COLOUR BARS ARE THE E-FIELD INTENSITY .................................................................................................. FIGURE 4-3: CAVITY GAP AND FIELD DEFINITION ADAPTED FROM [85]...................................................................... FIGURE 4-4: THE MEISSNER EFFECT ILLUSTRATED IN A VIDEO FROM [54] .................................................................. FIGURE 4-5: LONDON TWO-FLUID CHARGE CARRIER MODEL, PICTURE FROM [54] ..................................................... FIGURE 4-6: ILLUSTRATION OF THE VARIATION OF THE CRITICAL MAGNETIC FIELD AS A FUNCTION OF T, FROM [54] .......... FIGURE 4-7: FEYNMAN DIAGRAM OF THE ELECTRON-PHONON INTERACTION PROCESS ................................................. FIGURE 4-8: SECONDARY ELECTRON TRAJECTORIES IN A MULTIPACTING EMISSION FROM [61] ...................................... FIGURE 4-9: CAVITY POWER BALANCE................................................................................................................ FIGURE 4-10: BSC03 (LEFT) AND BSC04 (RIGHT) Q0 AND X-RAY CURVES AS A FUNCTION OF EACC ................................ FIGURE 4-11: CAVITY EQUIVALENT RLC CIRCUIT LOADED AND EXTERNAL CIRCUIT AND BEAM ........................................ FIGURE 4-12: S21 PARAMETER, MEASURED ON BSC04 AT 2 K IN THE MINERVA PROTOTYPE CRYOMODULE, AND REPRODUCED WITH THE CAVITY FIELD MODEL, AT +/-2000 HZ AROUND THE RESONANCE FREQUENCY .................. FIGURE 4-13: SMITH CHART OF THE CAVITY VOLTAGE AT THE ACCELERATOR GAP ....................................................... FIGURE 4-14: E FIELD ALONG THE BEAM AXIS IN BOTH GAPS PREDICTED BY CST (LEFT) AND RLC MODEL SMITH CHART (RIGHT) FOR THE SAME 𝑃𝑖 ................................................................................................................................. FIGURE 4-15 : THE CAVITY PHASER DIAGRAM FOR THE 2 CONFIGURATIONS UNDER/OVERCOUPLED CROSSED WITH THE2 CASES ON/OFF RESONANCE .............................................................................................................................. FIGURE 4-16: PHASER DIAGRAM AND TRANSMISSION CHART FOR A FIELD REGULATED AT 7 MV/M, WITH A 4-MA BEAM AND SYNCHRONOUS PHASE -30° .................................................................................................................... FIGURE 4-17: FPGA-BASED DIGITAL CAVITY FIELD AND RESONANCE CONTROL MAIN PHILOSOPHY ................................. FIGURE 4-18: THE PROTOTYPE CRYOMODULE RF SETUP AT THE IJCLAB TEST PIT ........................................................ FIGURE 4-19: SIMULINK MODEL OF THE RF PROCESS AND ITS CONTROL ................................................................... FIGURE 4-20: OSCILLATION OF THE CAVITY RESONANCE FREQUENCY DUE TO VIBRATIONS ............................................. FIGURE 4-21: GEOMETRICAL DEFORMATIONS INDUCED BY THE LORENTZ FORCES, FROM [174] .................................... FIGURE 4-22: MINERVA SPOKE CAVITY SENSITIVITY TO PRESSURE. EACH OF THE 3 FIGURES IS THE AVERAGE PRESSURE (IN BAR) IN THE ZONE INDICATED ON THE LEFT-HANDED SIDE. FOR INSTANCE, "010" MEANS 1 BAR IN THE CAVITY TANK AND 0 BAR FOR THE BEAM AND INSULATION VACUUMS; THE BOLD DIGITS ARE WERE THE PRESSURE VARIATION IS APPLIED FIGURE 4-23: MINERVA SLOW AND FAST TUNER OVERVIEW, ADAPTED FROM [178] ................................................ FIGURE 4-24: MINERVA PROTOTYPE SPOKE CAVITY S21 SHIFT DURING A 4 K COOL-DOWN ....................................... FIGURE 4-25: RESONANCE FREQUENCY SHIFT VERSUS FLANGE DISPLACEMENT, AT CRYOGENIC TEMPERATURE ON BSC04, FROM [94] .......................................................................................................................................... FIGURE 4-26: PHASER OF AN OVERCOUPLED CAVITY, WITHOUT BEAM AND 45 HZ DETUNING ....................................... FIGURE 4-27: MEASURED PIEZO DYNAMIC TRANSFER FUNCTIONS OF BSC01, BSC02 AND BSC04 ............................... FIGURE 4-28: BSC04 COLD TEMPERATURE PIEZO DYNAMIC CHARACTERIZATION MODEL-TO-TEST COMPARISON .............. FIGURE 4-29: CAVITY "WALL MODEL" ............................................................................................................... FIGURE 4-30: COAXIAL LOOP (LEFT) OR ANTENNA (RIGHT) COUPLING, FROM [165] ................................................... FIGURE 4-31: MINERVA PROTOTYPE COUPLER ENVIRONMENT MAIN FEATURES ....................................................... FIGURE 4-32: XFEL (LEFT) FROM [56] AND ESS FROM [181] MAIN COUPLER DESIGN FEATURES .................................. FIGURE 4-33: RF COUPLER ROOM TEMPERATURE CONDITIONING TEST BENCH AT IJCLAB ............................................ FIGURE 4-34: TEMPERATURE INSTRUMENTATION OF THE COUPLERS IN THE CONDITIONING TEST BENCH. EACH WINDOW IS EQUIPPED WITH A PT100, THE OTHER NUMBERS REFER TO THERMOCOUPLES. THE ANTENNA AND CUFF HEAT EXCHANGER WATER FLOWMETERS ARE NOT SHOWN. ................................................................................... FIGURE 4-35: LPSC'S MATLAB MODEL SCHEMATIC (TOP) AND CAD CORRESPONDENCE (BOTTOM) ............................ FIGURE 4-36: LPSC'S MODEL STRICTLY TRANSLATED INTO SIMSCAPE ...................................................................... FIGURE 4-37: THE LATEST VERSION OF THE SIMSCAPE COUPLER MODEL ...................................................................

  FIGURE 3-50: THE MINERVA OPEN-LOOP CRYOGENIC PROCESS MODEL .................................................................. FIGURE 3-51: MINERVA PROTOTYPE CRYOGENIC CONTROL ARCHITECTURE ............................................................. FIGURE 3-52: MINERVA CRYOGENIC PROCESS MODEL ARCHITECTURE, COMPLIANT WITH WHICHEVER CONTROL BOX ...... FIGURE 4-1: THE DIFFERENT HEAT LOADS GENERATED BY THE RF PARTS ................................................................... FIGURE 4-2: CAVITY RESONANT FIELD CLASS AS A FUNCTION OF THEIR FAMILY, ADAPTED FROM [80], [145] AND [146] -THE COLOUR BARS ARE THE E-FIELD INTENSITY .................................................................................................. FIGURE 4-3: CAVITY GAP AND FIELD DEFINITION ADAPTED FROM [85]...................................................................... FIGURE 4-4: THE MEISSNER EFFECT ILLUSTRATED IN A VIDEO FROM [54] .................................................................. FIGURE 4-5: LONDON TWO-FLUID CHARGE CARRIER MODEL, PICTURE FROM [54] ..................................................... FIGURE 4-6: ILLUSTRATION OF THE VARIATION OF THE CRITICAL MAGNETIC FIELD AS A FUNCTION OF T, FROM [54] .......... FIGURE 4-7: FEYNMAN DIAGRAM OF THE ELECTRON-PHONON INTERACTION PROCESS ................................................. FIGURE 4-8: SECONDARY ELECTRON TRAJECTORIES IN A MULTIPACTING EMISSION FROM [61] ...................................... FIGURE 4-9: CAVITY POWER BALANCE................................................................................................................ FIGURE 4-10: BSC03 (LEFT) AND BSC04 (RIGHT) Q0 AND X-RAY CURVES AS A FUNCTION OF EACC ................................ FIGURE 4-11: CAVITY EQUIVALENT RLC CIRCUIT LOADED AND EXTERNAL CIRCUIT AND BEAM ........................................ FIGURE 4-12: S21 PARAMETER, MEASURED ON BSC04 AT 2 K IN THE MINERVA PROTOTYPE CRYOMODULE, AND REPRODUCED WITH THE CAVITY FIELD MODEL, AT +/-2000 HZ AROUND THE RESONANCE FREQUENCY .................. FIGURE 4-13: SMITH CHART OF THE CAVITY VOLTAGE AT THE ACCELERATOR GAP ....................................................... FIGURE 4-14: E FIELD ALONG THE BEAM AXIS IN BOTH GAPS PREDICTED BY CST (LEFT) AND RLC MODEL SMITH CHART (RIGHT) FOR THE SAME 𝑃𝑖 ................................................................................................................................. FIGURE 4-15 : THE CAVITY PHASER DIAGRAM FOR THE 2 CONFIGURATIONS UNDER/OVERCOUPLED CROSSED WITH THE2 CASES ON/OFF RESONANCE .............................................................................................................................. FIGURE 4-16: PHASER DIAGRAM AND TRANSMISSION CHART FOR A FIELD REGULATED AT 7 MV/M, WITH A 4-MA BEAM AND SYNCHRONOUS PHASE -30° .................................................................................................................... FIGURE 4-17: FPGA-BASED DIGITAL CAVITY FIELD AND RESONANCE CONTROL MAIN PHILOSOPHY ................................. FIGURE 4-18: THE PROTOTYPE CRYOMODULE RF SETUP AT THE IJCLAB TEST PIT ........................................................ FIGURE 4-19: SIMULINK MODEL OF THE RF PROCESS AND ITS CONTROL ................................................................... FIGURE 4-20: OSCILLATION OF THE CAVITY RESONANCE FREQUENCY DUE TO VIBRATIONS ............................................. FIGURE 4-21: GEOMETRICAL DEFORMATIONS INDUCED BY THE LORENTZ FORCES, FROM [174] .................................... FIGURE 4-22: MINERVA SPOKE CAVITY SENSITIVITY TO PRESSURE. EACH OF THE 3 FIGURES IS THE AVERAGE PRESSURE (IN BAR) IN THE ZONE INDICATED ON THE LEFT-HANDED SIDE. FOR INSTANCE, "010" MEANS 1 BAR IN THE CAVITY TANK AND 0 BAR FOR THE BEAM AND INSULATION VACUUMS; THE BOLD DIGITS ARE WERE THE PRESSURE VARIATION IS APPLIED FIGURE 4-23: MINERVA SLOW AND FAST TUNER OVERVIEW, ADAPTED FROM [178] ................................................ FIGURE 4-24: MINERVA PROTOTYPE SPOKE CAVITY S21 SHIFT DURING A 4 K COOL-DOWN ....................................... FIGURE 4-25: RESONANCE FREQUENCY SHIFT VERSUS FLANGE DISPLACEMENT, AT CRYOGENIC TEMPERATURE ON BSC04, FROM [94] .......................................................................................................................................... FIGURE 4-26: PHASER OF AN OVERCOUPLED CAVITY, WITHOUT BEAM AND 45 HZ DETUNING ....................................... FIGURE 4-27: MEASURED PIEZO DYNAMIC TRANSFER FUNCTIONS OF BSC01, BSC02 AND BSC04 ............................... FIGURE 4-28: BSC04 COLD TEMPERATURE PIEZO DYNAMIC CHARACTERIZATION MODEL-TO-TEST COMPARISON .............. FIGURE 4-29: CAVITY "WALL MODEL" ............................................................................................................... FIGURE 4-30: COAXIAL LOOP (LEFT) OR ANTENNA (RIGHT) COUPLING, FROM [165] ................................................... FIGURE 4-31: MINERVA PROTOTYPE COUPLER ENVIRONMENT MAIN FEATURES ....................................................... FIGURE 4-32: XFEL (LEFT) FROM [56] AND ESS FROM [181] MAIN COUPLER DESIGN FEATURES .................................. FIGURE 4-33: RF COUPLER ROOM TEMPERATURE CONDITIONING TEST BENCH AT IJCLAB ............................................ FIGURE 4-34: TEMPERATURE INSTRUMENTATION OF THE COUPLERS IN THE CONDITIONING TEST BENCH. EACH WINDOW IS EQUIPPED WITH A PT100, THE OTHER NUMBERS REFER TO THERMOCOUPLES. THE ANTENNA AND CUFF HEAT EXCHANGER WATER FLOWMETERS ARE NOT SHOWN. ................................................................................... FIGURE 4-35: LPSC'S MATLAB MODEL SCHEMATIC (TOP) AND CAD CORRESPONDENCE (BOTTOM) ............................ FIGURE 4-36: LPSC'S MODEL STRICTLY TRANSLATED INTO SIMSCAPE ...................................................................... FIGURE 4-37: THE LATEST VERSION OF THE SIMSCAPE COUPLER MODEL ...................................................................

