E Merle-Lucotte

D Heuer

M Allibert

V Ghetta

C Le Brun

L Mathieu

R Brissot

E Liatard

Optimized Transition from the Reactors of Second and Third Generations to the Thorium Molten Salt Reactor

Molten salt reactors, in the configuration presented here and called Thorium Molten Salt Reactor (TMSR), are particularly well suited to fulfil the criteria chosen by the Generation IV forum, and may be operated in simplified and safe conditions in the Th/ 233 U fuel cycle with fluoride salts. Amongst all MSR configurations in the thorium cycle, many studies have highlighted the configurations with no moderator in the core as simple and very promising. Since 233 U does not exist on earth and is not being produced today, we aim at designing a critical MSR able to burn the Plutonium and the Minor Actinides produced in the current operating reactors, and consequently to convert this Plutonium into 233 U. This leads to closing the current fuel cycle thanks to TMSRs started with transuranic elements on a Thorium base, i.e. started in the Th/Pu fuel cycle, similarly to fast neutron reactors operated in the U/Pu fuel cycle. We will detail optimizations of this transition between the reactors of second and third generations to the Thorium cycle. Such a transition is based on a fleet of TMSRs with no moderator in the core, including TMSRs started with Plutonium and TMSRs directly started with 233 U. We developed parametric studies to optimize these TMSRs, amongst which the study presented here, based on one of the main TMSR parameters: the percentage of heavy nuclei in the fuel salt of the TMSR configuration, which modifies the moderation ratio of the reactor and thus influences both the initial fissile inventory and the spectrum of the reactor. We analyze the characteristics of each reactor configuration, in terms of deterministic safety parameters, fissile matter inventory, salt reprocessing, radiotoxicity and waste production, and finally deployment capacities.

I. INTRODUCTION

Molten Salt Reactors (MSRs), one of the systems selected by the Generation IV forum, may be operated in simplified and safe conditions in the Th/ 233 U fuel cycle and with fluoride salts, whether in a thermal or a fast neutron spectrum [START_REF] Mathieu | Proposition for a Very Simple Thorium Molten Salt Reactor[END_REF][START_REF] Mathieu | The Thorium Molten Salt Reactor: Moving on from the MSBR[END_REF][START_REF] Mathieu | Cycle Thorium et Réacteurs à Sel Fondu : Exploration du champ des Paramètres et des Contraintes définissant le Thorium Molten Salt Reactor[END_REF][START_REF] Nuttin | Potential of Thorium Molten Salt Reactors[END_REF] . These MSRs are called Thorium Molten Salt Reactors (TMSRs). More precisely, we consider in this study TMSRs with no moderator in the core, the moderation ratio depending only on the salt composition. It has been demonstrated [START_REF] Merle-Lucotte | Fast Thorium Molten Salt Reactors started with Plutonium[END_REF] that such reactors can operate with the relevant properties, in terms of breeding ratio, feedback coefficients, production of transuranic nuclei and material steadiness to irradiation.

In Section II, we introduce the TMSR concept with no moderator in the core, also the called non-moderated TMSR. We state the main criteria used to optimize these reactors, such as safety level, radiotoxicity and waste production, and deployment capacities. Regarding these criteria, we present some parametric studies of this TMSR concept: TMSR configurations are evaluated in section III by varying the reprocessing constraints and the proportion of heavy nuclei contained in the fuel salt.

One of the pending questions concerning TMSRs is the supply of the fissile matter, and as a consequence the deployment possibilities of a fleet of TMSRs, since 233 U does not exist on earth and is not yet produced in the current operating reactors. In order to optimize the transition between second and third generation reactors and TMSRs, we choose to produce 233 U directly in nonmoderated TMSRs started with Plutonium as fissile matter on a Thorium basis, and then operated in the Th/ 233 U cycle [START_REF] Merle-Lucotte | Fast Thorium Molten Salt Reactors started with Plutonium[END_REF] . We aimed at designing a reactor able to burn the Plutonium and the minor actinides produced in currently operating reactors to close their fuel cycle, and consequently to convert this Plutonium into 233 U. Such Pustarted TMSRs are studied and evaluated in section IV.

Finally, the full transition between the second and third generation reactors to the Thorium cycle is optimized in the second and third sections by considering the deployment capacities of each TMSR considered, both Pustarted and 233 U-started configurations. The main constraint of a sustainable power deployment is the saving of resources, i.e. in our case the best conversion of Plutonium in 233 U and the configurations allowing the shortest doubling times, as detailed in section V. This work is based on the coupling of a neutron transport code called MCNP [START_REF] Briesmeister | MCNP4B-A General Monte Carlo N Particle Transport Code[END_REF] with the materials evolution code REM [START_REF] Mathieu | Cycle Thorium et Réacteurs à Sel Fondu : Exploration du champ des Paramètres et des Contraintes définissant le Thorium Molten Salt Reactor[END_REF][START_REF] Nuttin | Potentialités du concept de réacteur à sels fondus pour une production durable d'énergie nucléaire basée sur le cycle thorium en spectre épithermique[END_REF] . The former calculates the neutron flux and the reaction rates in all the cells while the latter solves the Bateman equations for the evolution of the materials composition within the cells. These calculations take into account the input parameters (power released, criticality level, chemistry ...), by adjusting the neutron flux or the materials composition of the core on a regular basis. Our calculations rest on a precise description of the geometry and consider several hundreds of nuclei with their interactions and radioactive decay; they allow a thorough interpretation of the results.

II. GENERAL DESCRIPTION OF THE REACTORS, PARAMETERS AND CRITERIA

II.A. Description of the non-moderated MSR concept

The general concept of the Thorium Molten Salt Reactor (TMSR) is a 2500 MWth (1 GWe) reactor operated in the 232 Th/ 233 U fuel cycle. In this article, TMSRs may be started directly with 233 U or with Plutonium and minor actinides, mixed with Thorium. The reactor is composed of a single large fuel salt channel, 1.25m radius and 2.60m height, as shown on Fig. 1. One third of the 20 m 3 of fuel salt circulates in external circuits and, as a consequence, outside of the neutron flux (Fig. 2). This blanket has been designed such that it stops approximately 80 % of the neutrons, thus protecting external structures from irradiation while improving the system's breeding. The salt channel is also surrounded by two axial reflectors. These reflectors are made of ZrC in order to avoid the use of a moderator material. We assume that helium bubbling in the salt circuit is able to extract the gaseous Fission Products (FP) and the noble metals within 30 seconds [START_REF] Mathieu | Proposition for a Very Simple Thorium Molten Salt Reactor[END_REF][START_REF] Mathieu | Cycle Thorium et Réacteurs à Sel Fondu : Exploration du champ des Paramètres et des Contraintes définissant le Thorium Molten Salt Reactor[END_REF] . We also consider an off-line reprocessing of the total salt volume in a separate chemical unit, with a complete extraction of the FPs. In this reprocessing, the TRUs are not extracted but are reinserted in the core to insure their self-burning in the TMSR. This off-line reprocessing will be discussed in paragraph III.A. We also assume that the 233 U produced in the blanket is extracted within a 6 month period.

