Coline

Anaïs, Cécile Jorge Lara

Keywords: ALM Augmented Lagrangian Method, AVR Automatic Voltage Regulator, COI Center of Inertia, DAE Dierential and Algebraic Equations, EMT Electro-magnetic Transient, EU European Union, FACTS Flexible AC Transmission Systems, FFR Fast Frequency Response, FGM Fast Gradient Model Predictive Control Power Oscillation Damping, PSS Power System Stabilizer Virtual Rotor Control, VSC Voltage Source Converter, VSM Virtual Synchronous Machine, xli

you for your serenity, your kindness and your craziness. I knew that this PhD will make me learn a lot, but I could not expect to nd such a group of close friends while working on my thesis! You are awesome and the only regret I have is that I don't speak Spanish yet! Nevertheless, I could also count on the support of my friends I had met during my studies and who are now living all around the

essential to thank all the people who have been part of this experience and helped me through this journey up to the defense of my PhD. I have had the chance to defend my thesis in the city of Lyon where I had started to go to engineering school seven years ago, so we can say that the cycle is complete. In addition, half of my thesis happened during the Covid-19 pandemic and the support that I received from my loved ones, friends and colleagues was even more meaningful to me and is a signicant part of my PhD.

First, I would like to thank Prof. Houria Siguerdidjane and Prof. Mauro Carpita for having spent a lot of time in reading and reviewing my PhD thesis. I want also to express my sincere gratitude to Prof. Zhe Chen and Prof. Germain Garcia for their constructive feedback and taking part in my PhD thesis committee and Prof.

Gildas Besançon for having chaired my PhD thesis committee. Also, I would like to thank SuperGrid Institute and its management board for having funded my research project and having believed in me. Furthermore, I am grateful for the support I received from University of Grenoble Alpes, going from the administrative sta to the skilled researchers from G2Elab and GIPSA-lab whom I had the chance to discuss with.

Then, my thoughts go to my PhD thesis directors Prof. Didier Georges and Prof. Seddik Bacha. Didier, thank you for your diligence and your friendliness, you have always been consensual and gave me great feedback all along the thesis. Seddik, I really enjoyed all the discussions we had during these three years ranging from history and geopolitics to the control of MMCs, you always managed to make yourself available when needed. Also, I had the chance during my PhD to join Dr. Abdelkrim Benchaib's team, who was also one of my thesis supervisors. Thank you Abdelkrim, for having built such a great working environment and to believe in our work everyday, it was truly supportive to work with you. Finally, I would like to thank Dr. Jing Dai for supervising my work as well. All your remarks and feedback you gave me when I was writing papers or my PhD manuscript were i Acknowledgments always constructive. Thank you Jing, for all the discussions we had, your kindness and humility. I can say that it was an honor to work with the four of you. But a thesis is not only about a supervising team, and in my case, I had the chance to join an incredible research group to pursue my work. I would like to thank this team, the control and stability group, a.k.a the control guys, a.k.a the late risers: Juan Carlos, Kosei, Mathieu, Paul, Miguel, Lucas, Hind and Gianni and the past members of the group: Sjoerd, Roman, Laila, Jose, Abdelhak and Christopher.

Thank you for all the laughs, the coee-break discussions, the afterwork drinks and the Matlab talks. It was a pleasure to work with such talented and humble people.

All of you sincerely made this PhD an unforgettable experience and I am glad to have had you as colleagues and to count you now as friends.

Also, I want to thank Antoine and Louis for having helped me with the realtime platform to achieve the experimental validation of my work. It was at the same time truly instructive and really fun to work with you at the platform and this will remain one of my best memories from my PhD work. Thanks a lot! But, during these years of work at SuperGrid Institute, I also had the privilege to meet amazing people out of my working group. To Nicolas M., Pascal, Domagoj, Helena, Raga, Nicolas B., Shae, Léo and all the other people I met there: thank you! You all contributed to make my stay a great experience. Also, to the people from the soccer and basketball teams of SuperGrid, thank you for your constant smiles and your friendliness. It was really nice to meet you over sports during our lunch breaks. I think it is also important to express my gratitude to Prof. Jerey T. Glass and his team who welcomed me during my M.Sc as an intern into their lab at Duke University, NC, USA for ve months. You gave the 22-year old student I was one of his best life experiences. In particular, I want to thank Jimmy and Edgard for sparking my passion for research. I do not think that I would have started a PhD without meeting you guys, you were and still are a true source of inspiration.

Then, from a more personal point of view, by coming back to Lyon for my PhD, I knew that I was in familiar territory since I already had some friends living there who I could count on during this thesis. To my group of engineering friends in Lyon: Arthur, Paul, Chloé, Antoine, Lisa, Elise, Louis, Maud, Valentin, Elodie, Gaultier, Sébastien, Théophile and Carlota, I would like to say Merci beaucoup!.

It was great to have such a nice group of people to hang out after work or to have a cool dinner with. I am glad that I have shared these three years with you. Moreover, Dans la première partie de ce manuscrit, les diérents services systèmes fournis par les GS sont listés et un état de l'art des stratégies de contrôle du VSC existantes est donné, en incluant les notions de suiveurs (grid-following ) et de formeurs (gridforming ) du réseau. Ces deux types de contrôles sont comparés du point de vue de leur capacité à apporter au réseau des services similaires à ce que les GS pouvaient apporter. Ensuite, le rôle de ces deux contrôles est étudié an de déterminer le taux de pénétration maximal en EP tel que le réseau reste stable ainsi que les problèmes d'interactions potentielles avec les GS dans le cas d'un réseau hybride de référence.

Résumé court en français

Il a pu être démontré que passer d'un contrôle type grid-following au grid-forming permet d'améliorer le taux de pénétration maximal des VSC dans le réseau, même si dans ce dernier cas des interactions avec les GS existent et devront être prises en compte pour permettre l'intégration massive d'EP dans le réseau.

Dans la seconde partie de ce manuscrit, l'accent est mis sur l'application du gridforming aux Convertisseurs Modulaires Multi-niveaux (MMC) dédiés à l'intégration vii Résumé court en français de FEO dans les réseaux AC et l'impact des dynamiques du réseau DC sur la réponse de la station MMC contrôlée en grid-forming, quand le convertisseur doit également assurer la stabilité de la tension DC. Dans ce contexte, une méthode est proposée pour estimer les besoins en énergie des sous-modules du MMC pour fournir au réseau AC un support inertiel. Un contrôle associé à cet usage est proposé. Sa compatibilité avec le grid-forming issu de l'état de l'art est également validée dans un grand réseau multi-machines et multi-convertisseurs. Dans un second temps, une solution de contrôle basée sur une approche multivariable est nalement proposée Depuis quelques décennies, de plus en plus de projets visant à introduire du renouvelable dans le mix de la production d'électricité voient le jour. C'est le cas notamment au sein de l'Union européenne où les estimations d'évolution du taux de renouvelables prévoient une augmentation signicative de ces ressources. Ces sources de production reposant sur les énergies renouvelables ont de nombreuses caractéristiques qui les distinguent des centrales de production conventionnelles:

• distributivité : contrairement aux grandes infrastructures que sont les centrales de production, ces systèmes renouvelables peuvent être installés à l'échelle du réseau de distribution et même à l'intérieur des centres urbains.

• intermittence : là où la puissance produite par une centrale thermique peut être facilement contrôlée en jouant sur la quantité de combustible brûlé, l'énergie produite par les systèmes photovoltaïques et éoliens dépend des conditions météorologiques. Par conséquence, parvenir à faire correspondre la consommation à la production en temps réel représente un véritable dé dans ce type de congurations.

• interface au réseau : une centrale thermique conventionnelle est couplée au réseau électrique par le biais d'une interface électromécanique appelée le Générateur Synchrone (GS), qui convertit une puissance mécanique de rotation en électricité. Pour la plupart des systèmes éoliens et photovoltaïques cependant, le couplage au réseau est assuré par des convertisseurs à base d'Electronique de Puissance (EP) qui convertissent un type de tension donné (généralement une tension continue, dite DC en anglais) en une tension alternative (AC) triphasée. L'objectif de cette thèse de doctorat est d'étudier le contrôle et l'opération de convertisseurs de puissance de type VSC avec un focus porté sur les Convertisseurs 1 Source: https://www.hitachi.com/rev/archive/2020/r2020_04/gir/index.html.

x Chapitre 1: Introduction générale Modulaires Multi-niveaux (MMCs en anglais) intégrant des sources d'énergie renouvelables. Ce travail s'intéresse à la stabilité du réseau AC/DC quand le convertisseur est opéré en mode de contrôle formeur (grid-forming en anglais) et se focalise donc sur le contrôle à l'échelle du convertisseur Ce manuscrit de thèse est organisée de la façon suivante:

• Dans le chapitre 2, un état de l'art des stratégies de contrôle du convertisseur existantes est présenté. En particulier, les diérents modes de contrôle sont comparés et leur pertinence est évaluée vis-à-vis de leur capacité à fournir les principaux services systèmes que peuvent fournir les machines synchrones.

• Dans le chapitre 3, une analyse de stabilité de ces principaux modes de contrôle (plus précisément les modes grid-following et grid-forming) est proposée dans le but de déterminer le taux de pénétration maximal en EP dans des réseaux hybrides où les convertisseurs et générateurs synchrones cohabitent.

• Dans le chapitre 4, l'accent est mis sur le rôle de la technologie HVDC, et plus particulièrement l'application des MMCs à l'interconnection de Fermes Éoliennes Oshores (OWF en anglais) avec le réseau continental. L'originalité de ce travail consiste à prendre en compte les dynamiques du réseau DC et de proposer une nouvelles manière de dénir les besoins en énergie du MMC pour fournir un support inertiel au réseau AC et de l'utiliser avec une nouvelle loi de commande.

• Dans le chapitre 5, une Commande Prédictive à base de Model (MPC en anglais) est appliquée pour contrôler de manière optimale l'énergie interne du MMC ainsi que la tension du réseau DC. Cette solution de commande est synthétisée en utilisant un modèle identié du réseau AC et ses performances vis-à-vis du rejet de perturbations sont comparées au contrôle reposant sur deux correcteurs Proportionnel Intégral (PI) en parallèle. Le comportement du MPC et sa robustesse vis-à-vis des incertitudes de paramètres et des changements de points de fonctionnement sont établis dans un second temps.

• Dans le chapitre 6, le contrôle grid-forming ainsi que la couche de contrôle MPC présentée au chapitre 6 sont validés en utilisant un correcteur temps réel et appliqués à une maquette de MMC de 6 kVA. Plus particulièrement, le concept de grid-forming appliqué à la maquette de MMC permet au convertisseur de se synchroniser au réseau AC sans la Boucle à Vérouillage de Phase (PLL en anglais). Deuxièmement, le MPC proposé au chapitre 6 est implementé en C de manière à être compatible avec le reste de la structure de commande implémentée en Matlab/Simulink. xi Résumé de la thèse en français

• Dans le chapitre 7, une conclusion sur les principales contributions de cette thèse de doctorat est dressée et les hypothèses de départ considérées pour établir ces travaux sont discutées. Dans un second temps, d'éventuels nouveaux axes de recherche sont présentés an de pouvoir étendre les résultats de cette thèse.

Une synthèse des diérents chapitres est proposée dans la suite de ce résumé.

Chapitre 2: État de l'art des solutions existantes Pour assurer la stabilité de réseaux hybrides où GS et EP cohabitent, les convertisseurs vont devoir intégrer petit à petit les fonctions de soutien au réseau que proposent les GS de telle sorte que les réseaux restent opérables dans des conditions similaires, puisque les charges et les machines restantes sur le réseau imposent toujours des contraintes en fréquence, tension et angle sur le réseau AC.

L'objectif de ce chapitre est de présenter les commandes développées dans la littérature scientique an de fournir ces fonctions de soutien à l'aide de convertisseurs AC/DC de type VSCs, décrits dans la gure 2. Il s'agit d'un sujet vraiment d'actualité au niveau de la communauté scientique et qui attire même l'attention de la presse généraliste. pas passer par une PLL. Cela permet notamment aux convertisseurs opérés avec ce schéma de commande de se connecter aux réseaux faibles, voire aux réseaux passifs.

Du fait du changement de paradigme entre ces deux modes de contrôles, ce chapitre comprend une analyse comparative de ces deux modes, à la fois du point de vue de leur impact sur le comportement du convertisseur, tels que la méthode de synchronisation, le suivi de références de puissance et les actions pour limiter le courant, mais également sur les fonctionnalités qu'ils peuvent fournir au réseau AC, tels que l'inertie synthétique, le partage de charge, le support en puissance réactive, la contribution à la force du réseau, la contribution aux courants de court-circuit et l'amortissement des oscillations de puissance.

xiii Résumé de la thèse en français Chapitre 3: Analyse modale sur les limites d'intégration de l'EP dans les réseaux AC Une fois que les diérents modes de contrôle ont été présentés, une analyse sur le rôle que peuvent jouer ces diérents modes de contrôle sur les limites d'intégration de l'EP dans les réseaux AC est proposée. Cette analyse repose sur l'analyse modale d'un réseau de référence élémentaire composé d'un GS et d'un VSC interconnectés par une ligne de transmission de longueur variable et séparées par une charge. Ce réseau de référence est présenté dans la gure 5. An d'étudier l'évolution de la pénétration en électronique de puissance dans le réseau, une métrique dénommée ici le ratio de puissance et notée P R est utilisée. Il y a dans la littérature plusieurs manières de dénir ce ratio de puissance. Dans cette étude, la formulation retenue est:

P R = S vsc S sg + S vsc = S vsc S base (1)
où S vsc and S sg sont respectivement les puissances de base du VSC et du GS et S base la puissance de base du système, maintenue constante. Ce P R permet de comparer le ratio de la puissance installée du VSC avec la puissance du système total. Par conséquence, en faisant varier ce P R , il est possible de simuler des cas où le rapport entre convertisseur et générateur synchrone est diérent, et donc d'analyser le rôle de ce rapport sur la limite de stabilité petit signal.

Les résultats obtenus à partir de l'analyse modale sont donnés dans la gure 6. Il a ainsi été montré que les dynamiques de la ligne de transport généraient des interactions négatives avec la PLL du VSC lorsque celui-ci est en mode source de courant, conrmant une tendance observée dans la littérature. En revanche, pour ce qui est du VSC en mode grid-forming, la longueur de ligne semble avoir peu d'impacts sur le ratio de puissance maximal admissible.

Cependant, cette analyse a permis de mettre en évidence une interaction négative qui correspond aux modes instables entre le Régulateur Automatique de Tension xiv Chapitre 4: Application du grid-forming aux convertisseurs modulaires multi-niveaux Une fois cette méthodologie appliquée, une loi de commande basée sur la structure actuelle de contrôle du MMC est proposée. Cette loi de commande, dénommée commande du rotor virtuel, agit sur la référence d'énergie envoyée au contrôle du MMC. Elle permet de faire varier l'énergie stockée dans les sous-modules du MMC proportionnellement à la déviation du carré de la fréquence estimée du réseau AC.

Elle s'exprime de la manière suivante:

W * mmc = W * mmc,0 + k vr • (ω 1 2 -ω * 2 ) (2)
W min mmc ≤ W * mmc ≤ W max mmc (3)
où W * mmc,0 = 1 p.u. est la référence d'énergie correspondant à la fréquence de référence (50 Hz), ω * est la fréquence de référence normalisée et k vr est un coecient réglable qui permet d'émuler la réponse d'une machine tournante. W min mmc et W max mmc sont utilisées an de s'assurer que le correcteur de l'énergie interne respecte les limites physiques du MMC.

(a) AC power disturbance (b) DC power disturbance • Analyse de stabilité transitoire des MMCs:

3.6

Illustrative scheme of the selection of AVR and PLL specic dynamics for studies. . .

3.7

Modal analysis results for dierent values of P R with the VSC in grid-feeding (left-side) and grid-supporting (right-side) modes and x eq = 0.33 p.u. . . . . . . . . . . . . . . .

3.8

Participation factors of the λ 4,5 mode for dierent values of the power ratio P R whith the VSC in grid-feeding mode and x eq = 0.33 p.u. . . . . . . . . . . . . . . . . . . . .

3.9

Modal analysis results for dierent values of x eq with the VSC in grid-feeding (left-side) and grid-supporting (right-side) modes and P R = 0. The Industrial Revolution started in the mid-XVIII th century in Europe and more precisely in the British countryside, where famines were a recurring problem [START_REF]European Routes of Industrial Heritage (ERIH), Industrial History of European Countries[END_REF].

While in the early 1800's the mechanical machines were driven by hydro power and were installed alongside rivers, the emergence of steam engines have been a game changer in the development of the rst mass production factories in the United Kingdom (UK). Indeed, these machines were powered with a fuel, such as coal or oil, that could be stored and exchanged between communities and countries. This fuel was burnt in order to heat some water and turn it into steam. This relatively high-pressure steam was used to drive a wheel in mills or factories. Back then, these types of steam-powered machines were only used in the agricultural and industrial sectors to pump water, to mill grains and later to produce manufactured goods such as pieces of clothing or furniture.

Subsequently, a novel era began with the invention of the rst electrical machines, capable of converting a mechanical eort into electricity. Thanks to the numerous 1 Chapter 1. General Introduction research works of Nicolas Tesla on Alternative Current (AC) systems [2], the development of the rst AC electrical networks began in the last decades of the XIX th century. Fossil fuels then became increasingly more used in the production of electricity in power plants and the electrical network made it possible to transport this energy to the consumers connected to the grid. At the same time, thermal motors turned out to be more and more used in embedded applications, e.g. cars, buses, ferries and then airplanes. Thereafter, in the post-WWII era, the electrication of industrial, agricultural and tertiary sectors in most of the countries around the world led to a drastic increase of the consumption of fossil fuels in the world which became dependent on those resources for its economic development and for essential sectors. Because the oil reserves are not innite, there is a real risk of fuel shortage, causing many geopolitical conicts around the world and aecting the international relations.

Climate change and necessary energy transition

In addition to the problems related to the existing resources of fossil fuels, the quantity of fossil fuels that has been consumed since the beginning of the Industrial Revolution is tremendous and is now the main cause for the average rise of temperature of +0.87 o C in the atmosphere compared to the pre-industrial data [3].

Climate change has already been said to be responsible for several natural disasters such as the 2020 Australian wildres [4], or the cold storm that hit Texas in February 2021 causing week-long power shortages [5] and whose cause can be linked to the heating of the Arctic [6]. In the future, climate change could provoke greater natural disasters, such as ooding causing many human casualties and massive infrastructure damages [7].

In consequence, the Intergovernmental Panel on Climate Change (IPCC) considers that 70-85% of electricity should be produced by renewable energy sources [3] in 2050 to limit global warming to +1.5 o C. In order to carry out this energy transition, there is a need for a change of paradigm and an update of the current energy infrastructure. From now on, if we use the existing infrastructure until the end of its lifetime, it will emit 658 gigatonnes of CO 2 . If we add the proposed power plants using fossil fuel (planned, permitted or under construction), approximately 188 gigatonnes of CO 2 emitted in their liftetime will be added less than 580 gigatonnes of CO 2 . This means that all the planned fossil-fuel-based infrastructure projects have to be stopped, while some of the existing infrastructures will have to be decommissioned. This incurs a huge responsibility to the electricity sector, since the energy transition depends on, among others:

• the replacement of the current fossil fuel power plants by renewable energy sources, which must be controlled adequately;

• an electrication of some sectors that used to be dependent on fossil fuels, such as transport systems;

• the development of energy storage systems to compensate for the intermittency of wind and solar power in order to ensure the reliability of electricity.

A transition reliant on resources

The transformations previously mentioned require massive investments and new materials to develop those technologies. The problems related to the materials to build this type of infrastructure are discussed in this paragraph.

Most of the renewable energy systems, such as the photovoltaic panels and the wind turbines, require particular materials, even in small amounts: the rare earth metals. These minerals are elements of the periodic table that have really specic electronic and magnetic properties that make them key players of these systems [10]. However, their scarcity poses problems of shortages in the future of the energy transition.

In addition, the electrication of many economic sectors requires many lines and cables to transport and distribute electricity from the producers to the consumers.

The main conductor used in these cases is copper, whose demand should explode in the upcoming decades, creating concerns over the capacity of supply. Indeed, the cumulative primary copper demand between 2010 and 2050 is found to be 89.4%

of the copper resources known in 2010, and in 2018, 34% of consumed copper was used in power network projects [11].

Finally, the necessity of developing storage systems will increase the consumption of lithium material, with Li/ion batteries being nowadays one of the most mature solutions to store electricity. In addition to the multiple embedded applications, e.g. These renewable energy sources have many features that make them truly dierent from the conventional power plants:

• distribution : contrary to the huge power plants, these systems can be installed at the distribution level and incorporated in cities, solar PV panels being much more acceptable than tall smoky pipes;

• intermittency : while a thermal power plant's output power can be easily controlled by acting on the amount of fuel to be burnt, wind and solar powers are dependent on weather conditions. In consequence, matching the demand with this type of production at any time is a true challenge;

• grid interface : a thermal power plant is coupled with the grid using an electromechanical interface called a Synchronous Generators (SG), which In particular, the role that PE-based interfaces play in the introduction of renewable energy is the focus of this thesis and is developed in the next section.

More power electronics

As explained in Section 1.2.1, more renewable energy means more power electronics in the grid. This increasing share of PE is observed at the distribution level with the distributed generation units [18] and PE-based loads [START_REF] Martinez-Trevino | Synthesis of Constant Power Loads Using Switching Converters Under Sliding-Mode Control[END_REF] but also at the transmission level, where the development High Voltage Direct Current (HVDC)

projects increases [START_REF] Nishioka | Global rise of hvdc and its background[END_REF][START_REF] Obradovic | Assessment of HVDC Frequency Control Methods in the Nordic Test System[END_REF]. The main functions of HVDC are:

• to enhance the meshing of the grid so that there are more exchanged power capacity between countries. This is even more useful in the case of grids with a large proportion of renewable energy where the intermittancy can be partially compensated for by power exchange;

• to interconnect renewable energy sources such as oshore wind farms with the main grid.

A map of Europe with the present and future VSC-HVDC projects is given in Fig. 1.4. This map shows that even though VSC-HVDC projects were mainly developed in Northern countries in the rst place, they are now gaining popularity among public institutions over the entire continent.

1.2. Upcoming changes for the transmission grid to interconnect synchronous and asynchronous areas [START_REF] Nishioka | Global rise of hvdc and its background[END_REF].

In addition, there are also more and more stability-dedicated systems that are installed in the grids: the Flexible AC Transmission Systems (FACTS). Their function is not to inject active power but rather to improve the stability margins and the power transfer capacity of the grid [START_REF]Flexible AC Transmission Systems (FACTS)[END_REF]. These devices, such as Static Var Compensators (SVC) or STATCOM, are mainly based on PE as well.

In consequence, it is expected to see in the near future the emergence of PE-dominated systems that will replace the traditional SG-dominated grids. An illustration of such a network is given in Fig. In the near future, the reliability of such grids requires the proper operation of the AC and DC grids where the PE-based interfaced will be prevalent. The coordinated control of such grids is one of the main challenges in the power system community.

Scope and objectives of the thesis

The main objective of this PhD thesis focuses on the control of power converters in the transmission grid, when this grid contains a high proportion of renewable energy.

In particular, studying the concept of grid-forming control that makes it possible to operate a Voltage Source Converter (VSC) like a Virtual Synchronous Machine (VSM) was one of the objectives of this thesis. This concept had been widely known in the microgrid community to operate 100% grids where all the power could come from PE-based sources [START_REF] Rocabert | Control of power converters in AC microgrids[END_REF]. However, there was no consensus regarding its aspects in the power system community, leading to several EU-funded projects to investigate this type of solution, as the MIGRATE project [START_REF] Migrate | Deliverable 1.1: Current and arising issues caused by increasing power electronics penetration[END_REF].

This PhD thesis deals with the integration of power electronics in the transmission grid and the associated issues with the dismantlement of traditional power plants happening at the same time. It rst presents the limits of the classical 1.4. Main contributions converter control mode, i.e. the grid-following or grid-feeding control, in terms of dynamic behavior and expected support functions. Then, a comparison with the new grid-forming control is made and this latter class of control is considered for all the rest of the thesis. Its role in the integration of more VSC-based sources in the grid is studied, regardless of the VSC topology (two-level, three-level or multilevel)

nor the DC-side dynamics.

In the second part of the thesis, the focus is placed on MMCs interconnecting oshore wind farms using HVDC links. In consequence, the DC-grid dynamics are integrated in the study and the specicities of MMC technology are considered to operate the converter in grid-forming mode. The control of the MMC is studied in terms of the stability of the AC/DC grid and more specically the DC voltage and the AC frequency. Most of the disturbances studied in this work are power disturbances such as load steps or loss of parts of the DC-grid power. However, more serious disturbances such as line tripping or generator losses were also considered in this work to study the robustness of the proposed methods. On the contrary, the behavior of grid-forming controlled VSCs under AC faults [START_REF] Taul | An Overview of Assessment Methods for Synchronization Stability of Grid-Connected Converters Under Severe Symmetrical Grid Faults[END_REF] remains out of the scope of this PhD thesis.

To summarize, this thesis is about the control and operation of VSCs, with a focus on MMCs in HVDC systems integrating renewable energy sources. This work deals with the stability of the AC/DC system when operating a converter in gridforming mode and thus is about the control of the conversion unit. There is no considerations about the internal dynamics of the renewable sources, even though such considerations could extend the results of this work.

Main contributions

The main contributions of this thesis are summarized below:

• An overview of the existing control strategies has been given. In addition to the existing literature, a comparative analysis of the main grid-following and grid-forming controls has been performed regarding their potential from the converter and the grid perspectives;

• In Chapters 4 to 6, a simple constant power source model has been considered

to take into account the DC-side dynamics when operating a converter in grid-forming mode. This was novel compared to similar work in the literature.

The combination of the DC-voltage controller with the grid-forming control was another contribution;

• When the DC grid is considered as a constant power source, the power cannot be adjusted, which is a problem when AC frequency-related services are considered. In this case, we proposed a methodology to calculate the necessary internal energy from an MMC to provide similar inertial response as SGs. To complement, a control law has been designed and tuned to use this energy accordingly.

• In order to optimize the use of the MMC internal energy, an optimal control solution based on Model Predictive Control (MPC) has been proposed to control the MMC internal energy using an online optimization tool. This approach makes it possible to use the MMC internal energy to support the AC frequency and DC voltage in transient depending on the type of disturbance to the system;

• The main problem of sophisticated solutions such as MPC is that there are many technical and computing issues related to their usage in fast industrial systems such as power electronics. Another contribution of this thesis is to prove the feasibility of this kind of controller in a real-time target. This controller was synthetized in C and tested upon a physical mockup of an MMC in real-time conditions.

List of publications

During the three years of work on this project, several publications have been submitted, peer-reviewed and published in scientic journals and international conferences. Some of them were still in the peer-reviewing process when the manuscript was sent to the PhD thesis committee. The list of the publications can be found below:

Journal papers

Interdisciplinary report

• Credited as contributing author in [18]: Irina Oleinikova, Emil Hillberg. micro vs MEGA: trends inuencing the development of the power system, ISGAN report, 2020.

Organization of the manuscript

The manuscript is organized as follows:

• In Chapter 2, a state of the art of the existing converter control modes is presented. In particular, the dierent control modes are compared and their relevancy is assessed regarding the main ancillary services that used to be provided by synchronous generators;

• In Chapter 3, a stability analysis of the main converter control modes, namely grid-following and grid-forming, is presented in order to assess the power electronics penetration limits in hybrid grids with remaining synchronous generators.

• mainland [START_REF] Glasdam | Review on multilevel voltage source converter based HVDC technologies for grid connection of large oshore wind farms[END_REF]. At the distribution-grid level, Power Electronics (PE) converters are used to integrate renewable power sources [START_REF] Blaabjerg | Power Electronics as Ecient Interface in Dispersed Power Generation Systems[END_REF] and new types of loads, such as islanded systems [START_REF] Venkatesh | Frequency Excursion Mitigation in a Multi-source Islanded Energy System Using Meta-Heuristic Optimization Strategies[END_REF][START_REF] Entso-E | High Penetration of Power Electronic Interfaced Power Sources and the Potential Contribution of Grid Forming Converters[END_REF] and electric vehicles [START_REF] Entso-E | High Penetration of Power Electronic Interfaced Power Sources and the Potential Contribution of Grid Forming Converters[END_REF]. These changes will lead to an evolution of the operation of power systems, where a large number of Synchronous Generators (SG) will be replaced by PE-interfaced sources and loads [18].

Literature review

To ensure stability of such power systems in the future, the PE converters will have to play the roles of traditional SG. This topic has been discussed for years in the scientic community [START_REF] Kroposki | Can smarter solar inverters save the grid?[END_REF] and draws public attention [START_REF] Ambrose | Solar farms can keep UK's lights on even at night[END_REF]. In particular, the current methods we know to operate and secure the power systems will have to evolve. In order to understand the necessary conditions to operate these PEdominated grids, we need to analyze the elementary functions that are brought by synchronous generators. Traditionally, the grid stability and reliability is ensured by implicit actions from generators themselves and by specic actions which require additional controls, such as primary frequency support or Power System Stabilizers (PSS) or dedicated equipment such as synchronous condensers or FACTS. In various regions of the world, some of these functions are allocated using ancillary services market [3941]. In parallel, with the growing role of PE-based converters, more and more studies now focus on the potential contributions of VSCs to the operation of 100% PE-based grids, such as microgrids [START_REF] Rocabert | Control of power converters in AC microgrids[END_REF], oshore networks [START_REF] Ramachandran | AC grid forming by coordinated control of oshore wind farm connected to diode rectier based HVDC link -review and assessment of solutions[END_REF] and HVDC transmission grids [START_REF] Yu | A review of control methods for providing frequency response in VSC-HVDC transmission systems[END_REF][START_REF] Luscan | A vision of HVDC key role towards fault-tolerant and stable AC/DC grids[END_REF]. However, these papers only deal with the control options, instead of addressing the needs of the grid. In the last decades, a consistent classication of the power system stability issues has been used as a reference for transmission grid operability and stability [START_REF]Denition and Classication of Power System Stability IEEE/CIGRE Joint Task Force on Stability Terms and Denitions[END_REF]. But it is only recently that this classication has been adapted to take into account the increasing role of VSCs in the grid [START_REF]Power System Dynamic Performance Committee Task Force on Stability denitions and characterization of dynamic behavior in systems with high Bibliography penetration of power electronic interfaced technologies, Stability denitions and characterization of dynamic behavior in systems with high penetration of power electronic interfaced technologies[END_REF], but no classication of the existing control solutions for any of those issues is achieved.

More recently, the impacts of VSC control to provide specic support functions to the common system stability issues was discussed in [START_REF] Peng | On the stability of power electronics-dominated systems: Challenges and potential solutions[END_REF], which only focuses on vector-controlled, also named current-source controlled, VSCs and only briey mentions the role of grid-forming. On the other hand, some recent studies [START_REF] Tayyebi | Frequency Stability of Synchronous Machines and Grid-Forming Power Converters[END_REF][START_REF] Crivellaro | Beyond low-inertia systems: Massive integration of grid-forming power converters in transmission grids[END_REF] did compare the role of dierent grid-forming implementations in multi-machine systems but did not compare them to the most commonly used current-source mode. Indeed, even though there has been some research conducted on classifying those dierent grid forming controls [START_REF] Unruh | Overview on Grid-Forming Inverter Control Methods[END_REF][START_REF] Cheema | A comprehensive review of virtual synchronous generator[END_REF], there is still a missing piece between

Context and literature review

what can be achieved in terms of ancillary services, i.e. from the grid point-ofview, by the dierent VSC control modes. In [START_REF] Hartmann | Eects of decreasing synchronous inertia on power system dynamicsoverview of recent experiences and marketisation of services[END_REF], the authors study the evolution of power systems and how it will aect the existing ancillary services. Although it provides insightful review of the implementation of new ancillary services, it does not focus on the control aspects of the VSCs. In [START_REF] Zhao | Power system support functions provided by smart invertersa review[END_REF], the authors present a review of the support functions of VSC for microgrids, but this work is specic to low-voltage systems and do not discuss wide-area issues such as frequency support or power oscillation damping for instance. Some papers proposed a review of specic support functions of VSCs for transmission grids, such as inertia [START_REF] Fang | On the Inertia of Future More-Electronics Power Systems[END_REF],

short-circuit contribution for unbalanced faults [START_REF] Jia | A review on grid-connected converter control for short-circuit power provision under grid unbalanced faults[END_REF] and power sharing options, both active and reactive, hence integrating grid-forming controls [START_REF] Han | Review of active and reactive power sharing strategies in hierarchical controlled microgrids[END_REF]. In addition, more recent reports were published about the role of grid-forming controlled VSCs in grid support functions [START_REF] Entso-E | High Penetration of Power Electronic Interfaced Power Sources and the Potential Contribution of Grid Forming Converters[END_REF][START_REF] Jouini | New options for existing system services and needs for new system services[END_REF]. However, no comparison is clearly drawn with the most commonly-used current-source controls. In parallel, the role of the grid-forming controlled VSCs in enhancing the PE-based sources penetration limits using modal analysis have been the topic of several studies [START_REF] Mourouvin | Understanding the role of VSC control strategies in the limits of power electronics integration in AC grids using modal analysis[END_REF][START_REF] Siraj Khan | Stability Assessment of a System Comprising a Single Machine and a Virtual Oscillator Controlled Inverter with Scalable Ratings[END_REF][START_REF] Markovic | Understanding stability of low-inertia systems[END_REF]] but these papers are mainly based on simplied systems and it is hard to extract the precise role of each support function when applied to more complex systems. To the authors' best knowledge, there is no review in the literature which focuses on the dierent VSC control options for the needed support functions from the transmission grid perspective.

Chapter summary and contents

In this chapter, we gather information about the existing grid services that are currently provided by synchronous machines and that will be needed from the VSCs to ensure a smooth transition to hybrid grids with remaining synchronous units and a high proportion of PE-based converters. The novelty of this chapter is to propose a comprehensive comparison of the converter control modes, i.e. current-source and voltage source modes, not only in terms of converter-level control performances, but also in terms of the potential ancillary services they can provide to the grid.

The chapter is organized as follows. In Section 2.2, we describe the ongoing evolution of power system production and transmission structures and the role of SGs in power system stability. In Section 2.3, the VSC control modes are introduced and an updated classication is dened for the transmission grid. In Section 2.4, the role of the VSC in both current-source and voltage-source modes are explored in terms of the grid support functions. Finally, an overview of the dierences is drawn in Table 2.3 and discussed in Section 2.5.

Chapter 2. State-of-the-art of existing control solutions for Voltage Source Converters to support the Transmission Grid 1) is for inertial response, ( 2) is for primary frequency control and ( 3) is for secondary frequency control.

Impacts of the replacement of SGs by Power Electronics-based Stations

The role of the SGs in power systems has been studied for decades [START_REF] Kundur | Power System Stability and Control[END_REF][START_REF] Wang | Modern Power Systems Analysis[END_REF], and their services to the grid are summarized in this section, in particular, frequency support, reactive power support, contribution to system strength, short circuit contributions and power oscillation damping.

Frequency support

Traditionally, the grid frequency is mainly supported by the synchronous generators which provide a frequency response following a contingency in the form of an unbalance between the generation and the load. A typical frequency prole of such response is given in Fig. 2.1, where the dierent mechanisms are highlighted in terms of the time constant:

• Inertial response : The physical law imposes that, in case of a frequency deviation, the SG instantaneously releases the kinetic energy in its rotor, which reduces the Rate of Change of Frequency (RoCoF).

• Primary frequency control : The speed governor, usually in form of a droop, adjusts the mechanical power from the prime mover in the time scale of 30 seconds, in order to stabilize the frequency.

• Secondary frequency control : The secondary frequency regulator, with a time constant of 15 minutes, adjusts the mechanical power of the synchronous generators of the control area where the disturbance originates, in order to bring back the grid frequency to 50 Hz and restore the power exchange between the control areas.