FIGURE 4 -

 4 FIGURE 4-38: HEAT LOAD CONTRIBUTION BY THE THREE COUPLER CUFF LAYERS, AS A FUNCTION OF THE RRR AND AVERAGE TEMPERATURE (THICKNESS = 2 MM (SS) -1.5 µM (NI) -15 µM (CU) ........................................................... FIGURE 4-39: ORIGINAL SACRIFICIAL CUFF (LEFT) CUT INTO 5 STICKS BY PMB. SAMPLE #5 COPPER LAYER WAS REMOVED WITH ACID. SAMPLES # 1-2-3-4 WERE THEN CUT IN 45-MM SAMPLES FOR ANALYSIS ................................................. FIGURE 4-40: SAMPLE PUT IN RESIN AND INSTALLED ON A LAM PLAN GRINDING MACHINE AT IJCLAB ............................. FIGURE 4-41: LASER CONFOCAL IMAGE (LEFT) AND EDX CONTINUOUS COMPOSITION MEASUREMENT OVER AN SEM IMAGE (RIGHT) OF A CUFF CROSS-SECTION ........................................................................................................... FIGURE 4-42: RRR SAMPLE MEASUREMENT INSERT AT IJCLAB ............................................................................... FIGURE 4-43: SIMPLIFICATION OF THE COUPLER AS A LOSSLESS COAXIAL LINE ............................................................ FIGURE 4-44: VALIDATION OF THE E FIELD (IN V/M) FROM THE ANALYTICAL MODEL (RIGHT) IN PURE TRAVELLING WAVE MODE, 80 KW, AGAINST CST RESULTS (LEFT) ............................................................................................ FIGURE 4-45: VALIDATION OF THE AMPLITUDE OF THE E FIELD (IN V/M) FROM THE ANALYTICAL MODEL (RIGHT) IN PURE STANDING WAVE MODE, 20 KW, AGAINST CST RESULTS (LEFT) ..................................................................... FIGURE 4-46: H FIELD (A/M) IN THE CAVITY AND ALONG THE COUPLER @ PI = 1 W. LEFT = OFF-RESONANCE, MIDDLE = AT RESONANCE & MATCHED WITH BEAM, RIGHT = AT RESONANCE AND OVERCOUPLED ........................................... FIGURE 4-47: COUPLER H FIELD AMPLITUDE (LEFT) AND LINEIC JOULE HEAT LOADS (RIGHT) ......................................... FIGURE 4-48: THE UPDATED RF HEAT LOAD BLOCK, INCLUDING THE SIMPLE ANALYTICAL "FIELD" MODEL ........................ FIGURE 4-49: PLOTS OF THE RF FIELD SETTINGS (TOP LEFT), THE DETUNING EFFECTS (TOP RIGHT) AND THE POWER BALANCE (BOTTOM) ........................................................................................................................................... FIGURE 4-50: E AND H FIELD ALONG THE COUPLER FOR SEQUENCES 3, 4 AND 5. THE ABSCISE IS THE DISTANCE (NEGATIVE) FROM THE COUPLING PLANE, I.E. ABOUT THE ANTENNA TIP. THE WINDOW IS LOCATED AT ABOUT -0.42 M. ............FIGURE 4-51: RF HEAT LOAD SOURCES (LEFT) AND DISTRIBUTION ON THE HEAT SINKS (RIGHT) ..................................... FIGURE 5-1: CONFIGURATION MAIN DIFFERENCES BETWEEN TEST PHASE 1 AND TESTS PHASES 2 ................................... FIGURE 5-2: SCHEMATICS OF THE MASS EXCHANGES IN THE 2 K TANK ...................................................................... FIGURE 5-3: CAVITY HEATER ELECTRICAL SUPPLY SCHEME ...................................................................................... FIGURE 5-4: CV04 SIMULATED MASS FLOWRATE, WITH A POSITION OFFSET OF -7%, AGAINST THE LEVEL-CORRECTED MEASURED MASS FLOWRATE ................................................................................................................... FIGURE 5-5: CV05 SIMULATED MASS FLOWRATE, AGAINST THE LEVEL-CORRECTED MEASURED MASS FLOWRATE, WITH THE COMPRESSIBLE FLOW EQUATION FROM IJCLAB (LEFT) AND FROM THE ANSI STANDARD (RIGHT) ........................... FIGURE 5-6: 80 K LOOP CV601 AND CV651 SIMSCAPE MODEL COMPARISON VERSUS A TEST AT RT ............................ FIGURE 5-7: MINERVA PROTOTYPE HX TEMPERATURE INSTRUMENTATION ............................................................ FIGURE 5-8: HX MASS FLOWRATES (LEFT), HX POWER ON THE HP SIDE (CENTRE) AND ON THE LP SIDE (RIGHT) DURING WP20-23 FROM TEST PHASE 1 ............................................................................................................... FIGURE 5-9: MODEL OF THE HX DOWNSTREAM PIPE FOR TT030 DISCREPANCY INVESTIGATION .................................... FIGURE 5-10: HX MASS FLOWRATES (LEFT), HX POWER ON THE HP SIDE (CENTRE) AND ON THE LP SIDE (RIGHT) FOR TEST PHASE 2.2 WP11................................................................................................................................. FIGURE 5-11: DEFINITION OF THE HEAT FLOWS AROUND THE FEET .......................................................................... FIGURE 5-12: BALANCE BETWEEN THE TOTAL HEAT FROM THE FEET (LEFT) AND THE SUM OF THE HEAT DIFFUSING FURTHER AWAY (RIGHT), DURING A 4 K MODE STEADY-STATE SEQUENCE FROM TEST PHASE 2.1 ....................................... FIGURE 5-13: HEAT FROM THE TABLE FRAME TOWARDS THE 10 K LOOP INTERCEPT, DURING WP5 ............................... FIGURE 5-14: HEAT FROM THE 10 K LOOP INTERCEPTS TO THE 2 K TANK, DURING WP5 ............................................. FIGURE 5-15: TABLE SUPPORT TEMPERATURE INSTRUMENTATION MAPPING AND SIMSCAPE REPRODUCTION ................... FIGURE 5-16: HEAT FLOW DEFINITIONS IN THE AIR-SIDE AREA ................................................................................ FIGURE 5-17: BALANCE BETWEEN THE ELECTRICAL HEATERS AND THE HEAT TO THE 80K LOOP ...................................... FIGURE 5-18: HEAT FLOW DEFINITIONS AROUND IN THE 10K LOOP AREA ................................................................. FIGURE 5-19: 10 K LOOP COMPONENT-BY-COMPONENT HEAT LOAD, MEASURED BY THE FLUID SENSIBLE HEAT METHOD ... FIGURE 5-20: FLUID-SIDE VERSUS MATERIAL-SIDE COUPLER TO 10K LOOP HEAT LOAD CALCULATION ............................. FIGURE 5-21: NIOBIUM COUPLER PORT TEMPERATURE INSTRUMENTATION ............................................................... FIGURE 5-22: HEAT FLOW FROM 10 K TO 2 K (LEFT) COMPARED TO THE HEAT FROM 80 K TO 10 K (COPPER RRR = 30, CENTRE VERSUS RRR = 120, RIGHT), ALL COMPUTED WITH THE MATERIAL CONDUCTION METHOD .......................

FIGURE 5 -

 5 FIGURE 5-23: QM THERMAL SHIELD COOL-DOWN DURING TEST PHASE 1 (LEFT), AND SIMSCAPE REPRODUCTION AT ISO CONDITIONS (RIGHT) ............................................................................................................................. FIGURE 5-24: QVB THERMAL SHIELD COOL-DOWN DURING TEST PHASE 1 (LEFT), AND SIMSCAPE REPRODUCTION AT ISO CONDITIONS (RIGHT) ............................................................................................................................. FIGURE 5-25: ROOT-CAUSE ANALYSIS OF THE QM THERMAL SHIELD MASS FLOWRATE MODEL TO TEST MISMATCH ........... FIGURE 5-26: CV601 AND CV651 FLOW PATH IN THE TESTQVB ........................................................................... FIGURE 5-27: QM 80 K LOOP HEAT LOAD DISTRIBUTION OVER SEVERAL WORKING POINTS, DURING TEST PHASE 2.1 ........ FIGURE 5-28: QM CHIMNEY LAYOUT AND HEAT LOADS ........................................................................................ FIGURE 5-29: 2 K LOOP HEAT LOAD DISTRIBUTION AND BALANCE OVER TEST PHASE 2.1; EACH WP IS AVERAGED OVER 5 TO 15 MIN; THE CONDITIONS ARE DETAILED IN THE LEGEND ("2K" OR "4K" MEANS 2 K OR 4 K MODE, "TRF" MEANS TRANSFER THROUGH THE JUMPER, "L" MEANS 2 K TANK LEVEL AND "10K" MEANS 10 K LOOP MASS FLOW RATE ... FIGURE 5-30: CRYOGENIC PROCESS MODEL CONTROL AND MODELLING ARCHITECTURE ADOPTED FOR THE PRECALIBRATIONS OF THE PI CONTROLLERS ............................................................................................................................. FIGURE 5-31: GENERIC ARCHITECTURE OF THE PI CONTROL LOOPS IN THE SIMULINK AND IN THE PLC CONTROL BOXES ...... FIGURE 5-32: MODEL VALIDATION OF THE EHX43 PI CONTROLLER; LEFT = FLANGE TEMPERATURE TTX44, CENTRE = WINDOW TEMPERATURE TTX43 AND RIGHT = EHX43 HEATER VOLTAGE AND POWER ...................................................... FIGURE 5-33: MEASURED COUPLER FLANGE AND WINDOW TEMPERATURES, AT THE FIRST 80 K COOL-DOWN OF TEST PHASE 2.1, LEFT = FLANGE TEMPERATURES, CENTRE = WINDOW TEMPERATURES, RIGHT = QM THERMAL SHIELD NITROGEN TUBE TEMPERATURES (TT682 IS AT THE COUPLER 2 OUTLET, TT683 IS AT THE COUPLER 1 OUTLET) ..................... FIGURE 5-34: EXPERIMENTAL FAULT SIMULATION OF THE EH144/244 FLANGE TEMPERATURE CONTROLLERS ................. FIGURE 5-35: EXPERIMENTAL FAULT SIMULATION OF THE EH143/243 WINDOW TEMPERATURE CONTROLLERS .............. FIGURE 5-36: 80 K COOL-DOWN AT THE START OF TEST PHASE 2.2, LEFT, THE THERMAL SHIELD OUTPUT TEMPERATURES, CENTRE, THE NITROGEN MASS FLOWMETERS, RIGHT, THE NITROGEN INLET PRESSURE ......................................... FIGURE 5-37: PERFORMANCE OF THE 10 K LOOP MASS FLOW RATE CONTROL, OVER A START-UP AT 08:45 THEN SETPOINT CHANGE AT ABOUT 10:00; LEFT = THE CONTROL VALVE (CV02), CENTRE = THE CONTROLLED MASS FLOWRATE, RIGHT = THE BOUNDARY PRESSURE CONDITIONS; THE SEQUENCE COMES FROM TEST PHASE 2.2, IN 4 K MODE .................... FIGURE 5-38: TEST PHASE 1 (4 K MODE) EXPERIMENTAL COMMISSIONING AND FINE-TUNING OF THE 4 K TANK PRESSURE CONTROLLER, TOGETHER WITH THE 4 K AND 2 K TANK LEVEL CONTROLS ACTIVE ................................................ FIGURE 5-39: TEST PHASE 2.2 (2 K MODE) EXPERIMENTAL COMMISSIONING AND FINE-TUNING OF THE 2 K TANK PRESSURE CONTROLLER ........................................................................................................................................ FIGURE 5-40: TEST PHASE 2.2 STEADY-STATE POINT IN 2 K MODE, ALL REGULATIONS ACTIVE ....................................... FIGURE 5-41: THE PRINCIPLE OF THE LQ VERSUS A DOUBLE PI CONTROLLER STRATEGY ................................................ FIGURE 5-42: MATLAB MODEL FORM CONVERSIONS .......................................................................................... FIGURE 5-43: COMPARISON OF THE PROCESS NON-LINEAR MODEL TO THE LINEARIZED MODEL, LEFT, THE LEVEL CONTROL, RIGHT, THE PRESSURE CONTROL ............................................................................................................... FIGURE 5-44: A CONTROLLER BASED ON THE ERROR ............................................................................................. FIGURE 5-45: A CONTROLLER BASED ON A FULL-STATE FEEDBACK ........................................................................... FIGURE 5-46: A CONTROLLER BASED ON AN LQ COMBINED WITH A STATE OBSERVER .................................................. FIGURE 5-47: SIMULATED COMPARISON OF THE DOUBLE PI (TOP) VERSUS LQ (BOTTOM), IN DISTURBANCE REJECTION OF A 2.5 W HEAT INPUT AT 12:10 AND A 5 W INPUT AT 12:35 ........................................................................... FIGURE 5-48: CONTROL ARCHITECTURE OF THE PROTOTYPE CRYOMODULE ADOPTED FOR THE REAL TESTS OF THE LQ + OBSERVER CONTROLLER DURING TEST PHASE 2.2 ........................................................................................ FIGURE 5-49: TEST OF THE LQ + OBSERVER SIMULINK CONTROL BOX, AT 18:24 DURING TEST PHASE 2.2, IN 2 K MODE ... FIGURE 5-50: REPRODUCTION (RIGHT) AT ISO PI (LEFT) OF THE MEASURED TRANSMITTED POWER 𝑃𝑡 (CENTRE) AND ACCELERATING VOLTAGE 𝐸𝑎𝑐𝑐 ON BSC03 DURING THE FIRST 2 K TEST IN TEST PHASE 2.2 ................................. FIGURE 5-51: TEST PHASE 2.3 BSC03 DETUNING SWEEP, AT ISO 𝑃𝑖, LEFT = DETUNING RAMPS, CENTRE = MEASURED 𝑃𝑡 AND 𝐸𝑎𝑐𝑐, RIGHT = MODELLED 𝑃𝑡 AND 𝐸𝑎𝑐𝑐 ................................................................................................ FIGURE 5-52: TEST PHASE 2.3 BSC03 DETUNING SWEEP, AT ISO 𝑃𝑖, LEFT = DETUNING RAMPS, CENTRE = MEASURED 𝜙𝑖, 𝜙𝑟 AND 𝜙𝑡, RIGHT = MODELLED 𝜙𝑖, 𝜙𝑟 AND 𝜙𝑡 ............................................................................................ FIGURE 5-53: LORENTZ FORCE INDUCED FREQUENCY SHIFT ON BSC03, CHECKED AGAINST THE SENSITIVITIES MEASURED IN