II.B. Reactor parameters considered: Proportion of heavy nuclei in the fuel salt and Reprocessing

In this section, we aim at optimizing this TMSR concept by varying two reactor parameters:

-the composition of the fuel salt, more precisely the proportion of heavy nuclei in the fuel salt; -the reprocessing, which modifies the salt composition while modifying the reactor's poisoning by FPs. The modification of the fuel salt consists in varying the proportion of heavy nuclei in the fuel salt; this has an influence both on the initial fissile inventory and on the spectrum of the reactor. For HN proportions ranging from 20 mole % to 30 mole %, the salt used is a binary fluoride salt LiF-(HN)F 4 , whose melting temperature is around 570°C (Fig. 3). We thus choose an operating temperature of 630°C, with a corresponding 233 U proportion in heavy nuclei of about 3 %, a thermodynamic efficiency of 40 %, and a salt density around 4.3 with a dilatation coefficient 8 of 10 -3 /°C. For lower proportions of HN, this binary salt is no longer liquid at the operating temperature of 630°C previously selected. Two alternatives can be considered:

-Introducing Beryllium in the fuel salt to lower its eutectic point. The salt is then LiF-BeF 2 -(HN)F 4 at the same operating temperature as before. -Operating at a higher temperature (around 1000°C), still using a binary fluoride salt LiF-(HN)F 4 .

The first solution, Beryllium addition, has been chosen in this study to avoid some problems du to much higher operating temperatures, in particular with materials. This choice, in turn, raises new issues such as Plutonium solubility, which we discuss in Paragraph IV.D.

Regarding the second parameter, the off-line reprocessing, the amount of FPs produced in the reactor, and thus to be extracted by the reprocessing, depends only on the power produced. It may be difficult, even impossible, to extract the main FPs (lanthanides) in the presence of HN. As a consequence, and in order to have equivalent reprocessing in every reactor configuration, it is necessary to have the same amount of HN reprocessed per day whatever the HN proportion in the fuel salt. We will first study the influence of this parameter in paragraph III.A. Then, for all TMSRs considered in this article we fix a delayed reprocessing of a salt volume containing 200 kg of HN per day. This choice is detailed in paragraph III.A.

II.C. Criteria used for the TMSR evaluation

Our TMSR optimization is based on three main criteria detailed in this paragraph: the safety level, the radiotoxicity and waste production, and the deployment capacities.

II. C. 1. Safety level

The feedback coefficient is traditionally broken down into three sub-coefficients related to the different components of the core presented above:

heating graphite dilatation salt heating salt total dT dk dT dk dT dk dT dk _ _ _ ⎟ ⎠ ⎞ + ⎟ ⎠ ⎞ + ⎟ ⎠ ⎞ = ⎟ ⎠ ⎞
The third sub-coefficient is negligible in our case since there is no graphite moderator in core.

The uncertainties indicated in this article are a quadratic combination of the statistical and systematic uncertainties on the determination of the sub-coefficients. The statistical errors are precisely estimated by simulation, the quantification of the systematic errors is more difficult. More precisely, concerning the systematic uncertainties on the contribution of salt heating, the cross-sections concerned are well known, inducing only negligible uncertainties. The uncertainties on the salt density and its dilation lead to systematic errors lower than 20% on the contribution of salt dilatation, a value used for the estimation of uncertainties in the next sections.

II. C. 2. Waste production and radiotoxicity

Wastes result both from leakages occurring during the reprocessing and from the final transuranic elements inventories.

These transuranic elements inventories present at the end of a reactor's lifespan may be re-injected in a new reactor and are usually not considered as waste, as long as such reactors are operating. But these actinides inventories have to be taken into account for a sustainable waste management, where the entire life of a fleet of reactors, from launching to shutdown, has to be considered. Leakages are proportional to the integral of the transuranic elements inventories and to the amount of HN reprocessed per day (equivalent for all our TMSRs). Leakages occurring all along a reactor's lifespan are negligible compared to the final inventories. Only these final inventories of transuranic elements are thus given for each TMSR, at equilibrium. For Pu-started TMSRs, more precisely here for TMSRs started with the transuranic elements produced in current operating reactors and then operated in the thorium cycle, we estimate the reduction of radiotoxicity reached thanks to these TMSRs at equilibrium, one of our goals being to close the current fuel cycle.

II. C. 3. Deployment capacities

This has to be evaluated both for each reactor configuration and for a combination of Pu-started TMSRs producing 233 U and of 233 U-started TMSRs, as presented in section V. We first compare the 233 U production in a TMSR configuration with its own initial fissile ( 233 U), by considering the reactor doubling time. Then we compare the 233 U production by a Pu-started TMSR configuration with the initial 233 U inventory necessary to launch the corresponding 233 U-started TMSR.

III. THORIUM MOLTEN SALT REACTORS STARTED WITH 233 U

III.A. Study as a function of the off-line reprocessing

For each proportion of heavy nuclei in the salt, ranging from 6 mole % to 27.5 mole %, we evaluate by simulation the breeding ratio of each reactor configuration as a function of the amount of heavy nuclei reprocessed per day. The result of this study is displayed on Fig. 4, allowing the visualization of many parameters of interest of the non-moderated TMSR: reprocessing design, reactor deployment and breeding capacities. The 233 U initial inventory ranges from 2400 kg for a HN proportion in the salt of 6 mole % to 6300 kg for a HN proportion of 27.5 mole %. This corresponds to a variation of the neutron spectrum from an epithermal to a fast spectrum, as shown on Fig. 5.

We chose a reasonable reprocessing rate of 200 kg of HN per day, indicated by the orange horizontal line on Fig. 4. This choice disqualifies TMSRs with HN proportions lower than 7%, since they cannot then be breeder reactors (see Fig. 4).

III.B. Safety level

The total feedback coefficient at equilibrium is displayed on Fig. 6, together with its components, the contributions of the salt heating and salt dilatation, as a function of the HN proportion in the salt. All these safety coefficients are significantly negative for all HN proportions, including the density coefficient which can be viewed as a void coefficient. The total feedback coefficient ranging from -10 pcm/K to -5 pcm/K and thus insuring a very good level of deterministic safety 9 in all these 233 Ustarted TMSR configurations, safety is not a discriminating factor in our case to choose the optimal salt composition. The Pu inventory equal to around 300 kg and the Am inventory equal to 7 kg only slightly depend upon the HN proportion. The Np inventory ranges from 100 to 200 kg, and the Pa inventory from 70 to 100 kg. Note that the Cm inventory displayed on Fig. 7, equal to some kilograms, is not correctly evaluated for the higher HN proportions since this inventory does not yet reach equilibrium in these cases. The global behavior of this Cm inventory, however, is coherent with the results presented on Fig. 7. One tends traditionally to consider that the faster the neutron spectrum is (here the larger the HN proportion is), the lower the amounts of TRU are in a reactor at equilibrium, since the TRU burning is better. We do not observe such a behavior on Fig. 7 because the heavy nuclei inventories are significantly larger for large HN proportions.