2.2. Impacts of the replacement of SGs by Power Electronics-based Stations

Reactive power support

The reactive power support refers to the capacity of a given system to provide some reactive power which is used to maintain the grid voltage. For this, the SG are equipped with an Automatic Voltage Regulator (AVR), which observes the voltage magnitude and adjusts the reactive power accordingly. Indeed the reactive power compensation is used to maintain the grid voltage within acceptable limits. This support is often seen as an ancillary service from the TSO point of view [START_REF] Anaya | Reactive Power Management and Procurement Mechanisms: Lessons for the Power Potential Project[END_REF]. It can be separated in two types [START_REF] Anaya | Reactive Power Management and Procurement Mechanisms: Lessons for the Power Potential Project[END_REF][START_REF]Overview of ancillary services for the power grid, Belgian High Voltage Grid Operator[END_REF] according to the time scales:

• dynamic support : it refers to the capability to control the bus voltage magnitude following a disturbance. The Belgian TSO Elia denes it as a voltage control that is activated automatically by the user's production units [START_REF]Overview of ancillary services for the power grid, Belgian High Voltage Grid Operator[END_REF]. This is provided by the SGs' AVRs for instance;

• static support : it refers to changes of reactive power references in a given grid in order to support the AC grid voltage at all nodes. This corresponds to a centralized action dictated by the regional TSO. It is triggered either automatically or manually depending on the situation [START_REF]Overview of ancillary services for the power grid, Belgian High Voltage Grid Operator[END_REF].

To illustrate the role of reactive power in voltage control, a simple case system is given in Fig. 2.2 (a). The reactive power received at the bus 1, denoted by Q R can be calculated, with phasor assumptions [START_REF] Machowski | Power System Dynamics[END_REF], as:

Q R = V 1 X (V 2 cos δ -V 1 ) (2.1) Q R ≈ V 1 X (V 2 -V 1 ) (2.2) Fig. 2.2 (b) gives the evolution of Q R using (2.
2) and shows the importance of acting on the reactive power to control the AC voltage magnitude. In practice in traditional power systems, the operating points of SGs are determined in power ow studies in order to avoid line congestions and have acceptable voltage magnitude levels [START_REF] Xu | Valuation of reactive power support services based on sensitivity and risk analysis[END_REF] but the TSOs also rely on automatic voltage control actions to maintain the bus voltages following unexpected disturbances.

System Strength

The system strength, or voltage stiness, is a metric that quanties the capability of a given AC bus to maintain the bus voltage around its nominal setpoint, in terms of both voltage magnitude and phase angle. In its 2016 System Operability Framework [START_REF]System Operability Framework[END_REF], the British TSO National Grid gives the following denition in the voltage management section: System strength is a regional characteristic which can be expressed as short circuit level (SCL), measured in kA. It provides an indication of the local dynamic performance of the system and behaviours in response to a disturbance.

With this denition, it is clear that a bus connected to a voltage source, such as an SG, has a sti voltage since the generator controls its voltage magnitude thanks to its AVR. To quantify the system strength of a grid dominated by SG, the power system community has used for decades the notion of Short-Circuit Level (SCL)

or Short-Circuit Ratio (SCR) [START_REF]Short-circuit modeling and system strength[END_REF], dened as the amount of fault current that will be injected to the bus in case of a fault normalized with the bus base power. This means that for a given voltage, the higher the SCR is, the closer it is to SGs since the equivalent impedance to the faulty bus is smaller. Nowadays, the system strength is considered as a rising issue because a low value of SCR can create stability problems due to the PE-based source controllers [START_REF]Power System Dynamic Performance Committee Task Force on Stability denitions and characterization of dynamic behavior in systems with high Bibliography penetration of power electronic interfaced technologies, Stability denitions and characterization of dynamic behavior in systems with high penetration of power electronic interfaced technologies[END_REF].

Although the SCR is a single notion to study the system strength, it actually contains two dierent power system issues: the system strength at the given bus discussed in this subsection, and the contribution to the short-circuit current, discussed in the next subsection.

Short-circuit contributions

The voltage-source behavior of the machine and its large overcurrent capacity make it possible to inject a high short-circuit current in case of a fault, which is essential for the proper operation of the protective relays. The SGs inherently provide a Short-Circuit Contribution (SCC) when a fault occurs. Indeed, during the rst instants 2.2. Impacts of the replacement of SGs by Power Electronics-based Stations following the fault, the machine remains connected and injects a fault current which can be as high as 500% of the nominal current [START_REF]4/C6.35[END_REF]. This behavior is illustrated in Fig. 2.3. As shown in this gure, from the point of view of the faulty bus, all the fault currents from the SG and the other AC grid fault currents converge to it. These fault currents are valuable because they are used to detect and locate the fault in the AC grid. Note that, similar to the inertial response, the short-circuit contribution is also a natural behavior of the SG, i.e. no additional control is needed to realize this. However, this feature is based on the voltage-source behavior of the SG which still relies on the AVR performances during the fault.

Power oscillation damping

Power oscillations refer to low-frequency oscillations, in the range of 0.1 to 1 Hz, which are due to structural limits of the system when certain large-scale disturbances (synchronous generator loss, line tripping, ect) occur. They are mainly due to the poor damping of low-frequency modes which exist in power systems where the loads and production units are not regularly distributed or where the transmission lines are relatively long [START_REF] Liu | PSS design for damping of inter-area power oscillations by coherency-based equivalent model[END_REF]. These oscillations must be taken care of because they provoke power oscillations which generate more losses and can cause the tripping of lines.

To solve this problem, the Power System Stabilizer (PSS) [START_REF] Machowski | Power System Dynamics[END_REF] can be added to the AVR to improve the damping of power oscillations.

However, this problem is still an up-to-date topic because the locations of the SGs equipped with PSSs plays a major role on the damping factor of the whole system.

An illustrative case of inter-area oscillations which occured in Europe in 2017 [START_REF] Entso-E | Oscillation Event 03.12[END_REF] is given in Fig. This event shows the importance of the PSSs allocation, since the disturbance severity can be drastically dierent depending on the grid node, as described in Fig.

2.4.

Consequences of the SG dismantlement on system stability

As synchronous generators are being progressively replaced by PE converters, all the functions described earlier in this section will have to be fullled in transitional grids with high proportion of PE and the remaining SGs. This issue is gaining more and more attention from academia [START_REF] Milano | Foundations and challenges of low-inertia systems[END_REF] and industry [START_REF] Migrate | Deliverable 1.1: Current and arising issues caused by increasing power electronics penetration[END_REF] across the world.

In 2017, a list main challenges on power system stability were highlighted by

European TSOs [START_REF] Migrate | Deliverable 1.1: Current and arising issues caused by increasing power electronics penetration[END_REF], of which the following items are related to the replacement of SGs by PE-based generation units:

1. Decrease of inertia; European countries which still rely on coal-powered plants would have a much higher inertia. Finally, countries that have started to introduce renewables in their electrical systems but still rely on low-carbon power plants, either hydro or nuclear, would be in-between.

In consequence, there is a need to counterbalance the impacts of dismantling current power plants in order to maintain the system operable. In Fig. 2.5 we illustrate the consequences of decreasing inertia and the deployment of primary frequency reserves following an active power load step in the frequency deviation of a two-bus system with one SG and one load. The lack of inertia may cause the tripping of certain components due to an excessive value of RoCoF.

Chapter 2. State-of-the-art of existing control solutions for Voltage Source Converters to support the Transmission Grid

Opportunity of the use of power electronics in transmission grids

Even if many challenges arise due to this grid transition, PE-based sources may oer innovative solutions since these interfaces are much more controllable and faster than typical SGs. In addition, as illustrated in Table 2.2, the integration of renewable energy sources such as OWF, here denoted as wind power plants, the gridside converter rarely works at its maximum rated power due to wind intermittency mostly. The average active power setpoint can be derived as follows. The maximum active and reactive power headroom from this average setpoint are then calculated while assuming the initial reactive power setpoint is q * = 0:

P hr = S b × (1 -p * avg ) (2.4) Q hr = S b × 1 -p * avg 2 (2.5)
It appears that the active power at the operating point of the converter is quite low, here assumed to be at p * avg = 0.23 p.u.. In consequence, there would be a lot of power headroom, both active and reactive, available in average, to counterbalance the progressive dismantlement of SGs and the upcoming challenges listed above. This will still require to implement additional control options, which are summarized in Sections 2.3 and 2.4.

Current-source mode

The current-source control, also known as grid-following or grid-feeding control, is the most commonly used method, where the converter imposes both the active and the reactive power, independent of the grid frequency and voltage. The controller receives the power setpoints and yields the reference for the current to be injected into the grid. Thus, the converter works as a current source, i.e. the current (and hence the active and reactive power) is controlled while the voltage magnitude and the frequency is free to change. Note that the power reference itself can be varying due to other factors, e.g. the intermittency of renewable energy sources, or an overlayer controller providing some grid service.

A classical implementation of the grid-following control is shown in Fig. 2.7. The voltage at the point of common coupling (PCC) is denoted as v 0 , while the voltage synthesized by the converter as v i . A reactor l f is placed between the two voltages. With a properly synthesized v i , the power exchanged between the converter and the AC grid across the reactor l f follows its reference.

In Fig. 2.7, the current and the voltage in the natural abc frame are transformed in the synchronous dq frame, and a cascaded control is used. The control is composed of four blocks:

• Phase-Locked Loop (PLL): it observes the three-phase voltage at the PCC located between the converter and the AC grid, and estimates its angle θ, as illustrated in Fig. 2.8. This information will be used by the other blocks.

Chapter 2. State-of-the-art of existing control solutions for Voltage Source Converters to support the Transmission Grid • Outer power loop: based on p ref and q ref , the reference values of active and reactive power, the outer power loop calculates the current references, denoted by i d * L and i q * L , which are then sent to the inner current loop. By aligning the grid voltage to the d axis, p and q can be decoupled and thus controlled independently. In particular, p depends only on i d L while q only on i q L . To L . An alternative way to calculate the current references is to use closed-loop control, where PI controllers compare the power references with the measured and ltered power, and then adjusts the current references.

• Inner current loop: based on the current references i d * L and i q * L sent by the outer power loop, the inner current loop calculates v d * i and v q * i , the desired voltage to be synthesized by the converter, which are then sent to the PWM block.

The PI controllers compare the current references with the measured currents, and then adjusts the voltage references. This inner loop is used to control the current owing through the converter and limit potential overcurrents. More information about the tuning can be found in [2]. Figure 2.9: VSC cascaded structure control scheme in voltage-source mode.

The grid-following control is not capable of feeding a passive network, because it always needs a grid voltage to follow, whose frequency and magnitude should be regulated by synchronous machines or converters working in other control modes.

When left unregulated, the grid voltage and frequency may deteriorate and result in a voltage collapse, which is unacceptable from the demand-side point of view.

Voltage-source mode

This control mode, which is also referred to as grid-forming control, has emerged in islanded microgrids rst [START_REF] Rocabert | Control of power converters in AC microgrids[END_REF]. In this mode, the VSC operates as a voltage source and, in consequence, generates a voltage of desired magnitude and frequency at the connecting bus.

The general scheme is given in Fig. 2.9. The voltage at the PCC, denoted by v 0 , is also the voltage of the capacitance c f . This capacitance can be the physical capacitance of an LC lter [START_REF] Markovic | Stability Analysis of Converter Control Modes in Low-Inertia Power Systems[END_REF] or the capacitance of an AC component such as an overhead line or underground cable [START_REF] Sanchez-Sanchez | The role of the internal energy in MMCs operating in grid-forming mode[END_REF] if there is no lter. The control scheme is composed of the following blocks:

Chapter 2. State-of-the-art of existing control solutions for Voltage Source Converters to support the Transmission Grid

• Outer power loop: based on the voltage and current measurements at the PCC, the control loop calculates the active and reactive powers which are compared with the references. The control errors of active and reactive power are used to adjust, respectively, its frequency ω and voltage magnitude e * . This function reects the principles of grid-forming which will be detailed in Section 2.3.3.

• Virtual impedance : when several VSCs are working in voltage-source mode in parallel, this block can be used to eliminate the circulating currents between them. In fact, it changes the voltage reference e * sent by the outer power loop to emulate an additional impedance between the controlled voltage v 0 and the PCC to generate the voltage reference v dq * 0 which is used for the inner control loops, as described in [START_REF] Markovic | Stability Analysis of Converter Control Modes in Low-Inertia Power Systems[END_REF].

• Inner control loops : in addition to the inner current control similar to the one in Section III-A for the grid-following control, there is also an inner voltage control law that controls v dq 0 using PI controllers and generates current references i d * L and i q * L for the current controller. More information about the tuning can be found in [START_REF] Denis | From grid-following to grid-forming: The new strategy to build 100 % power-electronics interfaced transmission system with enhanced transient behavior[END_REF].

• Pulse-Width Modulation (PWM): similar to the one in Section 2.3.1.

There are in the literature dierent variants of grid-forming control, which depends on the way the power variations and the voltage references are coupled in the outer power loop. In particular, in Fig. 2.10 (c), the droop-based grid-forming control [7881] couples the active power deviation with that of the VSC estimated frequency ω. The virtual inertia and load sharing of the grid-forming control can be tuned by acting on the low-pass lter cuto frequency ω pq and the droop gain m pω given in Fig. 2.10 (c).

It is important to note that the vf control, also known as voltage and frequency control [START_REF] Cheema | A comprehensive review of virtual synchronous generator[END_REF], which basically imposes a constant voltage magnitude e * , usually 1 p.u., and a constant frequency ω = ω ref , usually 50 Hz or 60 Hz, may also be referred to as grid-forming control in microgrid literature [START_REF] Rocabert | Control of power converters in AC microgrids[END_REF]. In fact, vf mode can be seen as a particular case of the scheme given in Fig. 2.9 where the outer loops are bypassed and the references are xed values of e * and ω.

Classication in the context of transmission grids

Thanks to the small size of a microgrid, a grid-forming converter is able to impose by itself the frequency to the whole system. However, in the context of a transmission grid, one single converter, which is of much smaller rating than the whole synchronous area, would become incapable of imposing the frequency 2.3. Converter-level control modes on its own. Therefore, compared to its original denition in the microgrid where the converter behaves as a voltage source with imposed frequency and voltage magnitude, the grid-forming control in the transmission system is slightly dierent, and the following classication can be used instead:

• Grid-following : the converter works as a current source that injects power without consideration on the grid conditions.

• Grid-supporting : the converter works as a current source and modulates its power references with respect to the initial load ow conditions to provide voltage, frequency or angle support. In other words, the gridsupporting combines the current-source working mode with an external control modulating the power references. A commonly-used method for controlling the active and reactive powers using a PI control is given in Figs. 2.10 (a) 2.10 (b);

• Grid-forming : the converter works as a voltage source and provides voltage and frequency support. However, the frequency imposed by the converter will not be the nominal frequency, but such that the resulting power and the frequency are coupled to emulate the behavior of SM. The dierences between existing grid-forming controls lie in the dierent control designs of their active and reactive outer power loop and the coupling between them.

In addition to the droop-based control given in Figs. 2.10 (c) 2.10 (d), other options exist, such as emulation of Synchronous Machines (SM) [8287] and more advanced control-oriented strategies [START_REF] Groÿ | The eect of transmission-line dynamics on grid-forming dispatchable virtual oscillator control[END_REF][START_REF] Arghir | Grid-forming control for power converters based on matching of synchronous machines[END_REF]. Besides, some comparisons were drawn between dierent grid-forming options in microgrids [START_REF] Johnson | Comparison of virtual oscillator and droop control[END_REF] and in transmission grids [START_REF] Or | virtual inertia: Comparison from the perspective of converter operation mode[END_REF][START_REF] Tayyebi | Interactions of gridforming power converters and synchronous machines a comparative study[END_REF] and showed similar small-signal behaviors and compatible control laws between each other. However, the transient stability analysis showed dierent behaviors of the grid-forming options [START_REF] Yu | Comparative Transient Stability Assessment of Droop and Dispatchable Virtual Oscillator Controlled Grid-Connected Inverters[END_REF].

Note that the above classication, borrowed from [START_REF] Rocabert | Control of power converters in AC microgrids[END_REF], is somewhat an abuse of language, because the classication criterion is a mixture of the control method and control objective. On the one hand, in terms of the control method, the grid-supporting control resembles the grid-following since both have the converter working as a current source; on the other hand, in terms of the control objective, the grid-support control seems closer to the grid-forming control, since both make the converter to contribute to the voltage and frequency regulation. In other words, to support the frequency, one can either use the grid-supporting control where a ∆p * term is added in the active power controller of the converter working as a current source, as described in Fig. 2.10 (a), or alternatively, use the grid-forming control as described in Fig. 2.10 (c). In addition, the active power controller of the grid-forming scheme from Fig. 2.9 plays a double role as described in Fig. 2.10 (c).

Indeed, it provides a PLL-free synchronizing unit [START_REF] Zhang | Power-synchronization control of gridconnected voltage-source converters[END_REF] and makes the VSC contribute to frequency support as SGs do, i.e. by providing both synthetic inertia and primary frequency response [START_REF] Markovic | Stability Analysis of Converter Control Modes in Low-Inertia Power Systems[END_REF].

In terms of performances, grid-following control is designed to ensure a good power tracking of the power references p ref and q ref using a dedicated controller as in Figs. 2.10 (a),2.10 (b). However, the behavior of the VSC relies on strong grid assumptions, i.e. with a high SCR and the VSC can even become unstable if connected to a weak grid (SCR<3). On the other hand, the grid-forming control has a power tracking capability thanks to its power controller shown in Figs. 2.10 (c),2.10 (d) but since it behaves as a voltage source, its AC power response depends on the SCR. However, it is capable of working in weak grid conditions as well, on the contrary to grid-following controlled VSCs. These conclusions are shown in Fig. 2.11 where simulations were carried out using a simple benchmark containing a VSC model based on Fig. 2.6 which is connected to an innite bus through a transformer and a line modeled with a dynamic phasor model.

Grid-level control modes

With the current-source and voltage-source modes described at the converter level in Section 2.3, this section presents their services from a grid point of view.

2.4. Grid-level control modes

Frequency support

Contrary to SGs whose frequency response is a combination of the physical behavior and explicit control, the VSC output power is entirely decoupled from the grid frequency behavior in grid-following mode. However, there are dierent active power proles that can be provided by the VSC to support the grid frequency:

• Active power / frequency droop: developed in [9699], its aim is to provide an active power response that is proportional to the measured frequency deviation as described in Fig. 2.12 (a). This control scheme is inspired by the primary frequency control of traditional SGs given in Fig. 2.1. In consequence, the VSC takes part in the load sharing with the SGs.

• Virtual inertia : as in [START_REF] Duckwitz | Modeling and design of df/dt-based inertia control for power converters[END_REF][START_REF] Poolla | Placement and implementation of gridforming and grid-following virtual inertia and fast frequency response[END_REF], with this control, almost instantenously after the disturbance, the converter changes the active power reference proportionally to the RoCoF in order to emulate the kinetic energy delivered by the SG's rotor. It is known as the inertial response, and is illustrated in Fig. 2.12 (b). It is important to note that the implementation of a derivative term d∆ω dt would require an additional notch lter in practical conditions.

• Fast Frequency Response (FFR): this type of response has started to spread in islanded transmission grids [START_REF] Hartmann | Eects of decreasing synchronous inertia on power system dynamicsoverview of recent experiences and marketisation of services[END_REF] and provides an active power contribution deployed with a latency of less than 1s [102104]. In consequence, it can be seen as a trade-o between the inertial response and the primary frequency control, and the power prole of the FFR can be customized instead of following a certain pattern, as illustrated in Fig. 2.12 (c). Figure 2.12: Illustration of dierent potential supporting actions using a variation of power reference provided by a power converter.

As to a grid-forming converter, it provides inherent frequency support since it emulates the SG dynamic response, i.e. with virtual inertia and load sharing capability. Concerning this last item, the current-source controlled and voltagesource controlled VSCs provide similar responses if sized and tuned accordingly [START_REF] Zuo | Eect of voltage source converters with electrochemical storage systems on dynamics of reduced-inertia bulk power grids[END_REF]. However, if the DC-side of the converter is not adapted to provide this load sharing capability, this contribution may not be desired. This inherent frequency droop action of the grid-forming controller can thus be cancelled by using a highpass lter or a frequency estimation ω from a PLL [START_REF] Rokrok | Eect of using PLL-based grid-forming control on active power dynamics under various SCR[END_REF] instead of the nominal frequency ω ref in the active power control loop given in Fig. 2.10 (c). Another option is to replace the droop gain m pω by an IP controller [START_REF] Narula | Impact of steady-state grid-frequency deviations on the performance of grid-forming converter control strategies[END_REF].

About the inertia then, since the grid-forming controlled VSC is operated as a SG during the rst instants following any kind of disturbances, it does provide some inertia to the grid. On the other hand, the inertia provided by the virtual inertia controller in current source mode from Fig. 2.12 (b) is slightly dierent because it is an extra control loop that relies on the PLL speed, accuracy and the notch lter to deliver such inertia, meaning it will not be delivered at the very rst instants following the disturbance. which are also based on VSC technology. Indeed, by controlling the reactive current of a converter, wether it is an HVDC grid, a FACTS or a system connected to an energy source, the VSC is capable of adjusting its reactive power to support the PCC voltage as described in Section 2.2.2. In the literature, there are dierent ways the VSC can control its reactive power to support the AC voltage. In addition to the reactive power loop from Fig. 2.10 (b), there are other options which are given in Fig. 2.13. In Fig. 2.13 (a), another implementation of the reactive power loop is proposed to control the PCC bus voltage using PI control. In Fig. 2.13 (b), the support is a power droop similar to the active power frequency droop in Fig. 2.12 (a), which is used to provide a support proportional to the voltage magnitude deviation.

In Fig. 2.13 (c), the controller regulates the voltage magnitude at the PCC but the reference is changed dynamically with respects to the reactive power deviation.

However, the two implementations from Figs. implementing a virtual impedance as in Fig. 2.9. Also, it is important to note that the grid-forming control is not only about maintaining the AC voltage magnitude but also its phase, as it is described in the next section.

Contribution to system strength

In the literature, it has been shown that PE-based converters controlled in currentsource mode can be unstable when connected to a weak grid, i.e. where the SCR at the PCC is inferior to 3, due to the interactions between the line dynamics and the PLL [START_REF] Zhou | Impact of Short-Circuit Ratio and Phase-Locked-Loop Parameters on the Small-Signal Behavior of a VSC-HVDC Converter[END_REF]. To illustrate these interactions, a frequency analysis of the openloop PLL given in Fig. As shown in the Bode plots given in Fig. 2.14, there is a trade-o between the speed of the tracking capability of the PLL and its robustness: if the PLL is fast enough to track any frequency changes, it is also more sensitive to disturbances. The resonance frequency of the PLL is found to be around 100 Hz, which is the order of magnitude of the transmission line dynamics [START_REF] Markovic | Understanding stability of low-inertia systems[END_REF], hence the negative interactions with the lines causing instability when the lines are too long, which correspond to weak grid conditions.

Grid-level control modes

Nowadays, this weak grid issue is responsible for renewable energy curtailments in some power systems [START_REF] Matevosyan | Experience in Texas with declining inertia[END_REF]. In the literature, there are three main solutions to allow more VSCs to be connected to weak grids:

• Introduce synchronous condensers to provide system strength to the weak regions [START_REF] Marrazi | Allocation of synchronous condensers for restoration of system short-circuit power[END_REF]. However, this solution does not include any VSC-related function;

• Reinforce the weak grids, for example by grid-forming control in Fig. 2.9. This is dierent from the services by FACTS that improve the long-term voltage stability by reactive power compensation [START_REF]Denition and Classication of Power System Stability IEEE/CIGRE Joint Task Force on Stability Terms and Denitions[END_REF]. However, it is possible to control STATCOMs using grid-forming control [START_REF] Nguyen | Synchronverter-based operation of STATCOM to mimic synchronous condensers[END_REF] to make them contribute to the system strength;

• Develop novel PLL algorithms for weak and very weak grids, i.e with a SCR≤2 [START_REF] Molinas | Extended stability range of weak grids with voltage source converters through impedance-conditioned grid synchronization[END_REF][START_REF] Radwan | Improved vector control strategy for current-source converters connected to very weak grids[END_REF] and keep the current-source control structure presented in Fig. 2.7. Today, this current-source control has better performances than classical voltage-source controls in terms of power tracking for strong grid situations [START_REF] Rokrok | Classication and dynamic assessment of droop-based grid-forming control schemes: Application in HVDC systems[END_REF][START_REF] Harnefors | Referencefeedforward power-synchronization control[END_REF], as illustrated in Fig. 2.11. Therefore, the VSC owners and TSOs should reach an agreement on whether most, if not all, of the VSC should participate to some extent in this voltage control, as the SG used to do, or only some specic VSC should be required to ensure minimum service in this aspect.

Even though the notion of SCR was a consistent metric for short-circuit contributions and voltage stiness in SG-based systems, it is not sucient enough in PE-dominated systems since the fault-transient behavior and the system strength in normal operation are no longer coupled. Developing novel metrics to assess the system strength levels in PE-dominated grids is still an open-topic. Indeed, contrary to SGs, a VSC in grid forming does not maintain its voltage-source behavior at all time because of its limited overcurrent capability. Indeed, its contribution to the voltage stiness strongly depends on the nature of the disturbance and its operating point, in particular its active and reactive power references. In that way, dynamic analysis are required to assess the voltage stiness of a PE-dominated grid, while it was possible for SG-dominated grids to use only static analysis [START_REF] Marrazi | Allocation of synchronous condensers for restoration of system short-circuit power[END_REF].

Short-circuit contributions

Unlike SGs whose high overcurrent capability makes it possible to contribute to short-circuits during AC faults, as described in Section 2. is imposed, the conversion valves will be blocked and the VSC will be disconnected from the grid, hence interrupting its power conversion and making it unable to contribute to the short-circuit currents.

In consequence, we may require in the future that VSCs have a Fault-Ride Through (FRT) capability where they remain connected when encountering large disturbances, such as a fault or a generation unit loss. The FRT capability is usually assessed by TSOs using a FRT prole which corresponds to the minimum voltage response of the equipment following a given fault to remain connected to the grid.

There are many proles which depend on the TSO grid codes [START_REF] Díez-Maroto | Fault ride through capability of round rotor synchronous generators: Review, analysis and discussion of European grid code requirements[END_REF]. A typical FRT is given in Fig. 2.15 (a). In addition to the voltage deviations, the VSC can still lose synchronism even though the voltage magnitude remains within the FRT prole.

This transient stability needs to be assessed and is currently a topic of great concern by academia [START_REF] Taul | An Overview of Assessment Methods for Synchronization Stability of Grid-Connected Converters Under Severe Symmetrical Grid Faults[END_REF][START_REF] Wang | Grid-Synchronization Stability of Converter-Based Resources An Overview[END_REF].

Second, in addition to remaining connected, the VSC may also be asked to contribute to the short-circuit currents in order to help detecting the fault. In the literature, there have been some studies about developping strategies to make the VSCs in current-source mode remain connected and provide 1 p.u. of reactive current when the voltage drops below a certain level [START_REF] Keller | Understanding Fault Characteristics of Inverter-Based Distributed Energy Resources[END_REF][START_REF] Jia | Synchronous condenser allocation for improving system short circuit ratio[END_REF], 90% of nominal voltage in [START_REF] Taul | An Overview of Assessment Methods for Synchronization Stability of Grid-Connected Converters Under Severe Symmetrical Grid Faults[END_REF]. However, in several grid codes, these contributions are not mandatory anymore if the PCC voltage, v 0 in Fig. 2.15 (b), drops below 20% of its nominal value [START_REF]4/C6.35[END_REF] as it is illustrated in Fig. 2.15 (b).

In the voltage-source mode, as in Fig. 2.9, the FRT capability is more complex to achieve, because the branch current results directly from the voltage the VSC 2.4. Grid-level control modes imposes at its terminal. In consequence, the VSC control must act quickly to avoid a fault current which would damage the conversion devices or force the interruption [START_REF] Debry | Maximizing the penetration of inverter-based generation on large transmission systems: the MIGRATE project[END_REF]. In the recent literature, two main ways of achieving FRT with a VSC controlled in voltage-source mode have emerged:

• Use of a backup PLL: When the control detects a current rise, it switches to an emergency current-source mode and keeps the synchronization through a backup PLL. The new power references are chosen in order to inject a fault current of 1 p.u. [START_REF] Zhang | Power-synchronization control of gridconnected voltage-source converters[END_REF].

• Limitation of the voltage magnitude : The control keeps running in voltage source mode but the voltage reference is decreased to limit the branch current.

One solution could be the implementation of a fault detector with a virtual impedance control as in [START_REF] Denis | The Migrate project: the challenges of operating a transmission grid with only inverter-based generation. A grid-forming control improvement with transient current-limiting control[END_REF][START_REF] Qoria | Critical clearing time determination and enhancement of grid-forming converters embedding virtual impedance as current limitation algorithm[END_REF].

For both the current-source mode and the voltage-source mode, the shortcircuit contribution of a VSC is physically limited by the semiconductor devices.

In consequence, the current way the TSOs detect and clear AC faults has to be re-dened for large penetration of PE-based sources scenarii [START_REF] Debry | Maximizing the penetration of inverter-based generation on large transmission systems: the MIGRATE project[END_REF]. There are two tendencies that have emerged in the literature for AC fault detection:

• Enhance the fault response of PE-based sources : According to this conservative strategy based on the assumption that power grids will still rely on SGs, the fault detection algorithms are unlikely to change. On the other hand, the very limited overcurrent capability of VSCs, currently at approximately 110% as shown in Table 2.2, must be enhanced. One solution is to upgrade the thermal management system of the converter. There are examples in the literature of PE-based systems where the current is able to go up to 200% for less than 200µs [START_REF] Keller | Understanding Fault Characteristics of Inverter-Based Distributed Energy Resources[END_REF]. Another solution is to integrate synchronous condensers in parallel with the VSC to enhance the fault response at the point of coupling [START_REF] Biellmann | The benets of implementing Synchronous Compensators in grids with high penetration of Renewables[END_REF].

• Change the AC grid fault detection strategy : Today the AC fault detection algorithm is based on dierential functions, relying only on measurements of currents and Kirsho 's current law. However, with the recent advances in system identications, new methods based on Dynamic State Estimation (DSE) make it possible to monitor a given protection zone without requiring large overcurrents or coordination between the zones [START_REF] Paolone | Fundamentals of Power Systems Modelling in the Presence of Converter-Interfaced Generation[END_REF].

However the choice of relevant solutions requires the coordination of the dierent actors (TSOs, converter manufacturers, etc...) and is more a design problem rather than a control problem.

Chapter 2. State-of-the-art of existing control solutions for Voltage Source Converters to support the Transmission Grid Figure 2.16: Dierent manners of damping inter-area power oscillations in a single synchronized power system using synchronous generators, VSC-interfaced power sources and VSC-HVDC links.

Power oscillation damping

Following the development of several embedded HVDC links in some interconnected systems, several studies have shown that VSC-interfaced power sources and VSC-HVDC links operating in current-source mode can provide a Power Oscillation Damping (POD) function if the active power reference are adjusted accordingly [124127]. These power references can be changed according to the dierence of the two-area frequencies ∆p * = f (ω 2 -ω 1 ) or the dierence between the reference and the local frequency ∆p * = f (ω * -ω 2 ) for instance, as illustrated in Fig. 2.17 (a). In the former case, the system needs communication devices whereas the latter option only relies on local measurements. Both cases are illustrated in Fig. 2.16. The POD function can also be achieved using reactive power reference modulation [START_REF] Renedo | A communicationfree reactive-power control strategy in VSC-HVDC multi-terminal systems to improve transient stability[END_REF] or be applied to Multi-terminal HVDC (MTDC) embedded systems [START_REF] Gonzalez-Torres | A novel distributed supplementary control of multi-terminal vschvdc grids for rotor angle stability enhancement of ac/dc systems[END_REF].

For grid-forming VSCs, also given in Fig. 2.16, since they have a SG-like behavior, they add supplementary electromechanical-like modes in the system. However, these modes can still be damped by adding an additional damping controller though. In this case, to damp the power oscillations, the power references p ref and q ref , dened in Fig. 2.9, can be adjusted similar to a VSC working in current-source mode. In this way, the power-tracking capability of the grid-forming mode [START_REF] Rokrok | Classication and dynamic assessment of droop-based grid-forming control schemes: Application in HVDC systems[END_REF] is used to provide POD. Another option is to adapt the active power control loop to add a notch lter which eliminates the undesirable oscillations, as in [START_REF] Cortes | Virtual Controller of Electromechanical Characteristics for Static Power Converters[END_REF][START_REF] Baltas | Grid-Forming Power Converters Tuned through Articial Intelligence to Damp Subsynchronous Interactions in Electrical Grids[END_REF] and illustrated in Fig. 2.17 (b).

Discussions & Conclusions

(a) POD control structure using active power modulation [START_REF] Kotb | On the design and placement of a supplementary damping controller in an embedded VSC-MTDC network[END_REF].

(b) POD control structure using modied droop-based grid-forming control [START_REF] Cortes | Virtual Controller of Electromechanical Characteristics for Static Power Converters[END_REF].

Figure 2.17: Options for POD functions depending on the VSC control mode.

Nevertheless, all these types of POD impose some constraints on the VSC:

• Some power headroom must be dedicated to POD at all time to be capable of acting on the power references. In consequence, the VSC cannot operate at its maximum apparent power.

• It impacts the transferred power of the VSC, which can generate marketrelated and operational issues if the HVDC system is between two distinct areas/countries.

Discussions & Conclusions

The two VSC control modes, namely, the current-source control and the voltagesource control, were presented in Section 2.3, while the converter services to the grid were discussed in Section 2.4. As described in this last section, most of the grid support functions can be added to the grid-following scheme by slightly modifying the control, i.e. by adding current limitations or ∆p * ∆q * power references. As a consequence, the supporting action can be tuned accordingly to the available reserves from the DC side. On the other hand, voltage-source mode can provide most of the grid services since it is capable of emulating SG behavior. However, because of its voltage source behavior, the extracted power is a consequence of the imposed voltage. Thus, it requires a large energy source to provide functions such as frequency support, system strength contribution or POD. The short-circuit contribution is however limited by the semiconductor device overcurrents. Traditional SGs can Chapter 2. State-of-the-art of existing control solutions for Voltage Source Converters to support the Transmission Grid provide the short-circuit current thanks to their large overcucrrent capability, as shown in Section 2.2. These functions are included in the denition of grid-forming in some references [START_REF] Jouini | New options for existing system services and needs for new system services[END_REF] though, which can be misleading from a control point of view. To be properly integrated in the grid, the VSC control must consider the energy management algorithms of the power sources on the DC side that deliver this supplementary energy, which could be brought directly in DC or through another AC/DC converter. This extra energy can be provided by deloading PV arrays [START_REF] Rahmann | Fast frequency response capability of photovoltaic power plants: The necessity of new grid requirements and denitions[END_REF],

wind power plants [132134] or by storing it in an Energy Storage System (ESS)

through Li/ion batteries [START_REF] Zuo | Eect of voltage source converters with electrochemical storage systems on dynamics of reduced-inertia bulk power grids[END_REF][START_REF] Markovic | Optimal sizing and tuning of storage capacity for fast frequency control in lowinertia systems[END_REF] or supercapacitors [START_REF] Errigo | Optimal supercapacitor pack sizing for modular multilevel converter with integrated energy storage system[END_REF]. On the other hand, it is possible to control a VSC in grid-forming even though there is no dedicated storage unit, as for HVDC links interfacing two AC grids [START_REF] Zhang | Interconnection of two very weak AC systems by VSC-HVDC links using power-synchronization control[END_REF] or interconnecting oshore wind farms [START_REF] Mourouvin | AC/DC Dynamic Interactions of MMC-HVDC in Grid-Forming for Wind-Farm Integration in AC Systems[END_REF]. As a summary, the compatibility of the two control modes with the grid services is given in Table 2.3, where frequency support is split into virtual inertia and load sharing capability to highlight this.