FIGURE 5 -

 5 FIGURE 5-23: QM THERMAL SHIELD COOL-DOWN DURING TEST PHASE 1 (LEFT), AND SIMSCAPE REPRODUCTION AT ISO CONDITIONS (RIGHT) ............................................................................................................................. FIGURE 5-24: QVB THERMAL SHIELD COOL-DOWN DURING TEST PHASE 1 (LEFT), AND SIMSCAPE REPRODUCTION AT ISO CONDITIONS (RIGHT) ............................................................................................................................. FIGURE 5-25: ROOT-CAUSE ANALYSIS OF THE QM THERMAL SHIELD MASS FLOWRATE MODEL TO TEST MISMATCH ........... FIGURE 5-26: CV601 AND CV651 FLOW PATH IN THE TESTQVB ........................................................................... FIGURE 5-27: QM 80 K LOOP HEAT LOAD DISTRIBUTION OVER SEVERAL WORKING POINTS, DURING TEST PHASE 2.1 ........ FIGURE 5-28: QM CHIMNEY LAYOUT AND HEAT LOADS ........................................................................................ FIGURE 5-29: 2 K LOOP HEAT LOAD DISTRIBUTION AND BALANCE OVER TEST PHASE 2.1; EACH WP IS AVERAGED OVER 5 TO 15 MIN; THE CONDITIONS ARE DETAILED IN THE LEGEND ("2K" OR "4K" MEANS 2 K OR 4 K MODE, "TRF" MEANS TRANSFER THROUGH THE JUMPER, "L" MEANS 2 K TANK LEVEL AND "10K" MEANS 10 K LOOP MASS FLOW RATE ... FIGURE 5-30: CRYOGENIC PROCESS MODEL CONTROL AND MODELLING ARCHITECTURE ADOPTED FOR THE PRECALIBRATIONS OF THE PI CONTROLLERS ............................................................................................................................. FIGURE 5-31: GENERIC ARCHITECTURE OF THE PI CONTROL LOOPS IN THE SIMULINK AND IN THE PLC CONTROL BOXES ...... FIGURE 5-32: MODEL VALIDATION OF THE EHX43 PI CONTROLLER; LEFT = FLANGE TEMPERATURE TTX44, CENTRE = WINDOW TEMPERATURE TTX43 AND RIGHT = EHX43 HEATER VOLTAGE AND POWER ...................................................... FIGURE 5-33: MEASURED COUPLER FLANGE AND WINDOW TEMPERATURES, AT THE FIRST 80 K COOL-DOWN OF TEST PHASE 2.1, LEFT = FLANGE TEMPERATURES, CENTRE = WINDOW TEMPERATURES, RIGHT = QM THERMAL SHIELD NITROGEN TUBE TEMPERATURES (TT682 IS AT THE COUPLER 2 OUTLET, TT683 IS AT THE COUPLER 1 OUTLET) ..................... FIGURE 5-34: EXPERIMENTAL FAULT SIMULATION OF THE EH144/244 FLANGE TEMPERATURE CONTROLLERS ................. FIGURE 5-35: EXPERIMENTAL FAULT SIMULATION OF THE EH143/243 WINDOW TEMPERATURE CONTROLLERS .............. FIGURE 5-36: 80 K COOL-DOWN AT THE START OF TEST PHASE 2.2, LEFT, THE THERMAL SHIELD OUTPUT TEMPERATURES, CENTRE, THE NITROGEN MASS FLOWMETERS, RIGHT, THE NITROGEN INLET PRESSURE ......................................... FIGURE 5-37: PERFORMANCE OF THE 10 K LOOP MASS FLOW RATE CONTROL, OVER A START-UP AT 08:45 THEN SETPOINT CHANGE AT ABOUT 10:00; LEFT = THE CONTROL VALVE (CV02), CENTRE = THE CONTROLLED MASS FLOWRATE, RIGHT = THE BOUNDARY PRESSURE CONDITIONS; THE SEQUENCE COMES FROM TEST PHASE 2.2, IN 4 K MODE .................... FIGURE 5-38: TEST PHASE 1 (4 K MODE) EXPERIMENTAL COMMISSIONING AND FINE-TUNING OF THE 4 K TANK PRESSURE CONTROLLER, TOGETHER WITH THE 4 K AND 2 K TANK LEVEL CONTROLS ACTIVE ................................................ FIGURE 5-39: TEST PHASE 2.2 (2 K MODE) EXPERIMENTAL COMMISSIONING AND FINE-TUNING OF THE 2 K TANK PRESSURE CONTROLLER ........................................................................................................................................ FIGURE 5-40: TEST PHASE 2.2 STEADY-STATE POINT IN 2 K MODE, ALL REGULATIONS ACTIVE ....................................... FIGURE 5-41: THE PRINCIPLE OF THE LQ VERSUS A DOUBLE PI CONTROLLER STRATEGY ................................................ FIGURE 5-42: MATLAB MODEL FORM CONVERSIONS .......................................................................................... FIGURE 5-43: COMPARISON OF THE PROCESS NON-LINEAR MODEL TO THE LINEARIZED MODEL, LEFT, THE LEVEL CONTROL, RIGHT, THE PRESSURE CONTROL ............................................................................................................... FIGURE 5-44: A CONTROLLER BASED ON THE ERROR ............................................................................................. FIGURE 5-45: A CONTROLLER BASED ON A FULL-STATE FEEDBACK ........................................................................... FIGURE 5-46: A CONTROLLER BASED ON AN LQ COMBINED WITH A STATE OBSERVER .................................................. FIGURE 5-47: SIMULATED COMPARISON OF THE DOUBLE PI (TOP) VERSUS LQ (BOTTOM), IN DISTURBANCE REJECTION OF A 2.5 W HEAT INPUT AT 12:10 AND A 5 W INPUT AT 12:35 ........................................................................... FIGURE 5-48: CONTROL ARCHITECTURE OF THE PROTOTYPE CRYOMODULE ADOPTED FOR THE REAL TESTS OF THE LQ + OBSERVER CONTROLLER DURING TEST PHASE 2.2 ........................................................................................ FIGURE 5-49: TEST OF THE LQ + OBSERVER SIMULINK CONTROL BOX, AT 18:24 DURING TEST PHASE 2.2, IN 2 K MODE ... FIGURE 5-50: REPRODUCTION (RIGHT) AT ISO PI (LEFT) OF THE MEASURED TRANSMITTED POWER 𝑃𝑡 (CENTRE) AND ACCELERATING VOLTAGE 𝐸𝑎𝑐𝑐 ON BSC03 DURING THE FIRST 2 K TEST IN TEST PHASE 2.2 ................................. FIGURE 5-51: TEST PHASE 2.3 BSC03 DETUNING SWEEP, AT ISO 𝑃𝑖, LEFT = DETUNING RAMPS, CENTRE = MEASURED 𝑃𝑡 AND 𝐸𝑎𝑐𝑐, RIGHT = MODELLED 𝑃𝑡 AND 𝐸𝑎𝑐𝑐 ................................................................................................ FIGURE 5-52: TEST PHASE 2.3 BSC03 DETUNING SWEEP, AT ISO 𝑃𝑖, LEFT = DETUNING RAMPS, CENTRE = MEASURED 𝜙𝑖, 𝜙𝑟 AND 𝜙𝑡, RIGHT = MODELLED 𝜙𝑖, 𝜙𝑟 AND 𝜙𝑡 ............................................................................................ FIGURE 5-53: LORENTZ FORCE INDUCED FREQUENCY SHIFT ON BSC03, CHECKED AGAINST THE SENSITIVITIES MEASURED IN

  -64: CPL4 (BSC03) COUPLER ANTENNA AND INTERCEPT WATER HEAT LOADS MEASURED IN WARM CONDITIONING TEST BENCH (LEFT) AND REPRODUCED WITH SIMSCAPE (RIGHT) AT ISO 𝑃𝑖 ........................................................ 270 FIGURE 5-65: CPL2 (BSC04) CERAMIC WINDOW TEMPERATURE -IN CRYOMODULE COLD CONDITIONING, 4 K MODE, DETUNED -MEASURED (CENTRE) AND SIMULATED (RIGHT) AT ISO 𝑃𝑖 (LEFT) .................................................... 271 FIGURE 5-66: CPL2 (BSC04) HEAT LOAD TO WATER -IN CRYOMODULE COLD CONDITIONING, 4 K MODE, DETUNED -MEASURED (LEFT) AND SIMULATED (RIGHT) AT ISO 𝑃𝑖 .................................................................................. 272 FIGURE 5-67: CPL2 (BSC04) 10 K LOOP INLET/OUTLET TEMPERATURES (TOP) AND COUPLER/CAVITY FLANGE TEMPERATURES (BOTTOM) -IN CRYOMODULE COLD CONDITIONING, IN 4 K MODE, DETUNED -MEASURED (LEFT) AND SIMULATED (RIGHT) AT ISO 𝑃𝑖 ................................................................................................................................. 273 FIGURE 5-68: CPL4 (BSC03) CERAMIC WINDOW TEMPERATURE -IN CRYOMODULE COLD CONDITIONING, 2 K MODE, TUNED THEN DETUNED -MEASURED (CENTRE) AND SIMULATED (RIGHT) AT ISO 𝑃𝑖 (LEFT) ............................................. 274 FIGURE 5-69: CPL4 (BSC03) HEAT LOAD TO WATER -IN CRYOMODULE COLD CONDITIONING, 2 K MODE, TUNED THEN DETUNED -MEASURED (LEFT) AND SIMULATED (RIGHT) AT ISO 𝑃𝑖 .................................................................. 274 FIGURE 5-70: CPL4 (BSC03) 10 K LOOP INLET/OUTLET TEMPERATURES (TOP) AND COUPLER/CAVITY FLANGE TEMPERATURES (BOTTOM) -IN CRYOMODULE COLD CONDITIONING, IN 2 K MODE, TUNED THEN DETUNED -MEASURED (LEFT) AND SIMULATED (RIGHT) AT ISO 𝑃𝑖 ................................................................................................................. 275 FIGURE 6-1: THE PROTOTYPE COUPLER CUFF (TOP) VERSUS THE COUPLER SERIES DESIGN (BOTTOM), VIEWED IN CAD (RIGHT) AND SIMSCAPE MODEL (LEFT), COURTESY OF SCK•CEN ............................................................................... 281 FIGURE 6-2: LAYOUT OF THE SIL/HIL ARCHITECTURE DEVELOPED FOR THE MINERVA PROTOTYPE CRYOMODULE ........... 284 FIGURE 6-3: MIND-MAP OF THE COMBINED RF AND CRYOGENIC COMBINED MODELLING AND THEIR APPLICATIONS .......... 288

Figure 1 - 2 :

 12 Figure 1-2: nuclear waste classification in France, from[START_REF] Andra | Waste classification[END_REF]

Figure 1 - 3 :

 13 Figure 1-3: radioactive waste volume per economic sector (2018), adapted from [4] 1.1.2 From the discovery of the nuclear chain reaction to fission power plants

Figure 1 - 5 :

 15 Figure 1-5: 235 U fission principle

• 28 .

 28 5 tons 238 U • 0.3 ton 235 U remaining fissile material • 0.1 ton 236 U from sterile capture • 0.25 ton Pu • 1 ton Fission Products • 0.02 ton Np • 0.002 ton Am • 0.001 ton Cm Those quantities vary as a function of the reactor technology and fuel enrichment. For

Figure 1 -

 1 Figure 1-8: radiotoxicity of the fission products and actinides relative to the original uranium ore, as a function of the time after discharge from [8]

Figure 1 -

 1 Figure 1-10: example of Pu stock stabilization in a multiple-recycling strategy versus monorecycling and no recycling, from[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF]

Figure 1 -

 1 Figure 1-11: the possible minor actinide irradiation methods in the context of the Pu fuel closure cycle, adapted from[START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF]

Figure 1 -

 1 Figure 1-12: MYRRHA three-component coupling, from[START_REF] Bouly | R&D pour l'accélérateur du projet MYRRHA[END_REF]

Figure 1 -

 1 Figure 1-13: from the concepts evaluated during PDS-XADS to the EUROTRANS baseline; FP5 (1998-2002) and FP6(2002)(2003)(2004)(2005)(2006) are the EC framework programs, image from[START_REF] Bruyn | The MYRRHA project: Current design status & Evolution ADS to FR system[END_REF]

1. 3 . 3

 33 The 2010-2019 period: specifications towards the construction On the experimental side, EUROTRANS co-financed the GUINEVERE experiments, as a follow-up of the MUSE experiments[START_REF] Billebaud | The GUINEVERE Project for Accelerator Driven System Physics[END_REF], starting in 2006. The VENUS sub-critical leadcooled fast reactor at SCK•CEN was coupled to a new version of the GENEPI accelerator: GENEPI3C. Due to the needed commissioning time, the facility operation only started in 2010, during the next EC-funded project, FREYA[START_REF] Kochetkov | CURRENT PROGRESS AND FUTURE PLANS OF THE FREYA PROJECT[END_REF], coordinated by SCK•CEN.

Figure 1 -

 1  the VENUS reactor coupled to the GENEPI3C at SCK•CEN, from[START_REF] Kochetkov | CURRENT PROGRESS AND FUTURE PLANS OF THE FREYA PROJECT[END_REF]

Figure 1 - 16 :

 116 Figure 1-16: MYRRHA beam structure proposed in MYRTE

Figure 1 -Figure 1 -

 11 Figure 1-19: electrical consumption trade-off superconducting versus normal cavitiesFigure1-19 shows the ratio of the electrical21 to beam power with respect to the duty cycle. For duty-cycles larger than about 0.03 %, the superconducting technology is appealing. In practice, as a rule of thumb, superconducting technology is generally used

  is made of three consecutive sections based on a specific type of cavity: first, single-Spoke cavities up to 101.4 MeV, then double Spoke ones up to 172.3 MeV and ultimately Elliptical cavities up to 601.6 MeV (see section 4.1.1). The main linac specification are summarized in Annex 7.1, and the layout of the whole accelerator is presented in Figure 1-21.

Figure 1 -

 1  the MYRRHA accelerator layout, from[START_REF] Bouly | Superconducting LINAC Design Upgrade in View of the 100 MeV MYRRHA Phase I[END_REF]

Figure 1 -

 1 Figure 1-22: planned MINERVA staged implementation of the linac, as of November 2021, personal communication, by courtesy of SCK•CEN MINERVA will supply two dedicated target buildings: a Proton Target Facility (PTF) for ISOL experiments in view of medical applications and fundamental physics; a Fusion Target Station (FTS) for irradiation qualification tests of advanced material meant for fusion reactors.

