G Noguere

J Tommasi

E Privas

K.-H Schmidt

D Rochman

Systematics of Nd cumulative fission yields for neutron-induced

Systematics of cumulative fission yields of the Neodymium isotopes for thermal and fast neutron fission of 235 U, 238 U, 238 Pu, 239 Pu, 240 Pu and 241 Pu were obtained by combining integral results from the PROFIL experiments with theoretical calculations from the GEF code. The systematic behavior with the Neodymium mass number (A =143, 145, 146, 148, 150) deduced from the experimental trends is consistent with the smooth variation predicted by the GEF calculations, excepted for the 238 U(n,f) reaction. For this system, isotopic and isotonic effects in fission-fragment shell structures are not adequately taken into account in the theoretical calculation. The obtained results also confirm the weak energy dependence of the Nd cumulative fission yields in the energy range of interest for thermal and sodium fast reactors. They suggest an energy dependency comparable to the experimental uncertainty which lies below 3%, for the 235 U, 239 Pu and 241 Pu fissile isotopes.

I. INTRODUCTION

The irradiation experiments PROFIL-1, PROFIL-2A and PROFIL-2B were carried out in the fast reactor PHENIX (CEA Marcoule, France). Their interpretations with the deterministic code ERANOS [START_REF] Ruggieri | ERANOS 2.1: the international code system for GEN-IV fast reactor analysis[END_REF] have provided a large amount of integral trends [START_REF] Tommasi | Analysis of Sample Irradiation Experiments in PHENIX for JEFF-3.0 Nuclear Data Validation[END_REF][START_REF] Tommasi | Analysis of the PROFIL and PROFIL-2 Sample Irradiation Experiments in PHENIX for JEFF-3.1 Nuclear Data Validation[END_REF][START_REF] Privas | The use of nuclear data as nuissance parameters in the integral data assimilation of the PROFIL experiments[END_REF]. Some of them are sensitive to the cumulative fission yields of 143 Nd, 145 Nd, 146 Nd, 148 Nd and 150 Nd.

The Neodymium isotopes of interest for this work are all stable nuclides. In that case, the corresponding cumulative fission yield represents the total number of atoms of that nuclide remaining per fission after the decay of all precursors. Figure 1 illustrates the 148-decay chain to 148 Nd. The Nd isotopes have several neutronic advantages compared with other fission products. They are known to be characterized by a linear behavior with respect to burnup of nuclear fuel. Among the neodymium isotopes, 148 Nd is identified as an ideal burn-up monitor for determining the time-integrated neutron flux and scaling the reactor power history.

Effective cumulative fission yields obtained for the 235 U(n,f) reaction were already reported in Ref. [START_REF] Privas | Measurements of the effective cumulative fission yields of 143 Nd, 145 Nd, 146 Nd, 148 Nd and 150 Nd for 235 U in the PHENIX fast reactor[END_REF] with a preliminary investigation of the performances of the semi-empirical model implemented in the GEF code [START_REF] Schmidt | General Description of Fission Observables: GEF Model Code[END_REF]. As a result, GEF seems to be a suitable tool to handle the variation of the cumulative fission yields with the incident neutron energy.

The present work aims to complement the previous analysis by reporting Nd cumulative fission yields not only for 235 U(n,f) but also for 238 U, 238 Pu, 239 Pu, 240 Pu and 241 Pu from thermal (25.3 meV) to fast energy ranges ( 400 keV). The same methodology, as used for 235 U, was applied to extract experimental fission yields averaged over the fast neutron spectrum of the PHENIX reactor. The typical relative uncertainty that can be reached is lower than 3%, for 235 U, 239 Pu and 241 Pu.

For determining values at the thermal energy, we decided to normalize the cumulative fission yields calculated with the GEF code by using the PROFIL trends. The reliability of such a simplified approach was verified with a standard Integral Data Assimilation (IDA) procedure [8]. Due to the smooth and weak variation with the energy, strong correlations exist between the thermal and fast energy ranges. These correlations allowed to estimate the thermal cumulative fission yields of the Nd isotopes with relative uncertainties ranging from 2% to 5%, even for 238 U, 238 Pu and 240 Pu whose cumulative fission yields are poorly known.

FIG. 1: Example of decay scheme for isotopes with mass equal to 148 taken from Ref [START_REF] Terranova | Covariance evaluation for nuclear data of interest to the reactivity loss estimation of the Jules Horowitz material testing reactor[END_REF].

The consistency of our results expressed as a function of the mass number (A =143, 145, 146, 148, 150) and of the neutron energy are discussed throughout the paper by looking the behavior of the evaluated values compiled in the international libraries JEFF-3.1.1 [START_REF] Santamarina | The JEFF-3.1.1 Nuclear Data Library[END_REF] and ENDF/B-VII.1 [START_REF] Chadwick | ENDF/B-VII.1 nuclear data for science and technology: Cross sections, covariances, fission product yields and decay data[END_REF]. The overall good agreement achieved for the most important fissile targets ( 235 U, 239 Pu and 241 Pu) suggests that the combined analysis of the PROFIL results with the GEF calculations offers a set of valuable data for improving the theoretical parametrization around the Nd isotopes over a wide neutron energy range.

II. GOVERNING EQUATIONS

Since the methodology used in this work has been already applied on 235 U, only the most important features of the PROFIL experiments and of the theoretical background are briefly given in the present document. More detailed explanations can be found in Ref. [START_REF] Privas | Measurements of the effective cumulative fission yields of 143 Nd, 145 Nd, 146 Nd, 148 Nd and 150 Nd for 235 U in the PHENIX fast reactor[END_REF].

The principle of the PROFIL experiments consisted of irradiating samples containing a small amount of pure isotope in the fast neutron flux of the PHENIX reactor (CEA Mar-coule, France). Experimental results (E) are the isotopic composition of the samples before and after the irradiation period. The theoretical results (C) were calculated with the ERA-NOS code in association with the nuclear data library JEFF-3.1.1. In the present work, we are interested in the calculated-to-experimental ratios C/E X related to the Neodymium content produced in the studied samples after fission of a given actinide X. Since the ER-ANOS calculations were performed by using the fast cumulative fission yields Y X c J ( A Nd, E n ) of the JEFF-3.1.1 library given at E n =400 keV, effective cumulative fission yield Y X c P ( A Nd) can be determined from the PROFIL results as follows:

Y X c P ( A Nd) = Y X c J ( A Nd, E n = 400 keV) + ∆Y X c P ( A Nd), (1)
where ∆Y X c P ( A Nd) is related to the calculated-to-experimental ratios C/E X via the sensitivity coefficient S( A Nd) of C to the cumulative fission yields of the Neodymium isotope

A Nd: ∆ C/E X C/E X = S( A Nd) ∆Y X c P ( A Nd) Y X c J ( A Nd, E n ) , (2)
with the condition:

C/E X + ∆ C/E X = 1. (3)
By introducing Eqs (2) and (3) in Eq (1), we obtain:

Y X c P ( A Nd) = 1 + 1 -C/E X S( A Nd) C/E X Y X c J ( A Nd, E n = 400 keV), (4)
The GEF code, in association with the Q-matrix formalism [START_REF] Mills | Fission product yield evaluation[END_REF], is able to provide cumulative fission yields Y X c G ( A Nd, E) as a function of the incident neutron energy E n . The model implemented in GEF is empirical. Several parameters are adjusted to reproduce experimental data. However, it relies on physical notions, which permits to limit the amount of parameters, and to use the same set of parameters for all fissioning systems over a large excitation energy range. Results obtained for the 235 U(n,f) reaction are shown in Fig. 2. For the comparison with the PROFIL values, GEF results have to be averaged over the fission reaction rate: Assessment of cumulative fission yields at the thermal energy from PROFIL results consists in determining the energy dependent cumulative fission yields Y X c G ( A Nd, E n ) in order to satisfy the following relationship: The problem can be solved either by using standard Integral Data Assimilation procedures, for which reliable prior covariances between the fission yields are needed, or by scaling the GEF results via the normalization factor N . Results obtained by the two methods are discussed in section IV.