This distinction between the dierent objectives of a grid-forming controlled VSC are an up-to-date topic. In [START_REF] Gleadow | AC fault response options for VSC HVDC converters[END_REF], several categories of grid-forming were drawn such as:

• grid-forming: refers to VSCs controlled as constant voltage sources, which is equivalent to the vf control described in Section 2.3.2;

• synchronous grid-forming: refers to VSCs in grid-forming which are capable of synchronizing with other parallel-connected voltage sources (gridforming VSCs, SGs or SCs);

• Virtual Synchronous Machine (VSM): refers to synchronous grid-forming VSCs which are capable of providing power from an extra source of energy and a short-circuit current during faults.

In this classication, grid-forming and synchronous grid-forming refer to control aspects: the VSC is seen as a voltage source from the grid perspective and the outer loop control inputs are a voltage magnitude and angle, as in Fig. 2.9. On the other hand, Virtual Synchronous Machine refers to the grid aspects because it includes some of the support functions described in Section 2.2. This terminology is compatible with Table 2.3 and shows the need to distinguish in the grid-forming related literature the control aspects which makes it possible to have a PLL-free synchronization from the grid aspects which replaces the functions of the SGs with VSCs. This understanding makes us realize that some inherent functions of SGs, such as inertia, whose shortage in future power systems was considered as the worst issue for European TSOs in 2017 [START_REF] Migrate | Deliverable 1.1: Current and arising issues caused by increasing power electronics penetration[END_REF], are less important than others, such as Observes the variations of active power to estimate the grid frequency and phase angle. The converter and the grid are naturally coupled by the synchronizing torque.

power-tracking

Relies on strong grid assumptions. Quite good performances in power tracking when these conditions are met. See Fig. 2.11

Not as good as current-source mode for power tracking in terms of both time response and static error [START_REF] Rokrok | Classication and dynamic assessment of droop-based grid-forming control schemes: Application in HVDC systems[END_REF][START_REF] Harnefors | Referencefeedforward power-synchronization control[END_REF].

current-limiting action

Current limiters can be implemented in the current controller.

No inherent current controller at rst place. But cascaded structure with current controller is now widely used [START_REF] Markovic | Stability Analysis of Converter Control Modes in Low-Inertia Power Systems[END_REF][START_REF] Denis | From grid-following to grid-forming: The new strategy to build 100 % power-electronics interfaced transmission system with enhanced transient behavior[END_REF].

Grid-level virtual inertia

No. The VSC tracks power references.

Possible with virtual inertia control as in Fig. 2.12 (b). Not as well adapted as grid forming option [START_REF] Poolla | Placement and implementation of gridforming and grid-following virtual inertia and fast frequency response[END_REF].

Yes, due to the dynamics of the Low-Pass Filter (LPF) in the active power controller shown in Fig.

(c).

Yes, due to the LPF dynamics. Support is however limited by the quantity of available DC power.

load sharing capability

No. The VSC tracks power references.

Possible with active power / frequency droop control as in Fig. 2.12 (a).

Yes, as shown in [START_REF] Denis | From grid-following to grid-forming: The new strategy to build 100 % power-electronics interfaced transmission system with enhanced transient behavior[END_REF][START_REF] Or | virtual inertia: Comparison from the perspective of converter operation mode[END_REF].

Impossible if there is no dedicated storage system [START_REF] Mourouvin | AC/DC Dynamic Interactions of MMC-HVDC in Grid-Forming for Wind-Farm Integration in AC Systems[END_REF].

reactive power support No. The VSC tracks power references.

Possible with the supporting actions described in Fig. 2.13 Yes, the VSC controls the PCC voltage magnitude and adjusts its reactive power accordingly as long as it respects its overcurrent limitations.

system strength contribution Impossible. The VSC is operated as a current-source and the PCC voltage is seen as an external disturbance.

Yes. The VSC is operated as a voltage source and the PCC voltage magnitude and phase angle are seen as control variables [START_REF] Rocabert | Control of power converters in AC microgrids[END_REF].

short-circuit contribution

Can be implemented as in [START_REF] Jia | A review on grid-connected converter control for short-circuit power provision under grid unbalanced faults[END_REF]. The contribution is limited by the thermal limits [START_REF] Debry | Maximizing the penetration of inverter-based generation on large transmission systems: the MIGRATE project[END_REF].

FRT must be taken care of because of the voltage source behavior. See [START_REF] Zhang | Power-synchronization control of gridconnected voltage-source converters[END_REF] or [START_REF] Denis | The Migrate project: the challenges of operating a transmission grid with only inverter-based generation. A grid-forming control improvement with transient current-limiting control[END_REF][START_REF] Qoria | Critical clearing time determination and enhancement of grid-forming converters embedding virtual impedance as current limitation algorithm[END_REF] for control options.

power oscillation damping

No. The VSC tracks power references.

Possible using active or reactive power reference modulation as described in Fig. 2.17 (a). See [ Chapter 2. State-of-the-art of existing control solutions for Voltage Source Converters to support the Transmission Grid system strength reinforcement. In fact, it is clear now that weak grids is a signicant challenge right now for islanded power systems such as Texas [START_REF] Matevosyan | Experience in Texas with declining inertia[END_REF] or Australia [START_REF] Modi | Operational manifestation of low system strength conditions Australian experience[END_REF] where weak grid issues were recently reported. In conclusion, the desired behavior of VSCs in the future might not be to emulate SGs but could be less restrictive since those PE-based sources are much more controllable.

The contributions of this chapter can be listed as follows:

• a comprehensive introduction to the upcoming challenges of AC transmission grids due to the increasing amount of renewables and the ongoing dismantlement of synchronous generators;

• an overview of the main VSC control modes and a front-to-front comparison of the current-source and voltage-source modes, respectively the grid-following and grid-forming controls;

• a presentation of the main ancillary services that used to be realized by SGs and how the two types of VSC control modes can be adjusted and modied to provide some of these functions to the grid.

In future work, it should be primordial to investigate the compatibility of all the controllers with each other and how to dispatch the dierent control modes within a given grid. In particular, in the next chapters of the thesis, we will investigate the role of the main VSC control modes in the power penetration limits in AC grids In the introduction of this thesis, the evolution of the power system stability issues were discussed. In Chapter 2, a literature review was provided to identiy the role that VSC controls can play in providing the response to some of these issues such as frequency support, reactive power compensation or power oscillation damping.

These functions become more and more expected of the VSC. Indeed, by 2050, renewable energy is expected to produce 70-85% of electricity in an attempt to limit Chapter 3. Impacts of VSC control on the limits of Power Electronics Converters integration in AC grids global warming to 1.5 o C [3]. On the one hand, the conventional renewable energy sources, i.e. hydro power, which covered 14.4% of the annual power consumption in France in 2018 [140], would still be highly solicited. On the other hand, most of the other renewable energy sources, especially solar PV and wind, are interfaced with the grid through PE-based technology. In consequence, 100% PE-based grid scenarii will unlikely occur, but instead, hybrid systems with a high penetration of PE and the remaining SG including hydro power plants and synchronous condensers could be the future tendency. For such systems, it is essential to study the interactions between large PE-based stations and SGs. Hence, this chapter is about studying the interactions between the VSC in common control modes and the SG. A limit of power penetration is assessed using modal analysis tools.

Literature review

In the literature, some studies have been published to tackle the interactions between the VSC in the grid-feeding mode and SG and show penetration limit through modal analysis, with dierent notions of power ratio [START_REF] Lin | Stability assessment of a system comprising a single machine and inverter with scalable ratings[END_REF][START_REF] Collados-Rodriguez | Stability analysis of systems with high VSC penetration: Where is the limit[END_REF][START_REF] Pereira | Impact of synchronous machine dynamics on the stability of a power grid with high penetration of Variable Renewable Energies[END_REF][START_REF]Synchronous machine representations for stability studies of power systems with inverters[END_REF]. The notion of power penetration ratio has also been introduced in experimental setups recently [START_REF] Castillo | Experimental stability assessment of converter-dominated electrical grids[END_REF]. The impacts of critical dynamics such as the Phase-Locked-Loop (PLL) and transmission lines (characterized by the Short-Circuit Ratio, or SCR) have been highlighted for grid-feeding VSCs [START_REF] Zhou | Impact of Short-Circuit Ratio and Phase-Locked-Loop Parameters on the Small-Signal Behavior of a VSC-HVDC Converter[END_REF]. The impact of the transmission lines was also pointed out recently for grid-supporting strategies using frequency and voltage droop controllers [START_REF] Misyris | Grid Supporting VSCs in Power Systems with Varying Inertia and Short-Circuit Capacity[END_REF].

Following the development of grid-forming control strategies for transmission systems [START_REF] Zhang | Power-synchronization control of gridconnected voltage-source converters[END_REF][START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF], the grid-forming control has become an accepted solution for weak grids, i.e. high-impedance grids and low-inertia systems [START_REF] Milano | Foundations and challenges of low-inertia systems[END_REF]. The impact of the converter's electrical distance on the main grid [START_REF] Markovic | Stability Analysis of Converter Control Modes in Low-Inertia Power Systems[END_REF] and its interactions with existing machines when controlled as a voltage source [START_REF] Markovic | Understanding stability of low-inertia systems[END_REF] or operating in grid-forming mode [START_REF] Siraj Khan | Stability Assessment of a System Comprising a Single Machine and a Virtual Oscillator Controlled Inverter with Scalable Ratings[END_REF] have been recently studied using the notion of power ratio. Among dierent gridforming strategies, droop [START_REF] Markovic | Stability Analysis of Converter Control Modes in Low-Inertia Power Systems[END_REF] and Virtual Synchronous Machine (VSM) [?] controls are now the most commonly used in literature [START_REF] Markovic | LQR-based adaptive virtual synchronous machine for power systems with high inverter penetration[END_REF]. In addition, a comparison of existing grid-forming control strategies such as dispatchable virtual oscillator (dVOC) [START_REF] Groÿ | The eect of transmission-line dynamics on grid-forming dispatchable virtual oscillator control[END_REF] and matching synchronous machines [START_REF] Arghir | Grid-forming control for power converters based on matching of synchronous machines[END_REF] was conducted using timedomain simulations of multiple-bus systems [START_REF] Tayyebi | Interactions of gridforming power converters and synchronous machines a comparative study[END_REF]. However, to our best knowledge, there are no studies that compare current-source modes and voltage-source modes of VSC at the same operating conditions for assessing the stability limits of the system.

System modeling

Chapter summary and contents

In this chapter, the interactions between AC grid and existing VSC control modes are analyzed. In particular, three VSC control modes, namely grid-feeding, gridsupporting and grid-forming, are compared with each other in order to assess VSC/grid coupling level at the same grid conditions and power setpoints. In order to underscore the growing role of VSCs in transmission systems, an elementary benchmark network is considered. It includes an equivalent SG, a VSC, a load and a transmission line with a varying electrical distance. On this model, a modal analysis is conducted to study the interactions between the dierent VSC controls and the SG dynamics using a Matlab/Simulink environment. The converter penetration rate is represented by a variable power ratio between the converter and the machine, while the electrical distance by an RL dynamic impedance. This approach makes it possible to assess the stability limits on the VSC share imposed by the system strength. Negative interactions are identied and recommendations for novel converter control strategies proposed. Besides, the inuence of the AVR with the VSC in grid-forming as well as PLL-related stability issues with the VSC in gridfeeding or grid-supporting are shown to be the key limiting factors for increasing the penetration of PE-based sources. Last but not least, dierent scenarii are studied by simulations with a multi-machine system in order to deepen the modal analysis results.

This chapter is organized as follows: in Section 3.2, the test system and its nonlinear dierential equations are presented. In Section 3.3, the modal analysis method and results are presented, which are validated using a Matlab/Simulink state-space model and a multi-machine system developed in Modelica language in dierent scenarii in Section 3.4. Finally, the conclusions and future work are discussed in Section 3.5.

System modeling 3.2.1. Test system

For the study, an elementary benchmark is considered, which is composed of one VSC with its LC lter, one synchronous generator, the two connected by an AC line, and a load at the point of common coupling (PCC) located on the VSC side, as given in Fig. 3.1.

Chapter 3. Impacts of VSC control on the limits of Power Electronics Converters integration in AC grids In order to assess the system stability, the system is linearized around an operating point. A modal analysis is performed for dierent PE penetration levels and AC grid strengths, i.e. for dierent values of the line impedance. The penetration of PE is modeled by the Power Ratio, denoted by P R and dened as:

P R = S vsc S sg + S vsc = S vsc S base (3.1)
where S vsc and S sg are respectively the VSC and SG rated powers and S base is the system base power. As the SGs are currently the most commonly used for power generation, their dynamics need to be taken into account to highlight possible negative interactions [START_REF] Pereira | Impact of synchronous machine dynamics on the stability of a power grid with high penetration of Variable Renewable Energies[END_REF], [START_REF]Synchronous machine representations for stability studies of power systems with inverters[END_REF]. These interactions could not be tackled with the grid modeled as an equivalent voltage source [START_REF] Markovic | Stability Analysis of Converter Control Modes in Low-Inertia Power Systems[END_REF] or with only electromechanical models [START_REF] Misyris | Grid Supporting VSCs in Power Systems with Varying Inertia and Short-Circuit Capacity[END_REF]. Indeed, SG have several controllers among which the voltage regulator, or AVR, can have a 3.2. System modeling negative impact on the system. A conceptual scheme of the SG control model is given in Fig. 3.2. The SG model considered in this study is a reduced-order 1-axis model of a steam-powered SG that captures both electromechanical and electromagnetic dynamics. This model was developed in details in [START_REF] Kundur | Power System Stability and Control[END_REF] and the considered equations are given in Appendix A.

Synchronous machine model and control

Converter modeling & control

This section describes the converter model and their control schemes, which can be classied into two categories [START_REF] Rocabert | Control of power converters in AC microgrids[END_REF] and are adapted here for transmission grids. The details of this classication for transmission grids can be found in Chapter 2:

• Current source control: the VSC follows power references and injects a current in phase with the network thanks to the PLL synchronizing unit. It corresponds to the grid-feeding, also called grid-following mode. In the gridsupporting mode, the power references are adjusted to provide grid services;

• Voltage source control: the VSC works as a voltage source and is thus able to operate in islanded mode. It uses the power variations to synchronize with the grid so that no PLL is needed anymore. It corresponds to the grid-forming mode.

Both classes use cascaded loops including a current control loop with potential current limiting actions to protect the semiconductor devices. Indeed, the overcurrent capability of VSCs is only 110%, which is much smaller than traditional SGs [START_REF] Debry | Maximizing the penetration of inverter-based generation on large transmission systems: the MIGRATE project[END_REF]. Nevertheless, the impact of current limiting control on stability is not considered in this study. In both classes, the inner loop control uses PI controllers whose gains are tuned using pole placement methods [2], [START_REF] Denis | From grid-following to grid-forming: The new strategy to build 100 % power-electronics interfaced transmission system with enhanced transient behavior[END_REF] and whose method is described in Appendix B. The VSC is considered with an LC lter, as described in Fig. 3.3. In the current source mode, the control scheme regulates the line current i L by acting on the output voltage v i . To synchronize with the grid voltage v 0 , the scheme needs a PLL to capture the grid-frequency and the PCC electrical angle.

Chapter 3. Impacts of VSC control on the limits of Power Electronics Converters integration in AC grids

In the next paragraphs, the implemented VSC controls are described. These controls present some dierences with the general control schemes presented in Chapter 2. Also, each control block is explicited in order to ensure the repeatability of the presented method.

Physical model of VSC

The VSC model considered here is an ideal voltage source where the dynamics of the switches are neglected. In addition, the PWM control dynamics are neglected and thus the actual output voltage is considered to be equal to the reference voltage given by the control law:

v dq i = v dq i,ref (3.2)
Even though some recent works focus on the stability analysis of grid-connected converters taking into account the states of the switches [START_REF] Sanchez | On the Practical Stability of Hybrid Control Algorithm With Minimum Dwell Time for a DCAC Converter[END_REF], the use of averaged models, as in (3.2), is relevant to assess the small-signal stability of the grid [START_REF] Markovic | Understanding stability of low-inertia systems[END_REF][START_REF] Lin | Stability assessment of a system comprising a single machine and inverter with scalable ratings[END_REF][START_REF] Cossart | Model reduction of converters for the analysis of 100% power electronics transmission systems[END_REF]. Indeed, the switching dynamics of the VSC are much faster than the grid and SG dynamics that we focus on in this chapter.

Outer loop for current-source control

Current source control refers to the scheme that uses the VSC voltage output as the control variable to regulate the branch current passing through the LC lter.

The general scheme of this type of control is given in Fig. 3.4 and covers gridfeeding [START_REF] Lin | Stability assessment of a system comprising a single machine and inverter with scalable ratings[END_REF][START_REF] Zhou | Impact of Short-Circuit Ratio and Phase-Locked-Loop Parameters on the Small-Signal Behavior of a VSC-HVDC Converter[END_REF][START_REF] Pereira | Impact of synchronous machine dynamics on the stability of a power grid with high penetration of Variable Renewable Energies[END_REF] and grid-supporting [START_REF] Misyris | Grid Supporting VSCs in Power Systems with Varying Inertia and Short-Circuit Capacity[END_REF] strategies.

Figure 3.4: VSC control in current source mode: grid-feeding and grid-supporting.

With the grid-feeding control, the outer power loops use pure integral controllers, with k ip as the control gain, to regulate the ltered active and reactive powers to their references. In grid-supporting, an additional droop is added to change the 3.2. System modeling active and reactive power references to contribute respectively to frequency and voltage stability, which are given by:

p ref = p ref + K pω • (ω ref -ω pll ) (3.3) q ref = q ref + K qv • (v ref -||v dq 0 || 2 ) (3.4)
where p ref , q ref , ω ref and v ref are respectively the references of active power, reactive power, frequency and voltage in per unit obtained from the load ow solution.

The droop gains K pω and K qv are used to provide support to the grid, hence the name grid-supporting. As presented in Chapter 2, there are other ways to provide frequency and voltage support to the grid in current-source mode, but these other strategies are not studied here since the two droop methods are the most commonly used controls. In grid feeding mode, K pω = K qv = 0.

The power control loop which generates the current reference in dq-frame for the inner current controller is:

i dq L,ref = k ip • [ p , q ] T (3.5)
where ˙ p = p ref -p avg and ˙ q = q ref -q avg are respectively the active and reactive power control errors. The ltered measured powers are calculated as:

ṗavg qavg = ω c,power • p calc -p avg q calc -q avg (3.6)
where

p calc = v d 0 • i d L + v q 0 • i q L and q calc = v q 0 • i d L -v d 0 • i q
L are the calculated output powers for the current-source mode. Finally, the PLL is modeled by a PI controller with a low-pass lter, as described in [START_REF] Lin | Stability assessment of a system comprising a single machine and inverter with scalable ratings[END_REF][START_REF] Molinas | Extended stability range of weak grids with voltage source converters through impedance-conditioned grid synchronization[END_REF]:

vdq,pll 0 = ω c,pll • (v dq,pll 0 -vdq,pll 0 ) (3.7) ω pll = k i,pll • pll + k p,pll • ˙ pll (3.8) θpll = ω base • ω pll (3.9)
with k i,pll and k p,pll as the PI gains of the controller and ˙ pll = 0 -vdq,pll 0 , v0 the ltered signal of v 0 and ω c,pll the low-pass lter cuto frequency.

Outer loop for voltage-source control

In voltage-source mode, the control scheme regulates the PCC voltage v 0 , which is no longer an external disturbance from the control point of view. The scheme relies on cascaded loops, described in Fig. 3.5, where it uses the branch current i L Chapter 3. Impacts of VSC control on the limits of Power Electronics Converters integration in AC grids to act on v 0 and uses the VSC output voltage v i to control i L . The synchronization with the grid is insured by the outer-loop control, i.e. grid-forming, thus no PLL is needed any more. This control scheme for grid-forming is the most commonly spread in literature even though it requires a good enough estimation of both the lter inductance and capacitance to control the PCC voltage and a physical LC lter at the VSC output. To address this issue, some other grid-forming control schemes have emerged recently to avoid the voltage control loop and act as current sources [START_REF] Roldan-Perez | Design and analysis of a current-controlled virtual synchronous machine for weak grids[END_REF]. However, they remain out of the scope of this chapter. The outer loop controls use the variations of active power and reactive power to respectively synchronize with the grid and maintain a certain voltage amplitude.

The active power control for the droop and the VSM schemes are, respectively:

ω = ω ref + m pω • (p ref -p avg ) (3.10) ω = 1 2 • H c • (p ref -p calc -D c • ω) (3.11)
Then, the corresponding angle is calculated as

θ = ω base • ω (3.12)
The output powers of the VSC in voltage-source mode are

p calc = v d 0 • i d 0 + v q 0 • i q 0 and q calc = v q 0 • i d 0 -v d 0 • i q 0 .
It is important to note that the gains of the VSM control are chosen to give the same steady-state response as grid-forming droop control [START_REF] Markovic | Stability Analysis of Converter Control Modes in Low-Inertia Power Systems[END_REF].

The reactive power control is a droop controller and is similar for the droop (3.10) and VSM (3.11) controls, as in [START_REF] Markovic | Stability Analysis of Converter Control Modes in Low-Inertia Power Systems[END_REF]. The option of using the electromagnetic 3.2. System modeling state-space model of the SG [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF][START_REF] Roldan-Perez | Design and analysis of a current-controlled virtual synchronous machine for weak grids[END_REF] is discarded in this study. The control law is:

e * = v ref + m qv • (q ref -q avg ) (3.13) ev dq 0,ref = [e * , 0] T (3.14)

Inner control loops

In grid-forming mode, as in Fig. 3.5, the output terminal voltage and the line current are regulated using two cascaded loops. PI controllers are implemented for these loops and are tuned with pole placement method [2]. The voltage loop is designed to be ten times slower than the current loop to avoid potential interference [START_REF] Denis | From grid-following to grid-forming: The new strategy to build 100 % power-electronics interfaced transmission system with enhanced transient behavior[END_REF]. The inner voltage control law is:

i dq L,ref =k iv • dq v + k pv • ˙ dq v + ωc f • 0 -1 1 0 • v dq 0 + α i • i dq 0 (3.15) with ˙ dq v = v dq 0,ref -v dq 0 .
The current reference generated from the voltage controller is then used in the current control law as:

v dq i,ref = k ii • dq i + k pi • ˙ dq i + ωl f • 0 -1 1 0 • i dq l + α v • v dq 0 (3.16) with ˙ dq i = i dq L,ref -i dq L . The output signal of the current control loop, v dq i,ref , refers
to the desired voltage in dq-frame to be generated by the VSC internal actuators, which are not considered here.

The same current controller is used in all the grid-feeding, grid-supporting and grid-forming schemes. The inner current controllers based on PR (Proportional Resonant) control to regulate the VSC current in the stationary frame (αβ) [START_REF] Rocabert | Control of power converters in AC microgrids[END_REF][START_REF] Bacha | Power Electronic Converters Modeling and Control with Case Studies, advanced textbooks in control and signal processing[END_REF] are gaining more interest compared to the classical PI control [START_REF] Manoloiu | Comparison of PI and PR current controllers applied on twolevel VSC-HVDC transmission system[END_REF][START_REF] Nazeri | Design of Proportional-Resonant Controller with Zero Steady-State Error for a Single-Phase Grid-Connected Voltage Source Inverter with an LCL Output Filter[END_REF]. However, it remains out of this study where the most prevalent VSC control schemes are considered to assess the stability limits.

Network modeling & interfaces 3.2.4.1. Interconnecting line model

The network is modeled as a dynamic RL line which connects the SG bus with voltage v g to the load and VSC output lter bus with voltage v 0 , as described by:

di d eq dt = ω base l eq • (v d g -v d 0 -r eq • i d L + ω g • l eq • i q eq ) (3.17) di q eq dt = ω base l eq • (v q g -v q 0 -r eq • i q eq -ω g • l eq • i d eq ) (3.18)
The line impedance is dened in per unit, to take into account the changes of the SG power ratings when acting on the power ratio. Indeed, an electrical line of 100 km will not have the same inuence when connected to a 100 MVA machine or a 1000 MVA one. Then, considering a constant ratio X eq /R eq = 10, which is common in power system studies, the equivalent resistance can be calculated.

Load model

The load is modeled as a static R load //X load load connected at the PCC. The impedances are calculated using the expected load setpoints in S.I.:

R load = U 2 base P load ; X load = U 2 base Q load (3.19)

Global system & interfaces

From its description, the whole system appears to be a combination of several subsystems with their own dq-frames. To associate the equations at the load bus, an interface is needed between:

• {SG + line} in the SG's dq-frame, denoted by DQ;

• {VSC + LC lter} aligned with the load bus voltage's dq-frame, denoted by dq.

This association can be achieved using rotation matrices between the two dierent frames as:

x D x Q = sin(δ -θ) -cos(δ -θ) cos(δ -θ) sin(δ -θ) • x d x q
(3.20)

Modal analysis 3.3.1. Parameters and operating points

For the modal analysis, the parameters and input data of the system are given in Table C.1 in Appendix C. The VSC reactive power reference q ref and the SG active power reference P * m are not xed but are adjusted in order to have ||v 0 || = v * and ω g = ω ref . The tuning of the grid-forming outer loops (active and reactive power controllers) is based on [START_REF] Denis | From grid-following to grid-forming: The new strategy to build 100 % power-electronics interfaced transmission system with enhanced transient behavior[END_REF][START_REF] Vorwerk | Small-signal analysis of power systems with low rotational inertia[END_REF]. For the grid-feeding and grid-supporting schemes, the values of the power lter, PI power control and droop gains are based on [START_REF] Misyris | Grid Supporting VSCs in Power Systems with Varying Inertia and Short-Circuit Capacity[END_REF]. The feedforward gains α v and α i from (3.15) and (3.16) are boolean and 3.3. Modal analysis their values depend on the activation of the feedforward action for each controller.

For the current-source controlled strategies, α v = 1 as there is no voltage control loop. For grid-forming strategies, we choose α v = 1 and α i = 0 because the voltage controller's feedforward action proves to cause instability [START_REF] Denis | From grid-following to grid-forming: The new strategy to build 100 % power-electronics interfaced transmission system with enhanced transient behavior[END_REF].

Method for modal analysis

The system model gathers the state variables of the dierent components and can be rewritten in general form as:

ẋ = f (x, u) (3.21)
with x = x 1,SG , . . . , x n,SG , x 1,V SC , . . . , x m,V SC , i d eq , i q eq T as the state variable vector of the system, n the order of the SG subsystem with x i,SG as its state variables and m the order of the VSC subsystem with x i,V SC as its state variables. The vertical vector

u = p ref , q ref , v ref , P * m , v * , ω ref T
is the system input vector. Using this state-space model, we dene the setpoints of all the state variables corresponding to the load ow solution and we linearize the system around the operating point {u 0 , x 0 } as:

∆ ẋ =A • ∆x + B • ∆u (3.22) A := ∂f ∂x u 0 ,x 0 ∈ R (n+m+2)×(n+m+2) (3.23) B := ∂f ∂u u 0 ,x 0 ∈ R (n+m+2)×6 (3.24)
where A is the state matrix and corresponds to the structural properties of the system since it does not depend on the input/output signals. The matrix B is the input matrix. By extracting the eigenvalues of the A-matrix, we can analyze the stability of the system around the nominal point. The participation factors of some specic critical modes will also be used to enrich the results. The method for calculating the normalized participation factors is detailed in [START_REF] Wang | Modern Power Systems Analysis[END_REF].

A study over some selected dynamics is also carried out to highlight harmful interactions between the SG and the VSC modes. The impact of the AVR dynamics (both lead-lag lter and AC exciter) on the system stability is studied. We compare the modal analysis results of two cases: the system with the AVR dynamics considered or neglected, as illustrated in Fig. 3.6 (a). For PLL-based controls, i.e.

grid-feeding and grid-supporting, we compare two cases: one with the PLL dynamics, the other with an ideal PLL that captures the exact angle and exact frequency with no time-delay, as in Fig. 3.6 (b).

. Inuence of the power ratio

The results of the modal analyses for a given grid scenario with x eq = 0.33 p.u. are displayed in Fig. 3.7 for the grid-feeding and grid-supporting cases. In these results, we observe that the AVR has little impact on the system stability when the VSC is controlled in current-source mode. Indeed, the critical modes in Figs. as a limiting factor because the modes with an ideal PLL are all stable for P R ≤ 0.8

as described in Figs. 3.7 (e)-3.7 (f ). In addition, the grid-supporting control results in a higher penetration limit compared to grid-feeding.

For further investigation, we calculate the participation factors of the critical mode, which is denoted by λ 4,5 in Fig. 3.7 (a), to show the potential destabilizing role of the PLL. The results are given in Fig. 3.8. It is clear that the PLL is responsible for the observed instability due to high participation factors when the power ratio is low. However, the larger the P R , the smaller the PLL-variable participation. Indeed, as P R increases, the share of the PLL is gradually replaced by the state variables of the passive components, namely the RL dynamics of the line and the LC lter, which implies that we can reach the structural limits of the current-source controlled VSCs regardless of the tuning of the PLL. This result conrms the limits of controlling VSCs in current sources when increasing their number in the AC grid, a phenomenon that cannot be observed when modeling the lines using static models [START_REF] Collados-Rodriguez | Stability analysis of systems with high VSC penetration: Where is the limit[END_REF]. The role of the PLL in destabilizing the system with PE-based sources attracts more and more attention from TSOs [START_REF] Grid | Impacts of Declining Short-Circuit Levels[END_REF]. However, the nature of the oscillations appears to depend on the grid conditions [START_REF] Hou | Modeling and HIL Test of a D-PMSG Connected to Power System with Damping Control for Real Time Studies[END_REF] and the PLL structure [START_REF] Fan | Wind in Weak Grids: 4 Hz or 30 Hz Oscillations[END_REF] and needs further investigation in the future.

Inuence of the electrical distance

In this paragraph, we perform a modal analysis on the PLL-based controlled VSCs for a given power ratio P R = 0.2 and we change the line impedance. The results for the grid-feeding and grid-supporting are given in Fig. 3.9. As in the previous paragraph, it is clear that the AVR dynamics play no role on the small-signal Chapter 3. Impacts of VSC control on the limits of Power Electronics Converters integration in AC grids Figure 3.8: Participation factors of the λ 4,5 mode for dierent values of the power ratio P R whith the VSC in grid-feeding mode and x eq = 0.33 p.u. stability of the system when the VSC is controlled either in grid-feeding or in gridsupporting. The system stability is still too sensitive to the power ratio and does not allow us to increase the penetration of PE above 30%. Nevertheless, the use of dierent grid-feeding control structures based on PR controllers could enhance the stability limits, since PR controllers exhibit better performances in weak grid conditions compared to PI controllers [START_REF] Midtsund | Evaluation of current controller performance and stability for voltage source converters connected to a weak grid[END_REF]. In the next section, we study the role that the grid-forming control can play in increasing those limits.

3.3.4.

Results for PLL-free controls: grid-forming schemes

Inuence of the power ratio

Since the grid-forming droop and VSM exhibit the same small-signal behavior [START_REF] Or | virtual inertia: Comparison from the perspective of converter operation mode[END_REF],

we only plot the results for the droop control with respect to the power ratio in Fig. 3.10. For the case x eq = 0.33 p.u., we observe that the system is stable until around 45%, whereas it is stable for all the values of P R when neglecting the AVR dynamics. To understand the critical mode evolution, we take a look at the critical modes λ 12,13 and λ 18,19 dened in Fig. 3.10 (a).

We calculate the participation factors of each mode for dierent values of P R .

The results, given in Fig. 3.11, show that both modes are low-frequency modes, since the participation factors of the SG mechanical variables, denoted by x k,mecha , is high. Even if the role of the AVR dynamics is clear for modes λ 18,19 , its impact on modes λ 12,13 is less evident using the participation factors. From Fig. 3.10, it is clear that the inuence is weak but causes instability since the conjugate modes just slightly move beyond the imaginary axis, compared to the case where the AVR is neglected.

In addition, compared to the electromagnetic dynamics, the time constants of the AVR lead-lag lter are rather slow, which fall in the range of electromechanical dynamics. This can explain why the undamped oscillatory modes λ 12,13 interact with the electromechanical variables. This result may seem counter-intuitive since 100% PE-based generation with VSCs in grid-forming has been proved to be stable [START_REF] Denis | From grid-following to grid-forming: The new strategy to build 100 % power-electronics interfaced transmission system with enhanced transient behavior[END_REF]. In fact, the transition to 100% will go through a hybrid system where a high proportion of PE generators coexist with the remaining SGs. Because of their interactions, even with all converters working in grid-forming, this transition of the power systems should be more challenging than the 100% PE scenario [START_REF] Markovic | Understanding stability of low-inertia systems[END_REF].

Inuence of the electrical distance

In this part, we perform a modal analysis for a given grid scenario with P R = 0.45 for dierent values of the line reactance x eq . The results in Fig. 3.12 show the AVR dynamics move the modes to the right semi-plane. Even if the dependency of the modes on the electrical distance is small, bringing the SG closer to the VSC makes the system unstable. On the one hand, this result is counter-intuitive when compared to the PLL-controlled VSC. On the other hand, this result conrms the local interaction between the AVR and the synchronizing scheme of the grid-forming VSC.

Finally, for grid-forming controls (both droop and VSM), the interactions with AVR dynamics appear to act as a limiting factor for the penetration limit which cannot be observed when modeling the network using an innite bus.

Inuence of the active power droop gain

As described in Section 3.3.4.1, the maximum allowable power ratio is around 45% with x eq = 0.33 p.u. This value is similar to the 50% that was found in a previous study on the interactions between a similar SG and a VSC in grid-forming mode in a microgrid directly connected to the machine [START_REF] Siraj Khan | Stability Assessment of a System Comprising a Single Machine and a Virtual Oscillator Controlled Inverter with Scalable Ratings[END_REF]. In order to understand the role of the grid-forming control gains in the stability limit, we perform a parametric study with the m pω varying from 1% to 5%. The results are presented in Fig. 3.13.

It is clear that the droop gain has little inuence on the maximum allowable power ratio, because even though it moves the critical eigenvalues in the complex plan, the intersection between the imaginary axis and the rightmost eigenvalues is reached for P R ≈ 45%. On the other hand, when the AVR dynamics are neglected, the system remains stable for all the values of m pω we considered.

This result conrms the conclusions drawn in Sections 3.3.4.1 and 3.3.4.2 on the inuence of the outer power control loop and the AVR of the SG on the critical modes of the system. In addition, we also studied the impacts of the low-pass lter cuto frequency ω c,power . Since this parameter has no impacts on the eigenvalues, the results were not displayed in this study.

Penetration limit assessment

In order to summarize all the results we obtained for the dierent VSC control strategies, for dierent values of the line reactance on the SG and the power ratio, we plot in Fig. 3.14 the maximum allowable, i.e. stable, power ratio. This gives an indication about the inuence of x eq on the system stability. For PLL-based strategies (grid-feeding and grid-supporting), the longer the line, the lower the maximum allowable power ratio. This is an intuitive result since the system stability relies on the PLL algorithm, which is sensitive to the electrical distance to the voltage sources [START_REF] Zhou | Impact of Short-Circuit Ratio and Phase-Locked-Loop Parameters on the Small-Signal Behavior of a VSC-HVDC Converter[END_REF]. For grid-forming controls, the impact of the electrical distance is such that the closer the PE is to the SG, the less stable the system becomes. To the authors' best knowledge, this result has not been shown in the literature so far and which could drive future research on grid-forming algorithms for improving their penetration limits in power systems. One solution could be introducing a virtual impedance control, as in [START_REF] Markovic | Understanding stability of low-inertia systems[END_REF], to electrically move away the VSC. However, this control would weaken the PCC voltage stiness because it would bring farther the voltage source connected to it. Besides, since the electrical distance plays a small role in the stability limit, as described in Fig. 3.14, it may weaken the grid while bringing no real improvements in terms of small-signal stability.

In addition, for longer lines with x eq > 0.45 p.u., the grid-supporting strategy tends Figure 3.17: Evolution of the system state-variables when the VSC is in grid-forming droop mode following an active power load step at t = 20 s, with x eq = 0.33 p.u and P R = 0.5.

Power System Stabilizers (PSS) could enhance the compatibility of VSC with the remaining SG. The recent works on penetration limit assessment where the SG is considered with a PSS back this assumption [START_REF] Markovic | Understanding stability of low-inertia systems[END_REF].

Table 3.1: Dierent scenarii using the 4-machine, 2-area system VSC control mode Line reactance x eq Case 1 grid-feeding 0.005 p.u.