Finally the TRU inventories do not really depend upon the HN proportion in the fuel salt, so that this parameter is not discriminating either.

IV. THORIUM MOLTEN SALT REACTORS STARTED WITH PLUTONIUM

The idea in this section is to design a TMSR able to burn Pu while producing 233 U, without loosing the advantages of the 232 Th/ 233 U fuel cycle. We have thus simulated TMSRs started with Plutonium and Thorium, and then operated in the 232 Th/ 233 U fuel cycle, i.e. fed with Thorium. More precisely, to be more realistic, these TMSRs are started with the mix of Pu, Np, Am and Cm listed in Table I corresponding to the transuranic elements of an UOX fuel after one use in a standard PWR and five years of storage [ref] . Note that these reactors and their fuel salt are completely equivalent to the TMSRs presented in the previous section, except for the initial fuel load.

IV.A. Safety level

The feedback coefficients for Pu-started TMSRs have been evaluated after one year of operation and at equilibrium. The total feedback coefficients after one year of operation are displayed on Fig. 11 as a function of the HN proportion, together with their components, the contributions of salt heating and salt dilatation. They correspond to the initial safety behavior of the reactor, since the inventories after one year of operation are quite identical to the initial inventories, but we also take into account the effects of the fission products which are not present in the initial load of the core.

All these initial safety coefficients are still largely negative for any HN proportion, even if less negative (~25%) than for the 233 U-started TMSRs (see Fig. 6). The salt density contribution is equivalent, while the salt heating contribution is less negative because of the presence of Pu instead of 233 U.

The feedback coefficients evaluated at equilibrium are equivalent to the results presented on Fig. 6 for the 233 Ustarted TMSRs, since these reactors are identical at equilibrium (see paragraph IV.B). As detailed in Table II, the initial fissile inventories are higher for the Pu-started TMSR than for the 233 Ustarted TMSR, from 20% for the larger HN proportions to 40% for the lower HN proportions. Only the amount of fissile Pu is indicated for the Pu-started TMSRs, the amounts of other transuranic elements used to start the reactor being proportional according to the percentages given in Table I.

IV.B. Inventories and Waste reduction

The initial inventory in fissile Pu necessary for a Pustarted TMSR ranges from 4 to 8 metric tons, and the Thorium initial inventory from 14 to 42 metric tons. These reactors being studied to close the current fuel cycle, we have to estimate the reduction of radiotoxicity reached thanks to these TMSRs at equilibrium. We aim at burning all TRUs introduced to start the reactor, so as to have TRU inventories at equilibrium identical to the TRU inventories of 233 U-started TMSRs, and this is verified. Fig. 9 shows the burning rate obtained for all the transuranic elements after 25, 50, 100 and 200 years of operation. Operation times greater than 60 years (a reactor lifespan) are obtained by transferring the salt contained in the ending TMSR in a new TMSR. This is made easier thanks to the fact that the fuel is liquid. The lower HN proportion configurations allow faster reductions, since more than 80% of the TRUs are burned after only 25 years of operation, and nearly 95% after 50 years of operation. The larger HN proportion configurations allow higher burning rates, up to 97%, but on the longer run, after more than around 100 years of operation.

Finally, in terms of transuranic inventory, the TMSRs started with Plutonium become equivalent to TMSRs directly started and operated with 233 U after around forty years, where more than 85% of the initial TRU inventories are burned, the assets of the Thorium fuel cycle being then recovered.

IV.D. Plutonium Solubility

Pu having a lower solubility limit in a fluoride salt, compared to Uranium for example, this issue has to be examined for the Pu-started TMSRs. Fig. 10 presents the initial mole proportion in the different Pu-started configurations (red line), proportion ranging from 2.5% for the lower HN proportions to 6.5% for the larger HN proportions. According to many studies among which the MSBR studies, it is well attested that the Pu maximum solubility limit in a fluoride salt decreases as the Th proportion increases, deteriorates even faster as Be is added, and is improved if the temperature is increased.

The maximum solubility limit for Pu indicated on Fig. 10 together with its uncertainty (yellow area) give an idea of the typical shape of the solubility limit in a fluoride salt. The lowering of this limit for the smaller HN proportions is due to the addition of Beryllium in the salt. In our simulations, the maximal Pu solubility limit is reached for the TMSR with 20% of HN in the salt, with no Be added and with the minimal amount of Th in the salt. The reactor configurations with larger HN proportions may encounter Pu solubility problems. The yellow area states for the Pu solubility limit with its uncertainties.

While taking into account the uncertainties on the evaluation of the solubility limit and to be sure not to overshoot this limit, a good compromise could be to slightly increase the reactor operating temperature by around 100 degrees. This allows the use of a fuel salt with no Beryllium and with a lower proportion of HN, and thus a good solubility of Pu.

V. DEPLOYMENT CAPACITIES AND RESOURCES MANAGEMENT

Concerning 233 U-started TMSRs, the deployment capacities are based on the amount of 233 U produced compared to the initial fissile ( 233 U) inventory necessary to start such a reactor. An example of the amount of 233 U produced and extracted all along a TMSR lifespan is presented on Fig. 11 To optimize the deployment capacities of a reactor, it first seems logical to minimize the initial fissile inventory, especially in our case where the fissile matter used is not currently available. This means choosing preferably the lower HN proportions. As previously mentioned, a more complete parameter to consider is the reactor doubling time, which combines this initial fissile inventory constraint with the breeding capacities of the system, i.e. its own capacity to start itself. By generalizing the procedure illustrated on Fig. 11, we thus calculate the reactor doubling times for all HN proportions considered (see Fig. 12). We see again the result detailed on Fig. 11 of a reactor doubling time of 83 years for an HN proportion of 10%.

The higher deployment capacities allowed by the use of TRUs in the Pu-started TMSR are also visible on the reactor doubling times, displayed on Fig. 12 (dashed line), where the configurations with HN proportions larger than 15% have the lower reactor doubling times, around 30 years, to be compared with the doubling times of 45 years reached for the 233 U-started TMSRs. Finally, these configurations with HN proportions larger than 15% have the lowest reactor doubling times and consequently have the best deployment capacities. These findings that the TMSR configurations with large HN proportions have higher deployment capacities come into conflict with the simple analysis based only on the minimization of the initial fissile inventory, considered at the beginning of this section. Some neutronic behaviors of the core have indeed to be taken into account. For example, for an identical reprocessing, the poisoning of the core by the fission products is relatively more significant for a configuration with a lower HN proportion, as shown on Fig. 13 which presents the values of some parasitic capture rates for different HN proportions. This leads to a reduction of the breeding ratio during reactor operation and finally to smaller deployment capacities for the lower HN proportion configurations.