X c G ( A Nd) = N Emax 0 Y X c G ( A Nd, E n )σ f X (E n )φ(E n )dE n Emax 0 σ f X (E n )φ(E n )dE n , (5)
Y X c G ( A Nd) = Y X c P ( A Nd). (6)

III. CUMULATIVE FISSION YIELDS IN THE FAST ENERGY RANGE

A. Average calculated-to-experimental ratios from PROFIL Crucial ingredients in Eq. ( 4) are the calculated-to-experimental ratios C/E X calculated for a given actinide X. However, after the irradiation period, the quantity of Neodymium measured by Inductively Coupled Plasma Mass Spectrometry results from the fission of all actinides present in the studied sample. As a result, calculated-to-experimental ratios C/E account for all these contributions, noted X in the following expression:

C/E X ω X (φ) ( C/E X + δ f (X)) , (7)
with the condition:

X ω X (φ) = 1. ( 8
)
The weight ω X associated to each fissile isotope X depends on the fluence level φ. The correction term δ f (X) is introduced to account for possible biases on the calculations of the Neodymium content due to the theoretical fission cross sections used in the ERANOS calculations.

We have used ERANOS results to estimate the values of ω X as a function of the fluence. In In Eq. ( 7), the order of magnitude of the correction term δ f (X) depends on the evaluated fission cross sections tabulated in the JEFF-3.1.1 library (Fig. 3). It can be estimated with fission indices measured in fast critical mock-up reactors. Weighted averages of few calculated-to-experimental ratios measured in the MASURCA [START_REF] Ruggieri | JEFF-3.1 nuclear data validation for fast reactor physics[END_REF], PROTEUS [START_REF] Perret | Reanalysis of the Gas-cooled fast reactor experiments at the zero power facility Proteus -Spectral indices[END_REF][START_REF] Perret | Toward Reanalysis of the Tight-Pitch HCLWR-PROTEUS Phase II Experiments[END_REF] and FCA [START_REF] Rimpault | Preliminary results on FCA-IX fission chambers analysis[END_REF] facilities have led to the following correction terms for the 238 U(n,f), 238 Pu(n,f),

239 Pu(n,f), 240 Pu(n,f) and 241 Pu(n,f) reactions:

δ f ( 238 U ) = -0.009 ± 0.009, δ f ( 238 Pu) = +0.140 ± 0.020, δ f ( 239 Pu) = -0.018 ± 0.007, δ f ( 240 Pu) = +0.030 ± 0.014, δ f ( 241 Pu) = -0.004 ± 0.021.
All corrections terms lie below ±3%, excepted the one related to 238 Pu. According to the obtained result, the 238 Pu fission cross section of JEFF-3.1.1 is overestimated by 14%.

Consequently, cumulative fission yields determined from the 238 Pu samples should be considered with care. No correction is applied to the 235 U fission cross section because it was used to scale the fluence level. The corrected ratios C/E X are given in Table II. The reported uncertainties include the uncertainties of the weights ω X and correction terms δ f (X). They remain below ±3.5% for all the calculated-to-experimental ratios. Such rather low relative uncertainties will ensure an accurate estimation of the effective cumulative fission yields in the fast energy range.

B. Effective cumulative fission yields

Effective cumulative fission yields of 143 Nd, 145 Nd, 146 , Nd 148 Nd and 150 Nd have been deduced from the calculated-to-experimental ratios C/E X listed in Table II by using Eq. ( 4).

Results are given in Table III. The quoted uncertainties take into account the accuracy of S( A Nd), which is the sensitivity coefficient of C to the cumulative fission yields of the Neodymium isotope A Nd, and the uncertainty due to the fluence scaling procedure. The value of S( A Nd) is close to unity and the associated relative uncertainty is set to 2%. The fluence scaling procedure has been carefully studied in Ref. [START_REF] Privas | The use of nuclear data as nuissance parameters in the integral data assimilation of the PROFIL experiments[END_REF]. Since the well-known fission cross section of 235 U was taken as reference for the interpretation of the PROFIL experiments, the relative uncertainty is rather low (< 2%).

In the same Table, our results are compared with fast cumulative fission yields evaluated in the international library JEFF-3.1.1 and in the US library ENDF/B-VII.1. The differences between all the values depend on the fissile isotope and on the library. However, we can note that the agreement between most of the values extracted from the PROFIL results and those from the JEFF-3.1.1 and ENDF/B-VII.1 libraries remains within the limit of the uncertainties. Results obtained for 235 U were already discussed in Ref. [START_REF] Privas | Measurements of the effective cumulative fission yields of 143 Nd, 145 Nd, 146 Nd, 148 Nd and 150 Nd for 235 U in the PHENIX fast reactor[END_REF]. The PROFIL trends are in better agreement with the fission yields recommended in the US library, but our fission yields are systematically lower. This bias could be due to an unrecognized systematic source of uncertainties. As a result, the PROFIL trends can be introduced in the experimental data base used for producing evaluated fission yield libraries by keeping in mind such a systematic bias. For 238 Pu, fast cumulative fission yields are poorly known and the present results could significantly improve the content of these libraries.

C. Systematic variation with the mass number

The systematic variation of the fast cumulative fission yields as a function of Neodymium mass number are presented in Fig. 4. The results are normalized to Y c ( 148 Nd). Two isotopic chains with Z CN = 92 and Z CN = 94 are compared, corresponding to an isotonic series ranging from N CN = 144 to N CN = 148. For Plutonium isotopes, the observed variations follow "parabolic" trends, indicated by "eye-guide" curves. As expected for heavy fissionfragments, the mass yields for the studied fissioning systems decrease with different slopes.

The higher N CN is, the weaker is the variation with the Nd mass number. Cumulative fission yields for 235 U follow a similar trend, while those for 238 U have different behaviors FIG. 4: Fast cumulative fission yields as a function of Nd mass number deduced from the PROFIL experiments for Uranium (left-hand plot) and Plutonium (right-hand plot) isotopes normalized to Y c ( 148 Nd). Our results are compared with those obtained from the interpretation of the TACO experiment [START_REF] Robin | The importance of fission product nuclear data in burnup determination[END_REF][START_REF] Koch | Systematics of fast cumulative fission yields[END_REF]. from either side of A=146.

In view of confirming the trend for 238 U, we have compared the PROFIL results with cumulative fission yields extracted from the TACO experiment [START_REF] Robin | The importance of fission product nuclear data in burnup determination[END_REF][START_REF] Koch | Systematics of fast cumulative fission yields[END_REF]. Both experiments are based on the same principle. In the case of the TACO experiment, samples of separated isotopes were irradiated in the fast reactor RAPSODIE (CEA Cadarache, France). Even if the neutron spectrum in PROFIL is slightly softer than the TACO spectrum, results provided by the two experiments are in excellent agreement and confirm the singular variation observed for 238 U.