Figure 1 -

 1 Figure 1-23: the MINERVA site at SCK•CEN in Mol, personal communication, by courtesy of SCK•CEN

Figure 2 - 3 :

 23 Figure 2-3: cryomodules and auxiliary systems in the tunnel, personal communication by courtesy of SCK•CEN

Figure 2 - 5 :

 25 Figure 2-5: synoptic view of the QM model developed for SPIRAL2 from[START_REF]Accelerator-driven Systems (ADS) and Fast Reactors (FR) in Advanced Nuclear Fuel Cycles: A Comparative Study[END_REF]

2 . 5

 25 Perspective of a cryomodule control validation tool[START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF]

  , LEP, KEKB, CESR Cavities: ESS, XFEL, LCLS-II, SNS Magnets: HERA, Tevatron Magnets: TORE SUPRA, LHC Magnets: SSC

Figure 3 -

 3 Figure 3-2 summarizes the two steps of the superfluid helium production: • Isobaric subcooling of the incoming saturated liquid or supercritical helium down to around 2.5 K by the exiting cold vapor, in a heat exchanger (HX): (1) → (1') • Isenthalpic expansion of the helium from 1.1 bar (saturated) or 3 bar (supercritical) down to 30 mbar through a flow restriction: in this helium thermodynamic zone, the helium temperature decreases as the pressure drops: (1') → (2). A control valve, so-called the Joule-Thomson valve (J-T), ensures both the flow restriction and the control of the liquid level in the cavity tank

Figure 3 Figure 3 - 3 :

 333 Figure 3-2: subcooled JT expansion

Figure 3 - 9 :

 39 Figure 3-9: the fluid property P-u and P-𝑢 ̅ spaces from[START_REF] Peterson | Normal Cryogenic Fluid Behavior[END_REF]

Figure 3 -

 3 Figure 3-12: specific heat comparisonFor all the material, except niobium, the reference Excel functions extract the data from several sources:

Figure 3 -

 3 Figure 3-13: phonon (Kgs) and electron (Kes) thermal conductivity contributions in pure niobium, RRR = 30, AGS = 50 µm, fit "BCS theory"

  To compute the necessary fluid interactions with the walls (the friction and heat transfer), the Euler equations are complemented by macroscopic experimental relations, more detailed in the next section. The three Euler equations are solved by a new Simscape block that stands for a unit discretized fluid volume, as shown in Figure 3-15: I is the centre of the mesh element or the internal node, A and B the inlet and outlet nodes. The first Euler equation is the mass conservation equation (or transport equation) over the discretized element of volume V contained between node A and node B:

Figure 3 -

 3 Figure 3-15: fluid volume between A and B and node I

Figure 3 -

 3 Figure 3-16: the generic duct model Cryoscape icon

Figure 3 -

 3 Figure 3-17: the segmented generic duct

Figure 3 -

 3 Figure 3-19: segmented heat equation combined block, the thermal link

  -21.

Figure 3 -

 3 Figure 3-20: native Simscape radiative heat transfer block

Figure 3 -

 3 Figure 3-22: 1-zone tank model

Figure 3 - 23 :•

 323 Figure 3-23: relationships among individual physical models, from[START_REF] Li | Simulation of mass and heat transfer in liquid hydrogen tanks during pressurizing[END_REF]

Figure 3 -

 3  hydrogen tank temperature and velocity profiles from[START_REF] Grayson | CFD Modeling of Helium Pressurant Effects on Cryogenic Tank Pressure Rise Rates in Normal Gravity[END_REF]

Figure 3 -

 3 Figure 3-26: comparison of the different 2-zone tank models, the green arrows stand for the mass transfers, the yellow ones, for the heat transfers This model would be similar to the NASA lumped model, except 2 major changes:

  with warm vapor L-V interface phase change = vapor or liquid mass flow rate source Rollin film in superfluid He « Neck effect »: the vapor intercepts and evacuates the heat before reaching the liquid phase

Figure 3 - 28 :

 328 Figure 3-28: Cryoscape segmented fluid duct with wall mass and external radiation heat transfer 3.4.2 The control valve models 3.4.2.1 Valve flow models

  Part 1: the flow coefficient 𝐾𝑣 model as a function of the % 𝑜𝑝𝑒𝑛𝑖𝑛𝑔 • Part 2: the mass flowrate 𝑚̇ as a function of 𝐾 𝑣 and boundary conditions (P, T) About the flow model part 1, the 2 classical 𝐾 𝑣 = 𝑓(% 𝑜𝑝𝑒𝑛𝑖𝑛𝑔) characteristics are:

Equations ( 3 -

 3 24 ) and ( 3-26 ) are common formulae used by the valve manufacturers to size the valves, and are referenced by the International Society of Automation (ISA) in [140]. Another equation for compressible flow is actually used internally by IJClab cryogenic experts, seen in ( 3-27 ). This equation gives the save result as ( 3-26 ) at room temperature, but is significantly different at 4 K. 𝑚̇𝑐 𝑜𝑚𝑝 = 519 * 10 -5 * √ 𝑃 𝑖𝑛 𝑇 𝑖𝑛 * √1 -min(𝑋 ; 𝑋 𝐶 ) * 𝐾 𝑣 3600 * √𝜌 𝑖𝑛 * 𝑃 𝑖𝑛 * min(𝑋 ; 𝑋 𝐶 ) ( 3-27 ) This second equation for compressible flow is also implemented, to be used as a reference. To make those equations available in Simscape, a new Cryoscape block, which icon is seen in Figure 3-30 is implemented. In this block, the user can choose which equation set to use, ( 3-24 ) & ( 3-26 ) or ( 3-24 ) & ( 3-27 ). Then, the selection of the compressible or incompressible equation relies simply on the vapor mass fraction. In two-phase mixture, a blending is done between the compressible and incompressible equations, linearly with the vapor mass fraction.In this block, the user may also parametrize the flow model part 1, i.e. the flow coefficient

Figure 3 -

 3 Figure 3-30: the valve flow model

Figure 3 -

 3 Figure 3-31: MINERVA cryogenic valve main features

Figure 3 -

 3 Figure 3-32: combined thermal and flow model of a cryogenic valve (left) integratd in a sub-assembly (right)

Figure 3 -

 3 Figure 3-33: simulated data of IJClab B106 group pumping capacity as a function of the pressure The total volume of fluid between the cryomodule and the pumping groups comes from IJClab CAD files.Figure 3-34 displays the Simscape model (left) and the corresponding CAD model (for clarity, the infrastructure section with the flowmeter is skipped).

Figure 3 -

 3 34 displays the Simscape model (left) and the corresponding CAD model (for clarity, the infrastructure section with the flowmeter is skipped).

Figure 3 - 4

 34 Figure 3-34: 2 K pumping group Simscape model (left) and the corresponding CAD picture (right)

Figure 3 -

 3 Figure 3-35: the valve box "4 K" tank By contrast, the QM phase separator is made of two distinct tanks, the cavity tank (~72 L) and the buffer tank (~11 L); they are connected by means of two communication pipes: by the top communication tube, and by the bottom communication tube, as seen in Figure 3-36.

Figure 3 -

 3 Figure 3-36: QM phase separator main view and cross-sectionsBoth tanks are fundamentally different in shape, but the modelling approach is the same: the tank fluid volume is a 2-phase tank block from Cryoscape, and the entire wall mass is represented as a unique thermal mass. The thermal mass is punctual, but the external

•

  Figure 3-38: table layout (left) and corresponding Cryoscape model (right) A 2D discretization of the table, was chosen: a total of ~100 Cryoscape blocks are necessary to resolve the individual heat loads, reproduce the asymmetric temperature profiles, and comply with the instrumentation location. The model in shown in the righthanded side of Figure 3-38.

  from the valves (only in the QVB), the supports (tie rods in the QVB, table in the QM) and the instrumentation wires. The MINERVA prototype thermal shields are made of several folded and stamped 2 or 3-mm sheets of OFHC copper to form a roughly closed cylinder. They are cooled by active circulation of liquid nitrogen, running in a brazed serpentine copper tube.

Figure 3 -

 3 Figure 3-39: the testQVB thermal shield discretization (centre), the 80 K circuitry (right) and Simscape implementation (left) The chosen discretization is shown to the right in Figure 3-39 for the testQVB (15 elements) and in Figure 3-40 for the prototype QM (14 elements): the element borders are highlighted by the purple lines.

Figure 3 -

 3 Figure 3-40: the prototype QM thermal shield discretization (right), the 80 K circuit in yellow arrows and the Simscape model (left)

Figure 3 -

 3 Figure 3-41: the prototype of the "2.5 g/s" heat exchanger of the ESS Spoke cryomodule

  39 LMTD (Log Mean Temperature Difference) and NUT (Number of Transfer Units) are two heat exchanger performance calculation methods based on standard abacus

Figure 3 - 42 :

 342 Figure 3-42: Simscape model of the HX

Figure 3 -

 3 Figure 3-43 shows the Cryoscape model comparison of the exchanged power versus those datasets, under the same conditions and HX geometry. The agreement is within +/-5 %, proving the model can both reproduce simulations, although not surprising, as DATE likely uses a similar model, and real tests.

Figure 3 - 43 :

 343 Figure 3-43: Simscape against CERN tests or DATE simulations

Figure 3 -

 3 Figure 3-44: simulation with Simscape of different plate materials As an illustration of the HX model capability, the Cryoscape model is used to predict the expected temperature upstream of the J-T valve in the MINERVA prototype conditions.

Figure 3 -Figure 3 -

 33 Figure 3-45 shows the temperature profile along the HX. The HX exhibits an excellent performance, at 2.18 K at the HP outlet. One should keep in mind that this result is only theoretical: the jumper heat loads on the vapor return raise the LP inlet temperature to about 2.5-3 K, as seen in part 5.2.7, resulting in more vapor production by flash.

Figure 3 - 47 :

 347 Figure 3-47: schematics of the IJClab room temperature ΔP test bench

Figure 3 -

 3 Figure 3-48 shows the comparison of the test versus different deformed ΔP models, from left to right: no deformation, the supplier coefficients, the optimal coefficients found.

Figure 3 - 18 Figure 3 - 49 :

 318349 Figure 3-48: LP-side deformed core ΔP models versus the experiment, the solid lines are the linear regressions of the dotsThe ΔP shown in this figure is only the core ΔP; the fittings and the expansion/contraction participations are removed with correlations, and actually only stand for 5-10 %. One sees that the supplier coefficients match well but can be optimized: the optimal coefficients are actually more physical (more laminar and more quadratic friction). The influence of the static pressure on the HP side (thus the plate deformation) is fully negligible, at least at room temperature.

Figure 3 -Figure 3 - 51 :

 3351 Figure 3-50: the MINERVA open-loop cryogenic process modelAccelerator facilities feature a hierarchized and distributed cryogenic control architecture[START_REF] Vassal | Etude d'un réseau cryogénique multi-clients pour SPIRAL2[END_REF],[START_REF] Bradu | Modélisation, simulation et contrôle des installations cryogéniques du Cern[END_REF]: the cryomodule, that belongs to the lowest control layer (field layer), has a dedicated controller for its own process. Above the field layer are the different supervision layers. The MINERVA prototype control architecture is actually rather close to the accelerator field layer, with some differences in the number of I/Os and in the software environment used. The MINERVA prototype QM and QVB controller is a Programmable Logic Controller (PLC) from Siemens; it is fully separated from the infrastructure control (2 other distinct PLCs), and as said earlier, only a few variables are shared with the RF controls. This architecture is shown in Figure3-51.

6 ;

 6 the name of the sensors and actuators can be retrieved from the instrumentation diagram in 7.1.

Figure 3 - 52 :

 352 Figure 3-52: MINERVA cryogenic process model architecture, compliant with whichever control box This model architecture, where the open-loop Simscape process model is embedded in a sub-system enables to control it either using the Simulink control box, or the PLC control box. This latter object is actually an embedded MATLAB function that can read

  Cavity field model • Low-level RF (LLRF) model • High-power line RF model • RF field and resonance control algorithm • Cavity mechanical tuning and detuning models • Cavity heat transfer model

Figure 4 - 1 :•

 41 Figure 4-1: the different heat loads generated by the RF parts

Figure 4 - 2 :

 42 Figure 4-2: cavity resonant field class as a function of their family, adapted from [80], [145] and [146] -the colour bars are the E-field intensity

  -5 from[START_REF] Gurevich | General aspects of superconductivity[END_REF]. His theory considers both charge-carrier types, the normal and the superconducting electrons as fluids. The latter moves without friction, thus short-circuiting the normal ones and producing this observed almost perfect electrical conductivity.The renowned second London equation[149]

Figure 4 - 4 :

 44  the Meissner effect illustrated in a video from[START_REF] Xu | FRIB Cryomodule Design and Production[END_REF]

Figure 4 - 5 :

 45 Figure 4-5: London two-fluid charge carrier model, picture from[START_REF] Xu | FRIB Cryomodule Design and Production[END_REF]

Figure 4 - 6 :Figure 4 - 7 :

 4647  illustration of the variation of the critical magnetic field as a function of T, from[START_REF] Xu | FRIB Cryomodule Design and Production[END_REF]

Figure 4 -

 4 Figure4-8: secondary electron trajectories in a multipacting emission from[START_REF] Olry | Recent Progress of ESS Spoke and Elliptical Cryomodules[END_REF]

Figure 4 - 9 :

 49 Figure 4-9: cavity power balance

Figure 4 -

 4 Figure 4-11, adapted from [80] and [165], represents the cavity RLC equivalent circuit which voltage is connected to a matched source (the generator or amplifier) by a transmission line of characteristic impedance 𝑍 0 (the RF power lines) and by an idealized transformer (the coupler).