Case 2 grid-feeding 0.5 p.u.

Case 3 grid-forming 0.5 p.u.

Inuence of the power system topology

In order to check if the results about the penetration limit assessment given in Fig. 3.14 can be generalized to more complex systems, we simulate dierent scenarii with a multi-machine system. For the validation purpose, we choose to consider the much more complicated system shown in Fig. 3.18, obtained by replacing the SG in Fig. 3.1 by the 2-area Kundur system dened in [START_REF] Kundur | Power System Stability and Control[END_REF].

Figure 3.18: Modied Kundur's 4-machine, 2-area system, based on [START_REF] Kundur | Power System Stability and Control[END_REF].

The Kundur's 4-machine system is a 3-phase balanced system of symmetric topology with four 900 MVA SG which was used recently for studying the impacts of power electronics integration in power systems [START_REF] Pereira | Simplied approach for frequency dynamics assessment of 100% power electronics-based systems[END_REF]. For the multi-machine system, the VSC model was developed in Modelica language and the Kundur system is based on the iPSL library [START_REF] Vanfretti | iTesla Power Systems Library (iPSL): A Modelica library for phasor time-domain simulations[END_REF]. The whole system was simulated using Open Modelica software. The initial power ow data and the system bases are given in Appendix D. We considered a 1000 MVA VSC which corresponds to P R = 21.7% based on (3.1). The VSC control mode and the line reactance of the three cases simulated are summarized in Table 3.1, in the Kundur system base in order to assess the limits of stability of grid-feeding controlled VSCs when connected to a high impedance grid.

The line resistance r eq is calculated accordingly as in Section 3.2.4.1.

For the simulation cases, the VSC operates at p * = 0.7 p.u. to match the active power setpoint used so far. The load connected to the PCC is modeled as a constant-

Validation by simulations

current source load to consume 700 MW at the initial voltage. For the grid-feeding cases, the reactive power reference is adjusted to q * = -0.065 p.u. in order to match the absorbed reactive power of Case 3. For the VSC control gains in grid feeding, the control gains are similar to the values from the previous sections. For Case 3 in gridforming mode, the only dierences from Table C

ω COI = 1 4 i=1 H g i S g i • 4 i=1 H g i S g i • ω i (3.25)
where S g i = 900 MVA, H g i and ω i are respectively the rated power, the inertia constant and the rotor speed of the i th SG. The slight dierences between the voltage setpoints in Figs. 3.19 (a) to 3.19 (d) in grid-feeding and grid-forming modes are due to the changes between the line impedance (x eq ,r eq ).

With grid-feeding, we observe that the bus voltages settle at their steady-state values when x eq = 0.005 p.u., but show a limit cycle with oscillations at a high frequency when x eq = 0.5 p.u. Even though the disturbance was introduced at Bus 9 (see Fig. . This is in accordance with the claims in Section 3.5 for the case where the P R is around 20%. In grid-forming, all the bus voltages are brought back to stable operating points when x eq = 0.5 p.u., which shows the robustness of the control with respect to high-impedance grids.

For the frequency response given in Fig. 3.19 (e), we observe a smaller frequency dip for Case 2 than Case 1, which is due to the voltage-dependent nature of the load at the PCC. However, we still observe low frequency modes in the rst two cases, with the frequency of Case 2 less damped than Case 1. In Case 3, the grid frequency deviation exhibits a better response with a smaller frequency dip and a smaller steady-state error, thanks to the frequency support of the grid-forming controlled VSC, while in the grid-feeding cases, the VSC keeps injecting the same power despite the contingency.

Conclusions

In this chapter, stability issues of AC power network with PE based VSC converters using dierent control strategies have been highlighted. Rather than using an AC power system solely based on PE, a simplied network with both VSC and SG has been considered in order to show the impact of PE on power system stability. 64

Conclusions

Three VSC control strategies and AVR dynamics of the SG are considered against dierent values of the power ratio (VSC power rating over SG) and the transmission line length. The following concluding remarks can be pointed out:

• For grid-feeding control, it has been shown that, due to the PLL, stability is lost when the PE penetration ratio becomes too high. This issue can be partially mitigated when using the grid-supporting control, which tolerates a higher PE penetration ratio than the grid-feeding control. In future work, a complementary study could be pursued to take into account the impacts of the control gains, especially the PLL, in the stability limits. Additionally, the inuence of the electrical distance between the VSC and SG which has been studied using participation factors has also shown stability issues when the electrical distance is increased. For real power systems, this problem exists but with less impact since the grid buses are reinforced through redundant lines.

• For grid forming controls, i.e. droop and VSM, it has been highlighted that the interactions between the AVR and the VSC control seem to be almost independent of the electrical distance, where stability issues may be caused by the electromechanical resonance with VSC outer-loop controls. In this case, it would be desirable to extend the use of the PSS as a potential solution. This could be done in future work knowing that the interactions between VSCs and SGs may change when considering multiple-bus AC power systems with dierent types of SGs.

In Section 3.4.3, time domain simulations were carried out in a multi-machine system. However, the conclusions of this study could be tested in the future with even more realistic power systems where the distribution of the SGs and VSCs corresponds to the energy needs and resources. In future work, stability analysis methods based on attraction regions tools [START_REF] Gonzalez-Torres | Transient stability of power systems with embedded VSC-HVDC links: stability margins analysis and control[END_REF][START_REF] Hu | Large Signal Synchronizing Instability of PLL-Based VSC Connected to Weak AC Grid[END_REF] could provide complementary results for understanding the limits of power electronics penetration in AC grids.

Moreover, it will be essential to consider the DC-side dynamics of the VSC when studying the overall dynamics of the AC network with the constraints from these new power sources with the role of grid-forming, an aspect that has been up to now rarely addressed in literature.

forming strategies to relax the power electronics penetration limits in AC grids was highlighted. Similarly to distributed PE-based sources, High Voltage Direct Current (HVDC) stations have to fulll some additional requirements to make the grid operable and reliable, such as contributing to forming the grid voltage:

the grid-forming capability [START_REF] Entso-E | High Penetration of Power Electronic Interfaced Power Sources and the Potential Contribution of Grid Forming Converters[END_REF]. Such requirements concern especially islanded transmission grids such as the UK, where the TSOs experience at the same time a fast decommissioning of fossil-fuel power plants and their associated Synchronous Generators (SG). The remaining SGs will be more unevenly distributed geographically as shown in Fig. 4.1, thus creating weak regions in the grid. In Europe, a lot of research has been conducted so far on grid-forming control [START_REF] Jouini | New options for existing system services and needs for new system services[END_REF], where the VSC controlled in grid-forming is often studied with an ideal DC grid [START_REF] Taul | An Overview of Assessment Methods for Synchronization Stability of Grid-Connected Converters Under Severe Symmetrical Grid Faults[END_REF] and the interactions between AC and DC grids are still a topic that requires further investigation [START_REF] Peng | On the stability of power electronics-dominated systems: Challenges and potential solutions[END_REF]. This chapter focuses on the integration of Oshore Wind Farms (OWF) to the AC grid by a DC collector and a HVDC link with the onshore MMC working in DC voltage control mode, as described in [START_REF] Liu | Contribution of VSC-HVDC to Frequency Regulation of Power Systems With Oshore Wind Generation[END_REF][START_REF] Shinoda | DC voltage control of MMC-based HVDC grid with virtual capacitor control[END_REF] and illustrated in Fig. 4.2. The impacts of the interactions between DC-voltage control and grid-forming control on the system are discussed in the particular case of the MMC, which is able to decouple AC and DC in transient by acting on its internal energy. We highlight the problems related to the DC voltage and internal energy and propose possible solutions to these issues. Subsequently, we develop a method to calculate the energy requirements for MMC stations to provide AC-grid ancillary services. Then, an additional highlevel control is proposed to ensure the proper use of the MMC enhanced internal 4.1. Introduction energy. Its performance is assessed using a benchmark representing the case of the high-voltage OWF interconnection described below. The method developed in this chapter is applied to the UK National Grid scenario. Finally the controller is tested upon a multi-machine system to assess its compatibility with a hybrid grid where power electronics and SGs coexist.

Literature review

In the literature, many studies have been conducted on the control and operation of MMCs for OWF integration. In most of them, the onshore converter is in charge of regulating the DC voltage for the DC side while, on the AC side, it works as a current source converter which uses a classical Phase Locked-Loop (PLL) to synchronize with the AC grid. However, since coastal areas are often the weak parts of a power system, the synchronization through a PLL could be compromised, as highlighted in Chapter 3. One solution is to let the onshore converter work in the grid-forming mode, whereby it behaves like a self-synchronized converter controlled as a voltage source. The interferences between grid-forming control on the AC side with the DC voltage control were discussed for standard 2-level VSCs in [START_REF] Zhang | Interconnection of two very weak AC systems by VSC-HVDC links using power-synchronization control[END_REF]. On the other hand, a lot of studies have been carried out on low-inertia systems with more and more power-electronics (PE) converters connected to the grid [START_REF] Siraj Khan | Stability Assessment of a System Comprising a Single Machine and a Virtual Oscillator Controlled Inverter with Scalable Ratings[END_REF][START_REF] Markovic | Understanding stability of low-inertia systems[END_REF][START_REF] Markovic | Stability Analysis of Converter Control Modes in Low-Inertia Power Systems[END_REF], but many of them do not take into account the DC dynamics and the impacts of the AC power uctuations on the DC voltage.

In addition, while some studies focus on designing storage systems for VSCs dedicated to grid-forming function [START_REF] Markovic | Optimal sizing and tuning of storage capacity for fast frequency control in lowinertia systems[END_REF], the impacts of the grid-forming control on the behavior of HVDC systems, including Modular Multilevel Converter (MMC)

internal energy [START_REF] Sanchez-Sanchez | The role of the internal energy in MMCs operating in grid-forming mode[END_REF] and DC-grid voltage [START_REF] Mourouvin | AC/DC Dynamic Interactions of MMC-HVDC in Grid-Forming for Wind-Farm Integration in AC Systems[END_REF], have only been the focus of a few recent studies. However, the change in the high-level control from commonly used grid-following control to grid-forming control may have consequences on the energy management of the HVDC stations. It could even represent an advantage as the MMC submodules can act as a buer between AC and DC grids [START_REF] Freytes | Small-signal state-space modeling of an HVDC link with modular multilevel converters[END_REF] and thus avoid the propagation of disturbances. Moreover, there have been some studies that propose novel control solutions to use the MMC internal energy to support the AC grid frequency: in [START_REF] Spallarossa | Augmented inertial response of multi-level converters using internal energy storage[END_REF], the MMC energy is discharged using an energy reference prole, while in [START_REF] Kim | Exploiting redundant energy of MMC-HVDC to enhance frequency response of low inertia AC grid[END_REF] the MMC energy is discharged using a classical frequency droop. However, to our best knowledge, there are no studies that propose to couple the MMC internal energy with the AC grid frequency similarly to the SG's rotating mass kinetic energy as it is proposed in this chapter.

Chapter summary and contents

In this chapter, the operation of an MMC in grid-forming control is assessed with the DC-grid constraints taken into consideration. In particular, the high-level dynamics of a typical grid-forming droop control are studied. A trade-o is highlighted between supporting the AC grid through grid-forming control and ensuring the security of the DC grid due to the additional delays in power tracking brought by the grid-forming control. Subsequently, a method is presented to calculate the energy requirements of the MMC submodules to provide inertial response to the AC grid as SGs do.

This method is combined with a novel control law that makes it possible to use this internal energy to support the AC grid and that is compatible with the stateof-the-art grid-forming control. This control is then analyzed for dierent types of disturbances and dierent tuning parameters to show its eciency. The validation of this control and its compatibility with the grid-forming control is also another contribution of this work.

The structure of this chapter is as follows: In Section 4.2, the MMC technology is presented and the test system used in this chapter is introduced. In Section 4.3, the control structure of an MMC is presented, with a focus on the high-level structure.

In Section 4.4, the AC/DC dynamic interactions caused by the operation of the MMC in grid-forming are discussed. Then, the additional energy requirements to provide synthetic inertia to the grid are presented in Section 4. The most known topology of MMC-HVDC system is given in Fig. The voltage generated at the AC point of coupling is the sum of the submodule voltages connected at that moment. Thus, by acting on the gate signals sent to the submodules, it is possible to generate a stepped square sinusoidal waveform [START_REF] Deng | Voltage-Balancing Method for Modular Multilevel Converters Switched at Grid Frequency[END_REF] as Chapter 4. Operation of Modular Multilevel Converters in Grid-Forming Mode: High-level Control Constraints and Enhancements described in Fig. 4.4. This voltage waveform is much closer to the ideal sine wave compared to the two-level VSCs. In consequence, there is no need of the LC lter in the case of MMC. Thus, the physical AC capacitance is no longer available for implementing the cascaded control structure in Chapter 2 used in the grid-forming control. In the literature, while some works still assume the capacitive eects of AC grid in order to use this cascaded structure [START_REF] Sanchez-Sanchez | The role of the internal energy in MMCs operating in grid-forming mode[END_REF][START_REF] Arco | Operation of a Modular Multilevel Converter Controlled as a Virtual Synchronous Machine[END_REF], some other works highlight the necessary changes due to the absence of the physical LC lter in MMC [START_REF] Qoria | Grid-forming control to achieve a 100% power electronics interfaced power transmission systems[END_REF]. Thanks to its modular nature, the MMC technology becomes a more interesting option for future ohsore wind farm interconnection projects [START_REF] Glasdam | Review on multilevel voltage source converter based HVDC technologies for grid connection of large oshore wind farms[END_REF]. The main advantages of MMCs compared to classical two-level VSCs are summarized below:

• The conversion equipment generates smaller losses than the conventional VSCs;

• The MMC modular nature makes it possible to easily change the rating, if a higher voltage is needed for instance, without having to replace the entire arm/valve;

• Some extra energy is embedded in the capacitances of the submodules, which gives an extra degree of freedom from a control aspect. In particular, this last item is detailed in Section 4.2.2.

The MMC technology is still emerging and remains a more expensive solution when compared to commoner solutions, such as Line Commutated Converters (LCCs) or two-level VSCs [START_REF] Ramadan | Performance enhancement and robustness assessment of VSCHVDC transmission systems Bibliography controllers under uncertainties[END_REF]. As an example, the price of a 1000 MW MMC station is higher than a similar two-level VSC: 125 (±20%) million euros versus 90 (±20%) million euros [2]. The power from the OWF is modeled by an ideal power source with a delivered power P L . The DC-voltage dynamics are:

dv 2 dc dt = 2 C dc • (P L -P dc ) (4.1)
where C dc is the DC-side capacitor, v dc is the DC-grid voltage and P dc is the power received by the converter.

It is important to note that contrary to the classical 2-level VSCs, MMCs have an embedded storage capacity thanks to their large number of submodules which are used for the voltage modulation. As proposed in [START_REF] Freytes | Small-signal state-space modeling of an HVDC link with modular multilevel converters[END_REF], from a grid point of view, the MMC can be seen as a classical AC/DC converter plus an equivalent DC/DC converter which is the interface with the DC grid, as illustrated in Fig. 4.6. This is possible by assuming that the MMC submodules are well balanced at any time.

W mmc = 1 2 • C eq • v 2 Σ (4.3)
where P ac is the AC output power, v Σ is the voltage of the MMC equivalent capacitor, W mmc is the the MMC internal energy, and C eq is the equivalent capacitance of the MMC, which represents all the submodules in the three phases. C eq can be expressed as:

C eq = 6 N arm • C sub (4.4)
where N arm is the number of the submodules in each of the 6 arms of the MMC and C sub is the capacitance of one submodule.

The dynamics of the AC line between the equivalent AC grid and the MMC are given by:

di d eq dt = ω base l eq • (v d 2 -v d 1 -r eq • i d eq + ω sys • l eq • i q eq ) (4.5) di q eq dt = ω base l eq • (v q 2 -v q 1 -r eq • i q eq -ω sys • l eq • i d eq ) (4.6)
where ω sys is the grid frequency and ω base the system fundamental angular frequency, and l eq and r eq are respectively the line inductance and resistance in per unit.

The numerical values of the parameters are given in Table 4.1.

For the equivalent AC grid, as described in Fig. 4.5, a perfect voltage source is considered: v 2 = 1 p.u. For the frequency dynamics, a steam-powered synchronous generator described in [START_REF] Siraj Khan | Stability Assessment of a System Comprising a Single Machine and a Virtual Oscillator Controlled Inverter with Scalable Ratings[END_REF] and based on models from [START_REF] Kundur | Power System Stability and Control[END_REF] is used.

Control of the MMC

The general MMC control structure is adapted from the one used in grid-following [START_REF] Zama | High Dynamics Control for MMC Based on Exact Discrete-Time Model With Experimental Validation[END_REF]. For the control of the MMC, as described in Fig. 4.7 (a), the considered MMC is controlled in grid-forming mode on the AC side and in DC-voltage mode on the DC side. In this gure, the considered control dynamics are as follows:

• DC voltage controller which uses a PI control to regulate the DC voltage to its reference. Usually we choose v * dc = 1 p.u.;

Control of the MMC

• Energy controller, which uses a PI control to maintain the MMC total internal energy within an acceptable level;

• Active power controller, also known as power synchronization loop [START_REF] Zhang | Power-synchronization control of gridconnected voltage-source converters[END_REF], which uses the variations of active power to synchronize with the grid.

Quantity

p * dc = p * + α • ∆p * (4.7) p * ac = p * -(1 -α) • ∆p * (4.8)
where α is the weighting coecient. In this chapter, we only consider the cases where α = 0 and in consequence the MMC energy is only balanced by the AC power modulation. However, the energy control action can also be ensured by the DC power (α = 1) [START_REF] Shinoda | DC voltage control of MMC-based HVDC grid with virtual capacitor control[END_REF] or shared between the two (0 < α < 1) [START_REF] Sanchez-Sanchez | Optimal multivariable MMC energy-based control for DC voltage regulation in HVDC applications[END_REF]. The gridforming control structure is taken from [START_REF] Markovic | Stability Analysis of Converter Control Modes in Low-Inertia Power Systems[END_REF]. The design of the DC voltage and energy PI controllers are achieved using pole placement methods as described in [START_REF] Shinoda | DC voltage control of MMC-based HVDC grid with virtual capacitor control[END_REF] 1. The wind is blowing more strongly, hence a higher power P L is injected into the DC grid;

2. The DC voltage increases, due to (4.1);

3. The DC voltage controller adjusts the power reference p * ; 4. p * dc and p * ac are adjusted after the weighting factor block from (4.7) and (4.8).

The response time of the DC-voltage controller is not given here because it is used for parametric studies in the following section.

Impacts of grid-forming control on the DC-voltage loop 4.4.1. Time-domain analysis

As described in Section 4.2.2, the DC-voltage regulator keeps the DC voltage within certain acceptable limits, with a time constant of T V dc . Nevertheless, the AC-side active power loop should be chosen fast enough to ensure robust synchronization with the grid and its decoupling from the inner loops, while slow enough to lter the power harmonics [START_REF] Denis | From grid-following to grid-forming: The new strategy to build 100 % power-electronics interfaced transmission system with enhanced transient behavior[END_REF]. To satisfy these conditions, we chose:

ω LP F = 1 T LP F = ω base 10 (4.10)
We implement a simulation setup in Matlab/Simulink environment to illustrate the impact of poorly designed AC/DC controls with their time constants close to each other. Two types of simulations are carried out: one with identical time constants:

T V dc = T LP F , and the other with a slower DC voltage control loop:

T V dc = 4 • T LP F ,
where T V dc is the time constant of the PI controller.

It is important to note that T V dc is dierent from the response time, or the settling time, T rdc . Based on [START_REF] Shinoda | DC voltage control of MMC-based HVDC grid with virtual capacitor control[END_REF], we have: It is quite interesting to note that the AC power oscillations in Fig. 4.8 (a) are not visible on the DC voltage. This is due to the decoupling of AC and DC powers in transient, which is made possible by the MMC internal energy. In fact, the DCside disturbance directly causes variations in the MMC internal energy, as shown in Fig. 4.9 (b), which has an impact on the AC power reference due to the internal energy controller, as illustrated in Fig. 4.9 (a) and thus on the AC power as in Fig.

T rdc = 3 • T V dc
4

.8 (a). Last but not least, it is important to note that all these dynamics are fast compared to the grid frequency dynamics, and represent too little energy to have any signicant impact on the AC grid frequency, as given in Fig. 4.10.

Parametric study

In order to understand the cause of the oscillations observed on the AC power, we performed a modal analysis of the system for dierent values of T V dc :

T LP F 2 < T V dc < 10 • T LP F (4.12)
We dene a parametric DC-voltage time constant T V dc (k) as:

T V dc = T LP F k (4.13)
The eigenvalues are displayed in Fig. 4.11 for k = {0.1, 0.2, ..., 2}. As expected, slowing down the DC voltage controller results in a better damping on the oscillatory modes related to the controller. Indeed, when T V dc is increased, i.e. when k is reduced, the oscillatory modes have their imaginary part reduced and move towards the real axis, which corresponds to completely damped modes.

The modal analysis conrms the AC/DC interactions between the DC-voltage controller and the AC grid-forming loop. However, it is hard to extract any optimal controller gains from this gure since it does not take into account the constraints The squared max deviation of the simulation corresponding to a given T V dc is calculated as: As expected, for lower values of k, i.e. slower DC-voltage control, the DC-voltage deviation is relatively signicant whereas the MMC energy is kept within acceptable limits. The deviation of the MMC energy becomes greater than that of the DC voltage when k = 0.6, for which the sum of the deviations Σ = dc + mmc is also one of the most acceptable. However, if it had been decided to give dierent weights to the DC voltage and the MMC internal energy deviations, the results could have been dierent. In the future, it could be reasonable to expect this trade-o between the grid and the converter constraints to be made by the grid operators themselves. To summarize, this study analyzes the dynamic behavior of an MMC-HVDC controlled in DC voltage mode in DC and grid-forming in AC, to illustrate negative interactions that may occur when integrating oshore wind farms in the grid.

dc (k) = (v * dc 2 -v dc,min 2 ) 2
The importance of slowing down the DC voltage control loop is shown to avoid the interactions issues. However, this solution poses problems to the DC stability and some measures should be taken, such as adding a physical DC capacitance.

Concurrently, the MMC can also be used to provide additional services to the AC grid, especially frequency-related support. This last part is the focus of the next sections of this chapter.

Method to dene MMC internal energy requirements 4.5.1. Grid code requirements & frequency response

Frequency is one of the key metrics monitored by Transmission System Operators (TSOs) to ensure the stability of the grid, and all the loads and generators have to respect the voltage and frequency requirements imposed by the TSOs. These requirements are described in the grid code. In the rest of this chapter, we focus on the UK grid code, where National Grid imposes the operational frequency ranges given in Table 4.3.

In an SG-dominated system, inertia is the sum of all the SGs inertia connected

to the system at a given moment. This inertia is useful because it limits the Rate of Change of Frequency (RoCoF). In particular, a large RoCoF implies a high risk of SG disconnection or line tripping, while a small RoCoF makes it easy to counterbalance the disturbance within an acceptable amount of time. The UK grid frequency is currently well regulated and, even though its required range of normal operation is ± 1 Hz (see Table 4.3), it is in reality kept within an even narrower range. For example, Fig. 4.13 shows that the frequency remained within the range of ±250 mHz during the whole month of November 2019 in the UK.

Figure 4.13: Frequency monitoring in the UK transmission system in November 2019 [START_REF]Historic Frequency Data[END_REF].

However, with the ongoing dismantlement of SGs, HVDC stations, such as OWF onshore MMC, could be asked to provide frequency response to support the grid as SGs used to do. If the frequency containment reserves can be managed at the wind-farm level [START_REF] Shapiro | Wind farms providing secondary frequency regulation: evaluating the performance of model-based receding horizon control[END_REF], the inertial response of SGs may be too fast to be emulated by the wind farm control. Therefore, this work focuses on the inertial response of SGs using the internal energy of the MMC.

Synthetic inertia needs

The method presented in this chapter makes the MMC emulate the typical inertial response of synchronous generators following an active power disturbance, given in Fig. 4.14. In particular, the MMC submodules are required to provide as much energy as needed in order to emulate this inertial response, for example the rstswing response covered by the green hashed lines in Fig. 4.14. Since the frequency range in Table 4.3 in which any equipment (SG or VSC) is supposed to remain connected is ±1 Hz, only a fraction of the entire equivalent kinetic energy of the rotating mass would be enough to realize the frequency support. In the following, Since an MMC already stores some internal energy in its submodules, a primary comparison is required to see how much extra energy is needed. We assume that the submodules can tolerate a variation of 20% around their nominal energy [START_REF] Shinoda | Virtual Capacitor Control: Mitigation of DC Voltage Fluctuations in MMC-Based HVdc Systems[END_REF].

Table 4.4 gives a comparison of the available energy between the rotating mass of a typical SG and an MMC of the same power rating, based on the features in [START_REF] Sanchez-Sanchez | The role of the internal energy in MMCs operating in grid-forming mode[END_REF],

where the available energy is calculated as

∆W SG = H 0 • (ω 2 0 -ω 2 min ) • S nom for the SG and ∆W M M C = H 0 • (W 0 -W min ) • S nom

Energy requirement calculations

This part presents the method to calculate the energy requirements and the corresponding size of the MMC submodules to provide similar synthetic inertia as SGs. In particular, we seek to increase the size the submodules so that its released First, the kinetic energy variation of an SG when the frequency decreases by 1

Hz can be found as

∆W SG = H 0 • (ω 2 0 -ω 2 min ) • S nom = 3 • (1 -(49/50) 2 ) • 500 = 59.4MJ (4.16)
The internal energy of the original (not oversized) MMC, denoted by W mmc , can be expressed as W mmc = 1 2 C eq v 2 dc , where C eq = 6 Narm C sub is the equivalent capacitor of the original MMC. Using the values in Table 4.4, we have

W mmc = 1 2 C eq v 2 dc = 1 2 6 N arm C sub v 2 dc = 1
∆W mmc = (1 -0.8) • N oversize • W mmc (4.18)
If we require that when the MMC internal energy variation reaches its limit, the energy released or absorbed by the submodules is of the same amount that an SG of the same power rating would release or absorb when the frequency changes by ±1 Hz, then we must have

∆W SG = ∆W mmc = (1 -0.8) • N oversize • W mmc (4.19)
Numerically, The above method is used to design oversized MMC submodules or alternative ESS for dierent values of the inertia constant and the frequency deviation range of the SG, and the results are summarized in Table 4.5.

N oversize = round ∆W SG (1 -0.8) • W mmc = round 59,
In fact, the alternative ESS can also be considered as equivalent oversized capacitors because the associated DC/DC converter can be controlled using the control law proposed in [START_REF] Shinoda | Virtual Capacitor Control: Mitigation of DC Voltage Fluctuations in MMC-Based HVdc Systems[END_REF] which emulates the behavior of an additional parallelconnected capacitor. In consequence, in the rest of the study, only the oversized MMC submodules are considered to provide the extra energy.

Control design 4.6.1. Proposed control law

To provide frequency support using oversized submodules, we propose to use a dynamic energy reference controller [START_REF] Sanchez-Sanchez | The role of the internal energy in MMCs operating in grid-forming mode[END_REF][START_REF] Shinoda | Virtual Capacitor Control: Mitigation of DC Voltage Fluctuations in MMC-Based HVdc Systems[END_REF], whereby the internal energy reference W * mmc is changed in proportion to the deviation of the square of the grid frequency ω g . Since this variable is dicult to compute in real time, we consider rather the local frequency imposed by the grid-forming control and denoted by ω 1 . The control law is:

W * mmc = W * mmc,0 + k vr • (ω 1 2 -ω * 2 ) (4.22) W min mmc ≤ W * mmc ≤ W max mmc (4.23)
where W * mmc,0 = 1 p.u. is the energy reference corresponding to 50 Hz, ω * is the AC grid reference frequency in per unit, and k vr is a tunable coecient which emulates the response of a rotating machine. W min mmc and W max mmc are used to ensure that the energy controller respects the physical limits of the MMC and their values will be specied later. on [START_REF] Mourouvin | AC/DC Dynamic Interactions of MMC-HVDC in Grid-Forming for Wind-Farm Integration in AC Systems[END_REF]. The blocks in green are specic to this study.

Since this control law is based on emulating the kinetic energy of the rotating mass of a SG, it is named Virtual Rotor Control or VCR in this chapter where k vr is the associated virtual rotor coecient. In order to provide this energy to the AC grid, the energy controller is coupled to the AC power reference only. This AC power reference p * ac is used in the grid-forming control as detailed in Fig. 4.15. This control is based on the general control scheme presented in Fig. 4.7. Besides, the MMC also ensures the DC grid voltage control.

Controller tuning

In this subsection, the tuning of the coecient k vr is discussed. Since the aim of the control law is to use all the available energy in the MMC submodules when the maximum acceptable frequency is reached, W min mmc = 80% must be reached when the grid frequency reaches ω min = 49.0 Hz based on Table 4.3. Thus, k vr can be calculated as: To show the eectiveness of the virtual rotor control, we conduct simulations on a simple system where an MMC representing an OWF, modeled by an equivalent power source with DC-side voltage dynamics, is connected to an aggregated AC grid by an overhead line. Only the outer-loop control shown in Fig. 4.15 is considered 4.8. Extension to multi-machine systems for MMC, and the grid-forming control used is the droop-based option [START_REF] Rokrok | Classication and dynamic assessment of droop-based grid-forming control schemes: Application in HVDC systems[END_REF]. More details about this test system can be found in Section 4.2.

W min mmc =W * mmc,0 + k vr • (ω min 2 -ω * 2 ) = 80% (4.24) =⇒ k vr = W min mmc -W * mmc,0 ω min 2 -ω * 2 = 0.8 -1 0.98 2 -1 2 ≈ 5
The submodule capacitance considered in the simulations is C sub = 96 mF, based on the results from Table 4.5, in order to match the response of a SG of H 0 = 3 s in a range of ±1 Hz. The additional physical and control parameters of the test system are given in Tables 4.1 and 4.2. The parameters concerning the frequency dynamics of the aggregated AC grid can be found in [START_REF] Lin | Stability assessment of a system comprising a single machine and inverter with scalable ratings[END_REF].

Four types of disturbance are tested. Two of them are an AC power increase by 50 and 250 MW, respectively, which can represent a sudden load increase, while the other two are a DC power drop by the same values, which can represent the loss of some of the energy coming from the OWF for instance.

Results

The results are given in Fig. 4.16. They show that for all the 4 considered disturbances, the virtual rotor control always yields better results than the standard MMC energy controller in terms of the RoCoF and the frequency nadir. In addition, since W mmc never reaches 0.8 p.u., there is still some available energy to further support the frequency, which can be realized by choosing a larger k vr . Lastly, the virtual rotor control neither improves nor deteriorates the DC voltage, which suers from a severe dip when subject to the 250 MW disturbance from the DC grid.

To investigate the role of k vr , further simulations are conducted with k vr varying between 5 and 25. The time evolution of physical quantities are given in Fig. 4.17, and the dependency of the frequency nadir and the RoCoF upon k vr is given in Fig. 4.18. On the one hand, increasing k vr results in a smaller (or better) RoCoF, while on the other hand, the nadir is the highest (or the most desirable) when k vr = 9. These observations are similar for both the DC and AC disturbances. However, as k vr increases, W mmc may violates its lower limit W min mmc = 80%, which is less acceptable than the case where k vr = 5. This large overshoot in the evolution of W mmc may be attributed to the relatively poor power tracking performances of the grid-forming control compared to the classical grid-following control [START_REF] Harnefors | Referencefeedforward power-synchronization control[END_REF].

Extension to multi-machine systems

Grid model

So far in this chapter, all the results presented are based on an aggregated model of an AC grid, which is relevant for studying the frequency-related functions of the AC grid and connected converters. However, in a real power system, there are many In order to have a realistic model, we consider the Nordic 44 system which contains 80 synchronous generators. More details about this model can be found in [START_REF] Jakobsen | The Nordic 44 test network[END_REF]. We modied the existing power system by introducing three MMCs with dierent control modes, one in AC power control mode (p ac -mode) and two in DC- voltage mode (v dc -mode).

… … … … … … … … … … … … … … … … … … … … … …… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … L… … … … … … … … … … … … System
The modications with respect to the original case [START_REF] Jakobsen | The Nordic 44 test network[END_REF] are:

• Two grid-forming MMCs (1000 MVA each) in v dc -mode are connected to buses 7000 and 5500 through transmission lines (x l = 0.166 p.u. and r l = 0.0166 p.u.). On their DC side, an equivalent capacitor and a ltered DC current injector are connected. The initial power set-point of these stations is 0 MW.

• The generator at bus 3245 is replaced by a grid-forming MMC station in p acmode. The active power set-point of this station is the same as the replaced synchronous generator (i.e. 205MW).

• The transmission lines connecting the MMCs to the nearest generators are represented by dynamic π-sections (also known as hybrid network model [START_REF] Karawita | A hybrid network model for small signal stability analysis of power systems[END_REF]).

In the rest of the chapter, each MMC will be referred to by its bus number in order to avoid misunderstandings.

Test cases

In order to test the impacts of the AC grids dynamics on the method proposed in this chapter, we conduct two test cases:

• Case 1 (without VRC): the three MMCs are similar in terms of the number of submodules and ratings. Their submodules are not oversized and no VRC is implemented;

• Case 2 (with VRC): MMC 3245 is similar to case 1, but MMCs 7000 and 5500 have oversized submodules so that they can provide synthetic inertia to the system. For the latter two MMCs, VRC is implemented with k vr = 25.

The numerical values of the MMCs parameters can be found in Table 4.6. In both cases, we simulate the loss of a generator in the bus 5300 which results in a -800MW power unbalance in the system. This event is triggered at t = 5s. In terms of control, the grid-forming outer loop control is the same as the one presented in Section 4.3. The MMC inner loop control is based on [START_REF] Harnefors | Referencefeedforward power-synchronization control[END_REF]. The values of the control parameters which dier from Section 4.3 can be found below. In addition to those changes, compared to the numerical values given in Table 4.2, the following control parameters were also considered for the three MMCs:

MMC #

k vr = 25, m pω = 0.02 p.u., τ rdc = 0.15 s, τ rwh = 0.4 s (4.26)

Results and discussions

The two cases described in Section 4.8.2 are simulated using Dymola software [175] in DAE-mode, with the grid and the MMCs modeled using Modelica language. The results are given in Fig. 4.20.

It is interesting to note that the VRC works well even when we consider a large power system. Indeed, as described in Fig. 4. 20 (d) where the internal energy of MMCs 7000 and 5500 is used accordingly to the frequency deviation to support the AC grid. This contribution is also observable in terms of the injected AC power given in Fig. 4.20 (a). However, the VRC appears to have almost no impacts on the AC voltage magnitude or the reactive power, which justies our choice to use simplied models. Also, it is important to note that, similarly to the results given in Fig. 4.16 with an aggregated model, the AC-side disturbance does not propagate to the DC grid in either case, as expected.

From the grid perspective, we see that in both cases, we have low-frequency oscillations in the active power and the rotor speed of the generators, which are transmitted to the MMC internal energy, but not to the DC grid thanks to the buering action of the MMC submodules. Indeed, these oscillations are caused by the interactions between all the synchronous generators of the grid, and are transmitted to the MMCs because they are controlled in grid-forming mode and behave in consequence as virtual synchronous generators.

Finally, we observe in Fig. 4.20 (f ) that the VRC does provide some inertial support to the neighboring SGs by discharging the MMCs submodules, but this support is relatively small because the Nordic 44 grid is much bigger in terms of the installed power compared to the ratings of the two MMCs controlled with VRC.

However, the low-frequency oscillations are still present on the frequency and active power. Additional actions should be considered to damp these oscillations.

Conclusions

In this chapter, a method is proposed to quantify the energy requirements when increasing the size of the MMC submodules in order to provide the same AC frequency support as SGs. Then, a control law compatible with the state-of-theart MMC grid-forming control is proposed to emulate the inertial response of SGs by using the MMC internal energy. Simulations were performed and show a better response of the new controller in terms of the AC frequency response to AC and DC power disturbances.