As complementary results, the PROFIL trend for 235 U and 238 U is compared in Fig. 5 with effective cumulative fission yields Y c G ( A Nd) calculated with Eq. ( 5) by using results of the GEF code. The semi-empirical model implemented in GEF is able to follow the general trend of the experimental cumulative fission yields for 235 U. The theoretical calculations suggest a smooth variation with the Nd mass number which is confirmed by the PROFIL results. For 238 U, the behavior with the Nd mass number is not as smooth as for 235 U. The deviation of GEF from the experimental result is somewhat amplified due to the normalization of the yields at A = 148. On an absolute scale, the value for 143 Nd does not deviate so much, and GEF results seem to become too low towards the heavier Nd isotopes. Since cumulative fission yields results from a complex summation of all contributions over the entire decay time after a fission event, the origin of such a deviation from the experimental trend is therefore difficult to identify. It could reflect a general smooth of the GEF code, rather than a more complex structural problem such as shell closure effects that govern the odd-even staggering in the heavy fission-fragment peak. More work is probably needed for improving the description of the "super-asymmetric fission channel" introduced in the GEF code. In that case, the success of the model strongly depends on the accuracy of the experimental fission-fragment yield distributions. Promising results with an unprecedented mass resolution were already obtained in the frame of the SOFIA experiment, but for 238 U electromagnetic fission which could be related to fission of 237 U induced by neutrons of 8-9 MeV incident energies. A detailed discussion that allows to understand the difficulty of interpreting our results can be found in Ref. [START_REF] Pellereau | Accurate isotopic fission yields of electromagnetic-induced fission of 238 U in inverse kinematics at relativistic energies[END_REF]. were generated with the NJOY code [START_REF] Macfarlane | Methods for Processing ENDF/B-VII with NJOY[END_REF].

deduced from each PROFIL experiment. At the thermal energy, the cumulative fission yield of 148 Nd provided by the IDA procedure is:

Y 235 U c IDA ( 148 Nd, E th ) = 0.01656 ± 0.00040 (2.4%).
For comparison, the normalization of the GEF curve with the PROFIL data provides an equivalent result:

Y 235 U c norm ( 148 Nd, E th ) = 0.01657 ± 0.00041 (2.5%).
The excellent agreement between the two approaches and JEFF-3.1.1 on 148 Nd is an encouraging result for applying the normalization strategy instead of the more complex IDA procedure over all the PROFIL data. Table IV summarizes the cumulative fission yields obtained for 235 U at the thermal energy. The main difference between the two methods occurs for 143 Nd. The central values are nearly the same, but the relative uncertainty of ±11.1% obtained with the IDA procedure remains too close to the prior uncertainty of ±12%. Such a difference illustrates the non-negligible effect of the prior covariance matrix on the posterior uncertainty. We have to keep in mind this constraint in the interpretation of the results provided in the following sections. The thermal cumulative fission yields of the Neodymium isotopes obtained after normalization of the GEF calculations with the PROFIL data are reported in Table V. The comparison with the evaluated data, reported in JEFF-3.1.1 and ENDF/B-VII.1, confirms that our methodology can provide reliable thermal values for 235 U, 239 Pu and 241 Pu. As already observed in the fast energy range, the agreement remains within the limit of the reported uncertainties. In the case of the 235 U(n,f) reaction, our result for 148 Nd is 1.4% and 1% lower than the one recommended in JEFF-3.1.1 and ENDFB/-VII.1 libraries, respectively. These differences lie below the PROFIL uncertainty. The present work also provides valuable tendencies on the poorly known cumulative fission yields for the 238 Pu(n th ,f) and 240 Pu(n th ,f) reactions. For 238 Pu, the good agreement obtained between our results and JEFF-3.1.1 at A = 143 and A = 145 contributes to support the combine analysis of the PROFIL results with the GEF calculations. Cumulative fission yields at thermal energy for 238 U are only given for comparison. The present approach are probably not adequate in that case. As discussed in section III C, improved GEF calculations are needed for correctly described the fast cumulative fission yields for 238 U before extrapolating the theory in the thermal energy range.

Results reported in Table V are presented in Fig. 8 as a function of the Nd mass number.

The observed behavior is nearly similar to the one shown in Fig. 4. The "eye-guide" curves follow a "parabolic" shape, which is consistent with the behavior of the cumulative fission yields evaluated in JEFF-3.1.1. It is also correctly predicted by the GEF calculations.

For the 235 U(n,f) reaction, results obtained with the IDA and normalization procedures (Table IV) are reported in the same plot. Both procedures provide a similar trend. The singular behavior for 238 U is still visible. However, the abrupt variation with the Nd mass number is more pronounced than in the fast energy range and the differences with GEF (prior values) are rather large. As a result, our thermal cumulative fission yields for the 238 U(n th ,f) reaction should be taken with care.

C. Systematic variation from thermal to fast energy ranges Detailed discussions on the variation of cumulative fission yields with the incident neutron energy are scarce in the literature and the specific behavior of the Neodymium isotopes is not well documented. As indicated by Britt [START_REF] Britt | Review of the status of cumulative fission yields from 239 Pu(n,f) of interest to nuclear forensics[END_REF], it exists in the literature some experimental evidences of the weak energy-dependence of fission yields in the immediate vicinity of the heavy mass peak. The issue for the 235 U(n,f) and 239 Pu(n,f) reactions from the thermal to fast energy ranges was qualitatively addressed by Maeck [START_REF] Maeck | The correlation of 235 U thermal and fast reactor fission yields with neutron energy[END_REF][START_REF] Maeck | The correlation pf 239 Pu thermal and fast reactor fission yields with neutron energy[END_REF] in the beginning of the 80s by using the cumulative ratio 150 Nd/ 148 Nd as an energy index. More recently, two integral experiments in mostly thermal and epithermal neutron spectra have confirmed the fluctuations of the 235 U cumulative fission yields in the resonance range of the neutron cross sections [START_REF] Leconte | Fluctuation of mass yield distribution for the epithermal fission of 235 U and its impact on HCLWR calculations[END_REF]. However, for the Nd mass number, a variation lower than 1% is expected. As a result, since rather low relative uncertainties of about 1% are difficult to achieve, variations observed in the evaluated fission yield libraries between the thermal and fast energies could mainly reflect the accuracy and the dispersion of the experimental values introduced in the evaluation procedure. One of the recurrent problems is also the definition of the "fast energies". In JEFF-3.1.1, it is set to E n = 400 keV, while E n = 500 keV has been chosen in ENDF/B-VII.1. This difference of 100 keV is difficult to explain and probably not justified for fast cumulative fission yields. Indeed, data introduced in the evaluation procedure are usually measured in fast neutron spectra of different hardness, for which the mean excitation energy of fissioning events is scarcely reported in the literature. In the present work, our results are not affected by these considerations because we directly provide effective values valid for typical sodium fast reactors (SFR).

The variation of the cumulative yields of the Neodymium isotopes with the neutron energy is illustrated in Fig. V). Abscissa of bottom plot is related to results obtained from the normalization of the GEF calculations with the PROFIL trends (column (c) of Table V). The dispersion of the results around the first bisector provides valuable information on the correlation between the thermal and fast energy ranges as a function of the fission yields.

For 235 U(n,f), we can observe a systematic deviation from the first bisector which increases with the fission yields value. In the bottom plot, the deviation is in the limit of our experimental uncertainties. This trend could be a mathematical artefact coming from the normalization strategy applied to the GEF results. As indicated in Table IV, the difference with the IDA procedure is maximal for the cumulative fission yield of 143 Nd and reaches 2.5%. For 239 Pu(n,f), a weak energy dependence regardless of the fission yield value is suggested by all the three plots. A similar conclusion arises from our results for 241 Pu(n,f).