To conclude this paragraph, the use of Pu and minor actinides to start TMSRs not only allows closing the current fuel cycle but also improves the deployment of such reactors.

VI. CONCLUSIONS

We have presented here a very promising, simple and feasible concept of Molten Salt Reactor with no moderator in the core, operated in the Th/ 233 U fuel cycle, and called non-moderated Thorium Molten Salt Reactor (TMSR).

We have detailed in this article some parametric studies, related to the system reprocessing constraints, and the heavy nuclei composition of the salt which modifies the neutron spectrum of the reactor. We have thus demonstrated the very good characteristics of such reactors, in terms of deterministic safety parameters, waste production, deployment capacities… Since 233 U does not exist on earth and is not being produced today, we have examined a new way to produce 233 U directly in standard TMSRs started with the Plutonium and transuranic nuclei produced in the currently operating reactors as fissile matter on a Thorium base, and then operated in the Th/ 233 U cycle. We aimed at designing a critical reactor able to burn the Plutonium and the Minor Actinides produced in the currently operating reactors, and consequently to convert this Plutonium into 233 U. This leads to closing the current fuel cycle thanks to these Pustarted TMSRs, i.e. TMSRs started in the Th/Pu fuel cycle, similarly to fast neutron reactors operated in the U/Pu fuel cycle. The burning of transuranic elements in these Pustarted TMSRs results in high waste reduction rates, up to 95-97% for all TMSR configurations evaluated.

We particularly point out in our analyses the excellent level of deterministic safety of all the TMSR configurations studied, for the 233 U-started TMSRs as well as for the Pu-started TMSRs.

Finally, to optimize the transition from the second and third generation reactors to the Thorium cycle, we have considered the deployment capacities of each TMSR configuration, combining Pu-started and 233 U-started reactors. The main constraint of a sustainable power deployment was the saving of resources, i.e. in our case the higher breeding ratio and the best conversion of Plutonium in 233 U. This results in short reactor doubling times, around 30 years, especially for the TMSR configurations containing the proportions of heavy nuclei larger than 15%, i.e. with the faster neutron spectra. Crossing this criterion with the constraint of the Pu solubility, which leads to a preference for TMSR configurations with less than around 20% of heavy nuclei in the salt, the most promising TMSRs turn out to be the configurations with heavy nuclei proportions ranging from , 15% to 20%, typically 17.5% of heavy nuclei in the fuel salt. This typical TMSR configuration contains 4.5 mole % of Plutonium to launch the Pu-started version, with a reactor doubling time of 30 years and a feedback coefficient at equilibrium equal to -7 pcm/K both for the 233 U-started and the Pu-started reactors, allowing a high level of deterministic safety. Fig. 14 illustrates the evolution of the fuel salt composition all along the operation of this reactor, for the 233 U-started (solid lines) and for the Pu-started (dashed lines) TMSR.

Fig. 1 .

 1 Fig.1. Vertical section of the non-moderated TMSR simulated core A fertile radial blanket surrounds the core. It contains a binary fluoride salt LiF -ThF 4 , with 28 mole % of 232 Th.This blanket has been designed such that it stops approximately 80 % of the neutrons, thus protecting external structures from irradiation while improving the system's breeding. The salt channel is also surrounded by

Fig. 2 .

 2 Fig.2. Vertical schematic section of the whole TMSR, including pumps and heat exchangers

Fig. 3 .

 3 Fig.3. LiF -ThF 4 phase diagram

Fig. 4 .

 4 Fig.4. Amount of heavy nuclei reprocessed per day versus initial fissile ( 233 U) inventory, for different heavy nuclei proportions in the fuel In Fig. 4, the dark line represents breeder reactors, the red one the reactors producing their initial 233 U inventory in 100 years (i.e. what is called a doubling time of the reactor of 100 years), and finally the green line stands for reactors producing their initial 233 U inventory in 50 years (or a reactor doubling time of 50 years). Under-breeder reactor configurations, which are located under the black line (bottom of the figure), will not be considered in the following since they do not allow any sustainable reactor deployment.

Fig. 5 .

 5 Fig.5. Neutron spectrum of two TMSR configurations (6% and 27.5% of heavy nuclei in the salt) compared to the neutron spectrum in a pressurized water reactor (PWR) and in a fast neutron reactor (FNR)

Fig. 6 .

 6 Fig.6. Feedback Coefficients of 233 U-started TMSRs at equilibrium as a function of the HN proportion

Fig. 7 .

 7 Fig.7. Inventory of the transuranic elements of the 233 Ustarted TMSRs at equilibrium, as a function of the proportion of heavy nuclei in the salt

Fig. 8 .

 8 Fig.8. Feedback Coefficients of Pu-started TMSRs after one year of operation as a function of the HN proportion

Fig. 9 .

 9 Fig.9. Burning rate of all the transuranic elements after different operation times, as a function of the HN proportion

Fig. 10 .

 10 Fig.10. Initial Pu mole proportions in the fuel salt for the different HN proportion configurations (dark dots and red line).The yellow area states for the Pu solubility limit with its uncertainties.

  (red line) for a TMSR with 10% of HN in the fuel salt, other TMSR configurations with different HN proportions behaving identically: for TMSRs directly started with 233 U, the 233 U extraction follows a linear growth, of 37 kg per year in the case shown. This production is directly related to the breeding ratio of the configuration. Pu-started TMSRs allow the extraction of significantly larger amounts of 233 U during the first 20 years of operation (Fig. 11, black line), thanks to the burning of TRUs which saves a part of the 233 U produced in the core.This production reaches 100 kg per year in the example shown, i.e. 60% more than in the corresponding 233 Ustarted TMSR. After the first 20 years, the 233 U extraction rate is equivalent to that of the 233 U-started TMSR.

Fig. 11

 11 Fig.11. 233 U extracted during the operation of TMSR with 10% of HN in the fuel salt The operating time necessary to produce one initial fissile inventory is called the reactor doubling time. On Fig. 11, the reactor doubling time of a 233 U-started TMSR is equal to 83 years and corresponds to the crossing of the 233 U production line (red line) with the initial 233 U inventory (green dashed line). This reactor doubling time is reduced to 40 years when using Pu in the first load (Fig. 12. dashed line).

Fig. 12 .

 12 Fig.12. Reactor doubling time of 233 U-started (solid line) and Pu-started TMSRs (dashed line) for different HN proportion configurations

Fig. 13 .

 13 Fig.13. Examples of parasitic capture rates for different components of the fuel salt as a function of the HN proportion

Fig. 14 .