Chapter 4. Operation of Modular Multilevel Converters in Grid-Forming Mode: High-level Control Constraints and Enhancements

In the simulation section, we observed that the energy reference limits were not always respected when k vr > 5, which can be attributed to the relatively poor tracking capability of the grid-forming control compared to the grid-following control. In Section 4.8, the results obtained with an aggregated grid were extended to the case of the Nordic power system. This analysis showed the compatibility of the proposed control with more complex grids where PE-based sources and SGs coexist. In the future, the novel grid-forming controls with enhanced tracking performances [START_REF] Narula | Impact of steady-state grid-frequency deviations on the performance of grid-forming converter control strategies[END_REF][START_REF] Harnefors | Referencefeedforward power-synchronization control[END_REF] could be combined with the new control proposed in this chapter. In addition, it would be worthwhile investigating the possibility of using the MMC energy in order to provide complementary AC or DC grid services, such as the regulation of the DC voltage, which suered from a severe dip in the simulations shown in Section 4.7.2.

In future work, the following items should be investigated in order to consolidate the results obtained in this chapter:

• First, the novel grid-forming controls with enhanced tracking performances [START_REF] Narula | Impact of steady-state grid-frequency deviations on the performance of grid-forming converter control strategies[END_REF][START_REF] Harnefors | Referencefeedforward power-synchronization control[END_REF] could be combined with the new control proposed in this chapter;

• The dynamics of the power source, here the oshore wind farm, should be taken into account in order to assess the AC/DC stability of this more detailed system;

• In addition, the low-frequency oscillations observed in Fig. 4.20 for the Nordic 44 grid were not solved with the VRC. It could be worth investigating the use of complementary controllers, such as POD controllers [START_REF] Renedo | Active power control strategies for transient stability enhancement of AC/DC grids with VSC-HVDC multiterminal systems[END_REF][START_REF] Gonzalez-Torres | A novel distributed supplementary control of multi-terminal vschvdc grids for rotor angle stability enhancement of ac/dc systems[END_REF], to damp these oscillations;

• Finally, the use of Multi-Input Multi-Output (MIMO-)oriented control represents an interesting opportunity to stabilize the system while taking into account the AC/DC constraints. In particular, this item will be developed in Chapter 5.

Chapter 5. Application of Model Predictive Control to Operate an MMC in Grid-Forming Mode [START_REF] Mourouvin | AC/DC Dynamic Interactions of MMC-HVDC in Grid-Forming for Wind-Farm Integration in AC Systems[END_REF][START_REF] Sanchez-Sanchez | The role of the internal energy in MMCs operating in grid-forming mode[END_REF]. Indeed, contrary to the classical 2-level Voltage Source Converters (VSCs), this internal energy acts as a buer between the AC and DC grids and can be seen as a supplementary degree of freedom which can be used to provide additional services such as DC-grid voltage support [START_REF] Shinoda | Virtual Capacitor Control: Mitigation of DC Voltage Fluctuations in MMC-Based HVdc Systems[END_REF] or AC-grid inter-area oscillations damping [START_REF] Fradley | Assessment of the impact of MMC-VSC intrinsic energy on power system stability[END_REF][START_REF] Taese | Power oscillation damping with virtual capacitance support from modular multilevel converters[END_REF]. In addition, the MMC submodules can also be designed to store even more energy for potential AC grid frequency support [START_REF] Judge | Modular Multilevel Converter With Partially Rated Integrated Energy Storage Suitable for Frequency Support and Ancillary Service Provision[END_REF]. But these considerations focus on the submodule-level design and control and do not take into account the progress in high-level control such as grid-forming control, where the converter acts as a self-synchronized voltage source with respect to the AC grid. This new paradigm changes the way MMCs are currently operated [START_REF] Arco | Operation of a Modular Multilevel Converter Controlled as a Virtual Synchronous Machine[END_REF] compared to the classical grid-following control where the converter behaves as a current source, as presented in Chapter 2.

In this chapter, we propose a novel way of designing the upper layer of the highlevel MMCs control structure which is compatible with the concept of grid-forming control and ensures the proper operation of the DC grid, the AC grid and the converter submodules at the same time, based on Model Predictive Control (MPC) approach. Contrary to the control law presented in Chapter 4, this MPC relies on a semi-centralized approach to coordinate the AC and DC grids at the same time.

Literature review

Since its early development, MPC has been dedicated to slow dynamic systems such as chemical processes [START_REF] Camacho | Model Predictive Control[END_REF] or entire hydro power valleys [START_REF] Flórez | Decentralizedcoordinated model predictive control for a hydro-power valley[END_REF], mainly due to the required computation costs. However, with the constant improvement of the computational systems, it is now possible to apply MPC algorithms to the elds of power electronics and power systems.

Regarding HVDC applications, MPC has been used to enhance the operation of MTDC grids [START_REF] Sánchez | Secondary Model Predictive Control Architecture for VSC-HVDC Networks Interfacing Wind Power[END_REF] or applied to HVDC links to support the neighbouring AC grids [START_REF] Thau | Model Predictive Control of VSC-HVDC Embedded into AC Grid Subject to State and Control Constraints[END_REF]. However, these works are more about the system-level operation of AC/DC grids, meaning the MPC is a centralized solution that coordinates the behavior of several stations. In this chapter, even if we also deal with grid-level support, we focus here on converter-level operation, and especially its compatibility with grid-forming control. This last item was the topic of a recent study which focused on developping a centralized optimal controller which uses Linear Quadratic Regulator (LQR) method to update the grid-forming control gains of distributed VSCs [START_REF] Markovic | LQR-based adaptive virtual synchronous machine for power systems with high inverter penetration[END_REF]. However, this method is at the grid-level and does not consider any constraints. On the other hand, at the converter-level, while the use of MPC for 5.2. Problem formulation Figure 5.1: Energy exchange between the AC and DC grids via the MMC submodules, from [START_REF] Freytes | Small-signal state-space modeling of an HVDC link with modular multilevel converters[END_REF].

two-level VSCs controlled in grid-forming was the topic of a recent paper [START_REF] Stanojev | MPC-Based Fast Frequency Control of Voltage Source Converters in Low-Inertia Power Systems[END_REF],

there is no paper, to the best of our knowledge, which focuses on the role that can be played by the MMC in supporting the AC/DC grids under constraints using MPC. The application of the MPC in this case is the main contribution of this work.

Chapter summary and contents

In this chapter, an optimization problem is formulated and a complete control solution is proposed that uses the Model Predictive Control (MPC) for AC/DC energy management of a Modular Multilevel Converter (MMC) controlled in gridforming mode. First, for the implementation of the MPC, a relevant model of the grid is found by system identication. The identied model is then validated before its usage in the MPC. The simulation results show a better response to disturbances from AC grid and DC grid, as well as a better coordination in the management of the MMC internal energy and DC grid voltage compared to the commonly used dual PI structure.

The structure of this chapter is as follows: In Section 5.2, the problem of controlling the outer loops of the MMC is stated and the need to have a MIMOoriented control is explained. In Section 5.3, an identication method, namely the prediction error method, is described to obtain an equivalent linear model of the AC grid. In Section 5.4, the MPC algorithm is detailed, and the cost function and constraints are specied. In Section 5.5, the control solution is tested on a simplied system to show its eectiveness compared to the classical PI structure. Section 5.6 concludes.

Problem formulation

MMC interfacing AC and DC grids

The energy exchange between AC and DC grids realized by the MMC can be illustrated by Fig. 5.1, where the MMC is modeled by an AC/DC converter, a DC/DC converter and a single capacitance representing all the submodules. v 1 ∠θ 1 , v dc and v Σ are respectively the AC voltage, the DC voltage, and the MMC submodule equivalent capacitance voltage.

Problem formulation

The control of the internal energy of the MMC interfacing between the AC and DC grids can be seen a problem of 3 connected reservoirs, as shown in Fig. 5.2, where p * dc and p * ac are the energy exchanged with the DC and AC grids, respectively, p L the power injection/withdrawal imposed by the DC grid, and p M the power injection/withdrawal imposed by the AC grid. Naturally, the controller in charge of regulating the MMC internal energy can use either p * dc or p * ac or both of them as the control variables. For example, in [START_REF] Mourouvin | AC/DC Dynamic Interactions of MMC-HVDC in Grid-Forming for Wind-Farm Integration in AC Systems[END_REF], p * ac is used to regulate the MMC inernal energy, by means of a classical double PI structure shown in Fig. 5.3 (a). However, this oine choice of the control variables and the tuning of the PI controllers may turn out to be inadequate when the AC and DC grids experience drastic changes of operation conditions, such as the loss of part of the DC grid and the reduction of the inertia and frequency reserves due to high renewable energy shares.

Therefore, in this chapter, an online approach to MMC internal energy control is proposed where the AC and DC powers are optimally coordinated to enhance the system disturbance mitigation. Since the energy constraints of the system have to be taken into account, especially for the DC voltage and MMC internal energy, MPC is used. As shown in Fig. 5.3 (b), this MPC layer generates the DC and AC power references, p * dc and p * ac , to manage the DC voltage and MMC internal energy level and maintain them within acceptable limits while contributing to AC grid support.

To be optimal, this implementation could require a regular update of the grid model to take into account operational changes such as power ow updates. In Continental Europe for instance, these changes occur every hour [184]. The update of the model is however not considered in this work.

State-space representation

Based on simplied power exchanges of an MMC which are detailed in Chapter 4, the plant in Fig. 5.2 can be modeled in per-unit system as:

Σ =          ẋdc = 2 C dc • (p L -p dc ) ẋmmc = 2 Ceq • (p dc -p ac ) ẋac = f (x ac , p ac , p M ) (5.1)
where

x dc = v 2 dc , x mmc = v 2 Σ ,
x ac is the state variable vector of the AC system, C eq the equivalent MMC internal capacitance storing the total internal energy, and f describes the AC-grid dynamics. For the DC-side dynamics and the MMC internal dynamics, the simplied rst-order linear models in (5.1) are considered to be known Chapter 5. Application of Model Predictive Control to Operate an MMC in Grid-Forming Mode for the MPC implementation. On the contrary , the AC model is assumed to be unknown and will be identied in Section 5.3.

Control structure

The general structure of the MPC with state estimation is shown in Fig. 5.4. The aim of the MPC block is to come as a hierarchical control that sends power references, p * ac and p * dc to the lower control loops. In this case, p * ac is sent to the grid-forming control whose outer loop is considered in this paper since its dynamics are slow when compared to the inner loops and the DC current controller that receives the DC power reference p * dc . For the MPC implementation, a good estimation of the nonlinear system Σ is needed in order to predict its future behavior. This estimated model, denoted by Σ c , is obtained via a system identication method, which is detailed in Section 5.3. This model is also useful to estimate the current states of the AC/DC system that are observable but not directly measurable. Indeed, this model Σ c is also used in the Kalman lter described in Fig. 5.4. More information about the Kalman lter is given in Section 5.4.1. In particular, as illustrated in Fig. 5.5 explaining the grid-forming control of the MMC, the block from θ 1 to ω g and p ac is unknown.

As the implementation of MPC in Section 5.4 needs a linear system, we consider the equivalent linear transfer function G(s), which corresponds to the linearized dynamics of the AC grid around an equilibrium state provided by a power ow computation with reference frequency ω * :

∆ω(s) = G(s) • p * ac (s). (5.2)
where we choose to consider the frequency using ∆ω = ω -ω * as the output instead of ω g . Indeed, the former is the local estimation of the frequency deviation 100 5.3. System identication Figure 5.5: Block scheme of the AC system with the MMC in grid-forming control mode.

and is directly available while the latter, the grid frequency, is a global variable corresponding to the weighted sum of the frequencies measured at several distant buses of the entire system.

System identication

In this section, a method to estimate the linearized dynamics of the AC grid is proposed. To estimate G(s) dened in (5.2), we choose to rst estimate its discrete version, G(z), which will be later converted to a continuous-time model, using a probing signal. This choice is made because the prediction error method used for identication in this chapter is said to give a more relevant identied model than the classical subspace identication for linear MPC applications [START_REF] Jorgensen | Continuous-discrete time predictionerror identication relevant for linear model predictive control[END_REF].

Other identication techniques, such as ambient identication can be applied to AC grid parameter identication [START_REF] Wang | Online Inertia Estimation Using Electromechanical Oscillation Modal Extracted from Synchronized Ambient Data[END_REF]. However, they are out of the scope of this chapter.

Identication method

The general structure of the model estimation is shown in Fig. 5.6, where u id = p * ac is the model input, ŷid = ∆ω is the estimated model output, and d is the disturbance of the system, not identied here, which models the power disturbances such as p M from (5.1). The AC grid model in Fig. 5.6 used to generate y id under the excitation u id is described in Section 5.5.1. Ĝ(z) is related to these variables as:

ŷac (z) = Ĝ(z) • u ac (z) + d(z) (5.3)
To identify Ĝ(z), the nonlinear system is excited with the following multisine probing signal u id and the output y id is obtained:

u ac = u id = Nω-1 i=0 A i sin(ω i t + φ i ) (5.4)
Figure 5.6: Detailed method for estimating the transfer function Ĝ with a multisine probing signal using prediction error method [START_REF] Ljung | System Identication: Theory for the User, ser. Information and system sciences series[END_REF].

where N ω = 1000, ω i = 0.001 • (i + 1) Hz, A i = 2 × 10 -4 p.u. ∀i, and φ i is a phase shift randomly chosen. A frequency grid from 0.001 to 1 Hz is chosen to scan the AC grid, because the time constants of the frequency-related dynamics of the latter typically lie between 1s and 10s [START_REF] Markovic | Understanding stability of low-inertia systems[END_REF]. An optimization of the probing signal could be performed to minimize the energy of the whole excitation [START_REF] Boersma | Probing Signal Design for Enhanced Damping Estimation in Power Networks[END_REF], but this is not the focus of the chapter. Ĝ(z) is obtained using the prediction error method [START_REF] Ljung | System Identication: Theory for the User, ser. Information and system sciences series[END_REF] and assuming the output error model structure in the System Identication toolbox in Matlab [START_REF]Mathworks, System identication toolbox[END_REF]. More information about this method can be found in Appendix E.

A 4th-order model is found to match the nonlinear model dynamics with a certain accuracy, which is dened in Section 5.3.3. The parameters used for the output error function are given in Table 5.1.

Choice of the system order

For most of the MPC applications, the size of the computed model is a key gure since it is directly related to the computation costs of the online optimization. In this case, we try to have a proper AC grid model which captures the behavior of the real nonlinear system in the frequency range described in the previous section. In particular, the order of the identied system, characterized by the couple {n f , n b } is a tunable parameter from the point of view of control. Indeed, we want to obtain a model that is small enough to lower the computation costs while accurate enough to capture the grid relevant dynamics.

System identication

Figure 5.9: Scheme of the validation experiment to check the reliability of the identied model Ĝ.

Validation of the identied model

In order to check the validity of our identied model Ĝ(z), we test it, as shown in Fig. 5.9, with another excitation signal u exp dened as:

u ac = u exp = Nω-1 i=0 A i sin(ω i t + φ i ) = u id (5.6)
whose only dierence from u id is the values of A i , which is equal to 0 if i/10 is an integer and 1 otherwise. The aim of this dierent choice of u id is to check the accuracy of Ĝ(z) in the low-frequency zone corresponding to AC frequency dynamics. The results in Fig.

Σ AC =    ẋac = Â • x ac + B • u ac y ac = Ĉ • x ac + D • u ac (5.7)
where x ac ∈ R 4 is the state variable vector of the AC grid, u ac = p * ac , y ac = ∆ω, and ( Â, B, Ĉ, D) the state-space matrices.

Based on Σ AC , a 6th-order model for the whole system, denoted by Σ c , is obtained as: To implement the MPC as given in Fig. 5.4, knowledge of all the states at each time step is also necessary, which requires the use of sensors or state observers if the variables are not measurable. For the state observation, xdc = x dc and xmmc = x mmc are considered measurable, as described in Section 5.2.2, but the AC grid states x ac are not measurable, for which we use a state observer, the Kalman lter in Fig. 5.4, to obtained its estimation xac . Since we obtained an identied linear state-space model for the AC grid, as described in (5.7), we can use linear state observers. However, it would be possible to extend this work to nonlinear state observers [START_REF] Besançon | Nonlinear observers and applications, ser. Lecture notes in control and information sciences[END_REF] but this remains out of the scope of this chapter. This Kalman lter is a continuous-time observer which uses the model Σ AC that is observable. We dene the process noise covariance Q = 1 and the sensor noise covariance R = 0.1 in order to correct the dierences between the linear model Σ AC and the nonlinear function f using the

Σ c =    ẋc = A • x c + B • u c y c = C • x c + D • u c

Controller formulation

The system to be controlled with the MPC, Σ, is a nonlinear continuous-time 2-input/3-output MIMO system, where it is impossible to control each output independently with zero steady-state error [START_REF] Wang | Model Predictive Control System Design and Implementation Using MATLAB®, ser. Advances in Industrial Control[END_REF]. To apply MPC, a discretetime system is built using a zero-order hold with a sampling time T s and the continuous-time linear model Σ c . In the following, we denote by x d the discrete state variable vector and by y the discrete output vector. The discrete-time system

Σ d = {A d , B d , C d , D d } is dened such that:
A d =e ATs

x d (k + 1) = A d • x d (k) + B d • u(k)
(5.12)

y(k) = C d • x d (k) (5.13)
To ensure zero steady-state error, the technique of constructing a state-space model with an embedded integrator. One of the advantages of this technique is that we work on the control input variation sequence ∆u(k) = u(k) -u(k -1) instead of the control signal sequence u(k) itself. This makes the control design more general since it does not require any steady-state information [START_REF] Wang | Model Predictive Control System Design and Implementation Using MATLAB®, ser. Advances in Industrial Control[END_REF]. The embedded integrator is used to nd an augmented model of the system as:

x(k + 1) = A e • x(k) + B e • ∆u(k) (5.14) y(k) = C e • x(k) (5.15) with ∆x d (k) = x d (k) -x d (k -1), x(k) = ∆x d (k) T y(k) T T ∈ R 9 , A e = A d 0 C d A d I 3 • , B e = B d C d B d
, C e = 0 0 0 I 3 .

(5.16)

The MPC predicts the future states at time horizon i = 0, 1, . . . , N p :

x(k + (i + 1)|k) = A e • x(k + i|k) + B e • ∆u(k + i|k) (5.17)
where N p is the prediction horizon and N c is the control horizon which should be chosen smaller than or equal to N p .

The problem can be formulated in matrix form:

Y (k) = y(k + 1|k) . . . y(k + N p |k) T ∈ R 3Np (5.18) ∆U (k) = ∆u(k) . . . ∆u(k + N c -1) T ∈ R 2Nc (5.19) ∆Y (k) = ∆y(k) . . . ∆y(k + N c -1) T ∈ R 3Np (5.20)
where Y (k) is the predicted augmented-system output vector at a given time t = k, and ∆u(k

+ i) = u(k + i) -u(k + i -1) and ∆y(k + i) = y(k + i) -y(k + i -1)
are the dierence of the control variables and outputs respectively. Y (k) can be calculated using:

Y (k) = F • x(k) + Φ • ∆U (k) (5.21)
where F and Φ are: The whole linear MPC function was implemented in a Matlab environment using the YALMIP toolbox [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF].

F = C e A e C

Optimization problem and cost function

To take into account the DC voltage, the MMC internal energy and the AC grid frequency, we choose the following cost function J for the optimization problem:

J(k) = ( Rs -Y (k)) T • Q • ( Rs -Y (k)) + ∆U (k) T • R • ∆U (k) + ∆Y (k) T • P • ∆Y (k) (5.24)
where Rs ∈ R 3Np is the vector of the desired output setpoint in the prediction horizon N p , and the weighting coecients (5.25) For this MPC implementation, there is no discount factor over time between the observed states/control variables and the predicted states/calculated control variables, i.e. qi = qj , ri = rj , pi = pj , ∀i = j. In addition, the magnitude of the control variable is not considered, i.e. ri = 0, ∀i. The weighting matrice qi is dened as qi = diag{α dc , α mmc , α ω } where α dc , α mmc and α ω represent respectively the weighting factors of the tracking of v 2 dc , v 2 Σ and ∆ω. Although the tracking of the AC grid frequency is not the objective of the MPC, α ω can be used in order to provide potential frequency support. In fact, regulating the frequency to 50 Hz following an active power disturbance would need an innite source of energy from the DC grid, which we considered rather nite in this study. The weighting matrice pi is dened as pi = diag{0, 0, α r } where α r represents the weighting factor for minimizing the time dierence ∆ω(k + 1) -∆ω(k), hence the RoCoF. The other diagonal terms in pi are equal to 0 because we are not interested in the rate of change of the DC voltage or the internal energy.

Q ∈ R 3Np×3Np , R ∈ R
q 3i+1 0 0 0 q 3i+2 0 0 0 q 3i+3    ; rj = r 2j+1 0 0 r 2j+2 ; pi =    p 3i+1 0 0 0 p 3i+2 0 0 0 p 3i+3   
All these control parameters will be dened in Section 5.5.1.

Constraints

To ensure that the levels of MMC internal energy and DC grid are kept within acceptable limits, the following constraints are considered: 0.8 p.u. < y d (1) < 1.2 p.u.

Case study and simulation results

As presented in Fig. 5.4, the MPC presented is the last section is then tested on the case study which is described in this section.

Simulation results are carried out for dierent types of contingencies to show the eectiveness of the MPC. The optimization algorithm used is a the quadratic programming algorithm quadprog from the Optimization toolbox in Matlab [START_REF]Quadratic programming[END_REF].

System considered

We consider a two-bus system where an MMC, modeled with its outer loop control, is connected to an aggregated AC grid with the mechanical modes considered, as in [START_REF] Mourouvin | AC/DC Dynamic Interactions of MMC-HVDC in Grid-Forming for Wind-Farm Integration in AC Systems[END_REF]. The equivalent scheme of the system is given in Fig. 5.11. More information about the assumptions to build this kind of models can be found in Chapter 4. The physical parameter values are given in Table 5.2. as in [START_REF] Errigo | Optimal supercapacitor pack sizing for modular multilevel converter with integrated energy storage system[END_REF][START_REF] Cupertino | High Performance Simulation Models for ES-STATCOM Based on Modular Multilevel Converters[END_REF]. A method to calculate the needed extra energy to provide inertial support with the MMC submodules is presented in Chapter 4 and is used in this case to test the MPC solution.

The MMC control parameters are given in Table 5.3.

Test cases

To check the eciency of the MPC described in Section 5.4.2 in controlling the energy between the DC grid, the MMC submodules and the AC grid, two dierent disturbances are introduced:

• Case 1 : a DC-side disturbance in p L of -150MW, representing e.g. the loss of part of a wind farm.

• Case 2 : an AC-side disturbance in p M of -150MW, representing e.g. a load step in the AC grid.

Simulation results

The post-contingencies responses for the two cases are given in Fig. 5.12. They show that following the DC grid disturbance, the MPC is able to control the DC grid voltage v dc as eciently as the dedicated PI control, with no overshoot, while providing support to the AC grid frequency ω g using the stored energy from the submodule capacitances. This support smoothens the AC power response p ac .

Case study and simulation results

Indeed, in Fig. 5.12 (a), the AC grid frequency exhibits a better response with a smaller rate of change thanks to the weighting factor α r . It is also interesting to note that in Fig. 5.12 (b), the MPC deploys part of the MMC internal energy reserves W mmc , described in Section 5.5.1, to support the AC grid frequency ω g and recharges them thereafter when ω g is coming back to the steady-state value. However, since the weighting factor α mmc is quite small compared to the other normalized weighting factors, the tracking of the MMC internal energy to its reference becomes less prioritized than the control of the DC voltage and the support of the RoCoF, which is desired here. Therefore, no convergence towards the reference value is observed with the dual PI solution in 5.12.

Inuence of controller tuning and varying parameters

In the previous section, the performances of the MPC were demonstrated. However, in that case, the estimated grid model was based on the same system the control solution was tested on. In real-life conditions, the grid changes regularly, due to both planned actions (power reference dispatching, ect.) and unplanned actions (line tripping, generation loss, etc.). In consequence, it is important to check the robustness of this control solution in these conditions. This is the aim of this section.

In addition, if not otherwise specied, the values of the rest of the system and control parameters are based on the nominal case whose values are given in Table 5.2 and 5.3 respectively.

Inuence of MPC tuning

In control theory, the MPC response can be easily tuned by acting on:

• the horizon, dened by the prediction horizon N p and the control horizon N c ;

• the cost function J;

In this application, the choice of the cost function J has been discussed in Section 5.4.3. We will only look at the prediction horizon in this subsection. As stated in Section 5.3.1, the AC grid frequency dynamics lie between 1-10s. In consequence, since the sampling time of the MPC is T s = 75ms, we can dene three dierent scenarii as follows:

• short-term prediction: N c = 4 and N p = 5;

• middle-term prediction: N c = 8 and N p = 10, nominal case;

• long-term prediction: N c = 16 and N p = 20;

The results of these dierent test cases are given in Fig. 5.13. These results show that for both AC and DC disturbances, the prediction horizon does have an impact on the response of the controller. In particular, the larger the prediction window is, the smaller the MMC contribution the AC grid is. On the other hand, for the short-term prediction case, the response is abrupt which is in accordance with the theory.

In our test cases, the MPC is used to provide support function to the AC grid inertia whose dynamics are in the timescale of 0.1-1s. The middle-term prediction which was considered as the nominal case in this study is in consequence the most adequate to provide this kind of support and anticipate the behavior of the grid on a relevant prediction window since T s × N c = 600ms and T s × N p = 750ms.

Inuence of the power setpoint

From time to time, the power references sent to the HVDC station can change due to the power dispatch of the TSO, which would change the setpoint of the system [START_REF] Gionfra | A novel distributed particle swarm optimization algorithm for the optimal power ow problem[END_REF].

In this case, we want to analyze the impacts when the identied model is not updated and thus does no longer correspond to this new power setpoint. It is then relevant to analyze the inuence of this power setpoint on the accuracy and the robustness of the proposed MPC.

Three scenarii are investigated in this section:

• p * = 0p.u., nominal case;

• p * = 0.5p.u.;

• p * = 0.8p.u.;

The simulation results are given in Fig. 5.14 for the same power disturbances that were presented in Section 5.5.2. These results show that the active power setpoint p * appears to have almost no impact on the MPC results. Indeed, the three simulations yield similar values of the optimized cost function and the same grid responses.

These results are interesting because it conrms that the methodology developed in this chapter can be used to operate an MMC at dierent power setpoints without having to re-identify the AC system. However, when there is a grid change, the power ow is not the only parameter to vary. Indeed, the frequency reserves as well as the inertia of the grid can encounter massive changes. This impact can be huge, especially if the identied system does not take this into account. This is investigated in the following section. As stated in the previous subsection, the grid can experience drastic changes which can aect the primary frequency control and the inertial response of the grid, determined respectively by the droop r and the inertia H of the system. This can be even more problematic when the identied model used in the MPC have not been updated when those changes occured. In consequence, it is important to analyze the impact of the model parameter changes on the robustness of the control solution.

In this part, we will change some parameters of the AC/DC grid such as the inertia constant H and the frequency droop coecient r and study in particular the impacts of decreasing H and r, which is a tendency that could occur more often in future power grids with less SGs and more renewable energy sources. The results for the variation of the inertia H and frequency droop r are respectively given in Fig. 5.15 and Fig. 5.16. It is interesting to note that in the two cases, the contributions of the MMC to the DC grid are similar, meaning the change of the parameters of the AC grid does not aect the DC-grid support, which is desirable. In addition, we observe in Fig. 5.15 that the inertial contribution to the AC grid depends on its existing inertia. In particular, the MMC would provide power faster to support the grid frequency if the grid inertia is lower, which is an useful feature for grid applications. Finally, as shown in Fig. 5.16 (b), the contribution of the MMC also depends on the amount of the deployed primary reserves by the rest of the grid. In the case of r = 10%, we see that the MPC does not recharge the MMC submodules following the AC disturbance since the frequency deviation is already signicant.

Conclusions

The main contributions of this chapter can be listed as follows:

• A full control solution including system identication and MPC has been proposed to control an MMC system;

• This control, as an upper layer of the existing structure, is compatible with the state-of-the-art grid-forming control of MMC;

• The use of MPC in the case of MMC with additional internal energy has shown a better response compared to the most commonly-used structure for both AC and DC grid disturbances;

• The robustness of the MPC was assessed regarding variations of the power setpoints and some AC grid parameters. important modes of the AC system. Using this system identication method, an MPC has been implemented to optimally generate the AC and DC power references Chapter 5. Application of Model Predictive Control to Operate an MMC in Grid-Forming Mode used in the MMC control. Indeed, in the future power systems where the traditional synchronous generators coexist with the PE-based power sources, diverse frequency control mechanisms acting jointly may induce additional power oscillations, whose modes are ever-changing with time. To identify such modes, the identication method used in this chapter can be applied, so that the converter can eectively damp power oscillations. In addition, in Section 5.5.4, we investigated the role of the controller tuning and the impacts of parameter changes on the AC grid. This study showed that the MPC tuning is adequate to our application and that it is robust to changes in the AC power setpoint, grid inertia or frequency droop values.

Based on these results, the following points could be investigated to enhance the response of the system and to prove its performance for more realistic power systems:

• A dierent tuning of the cost function weighting factors could enhance the dynamic response of the system;

• An identication method better adapted to the MPC application could be considered. Indeed, the identied model was found to be optimal for a one-step-ahead prediction. However, the ideal identication method should minimize the error over the prediction horizon used in the MPC in order to give even better results;

• The identication method can be also applied to the DC system and the MMC, if their parameters are not known a priori or if more complicated models are to be used;

• A real-time implementation of the proposed control could conrm the feasibility of MPC in these kinds of applications. In particular, this last item will be the topic of Chapter 6. proposed in Chapter 5 but is about replacing the classical power synchronization control [START_REF] Zhang | Power-synchronization control of gridconnected voltage-source converters[END_REF] of grid-forming by an MPC-based approach. MPC has also been applied

Experimental Validation of The

to MMC control in experimental conditions, but the MPC layer was dedicated to cancelling circulating currents [START_REF] Bocker | Experimental Comparison of Model Predictive Control and Cascaded Control of the Modular Bibliography Multilevel Converter[END_REF] and State of Charge (SOC) balancing of the MMC arms [START_REF] Yuan | A Novel Hybrid Control Strategy for the Energy Storage Modular Multilevel Converters[END_REF], and its application to MMC regarding grid-level considerations was not studied. In conclusion, there are no studies, to our best knowledge, that apply MPC to the energy management level of the MMC operated in grid-forming.

Chapter summary and contents

In this chapter, an experimental validation of the hierarchical MPC is proposed.

This outer control layer comes as an extra control loop whose function is to ensure the AC/DC energy management of an MMC. Contrary to Chapter 5 where averaged models of the MMC were considered, in this chapter, the MPC is tested on a detailed mockup of a 6kVA MMC which contains 60 submodules and interfaces a 400V DC grid and a 120V (line to ground) three-phase AC grid. In addition, the MPC is tested with a recent grid-forming control [START_REF] Harnefors | Referencefeedforward power-synchronization control[END_REF] compatible with the specicities of MMCs compared to two-level VSCs. The MPC solution shows similar tracking performance with the dual PI control solution but appears to have a better disturbance rejection when compared to this dual PI.

The structure of this chapter is as follows: in Section 6.2, the procedure of experimental verication is presented, including a workow to test the controllers in real-time conditions and the mockup used for the experimental validation. In Section 6.3, we present how the GFC was implemented on this existing structure.

In Section 6.4, the hierarchical MPC implementation is detailed. In Section 6.5, the results obtained in PHIL are presented and compared to the dual PI solution and Section 6.6 nally concludes.

Procedure of experimental validation

This part presents the methodology used to test our controller in real-time conditions.

Presentation of the mockup

The setup we will be using is presented in [START_REF] Zama | Linear feedback Dead-Beat Control for Modular Multilevel Converters: Validation under Faults Grid Operation mode[END_REF][START_REF] Zama | High Dynamics Control for MMC Based on Exact Discrete-Time Model With Experimental Validation[END_REF][START_REF] Zama | Modeling and Control of Modular Multilevel Converters (MMCs) for HVDC applications[END_REF] The electrical equivalent scheme of this setup is given in Fig. 6.2 (b). It is important to note that in addition to the dierent components presented above, there are also some passive components to recreate the transmission grid conditions.

In particular, there is a physical inductance L eq between the MMC and the AC power amplier which represents the inductive eect of a power transformer between the AC grid and the MMC. In addition, there is a physical DC capacitance C dc between the MMC and the DC power amplier that represents the equivalent DC grid inertia of this system. The numerical values of the mockup are given in Table 6.1.

Also, it is important to note that with this kind of setup it is possible to consider simulated AC/DC grids in real time instead of perfect sources [START_REF] Qoria | Grid-forming control to achieve a 100% power electronics interfaced power transmission systems[END_REF][START_REF] Taese | Power oscillation damping with virtual capacitance support from modular multilevel converters[END_REF].

However, since this work focuses on the synthesis of an online MPC dedicated to the converter real-time control, we will only consider the system described in Fig. 6.2 Chapter 6. Experimental Validation of The Online Hierarchical MPC applied to a Modular Multilevel Converter in Grid-Forming Figure 6.3: Existing MMC control with the PHIL setup before starting the project [START_REF] Zama | Linear feedback Dead-Beat Control for Modular Multilevel Converters: Validation under Faults Grid Operation mode[END_REF][START_REF] Zama | High Dynamics Control for MMC Based on Exact Discrete-Time Model With Experimental Validation[END_REF].

complex and can be divided into three categories, corresponding to the three control layers:

• AC/DC level : This control layer corresponds to the management of the AC/DC interface and the MMC energy. Since the DC grid is a current source, the MMC, as in Chapter 4 and Chapter 5, has to control the DC voltage. At the same time, the controller must ensure that the internal energy of the MMC, denoted by W g , is well controlled to make sure that the MMC submodules are operable. These two controllers are coupled using a weighting factor block that was introduced in Chapter 4. In addition to these two controllers, there are also the energy sum (W Σ ) and energy dierence (W dif f ) controllers which ensure that the energy is well balanced between the phases and the arms.

These controllers were not considered in Chapter 4 and Chapter 5 because the MMC internal energy was modeled using an average model.

• High level : This control layer corresponds to the AC and DC power and current controls of the MMC. In Fig. 6.3, the two controls are divided in AC and DC sides because the MMC is capable of controlling them independently. For the AC side, this control structure is the most commonly used grid-following control, which uses a PLL to synchronize with the grid, and where the AC current control generates the voltage reference v * Σ sent to the low-level control. For the DC side, the DC current control similarly generates a voltage reference v * dif f sent to the low-level control.

• Low level : This control layer corresponds to the activation and deactivation of the submodules to generate the desired voltage waveform. A Balancing Control Algorithm (BCA) is implemented to specify which submodules to be inserted. This BCA uses measurements of the submodules voltages and activates or desactivates the submodules so that their voltages converge to their mean value.

In our case, we want to implement the grid-forming control (GFC) and the MPC similarly to the control structure presented in Chapter 5. This requires to make changes in the AC/DC-level control and the High-level control.

Choice of hardware conguration

The two hardware congurations considered are given in Fig. 6.4. In Fig. 6.4 (a), the conguration consists in compiling the MPC layer in a dierent hardware ship than the OP4510 where the main MMC control is compiled. This makes the solution more modular but requires to establish communication protocols between the two chips since they need to communicate with each other in real time. In Fig. 6.4 (b), the solution implements the MPC coded in C in an S-function so that all the MMC control is in Matlab/Simulink and is then compiled in a single chip. One of the main advantages of the latter hardware conguration is that it does not require any communication protocol. Furthermore, since the MPC layer remains a 100% local controller dedicated to the converter control, it makes sense to compile all the MMC control into a single hardware system. However, for controls dedicated to the operation of AC/DC systems as a whole and to provide support to the AC grid by coordinating the response of an HVDC system [START_REF] Gonzalez-Torres | Transient stability of power systems with embedded VSC-HVDC links: stability margins analysis and control[END_REF][START_REF] Sánchez | Secondary Model Predictive Control Architecture for VSC-HVDC Networks Interfacing Wind Power[END_REF], other congurations should be investigated.