This trend is fully confirmed by the US library, but in contradiction with JEFF-3.1.1. This raises problems related to the reliability of the theoretical calculations and of the normalization strategy. Since we have used the GEF data as shape data, the behavior between the fast and thermal energy ranges are fully correlated and depend on the semi-empirical model implemented in GEF. The satisfactory agreement with the evaluated cumulative fission yields of the Neodymium isotopes for 235 U, 239 Pu and 241 Pu (ratio (a)/(c) and (b)/(c)

FIG. 2 :

 2 FIG.2:The black lines are the Nd cumulative fission yields as a function of the incident neutron energy for 235 U calculated with the GEF code[START_REF] Schmidt | General Description of Fission Observables: GEF Model Code[END_REF]. They are compared to a neutron spectrum (grey line), in arbitrary units, representative of the PROFIL experiments[START_REF] Tommasi | Analysis of Sample Irradiation Experiments in PHENIX for JEFF-3.0 Nuclear Data Validation[END_REF][START_REF] Tommasi | Analysis of the PROFIL and PROFIL-2 Sample Irradiation Experiments in PHENIX for JEFF-3.1 Nuclear Data Validation[END_REF][START_REF] Privas | The use of nuclear data as nuissance parameters in the integral data assimilation of the PROFIL experiments[END_REF]. The open circles indicate the position of the thermal energy (25.3 meV) and fast neutron energy (400 keV).

Y

  in which φ(E n ) stands for the fast neutron spectrum representative of the PROFIL experiment. It is shown in Fig. 2 as a function of the incident neutron energy E n . The fission cross sections σ f X (E n ) of interest for this work are shown in Fig 3. The maximum energy E max is set to 20 MeV. The normalization factor N is equal to unity.

FIG. 3 :

 3 FIG. 3: Fission cross sections of 235 U, 238 U, 238 Pu, 239 Pu, 240 Pu and 241 Pu recommended in the Joint Evaluated Fission and Fusion library JEFF-3.1.1 [9].

  the 238 U samples, the Neodymium isotopes are mainly produced via the 238 U(n,f), 239 Pu(n,f) and 235 U(n,f) reactions with weights equal to 0.659, 0.326 and 0.015, respectively, for an average fluence of φ = 10.0 × 10 22 n/cm 2 . In the 238 Pu sample, the main contributions come from the 238 Pu(n,f) and 239 Pu(n,f) reactions. The weights are close to 0.85 and 0.15, respectively, for φ = 9.2 × 10 22 n/cm 2 . In the 240 Pu sample, the weights associated to the 240 Pu(n,f), 241 Pu(n,f), 238 Pu(n,f) and 239 Pu(n,f) reactions are 0.805, 0.177, 0.012 and 0.006 (φ = 17.3 × 10 22 n/cm 2 ). For the 235 U, 239 Pu and 241 Pu samples, contributions from other fission reactions are negligible and the weights ω X are assumed to be close to unity. The uncertainty on each ω X is set to ±0.01.

FIG. 5 :

 5 FIG. 5: Comparison of the fast cumulative fission yields deduced from the PROFIL experiments and calculated with the GEF code for 235 U(n,f) and 238 U(n,f). Results are given as a function of the Nd mass number and normalized at A = 148.

FIG. 6 :

 6 FIG. 6: Cumulative fission yields of 148 Nd for 235 U calculated with the GEF code as a function of the incident neutron energy, together with the relative uncertainties and correlation matrix, before (left-hand plot) and after (right-hand plot) integral assimilation of the PROFIL data. Uncertainty propagation calculations were performed with the CONRAD code [21]. The two plots

FIG. 7 :

 7 FIG. 7: Comparison of the values and uncertainties of Y 235 U c

FIG. 8 :

 8 FIG.8: Variation of the thermal cumulative fission yields with the Nd mass number. The posterior results were obtained after normalization of the GEF trends with the PROFIL data. They are compared with the prior GEF calculations and the evaluated values reported in JEFF-3.1.1. For 235 U, the thermal cumulative fission yields obtained from the integral assimilation of the PROFIL data is also shown.

FIG. 9 :

 9 FIG. 9: Effective cumulative fission yields of the Neodymium isotopes determined with the PROFIL trends in the fast energy range (column (c) of Table III) as a function of the thermal values reported in columns (a), (b) and (c) of Table V. The solid line represents the first bisector. The dotted lines are "eye-guide" lines that follow the trends obtained for 235 U(n,f) and 238 U(n,f).

9 .

 9 We decided to display the effective values determined with the PROFIL trends in the fast energy range (column (c) of Table III) as a function of the thermal values. Abscissa of top and middle plots represents the thermal values evaluated in the JEFF-3.1.1 and ENDF/B-VII.1 libraries (column (a) and (b) of Table

Table I

 I reports the calculated-to-experimental ratios C/E of the Neodymium buildup averaged over the individual C/E ratios obtained from the PROFIL-1, PROFIL-2A and PROFIL-2B experiments. They were calculated with the ERANOS code by using the evaluated nuclear data of the JEFF-3.1.1 library. The quoted uncertainties are only the statistical uncertainties coming from the Inductively Coupled Plasma Mass Spectrometry.

TABLE I :

 I Average calculated-to-experimental ratios C/E of the Neodymium buildup for the PROFIL-1, PROFIL-2A and PROFIL-2B experiments. The numbers in the brackets indicate the number of experimental values used in the average. The quoted uncertainties take only into account the isotopic ratio uncertainties provided by Inductively Coupled Plasma Mass Spectrometry.

	Sample	143 Nd	145 Nd	146 Nd	148 Nd	150 Nd
	235 U (9) 0.985±0.003 1.022±0.003 1.032±0.003 1.034±0.002 1.057±0.003
	238 U (3) 0.963±0.008 0.964±0.008 0.981±0.008 1.012±0.009 0.968±0.008
	238 Pu (1) 1.193±0.012 1.324±0.013 1.361±0.014 1.322±0.013 1.185±0.012
	239 Pu (2) 0.965±0.003 0.993±0.005 1.011±0.003 1.011±0.002 1.003±0.003
	240 Pu (1)	-	-	-	1.023±0.020	-
	241 Pu (2) 0.989±0.007 0.990±0.007 1.003±0.007 1.008±0.007 0.991±0.007
	Sample	143 Nd	145 Nd	146 Nd	148 Nd	150 Nd
	235 U	0.985±0.003 1.022±0.003 1.032±0.003 1.034±0.002 1.057±0.003
	238 U	0.971±0.028 0.958±0.028 0.974±0.029 1.021±0.029 0.957±0.029
	238 Pu 1.094±0.030 1.243±0.031 1.282±0.032 1.237±0.031 1.077±0.030
	239 Pu 0.982±0.008 1.011±0.008 1.029±0.008 1.029±0.007 1.021±0.007
	240 Pu	-	-	-	0.993±0.037	-
	241 Pu 0.993±0.022 0.994±0.022 1.007±0.022 1.011±0.022 0.995±0.022

TABLE II: Calculated-to-experimental ratios C/E X determined from values reported in Table

I

after subtraction of the fission contributions from other actinides and corrected by the term δ f (X) (Eq. (

7

)). The quoted uncertainties account for the isotopic ratio uncertainties provided by Inductively Coupled Plasma Mass Spectrometry, the uncertainties of the correction terms δ f (X) and of the weights ω X .