Figure 4 -

 4 Figure 4-11: cavity equivalent RLC circuit loaded and external circuit and beam This equivalent circuit actually represents the whole RF process; it can be compared with the real hardware presented in section 4.4.3.2. It can be used from two different points of view: (i) to express the cavity impedance as seen from the transmission line and presented to the generator; (ii) or from the cavity point of view, to calculate the cavity voltage induced by the generator currents.

Figure 4 -

 4 Figure 4-12: S21 parameter, measured on BSC04 at 2 K in the MINERVA prototype cryomodule, and reproduced with the cavity field model, at +/-2000 Hz around the resonance frequency

Figure

  Figure 4-16 (bottom) shows a crossed 𝑄 𝑖 & ∆𝑓 sweep with the cavity field model, in a transmission chart. This chart shows in iso colours the ratio

Figure 4 -

 4 Figure 4-17: FPGA-based digital cavity field and resonance control main philosophy

Figure 4 -

 4 Figure 4-22: MINERVA Spoke cavity sensitivity to pressure. Each of the 3 figures is the average pressure (in bar) in the zone indicated on the left-handed side. For instance, "010" means 1 bar in the cavity tank and 0 bar for the beam and insulation vacuums; the bold digits are were the pressure variation is applied In this figure, 𝐾 𝑃 is the detuning pressure sensitivity, in Hz/mbar, that follows: ∆𝑓 = 𝐾 𝑃 * ∆𝑃 ( 4-44 )

Figure 4 - 24 :

 424 Figure 4-24: MINERVA prototype Spoke cavity S21 shift during a 4 K cool-down

Figure 4 -

 4 Figure 4-25: resonance frequency shift versus flange displacement, at cryogenic temperature on BSC04, from[START_REF] Bonne | Modélisation et contrôle des grands réfrigérateurs cryogéniques[END_REF]

Figure 4 -

 4 Figure4-27 displays the two MYRTE measurements and the MINERVA one. The amplitude of the detuning at cold temperature is rather comparable between MINERVA and MYRTE. At warm temperature, the spectrum is really different, showing the transfer function between warm and cold is not a simple coefficient.

Figure 4 -

 4 Figure 4-26: phaser of an overcoupled cavity, without beam and 45 Hz detuning

Figure 4 -

 4 Figure 4-29: cavity "wall model"

Figure 4 -

 4 Figure 4-34: temperature instrumentation of the couplers in the conditioning test bench. Each window is equipped with a PT100, the other numbers refer to thermocouples. The antenna and cuff heat exchanger water flowmeters are not shown.

Figure 4 -

 4 Figure 4-35: LPSC's MATLAB model schematic (top) and CAD correspondence (bottom)4.7.2.2 A new Simscape heat transfer model

Figure 4 -•

 4 Figure 4-36: LPSC's model strictly translated into SimscapeA comprehensive comparison of the model outputs (temperature ranges, exchanged heat flows) was done. The deviations are within +/-5%, which is satisfactory enough, considering the wide operating condition range (cryogenic and room temperature); they are mainly due to differences in the material property.

Figure 4 -

 4 Figure 4-37: the latest version of the Simscape coupler modelIn this figure, the three layers of the coupler cuff, the stainless-steel, the copper and the intermediate adherence layer in nickel are modelled by three parallel layers of thermal conduction blocks. In the following, those layers are respectively noted "SS", "Cu" and "Ni". Although negligible in thickness, the copper layer in fact dominates the conduction at temperature below 50 K, especially at high copper RRR as highlighted in Figure4-38.

Figure 4 -

 4 Figure 4-38: heat load contribution by the three coupler cuff layers, as a function of the RRR and average temperature (thickness = 2 mm (SS) -1.5 µm (Ni) -15 µm (Cu)

  network for on-line RF heat load update

Figure 4 -

 4 Figure 4-41: laser confocal image (left) and EDX continuous composition measurement over an SEM image (right) of a cuff cross-section

Figure 4 3 . 1

 431 Figure 4-42: RRR sample measurement insert at IJClab

Figure 4 -

 4 Figure 4-43: simplification of the coupler as a lossless coaxial line

Figure 4 -Figure 4 -

 44 Figure 4-44: validation of the E field (in V/m) from the analytical model (right) in pure travelling wave mode, 80 kW, against CST results (left)

Figure 4 -

 4 Figure 4-46: H field (A/m) in the cavity and along the coupler @ Pi = 1 W. Left = offresonance, middle = at resonance & matched with beam, right = at resonance and overcoupled To conclude, the needed 4 coupler field analytical model boundary conditions 𝑃 𝑖 , 𝑃 𝑟 , 𝜙 1 and 𝜙 2 are now related to the cavity field model variables, to predict the field along the coupler in whatever beam current, cavity coupling 𝑄 𝑖 and tuning.

Figure 4 -

 4 Figure 4-48: the updated RF heat load block, including the simple analytical "field" model

Figure 4 -

 4 Figure 4-49: plots of the RF field settings (top left), the detuning effects (top right) and the power balance (bottom)

Figure 4 -

 4 Figure 4-50: E and H field along the coupler for sequences 3, 4 and 5. The abscise is the distance (negative) from the coupling plane, i.e. about the antenna tip. The window is located at about -0.42 m.

Figure 4 -

 4 Figure4-51 (left-handed side) shows the variation of the RF and non-RF heat load sources in the coupler and in the cavity; the coupler heater controls (especially the window) react to the RF heat loads by decreasing the electrical power. In the same chart, to the righthanded side, the variation of the heat distribution to the heat sinks is shown. It appears that, the cavity RF loads58 and the antenna loads, which have the highest amplitude almost immediately reach their respective heat sinks (helium bath and water).

Figure 5 - 2 :

 52 Figure 5-2: schematics of the mass exchanges in the 2 K tank

Figure 5 - 5 :

 55 Figure 5-5: CV05 simulated mass flowrate, against the level-corrected measured mass flowrate, with the compressible flow equation from IJClab (left) and from the ANSI standard (right)

Figure 5 - 6 :

 56 Figure 5-6: 80 K loop CV601 and CV651 Simscape model comparison versus a test at RT

Figure 5 -

 5 7 is unphysically too warm by some tenths of a K, likely due to interactions with nearby warmer parts, meaning the heat exchange on this side is not reliable During test phase 2.1, a characterization of the HX was performed. This test, called WP20-23 (in Figure5-29, explicated in part 5.2.8), consists in a dynamic heat load simulation of 4 heater electrical power steps (0 -9 -19 -27 W), equally spread over 1 hour, in 2 K mode. A Simscape model combining the heat exchanger and the CV05 valve was setup: the comparison versus the model at iso conditions is shown in Figure5-8.

Figure 5 -

 5 Figure 5-8: HX mass flowrates (left), HX power on the HP side (centre) and on the LP side (right) during WP20-23 from test phase 1

Figure 5

 5 Figure 5-7: MINERVA prototype HX temperature instrumentation

Figure 5 - 9 :

 59 Figure 5-9: model of the HX downstream pipe for TT030 discrepancy investigation A second assumption was tested: a direct heating of the sensor by a parasitic thermal contact, for instance by conduction through the MLI or by a poor wire thermalization.

Figure 5

 5

Figure 5 - 10 :

 510 Figure 5-10: HX mass flowrates (left), HX power on the HP side (centre) and on the LP side (right) for test phase 2.2 WP11

Figure 5 -

 5 Figure 5-11: definition of the heat flows around the feet

Figure 5 -

 5  balance between the total heat from the feet (left) and the sum of the heat diffusing further away (right), during a 4 K mode steady-state sequence from test phase 2.1A second cross-check is performed, this time around the cavity support arms, during the working point WP5 from test phase 2.1 (Figure5-29 from section 5.2.8): the heat from the cavity frame to the 10 K loop intercepts is shown in Figure5-13 for both the cryomodule left and right sides. The asymmetry between them is due to a permanent malfunction in the 10 K loop: the right-handed parallel intercept line does not circulate, probably due to a vapor lock.

Figure 5 -

 5 Figure 5-13: heat from the table frame towards the 10 K loop intercept, during WP5

Figure 5 -

 5 14 (left-handed chart), and is fully intercepted by the 10 K loop: the fluid sensible heat method from Figure 5-19 indicates 0.6 W, a 30 % discrepancy then, but acceptable again.

Figure 5 -

 5 Figure 5-14: heat from the 10 K loop intercepts to the 2 K tank, during WP5

Figure 5 -

 5 Figure 5-16: heat flow definitions in the air-side area

Figure 5 -

 5 Figure 5-18: heat flow definitions around in the 10K loop area

  -20: the lefthanded side shows the fluid-side measurement, while the 2 right-handed side charts show the material-side at respectively RRR = 30 and RRR = 120 (the average measured one).

Figure 5 -

 5 Figure 5-20: fluid-side versus material-side coupler to 10K loop heat load calculation

Figure 5 -

 5 Figure 5-19: 10 K loop component-bycomponent heat load, measured by the fluid sensible heat method

Figure 5 -

 5 [START_REF] Saugnac | THE MYRRHA SPOKE CRYOMODULE[END_REF], those 2 should be equal. This is indeed what is observed within +/-20 % tolerance in Figure5-22.

Figure 5 -Figure 5 -

 55 Figure 5-22: heat flow from 10 K to 2 K (left) compared to the heat from 80 K to 10 K (copper RRR = 30, centre versus RRR = 120, right), all computed with the material conduction methodDuring this sequence, the 80 K loop was actually not running at its nominal level, due to an accidental transient. Consequently, the temperatures at stake in 𝑄 ̇80→10𝐾 were

Figure 5 -

 5 Figure 5-23: QM thermal shield cool-down during test phase 1 (left), and Simscape reproduction at iso conditions (right)

Figure 5 -

 5 Figure 5-24: QVB thermal shield cool-down during test phase 1 (left), and Simscape reproduction at iso conditions (right)Overall, in both thermal shields, the competition between the material diffusion and the fluid convection is, in average, correctly reproduced; this is a good point for the controloriented tasks, where a faithful transient behaviour is required.

Figure 5 -mismatch

 5 Figure 5-25: root-cause analysis of the QM thermal shield mass flowrate model to test

Figure 5 -

 5 Figure 5-26: CV601 and CV651 flow path in the testQVB

Figure 5 - 28 :

 528 Figure 5-28: QM chimney layout and heat loads

Figure 5 -

 5 Figure 5-29: 2 K loop heat load distribution and balance over test phase 2.1; each WP is averaged over 5 to 15 min; the conditions are detailed in the legend ("2K" or "4K" means 2 K or 4 K mode, "TRF" means transfer through the jumper, "L" means 2 K tank level and "10K" means 10 K loop mass flow rate All the heat load contributions in this chart were presented earlier in the component model parts, except the green and the yellow ones. The green contributions are the two CTS transient heat loads (the cool-down is slow due to the large poorly thermallyanchored masses), estimated by a material conduction method on the CTS support frames; the total features a coherent tendency, decreasing over time during test phases 2.1 and 2.2.

Figure 5 -

 5 Figure 5-30: cryogenic process model control and modelling architecture adopted for the precalibrations of the PI controllersEach cryogenic controller has specific requirements; from the experience in other accelerator facilities, and in order to achieve minimum standardization, it was evaluated that PI controllers could be sufficient for all of them. The generic architecture of the PI control loops is illustrated in Figure5-31.

Figure 5 -

 5 Figure 5-31: generic architecture of the PI control loops in the Simulink and in the PLC control boxesThe role of the PI controller is to build a correction of the nominal output steady-state command, the pre-command, to compensate for the error between the setpoint and the

Figure 5 -

 5 Figure 5-32: model validation of the EHx43 PI controller; left = flange temperature TTx44, centre = window temperature TTx43 and right = EHx43 heater voltage and powerThe reason of those fault simulations is double: to check that the associated temperature drops are not too severe and to see how the controller catches-up, once back in order. The gains and pre-command (about 28 V) were copied into the PLC CPU; the experimental validation occurred during the first 80 K cool-down of test phase 2.1, illustrated in Figure5-33.

Figure 5 -

 5 Figure 5-33: measured coupler flange and window temperatures, at the first 80 K cooldown of test phase 2.1, left = flange temperatures, centre = window temperatures, right = QM thermal shield nitrogen tube temperatures (TT682 is at the coupler 2 outlet, TT683 is at the coupler 1 outlet)

Figure 5 -

 5 Figure 5-34: experimental fault simulation of the EH144/244 flange temperature controllers

Figure 5 -

 5 Figure 5-36: 80 K cool-down at the start of test phase 2.2, left, the thermal shield output temperatures, centre, the nitrogen mass flowmeters, right, the nitrogen inlet pressure

Figure 5 -

 5 Figure 5-37: performance of the 10 K loop mass flow rate control, over a start-up at 08:45 then setpoint change at about 10:00; left = the control valve (CV02), centre = the controlled mass flowrate, right = the boundary pressure conditions; the sequence comes from test phase 2.2, in 4 K mode

Figure 5 -

 5 Figure 5-39: test phase 2.2 (2 K mode) experimental commissioning and fine-tuning of the 2 K tank pressure controller

Figure 5 -

 5 Figure 5-40: test phase 2.2 steady-state point in 2 K mode, all regulations active

Figure 5 -Figure 5 - 42 :

 5542 Figure 5-41: the principle of the LQ versus a double PI controller strategyIn this section, after a brief introduction to the state-space representation, the cryogenic process model linearization is presented; then a comparison LQ versus PI is carried out on the process model, and finally, a real test is attempted on the MINERVA prototype cryomodule.5.3.2.1 State-space representation

Figure 5 -

 5 Figure 5-47: simulated comparison of the double PI (top) versus LQ (bottom), in disturbance rejection of a 2.5 W heat input at 12:10 and a 5 W input at 12:35

Figure 5 -

 5 Figure 5-48: control architecture of the prototype cryomodule adopted for the real tests of the LQ + observer controller during test phase 2.2

Figure 5 -

 5 Figure 5-49: test of the LQ + observer Simulink control box, at 18:24 during test phase 2.2, in 2 K mode

Figure 5 -

 5 Figure 5-50: reproduction (right) at iso Pi (left) of the measured transmitted power 𝑃 𝑡 (centre) and accelerating voltage 𝐸 𝑎𝑐𝑐 on BSC03 during the first 2 K test in test phase 2.2

Figure 5 -

 5 Figure 5-51: test phase 2.3 BSC03 detuning sweep, at iso 𝑃 𝑖 , left = detuning ramps, centre = measured 𝑃 𝑡 and 𝐸 𝑎𝑐𝑐 , right = modelled 𝑃 𝑡 and 𝐸 𝑎𝑐𝑐 Over this detuning sweep, that covers roughly +/-5 bandwidths, the cavity field model response matches +/-5 % versus the test, except at the starting point where the cavity was not precisely at the resonance. This is consistent with what was observed in the model pre-validation against VNA measurement, and again not surprising.