 14 Fig.14. Heavy nuclei inventory for the 233 U-started TMSR (solid lines) and for the Pu-started TMSR (dashed lines)

TABLE 1

 1 Proportions of transuranic nuclei in UOX fuel after one use in PWR without multi-recycling (burnup of 60 GWd/ton) and after five years of storage[START_REF] De Saint Jean | Scénarios CNE : réacteurs classiques, caractérisation à l'équilibre[END_REF]

	Element	Proportion in the mix
	Np 237	6.3 %
	Pu 238	2.7 %
	Pu 239	45.9 %
	Pu 240	21.5 %
	Pu 241	10.7 %
	Pu 242	6.7 %
	Am 241	3.4 %
	Am 243	1.9 %
	Cm 244	0.8 %
	Cm 245	0.1 %

Table II .

 II Initial inventories (kilograms) of Th and fissile matter for the Pu-started TMSRs and for the 233 U-started TMSRs

	HN	233 U started TMSR	Pu-started TMSR
	proportion	Th	233 U	Th	239 Pu+ 241 Pu
	7.5%	19760	2550	13850	4887
	10%	24050	3105	17560	5524
	12.5%	27790	3575	20890	6022
	15%	33060	4140	25380	6786
	17.5%	37230	4650	29150	7297
	20%	42380	5170	33640	7968
	22.5%	46100	5580	37040	8378
	25%	48640	5820	39320	8668
	27.5%	52190	6180	42570	9037

ACKNOWLEDGMENTS

We are thankful to Elisabeth Huffer for her help during the translation of this paper.


Jean-Thomas Camino

Christian Artigues

Laurent Houssin

Stéphane Mourgues

Mixed-Integer Linear Programming for Multibeam Satellite Systems Design: Application to the Beam Layout Optimization

Keywords: Satellite Telecommunications, Systems Design, Mixed-Integer Linear Programming, Operations Research

In a society where the demand for multimedia applications and data exchange is experiencing an unstoppable growth, multibeam systems have proven to be one of the most relevant solutions for satellite-based communication systems. Though already well represented among the geostationary satellites today, there are still several unresolved design optimization challenges for these complex systems that could lead to improved performances and to better system costs. The satellite platform, the repeater, and the antennas are examples of subsystems that should be designed jointly in order to reach an optimized technical solution that fulfills the service requirements. Traditionally, such complex tasks are addressed through a decomposition of the overall system design into a sequence of smaller decision problems. In this article, we propose to rely on operations research techniques to, on the one hand, take into account explicitly the interdependencies of these decomposed problems, and on the other hand, to handle the own constraints of each subsystem and their interactions. In this paper, the focus is laid on the optimization of the beam layouts of the multibeam satellites. Indeed, in addition to being a perfect example of the aforementioned importance of dealing with subsystem constraints, this problem appears early in the chain of design of a multibeam satellite system and is therefore critical for the quality of the telecommunication system: the weaknesses of a beam layout cannot be made up for later on in the system design. For this crucial optimization phase, the strength of the methodology we propose in this paper is to use mixed-integer linear programming to incorporate explicitly technological feasibility constraints of the subsystems involved, while preparing at best the subsequent design problems. Most importantly, our approach allows to overcome several resisting flaws of the already existing algorithms.

I. INTRODUCTION

On the service area of a given satellite system, any user can be reached regardless of its location. This is particularly interesting for rural areas for which deploying a terrestrial network is hard or even impossible. This unique characteristic of the satellite systems combined with their ability to offer all kinds of telecommunication services (television, radio, telephone, internet...) have made them indispensable in our ultra-connected society. A multibeam satellite is a particular type of telecommunication satellite characterized by a plurality of relatively narrow beams used to provide coverage to its service area. As discussed further in the article, compared to systems that use only one large beam to cover the entire service area, Fig. 1. Forward link of a multibeam satellite system the antenna gains are significantly improved with the multibeam solution. Plus, since two beams that are sufficiently separated spatially are allowed to reuse the same frequency, opting for a multibeam coverage is a way to make the most of the available system bandwidth which is a really expensive resource in any telecommunications system. As shown in Fig. 1, one or several gateways (ground stations) are connected through the terrestrial network to producers of multimedia content or to the internet backbone, depending on the service provided by the satellite. Radio frequency signals are generated from these gateways in direction of the satellite in the uplink, denoted by [START_REF] Angeletti | Non-regular multibeam coverage payloads for non-uniform traffic demand[END_REF] in Fig. 1. The role of the satellite payload (2) (composed of "receive" and "transmit" radiofrequency antennas and of the so-called "repeater") is then to receive, downconvert in frequency, amplify in power, and retransmit these signals in the different beams of the downlink (3) where the end-users are located and equipped with terminals allowing them to receive the content delivered by the satellite. This is called a "forward link", which in some cases (internet service by satellite for instance) is completed by a "return link" from the users to the gateways. This standard system architecture of Fig. 1 is the one we chose to consider in this study.

Reaching the high data rates aimed in the telecommunications services today raises several system level challenges for multibeam satellites, and has therefore motivated a significant amount of research. In [START_REF] Vidal | Next generation high throughput satellite system[END_REF], the state-of-the-art of these systems is summarized and promising techniques to improve the system performances are analyzed: wider spectrum allocation, frequency reuse schemes, feeder link design (between the gateways and the satellite) and interference mitigation techniques. In [START_REF] Poulenard | High altitude clouds impacts on the design of optical feeder link and optical ground station network for future broadband satellite services[END_REF], the authors highlight the interest of relying on optical communications instead of traditional radiofrequency signals for the feeder link of geostationary satellite systems, and propose a methodology and algorithms to tackle the several design issues that these innovative system architectures raise. The authors of [START_REF] Christopoulos | Coordinated multibeam satellite co-location : The dual satellite paradigm[END_REF] analyse multibeam systems that use two co-localized multibeam satellites that are coordinated to provide enhanced and more robust performances. Some research papers focus exclusively on one of the satellite subsystems: for instance, the authors of [START_REF] Touya | Control and topological optimization of a large multibeam array antenna[END_REF] investigate algorithms to optimize the design of multibeam array antennas. For a given system architecture and a given satellite payload, literature is also rich on the question of how to optimize the allocation of resources such as the satellite power and the system bandwidth: [START_REF] Park | A dynamic bandwidth allocation scheme for a multi-spot-beam satellite system[END_REF], [START_REF] Kiatmanaroj | On scheduling models for the frequency interval assignment problem with cumulative interferences[END_REF], [START_REF] Kiatmanaroj | Frequency assignment in a SDMA satellite communication system with beam decentring feature[END_REF], [START_REF] Wang | Optimization of power allocation for a multibeam satellite communication system with interbeam interference[END_REF] and [START_REF] Touati | Radio planning in multibeam geostationary satellite networks[END_REF] for instance. On the other hand, the stress is rarely put on the strong interdependencies of all the subsystems of a multibeam satellite system in the design phase. Under the assumption that the system architecture has already been chosen and is the one of Fig. 1, the objective of this article is indeed to demonstrate that, when some crucial design problems are not solved jointly for the different subsystems involved, then, whatever the quality of the following allocation of resources and individual subsystem optimization processes, the overall system solution will be either technologically infeasible or perfectible in terms of cost or performances. We also show that in the case of the complex systems that are the multibeam satellite networks, the process of making optimized technical decisions can be greatly enhanced if one relies on operations research techniques to handle the numerous constraints and criteria of all the different subsystems that interact. In this context, the beam layout optimization problem is presented and an example of algorithmic solution is provided and compared to the literature on the subject.