TABLE III :

 III Comparison of the effective cumulative fission yields obtained in this work with the cumulative fission yields reported in the JEFF-3.1.1 and ENDF/B-VII.1 libraries at 400 keV and 500 keV, respectively.

	fissile Neodymium	JEF-3.1.1	ENDF/B-VII.1	This work	Ratio	Ratio
	isotope	isotope	(a)	(b)	(c)	(a)/(c) (b)/(c)
	235 U	143 Nd	0.05533 (1.0%) 0.05731 (0.5%) 0.05605 (2.1%) 0.987	1.022
		145 Nd	0.03796 (1.8%) 0.03776 (0.5%) 0.03703 (2.2%) 1.025	1.020
		146 Nd	0.02927 (1.8%) 0.02921 (0.5%) 0.02826 (2.4%) 1.036	1.034
		148 Nd	0.01697 (1.2%) 0.01683 (0.5%) 0.01636 (2.4%) 1.037	1.029
		150 Nd	0.00702 (2.4%) 0.00686 (0.5%) 0.00660 (2.4%) 1.063	1.039
	238 U	143 Nd	0.04680 (2.4%) 0.04622 (0.7%) 0.04824 (3.5%) 0.970	0.958
		145 Nd	0.03883 (5.3%) 0.03809 (0.7%) 0.04059 (3.6%) 0.957	0.938
		146 Nd	0.03573 (5.1%) 0.03446 (0.7%) 0.03672 (3.8%) 0.973	0.938
		148 Nd	0.02296 (1.6%) 0.02112 (0.7%) 0.02249 (3.8%) 1.021	0.939
		150 Nd	0.01311 (3.7%) 0.01273 (1.0%) 0.01371 (3.8%) 0.956	0.929
	238 Pu	143 Nd	0.04517 (30.5%) 0.04535 (16.0%) 0.04122 (3.7%) 1.096	1.100
		145 Nd	0.03394 (31.9%) 0.03236 (16.0%) 0.02717 (4.0%) 1.249	1.191
		146 Nd	0.02782 (32.5%) 0.02767 (16.0%) 0.02157 (4.2%) 1.290	1.277
		148 Nd	0.01689 (34.3%) 0.01757 (23.0%) 0.01359 (4.1%) 1.243	1.283
		150 Nd	0.00860 (37.5%) 0.00978 (32.0%) 0.00797 (4.0%) 1.079	1.227
	239 Pu	143 Nd	0.04296 (1.3%) 0.04328 (0.4%) 0.04374 (2.3%) 0.982	0.989
		145 Nd	0.03037 (2.2%) 0.02999 (0.5%) 0.03004 (2.4%) 1.011	0.998
		146 Nd	0.02527 (2.1%) 0.02455 (0.5%) 0.02455 (2.6%) 1.029	1.000
		148 Nd	0.01696 (1.7%) 0.01658 (0.4%) 0.01647 (2.5%) 1.030	1.007
		150 Nd	0.01005 (1.9%) 0.00993 (0.5%) 0.00984 (2.6%) 1.021	1.009
	240 Pu	148 Nd	0.01798 (5.0%) 0.01774 (2.0%) 0.01812 (4.4%) 0.992	0.979
	241 Pu	143 Nd	0.04593 (2.6%) 0.04671 (1.0%) 0.04627 (3.1%) 0.993	1.010
		145 Nd	0.03272 (2.6%) 0.03327 (1.0%) 0.03291 (3.2%) 0.994	1.011
		146 Nd	0.02740 (2.6%) 0.02806 (1.0%) 0.02721 (3.3%) 1.007	1.031
		148 Nd	0.01945 (2.5%) 0.01956 (1.0%) 0.01922 (3.3%) 1.012	1.018
		150 Nd	0.01199 (2.7%) 0.01230 (1.0%) 0.01205 (3.3%) 0.995	1.021

TABLE IV :

 IV Cumulative fission yields of the Neodymium isotopes for 235 U(n th ,f) obtained at E th =25.3 meV from the integral assimilation of the PROFIL data and from the normalization of the GEF results with the PROFIL data.

	Neodymium	prior	Integral Data Assimilation Normalization GEF results ratio
	isotope	value	(a)	(b)	(a)/(b)
	143 Nd	0.05246 (12.0%)	0.05598 (11.1%)	0.05742 (2.1%)	0.975
	145 Nd	0.03678 ( 2.3%)	0.03797 ( 2.3%)	0.03814 (2.3%)	0.996
	146 Nd	0.02783 ( 9.6%)	0.02881 ( 2.5%)	0.02898 (2.5%)	0.994
	148 Nd	0.01792 (14.3%)	0.01656 ( 2.4%)	0.01657 (2.5%)	0.999
	150 Nd	0.00746 (22.8%)	0.00657 ( 3.5%)	0.00661 (3.2%)	0.994
	B. Systematic variation with the mass number

TABLE V :

 V Comparison of the thermal cumulative fission yields obtained in this work with those reported at 23.5 meV in the JEFF-3.1.1 and ENDF/B-VII.1 libraries.

	fissile Neodymium	JEFF-3.1.1	ENDF/B-VII.1	This work	Ratio	Ratio
	isotope	isotope	(a)	(b)	(c)	(a)/(c) (b)/(c)
	235 U	143 Nd	0.05954 (1.4%) 0.05956 (0.4%) 0.05742 (2.1%) 1.037	1.037
		145 Nd	0.03955 (1.1%) 0.03933 (0.4%) 0.03814 (2.3%) 1.037	1.031
		146 Nd	0.02987 (1.0%) 0.02997 (0.4%) 0.02898 (2.5%) 1.031	1.034
		148 Nd	0.01681 (0.7%) 0.01674 (0.4%) 0.01657 (2.5%) 1.014	1.010
		150 Nd	0.00651 (1.0%) 0.00653 (0.5%) 0.00661 (3.2%) 0.985	0.989
	238 U	143 Nd			0.05580 (3.5%)
		145 Nd			0.05112 (3.6%)
		148 Nd			0.02949 (3.8%)
		150 Nd			0.01637 (3.9%)
	238 Pu	143 Nd	0.03911 ( 5.7%)		0.04119 (4.0%) 0.950
		145 Nd	0.02704 ( 6.0%)		0.02714 (4.2%) 0.996
		146 Nd	0.01277		0.02155 (4.4%) 0.593
		148 Nd	0.00863		0.01359 (4.3%) 0.635
		150 Nd	0.00586 (71.2%)		0.00797 (4.4%) 0.735
	239 Pu	143 Nd	0.04476 (1.1%) 0.04413 (0.5%) 0.04502 (2.3%) 0.994	0.980
		145 Nd	0.03036 (1.1%) 0.02982 (0.4%) 0.03062 (2.4%) 0.992	0.974
		146 Nd	0.02496 (1.0%) 0.02458 (0.4%) 0.02501 (2.6%) 0.998	0.983
		148 Nd	0.01658 (1.0%) 0.01642 (0.5%) 0.01683 (2.6%) 0.985	0.976
		150 Nd	0.00975 (1.3%) 0.00966 (0.4%) 0.00981 (2.8%) 0.994	0.985
	240 Pu	148 Nd		0.01772 (23.0%) 0.01860 (4.5%)		0.953
	241 Pu	143 Nd	0.04380 (2.1%) 0.04578 (0.7%) 0.04681 (3.1%) 0.935	0.978
		145 Nd	0.03141 (2.9%) 0.03263 (1.0%) 0.03337 (3.2%) 0.941	0.978
		146 Nd	0.02657 (2.6%) 0.02766 (0.7%) 0.02751 (3.3%) 0.966	1.005
		148 Nd	0.01881 (3.4%) 0.01932 (0.7%) 0.01927 (3.3%) 0.976	1.002
		150 Nd	0.01154 (2.8%) 0.01209 (1.0%) 0.01195 (3.4%) 0.966	1.012

The authors would like to thanks Virginie Huy, from CEA of Cadarache, for her contribution in the interpretation of the spectral indices measured in the FCA facilities.