Figure 5 -

 5 Figure 5-52: test phase 2.3 BSC03 detuning sweep, at iso 𝑃 𝑖 , left = detuning ramps, centre = measured 𝜙 𝑖 , 𝜙 𝑟 and 𝜙 𝑡 , right = modelled 𝜙 𝑖 , 𝜙 𝑟 and 𝜙 𝑡

Figure 5 - 53 :

 553 Figure 5-53: Lorentz force induced frequency shift on BSC03, checked against the sensitivities measured in the IJClab vertical cryostat (CV800)

Figure 5 -

 5 Figure 5-54: BSC03 𝑄 0 and X-ray measurements as a function of 𝐸 𝑎𝑐𝑐 during test phase 2.3, in 2 K mode

Figure 5 -

 5 [START_REF] Ghribi | Spiral 2 Cryogenic System for The Superconducting LINAC[END_REF], along with the Simscape reproduction; all the 𝑃 𝑖 values mentioned in the following sections are either CW, or PW-averaged, thus comparable. The coupler instrumentation is summarized in Annex 7.10.

Figure 5 -

 5 Figure 5-55: test phase 2.2 CPL2 (BSC04) coupler warm conditioning in cryomodule; left = imposed 𝑃 𝑖 , centre = measured cuff temperatures, right = Simscape reproduction

Figure 5 -Figure 5 -

 55 Figure 5-56: Simscape reproduction of CPL2 (BSC04) coupler warm conditioning in deliberately wrong tuning conditions

Figure 5 -

 5 Figure 5-58: test phase 2.2 CPL4 (BSC03) coupler warm conditioning in cryomodule; left = imposed 𝑃 𝑖 , centre = measured cuff temperature, right = Simscape reproduction with the multipacting model

Figure 5 -

 5 Figure 5-59: Simscape reproduction of CPL4 (BSC03) coupler warm conditioning without multipacting model

Figure 5 -

 5 Figure 5-61: CPL2 (BSC04) coupler cuff temperatures measured in warm conditioning test bench (centre) and reproduced with Simscape (right) at iso 𝑃 𝑖 (left)

Figure 5 -

 5 Figure 5-62: CPL2 (BSC04) coupler antenna and intercept water heat loads measured in warm conditioning test bench (left) and reproduced with Simscape (right) at iso 𝑃 𝑖 Switching now to CPL4 (BSC03), a similar sequence is used for the Simscape comparison, shown in Figure 5-63. The exact same multipacting model (adjustment #3) as in the previous section was implemented at the location of the window.

Figure 5 -

 5 Figure 5-63: CPL4 (BSC03) coupler cuff temperatures measured in warm conditioning test bench (centre) and reproduced with Simscape (right) at iso 𝑃 𝑖 (left)

Figure 5

 5

Figure 5 -

 5 Figure 5-64: CPL4 (BSC03) coupler antenna and intercept water heat loads measured in warm conditioning test bench (left) and reproduced with Simscape (right) at iso 𝑃 𝑖

2 K•

 2 The cryogenic intercept loops are running at the nominal settings • The flange/window heaters are active

Figure 5 -

 5 Figure 5-65: CPL2 (BSC04) ceramic window temperature -in cryomodule cold conditioning, 4 K mode, detuned -measured (centre) and simulated (right) at iso 𝑃 𝑖 (left)The comparison of the ceramic window temperature evolution is first shown in Figure5-65. The evolutions are clearly different, consistently with the excessive heat loads (yet minor compared to the multipacting on CPL4) due to the TiN coating (adjustment #1), observed during the 2 warm conditionings. Again, this effect is especially stressed in this configuration, where the E field is maximum at the window.

Figure 5 -

 5 Figure 5-66: CPL2 (BSC04) heat load to water -in cryomodule cold conditioning, 4 K mode, detuned -measured (left) and simulated (right) at iso 𝑃 𝑖The next charts, in Figure5-67, expose 2 comparisons to the model: the coupler inlet / outlet 10 K loop temperatures as well as the coupler / cavity flange temperatures. The former shows a yet unseen point: the impact of the coupler dynamic heat loads on the 10 K loop. These Joule effect loads were predicted by the model to be tiny, in the order of 0.2 W in these conditions, and indeed, are hardly seen in the experiments, lost in the measurement noise.

Figure 5 -

 5 Figure 5-67: CPL2 (BSC04) 10 K loop inlet/outlet temperatures (top) and coupler/cavity flange temperatures (bottom) -in cryomodule cold conditioning, in 4 K mode, detunedmeasured (left) and simulated (right) at iso 𝑃 𝑖

Figure 5 -

 5 Figure 5-68: CPL4 (BSC03) ceramic window temperature -in cryomodule cold conditioning, 2 K mode, tuned then detuned -measured (centre) and simulated (right) at iso 𝑃 𝑖 (left)

Figure 5 -

 5 Figure 5-69: CPL4 (BSC03) heat load to water -in cryomodule cold conditioning, 2 K mode, tuned then detuned -measured (left) and simulated (right) at iso 𝑃 𝑖 Then, the same chart around the 10 K loop and the cavity port as for the other coupler is shown in Figure 5-70.

Figure 5 -

 5 Figure 5-70: CPL4 (BSC03) 10 K loop inlet/outlet temperatures (top) and coupler/cavity flange temperatures (bottom) -in cryomodule cold conditioning, in 2 K mode, tuned then detuned -measured (left) and simulated (right) at iso 𝑃 𝑖 In detuned conditions, the temperature rises are consistent with what was observed on CPL2 (adjustment #4). In tuned conditions, sharp and irregular heat loads are observed, both on the temperatures around the coupler/cavity interface and in the 10 K loop; those are actually irrelevant, being due to transient multipacting spots, that disappeared in the following tests.

Figure 6 - 1 :•

 61 Figure 6-1: the prototype coupler cuff (top) versus the coupler series design (bottom), viewed in CAD (right) and Simscape model (left), courtesy of SCK•CENThe iterative optimization of the "1 loop" heat exchanger required evaluating performance trade-offs while accounting for manufacturing, mechanical resistance, and packaging constraints in parallel. The Simscape model assessed the following performance factors:

Figure 6 - 2 :

 62 Figure 6-2: layout of the SIL/HIL architecture developed for the MINERVA prototype cryomodule

7. 8 7 . 9 2 299 7 . 9

 879279 The MINERVA prototype QCELL thermal shield instrumentation during test phase 1 Heat load distribution and balance on the 2 K tank during test phase 2.HEAT LOAD DISTRIBUTION AND BALANCE ON THE 2 K TANK DURING TEST PHASE 2.2

  problem vi nu står inför. » Quelqu'un peut-il prévoir comment sera l'avenir d'ici un demi-siècle ? [à propos de la dégradation généralisée de l'environnement]. L'on est, soit trop jeune pour apprécier combien le temps presse, soit trop âgé pour se sentir encore concerné. Cette dualité

	Ulf Danielsson, fysiker i Uppsala Universitet
	"Vårt klot så ömkligt litet", 2016
	«

s'explique peut-être par l'immensité du problème auquel nous faisons face. » Ulf Danielsson, physicien à l'Université d'Uppsala "Notre monde si misérablement petit", 2016

  . FIGURE 1-7: 235 U FISSION CROSS-SECTION (BLACK) AND 238 U NEUTRON CAPTURE CROSS-SECTION (PINK) AS A FUNCTION OF THE NEUTRON ENERGY [7] ........................................................................................................................ FIGURE 1-8: RADIOTOXICITY OF THE FISSION PRODUCTS AND ACTINIDES RELATIVE TO THE ORIGINAL URANIUM ORE, AS A FUNCTION OF THE TIME AFTER DISCHARGE FROM [8] ..................................................................................... FIGURE 1-9: THERMAL REACTOR WASTE INVENTORY SCHEMATICS ADAPTED FROM [12] ................................................ FIGURE 1-10: EXAMPLE OF PU STOCK STABILIZATION IN A MULTIPLE-RECYCLING STRATEGY VERSUS MONO-RECYCLING AND NO RECYCLING, FROM [9] .............................................................................................................................. FIGURE 1-11: THE POSSIBLE MINOR ACTINIDE IRRADIATION METHODS IN THE CONTEXT OF THE PU FUEL CLOSURE CYCLE,

ADAPTED FROM

[START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF]

............................................................................................................................... FIGURE 1-12: MYRRHA THREE-COMPONENT COUPLING, FROM

  3-50: THE MINERVA OPEN-LOOP CRYOGENIC PROCESS MODEL .................................................................. FIGURE 3-51: MINERVA PROTOTYPE CRYOGENIC CONTROL ARCHITECTURE ............................................................. FIGURE 3-52: MINERVA CRYOGENIC PROCESS MODEL ARCHITECTURE, COMPLIANT WITH WHICHEVER CONTROL BOX ...... FIGURE 4-1: THE DIFFERENT HEAT LOADS GENERATED BY THE RF PARTS ................................................................... FIGURE 4-2: CAVITY RESONANT FIELD CLASS AS A FUNCTION OF THEIR FAMILY, ADAPTED FROM

  .. FIGURE 5-29: 2 K LOOP HEAT LOAD DISTRIBUTION AND BALANCE OVER TEST PHASE 2.1; EACH WP IS AVERAGED OVER 5 TO 15 MIN; THE CONDITIONS ARE DETAILED IN THE LEGEND ("2K" OR "4K" MEANS 2 K OR 4 K MODE, "TRF" MEANS TRANSFER THROUGH THE JUMPER, "L" MEANS 2 K TANK LEVEL AND "10K" MEANS 10 K LOOP MASS FLOW RATE ... FIGURE 5-30: CRYOGENIC PROCESS MODEL CONTROL AND MODELLING ARCHITECTURE ADOPTED FOR THE PRECALIBRATIONS OF THE PI CONTROLLERS ............................................................................................................................. FIGURE 5-31: GENERIC ARCHITECTURE OF THE PI CONTROL LOOPS IN THE SIMULINK AND IN THE PLC CONTROL BOXES ...... FIGURE 5-32: MODEL VALIDATION OF THE EHX43 PI CONTROLLER; LEFT = FLANGE TEMPERATURE TTX44, CENTRE = WINDOW TEMPERATURE TTX43 AND RIGHT = EHX43 HEATER VOLTAGE AND POWER ...................................................... FIGURE 5-33: MEASURED COUPLER FLANGE AND WINDOW TEMPERATURES, AT THE FIRST 80 K COOL-DOWN OF TEST PHASE 2.1, LEFT = FLANGE TEMPERATURES, CENTRE = WINDOW TEMPERATURES, RIGHT = QM THERMAL SHIELD NITROGEN TUBE TEMPERATURES (TT682 IS AT THE COUPLER 2 OUTLET, TT683 IS AT THE COUPLER 1 OUTLET) ..................... FIGURE 5-34: EXPERIMENTAL FAULT SIMULATION OF THE EH144/244 FLANGE TEMPERATURE CONTROLLERS ................. FIGURE 5-35: EXPERIMENTAL FAULT SIMULATION OF THE EH143/243 WINDOW TEMPERATURE CONTROLLERS .............. FIGURE 5-36: 80 K COOL-DOWN AT THE START OF TEST PHASE 2.2, LEFT, THE THERMAL SHIELD OUTPUT TEMPERATURES, FIGURE 5-37: PERFORMANCE OF THE 10 K LOOP MASS FLOW RATE CONTROL, OVER A START-UP AT 08:45 THEN SETPOINT CHANGE AT ABOUT 10:00; LEFT = THE CONTROL VALVE (CV02), CENTRE = THE CONTROLLED MASS FLOWRATE, RIGHT =

CENTRE, THE NITROGEN MASS FLOWMETERS, RIGHT, THE NITROGEN INLET PRESSURE ......................................... THE BOUNDARY PRESSURE CONDITIONS; THE SEQUENCE COMES FROM TEST PHASE 2.2, IN 4 K MODE ....................