The rest of the article is structured as follows. Section II defines precisely the telecommunication mission requirements that are at the start of any multibeam system. They are derived into a set of design challenges, the stress being put on the space segment optimization for which a design process is proposed. In section III, the crucial beam layout optimization step is tackled with operations research techniques and more particularly mixed-integer linear programming: the assumptions made and the mathematical models are detailed. Section IV presents experimental results and comparisons with other algorithmic solutions. The conclusion and the way forward are discussed in section V.

II. TELECOMMUNICATION MISSIONS AND CORRESPONDING DESIGN CHALLENGES

The requirements of the telecommunication missions can be of different types. In their most general form, they consist of a series of specific services that should be provided to designated regions of the Earth: television, radio, telephone, broadband applications (internet, videoconferencing...) to name a few. Both the quality and the quantity of these services can be specified. In addition, there might be for instance cost constraints on the overall system. Here, we will consider requirements inspired by broadband applications, where each point defined on the service area has its own traffic request expressed in Megabits per second, as illustrated with the fictitious scenario of Fig. 2 (generated randomly). From this specification of needs, the satellite system manufacturers must propose a whole solution including details on the ground segment (the gateways and the control centers), the user segment (satellite terminals), and the space segment (one multibeam satellite in the case considered here). Each of these three complex systems can be split in several subsystems with their own particularities and design challenges. It is impossible today, either numerically or manually by experienced engineers, to design globally the three segments of a multibeam satellite system: there are too many decisions to make on numerous interacting subsystems, and computationally heavy system simulations to assess the performances of the selected designs. The standard way to deal with such an issue Fig. 2. Example of traffic demand map is to divide the problem into simpler design questions that are chosen as to have a series of subproblems to solve that are as independent as possible. This very principle naturally leads to a division of the system design into the separate designs of the ground, user, and space segments. In this article, we assume that the ground segment and the user segment have already been designed: the number of gateways, their power, their position, the aggregate available system bandwidth, and the types of user terminals are known. As a result, it is possible to focus exclusively on the space segment and to discuss design approaches that can handle the various constraints (technological feasibility) and criteria (complexity, cost, weight) of the satellite payload, aswell as those dictated by the system requirements (see Fig. 3). Ideally, they should be handled all at once, but this global optimization would also be too complex to be effective in practice. Therefore, it is once again necessary to cleverly decompose the Fig. 3. Overview of the space segment design system design. In the following section, such a decomposed process is proposed with the stress being put on one crucial resulting design problem: the beam layout optimization.

III. BEAM LAYOUT OPTIMIZATION THROUGH MIXED-INTEGER LINEAR PROGRAMMING

In Fig. 1, we have an example of multibeam coverage with 13 beams over European countries. Note that, for a radiofrequency antenna, the antenna gain is a measure of the power concentration of the radiated electromagnetic signals, value therefore defined for each direction in three dimensions. Each beam is the representation of a certain antenna gain loss (-3dB is a commonly encountered threshold for instance) relatively to the peak gain value for the corresponding satellite radio source: as a first approximation, the rule is that the gain decreases when the angular distance between the considered direction and the peak direction increases. Also note that the bigger the beam, the less the peak antenna gain for this beam, which explains why multibeam systems are attractive compared to single beam systems that cover the same regions with only one large beam. From the point of view of the satellite, and looking in direction of the Earth, common satellite antenna designs lead to beams that can be represented disks on the surface of the Earth. This results in the definition of the "beamwidth" which is simply the diameter of such disks. We will say that a station is covered by beam if its disk representation contains this station. For a given service area, defining a beam layout means deciding what will be the number of beams, the position of the beam centers on the surface of the Earth, the beam diameters, the mapping of the user stations to the different beams, and the satellite reflector antenna associated to each beam. Note that the performances and cost of a broadband satellite system are expected to differ notably from one beam layout to another according to how the heterogeneity of the traffic demand (case considered) over the regions to cover is handled. One key remark is that the common beam layout solutions are not well suited for heterogeneous traffic demands (see III-B). On the other hand, potentially more appropriate solutions can cause a significant increase on the payload complexity or can compromise the technological feasibility, which needs to be controlled from the start when the beam layout is defined. This is why we decided to devote particular attention to this already very complex optimization: the beam layout is the first step of the space segment design we propose in Fig. 4. As discussed earlier, the only way this process can be effective is by making explicit the links between the different design steps. To that end, we chose to propose an algorithmic solution that defines the beam layout under antenna and repeater constraints to create favorable conditions for the subsequent design optimization phases: antenna and repeater design. Of course, the required system performances and cost are already targeted during this first optimization. These constraints and objectives are detailed in the following section.

A. Considered constraints and objectives 1) Platform constraints :

The main platform constraint we will consider is the limit on the number of beams that can be embarked on the spacecraft. This can be due to either power, cost, mass or accommodation limitations. In any case, we will explicitly integrate this upper-bound on the number of beams in our beam layout optimization algorithm.

2) Antenna constraints :

As illustrated in Fig. 1, each beam is transmitted by exactly one satellite reflector antenna (4 of them are represented, each one characterized by its own color). Under the assumption adopted throughout this study that Single Feed Per Beam antennas are used (one radio source per beam), the main technological constraint is that, for each layout considered, there must exist a mapping of the beams to the different reflector antennas for which there is no overlapping in the corresponding blocks of sources. This leads to a minimum angular distance from the satellite point of view between two beams associated to the same reflector antenna, that distance being a function of the size of the sources. This technological constraint is represented in Fig. 5 where some elements of the antenna geometry are represented: the focus O, the focal length F, the slant focal length F s , the diameter D (projection of the parabola diameter on the focal plane), and the angular distance between the beams ∆ θ caused by the inter-source distance ∆ i, j . The important property is that using larger beamwidths requires larger sources to concentrate the illumination on a reduced reflector surface, which leads to greater minimum inter-sources distances and, therefore, to greater minimum separation distances. In practice, the rule we decided to use for the remainder of the study was to assume that the minimum angular separation for two beams transmitted by the same reflector is directly proportional to the mean of their two beamwidths, according to a

coefficient κ ∈ R + such that 3 2 ≤ κ ≤ √ 3 (1)
This range is representative of commonly encountered configurations for such Single Feed Per Beam satellite systems.

3) Repeater limitations constraints :

Note that the traffic demanding stations are assigned to the beams leading to an aggregate traffic request per beam. While these demands are a result of the beam layout optimization procedure, they are strongly connected to the following payload design phases.