The originality of the present work consists in estimating cumulative fission yields at the thermal energy (E th =25.3 meV) by combining experimental results provided by the PROFIL experiments with theoretical calculations from the GEF code. The problem can be reduced to Eq. [START_REF] Privas | Measurements of the effective cumulative fission yields of 143 Nd, 145 Nd, 146 Nd, 148 Nd and 150 Nd for 235 U in the PHENIX fast reactor[END_REF]. For solving this equation, we can use standard Integral Data Assimilation (IDA) procedures [8]. In that case, the unknown free parameters are the energy dependent cumulative fission yields Y X c G ( A Nd, E). They have to be adjusted with a least-square method on the PROFIL data to satisfy Eq. ( 6). The IDA procedure requires prior information on the variances and covariances between the free parameters. The weakness of the IDA procedure is often the dependence of the posterior results to the prior covariance matrix.

The alternative solution consists in determining the normalization factor N of Eq. ( 5) by considering the GEF results as shape data. This second approach is valid if the thermal and fast cumulative fission yields are strongly correlated.

The two approaches were used to determine the thermal cumulative fission yields of the Neodymium isotopes for 235 U. Prior covariance matrix was generated by randomly varying model parameters of the GEF code [START_REF] Leray | Fission yield covariances for JEFF: A Bayesian Monte Carlo method[END_REF]. The obtained covariance matrix was then modified by adding defect model contributions calculated as the difference between the evaluated values from JEFF-3.1.1 and the GEF calculations at 23.5 meV, 400 keV and 14 MeV. Prior relative uncertainties and correlation matrix for Y 235 U c G ( 148 Nd, E) are shown in the left-hand plot of Fig. 6. In the energy range of interest for this work, large relative uncertainties of about 14% are reached and correlation coefficients are close to unity. The least-square model of the CONRAD code [START_REF] Archier | CONRAD evaluation code: development status and perspectives[END_REF] was used to produce the posterior results presented in the right-hand plot of Fig. 6. The integral assimilation of the PROFIL data lead to a significant decrease of the relative uncertainties over a wide energy range. Below 1 MeV, the final uncertainty (< 3%) is consistent with the uncertainties reported in column (c) of Table III. One of the remarkable results is also emphasized in Fig. 7. The arrows at 23.5 meV, 400 keV and 14 MeV show the good agreement of our posterior CONRAD calculations with the cumulative fission yields of JEFF-3.1.1. This agreement is confirmed by the detailed comparison given in the same figure between the evaluated values and results in Table V) could be seen as a probe of the capabilities of GEF to correctly predict the weak energy-dependence up to neutron energies of interest for SFR.

Once more, questions emerge from the singular behavior of the cumulative fission yields for 238 U(n,f). Further advanced theoretical investigations are needed, such as those conducted in Refs [27][START_REF] Moller | Evolution of uranium fission-fragment charge yields with neutron number[END_REF][START_REF] Sierk | Langevin model of low-energy fission[END_REF]. Models which calculate the mass distribution theoretically, and do not rely on a fit to experimental data, should provide informative systematics for a series of systems along isotopic chains as a function of the compound-nucleus excitation energy.

V. CONCLUSIONS

The present work demonstrates the possibility of combining theoretical calculations from the GEF code with integral trends measured in a fast neutron spectrum to determine thermal values consistent with those evaluated in international nuclear data libraries. The accuracy of the obtained cumulative fission yields ranges from 2% to 5%. Such rather low relative uncertainties are mainly driven by the accurate calculated-to-experimental ratios deduced from the PROFIL experiments.

The cumulative fission yields of the Neodymium isotopes for neutron-induced fission of 235 U, 238 Pu, 239 Pu, 240 Pu and 241 Pu exhibit a smooth variation as a function of the Nd mass number and a weak energy-dependence from thermal to fast energy ranges. For the isotopic chain Z CN = 94, the energy-dependence is even negligible in the case of sodium fast reactor applications.

For the 238 U(n,f) reaction, the behavior of the Nd cumulative fission yields is more complex and needs deeper theoretical works to be correctly understood. Improved modelisation of the fission process depends on the accuracy of the experimental yield distributions. Recent data obtained in the frame of the SOFIA experiment with an unprecedented mass resolution are promising results for solving the deviation observed between our PROFIL results and GEF calculations.

Acknowledgment


H Ferrasse

A Grange
email: avent.grange@cea.fr

F Bertrand

O Boutin

J B Droin

J H Ferrasse

A Marrel

Loss Of Off-site Power transient analysis for a Sodium-cooled Fast Reactor equipped with a gas Power Conversion System and preliminary optimisation of its operation

Keywords: Sodium-cooled Fast Reactor (SFR), gas Power Conversion System (PCS), Loss Of Off-site Power (LOOP), Multiobjective Optimisation Problem (MOP)

Loss of off-site power transient analysis for a sodium-cooled fast reactor equipped with a gas power conversion system and preliminary optimisation of its operation

Abstract

The French Commission for Atomic Energy and Alternative Energy (CEA) in collaboration with its industrial partners develops Sodium-cooled Fast Reactors (SFR) as industrial-scale demonstrators mainly guided by safety and operability objectives. In this paper, a SFR reactor associated to a nitrogen closed Brayton cycle for the Power Conversion System (PCS) is considered. In incidental and accidental conditions, the operation of the reactor must be defined to keep it under control and to fulfil safety requirements. This paper is dedicated to an alternative procedure to control a Loss Of Off-site Power (LOOP). Usually, in case of LOOP, the SFR standard procedure relies on passive Decay Heat Removal (DHR) systems to cool down the primary circuit. In this paper, an alternative solution substitutes the latter by the gas Power Conversion System (PCS). This aims at reducing the delay to reach the cold shutdown state while fulfiling safety criteria dealing with thermal stress issues. The operating of the gas PCS required three regulations:

-The regulation of the Turbo-Machinery (TM) rotation speed to keep a gas flow in the PCS; -The sodium temperature regulation of the secondary circuit, once the cold shutdown state is reached, to adapt the heat removed by the gas PCS to the decay heat; -The regulation of the gas temperature at the inlet of the compressors, in the PCS, to keep an efficient heat sink.

The actuator of the sodium temperature regulation of the secondary circuit is chosen among three different actuators thanks to a global sensitivity analysis performed with a metamodel-based methodology. The setting of the controller associated to the TM rotation speed regulation is justified by the study of different proportional-integral-derivative (PID) controllers.

A comparison of this alternative sequence with the reference one, based on simulations with the system thermalhydraulic code CATHARE2, is presented in this paper. The study indicates that the passive DHR systems do not allow the reactor to reach the cold shutdown state after 24 hours, whereas the procedure with the gas PCS required few hours to lead the reactor in this safety state, without increasing thermal stresses on the main vessel. Based on this first result, a Multiobjective Optimisation Problem (MOP) is solved in order to minimize simultaneously the delay to reach the cold shutdown state and the thermal stresses on the main vessel during the alternative procedure. These two objectives are conflicting, thus optimal compromises between them are required to solve the MOP and define a Pareto front. The multiobjective optimization step is supported by the study of different TM rotation speed targets. A Latin hypercube design of experiments is performed with the CATHARE2 code and is used to build the Pareto front. In this way, the alternative procedure allows the reactor to reach the cold shutdown state between 30 minutes and 4 hours. A short delay to reach the safety state induces thermal gradients through the main vessel about twice higher than the standard procedure, whereas a long delay to reach the safety state can divide the thermal gradients through the main vessel by four. To favour a specific objective of optimisation, this study highlights which TM rotation speed target must be chosen.