In the rest of this chapter, the single hardware conguration described in Fig. 6.4 (b) is considered.

Controller validation workow

The proposed setup and workow to demonstrate the feasability of the control using MPC technique is given in Fig. 6.5. As discussed in Section 6.2.3, all the MMC control is compiled on a single hardware system, which is the OP4510 (real-time target in Fig. 6.5). This system includes

• a CPU that executes the code in real time;

• an FPGA board that sends the control inputs (the gate signals) to the MMC and converts the console signals to analog outputs sent to the AC and DC ampliers. The FPGA board also collects the analog inputs from the mockup which are then sent to the CPU of the OP4510.

This workow is separated into two parts:

• Local Host part : it includes all the tasks that are executed on the local host, i.e. the physical computer, and are not executed in real time. This is where the control is developed in the Matlab/Simulink environment, compiled and sent to the target using RT-lab. As specied in the scheme, the control can be implemented and debugged directly in Matlab/Simulink without the need 6.3. Grid-Forming Control (GFC) implementation to pass by RT-lab. In addition, the local host acts as the console which makes it possible to send reference changes or new setpoints to the mockup and to display the measurements online.

• Real-Time Simulation part : it includes all the operations executed in real time, such as the interface between the FPGA of the OP4510 and the mockup and between the CPU of the OP4510, where the control is compiled, and its FPGA. Also this is where the MMC control is tested in real time since the MMC mockup and the AC/DC ampliers directly receive orders from the FGPA board of the OP4510.

This setup makes it possible to test any MMC control developed in Matlab/Simulink.

As detailed in Section 6.2.3, since all the tested controls will be sent to the OP4510 target, this setup will be considered in the rest of this chapter.

Grid-Forming Control (GFC) implementation

This grid-forming control is implemented taking into account the complexity of existing MMC control detailed in Fig. 6.3. The whole MMC control structure with the grid-forming control is updated as described in Fig. 6.7, which is known as the direct voltage control. The outer loop corresponds to the classical outer loop control used in the previous chapters. For the inner loop, there is a saturation block SAT which is used to saturate the ltered measure of the dq-current (i dq g ), denoted by i dq * g .

The damping resistor term R a is used to damp the variations in i dq g . The decoupling term jωL is used to compensate for the physical inductance in the MMC and thus to control the voltage at the PCC instead of the voltage at the MMC terminals.

However, in our case, since an innite bus is connected to the PCC, as described in Fig. 6.2 (b), it is not possible to control the PCC voltage, thus jωL = 0.

The above structure is based on recent works on grid-forming control where there is no inner voltage loop [START_REF] Harnefors | Referencefeedforward power-synchronization control[END_REF][START_REF] Harnefors | A Universal Controller for Grid-Connected Voltage-Source Converters[END_REF]. Indeed, because of the physical LC lter, this type of cascaded control, presented in Chapter 2, is better suited for two-level VSCs than for MMCs [START_REF] Qoria | Grid-forming control to achieve a 100% power electronics interfaced power transmission systems[END_REF]. However, there has been no consensus yet in the literature to dene a control structure of the MMC in grid-forming mode and some recent works still use the cascaded structure when considering the capacitive eects of the AC grid to design the inner voltage loop [START_REF] Sanchez-Sanchez | The role of the internal energy in MMCs operating in grid-forming mode[END_REF][START_REF] Arco | Operation of a Modular Multilevel Converter Controlled as a Virtual Synchronous Machine[END_REF].

The parameter values considered for this control are given in Table 6.2. In order to validate the GFC in Fig. 6.6, we investigate the activation of this control when the MMC is operating in a steady state in grid-following mode. To achieve such a steady Chapter 6. Experimental Validation of The Online Hierarchical MPC applied to a Modular Multilevel Converter in Grid-Forming state, we rst carry out the start-up sequence where the MMC submodules and the DC grid are energized by the AC grid. Indeed, since the grid-forming control relies on the synchronization based on the variations of active power, it has to start from an equilibrium point so that it remains synchronized. In consequence, we keep the startup sequence developed in [START_REF] Zama | Modeling and Control of Modular Multilevel Converters (MMCs) for HVDC applications[END_REF] in grid-following mode and switch to the gridforming mode when v dc = v * dc and W g = W * g . In order to smoothen the transients caused by the switching, we use the estimated angle from the PLL to initialize the three GFC integrators, contained in the low-pass lters and in the outer loop as described in Fig. 6.6. We performed two types of experiments: i ) switching from grid-following to grid-forming when i L = 0; ii ) switching from grid-following to grid-forming when i L = 0.3 p.u. The results given in Fig. 6.8 and Fig. 6.9 show that in both cases, the MMC manages to synchronize with the grid thanks to the power synchonization loop. However, even though the GFC activation does not aect the DC voltage and DC current, we observe a larger MMC internal energy transient when i L = 0.3 p.u.

MPC implementation

Also, we see that in this case the MMC aborbs some reactive power from the AC grid when the GFC is activated. This is because the converter, now controlled as a voltage source, adjusts its reactive power injection or absorption in order to regulate the AC voltage.

To summarize, these results are really encouraging because they demonstrate that we are capable of activating a grid-forming control of a physical mockup in real-time conditions.

6.4. MPC implementation 6.4.1. Choice of the software solution

The MPC was implemented in Chapter 5 using the Matlab-based tool YALMIP [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF] which exclusively works in a Matlab environment. In this chapter, we implement the MPC in C which, as a programming language close to the machine target language, can be easily complied into industrial systems. In consequence, we need to nd another software solution. In order for the solution to be compatible with dierent hardware systems, we consider the solutions based on C code exclusively.

In fact, there are several free software solutions [START_REF] Mattingley | CVXGEN: a code generator for embedded convex optimization[END_REF][START_REF] Zometa | μAO-MPC: a Free Code Generation Tool for Embedded Real-Time Linear Model Predictive Control[END_REF] to generate C-based linear MPC. There has been also some academic work to compare dierent linear MPC implementations in C for industrial processes [START_REF] Ibanez | Implementation of optimization-based controllers for industrial processes[END_REF]. In our case, we decided to use the μao-MPC solution [START_REF] Zometa | μAO-MPC: a Free Code Generation Tool for Embedded Real-Time Linear Model Predictive Control[END_REF] because, as a solution initially designed for microcontroller, it is highly portable and is an open-source software. .9: GFC activation on-load, when i L = 0.3 p.u.

μao-MPC embedded QP solver

This section presents the embedded QP solver of μao-MPC. In the literature, most of the solvers use condensed forms of the Quadratic Problem (QP) to solve it more eciently [START_REF] Jerez | A Condensed and Sparse QP Formulation for Predictive Control[END_REF]. This condensed form of the cost function, denoted by J QP , can be expressed as:

J QP (θ, x) = 1 2 θ T Hθ + h(x) T θ (6.1)
Chapter 6. Experimental Validation of The Online Hierarchical MPC applied to a Modular Multilevel Converter in Grid-Forming

The aim of the QP solver is to minimize this cost function:

min θ∈Θ J QP (θ, x) (6.2) subject to z(x) ≤ Eθ ≤ z(x) (6.3)
where θ is the control input vector, x is the state variable vector, H is the Hessian matrix of the function describing the evolution of the state variables θ, h is the gradient vector, E is a constant matrix to express the state constraints, z(x) and z(x) are respectively the state minimum and maximum constraints and

Θ = {θ|θ ≤ θ ≤ θ} (6.4)
where θ and θ are respectively the control input minimum and maximum constraints.

In order for the problem (6.2) to have a global minimum, the problem must be convex, i.e. the Hessian matrix must be positive-denite. In the case of μao-MPC, the solver was designed to only rely on simple mathematical operations to solve the QP problem dened in (6.2) and (6.3).This feature makes it compatible with most of the microcontrollers and hardware systems used for real-time controllers.

This solver uses the augmented Lagrangian method and Nesterov's fast gradient method [START_REF] Zometa | μAO-MPC: a Free Code Generation Tool for Embedded Real-Time Linear Model Predictive Control[END_REF], which are presented respectively in this part.

Augmented Lagrangian Method (ALM)

This method is used to convert the constrained problem into a serie of unconstrained optimization problems that are much simpler to solve:

min θ∈Θ J u (θ, λ * , x) (6.5)
where J u , called the condensed cost function, is the equivalent unconstrainted cost function of the optimization under constraints problem (6.2) (6.3) such that:

J u (θ, λ, x) = 1 2 θ T Hθ + h T θ + T (θ, λ, x) (6.6)
where T is dened as:

T (θ, λ, x) = min s λ T s + µ 2 s 2 2 (6.7) subject to z(x) ≤ Eθ + s ≤ z(x) (6.8)
where µ is the penalty parameter of the augmented Lagrangian method. Let λ * be a Lagrange multiplier that minimizes the condensed cost function J u . The augmented Lagrangian method is used to estimate this λ * by successive iterations.

λ i+1 = S(θ * i , λ i , x) (6.9)
where θ * i is the solution minimizing (6.5) but with λ * replaced by λ i and i is the iteration index:

θ * i = arg min θ∈Θ J u (θ, λ i , x) (6.10) and S(θ, λ, x) = min(λ(θ, λ, x), 0) + min( λ(θ, λ, x), 0)

(6.11) where λ(θ, λ, x) =λ + µ × (Eθ -z(x)) (6.12) λ(θ, λ, x) =λ + µ × (Eθ -z(x)) (6.13)
In order to take into account the constraints in (6.3) represented by the term T (θ, λ, x), we need to estimate λ i+1 , and in consequence, to have a good estimation of the optimal control input θ * i . This estimation is obtained by using the Fast Gradient Method (FGM) described in the next subsection.

Fast Gradient Method (FGM)

The fast gradient method is a technique to estimate the local minimum of a convex function based on the observation of its gradient. In our case, we assume that J u is a convex function, so we can apply the FGM. The gradient step projected onto the set Θ can then be calculated as:

G Θ (w, λ i , x) = P Θ (w - 1 L ∇J u (w, λ i , x)) (6.14)
where, knowing that θ and θ are respectively the control input minimum and maximum constraints:

P Θ (α) = max(θ, min( θ, α)), (6.15)
and L = H + µE T E is assumed to be the Lipschitz constant for ∇J u ,the gradient of J u . More details about these assumptions can be found in [START_REF] Zometa | μAO-MPC: a Free Code Generation Tool for Embedded Real-Time Linear Model Predictive Control[END_REF]. We can then calculate the approximation of the optimum θ j+1 and compute the extra step w used for the iterative loop:

θ j+1 = G Θ (w, λ i , x) (6.16) w = θ j+1 + ν × (θ j+1 -θ j ) ( 6
ν = ( √ L -φ)( √ L + φ) -1 , (6.18)
where φ is a strong convexity parameter for J u given by the smallest eigenvalue of H (the Hessian of the QP problem) [START_REF] Zometa | μAO-MPC: a Free Code Generation Tool for Embedded Real-Time Linear Model Predictive Control[END_REF]. As a reminder, a function f is said to be strongly convex with respect to a parameter m if and only if the function g,

dened such that g : x → f (x) - m 2 x 2 (6.19)
is convex. More information about this denition can be found in [START_REF] Veliov | Gradient Methods on Strongly Convex Feasible Sets and Optimal Control of Ane Systems[END_REF].

Finally, it is possible to calculate u j+1 which can then be used in the ALM described in Section 6.4.2.1.

Structure and tuning of the solver

The global structure of the solver combines the ALM and the FGM, and is described in Fig. 6.10. There are three tuning parameters for this solver:

• ex iter : the number of external interations, which corresponds to the number of times the ALM algorithm is called per MPC step;

• in iter : the number of internal interations, which corresponds to the number of times the FGM algorithm is called per ALM step;

• µ: the penalty parameter of the augmented Lagrangian method, dened in (6.7), which corresponds to the weight put on the state constraints dened in the QP optimization problem (6.2) (6.3). In practice, the developers of μao-MPC have set the software so that it nds an appropriate value of µ depending on the MPC problem.

The solver must do at least one external iteration and one internal iteration.

However, since the ALM is used to make the unconstrained problem converge, while taking into account the term T (θ, λ, x), the number of external iterations ex iter is only useful when we consider state constraints, which is our case for the application the MMC control presented in Chapter 5.

Validation of μao-MPC solution

In this section, we try to validate μao-MPC before its implementation on the mockup. We decided to implement the linear MPC using the C-based μao-MPC [START_REF] Zometa | μAO-MPC: a Free Code Generation Tool for Embedded Real-Time Linear Model Predictive Control[END_REF] and a Matlab/Simulink S-function to compare the controller performances with the well-known Matlab-based YALMIP [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF]:

6.4. MPC implementation Figure 6.10: Flowchart of the used embedded QP solver. Scheme based on [START_REF] Zometa | μAO-MPC: a Free Code Generation Tool for Embedded Real-Time Linear Model Predictive Control[END_REF].

• For the YALMIP solution, it was already implemented in the Simulink examples available online [START_REF] Lofberg | Simulink models with yalmip components[END_REF]. This solution relies on the Matlab interpreted function block [219] that forces the function to be computed exclusively in Matlab. In consequence, it is impossible to use it for real-time applications.

• For the μao-MPC solution, we generated the C-code of the control oine and then embedded it in an S-function [START_REF]S-function examples[END_REF], which makes it possible to compile C-code in a Matlab/Simulink environment.

The two tools use dierent solvers to calculate the optimal control solution. In YALMIP, we use Matlab's quadprog algorithm [START_REF]Quadratic programming[END_REF] introduced in Chapter 5

whereas for μao-MPC, there is an dedicated C-coded solver that relies only on matrix-vector operations. This latter solver is based on the augmented Lagrangian method (also called method of multipliers) and Nesterov's fast gradient method [START_REF] Zometa | μAO-MPC: a Free Code Generation Tool for Embedded Real-Time Linear Model Predictive Control[END_REF] presented in Fig. 6.10, with in iter = 5 and ex iter = 1.

The test system under consideration [START_REF] Lofberg | Simulink models with yalmip components[END_REF] is:

Σ example = ẋ = Ax + Bu y = Cx + Du (6.

20)

where:

A = 0 0 1 0 , B = 1 0 , C = 0 1 , D = 0. (6.21)
And the cost function J is dened as: YALMIP [START_REF] Lofberg | YALMIP : a toolbox for modeling and optimization in MATLAB[END_REF] versus μao-MPC [START_REF] Zometa | μAO-MPC: a Free Code Generation Tool for Embedded Real-Time Linear Model Predictive Control[END_REF].

J = (y * -y) T • Q • (y * -y) + u T • R • u (6.
The optimal control input is calculated based on minimizing J such that:

∂J ∂u = 0 (6.24) subject to -5 ≤ u ≤ 5 (6.25)
The results following the change of the reference y using a step function are given in Fig. 6.11. In Fig. 6.11 (a), the two MPCs have similar parameters and are supposed to give similar results. This is conrmed by the simulation results

where only slight dierences between the two implementations are observed. These dierences are mainly due to the dierence of QP solvers. The equations of the MPC must be of low order in order to be compatible with the real-time constraints. In Chapter 5, we had a 6 th -order model where 4 states were used to capture the AC grid dynamics. In this chapter, since the AC grid is modeled as a perfect voltage source, these dynamics are neglected, which means 6.5. Experimental results and discussions that the MPC can use a second order model as:

Σ c =    ẋc = Ax c + Bu c y c = Cx c + Du c (6.26) where x c = x dc x Σ T ∈ R 2 , u c = p * dc p * ac T ∈ R 2 , y c = x c , x dc = v 2 dc , x Σ = v 2 Σ
(v Σ being the average of the six arm voltages of the MMC),

A = 0 0 0 0 , B = -2 C dc 0 2 Ceq -2 Ceq , C = I 2 , D = 0. (6.27)
This system is discretized to be implemented in the MPC. The sampling time is T s = 0.015s and the prediction horizon is N = 6. The MPC takes also into account the following state and control input constraints: 0.8 p.u. < v The cost function J that is considered is dened as:

J = N i=0 (y * -y(k|k + i)) T • Q • (y * -y(k|k + i)) + ∆y(k|k + i) T • S • ∆y(k|k + i) (6.32)
with the normalized weighting factor matrices Q and T being:

Q = 1 0 0 10 , S = 0.1 0 0 0 (6.33)
In practice, the technique of the embedded integrator is used to build the augmented model of the system that is compiled into the target [START_REF] Wang | Model Predictive Control System Design and Implementation Using MATLAB®, ser. Advances in Industrial Control[END_REF]. For the solver, it appears that the higher ex iter and in iter , the better its performances. However, because of computational burden, we had to limit the maximum number of iterations, which of course also depends on the prediction horizon N and the time step T s . For such conguration, we found that in iter = 6 and ex iter = 4 yield the best compromise. 2), we want to assess the performances of the MPC. Also, in order to validate the control executed with the MMC mockup, a similar experiment is simulated using a Software-in-Loop (SIL) system, where the embedded code is compiled on an Opal RT 5700 target (OP5700). Contrary to the workow presented in Fig. 6.5 for the PHIL experiment, SIL relies on two cores of the OP5700 system to simulate the experiment in real time. One core is dedicated to simulate a detailed model of the mockup and another core is used to execute the same control that is embedded in the OP4510 system in PHIL. More information about the detailed model of the mockup and its validation can be found in [START_REF] Zama | Modeling and Control of Modular Multilevel Converters (MMCs) for HVDC applications[END_REF].

For the AC and DC disturbances, we dene them as follows:

• Case 3 is a DC power step which can represent the loss of parts of the generation from the DC grid or a DC load step;

• Case 4 is a negative step in the voltage magnitude which can represent a voltage sag or a short-circuit happening somewhere in the grid;

• Case 5 is a phase jump that can be used to model grid disturbances such as a line tripping. This type of disturbance is also used to characterize voltage sags in some works [START_REF] Bollen | Behavior of AC and DC drives during voltage sags with phase-angle jump and three-phase unbalance[END_REF][START_REF] Andrade | Fast Phase Angle Jump Estimation to Improve the Convergence Time of the GDSC-PLL[END_REF].

Experimental results and discussions

The rest of this section compares the two control solutions with the PHIL experiment.

Results regarding tracking

The results regarding the capability of both the dual PI and the MPC to control the internal energy W g and the DC voltage v dc are given in Fig. 6.14. First, we note that there is a small static error in PHIL between the reference of MMC internal energy W * g and the measurement of W g , which is not observed with the dual PI.

In addition, the MMC control is exactly the same between the SIL and the PHIL experiments, whereas this static error is only observed for the PHIL case. This error is dicult to interpret:

• it does not seem to be related to an error in the MPC code itself since it is the same code that is compiled into the OP5700 for the SIL experiment and into the OP4510 for the PHIL experiment. Yet, there is no steady-state error in SIL whereas there is a less than 1% error in PHIL;

• it cannot be caused by an oset or noise of the mockup sensors because the dual PI control solution in the PHIL experiment does not have any static error while it uses the same measurements for the control.

Despite the static error of less than 1%, the two sets of results are still quite similar, which means we can argue that the PHIL setup is valid against the SIL results. This oset could be caused by the numerical error of the QP solver combined with the sensor noise of the mockup. Correcting this oset will be one of the future tasks in the development of this PHIL setup.

Also, it is important to note that there is a small dierence between the DC voltage dynamic response of the SIL and the PHIL experiments. In both Figs. 6.14 (a) and 6.14 (e), the DC voltage response with the MPC exhibits an overshoot but the time response is slightly slower in the SIL experiment. On the other hand, the response of the dual PI in PHIL and SIL appear to be similar. For the energy tracking however, the dynamic response of the MPC in SIL (Fig. 6.14 (d)) and in PHIL (Fig. 6.14 (h)) are almost identical, except the static error.

Additional experimental data obtained with the PHIL setup for Case 1 and Case 2 are given in Fig. 6.15 and Fig. 6.16 respectively. We observe that the AC power and the DC current are well controlled. Also, the capacitor voltages of the MMC submodules are within a ±5% envelop around the nominal setpoint. It is important to note though that this distribution is the result of the BCA in the low-level control, since the MPC is only responsible for controlling the internal energy stored in the converter. This means that the total control structure is ecient because all the measurements reach acceptable values.

Results regarding disturbance rejection

Now that the MMC control given in Fig. 6.13 has been validated with a SIL simulation, we compare in this part the two controllers in terms of disturbance rejection, in PHIL only. The results for the disturbance rejection cases (Case 3 to Case 5) are given in Fig. 6.17. For Case 3 with the DC current disturbance, the MPC and the dual PI solutions give comparable results, even though the dual PI appears to have a smoother transient regarding the DC voltage. For the voltage magnitude negative step (Case 4), we see that MPC and dual PI give similar responses, as shown in Figs. 6.17 (c) and 6.17 To summarize, the results presented in Fig. 6.17 show that on the one hand, the MPC does not let v dc and W g track exactly their references but these variables respect the acceptable limits. On the other hand, MPC shows better performances regarding disturbance rejection. This tendency is really encouraging and justies the use of MPC-based solutions for the MMC control since in real-life conditions the AC/DC system encounters that will probably be handled by the MMC in the future.

The state variable responses with the MPC for Cases 3 to 5 are described in Figs. 6.18, 6.19 and 6.20 respectively. It is interesting to note that since the MMC is operated in grid-forming mode, the rst two disturbances generate some reactive power absorption/injection so that the MMC output voltage is maintained at 1 p.u.

Regarding Case 5, we observe active and reactive power transient oscillations with abnormally large amplitudes but the system comes back to a normal state in less than 500 ms. The main contributions of this chapter can be listed as follows:

• implementation of a recent grid-forming control [START_REF] Harnefors | Referencefeedforward power-synchronization control[END_REF][START_REF] Harnefors | A Universal Controller for Grid-Connected Voltage-Source Converters[END_REF] to operate the MMC as a voltage source in grid-connected mode;

• demonstration of the feasibility of the MPC layer to control a power converter operated in grid-forming in real time;

• use of an averaged model of the MMC, presented in Chapter 5, in the MPC algorithm to generate the optimal control inputs sent to a real mockup of an MMC with its complete set of controls.

• the experimental validation of the better performance of MPC regarding disturbance rejection than the state-of-the-art dual PI solution.

However, the MPC solution could still be enhanced by acting on the solver of the optimization, which must be embedded in the target as well, since the current solver shows some limits when we tried to enhance its accuracy. In future work, the following items could be investigated in order to strengthen the obtained results: • In its current form, the MPC only considers constant constraints, as detailed in (6.29). However, recent works suggested that the limits of the MMC internal energy depend on the system setpoints and the energy levels of each arm.

These limits can be calculated using analytical expressions [START_REF] Shinoda | Analysis of the Lower Limit of Allowable Energy in Modular Multilevel Converters[END_REF] and could be integrated in the MPC and would be an added value in the choice of MPCbased control versus classical PI solutions;

• In this chapter, the model of the DC grid and the MMC internal energy is based on averaged models. However, the dynamics of the MMC are more complex and a better knowledge of these dynamics could enhance the performances of the MPC. In consequence, it would be interesting to investigate the use of system identication techniques to identify an equivalent model of the MMC which will then be used by the MPC;

• As stated in Section 6.2.1, the AC and DC ampliers of the mockup were operated as perfect sources in this setup. However, it is possible to emulate multi-machine AC grids [START_REF] Qoria | Grid-forming control to achieve a 100% power electronics interfaced power transmission systems[END_REF] or multi-terminal DC grids [START_REF] Taese | Power oscillation damping with virtual capacitance support from modular multilevel converters[END_REF] would make it possible to study the impact of the proposed MMC control in even more realistic conditions;

• The MPC-based MMC control proposed in this PhD thesis was also quite suited for providing inertial support the AC grid, as shown in Chapter 5.

However, since the current MMC mockup represents a standard MMC, it was not possible to analyze how its internal energy could support the AC grid frequency. In a long-term range, we could consider to apply this controller to an MMC with a Large Buer Storage (LBS) that is designed for AC grid related ancillary services [START_REF] Errigo | Power converters with integrated energy storage systems for high voltage direct current systems[END_REF].

C h a p t e r This thesis focused on the application of grid-forming concept to VSCs and in particular MMCs in the context of power systems with high penetration of renewables, which are low inertia and weak grid systems. The conclusions of each chapter of the thesis is recalled in the next paragraphs.

In Chapter 2, a compehensive overview of the recent control strategies to operate a VSC are presented. In particular, the so-called grid-following and grid-forming controls are compared in terms of their behavior from the converter and the grid perspectives. The denition of these two controls as well as their contributions to grid services from the literature are assessed and discussed in this chapter.

In Chapter 3, the main VSC control strategies, namely grid-following/gridfeeding, grid-supporting and grid-forming, are compared using an elementary benchmark that models a hybrid grid with a VSC, an SG and a load, interconnected by a variable AC transmission line. Using modal analysis tools, the maximum allowable penetration limit of power electronics is assessed in the dierent cases and shows the advantage of grid-forming control for introducing more power electronics in the grid. However, because of its SG-like behavior, some interactions between the generator's AVR and the grid-forming control are expected and should be taken care of.

In Chapter 4, the grid-forming control is applied to HVDC systems and more specically to onshore MMCs in the case of oshore wind farm interconnection.

After having highlighted the stability issues due to the interactions between the DC voltage and the grid-forming controls, we propose a methodology to determine the energy of the MMC submodules to support the AC grid inertia. The associated control, named virtual rotor control, is proposed, and its tuning and the application to multi-machine systems are discussed. While the compatibility of this contro with grid-forming controlled MMC and neighboring SGs is proved, the problems related to the inter-area oscillations and to the support of the DC grid voltage remain an open topic.

In Chapter 5, a Multi-Input Multi-Output (MIMO) approach is proposed to support the AC and DC grids of the MMC using an MPC solution to replace the DC voltage and the MMC internal energy control loops. The MPC is implemented using an averaged model of the MMC, a simplied model of the DC grid and an identied model of the AC grid. This control solution provides better disturbance rejection and a better handling of the MMC internal energy than the classical dual PI control structure. Also, its robustness regarding setpoint changes as well as parameter uncertainties is veried with dierent test cases. Nevertheless, the applicability of such advanced control techniques to detailed models of MMCs as well as its compatibility with real-life hardware systems still requires further investigation.

In Chapter 6, the linear MPC, as presented in Chapter 5, is applied to a physical mockup of an MMC. This experimental validation is performed using a Power

Hardware-in-the-Loop (P-HIL) benchmark to prove the industrial feasibility of such concept. The MPC is implemented in C-code and coupled with the rest of the MMC grid-forming control implemented in Matlab/Simulink. All the control is compiled using Opal-RT software onto an OP4510 hardware system to control an MMC 7.1. General conclusions mockup of 6kVA and 60 submodules connected to AC and DC power ampliers.

The proposed control shows similar tracking performances when compared to the dual PI structure but a better disturbance rejection. This conrms the interest of MPC-based techniques for MMC operation in the future.

Concluding remarks

To synthetize the main conclusions of this PhD thesis, the present work has proven the interest to have more grid-connected VSCs operated in grid-forming mode.

Indeed, this grid-forming feature avoids weak grid problems and can even support islanding from the grid. However, the main function of a converter is to inject power to the grid and this control brings several problems such as: DC voltage mitigation, internal energy control, AC/DC interactions, etc. due to the relatively slow dynamics of the control and its poor power tracking. After having discussed the need of slowing down the DC-voltage controller to solve this problem, this PhD thesis investigated the application of grid-forming to MMCs and developed a methodology based on MPC to solve the problem. The application of MPC to the MMC-level control was demonstrated with a physical MMC mockup which makes a contribution towards applying this kind of advanced control solutions in industrial power electronics systems. In the case of the MPC proposed in this thesis, this control could be apply to real-life MMCs regardless of their converter control modes.

Scientic contributions

The scientic contributions of this dissertation can be summarized as follows:

• Chapter 2

• Overview of the existing VSC control strategies to support the AC grid.

• Comparative analysis of the behavior of the grid-following and gridfollowing VSC controls from the converter and the grid perspectives.

• Chapter 3

• Development of an elementary benchmark to study the impacts of the power ratio (the ratio of the VSC rated power to the rating of the whole system), the electrical distance and the control mode in the stability limits of a system comprising a VSC and an SG.

• Modal analysis showing the importance of grid-forming control to allow more power electronics to be integrated in the grid.

• Validation of the conclusions using time-domain simulations in a multimachine system.

• Chapter 4

• Analysis of the impacts of the grid-forming control upon the control of the MMC internal energy and the DC voltage.

• Proposition of a methodology with its associated control design to calculate the amount of energy from the MMC submodules to provide inertia-like response to the AC grid.

• Validation of the method and control using time-domain simulations in a modied version of the Nordic 44 system that includes 80 SGs.

• Chapter 5

• Proposition of a MIMO-based MPC to replace the MMC internal energy and DC voltage controllers.

• Identication of the AC grid model from the MMC perspective using system identication methods.

• Validation of the proposed MPC when compared to the classical dual PI control.

• Analysis of the robustness of the proposed control in terms of setpoint changes and parameter uncertainties.

• Chapter 6

• Practical implementation of grid-forming control using recent controls from the literature.

• Implementation of the MPC in C to make it compatible with hardware systems.

• Validation of the MPC proposed in Chapter 4 in P-HIL using a 6kVA mockup of an MMC and two AC and DC power ampliers.

Perspectives 7.2.1. Integration of the DC source dynamics

In the rst chapters of this work, the DC side was considered as an ideal voltage source. This is a well-known assumption to study the interactions between a VSC controlled in grid-forming and the AC grid but it cannot integrate all the constraints of future AC/DC grids, as shown in the later chapters of this thesis. In particular, in these chapters, the DC grid was modeled as a constant power source and a DC capacitance which takes into account the DC-voltage dynamics. However, it does 7.2. Perspectives not integrate all the dynamics of a realistic HVDC grid and its associated sources and loads. Indeed, if the DC grid is powered by a renewable energy source [START_REF] García-Ceballos | Integration of distributed energy resource models in the vsc control for microgrid applications[END_REF], its response to AC grid disturbances would depend on the dynamics of wind [START_REF] Boukhezzar | Nonlinear Control of a Variable-Speed Wind Turbine Using a Two-Mass Model[END_REF][START_REF] Sandoval-Moreno | Observer-based maximum power tracking in wind turbines with only generator speed measurement[END_REF] and solar PV [START_REF] Lopez-Santos | Sliding-mode control of a transformer-less dual-stage grid-connected Bibliography photovoltaic micro-inverter[END_REF] systems, including the dynamics of the power source itself and their associated controls. Moreover, if the DC grid is powered by a storage system such as a lithium/ion battery [START_REF] Markovic | Optimal sizing and tuning of storage capacity for fast frequency control in lowinertia systems[END_REF] or supercapacitors [172], its response would depend on the state of the charge of this system as well. In the future, such constraints should in consequence also be taken into account in the control and simulation of power electronics converters.

Articial Intelligence (AI) techniques for MPC

In Chapter 6, the solution of embedding an online optimal control in a real-time hardware system was studied by using μao-mpc code generation [START_REF] Zometa | μAO-MPC: a Free Code Generation Tool for Embedded Real-Time Linear Model Predictive Control[END_REF]. However, the computational limits of the CPU impose constraints on the solver number of iterations, causing potential convergence problems as well as small osets in the results obtained with the mockup when compared to the simulation data. With the growing role of articial intelligence (AI) based techniques in the domain of power electronics [START_REF] Cao | Reinforcement Learning and Its Applications in Modern Power and Energy Systems: A Review[END_REF][START_REF] Zhao | An Overview of Articial Intelligence Applications for Power Electronics[END_REF], it would be possible in our case to use machine learning techniques to determine an oine optimal solution of the MPC optimization problem. This solution would be possible by compiling data from the simulation le and using regression techniques [START_REF] Georges | A Simple Machine Learning Technique for Model Predictive Control[END_REF]. This approach could be much lighter in terms of online computational burden for the controller than the solution with the embedded solver in real time.

Transient stability analysis of MMCs

In Chapter 3, we performed a small-signal stability analysis of a hybrid system with a VSC and an SG. The conclusions of this analysis only stand for small disturbances and do not consider large ones, such as short circuits or generator loss for instance. In these cases, it is worth investigating the transient stability of the system. Contrary to SGs whose electrical angle is a physical variable linked to the rotor speed, the VSC synchronizing capability depends on the calculation of the electrical angle, which is given by a PLL or the power controller depending on the control mode. The VSC has a better capacity than SGs to remain synchronized with the grid because the angle is a digital gure that is not directly coupled to the physical conversion devices [START_REF] Pico | Transient Stability Assessment of Multi-Machine Multi-Converter Power Systems[END_REF].

However, it is still possible to observe a Loss of Synchronism (LoS) in the case of large faults in grid-following as well as in grid-forming [START_REF] Taul | An Overview of Assessment Methods for Synchronization Stability of Grid-Connected Converters Under Severe Symmetrical Grid Faults[END_REF]. While there have been some works on the transient stability enhancement of two-level VSCs [START_REF] Wang | Grid-Synchronization Stability of Converter-Based Resources An Overview[END_REF], the transient stability of an MMC still needs to be assessed. In particular, the absence of LC lter makes the current limitation algorithm, necessary in order to control the transient states of the MMC, more challenging. There are some recent works in developing current limitation controls for MMCs [START_REF] Qoria | Grid-forming control to achieve a 100% power electronics interfaced power transmission systems[END_REF][START_REF] Freytes | Grid-Forming Control with Current Limitation for MMC under Unbalanced Fault Ride-Through[END_REF] but their role in the transient stability analysis of hybrid grids still needs to be assessed.

Coordinated control of multiple converters

In this PhD thesis, we focused on the converter control of a VSC in a hybrid grid but without considering the opportunity of coordinating these dierent PE-based interfaces at a system level. Indeed, with the high controllability of these devices and adequate communication tools, it is possible to coordinate the action of several converters with various and decoupled DC sources [START_REF] Markovic | LQR-based adaptive virtual synchronous machine for power systems with high inverter penetration[END_REF], or from a multi-terminal DC (MTDC) grid embedded in an AC grid [START_REF] Gonzalez-Torres | A novel distributed supplementary control of multi-terminal vschvdc grids for rotor angle stability enhancement of ac/dc systems[END_REF] connected to a single synchronous AC grid. It is also possible to coordinate the support action of the dierent converters in the case of an MTDC interfacing several AC grids [START_REF] Taese | Power oscillation damping with virtual capacitance support from modular multilevel converters[END_REF][START_REF] Rodriguez-Cabero | Virtual Friction for Oscillation Damping and Inertia Sharing from Multi-Terminal VSC-HVDC Grids[END_REF] to provide a more where t r,5% is the desired settling time to reach the steady-state value in a ±5%

bandwidth.

In addition, it is possible to add a zero-compensation term in the PI structure, which can be used to make H(s) and G F (s) match perfectly.

B.3. Main applications and dierences from state-feedback pole placement method

It is important to note that the vector control, in the dq-frame, makes it possible to have time-invariant variables for the AC currents and voltages instead of sine waveforms [2]. In consequence, the tuning presented in this appendix can be used to tune the controller in the dq frame.