For instance, we can define for each beam a maximum traffic demand (that would depend on the beam diameter) above which we lose the guarantee that there will exist a feasible repeater architecture (hardware, frequency plan and bandwidth allocation) that can answer properly to that demand. This is exactly the constraint we implemented to make sure that the following payload design challenges can be resolved.

4) Objective :

Each station is characterized by its own traffic demand. The objective we decide to target is the maximization of the aggregate covered traffic, the ideal goal being 100%. Note that the traffic of a given station is considered covered when this station is covered by at least one beam and assigned to one of them.

B. Example of solutions to the beam layout problem

In this section, a concrete scenario is analyzed to understand why there is a need for non-uniform beam layouts. In Fig. 6, the same traffic demanding service area has been covered by three distinct beam layouts. The first two are regular layouts with respectively 117 beams of 0.30 • , and 50 beams of 0.5 • . The third layout is a nonuniform one with 48 beams that combine the two beamwidths. We assumed that 4 antenna reflectors were available on the spacecraft and that κ = √ 3 for the angular separation constraint of equation [START_REF] Angeletti | Non-regular multibeam coverage payloads for non-uniform traffic demand[END_REF]. For the feasibility of the frequency plan and of the repeater, a maximum traffic per beam has been defined per beamwidth (higher threshold in the case of the smaller beams). In all three cases, more than 99.5% of the traffic is covered, so as far as what concerns the objective, there is no way to differentiate the three solutions. On the left (subfigures (a), (c) and (e) of Fig. 6), the mapping of the beams to the satellite reflectors is represented (one color per reflector) to examine the angular separation constraint. We observe that in the case of the regular layouts, the antenna constraint is fully satisfied: the mapping of the beams is natural and follows a repeated pattern. On the other hand, for the irregular beam layout, this assignment of reflectors is a real challenge and we can observe that some couples of large beams using the same reflector are closer than in the case of the regular layouts: this solution is not fully compliant with the antenna constraints. Yet, if we had managed to deal properly with the full set of antenna constraints, the irregular layout would have been way more interesting. Indeed, it uses less beams that the two regulars solutions, but most importantly, the balancing of the traffic is more appropriate and allows to satisfy all the maximum load per beam constraints. On that regard, the subfigures (b), (d) and (f) of Fig. 6 provide the load of each beam for each layout (blue is for the low values and red for the high values). The first regular layout with 117 small beams uses too many beams and most of them have really low demands (which is not cost-effective for the hardware embarked), and one of them exceeds its maximum value allowed (the most red beam). With the regular layout of 50 beams, this time the number of beams is acceptable and the traffic is well balanced in average, but several beams exceed their maximum load limits, leading to impossibilities to find a bandwidth allocation and a corresponding payload architecture that could satisfy these demands. In the irregular

C. Existing work

The previous section showed that the common technique that uses a single beamwidth and the hexagonal lattice to cover the entire region is not appropriate for heterogeneous traffic demands. Yet, this pattern is known to maximize the density of circle packing (see [START_REF] Chang | A simple proof of thue's theorem on circle packing[END_REF] for instance), and would be optimal for a traffic demand perfectly homogeneous in space and in intensity. Also, the antenna feasibility (see III-A2) has been proven for these regular layouts for a spacecraft equipped with 3 or 4 reflectors which is a strong enough reason to continue to rely on them regularly. On the topic of breaking this regularity of the layouts, some litterature can be found. The authors of [START_REF] Angeletti | Non-regular multibeam coverage payloads for non-uniform traffic demand[END_REF] suggest to aim at a load-balanced distribution of these demands within the different beams through an adjustment of their widths, the main idea being that imbalances of traffic demands inside the beams can lead to user stations being in the end either overloaded or underloaded in terms of offered capacity. The method proposed uses a partitionning of the region into evenly loaded polygonal subparts, each one of these being then treated individually to determine what beamwidth should be used in it and where the beam should be placed through an analysis of its shape and surface, thus defining the full layout. In that work, the question of the mapping of the beams to the antenna reflectors is not addressed. Furthermore, in [START_REF] Kyrgiazos | Irregular beam sizes and non-uniform bandwidth allocation in hts multibeam satellite systems[END_REF] the authors observed that a drawback of this last method is that several zones remain uncovered or poorly covered. To overcome this issue, they use artificial neural networks and more precisely self-organizing maps that treat all the traffic demands in order to define sub-areas, each sub-area corresponding to one of the available beamwidths. The reasoning behind this method is that using narrow beams on the densest zone is a way to increase the antenna gains and the frequency reuse where it is the most needed, while using larger beams in the least dense areas can be sufficient to reach compliance. Beams are then placed in each sub-area in such a way as to minimize the uncovered areas, the regular lattice being used locally. Once the beams are placed over the service zone, the authors tackle the issue of the antenna configuration with a reflector allocation to the beams that includes a maximization of the angular distances between the beams coming from the same antenna, that NP-hard problem being solved heuristically. But the major drawback of treating the placement of the beams and their allocation to the antenna reflectors sequentially is that one is very likely to discover that, with a layout built regardless of the antenna constraint, even the optimal mapping of the beams to the reflectors can turn out to be technologically infeasible for the Single Feed Per Beam antennas. Guided by a combinatorial model of the problem, [START_REF] Camino | A greedy approach combined with graph coloring for non-uniform beam layouts under antenna constraints in multibeam satellite systems[END_REF] proposes a heuristic procedure combining a randomized multi-start greedy approach and graph coloring to find jointly interesting positions for the beams and an acceptable beam mapping to the reflectors, with an upper-bounded number of beams and a pre-processing of the beamwidths to use.

D. Mixed-integer linear programming model

We propose in Fig. 4 a space segment design process in which sizing problems of acceptable complexity emerge. Reducing the design to a series of simpler problems is numerically interesting but is risky in the sense that it is necessary to make sure that the subsequent design problems in the process flow will be feasible, and in the best conditions given all the decisions already made.

To that end, we decided to rely on operations research and more particularly on mixed-integer linear programming to solve the beam layout optimization problem with the constraints and objective of section III-A. The strength of the model we propose, besides being able to handle the subsystems interactions, is to handle actively the positioning of the beams: for each beam, there are variables defining its center. Most importantly, as opposed to what we can find in the litterature on the subject, we propose a model that allows to keep the domain of centers continuous. To do so, we observed that the beam positioning can be entirely done in the Euclidean plane in the right coordinate system (the true view angles coordinate system, see [START_REF] Bousquet | Satellite Communications Systems : Systems, Techniques and Technology[END_REF]), and we rely on results on the inner product in the plane to handle the angular distances through a discretization of the directions [0, 2π[. The mathematical model is presented in Fig. 7, voluntarily without too many details on how we reached it. The goal is really to show that mathematical expressions can capture the interconnections of the satellite subsystems and, therefore, that operations research can be a good tool for the system engineer facing complex design problems. In the model of Fig. 7,

B = {1, • • • , N B }
is the set of beams that can be placed on the service area, N B being the maximum number of beams allowed by the satellite platform. w ∈ W , we define an upper-bound for the traffic assigned to a given beam: Γ w ∈ R + . α s,b , ω b,w , ρ b,r , a b ∈ {0, 1} are binary variables that represent respectively: the covering ("1") or not ("0") of station s by beam b, the choice ("1") or not ("0") of beamwidth w by beam b, the choice ("1") or not ("0") of reflector r by beam b, and the fact that beam b is active ("1") or not ("0"). There are further binary variables (β b,b , γ b,b ,u ) and real positive coefficients (M s ,N s ,N) that are only here to express correctly, and relax when needed, all of our system constraints.