  ..... 266 FIGURE 5-58: TEST PHASE 2.2 CPL4 (BSC03) COUPLER WARM CONDITIONING IN CRYOMODULE; LEFT = IMPOSED 𝑃𝑖, CENTRE = MEASURED CUFF TEMPERATURE, RIGHT = SIMSCAPE REPRODUCTION WITH THE MULTIPACTING MODEL ............... 266 FIGURE 5-59: SIMSCAPE REPRODUCTION OF CPL4 (BSC03) COUPLER WARM CONDITIONING WITHOUT MULTIPACTING MODEL ............................................................................................................................................... 267 FIGURE 5-60: CPL4 (BSC03) COUPLER WARM CONDITIONING ANTENNA WATER HEAT LOADS, MEASURED (LEFT) AND IN SIMSCAPE WITH THE MULTIPACTING MODEL (RIGHT) .................................................................................... 268 FIGURE 5-61: CPL2 (BSC04) COUPLER CUFF TEMPERATURES MEASURED IN WARM CONDITIONING TEST BENCH (CENTRE) AND REPRODUCED WITH SIMSCAPE (RIGHT) AT ISO 𝑃𝑖 (LEFT) ............................................................................... 269 FIGURE 5-62: CPL2 (BSC04) COUPLER ANTENNA AND INTERCEPT WATER HEAT LOADS MEASURED IN WARM CONDITIONING TEST BENCH (LEFT) AND REPRODUCED WITH SIMSCAPE (RIGHT) AT ISO 𝑃𝑖 ........................................................ 269 FIGURE 5-63: CPL4 (BSC03) COUPLER CUFF TEMPERATURES MEASURED IN WARM CONDITIONING TEST BENCH (CENTRE) AND REPRODUCED WITH SIMSCAPE (RIGHT)

AT ISO 𝑃𝑖 (LEFT) ............................................................................... 270 FIGURE 5

TABLE 1 -

 1 1: EUROPEAN COMMISSION CO-FUNDED MYRRHA R&D PROGRAMS .......................................................... TABLE 1-2: DEMONSTRATOR AND SERIAL MINOR ACTINIDE BURNER SPECIFICATION FOLLOWING EUROTRANS ................. TABLE 1-3: MYRRHA BEAM AND RF PARAMETERS, COMPARED TO XFEL

TABLE 3 -

 3 1: COMPARISON OF THE FLUID THERMAL PROPERTIES OF THREE SUBSTANCES ..................................................TABLE 3-2: DATASHEET OF THE CONTROL VALVES OF THE MINERVA PROTOTYPE ...................................................... TABLE 3-3: SUMMARY OF THE WORKING CONDITIONS OF THE DATE SIMULATIONS OVER ALL THE DIFFERENT PROJECTS ..... TABLE 3-4: DEFORMATION OF THE DARCY FRICTION FACTOR FROM DATE ON THE ΔP TESTS WITH WATER ...................... TABLE 3-5: MINERVA PROTOTYPE PI CONTROL LAWS ......................................................................................... TABLE 3-6: MINERVA PROTOTYPE ON/OFF CONTROL LAWS ............................................................................... TABLE 4-1: COUPLER CONDITIONING TEST OVERVIEW ........................................................................................... TABLE 5-1: MINERVA PROTOTYPE TEST PHASE OVERVIEW ................................................................................... TABLE 5-2: REPRODUCTION OF THE COUPLER HEAT LOADS VERSUS WP5 AND WP8 FROM TEST PHASE 2.1 .................... TABLE 5-3: CRYOSCAPE REPRODUCTION OF THE COUPLER HEAT LOADS AND COMPARISON TO TEST IN THE CASE OF THE WINDOW HEATER FAILURE TEST ............................................................................................................... TABLE 5-4: QM CHIMNEY HEAT LOADS TO THE 10 K AND 2 K LOOP, SPLIT BETWEEN THE INLET (FROM THE QVB) AND OUTLET (FROM

THE QM) FLOW LINES ..................................................................................................................

TABLE 5 -

 5 5: PHYSICAL MODEL PARAMETER ADJUSTMENTS ...................................................................................... TABLE 5-6: THE PROTOTYPE QCELL PI CALIBRATIONS AFTER EXPERIMENTAL FINE-TUNING .......................................... TABLE 7-1: THE CUFF LAYER THICKNESS STATISTICAL MEASUREMENT RESULTS ............................................................ TABLE 7-2: THE CUFF COPPER LAYER RRR MEASUREMENT RESULTS .........................................................................

	LIST OF ACRONYMS
	HLW
	LL
	LL-HLW
	FP
	MOX
	PWR
	BWR
	GENIV
	EPR
	IFR
	P&T
	MA
	MYRRHA
	RIB
	EC
	ADS
	ADONIS
	GENEPI
	AGLAE
	ESS
	HIPA
	RF
	PW
	CW
	COP
	MTBF
	SNS
	LEBT
	MEBT
	LLRF
	PTS
	FTS
	ISOL
	SRF
	TS
	QM
	QVB
	QCELL
	LT
	HT
	AGS
	SS
	BCS theory
	HT
	L
	V
	HTC

FROM THE ADS CONCEPT TO THE CONSTRUCTION OF MINERVA 1.1 THE NUCLEAR INDUSTRY WASTES

  Discussed during the French Senate colloquium on transmutation « Déchets nucléaires, incinération et transmutation », organized by Jean-Claude Le Scornet from ACS and the former French Environment Minister Brice Lalonde, on May the 25 th , 20231.2.3 The debate about the need for dedicated burners for transmutation

	7 Autorité de sûreté nucléaire
	8

Beam pulse width (ms) CW RF wave amplitude PW Duty cycle 10 % RF buckets @ 352.2 MHz Bunch presence bunch RF bucket

	ESS, is also a pulsed accelerator, at 62.5 mA peak current concentrated in 2.86 ms wide
	pulses repeated at 14 Hz; the final energy of 2 GeV, gives an average beam power of
	5 MW. But ESS is a superconducting accelerator. LINAC4 and ESS beam specifications are
	summarized, with the one of XFEL, in Table 1-3.
	Table 1-3: MYRRHA beam and RF parameters, compared to XFEL
	RF pulse width (ms)	RF pulse width (ms)
		t
	PW Duty cycle 4 %	Bunch spacing (ns)	Bunch
			length
			(fs)
		t

1 𝐶𝑂𝑃 Cavity technology Repetition rate RF Pulse length RF duty-cycle Beam peak current Beam pulse length Beam duty-cycle Beam average current Beam energy Beam power

		-	hz	ms	%	mA	ms	%	mA	MeV	MW
	LINAC4	Normal	2	0,4	0,08	40	0,4	0,08	0,03	160 0,0051
	ESS	Supra	14	2,86	4,004	62,5	2,86	4,004	2,50	2000	5,0
	XFEL	Supra	10	1,38	1,38	5	0,65	0,65	0,03	20000 0,650
	MYRRHA	Supra	250	4	100	4	4	100	4,00	600	2,40

MYRRHA Phase 1 = MINERVA Reactor 600 MeV BR2

	• A wide partial load operating range for the reactor control
	• Fast RF transients to enable fault-compensation strategies
	• Electricity consumption constraints in the present energy shortage

Sensible heat (Cp) Sat° vapor @1013 mbar

			Latent heat
			@1013 mbar
		J/kg.K	J/kg
	Helium	9142	23413
	Nitrogen	1124	199179
	Water	2080	2256490

Table 3 -

 3 5: MINERVA prototype PI control laws

Table 3 -

 3 6: MINERVA prototype ON/OFF control laws

PLC « control box » Manual switch Actuator vector Sensor vector Open-loop Simscape « process model » and

  write process variables directly in the PLC CPU, through the modern OPC-UA40 protocol. In return, it is also possible to control the real process with the Simulink control box; it was done in practice, as detailed in part 5.3.2.4. Build pre-calibration of the PI gains for the prototype tests, in part 5.3.1• Synthetize and test an advanced controller for the prototype, in part 5.3.2• Prepare an advanced PLC virtual commissioning platform, in part 6.2.240 OPC-UA stands for Open Platform Communications Unified Architecture, and enables to make different machines and platform communicate with each other

					Name			Actuator	Sensor	Typ° target
	QVB TS temperature	CV601	TT620	90 K
	QVB 4 K tank level	CV01	LT010	35 cm
	QVB 4 K tank pressure without supply QVB 4 K tank pressure with supply This flexibility offered by the OPC-UA protocol was used to perform several control CV06 PT01 1080 mbar CV07 PT01 1080 mbar
	QM TS temperature optimization tasks:	CV651	TT651	90 K
	QM 2 K tank level in "4 K mode"	CV04	LT051	10 cm
	QM 2 K tank level in "2 K mode" QM 2 K tank pressure in "2 K mode" •	CV05 MKS	LT051 PT50/51	10 cm 30 mbar
				10 K loop MF		CV02	FT 10 K	240 mg/s
	Coupler window temperature	EH143/243	TT143/243	293.15 K
					Name			Actuator	Sensor	ON	OFF
	Coupler	flange	temperature	EH144/244	TT144/244	290	K	296	K
	Helium	mobile	Dewar	pressure	EH01	PTLHE	1160	mbar	1170	mbar
	Simulink	«	control	box	»

  2.2.3, once in the cryomodule 𝑄 𝑖 and 𝑄 𝑡 needed to be measured. The couplings were measured at 2 K, as this overcoupled configuration gives the highest accuracy, since the factor 1 + √ 𝑃 𝑟 and 𝑃 𝑖 . In that configuration, one notices that 𝑄 𝑖 ≈ 𝑄 𝐿 since 𝑄 0 ≫ 𝑄 𝑖 and 𝑄 𝑡 ≫ 𝑄 𝑖 . At room temperature, i.e. in undercoupled configuration, one can also get 𝑄 𝑖 and 𝑄 𝑡 but with a much lower accuracy.

	𝑃 𝑟 𝑃 𝑖	becomes almost independent from the
	measurement errors between

  |𝑆[START_REF] Chevret | Mesure de la réactivité de réacteurs sous-critiques pilotés par accélérateur par l'analyse d'expériences d'interruptions de faisceau programmées[END_REF] | was measured on BSC04, in the prototype cryomodule at 2 K with a VNA; this measurement is compared to the RLC model, in Figure4-12. Since the coupling 𝑄 𝑖 and 𝑄 𝑡 that feed this model actually come from this VNA measurement (the bandwidth and the maximum value), this first comparison is finally just a check of the correct |𝑆 21 | shape as a function of the excitation frequency, +/-2000 Hz around the resonance.

	21 =	𝑏 2 𝑎 1	=	𝑉 ̃𝑡 𝑉 ̃𝑖𝑛𝑐	⇒ |𝑆 21 |(𝑑𝐵) = 10 * 𝑙𝑜𝑔 (	|𝑃 𝑡 | |𝑃 𝑖 | )	( 4-32 )

  As a comparison, the needed power without beam to maintain the field at resonance, is 𝑃 𝑔 = 2.43 𝑘𝑊.The previous part illustrated a particular cavity phaser (case #1), where the complex cavity voltage satisfies the required 𝐸 𝑎𝑐𝑐 𝑡𝑎𝑟𝑔𝑒𝑡 and 𝜙 𝑠 . These results were obtained by computing the latter targets in the model equation, that gives the needed amplifier outputs and 𝑃 𝑖 (or 𝑃 𝑔 ), 𝜙 𝑖 (or 𝜙 𝑔 ).Practically, it works the other way round: the true actuators are 𝑃 𝑖 and 𝜙 𝑖 and they must be synthetized by control laws which are implemented in the LLRF hardware (see part 4.4.3.1 for description) so that 𝑃 𝑡 and 𝜙 𝑡 (the image of 𝐸 𝑎𝑐𝑐 and 𝜙 𝑐𝑎𝑣 ) satisfy 𝐸 𝑎𝑐𝑐 𝑡𝑎𝑟𝑔𝑒𝑡 and 𝜙 𝑠 . For MINERVA, the errors on 𝐸 𝑎𝑐𝑐 and 𝜙 𝑐𝑎𝑣 are controlled by two PI controllers.Lastly, as seen, in parallel to the cavity field control, the cavity resonance must be controlled too. In practice, neither 𝜓 nor ∆𝑓 is actually measured. Looking back at Figure4-16, it is clear that 𝜓 𝑜𝑝𝑡 is achieved when 𝐼 ̃𝑔 and 𝑉 ̃𝑐𝑎𝑣 are aligned, in other words, when 𝜙 𝑔 = 𝜙 𝑠 . Thus, while two PI controllers regulate 𝐸 𝑎𝑐𝑐 and 𝜙 𝑐𝑎𝑣 on their targets by acting on 𝑃 𝑖 and 𝜙 𝑖 , a third controller acting on 𝑓 0 makes 𝜙 𝑔 converge towards 𝜙 𝑠 .

	#1	#2	#3
	#2
	#1
	#3
	4.4.2.1 Normal operation
	• Case #1: 𝑃 𝑔 = 7.75 𝑘𝑊
	• Case #2: 𝑃 𝑔 = 8.27 𝑘𝑊
	• Case #3: 𝑃 𝑔 = 8.25 𝑘𝑊

Figure 4-16: phaser diagram and transmission chart for a field regulated at 7 MV/m, with a 4-mA beam and synchronous phase -30° 4.4.2 Principle of the cavity field and resonance control

Table 4 -

 4

	Antenna water
	Antenna 1&2 in
	216
	Antenna 1 out	Antenna 2 out		216
	212	213
	Pos 1	Pos 2	213
			Intercept	116
			water	115

1: coupler conditioning test overview

111 112 113 114 115 116 211 109 110 PT100-2 PT100-1 204 205 206 207 208 209 203 Short name Coupler #

	4.7.2 Coupler « heat transfer » model
	4.7.2.1 Coupler heat transfer model from the design phase
				Post 1			Post 2
	Iteration	Date
					Cuff #	Short name Coupler #	Cuff #
	1	Jul -Sept 2021	CPL1	P5-60-AD	P5-50 / PX-50-05	CPL2	P6-60-AD P6-50 / PX-50-07
	2	Oct -Nov 2021	CPL2	P6-60-AD	P6-50 / PX-50-07	CPL3	P4-60-AD P5-50 / PX-50-05
	3	Mar -Apr 2022	CPL4	P2-60-SD	P3-50 / PX-50-03	CPL5	P3-60-SD P2-50 / PX-50-01
	4	Jun-Jul 2022	CPL2	P6-60-AD	P6-50 / PX-50-07	CPL3	P4-60-AD P5-50 / PX-50-05
	5	oct-22	CPL6	P1-60-SD	P1-50 / PX-50-02	CPL1	P5-60-AD P5-50 / PX-50-05

  ⟨𝐻 𝜑 (𝑟, 𝑧, 𝑡) 2 ⟩

	𝑆	=	2(𝑃 𝑖𝑛𝑐 ̅̅̅̅̅ + 𝑃 𝑟𝑒𝑓 ̅̅̅̅̅ ) 𝑟ξ 0 𝑙𝑛 ( 𝑏 𝑎 )	∆𝑧 -4 *	√𝑃 𝑟𝑒𝑓 ̅̅̅̅̅ * 𝑃 𝑖𝑛𝑐 ̅̅̅̅̅ 𝑟ξ 0 𝑙𝑛 ( 𝑏 𝑎 )	( 𝛽 1	𝑐𝑜𝑠(𝛽(𝑧 2 + 𝑧 1 ) + 𝜙 2 -𝜙 1 )𝑠𝑖𝑛(𝛽∆𝑧))

Table 5 -

 5 2: reproduction of the coupler heat loads versus WP5 and WP8 from test phase 2.1

Table 5 -

 5

	5: physical model parameter adjustments
	Parameter adjustment	Component or sub-component	Deviation from physical parameters
	% Opening offset	All valves	Small
	Permeability	CV651	Very high
	Permeability	MKS valve	Very high
	HTC multiplier	HX	Small
	Dependency on the He level	QM 2 K buffer tank	Moderate
	Contact resistances	All relevant parts	Small
	Reference radiative heat load	QM thermal shield	Very high
	Reference radiative heat load	QM 2 K buffer tank	Very high
	Reference radiative heat load	2 K loop external transfer line	Moderate
	Reference radiative heat load 80 K loop external transfer line	Very high
	Those adjustments must be rigorously stored together with the other cryogenic process
	model parameters, and should be continuously challenged along the coming project
	phases: by further post-treatment of existing tests, by new prototype cryomodule
	parameter sweeps or configuration test, or even, by 3D model evaluations or separate
	principle tests. For instance, simple mock-up tests could be performed to evaluate the
	radiative heat load sensitivity, in particular, to geometry as well as to MLI fitting and
	emissivity dispersion.