Moreover, in the control systems community, the pole placement method refers to a technique that uses state feedback action to place the poles of the whole closedloop system [235]. While in the case presented in the appendix, a PI controller can only place the poles of a second-order closed-loop system, this state-feedback pole placement method is more general. However, it requires an accurate model of the system to be controlled, which is in practice uncommon. On the other hand, with the PI controller, the structure of a control system is such that the controller received the control error between an output (the variable to be controlled) and its setpoint. The initial power ow is given below in the base system where U base = 400kV, S base = 100MVA and f base = 60Hz. The inertia constants are H g 1 = H g 2 = 6.5s and H g 3 = H g 4 = 6.175s. More details about the simulation parameters can be found in [START_REF] Kundur | Power System Stability and Control[END_REF]. The synchronous generator model is based on a round rotor model from the iPSL library [START_REF] Vanfretti | iTesla Power Systems Library (iPSL): A Modelica library for phasor time-domain simulations[END_REF]. The AVR is modeled with an excitation system SEXS and coupled with a type II PSS. The governor is a G1 type model.

  photovoltaïques résidentiels se développe rapidement. C'est également le cas au niveau du réseau de transport où d'importantes sources d'énergie telles que les Fermes Eoliennes Oshore (FEO) commencent à être interconnectées au réseau grâce à la technologie de Haute Tension à Courant Continu (HVDC). Cette thèse de doctorat s'adresse à certains des futurs dés du réseau dans lequel les Générateurs Synchrones (GS) des centrales thermiques traditionnelles sont remplacés par des sources à base d'EP. Ce travail se concentre plus spéciquement sur la technologie des Convertisseurs Source de Tension (VSC) et de leurs commandes associées. En outre, le contrôle de ces convertisseurs et leurs impacts au niveau de réseaux hybrides, où l'EP et les GS coexistent, sont étudiés.

  pour améliorer le support des réseaux AC et DC qu'interconnecte le MMC. Cette solution, basée sur la synthèse d'une Commande Prédictive à base de Modèle (MPC), montre une meilleure gestion de l'énergie interne du MMC que la structure classique qui utilise des correcteurs PI et qui repose sur des hypothèses de découplages entre les grandeurs AC et DC. La robustesse du MPC face aux variations paramétriques du système est également démontrée par simulation. Enn, dans un dernier temps, la stratégie basée sur le MPC proposée pour opérer le MMC est validée en utilisant un banc expérimental qui inclut une maquette physique de 6 kVA d'un MMC de 60 sous-modules. Le contrôle grid-forming ainsi que le MPC proposé sont implémentés dans une cible Opal RT et testés avec les contraintes temps réel et permettent de valider l'approche développée dans ces travaux de thèse. viii Résumé de la thèse en français Chapitre 1: Introduction générale

  Aujourd'hui, cette intégration rapide de sources renouvelables conduit à une augmentation signicative du nombre d'interface à base d'EP dans le réseau électrique. Leur présence s'illustre au niveau des réseaux de distribution avec notamment de nouvelles charges conçues à base d'EP mais également à l'échelle de la très haute tension où de nouvelles installations Haute Tension à Courant ix Résumé de la thèse en français Continu (HVDC en anglais) voient le jour, comme illustré sur la carte donnée dans la gure 1. L'objectif de ces projets HVDC est de permettre d'améliorer le maillage du réseau AC existant et ainsi d'augmenter les échanges de puissance entre les pays. Cela serait d'autant plus utile dans les réseaux à forte proportion d'énergie renouvelable où l'intermittence de ces sources pourrait être partiellement compensée par une augmentation des échanges. Egalement, ces liaisons HVDC permettraient d'intégrer plus de sources renouvelables et notamment les fermes éoliennes oshore dans le réseau.

Figure 1 :

 1 Figure 1: Carte de l'Europe incluant les projets VSC-HVDC existants et futurs 1 .

Figure 2 :Figure 3 :

 23 Figure 2: Circuit électrique équivalent d'un VSC interfaçant un réseau AC et DC.

Figure 4 :

 4 Figure 4: Commande cascadée du VSC en mode source de tension.

Figure 5 :

 5 Figure 5: Réseau de référence élémentaire considéré pour l'étude.

Figure 6 :Chapitre 4 :

 64 Figure 6: Ratio de puissance maximal admissible pour diérentes valeurs de x eq .

Figure 7 : 4 :Figure 8 :

 748 Figure 7: Inuence de k vr sur le nadir et le RoCoF à la suite d'une perturbation de charge de 250 MW.

Figure 9 :

 9 Figure 9: Trois réservoirs interconnectés illustrant les échanges d'énergie entre le MMC et les réseaux AC et DC.

Chapitre 6 :

 6 Validation expérimentale Un des verrous majeurs qui bloquent aujourd'hui le déploiement de solutions de commande type MPC dans les systèmes industriels est les performances de calcul. En eet, le MPC repose sur la synthèse d'une commande optimale qui est calculée à partir de la projection d'un modèle déni dans le MPC sur un horizon de temps déni dans le correcteur. Cela nécessite donc à chaque pas de temps de réitérer xix Résumé de la thèse en français (a) Résultats de simulation à la suite d'une perturbation de puissance de -0.3 p.u. côté DC. (b) Résultats de simulation à la suite d'une perturbation de puissance de -0.3 p.u. côté AC.

Figure 11 :

 11 Figure 11: Comparaison des structures dual PI et MPC à la suite de deux types de perturbation.

Figure 12 :

 12 Figure 12: Installation expérimentale Power-Hardware-In-the-Loop (PHIL) à SuperGrid Institute à Villeurbanne, France.

Finalement, ce chapitre

  Figure 13: Comparaison des solutions dual PI et MPC en PHIL en termes de suivi de référence (Cas 1 et 2).

  [8]. The data regarding these emissions per industrial sector and per country are detailed in Figs. 1.1 (a) and 1.1 (b) respectively. On the other hand, if we want to respect the limit of +1.5 o C above the pre-industrial level xed by the Paris Agreement [9], we have to emit 1.1. Context (a) Estimated emissions of CO 2 per industry. (b) Estimated emissions of CO 2 per country.

Figure 1 . 1 :

 11 Figure 1.1: Annual CO 2 emissions from the existing and under-construction/under- consideration energy infrastructure in 2019 [8].

Figure 1 . 2 :

 12 Figure 1.2: Lithium reserves and resources in the world in 2020. Reserves corresponds to the economically exploitable part and resources include all the available minerals in the subsoil. Data from [13].

Fig. 1 .

 1 3 (a), and at the transmission level, with solar farms or oshore wind farms, as illustrated in Fig.1.3 (b).

Chapter 1 .

 1 General Introduction(a) Installed rooftop PV in Freiburg im Breisgau, Germany. (b) Oshore wind farm.

Figure 1 . 3 :

 13 Figure 1.3: Examples of renewable energy sources that use power electronics converters.

Figure 1 . 4 :

 14 Figure 1.4: Map of Europe including the existing and expected future VSC-HVDC projects

1 . 5 .Figure 1 . 5 :

 1515 Figure 1.5: Illustrative example of a hybrid power system incorporating synchronous generation and power electronics based sources [18].

C h a p t e r 2 State 2 . 1 .

 221 In Chapter 4, we focus on the role of HVDC technology, especially the applications of the high-voltage Modular Multilevel Converters to Oshore Wind Farm (OWF) interconnection with AC grids. The originality of this work is to take into account the DC grid dynamics and to propose a novel way to calculate the energy requirements of the MMC to provide inertial support to the AC grid; • In Chapter 5, Model Predictive Control (MPC) is applied to optimally control the MMC internal energy and the DC grid voltage. This control solution is implemented using an identied model of the AC grid and its performances regarding disturbance rejections are compared to the state-of-the-art control structure which relies on two PI controllers. Its behavior and robustness with respect to parameter uncertainties and setpoint changes are assessed in a second time; • In Chapter 6, the application of grid-forming control to MMCs as well as the MPC layer developed in Chapter 4 and Chapter 5 are validated using a real-time controller which is applied to a physical mockup of an MMC. In particular, the concept of grid-forming is applied to the MMC mockup which makes it possible to synchronize with the grid without any PLL. Second, the hierarchical MPC approach developed in Chapter 5 is implemented in C and made compatible with the rest of the control structure developed in Matlab/Simulink. • In Chapter 7, a conclusion of the main contributions of this PhD thesis is drawn and the assumptions considered in this work are discussed. In a second time, potential novel lines of research are presented to extend the results of this thesis. -of-the-art of existing control solutions for Voltage Source Converters to support the Transmission Grid This chapter is based on the journal publication (under review) Journal II. Context and literature review 2.1.1. Context Power systems now encounter massive changes in terms of generation, transmission and distribution of electricity. In the European Union, the share of renewable energy in electricity generation is expected to rise constantly for the next decades [17], in a worldwide attempt to tackle climate change [9]. In particular, at the transmissiongrid level, especially in Europe, many HVDC systems have been built to exchange energy across long distances and interconnect Oshore Wind Farms (OWF) to the Chapter 2. State-of-the-art of existing control solutions for Voltage Source Converters to support the Transmission Grid

Figure 2 . 1 :

 21 Figure 2.1: Dierent mechanisms with distinctive time scales of the grid frequency when encountering an active power disturbance. (1) is for inertial response, (2) is for primary

Chapter 2 .

 2 State-of-the-art of existing control solutions for Voltage Source Converters to support the Transmission Grid (a) Simplied view of a twobus system interconnected with a purely inductive line. (b) Reactive power evolution around small disturbances (δ = 0).

Figure 2 . 2 :

 22 Figure 2.2: Approximation of the reactive power steady-state value depending on the voltage magnitude [64].

Figure 2 . 3 :

 23 Figure 2.3: Grid-connected synchronous generator response when encountering a fault at its point of coupling.

2 . 4 .Figure 2 . 4 :

 2424 Figure 2.4: Illustration of the inter-area oscillation event of the 3 rd of Dec. 2017 in the

Figure 2 . 7 :

 27 Figure 2.7: VSC cascaded structure control scheme in current-source mode.

  calculate i d * L and i q * L based on p ref and q ref respectively, the control structure in Fig. 2.7 uses the measurement of the PCC voltage to generate the current references, where p ref and q ref are simply divided by the grid voltage to obtain i d * L and i q *

Figure 2 . 8 :

 28 Figure 2.8: Detailed control scheme of a classical PLL [75].

Chapter 2 .

 2 Figure 2.10: Examples of the outer power control loops of the VSC in current and voltage source modes respectively.

Figure 2 .

 2 Figure 2.11: Inuence of the SCR on the performances of the VSC in terms of power tracking, following a 0.5 p.u step of p ref . The test-case is based on[START_REF] Rokrok | Classication and dynamic assessment of droop-based grid-forming control schemes: Application in HVDC systems[END_REF] with PLL gains taken from[START_REF] Molinas | Extended stability range of weak grids with voltage source converters through impedance-conditioned grid synchronization[END_REF].

Chapter 2 .

 2 State-of-the-art of existing control solutions for Voltage Source Converters to support the Transmission Grid (a) Active power / frequency droop response (b) Virtual inertia response (c) Fast Frequency Response (FFR)

2. 4 .

 4 Grid-level control modes (a) AC voltage magnitude control [109].(b) AC voltage droop control[START_REF] Rocabert | Control of power converters in AC microgrids[END_REF].(c) AC voltage magnitude control with reactive power droop[START_REF] Misyris | Grid Supporting VSCs in Power Systems with Varying Inertia and Short-Circuit Capacity[END_REF].

Figure 2 .

 2 Figure 2.13: Dierent implementations of AC voltage support using reactive power modulation in current-source mode.

  2.8 is performed for dierent PLL gains. The analysis is carried out considering k α p,pll = α • k p,pll and k α i,pll = α • k i,pll for dierent values of α = {1, 2, 3, ..., 20}, where k p,pll = -0.05 p.u., k i,pll = -2.53 p.u., and ω c,pll = 200 rad/s, based on values from [95].

Figure 2 .

 2 Figure 2.14: Bode plots of the linearized PLL for dierent values of gain k α p,pll and k α i,pll

  2.4, VSCs are highly sensitive to overcurrents due to their semiconductor devices. If no current limit Chapter 2. State-of-the-art of existing control solutions for Voltage Source Converters to support the Transmission Grid (a) Typical Low Voltage Fault-Ride-Through (LVFRT) prole imposed by a grid operator. (b) Reactive fault current requirements depending on the voltage level.Based on[START_REF]4/C6.35[END_REF] and[START_REF] Taul | An Overview of Assessment Methods for Synchronization Stability of Grid-Connected Converters Under Severe Symmetrical Grid Faults[END_REF].

Figure 2 .

 2 Figure 2.15: Synthesis of the expected voltage and current responses of a VSC following an AC-side fault.

  PLL to capture the grid frequency and phase angle.

  and the specicities related to HVDC and Modular Multilevel Converters (MMC) technology. C h a p t e r 3 Impacts of VSC control on the limits of Power Electronics Converters integration in AC grids The contents of this chapter is based on the peer-reviewed publication Journal I.

Figure 3 . 1 :

 31 Figure 3.1: Considered benchmark for the study.

Figure 3 . 2 :

 32 Figure 3.2: Synchronous Generator (SG) with its voltage and frequency controls.

Figure 3 . 3 :

 33 Figure 3.3: Equivalent electrical scheme of VSC with LC lter.

Figure 3 . 5 :

 35 Figure 3.5: VSC control in voltage source mode: grid-forming.

  Transfer function of the AVR and the switch when neglecting the AVR dynamics. (b) Equivalent scheme of the PLL and the switch when neglecting the PLL dynamics.

Figure 3 . 6 :

 36 Figure 3.6: Illustrative scheme of the selection of AVR and PLL specic dynamics for studies.

  3.7 (a)-3.7 (c) and in Figs. 3.7 (b)-3.7 (d) are similar. On the other hand, the PLL appears to act

3. 3 .Figure 3 . 7 :

 337 Figure 3.7: Modal analysis results for dierent values of P R with the VSC in grid-feeding (left-side) and grid-supporting (right-side) modes and x eq = 0.33 p.u.

3. 3 .

 3 Figure3.9: Modal analysis results for dierent values of x eq with the VSC in grid-feeding (left-side) and grid-supporting (right-side) modes and P R = 0.2.

Chapter 3 .Figure 3 .

 33 Figure 3.10: Modal analysis results for dierent values of P R with the VSC in grid-forming (droop or VSM) mode and x eq = 0.33 p.u.

3. 3 .Figure 3 .

 33 Figure 3.11: Participation factors of the λ 12,13 and λ 18,19 modes for dierent values of P R with the VSC in grid-forming droop mode and x eq = 0.33 p.u.

Chapter 3 .Figure 3 .

 33 Figure 3.12: Modal analysis results for dierent values of x eq with the VSC in grid-forming (droop or VSM) mode and P R = 0.45.

Figure 3 .

 3 Figure 3.13: Modal analysis results for dierent values of P R and dierent droop gains m pω with the VSC in grid-forming droop mode and x eq = 0.33 p.u. The left-side gure is the nominal state-space model whereas the right-side gure is the state-space model with no AVR dynamics.

3. 4 .

 4 Figure 3.16: Evolution of the PCC voltage and grid frequency following a line impedance step at t = 30 s with P R = 0.25 and x eq = 0.25 p.u.

  3.19 (c)) which has the largest voltage dip, its voltage does not have the largest oscillation magnitude. In particular, the magnitude of voltage oscillations gets larger when the bus is located closer to the VSC (see Bus 8 voltage in Fig. 3.19 (b)) which suggests they are caused by the VSC operating in grid-feeding mode. In addition, these oscillations become less signicant near the SGs (see Bus 2 and Bus 11 voltages in Figs. 3.19 (a) and 3.19 (d))

Figure 3 . 19 :

 319 Figure 3.19: Evolution of selected bus voltage magnitudes and COI frequency following an 400 MW active power load step at t = 30s at Bus 9 for dierent VSC control modes and x eq .

Figure 4 . 1 :

 41 Figure 4.1: Distribution of synchronous generation at the maximal inequality moment in Great Britain [156].

Figure 4 . 2 :

 42 Figure 4.2: Studied architecture for the integration of oshore wind farms in AC grids.

  4.3. the MMC is operated by either activating or deactivating the submodules of the dierent arms to generate a three-phase AC voltage while maintaining a constant DC voltage on the other side of the converter.

Figure 4 . 3 :

 43 Figure 4.3: Conventional topology of a Modular Multilevel Converter. Figure based on [163, 164].

Figure 4 . 4 :

 44 Figure 4.4: AC voltage generation using the MMC submodules to generate a stepped square sinusoidal waveform.

4. 2 .

 2 Modeling of the system4.2.2. Modeling of the test systemWe use an average MMC model which integrates the DC-side dynamics and the AC-side active power control to highlight the interactions that may arise from their coupling. The study is based on the benchmark given in Fig.4.5 that represents the system of Fig.4.2.

Figure 4 . 5 :

 45 Figure 4.5: Studied equivalent system.

Figure 4 . 6 :

 46 Figure 4.6: System-level view of an MMC, based on [158, 159].

1 :

 1 Parameters of the test system.In this chapter, the AC voltage control dynamics of a reactive power controller are neglected because the focus of the study is the active power related with the interactions between grid-forming synchronization control and DC voltage controller. In addition, the inner-loop controls of the AC and DC sides are not considered, which are, by denition, much faster than the considered outer loop controls[START_REF] Markovic | Understanding stability of low-inertia systems[END_REF][START_REF] Denis | From grid-following to grid-forming: The new strategy to build 100 % power-electronics interfaced transmission system with enhanced transient behavior[END_REF]. The three considered control loops, with their detailed schemes, are shown in Fig.4.7 (b). The Weighting Factors function allows to distribute the regulation of energy between the AC and DC sides:

  Control scheme of the considered outer-loop control dynamics.

Figure 4 . 7 :( 4 2 :

 4742 Figure 4.7: Main control scheme of the MMC with the dierent levels of control.

( 4 .Figure 4 . 9 :

 449 Fig.4.8 (a), it is observed that slowing down the DC-voltage loop has a positive impact on the damping of the oscillations in AC due to the DC-side disturbance.

Figure 4 .

 4 Figure 4.10: Evolution of the AC grid frequency following a disturbance in DC power p L at t = 15s.

Figure 4 .

 4 Figure 4.11: Evolution of the eigenvalues of the linearized system when increasing the time-constant of the DC-voltage regulator, denoted by T V dc .

( 4 . 2 ( 4

 424 14) mmc (k) = (W * mmc -W mmc,max ) .15) where v dc,min is the minimum DC voltage following the DC-side disturbance and corresponds to the voltage nadir illustrated in Fig. 4.8 (b) and W mmc,max is the maximum MMC internal energy transient level following the DC-side disturbance and corresponds to the peak observed in Fig. 4.9 (b). The results for the dierent values of k are given in Fig. 4.12.

4. 5 .

 5 Method to dene MMC internal energy requirements

Figure 4 .

 4 Figure 4.12: Evolution of the squared max deviation of the DC voltage and MMC internal energy when increasing the time constant of the DC-voltage regulator, denoted by T V dc .

4. 5 .

 5 Method to dene MMC internal energy requirements we determine the MMC required energy that can provide a SG-like inertial response within the frequency range of ±1 Hz.

Figure 4 .

 4 Figure 4.14: Typical inertial response of synchronous generators.

Chapter 4 .

 4 Operation of Modular Multilevel Converters in Grid-Forming Mode: High-level Control Constraints and Enhancements or absorbed energy is of the same amount of what a SG with the same power rating would release or absorb when the frequency changes by a certain level.

Figure 4 .

 4 Figure 4.15: General control scheme of the MMC controlled in grid-forming mode, based

  (a) Dynamic response of the system subject to AC power disturbances of 50 and 250 MW.(b) Dynamic response of the system subject to DC power disturbances of 50 and 250 MW.

Figure 4 .

 4 Figure 4.16: Simulation results for the MMC operated with the proposed control law compared to the standard internal energy controller, under the 4 types of disturbance.

Chapter 4 .Figure 4 .

 44 Figure 4.18: Dependency of the frequency nadir and the RoCoF on the value of k vr following a disturbance of 250 MW.

Figure 4 . 19 :

 419 Figure 4.19: Modied transmission grid model of the Nordic 44 system representing the Nordic synchronous area.

2 Table 4 . 6 : 3 Chapter 4 .

 24634 Control mode SM capacitance (C sub ) # of SM per arm (N arm ) mode SM capacitance (C sub ) # of SM per arm (N arm ) Parameters of the MMCs incorporated in the Nordic 44 test system. 4.8.3. Modeling and control of the MMCs The three MMCs are modeled by (4.2) and (4.3), presented in Section 4.2.2. The AC transformer inductance and resistance, L f and R f respectively, shown in Fig. 4.Operation of Modular Multilevel Converters in Grid-Forming Mode: High-level Control Constraints and Enhancements are taken into account. For the three MMCs, we take L f = 0.18 p.u. and R f = 0.001 p.u.

Figure 4 . 20 :

 420 Figure 4.20: Evolution of relevant grid state variables following a -800MW generation loss at bus 3245 when t = 5s and comparison between Cases 1 and 2.

Figure 5 . 2 :

 52 Figure 5.2: Three interconnected reservoirs showing the energy exchange between MMC, AC and DC grids.

Figure 5 . 3 :

 53 Figure 5.3: General MMC high-level control structure.

Figure 5 . 4 :

 54 Figure 5.4: Control and state estimation general structure

Chapter 5 .∈ R 2 , 1 .

 521 Figure 5.10: Excitation signal and output signal from the identied model and measurement collected during the validation experiment, in per unit.

5. 4 .

 4 MPC implementation measurements of ∆ω. The continuous-time Kalman lter was implemented using the function in Matlab [191].

e

  At dt • B (5.11) Knowing that C d = C and assuming D d = 0, the discrete-time system Σ d can be written as:

  2Nc×2Nc and P ∈ R 3Np×3Np are diagonal matrices: Q = diag{ q1 , . . . , qNp }, R = diag{r 1 , . . . , rNc }, and P = diag{ p1 , . . . , pNp }. The elements of Q, R and P are themselves matrices 5.5. Case study and simulation results which are dened ∀i ∈ [[0, N p ]] and ∀j ∈ [[0, N c ]] as:

  p.u. < y d (2) < 1.2 p.u.(5.27) -0.2 p.u. < ∆u(1) < 0.2 p.u.(5.28) -0.05 p.u. < ∆u(2) < 0.05 p.u.

Figure 5 .Table 5 . 2 :

 552 Figure 5.11: Equivalent scheme of the considered system, from[START_REF] Mourouvin | AC/DC Dynamic Interactions of MMC-HVDC in Grid-Forming for Wind-Farm Integration in AC Systems[END_REF].

( a )

 a Simulation results following a -0.3 p.u. DC power disturbance. (b) Simulation results following a -0.3 p.u. AC power disturbance.

Figure 5 .

 5 Figure 5.13: Comparison of the MPC time-domain responses for dierent values of the prediction horizons.

5. 6 . Conclusions 5 . 5 . 4 . 3 .

 65543 Inuence of the model parameter changes

( a )

 a Simulation results following a -0.3 p.u. DC power disturbance. (b) Simulation results following a -0.3 p.u. AC power disturbance.

Figure 5 .

 5 Figure 5.15: Comparison of the MPC time-domain responses for dierent values of the AC grid inertia constant H.

Figure 5 .

 5 Figure 5.16: Comparison of the MPC time-domain responses for dierent values of the AC grid primary frequency reserves droop value r.

  Online Hierarchical MPC applied to a Modular MultilevelConverter inGrid-Forming The contents of this chapter is based on the publication Journal III (in submission process). 6.1. Introduction 6.1.1. Motivation and context In Chapter 4, we studied the impacts of the grid-forming control on the MMC dynamics. In Chapter 5, we showed that applying MPC to the outer layer of the MMC control could be a promising solution to manage more eciently the AC/DC energy exchanges. However, in these two chapters, the new concepts and methods were veried only by simulations. Moreover, the MMC model used to represent the controlled plant in Chapter 5 was an averaged one with perfect inner control and low-level control loops. The aim of this chapter is to validate the proposed control solution developed in Chapter 5 by implementing it in a Power Hardware in the Loop (PHIL) facility based 6.2. Procedure of experimental validation

  the MMC mockup[START_REF] Opal | Modular Multilevel Converters[END_REF] as well as for the harware embedding the controllers and the measurement units. This benchmark makes it possible to test our developed control solutions, compiled from a Matlab/Simulink le into C-code, as further detailed in Section 6.2.2. This compiled code is then built on a hardware system and interfaced with Opal RT software RT-lab [211] to have a real-time display of the results and interact with the target during the experiment. The full experimental setup is given in Fig.6.2. The MMC with its 60 submodules is at the center. At the top, there is the Opal RT 4510 (OP4510) hardware that contains the controller and sends the gate signals to the MMC. At the edges of the photograph, there are the AC and DC ampliers, which emulate respectively a voltage source and a current source. There are both connected to the OP4510.

6. 2 .Figure 6 . 4 :

 264 Figure 6.4: Considered hardware congurations to implement the control solution.

Chapter 6 .Figure 6 . 5 :

 665 Figure 6.5: Power Hardware-In-the-Loop experimental setup and workow.

Figure 6 . 6 :

 66 Figure 6.6: Modied MMC control to integrate the grid-forming control in the high-level layer: dual-PI control structure.

Figure 6 . 7 :

 67 Figure 6.7: Implemented grid forming control. Scheme based on [115, 212].

Figure 6

 6 Figure 6.8: GFC activation o-load, when i L = 0 p.u.

  Figure 6.9: GFC activation on-load, when i L = 0.3 p.u.

Chapter 6 .

 6 Experimental Validation of The Online Hierarchical MPC applied to a Modular Multilevel Converter in Grid-Forming (a) Similar MPC implementations (b) Change in u min in μao-MPC from -5 to -3 at t = 5s.

Figure 6 .

 6 Figure 6.11: Comparison of similar MPC implementations using two dierent tools:

Chapter 6 .Figure 6 .Fig. 6 . 3 . 4 .

 66634 Figure 6.13: Modied MMC control to integrate the grid-forming control in the high-level layer and the MPC in the AC/DC-level layer: MPC control structure.

Chapter 6 .

 6 Figure 6.14: Comparison of the tracking performances of the MPC and dual PI solutions in SIL and PHIL experiments.

6. 5 .Figure 6 .

 56 Figure 6.15: PHIL mockup data: MPC and Case 1

Figure 6 .

 6 Figure 6.16: PHIL mockup data: MPC and Case 2

6. 6 Figure 6 .

 66 Figure 6.17: Comparison of the dual PI and MPC solutions in PHIL following DC and AC grid disturbances (Case 3 to Case 5).

Figure 6 . 19 :

 619 Figure 6.19: PHIL mockup data: MPC and Case 4

Figure 6 . 20 :

 620 Figure 6.20: PHIL mockup data: MPC and Case 5

1 .

 1 Conclusions per chapterWith the ongoing changes in the electricity production, transmission and distribution, power systems are expected to endure drastic changes in their operation and control. The increasing share of renewable energy worldwide drives towards PE-dominated grids where converters are expected to play a major role at the distribution and transmission levels. To ensure the stability and reliability of such grids, these PE interfaces must provide support functions to the system as synchronous generators do. These expectations also apply to HVDC systems and in particular MMCs which are expected to play a growing role in the operation of large power corridors and in the interconnection of oshore wind farms.

  adapted system-level response to the AC and DC grid disturbances. This aspect represents one of the main advantages of the PE-based sources compared to the SGbased systems and requires grid-level analysis of the contributions of the converters. o C. An IPCC Special Report on the impacts of global warming of 1.5 o C above pre-industrial levels and related global greenhouse gas emission pathways, in the context of strengthening the global response to the threat of

L f df d 1 .

 1 It is then possible to place the poles of a simplied rst-order plant model where the PI controller regulates the single state variable. With the timescale separation in any system, we can design the whole control structure as a cascaded structure with cascaded loops, where we use the dynamic decoupling between dierent variables with dierent time scales. In such cascaded structure, each loop can use a PI controller to form the whole control system. Because of the inherent time-scale separation in power systems[START_REF] Markovic | Understanding stability of low-inertia systems[END_REF], among other reasons, this type of control structure is a common practice in the eld of VSC control. VI A p p e n d i x C Parameter values of the test system The parameter values of the test-system from in Section 3.2 are given in Table C.1.Table C.1: Parameter values of the system for the simulations. 825 p.u. ω c,power 31.42 rad/s L d 1.81 p.u.

ω

  base 314.16 rad/s k p,pll -120 p.u.

  le support au réseau AC que peut aider à fournir le MPC. Ici, l'enjeu est d'avantage de valider la structure de commande proposée dans cette thèse avec les contraintes temps réel et notamment d'étudier l'intérêt en termes de temps de réponse et de rejet des perturbations que peut représenter le MPC dans ce cas.

	Après avoir implémenté la commande grid-forming issue de travaux récents de
	la littérature, la couche MPC est implémentée directement en langage C à partir
	de l'outil μao-MPC qui permet de générer le code avec un solutionneur (solver en
	anglais) adapté au temps réel. Une fois le contrôle développé en Matlab/Simulink et
	embarqué sur la cible OP4510, diérents cas d'étude sont étudiés pour comparer les
	performances du dual PI et du MPC. Ces cas d'étude sont présentés dans le tableau
	1.
	Cas # Caractéristique	Description
	Cas 1	suivi de tension DC	Changement de référence de 1 à 0.95 p.u. sur v * dc
	Cas 2	suivi de l'énergie interne	Changement de référence de 1 à 0.9 p.u. sur W * g
	Cas 3	perturbation côté DC	Créneau de courant DC de 0 à 0.1 p.u. sur i L
	Cas 4	perturbation côté AC

Les réseaux AC et DC sont décrits par des modèles simpliés, ce qui ne permet pas de tester tous les types de perturbation et notamment les pertes de génération ou les sauts de charge côté AC. De plus, le MMC étant dimensionné pour émuler une station HVDC conventionnelle, ses sous-modules ne sont pas conçus pour fournir un support inertiel tel que déni dans le chapitre 4. Par conséquence, il n'est pas xxi Résumé de la thèse en français possible de tester Créneau de tension AC de 1 à 0.9 p.u. sur v 2 Cas 5 perturbation côté AC Créneau d'angle AC de 0 à 10 o sur θ 2

Table 1: Cas d'étude pour la validation expérimentale en PHIL. Les résultats concernant les performances en termes de suivi de référence (Cas 1 et 2) des correcteurs dual PI et MPC sont donnés dans la gure 13. À l'exception de l'erreur en régime permanent concernant le MPC, qui demeure inférieure à 1%, les deux correcteurs orent des performances similaires. La solution MPC est légèrement plus rapide que la solution PI mais également plus oscillante. Les résultats concernant les performances en termes de rejet de perturbation (Cas 3 à 5) sont donnés dans la gure 14. Si pour les Cas 3 et 4 les réponses des correcteurs se rapprochent à nouveau, pour le Cas 5, il est clair que le MPC fournit une meilleure solution de contrôle puisqu'il permet au MMC de rester connecté là où le MMC se bloque suite à la perturbation avec le dual PI.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Comparison of the characteristics of a SG and a VSC in terms of available power headroom and energy for support function provisions. . . . . . . . . . . . . . . . . . . 22 Commande cascadée du VSC en mode source de courant. . . . . . . . . . . . . . . . . xiii 4 Commande cascadée du VSC en mode source de tension. . . . . . . . . . . . . . . . . xiii 5 Réseau de référence élémentaire considéré pour l'étude. . . . . . . . . . . . . . . . . . xiv 6 Ratio de puissance maximal admissible pour diérentes valeurs de x eq . . . . . . . . . xv Illustration of the inter-area oscillation event of the 3 rd of Dec. 2017 in the European grid. Based on [70]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eects of decreasing inertia and primary frequency reserves in AC grid frequency ω g . H is the inertia constant and K f the primary frequency droop gain. The nominal case (blue line) is H = 6s and K f = 20 p.u. . . . . . . . . . . . . . . . . . . . . . . . . . . Detailed control scheme of a classical PLL [75]. . . . . . . . . . . . . . . . . . . . . . 2.9 VSC cascaded structure control scheme in voltage-source mode. . . . . . . . . . . . . 2.10 Examples of the outer power control loops of the VSC in current and voltage source modes respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.11 Inuence of the SCR on the performances of the VSC in terms of power tracking,

	Contents List of Figures	Contents
	2.4	Bibliography	157
	2.5	Cependant, il serait possible de proposer une réponse coordonnée de dif-férents convertisseurs connectés au même réseau AC avec des sources DC Appendix
		découplées et variées ou d'utiliser un réseau MTDC intégré dans un même A SG modeling	I
	2.6	réseau synchrone. Ce type de solutions nécessiterait cependant d'établir une B.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . communication entre les stations. Contents III B PI Controller Tuning using Pole Placement Method III List of Tables List of Figures Equivalent electrical circuit of a Voltage Source Converter (VSC) between its AC and
	2.7 2.8	Acknowledgments B.2 Example with a simple system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Cas d'étude pour la validation expérimentale en PHIL. . . . . . . . . . . . . . . . . . xxii IV 1 Carte de l'Europe incluant les projets VSC-HVDC existants et futurs. . . . . . . . . . x VSC cascaded structure control scheme in current-source mode. . . . . . . . . . . . . i Abstract Résumé court en français vii Résumé de la thèse en français ix Contents xxvii List of Tables xxxiii List of Figures xxxv 1 General Introduction 1 . 8 C Parameter values of the test system VII D Simulation conditions of the Kundur 4-machine, 2-area system IX E Denition of system identication tools XI F Kalman ltering XIII . 8 [36]IX v B.3 Main applications and dierences from state-feedback pole placement method . . 2.1 3 Inertia contributions of dierent countries in Europe following ENTSO-E 2030 scenario VI 2 Circuit électrique équivalent d'un VSC interfaçant un réseau AC et DC. . . . . . . . xii
			1.4	Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	9
		2.1	1.5 Dierent mechanisms with distinctive time scales of the grid frequency when encoun-List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	10
			1.6 tering an active power disturbance. (1) is for inertial response, (2) is for primary Organization of the manuscript . . . . . . . . . . . . . . . . . . . . . . . . . . . .	12
			frequency control and (3) is for secondary frequency control. . . . . . . . . . . . . . .	16
		2 State-of-the-art of existing control solutions for Voltage Source Converters 2.2 Approximation of the reactive power steady-state value depending on the voltage
			to support the Transmission Grid	13
				13 19
				xxvi	xxvii xxxi xxxiii xxxv

Dans le chapitre 3, une analyse de stabilité petit signal a été proposée en étudiant un réseau hybride composé d'un VSC et d'un GS. Les conclusions de cette étude n'étant valables que pour de petites perturbations, il serait intéressant d'étendre ce type d'études à des analyses large signal (aussi appelées analyses transitoires).

Ce type d'études a déjà été conduit dans le cas de réseaux hybrides multimachines et multi-convertisseurs, où les VSCs étaient pilotés en grid-following.

Cependant, ce type d'analyses n'ont pour le moment que très rarement été appliquées au MMC et au rôle qu'il peut jouer dans la stabilité transitoire de réseaux hybrides. Une telle étude serait d'autant plus pertinente qu'elle nécessiterait de s'intéresser aux manières de limiter le courant dans les bras du MMC lorsque celui-ci est piloté en grid-forming, un sujet d'actualité.