Thanks to the regulation of the TM rotation speed, the gas PCS is hence an adaptable system to optimize the thermalhydraulic behaviour of a SFR during a LOOP. Moreover, this alternative procedure strengthens the diversification of the systems to fulfil the DHR function.

Introduction

In 2006, the French government decided the commissioning of a new nuclear reactor to contribute to a sustainable management of radioactive material and wastes. This project have to respect the framework developed by the Generation IV International Forum (GIF) (Nuclear Energy Agency, 2016), and hence to participate to the development of the next nuclear energy system generation.

In accordance with these decisions, the French Commission for Atomic Energy and Alternative Energy (CEA) in collaboration with its industrial partners investigates the Sodium-cooled Fast Reactor (SFR) technology. In this paper, an industrial-scale SFR with a nitrogen closed Brayton cycle for the Power Conversion System (PCS) is studied to investigate its operability.

For specific incidents and accidents, control procedures must be defined to keep the reactor under control and to fulfil safety requirements. A focus is made on an alternative procedure to deal with a Loss Of Off-site Power (LOOP). Usually, in case of LOOP, SFR rely on specific Decay Heat Removal (DHR) systems to cool down the core. Here, the alternative procedure substitutes specific DHR system by the Nitrogen PCS to remove the decay heat, to reach a cold shutdown state and to maintain this state. The gas PCS requires regulations to operate efficiently the cooling of the reactor. This paper implements efficient statistical methods to define the associated regulations to the alternative procedure and to optimize this one. After a presentation of the SFR with the Nitrogen PCS option, the modelling of this system with the thermalhydraulic system code CATHARE2 is detailed. Then, the standard procedure with DHR systems and the alternative procedure in case of LOOP accident are described. In the next part, the regulations associated to the alternative procedure are defined using statistical methods. Finally, a comparison of the two procedures is proposed and an optimisation of the alternative procedure is performed to improve the capability of the PCS to cool down the reactor without inducing severe thermal stresses on the main vessel. This optimisation step is supported by an exploration of several turbomachinery (TM) rotation speed targets and a Pareto front is built to select the optimal procedures.

Sodium-cooled Fast Reactor design

The aim of this part is to describe the core and the systems of the studied SFR. A focus will be made on the systems used to cool down the reactor in case of LOOP transients.

Core, primary circuit and secondary circuit

An innovative CFV1 core design for SFR characterized by a low global sodium void worth effect [START_REF] Chenaud | Status of ASTRID core design studies at the end of predesign phase 1[END_REF] has been developed by CEA. This concept takes part of the main innovations tested on this kind of reactor. For the CFV core design, the sodium expansion and voiding reactivity feedback increases the margin up to sodium boiling in case of unprotected transients and reduces the severity of a primary power excursion in case of severe accidents (Bertrand et al., 2016a). The CFV core thermal power is equal to 1500 MW and all the primary components are included in a main vessel (Figure 1). A single conical "redan" (inner vessel) separates the primary sodium in two areas: the hot pool, filled with sodium at 550 °C from the cold pool, filled with sodium at 400 °C. Three mechanical Primary Pumps (PPs) are immerged in the primary sodium to ensure the sodium circulation from the cold pool to the hot pool by crossing the core. In steady-state operating conditions, the thermal power is extracted from the primary circuit to the secondary circuit through four Intermediate Heat Exchangers (IHX) (Figure 1).

Figure 1. Primary system arrangement

Control rods located above the core and maintained by electromagnets control the core reactivity.

In case of incidental or accidental transient, once a protection threshold is reached, these rods fall by gravity into the core, inducing a reactor scram. After such a scram, the DHR systems remove decay heat to ensure the safety and stability of the reactor.

Decay Heat Removal systems

One of the main innovations investigated for the SFR is the enhancement of the reliability of the DHR systems, based on a diversification strategy [START_REF] Hourcade | ASTRID Nuclear Island design: advances in French-Japanese joint team development of Decay Heat Removal systems[END_REF]. Three different systems are used to extract the decay heat (Figure 2):

-Two Direct Reactor Auxiliary Cooling Systems (DRACS) named RRB that use air as heat sink. These systems consist in a sodium/sodium heat exchanger located in the hot pool, a sodium circuit and a sodium/air heat exchanger out of the main vessel. The RRB remove the decay heat by natural convection (passive system).

Inner vessel Core

Main vessel Hot pool

Cold pool

-Two other DRACS named RRA, also using air as heat sink. These systems are composed of a sodium/sodium heat exchanger located in the cold pool, a sodium circuit equipped with a mechanical pump and a sodium/air heat exchanger out of the main vessel. -Two Reactor Vessel Auxiliary Cooling Systems (RVACS) named RRC, which remove the decay heat directly from the main vessel thanks to radiative heat transfer. These systems are located around the main vessel and are composed of: an oil circuit with a mechanical pump, an oil/water heat exchanger, a water circuit and a water/air heat exchanger with air as heat sink.

Figure 2. Decay Heat Removal systems location

Secondary circuits

These intermediate circuits remove the heat from the primary circuit and transfer it to the PCS through the IHX and the Sodium Gas Heat Exchangers (SGHE). The secondary side consists in four independent sodium loops, each one being equipped with a mechanical pump. They separate the primary circuit from the PCS in order to reduce the risk of gas inlet in the primary circuit and in the core.

Power conversion system

The PCS is a closed Brayton cycle, with two compression stages, in pure nitrogen at 180 bar in the high-pressure line (Figure 3). The plant is composed of two shaft lines with a thermal power of 750 MW each. A turbine, a low-pressure compressor, a high-pressure compressor and a generator are placed on the same shaft line and compose a turbine generator set. For normal operating, the gas temperature is decreased down to 27 °C through the two coolers before each compressor inlet to limit the compression work. The pre-cooler is located upstream of the lowpressure compressor and the intercooler is located upstream of the high-pressure compressor.

The heat sink is a water flow through the two coolers. A so-called recuperator Heat Exchanger (HX) allows gas temperature to raise up before its return to the SGHE. The recuperator HX enhances the net efficiency of the power plant that reaches 37.4% with the Gas-PCS.

For regulation and safety issues, by-pass lines equipped of valves are integrated in the Gas-PCS:

-The by-pass 1 line (BP1) (red line on the Figure 3) connects the high-pressure compressor outlet to the pre-cooler inlet. A valve located on this line enables to modify the mechanical balance on the shaft by reducing the gas flow rate through the high-pressure part of the cycle.

-The recuperator by-pass line (orange line on the Figure 3) bypasses the high-pressure side of the recuperator HX by connecting its upstream to its downstream; therefore, the gas which crosses this by-pass line is no more heated up. An opening of the valve located on this line induces a temperature decrease of the gas at the SGHX inlet. -The SGHX by-pass line: for the gas side of the SGHX (green line on the Figure 3); it connects the upstream of the SGHX to its downstream. An opening of the associated valve decreases the gas flow rate through the SGHX and hence the exchanged power in this HX.