  𝑟 𝑃 𝑖ratio and, in parallel, the amplifier power saturation. The problem of the noise trigger, however, remains; dealing with this problem is all the more difficult that the quench detection trigger levels depend on the numerous working points and parameters: with or without beam, RF field open-loop or closed-loop, nominal 𝑄 𝐿 , etc. For that matter, the RF team of the Advanced Rare Isotope Laboratory (ARIEL, a CW machine at TRIUMF, Canada[193]) are investigating a technique to discriminate real quenches from other normal RF transients. It is based on off-line transient simulations of the 𝑃 𝑟 𝑃 𝑖 ratio by a cavity equivalent electrical model associated to a physical modelling of cavity quench propagation [194]. The difference in 𝑃 𝑟 𝑃 𝑖 ratio observed during quenches versus during normal beam loading are used to calibrate the trigger levels of the quench detector in several working conditions.

THE MINERVA PROTOTYPE CRYOMODULE COMPLETE PID DURING TEST PHASES 2 7.5 THE MINERVA PROTOTYPE CAVITY MEASURED AND ESTIMATED PARAMETERS

  • Develop libraries and standards • User-friendly parametrization interface • Widespread modelling environment A way to share the costs of the code maintenance and improvement is to develop cooperative applications and data sharing; the latter is particularly interesting to extend the modelling validation datasets.

	7.2 The MINERVA prototype cryomodule complete PID during test phases 2 7.7 The MINERVA prototype coupler cuff material characterisation
	7 7.1 THE MYRRHA MAIN LINAC SPECIFICATIONS ANNEXES Unit Value Parameter Unit 7.2 Data 7.6 THE MINERVA SLOW TUNER PARAMETERS 7.7	Source/condition Value
	Section # Excitation f Motor steps per revolution #1 MHz E in (MeV) 16.6 E out (MeV) 101.4 Cavity type Single Spoke Operating frequency 352.2 f 0 BSC03 init MHz Gearhead reduction ratio f 0 BSC04 init MHz Lever arm Q t BSC03 -Srew pitch Optimal β 0.375 Cells per cavities 2 Q i BSC03 -Q t BSC04 -Sentitivity	#2 352,21 step/rev 101.4 172.3 Double Spoke 352.2 352,084 rev motor / rev gearbox #3 172.3 Requirement 200 601.6 Elliptical 6,25 VNA (4 K) 352,122 mm/mm 20 VNA (4 K) 5,05E+11 mm/gearbox rev 2 VNA (4 K) kHz/mm (flange) 128 704.4 0.495 3 5 9,80E+05 VNA (4 K) Hz/motor step 10,24 0.705 4,81E+11 VNA (4 K) Hz/motor rev 2048
	Cavities per cryomodule Q i BSC04	2 -		2 8,40E+05	4	VNA (4 K)
	Nb cryomodules BW BSC03	30 Hz		9 360	18	From Q L (4 K)
	Nominal E acc (MV/m) Fault recovery E acc (MV/m) BW BSC04 Section length (m) E acc target nominal	7.0 9.1 Hz 91.2 MV/m		6.8 9.0 420 36.3 7,0	11.0 14.3 121.0 From Q L (4 K) Design
	E acc target fault recovery	MV/m		9,1		Design
	φ s target (average)	°-30		Design
	Q 0 (RT, no field) BSC03/BSC04	-		7500		VNA (300 K)
	G	Ω		107	Q 0 measured at RT
	G	Ω		109	CST Microwave Studio
	r/Q	Ω		217	CST Microwave Studio
	β opt	-		0,37	CST Microwave Studio
	B peak /E acc	mT/(MV/m)	7,32	CST Microwave Studio
	E peak /E acc	-		4,29	CST Microwave Studio
	L acc	m		0,315		Definition
	K L (BSC03/BSC04/mean) Hz/(MV/m) 2 -10,2 / -8 / -9,1	Measured in CV800
	K P	Hz/mbar		5,1	Measured in CV800
			292 297

THE MINERVA PROTOTYPE COUPLER CUFF MATERIAL CHARACTERISATIONTable 7 -

 7 1: the cuff layer thickness statistical measurement results

Table 7 -

 7 2: the cuff copper layer RRR measurement results

	Sample
		Cu		Au	Ni
	1x	16.58 (0.35)	0.62 (0.19)	0.53 (0.02)
	1	15.47 (0.44)	0.68 (0.15)	0.48 (0.04)
	3xx	15.09 (0.41)	0.63 (0.14)	0.49 (0.02)
	4	15 (0.51)		0.66 (0.22)	0.44 (0.1)
		RRR Cu run #1 RRR Cu run #2
	1 -polished		99	104
	1x -polished		156	161
	3 -polished		123	131
	4 -non-polished	125	134

1.[START_REF] Irsn | Les unités de mesure[END_REF] The ADS as an answer to the minor actinides accumulation

The Advanced Sodium Technological Reactor for Industrial Demonstration (ASTRID), a 600-MWe sodium-cooled fast reactor was developed by a consortium led by CEA

The generation 4 international forum (GENIV) was created in 2001 as a cooperative effort to set the requirements and identify the technology of the EPR reactor (generation 3) successor

EPR for European Pressurized Reactor, developed mainly by Framatome and EDF

As heard during the French Senate colloquium on transmutation « Déchets nucléaires, incinération et transmutation »

The change in reactivity of the nuclear fuel per degree change in the fuel temperature (Wikipedia)

How much the reactivity of a nuclear reactor changes as voids (typically steam bubbles) form in the reactor moderator or coolant (Wikipedia)

Studiecentrum voor Kernenergie, Centre d'étude de l'énergie nucléaire, Mol, Belgium 1.3 ADS demonstrator specifications

1.4 MYRRHA accelerator high-level specifications

Discussed during the French Senate colloquium on transmutation « Déchets nucléaires, incinération et transmutation », May the 25 th 2023

Experience of the author from PSA Peugeot-Citroën and TitanX (former Valeo subsidiary) in the 2011-2019 period

Although the optimum in terms of RF-to-beam energy transmission is 𝜙 𝑠 = 0 (the particles arrive at the crest), for non-relativistic particles this phase is always set negative (the particles arrive earlier than the crest) to focus the bunch longitudinaly, as faster particles arriving sooner are less accelerated and vice-versa[START_REF] Wangler | RF Linear Accelerators[END_REF].

This convenient conversion factor gives 1.08 W/(m 3 /h) at 2 K, and 0.96 W/(m 3 /h) at 4 K, considering the respective latent heats and the warm vapor density at 293.15 K and 1013 mbar.

Even after choosing a modelling environment, one should continue watching for the competitors, for potential evolutions in the future: for instance, Python starts to offer competitive functionalities with MATLAB, including physical modelling and machinelearning libraries like Tensorflow.

In the literature, the Darcy friction factor is noted fD; in this study, it is deliberately noted f, but should not be mixed with the fanning factore (fD = 4f)

K connection with the couplers

Emissions close to the cryomodule, inside the concrete wall protection perimeter can reach up to 100 mSv/h, but outside, the dose rate should not exceed 1 µSv/h.

4.3 The cavity "field" model

A VNA measures the Scattering parameters (S-Parameters) of a network, i.e. the ratio of complex voltages. The power ratios are given by the module of the S-Parameters.

The « GO » decision for the continuous coupler HX was taken by end of 2022

Discussed during dedicated accelerator modelling workshops, in the context of the M4CAST collaboration

ACKNOWLEDGEMENTS

Je tiens à remercier les personnes qui m'ont accueilli dans la communauté des Accélérateurs, en particulier, Tomas Junquera, Jean-Claude Le Scornet et toute l'équipe d'ACS pour leur confiance en ce projet de reconversion.

8

CRYOGENIC PROCESS MODEL

This section reviews the cryogenic models presented in part 3, against relevant cryomodule test sequences. Each component features a specifically tailored instrumentation to measure relevant outputs such as heat flows and mass flow rates. The mass flowmeters were already presented in 3.1.3; this section starts by introducing the heat flow measurement methods. Then, each component model is reviewed, with an emphasis on the identification of a reference measured output, and on cross-checks against other independent measurements.

Thermal model evaluation methods

Heat flow measurement methods

Except for the valve hydraulic model, reviewed in the next section, the cryogenic process model essentially consists in thermal models. Their performance versus the tests is evaluated, either based on temperature or on heat flow indicators. Heat flows are seamlessly available in Simscape, but not in the tests, where they must be derived in different ways.

The first method is the "material conduction method": it relies simply on Fourier's law, i.e., the difference in conductivity integral between 2 points, as in [START_REF] Chen | A Study on Properties of Novel Metallic Foam for Nuclear Applications[END_REF][START_REF] Edf | Les déchets radioactifs[END_REF][START_REF]Valley of stability[END_REF][START_REF]Laradioactivite[END_REF][START_REF] Joliot-Curie | [END_REF](8)[START_REF] David | Quelle filière nucléaire pour quel cycle de combustible ?[END_REF][START_REF] Verma | Spent Nuclear Fuel and Accelerator-Driven Subcritical Systems[END_REF][START_REF] Ruzic | The Real Bad Stuff (High-Level Wastes)[END_REF][START_REF] Hargraves | Liquid Fluoride Thorium Reactors[END_REF][START_REF] Cea | Le cycle du combustible des réacteurs à neutrons rapides[END_REF][START_REF]Centre d'informations et de ressources sur CIGEO[END_REF][START_REF]Réacteur nucléaire naturel d'Oklo[END_REF]. At each point, the conductivity integral is known from a temperature measurement, and interpolation in the material table. When using this method, a crucial point is to have the simplest geometry between the 2 points, to reduce the errors in the conductive section; a second point is to maximize the distance between them, to make the temperature measurement To compare this linearized versus the non-linear process model, both were controlled in open-loop with the same input excitations, i.e. +/-5 % opening of CV05 and the MKS valve, around their respective steady-state positions. The comparison is displayed in Figure 5-43.

Figure 5-43: comparison of the process non-linear model to the linearized model, left, the level control, right, the pressure control

There are indeed some differences in the level and pressure outputs, as expected since both valves and the level reading are highly non-linear; overall, the tendencies are respected. This model is used in the next section to synthetize an LQ controller.

LQ synthesis and test on the process model

The method to synthetize the LQ controller is referenced in [START_REF]Laradioactivite[END_REF] from the L2S laboratory at Paris-Saclay, and in the MATLAB documentation [START_REF] Joliot-Curie | [END_REF]; a brief summary is first given here.

The principle of a PI controller is to make an output y converge to 𝑦 𝑐 , by building a command u based on the error 𝑦 𝑐 -𝑦, as in Figure 5-44. In the following, the term "plant" refers either to the real process, or to the process model.

Figure 5-44: a controller based on the error

Another method, the full-state feedback, illustrated in Figure 5-45, by contrast, uses the states x (instead of the outputs y) multiplied by a gain matric K in a direct comparison with the reference, with a scaling term M for no steady-state error.

Plant Controller y c

+ -u y Error The maths behind this method is based on the modification of the dynamic of the system, originally contained in the open-loop state matrix A so that the closed-loop state matrix (A-B*K) has its eigenvalues at a better location in the s-plane, by trimming the K matrix. Optimally choosing the K matrix is the role of the LQ method. An optimal command would be the one minimizing the quadratic cost function J:

The matrices Q and R are the weight matrices that respectively penalize the bad performance and the actuator efforts; they are the parameters to tune if the performance of the LQ controller is not satisfactory. This is equivalent to the pole-placement method. Then, the LQR function from MATLAB Control System Toolbox computes the gain 𝐾 which minimizes J, simply writing 𝐾 = 𝑙𝑞𝑟(𝐴, 𝐵, 𝑄, 𝑅).

There is only one problem left: in the real process, the states are not accessible; they should thus be estimated. The estimate is done by a common mathematical object from the control theory, the state observer; the estimated states are noted 𝑥 ̂ and the estimated outputs are noted 𝑦 ̂, by extension.

The most straightforward method to build the observer for MINERVA, is to use a model of the states, more precisely, the linearized model itself; this approach is widespread in the industry, and was also the one developed for the commissioning of SPIRAL2, in [8].

This model acts as a parallel plant, the "Observer plant", taking the same input command u, but giving access to its internal states 𝑥 ̂. The problem being that the linearized model is not a perfect representation of the reality, calling for a method to minimize the difference between 𝑥 and 𝑥 ̂. This method consists in adding a corrector gain L on the difference between y (measured) and 𝑦 ̂ (estimated): the observer plant is forced to converge toward the real plant, i.e. the process model or the real process. The schematic of this technique is shown in Figure 5