• Commande coordonnée de plusieurs convertisseurs: Dans l'ensemble de ce manuscrit de thèse, l'accent a été mis sur la manière de contrôler un xxv Résumé de la thèse en français convertisseur pour supporter un réseau hybride où EP et GS cohabitent. Chapitre 1: Introduction générale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Chapitre 2: État de l'art des solutions existantes . . . . . . . . . . . . . . . . . . . . . . xii Chapitre 3: Analyse modale sur les limites d'intégration de l'EP dans les réseaux AC . xiv Chapitre 4: Application du grid-forming aux convertisseurs modulaires multi-niveaux . xv Chapitre 5: Synthèse du MPC pour améliorer la réponse du MMC en grid-forming . . . xviii Chapitre 6: Validation expérimentale . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Chapitre 7: Conclusion générale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii 1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 A brief history of fossil fuels . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Climate change and necessary energy transition . . . . . . . . . . . . . . 2 1.1.3 A transition reliant on resources . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Upcoming changes for the transmission grid . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 More renewable energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 More power electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Scope and objectives of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Context and literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Compatibility of the two main VSC control modes with the dierent grid support functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1 Dierent scenarii using the 4-machine, 2-area system . . . . . . . . . . . . . . . . . . 62 4.1 Parameters of the test system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Control parameters of the test system. . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3 Frequency range requirements in the UK grid code [170]. . . . . . . . . . . . . . . . . 82 4.4 Comparison of standard operational quantities of a synchronous generator and an MMC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.5 Comparison of the energy requirements for the MMC to emulate dierent inertial responses. The requirements are given in terms of an oversized submodule capacitance and in terms of power and energy if an additional ESS is needed. . . . . . . . . . . . 85 4.6 Parameters of the MMCs incorporated in the Nordic 44 test system. . . . . . . . . . 91 5.1 Parameters used in the prediction error algorithm. . . . . . . . . . . . . . . . . . . . . 102 5.2 Parameters of the considered system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 5.3 Control parameters of the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.1 Parameters of the mockup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.2 Control parameters of the MMC control. . . . . . . . . . . . . . . . . . . . . . . . . . 131 6.3 Case studies for the PHIL experimental validation. . . . . . . . . . . . . . . . . . . . 142 C.1 Parameter values of the system for the simulations. . . . . . . . . . . . . . . . . . . . VII D.1 Power Flow data of the Kundur 4-machine, 2-area system . . . . . . . . . . . . . . . 7 Inuence de k vr sur le nadir et le RoCoF à la suite d'une perturbation de charge de 250 MW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi 8 Modèle modié du réseau nordique européen (Nordic 44-bus). . . . . . . . . . . . . . xvii 9 Trois réservoirs interconnectés illustrant les échanges d'énergie entre le MMC et les réseaux AC et DC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii 10 Structure générale de la commande MPC et estimation d'état. . . . . . . . . . . . . . xix 11 Comparaison des structures dual PI et MPC à la suite de deux types de perturbation. xx 12 Installation expérimentale Power-Hardware-In-the-Loop (PHIL) à SuperGrid Institute à Villeurbanne, France. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi 13 Comparaison des solutions dual PI et MPC en PHIL en termes de suivi de référence (Cas 1 et 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii 14 Comparaison des solutions dual PI et MPC en PHIL en termes de réjection de perturbation (Cas 3 à 5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiv 1.1 Annual CO 2 emissions from the existing and under-construction/under-consideration energy infrastructure in 2019 [8]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Lithium reserves and resources in the world in 2020. Reserves corresponds to the economically exploitable part and resources include all the available minerals in the subsoil. Data from [13]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Examples of renewable energy sources that use power electronics converters. . . . . . 6 1.4 Map of Europe including the existing and expected future VSC-HVDC projects to interconnect synchronous and asynchronous areas [20]. . . . . . . . . . . . . . . . . . 7 1.5 Illustrative example of a hybrid power system incorporating synchronous generation and power electronics based sources [18]. . . . . . . . . . . . . . . . . . . . . . . . . magnitude [64]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Grid-connected synchronous generator response when encountering a fault at its point of coupling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DC interfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . following a 0.5 p.u step of p ref . The test-case is based on [94] with PLL gains taken from [95]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.12 Illustration of dierent potential supporting actions using a variation of power reference provided by a power converter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.13 Dierent implementations of AC voltage support using reactive power modulation in current-source mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.14 Bode plots of the linearized PLL for dierent values of gain k α p,pll and k α i,pll with respect to the two outputs ω (left-side) and θ (right-side). . . . . . . . . . . . . . . . . . . . . 2.15 Synthesis of the expected voltage and current responses of a VSC following an AC-side fault. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.16 Dierent manners of damping inter-area power oscillations in a single synchronized power system using synchronous generators, VSC-interfaced power sources and VSC-HVDC links. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.17 Options for POD functions depending on the VSC control mode. . . . . . . . . . . . 3.1 Considered benchmark for the study. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Synchronous Generator (SG) with its voltage and frequency controls. . . . . . . . . . 3.3 Equivalent electrical scheme of VSC with LC lter. . . . . . . . . . . . . . . . . . . . 3.4 VSC control in current source mode: grid-feeding and grid-supporting. . . . . . . . . 3.5 VSC control in voltage source mode: grid-forming. . . . . . . . . . . . . . . . . . . . .

  2. . . . . . . . . . . . . . . . . . . .12 Modal analysis results for dierent values of x eq with the VSC in grid-forming (droop or VSM) mode and P R = 0.45. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.13 Modal analysis results for dierent values of P R and dierent droop gains m pω with the VSC in grid-forming droop mode and x eq = 0.33 p.u. The left-side gure is the nominal state-space model whereas the right-side gure is the state-space model with no AVR dynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.14 Maximum allowable power ratio with respect to the line reactance of the system. . . 59 3.15 Evolution of the PCC voltage magnitude and grid frequency following an active power load step at t = 20 s when the VSC is controlled in grid-feeding mode for dierent P R and x eq = 0.25 p.u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.16 Evolution of the PCC voltage and grid frequency following a line impedance step at t = 30 s with P R = 0.25 and x eq = 0.25 p.u. . . . . . . . . . . . . . . . . . . . . . . . 61 3.17 Evolution of the system state-variables when the VSC is in grid-forming droop mode following an active power load step at t = 20 s, with x eq = 0.33 p.u and P R = 0.5. Dependency of the frequency nadir and the RoCoF on the value of k vr following a disturbance of 250 MW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.19 Modied transmission grid model of the Nordic 44 system representing the Nordic synchronous area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flowchart to calculate the identied transfer functions Ĝ and G lin for dierent model orders. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bode plots of the linear transfer function G with respect to the identied models Ĝ and G lin . The i th -line gives the Bode plot (magnitude and phase) of G and the i th -order identied models Ĝ and G lin dened by n t . . . . . . . . . . . . . . . . . . . . . . . . . L = 0.3 p.u. . . . . . . . . . . . . . . . . . . . . . . . 133 6.10 Flowchart of the used embedded QP solver. Scheme based on [214]. . . . . . . . . . . 137 6.11 Comparison of similar MPC implementations using two dierent tools: YALMIP [193] versus μao-MPC [214]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

	List of Figures	List of Figures
	4.18 4.20 Evolution of relevant grid state variables following a -800MW generation loss at bus 6.9 GFC activation on-load, when i 6.12 Comparison of the elapsed time of the two control implementations in a Mat-
		3245 when t = 5s and comparison between Cases 1 and 2. . . . . . . . . . . . . . . . lab/Simulink environment with the standard tuning parameters of the solver. . . . . 139
	5.1 5.2	Energy exchange between the AC and DC grids via the MMC submodules, from [159]. Three interconnected reservoirs showing the energy exchange between MMC, AC and DC grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Modied MMC control to integrate the grid-forming control in the high-level layer and the MPC in the AC/DC-level layer: MPC control structure. . . . . . . . . . . . . . . 140 6.14 Comparison of the tracking performances of the MPC and dual PI solutions in SIL and PHIL experiments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 1
	5.3	General MMC high-level control structure. . . . . . . . . . . . . . . . . . . . . . . . . 6.15 PHIL mockup data: MPC and Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
	5.4 5.5	Control and state estimation general structure . . . . . . . . . . . . . . . . . . . . . . Block scheme of the AC system with the MMC in grid-forming control mode. . . . . 6.16 PHIL mockup data: MPC and Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.17 Comparison of the dual PI and MPC solutions in PHIL following DC and AC grid General Introduction
	5.6	Detailed method for estimating the transfer function Ĝ with a multisine probing signal disturbances (Case 3 to Case 5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
		using prediction error method [187]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.18 PHIL mockup data: MPC and Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
	5.7	6.19 PHIL mockup data: MPC and Case 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
		6.20 PHIL mockup data: MPC and Case 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
	5.8
		A.1	Transfer function of the governor and turbine model from [75]. . . . . . . . . . . . . .	II
	5.9	69 III RL electrical circuit with a DC voltage source v in considered as control input. . . . . IV Closed-loop system with a PI controller and a linear plant. . . . . . . . . . . . . . . . Scheme of the validation experiment to check the reliability of the identied model Ĝ. B.1 B.2
	4.3 5.10 Excitation signal and output signal from the identied model and measurement Conventional topology of a Modular Multilevel Converter. Figure based on [163, 164]. B.3 Illustrative dynamic response of the branch current i out following a DC voltage (v in )	71
		4.4 collected during the validation experiment, in per unit. . . . . . . . . . . . . . . . . . AC voltage generation using the MMC submodules to generate a stepped square step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	V
	sinusoidal waveform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Studied equivalent system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Equivalent scheme of the considered system, from [29]. . . . . . . . . . . . . . . . . . 4.5 5.12 Comparison of the dual PI and MPC structures following two kinds of disturbances. . 1.1. Context	72 73
	4.6 5.13 Comparison of the MPC time-domain responses for dierent values of the prediction System-level view of an MMC, based on [158, 159]. . . . . . . . . . . . . . . . . . . . . 4.7 Main control scheme of the MMC with the dierent levels of control. . . . . . . . . . 4.8 Simulation results of MMC external outputs in response to a disturbance in DC power 5.14 Comparison of the MPC time-domain responses for dierent active power reference p * horizons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1. A brief history of fossil fuels	73 76
		p L at t = 15s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 pre-disturbance values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	5.15 Comparison of the MPC time-domain responses for dierent values of the AC grid
		4.9 inertia constant H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulation results of relevant system state variables in response to a disturbance in DC power p L at t = 15s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
	5.16 Comparison of the MPC time-domain responses for dierent values of the AC grid 4.10 Evolution of the AC grid frequency following a disturbance in DC power p L at t = 15s. 79 primary frequency reserves droop value r. . . . . . . . . . . . . . . . . . . . . . . . . .
		4.11 Evolution of the eigenvalues of the linearized system when increasing the time-constant
	6.1	of the DC-voltage regulator, denoted by T V dc . . . . . . . . . . . . . . . . . . . . . . . 80 Proposed MPC for AC/DC energy management of an MMC. More details are available
		4.12 Evolution of the squared max deviation of the DC voltage and MMC internal energy in Chapter 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	6.2	when increasing the time constant of the DC-voltage regulator, denoted by T V dc . . . . 81 Power-Hardware-In-the-Loop (PHIL) experimental setup at SuperGrid Institute in
		4.13 Frequency monitoring in the UK transmission system in November 2019 [171]. . . . . Villeurbanne, France. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	82
	6.3	4.14 Typical inertial response of synchronous generators. . . . . . . . . . . . . . . . . . . . Existing MMC control with the PHIL setup before starting the project [74, 164]. . . .	83
	6.4	4.15 General control scheme of the MMC controlled in grid-forming mode, based on [29]. Considered hardware congurations to implement the control solution. . . . . . . . .
	6.5	The blocks in green are specic to this study. . . . . . . . . . . . . . . . . . . . . . . Power Hardware-In-the-Loop experimental setup and workow. . . . . . . . . . . . .	86
	6.6	4.16 Simulation results for the MMC operated with the proposed control law compared to Modied MMC control to integrate the grid-forming control in the high-level layer:
		the standard internal energy controller, under the 4 types of disturbance. . . . . . . . dual-PI control structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .	88
	6.7	4.17 Simulation results following a power step disturbance of 250 MW for dierent values Implemented grid forming control. Scheme based on [115, 212]. . . . . . . . . . . . . .
	6.8	of k vr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 GFC activation o-load, when i L = 0 p.u. . . . . . . . . . . . . . . . . . . . . . . . .
			xxxvii xxxviii xxxix

3.10 Modal analysis results for dierent values of P R with the VSC in grid-forming (droop or VSM) mode and x eq = 0.33 p.u. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Participation factors of the λ 12,13 and λ 18,19 modes for dierent values of P R with the VSC in grid-forming droop mode and x eq = 0.33 p.u. . . . . . . . . . . . . . . . . . . xxxvi List of Figures 3. . 61 3.18 Modied Kundur's 4-machine, 2-area system, based on [60]. . . . . . . . . . . . . . . 62 3.19 Evolution of selected bus voltage magnitudes and COI frequency following an 400 MW active power load step at t = 30s at Bus 9 for dierent VSC control modes and x eq . . 64 4.1 Distribution of synchronous generation at the maximal inequality moment in Great Britain [156]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.2 Studied architecture for the integration of oshore wind farms in AC grids. . . . . . .

Table 2 .

 2

	Features	symbol	Wind Power	Thermal Power
					Plant	Plant
	Installed capacity, from [72]		S b		2500 MVA	1000 MVA
	Operating hours, from [72]		t op		2000 h	5000 h
	Yearly produced energy, from [72]		W y		5000 GW.h	5000 GW.h
	Overcurrent capability, from [73]	i max		110%	500%
	Average power setpoint		p * avg		0.23 p.u.	0.57 p.u.
	Max. active power headroom		P hr		0.77 p.u.	0.43 p.u.
	Max. reactive power headroom		Q hr		0.97 p.u.	0.82 p.u.
	p * avg =	t op t year	=	t op (h) 8670	(2.3)

2: Comparison of the characteristics of a SG and a VSC in terms of available power headroom and energy for support function provisions.

Table 2 . 3 :

 23 124127]. Compatibility of the two main VSC control modes with the dierent grid support functions.

		POD actions
	Inherent damp-	can be added
	ing based on	as in Figs. 2.17
	SG emulation	(a) ( [124127])
	but can disturb	-	2.17	(b)
	the system as	( [130]). Needs
	well. Compati-	some available
	ble with SG's	energy	for
	PSS.	active	power
		damping.

  .1 are the inner voltage controller gains which correspond to a 1000 MVA VSC [94] and m qv = 0 in this case study. The VSC in grid-forming mode is operated with a voltage reference v ref = v 0 8 where v 0 8 is the initial value of Bus 8 voltage given in Appendix D. At t = 30s, we introduce a 400 MW load step at Bus 9. The corresponding results for the 3 cases are given in Fig. 3.19. v i corresponds to the i th bus voltage magnitude and ∆ω COI = ω COI -ω ref is the frequency deviation at the Center of Inertia (COI) which is dened as:

Table 4 .

 4 3: Frequency range requirements in the UK grid code [170].

	Frequency Range	Requirements
	51.5 Hz to 52 Hz	Operation for a period of at least 15 minutes
	51 Hz to 51.5 Hz	Operation for a period of at least 90 minutes
	49 Hz to 51 Hz	Continuous operation is required
	47.5 Hz to 49 Hz	Operation for a period of at least 90 minutes
	47 Hz to 47.5 Hz	Operation for a period of at least 20 seconds

  for the MMC.

	Quantity	Synchronous Generator	Modular Multilevel Converter [77]
	Rated power S nom	500 MVA	500 MVA
	DC grid rated voltage V DC b	not relevant	640 kV
	Submodule capacitance C sub	not relevant	8 mF
	# of submodules N arm	not relevant	400 per arm
	Inertia constant H 0 (s)	3 s	49.2 ms
	Minimum frequency ω min	49.0 Hz	not relevant
	Minimum energy W min [163]	/	0.8 p.u.
	Available energy ∆W	59.4 MJ	4.915 MJ
	Table 4.4: Comparison of standard operational quantities of a synchronous generator and
	an MMC.
	As shown in this table, even though the ratio between SG and MMC stored
	energy is almost 100 (3s vs 49.2ms), their ratio in terms of available energy is only
	slightly above 10 (59.4 MJ vs 4.915 MJ).

  Denote by C sub the equivalent capacitance of the oversized MMC submodules and introduce N oversize as the oversizing coecient of the submodule capacitance dened such that C sub = N oversize • C sub . For simplicity, N oversize is assumed to only take integer values. Obviously, the internal energy of the oversized MMC, denoted by

	2	6 400	0.008 * 640 2 = 24.58MJ	(4.17)

W mmc , can be expressed as W mmc = N oversize • W mmc . Assuming that W mmc can tolerate a minimum value of 0.8 p.u., the energy variation of the oversized energy can be obtained as

Table 4 .

 4 5: Comparison of the energy requirements for the MMC to emulate dierent inertial responses. The requirements are given in terms of an oversized submodule capacitance and in terms of power and energy if an additional ESS is needed.

	4 (1 -0.8) • 24, 58	= 12 (4.20)

[172]

. The required energy for such an ESS can be calculated as ∆W ESS = ∆W SG -0.2 × W mmc = 59.4 -0.2 × 24.58 = 54.5MJ

Table 5 .

 5 1: Parameters used in the prediction error algorithm.

	Quantity	Notation Value
	Sampling time	h	0.05 s
	Number of poles	n f	4
	Number of zeros	n b	4
	Experiment duration	t id	50 s

  5.10 give a Normalized Root Mean Square Error (NRMSE) t Φ N RM SE = 1 -|y-ŷ| |y-ȳ| > 99.3%. It is also important to note that this signal is designed to have a low impact in terms of disturbances from the AC grid point of view, here with u exp < 0.005 p.u. The identied model appears to be accurate enough for the MPC implementation. 5.3.4. Conversion of Ĝ(z) into a state-space model Once the transfer function Ĝ(z) is obtained, we construct the continuous-time statespace model of its minimum realization, denoted by Σ AC , such that:

Table 5 .

 5 3: Control parameters of the system.

	Quantity	Notation	Value
	Active power droop gain	m pω	0.05 p.u.
	LPF cuto frequency	ω pq	15.7 rad/s
	Energy controller response time

  . This experimental benchmark has been developed over years at SuperGrid Institute and relies on Opal RT solution 123 Chapter 6. Experimental Validation of The Online Hierarchical MPC applied to a Modular Multilevel Converter in Grid-Forming

	Table 6.1: Parameters of the mockup
	Quantity	Notation Value
	MMC rated power

Table 6 .

 6 2: Control parameters of the MMC control.

	Quantity	Notation	Value
	Active power droop gain	m pω	0.05 p.u.
	Reactive power droop gain	m qv	0 p.u.
	Outer loop lter cut-o frequency	ω pq	62.8 rad/s
	Inner loop lter cut-o frequency	ω ir	314 rad/s
	Damping resistance	R a	0.2 p.u.
	Inductance compensation	L	0 p.u.
	Energy control response time	τ rwh	0.5 s
	W Σ control response time	τ W Σ	1 s
	W dif f control response time	τ W dif f	1 s
	Energy reference	W * g	1 p.u.
	DC voltage reference	v * dc	1 p.u.
	Energy distribution coecient	α	1
	DC-voltage control response time	τ rdc	0.15 s
	DC-current control response time	τ idc	0.003 s
	AC-current control response time	τ iac	0.005 s
	Damping ratio of all control loops	ζ	0.707

  .17) Chapter 6. Experimental Validation of The Online Hierarchical MPC applied to a Modular Multilevel Converter inGrid-Forming where ν is the extra-step constant dened as:

Table 6 .

 6 3: Case studies for the PHIL experimental validation.6.5.1. Test casesThe aim of this section is to discuss the interest of replacing the dual PI controllers by a MPC when operating the MMC in grid-forming. Indeed, even if the MPC-based solution in Chapter 5 exhibited better performances, this type of solutions are still marginal for industrial power electronics systems. This subsection introduces the test cases we considered to compare the two control solutions in PHIL.

	6.5. Experimental results and discussions

The idea is to compare the two controllers in terms of tracking and disturbance rejection from either the AC or the DC grid. The dierent test cases that are considered are presented in Table

6

.3. For the tracking cases (Case 1 and Case

  climate change, sustainable development, and eorts to eradicate poverty, Summary for Policymakers, 2018. Cohen, X. Zhang, J. Francis, T. Jung, R. Kwok, J. Overland, T. Ballinger, U. Bhatt, H. Chen, D. Coumou et al., Divergent consensuses on arctic amplication inuence on midlatitude severe winter weather, Nature Climate Change, vol. 10, no. 1, pp. 2029, 2020. [7] F. Dottori, W. Szewczyk, J.-C. Ciscar, F. Zhao, L. Aleri, Y. Hirabayashi, A. Bianchi, I. Mongelli, K. Frieler, R. A. Betts et al., Increased human and economic losses from river ooding with anthropogenic warming, Nature Climate Change, vol. 8, no. 9, pp. 781786, 2018.

	[4]	H. Fountain, Climate Change Aected Australia's Wildres, Scientists
		Conrm, NY Times, Mar. 2020. [Online]. Available: https://www.nytimes.
		com/2020/03/04/climate/australia-wildres-climate-change.html
	[5]	The	Economist	explains,	Why	is	Texas	experiencing
		a	winter	freeze?	the	Economist,		Feb.	2021.	[Online].
		Available: https://www.economist.com/the-economist-explains/2021/02/17/
		why-is-texas-experiencing-a-winter-freeze
	[6]	J.

Table D .

 D 1: Power Flow data of the Kundur 4-machine, 2-area system

	bus #	1	2	3	4	5	6	7	8	9	10	11
	v (p.u.)	1.03	1.01	1.03	1.01	1.007	0.98	0.96	0.95	0.97	0.98	1.009
	θ( o )	20.27	10.51	-6.8	-17.0	13.8	3.71	-4.69	-18.5	-32.15	-23.7	-13.4
	P (MW)	700	700	719.1	700	0	0	-967	0	-1767	0	0
	Q(MVAR) 185.0 234.6 176.0 202.1	0	0	+84.7	0	+230.2	0	0
						IX

(a) AC voltage (b) abc current

(a) AC voltage

Acknowledgments

Before jumping into the main content of this PhD thesis manuscript, I believe it is Acknowledgments I also had the chance to meet outstanding people during my PhD, including my

Validation by simulations

Figure 3.14: Maximum allowable power ratio with respect to the line reactance of the system. to be less stable than the grid-feeding one. As the equivalent voltage source becomes farther, the PLL synchronizes to the grid with diculty. In fact, the droop actions from (3.3) is responsible for destabilizing the system.

Validation by simulations

In Sections 3.4.1 and 3.4.2, we run time-domain simulations using the nonlinear state-space model developed in Section 3.2 to verify the modal analysis results given in the previous section. In Section 3.4.3, we study the stability limit assessments in the case of a multi-machine grid.

Inuence of the power ratio and the electrical distance

Fig. 3.15 shows the time-domain simulation results on the system when the VSC is controlled in grid-feeding for P R = 0.1 and P R = 0.3 with x eq = 0.25 p.u., following a 5% step of active power load. They show that the higher value of P R results in unacceptable high-frequency voltage oscillations, as predicted by the modal analysis on the stability limits in Fig. 3.7. Next, we focus on the dependency of the maximum allowable P R on x eq to quantitatively conrm the results of the modal analysis. We launch a simulation for P R = 0.25 and x eq = 0.25 p.u. and introduce a line disturbance such that the line inductance increases up to x eq = 0.3 p.u.

The evolution of PCC voltage and frequency following this contingency is given in Fig 3.16. It shows that only the system with the VSC in grid-feeding becomes unstable, which conrms the modal analysis results in Fig. 3.14. In addition, a large Chapter 3. Impacts of VSC control on the limits of Power Electronics Converters integration in AC grids Figure 3.15: Evolution of the PCC voltage magnitude and grid frequency following an active power load step at t = 20 s when the VSC is controlled in grid-feeding mode for dierent P R and x eq = 0.25 p.u.

overshoot and a longer settling time can be observed when the VSC is controlled in grid-forming than in grid-supporting. This can result from the self-synchronizing capability of grid-forming techniques. Indeed, with grid-forming, we observe two equivalent oscillators, i.e. the machine and the converter that oscillate together to reach a consensus, whereas in the grid-supporting case, the two oscillators show a leading-following relation: the VSC follows the SG oscillations, hence much less interferences. On the other hand, regarding voltage stability, thanks to its explicit inner voltage control loop, the grid-forming ensures better performances than the grid-supporting, which uses the reactive power/voltage droop given in (3.4).

Inuence of the AVR dynamics

To clarify the relationship between the AVR dynamics and the instability of the system when the VSC is in grid-forming mode, we perform two time-domain simulations for P R = 0.5 and x eq = 0.33 p.u., with a 5% step in the load's active power as the disturbance. The simulation results given in Fig. 3.17 show that the system is unstable when the AVR is taken into account, whereas it remains stable when the AVR dynamics are neglected.

This conrms the modal analysis in Fig. 3.10. Besides, the frequency of the oscillations is rather slow and may correspond to the undamped low-frequency modes in Fig. In Chapter 4, we presented the MMC system, which is more complex than the classical two-level VSC and is dedicated to high-voltage systems. The MMC control is complex and requires the coordination of the low-level control, which makes sure that all the submodules are equally sollicitated in the generation of the desired waveform, and the high-level control, which focuses on the energy ows between the AC and DC grids. This chapter deals with the high-level control.

The impact of the MMC internal energy dynamics and its control on the gridlevel behavior is the topic of some recent studies in the power system community

System identication

With this in mind, we perform a system identication analysis based on dierent values of n t = n f = n b . As a reminder, the discrete-time transfer function Ĝ(z) is dened as:

The aim of this analysis is to compare the dierent identied models Ĝ(z) with the exact linear transfer function G. Denote by G lin (z) the model obtained by exciting, with the same excitation signal, the linearized AC grid model G. This entire procedure is described in Fig. 5.7. The comparison of the Bode plots of the dierent transfer functions is given in Fig. 5.8 for n t = {1, 2, . . . , 6}. In this gure, it is clear that the identied system is brought closer to the linearized model when n t increases. However, when n t ≥ 4, we observe no real improvement between the identied model and the linearized one.

In addition, since Ĝ(z) and G lin (z) have the same frequency-domain responses, it is conrmed that the output error algorithm works well when the output information is obtained using the nonlinear system. However, those results are based on noiseless measurements. In future work, it would be desired to take into account the noise of the sensor units to extend these results.

However, this procedure requires to know the linearized function G(s), which can only be calculated in simulation conditions when the model is available for linearization. In real-life conditions, the AC grid is unknown, as described in Section 5.2.3. As a consequence, another method, applicable to real-life conditions, is described in the next section to validate the identied model. As a reminder, the proposed control scheme developed in Chapter 5, as recalled in Fig. 6.1, represents a MIMO-based optimal approach to control the AC and DC powers owing through the MMC while taking into account the MMC and DC grid constraints.

Literature review

The main obstacle to applying MPC in power electronics and power systems is the real-life constraints (computational costs, communication delays, sensor noises, etc.). However, the evolution of hardware systems, especially their computational power, makes it possible to apply MPC to a larger eld of applications [197200].

But power electronics systems, due to their fast dynamics, are still an open eld for MPC applications.

Regarding the use of real-time setups for grid-connected VSCs, there have been many works that use experimental mockups of a VSC controlled in grid-following mode [201204]. However, the application of grid-forming to such converters is more recent and mainly concerns two-level VSCs [START_REF] Taul | An Overview of Assessment Methods for Synchronization Stability of Grid-Connected Converters Under Severe Symmetrical Grid Faults[END_REF][START_REF] Qoria | Grid-forming control to achieve a 100% power electronics interfaced power transmission systems[END_REF][START_REF] Pirsto | Multifunctional Cascade Control of Voltage-Source Converters Equipped With an LC Filter[END_REF] while operating MMC mockups in grid-forming remain marginal [START_REF] Arco | Operation of a Modular Multilevel Converter Controlled as a Virtual Synchronous Machine[END_REF]. This makes the available setup at SuperGrid Institute [START_REF] Zama | Linear feedback Dead-Beat Control for Modular Multilevel Converters: Validation under Faults Grid Operation mode[END_REF] even more interesting for our research.

Taking into account the scarcity of such setups, little research has been conducted on implementing MPC for grid-forming controlled VSCs. There has been some recent practical implementation of MPC for VSC controlled in grid-forming mode [START_REF] Labella | An MPC Approach for Grid-Forming Inverters: Theory and Experiment[END_REF].

However, this work is not about hierarchical MPC for energy management as it was (b). Nevertheless, simulating more complex AC/DC systems would be a promising way of extending these results and will be discussed in the conclusions.

Existing controls and required changes

The control structure that had already been implemented before we started the project is given in Fig. 6.3. It is based on previous work at SuperGrid Institute [START_REF] Zama | Linear feedback Dead-Beat Control for Modular Multilevel Converters: Validation under Faults Grid Operation mode[END_REF][START_REF] Zama | High Dynamics Control for MMC Based on Exact Discrete-Time Model With Experimental Validation[END_REF] and is implemented in Matlab/Simulink environment which is then compiled on the target using RT-lab as presented in Section 6.2.1. This MMC control is 6.4. MPC implementation Figure 6.12: Comparison of the elapsed time of the two control implementations in a Matlab/Simulink environment with the standard tuning parameters of the solver.

In Fig. 6.11 (b), we analyzed the eciency of online updates of the input constraints of the μao-MPC solution. The minimum value of the input u, denoted by u min , is changed from -5 to -3 at t = 5s. We observe that the MPC does consider this change in this gure, and that in consequence the tracking of the output reference is worse than the nominal solution.

The μao-MPC appears to oer similar performances as YALMIP in terms of synthetizing a linear MPC for linear systems. To compare the computation costs between the two options, we performed 10 consecutive simulations of this example for each solution. The respective elapsed times are given in Fig. 6.12. It is clear that the MPC implemented in C is much more ecient in terms of computational costs, up to 100 times faster, than the YALMIP option. This is an interesting feature since the computation cost of the algorithm will be critical for the implementation on a real hardware system.

The C-based μao-MPC solution appears to have sucient performances to be used for our MPC implementation.

Practical implementation of the MPC in real-time target

To implement the hierarchical MPC in the MMC mockup, we desired to keep the workow dened in [START_REF] Zama | Modeling and Control of Modular Multilevel Converters (MMCs) for HVDC applications[END_REF] and described in Section 6. A p p e n d i x

A SG modeling

In the following, the dierential equations are presented in the dq-frame with the q-axis aligned with the rotor.

Voltage dynamics

The SG is supposed to regulate its voltage v g at its reference v * at the connection point. This function is realized by the AVR, which is composed of a lead-lag lter controller and an AC exciter model, as:

where ∆v = v * -v g 2 , T b and T c are the time constants of the lead-lag lter, T a the exciter time constant, k a the exciter gain and v f d the eld excitation voltage.

It is important to note that the dynamics of PSS (Power System Stabilizers) could also have been taken into account, as in [START_REF] Markovic | Understanding stability of low-inertia systems[END_REF]. But, since the PSS enhances the system stability and does not reect the behavior of standard SG with classical AVR schemes, it is not included in our study.

To excite the machine, the AVR uses the eld excitation voltage, v f d . The dynamics are modeled as: rotor and the stator, the eld current i f d is related to the stator armature current i dq g , via the stator ux λ dq as:

The relationship between the stator ux and the terminal voltage v dq g , as described in [START_REF] Wang | Modern Power Systems Analysis[END_REF], with the copper losses, modeled with the resistance r a is given as:

The electrical power is given by P e = λ d • i q -λ q • i d , which neglects the copper losses in the stator.

Frequency dynamics

The swing equation is related to the dynamics of the rotor speed and thus of the grid frequency, ω g . The rotor angle, denoted by δ, corresponds to the integral of the frequency:

where ω ref is the grid reference frequency in per unit. The primary frequency control of the machine is realized by a droop controller as:

where P m,gov is the mechanical power at the governor input. The SG mechanical dynamics, which include the governor and turbine dynamics, are given in The aim of this appendix is to present the pole placement method [2] that is used in the power system community to tune the gains of PI controllers.

B.1. Theory

First, the classical structure of the PI controller is recalled in In this case, the PI controller's transfer function, denoted by C P I , can be dened as:

where K p and K i are respectively the proportional and integral gains of the controller.

III

And the closed-loop transfer function, denoted by G F , is:

The modes of the closed-loop system correspond to the poles of the transfer function G F (s). Applying a pole placement method here means that we choose adequate values of the PI gains, K p and K i , to shape the denominator of G F (s).

However, because the PI has only two parameters, it is not possible to place the poles of a closed-loop system with an order superior or equal to 3. But, in the case of VSC control, the cascaded control structure [START_REF] Denis | From grid-following to grid-forming: The new strategy to build 100 % power-electronics interfaced transmission system with enhanced transient behavior[END_REF], which is one of the most commonly-used control structures for the VSC, makes it possible to decouple the dynamics of the VSC, thus reducing the plant dynamics to a low order model.

B.2. Example with a simple system

In order to illustrate the pole placement method presented in the above section, we apply this methodology to a simple linear rst-order system representing an RL The control input of this plant is the DC voltage u = v in and the plant output is y = i out . Thanks to the Kirchho 's law, we obtain the following equation:

which is, in the frequency domain:

And the corresponding transfer function G(s) can be calculated as: In order to control this system and track the reference y * = i * out , we need to add a controller before the control input of the plant (v in ) as in Fig. B.1. To illustrate the pole placement method, we choose a PI controller whose gains will be tuned in order to design the denominator of the closed-loop transfer function G F (s). This transfer function is equivalent to:

As we added one order to the system when we introduce the PI controller dynamics, the closed-loop system is of second order. Then it is possible to choose the gains of the PI to obtain the transfer function of a typical second-order system:

where ω n is the undamped natural frequency and ξ the damping ratio.

To make these two G F (s) and H(s) match, we compare their charactistic polynomial:

And if we choose ω n and ξ properly we can then determine the values of K p and K i .

A common choice is:

, ξ = 0.707

Error structures

In this appendix, we present the dierent types of error structures used in the eld of system identication. More information can be found in the following PhD thesis [START_REF] Tohme | Initialization of Output Error Identication Algorithms[END_REF].

We consider the following simple system, dened as:

System identication techniques rely on the minimization of a certain error. There are two main error structures in the literature, namely Output Error (OE) and Equation Error (EE) structures [START_REF] Tohme | Initialization of Output Error Identication Algorithms[END_REF], respectively denoted by OE and EE and dened for the simple system (E.1) as:

where B(s) and Â(s) are respectively the estimations of B(s) and A(s) dened in (E.1). In this study, the identication method relies on the output error structure given in (E.2). This means that the identied model will be obtained by minimizing the error between the measured output y and the estimated model output ŷ.

Prediction error method

This section presents the system identication method named prediction error method and is based on [START_REF] Ljung | System Identication: Theory for the User, ser. Information and system sciences series[END_REF][START_REF] Tohme | Initialization of Output Error Identication Algorithms[END_REF][START_REF] Eckhard | Cost function shaping of the output error criterion[END_REF].

XI

Appendix E. Denition of system identication tools First, we consider a single-input/single-output discrete-time system:

where q is the forward shift operator, u(k) and y(k) are respectively the system inputs and outputs, v a white noise measurement noise and G the transfer function of the system.

Then, the corresponding identied transfer function Ĝ, that we want to match the response of G, is:

where ν(k) is supposed to be white noise and θ is the set of parameters that will be determined by the identication method.

The prediction error using the output-error structure (E.2) is here denoted as (k, θ) = y F -ŷF , where y F (k) = L(q)y(k) and ŷF (t, θ) = L(q)ŷ(k, θ) are respectively the ltered system output and optimal one-step-ahead predicted output calculated using (E.5).

The prediction error method is based on nding an optimal set of parameters θN which minimizes the error over an N-long selected set of data Z N , dened such that:

The algorithm minimizes the quadratic function V N :

And by using this method, we obtain a set of parameters θN of Ĝ that minimizes this criterion:

This set of parameters should converge to a solution θ 0 such that lim N →+∞ θN = θ 0 .

XII

A p p e n d i x

F Kalman filtering

A Kalman lter is a tool that belongs to the family of state estimators. Its main function is to estimate the non-measurable but observable states of a given system by using an estimated model of the real system and available measurements. The details about the implementation in Matlab can be found in [191].

The classical implementation of the continuous-time Kalman lter can be found below and is based on the control theory course given at ISAE-Supaero [START_REF] Alazard | Introduction to Kalman ltering[END_REF].

First, we consider the following continuous-time linear state-space system:

where v and w are the Gaussian noises on the input and the measurement respectively. The objective of the Kalman lter is to obtain an estimated state vector x(t) close enough to x(t). The following assumptions are also made:

• the pair (A,C) is supposed to be detectable. This means that all unobservable states are stable in the model;

• ω(t) and v(t) are centered gaussian white noises, dened by their covariances

The two matrices Q and R are respectively the process noise and the sensor noise covariances.

• R is invertible.

Then the state space realization of the Kalman lter is dened as: • the rst part (Ax + Bu) corresponds to the estimation of the future state based on the knowledge of the process;

• the second part corresponds to the corrective action based on the dierence between the process ŷ = C x -Du and the measurement y.

In consequence, the value of K F will depend on the condence we put in the model.

We can choose an arbitrary value or we can calculate it using an optimization method. It is this second option that is implemented in Matlab, where K F is chosen to minimize the steady-state covariance of the estimation error [191].

The value of K F can be determined by minimizing the cost function J K (t) dened as: This value corresponds to the steady-state covariance of the estimation error such that:

The details to obtain (F.8) using (F.7) can be found in [START_REF] Alazard | Introduction to Kalman ltering[END_REF].

To conclude, the user can play on the two covariance matrices Q and R to change the Kalman lter performances, which respectively represent the condence of the user in the state estimation and the measurement. In addition to these two parameters, it is also possible to act on the condence of the state estimation initialization, which is not detailed here. More information about the condence in the state estimation initialization can be found in [START_REF] Alazard | Introduction to Kalman ltering[END_REF].

XIV