S = {1, • • • , N S }
Maximize ∑ (s,b)∈S ×B T s α s,b (2)
under the following constraints In Fig. 7, equations ( 2),( 3),( 4),(5) represent system level decisions and constraints: maximization of the covered traffic, a station is served at most by one beam, a station can be served by a beam only if the beam covers the station geometrically, and a non-active beam cannot serve any station. ( 6),( 7),( 8),( 9), [START_REF] Poulenard | High altitude clouds impacts on the design of optical feeder link and optical ground station network for future broadband satellite services[END_REF] are antenna considerations: each beam is assigned exactly one reflector and one beamwidth, and two active beams assigned to the same reflector must be spatially separated according the equation (1). Finally, equation ( 11) is the maximum load per beam defined per beamwidth to ensure the feasibility of the following repeater design phases. Even if each equation can be categorized, they well capture the interconnections of all the subsystems involved. For instance, the beamwidths intervene at all levels: the geometrical covering of the stations, the spatial separation antenna constraint, and the maximum load per beam repeater constraint.

∀s ∈ S, ∑ b∈B α s,b ≤ 1 (3) ∀s ∈ S,∀b ∈ B,∀u ∈ U, x b -X stations,s y b -Y stations,s T U u,x U u,y ≤ 1 2 ∑ w∈W W w ω b,w + (1 -α s,b )M s (4) ∀b ∈ B, ∑ s∈S α s,b ≤ N s a b (5) ∀b ∈ B, ∑ w∈W ω b,w = 1 (6) ∀b ∈ B, ∑ r∈R ρ b,r = 1 (7) ∀b, b ∈ B such that b > b, β b,b + ∑ u∈U γ b,b ,u ≥ 1 (8) ∀b, b ∈ B such that b > b, ∀u ∈ U,

IV. FIRST EXPERIMENTAL RESULTS

Experiments were conducted on the linear model of Fig. 7 to assess its performances. We generated a series of instances characterized by a growing number of user stations in input (progressive expansion of the service zone). In the results of Fig. 8, 168 instances were tested, with a total number of stations per instance growing from 190 to 858. There were 4 available satellite reflectors, with a separation coefficient κ = √ 3. The traffic request of each station has been generated randomly and the upper-bounds of load per beam adjusted accordingly. Five beamwidths were defined: W = {0.35 • , 0.40 • , 0.45 • , 0.50 • , 0.55 • }. The number of discretized directions has been set to n directions = 12. The maximum number of beams has been adjusted three times according to the size of the instance (which is visible with the steps in figure (a) of Fig. 8). For all the runs, the MILP solver Gurobi has been used and we set a timeout to 600 seconds: the relative gap between the best solution and the best bound that the model behaves extremely well on small instances with several instances being solved optimally (gap=0%) in 600 seconds. On larger instances, the results are more mitigated since there are cases were the solution reached is relatively far from the estimated best possible solution. Yet, the good results on small instances have motivated a comparison with the greedy heuristic solution of [START_REF] Camino | A greedy approach combined with graph coloring for non-uniform beam layouts under antenna constraints in multibeam satellite systems[END_REF]. This comparison is natural since it is the only other known algorithm that considers the SFPB antenna separation constraints and that handles the beam centers actively. Fig. 9 summarizes their respective characteristics. An important remark is that, while our mixed-integer linear programming is exponentially impacted by the size of the instances (see subfigure (b) of Fig. 8), the complexity for the greedy heuristic grows linearly with respect to this input data size and allows to solve bigger instances. On the other hand, on small instances, our algorithm reaches and proves the optimality of the solution reached where the greedy algorithm only provides good but suboptimal solutions. This result should encourage the research on mixed-integer linear programming inspired techniques for the beam layout optimization.

V. CONCLUSION AND WAY FORWARD

The article allowed to highlight the complexity of designing multibeam satellite systems. A decomposition of the system design is therefore necessary and we proposed a detailed process for the space segment. The stress was put on the need to have explicit links between the different design steps, otherwise the solution produced might be infeasible or suboptimal in the end. To do so, we proposed to rely on mixed-integer linear programming to assist the design and the example analyzed here was the beam layout optimizaiton, for which a novel mathematical formulation has been presented and compared to an existing algorithmic solution. Excellent results were reached on part of the instances tested, but there is still work to do to overcome the numerical difficulties observed on the biggest Fig. 9. Comparison of the greedy heuristic of [START_REF] Camino | A greedy approach combined with graph coloring for non-uniform beam layouts under antenna constraints in multibeam satellite systems[END_REF] and of the mixed-integer linear programming approach for non-uniform beam layouts instances. Therefore, the natural way forward is further research to enhance the current mixed-integer linear programming algorithm. The goal is either to solve higher instances more rapidly or to take advantage of that computational gain to solve in an even more integrated manner the different space segment design steps: the beam layout, the antenna, and the repeater design. To do so, there are promising advanced techniques worth investigating in operations research: Benders' decompositions, branch-and-price formulations and metaheuristics for instance.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Decomposition of the space segment design and corresponding process

Fig. 6 .

 6 Fig. 6. (a),(c),(e) Satellite reflector mapping: 117 regular beams, 50 regular beams, 48 irregular beams (b),(d),(f) Load per beam: 117 regular beams, 50 regular beams, 48 irregular beams (blue = low values, red = high values)

  denotes the set of user ground stations, each station s ∈ S being characterized by its traffic demand T s ∈ R + and its coordinates X stations,s ,Y stations,sT ∈ R 2 . U = {1, • • • , n directions }is the set of discretized directions of the Euclidean plane, and for each u ∈ U u,x ,U u,y T ∈ R 2 are the coordinates of a unit vector characterizing that direction. R = {1, • • • , N R } is the set indexing the N R reflector antennas, and W = {1, • • • , N W } is the (finite) set of possible beamwidths: W 1 , • • • ,W N W ∈ R + . For each beamwidth

Fig. 7 .

 7 Fig. 7. Mixed Integer Linear Programming model devised for the beam layout optimization problem with linear Euclidean norm approximation

Fig. 8 .

 8 Fig. 8. (a) Growing numbers of variables and constraints (b) Gap with respect to the best bound after 600 seconds