The so-called nitrogen inventory control system called Nitrogen Supply System (NSS) enables to control the nitrogen inventory and consequently to modify the pressure in the PCS. It is composed of a gas extraction line connected to the high-pressure compressor outlet (highest PCS pressure point, blue arrow on the Figure 3) and a gas injection line connected to the pre-cooler inlet (blue arrow on the Figure 3).

CATHARE2 modelling of the reactor

To study the SFR behaviour in case of a LOOP transient, the whole reactor presented in section 2 is modelled with the thermalhydraulic system code CATHARE2 [START_REF] Geffraye | CATHARE 2 V2.5_2: A single version for various applications[END_REF].

CATHARE2 code description

CATHARE2 is a thermalhydraulic system code developed for the last thirty years by the CEA, Electricity of France (EDF), Framatome and the Radioprotection and Nuclear Safety Institute (IRSN). This code, originally devoted to best-estimate transient calculations in pressurized watercooled reactors, includes six equations (mass, momentum and energy balance for liquid and vapour phases).

The properties of sodium have been implemented, the friction and heat transfer correlations being the same as in the water standard version [START_REF] Geffraye | CATHARE 2 V2.5_2: A single version for various applications[END_REF]. In this version of the code, nitrogen gas is considered as an ideal gas.

To compute the neutron power evolution of the core, the following reactivity feedbacks are taken into account: Doppler effect, sodium density, cladding expansion, fuel expansion, hexagonal can expansion, diagrid expansion and finally the reactivity feedback resulting from the relative position of the control rods compared to the core and to the main vessel. All these reactivity coefficients are included in the point kinetics of neutron physics module of CATHARE2 including eight groups of delayed neutrons and four groups of fission products to model the decay heat.

All the computations presented in this article are performed with CATHARE2 v25_3 mod 5.1.

Input deck content

Primary circuit

The core channels, the IHX, the upstream and downstream of the pumps are modelled as 1-D CATHARE2 axial elements. The flow distribution within the cold and hot pools is modelled thanks to several dedicated 0-D volumes.

Decay Heat Removal systems

Among the three DHR systems, only the RRB are present in the input deck and modelled by a heat sink.

The removed power is considered as a linear function depending of the sodium temperature in the hot pool. This behaviour was observed for reactor tests simulating decay heat removal situations performed in Phenix and Superphenix reactors [START_REF] Tenchine | Some thermal hydraulic challenges in sodium cooled fast reactors[END_REF]. The RRB is designed to extract a thermal power of 50 MW for a sodium temperature in the hot pool of 530 °C and an air temperature of 35 °C. We considered the heat removal as preliminary following the analytical law (1):

𝑄 𝐷𝐻𝑅 ̇= 50 * 𝑇 𝐻 -35 530-35
(1) With: 𝑄 𝐷𝐻𝑅 ̇: Power extracted by the RRB (MW); 𝑇 𝐻 : Hot pool temperature (°C).

Power conversion system

The PCS system is fully modelled in the CATHARE2 input deck. Each of the heat exchangers (pre-cooler, intercooler, recuperator) is modelled thanks to two axial elements as a counter flow HX. The water flow in the coolers is modelled thanks to boundary conditions (water-inlet temperature fixed to 21 °C, water-feed flow rate respectively fixed to 12000 kg/s and 9000 kg/s for normal operating conditions in the pre-cooler and the intercooler and the water-outlet pressure fixed to one bar). The rotating mass equation is solved on the shaft including the turbine, the compressors and the resisting torques of the generator when it is connected to the grid. Comprehensive TM performance maps are provided as input data. The recuperator, the SGHX and the BP1 by-pass lines are modelled thanks to 1-D axial elements and their associated valves can be operated.

For the NSS:

-The extraction line is modelled by an orifice, with a diameter of 10 centimetres, a valve and the gas pressure boundary conditions of 1.013 bar; -In case of a gas removal, the difference between the high-pressure inside the PCS and the atmospheric conditions induces a choked flow. From the conservation equation of the mass, the definitions of the Mach number and of the sound speed, coupled to the isentropic flow relation and the ideal gas equation of state, a compressible form of the mass flow rate is obtained and given by the equation ( 2).

𝑄 𝑐𝑟 = 1 2 . 𝜋. 𝐷 2 4 . 𝑃 𝐶𝐻𝑃 . √ 𝑀 𝑁 2 .𝛾 𝑅.𝑇 𝐻𝐶𝑃 . ( 2
𝛾+1 ) 𝛾+1 2.(𝛾-1)
(2) With:

-𝑄 𝑐𝑟 : Choked flow (kg/s); -𝐷: Diameter of the extraction line section (m); -𝑃 𝐶𝐻𝑃 : Pressure at the outlet of the high-pressure compressor (Pa); -𝑀 𝑁 2 : Nitrogen molar mass (g/mol); -𝛾: Laplace coefficient (ratio of specific heats, 1.4 for diatomic gas); -𝑅: Ideal gas constant (J/mol/K); -𝑇 𝐻𝐶𝑃 : Temperature at the outlet of the high-pressure compressor (K).

Description of the two LOOP sequences

In case of a LOOP event, a major safety issue is to remove the decay heat of the reactor until the reaching of a stable cold shutdown state (defined by a homogeneous sodium temperature of the primary circuit close to 200 °C). Two procedures are studied in this paper:

-The standard procedure for SFR which relies on the RRB system to remove the decay heat. The Initiating Event (IE) induces a loss of the pumps associated to the RRA and RRC systems, which can not be employed [START_REF] Tenchine | Some thermal hydraulic challenges in sodium cooled fast reactors[END_REF]; -The alternative procedure, proposed in this paper, substitutes the RRB system by the gas PCS in case of a LOOP. This strategy has preliminary been investigated (Bertrand et al., 2016b). First, it was demonstrated that even if the TM rotation speed decreases to zero (normal behaviour for a LOOP accident), the recuperator HX allows the PCS to conserve a low natural circulation of gas which slows the heating up of the core. Furthermore, in this previous study, it was shown that the operating of the gas PCS can keep running the TM. The forced flow of gas coupled to different back-up flow rates of water in the coolers cools the reactor down. In the present paper, the PCS is also employed to allow the reactor to reach the cold shutdown state, but the TM rotation speed regulation and the water flow rate through the coolers are adapted to this purpose.

In the following, the strategy with RRB and the alternative strategy are described.

Initiating events and induced effects

A generator disconnection from the electrical grid initiates the LOOP. The resisting torque in the shaft induced by the generator disappears and involves a turbomachinery overspeed. The detection of the LOOP transient by the control systems causes the full opening of the by-pass 1 valve, which aims at limiting this overspeed. To save the turbomachinery integrity, its rotation speed must not exceed 3300 rpm. In the other part, the LOOP is initiated by the loss of the electrical supplies of the primary and secondary pumps. This loss induces the decrease of the primary and secondary flow rates to 25% of their nominal value in 30 seconds. The water feeding of the coolers is stopped. A few seconds after the IE, the electromagnets maintaining the control rods are switched off and the reactor scram occurs.

Standard procedure

For this sequence, 30 seconds after the IE, emergency electrical supplies are available to ensure a primary mass flow rate of 25% of its nominal value during the whole transient; however, the secondary pumps are stopped. Then, the RRB system is switched on the excess of the 470 °C sodium temperature threshold at the core inlet, this time is called tRRB. A summary of this standard procedure is provided in the table 1.

Alternative procedure

For the alternative sequence, the PCS substitutes the RRB to the decay heat removal. After the IE and the first consequences (cf. section 4.1), several actions are required to keep an efficient heat removal with the water heat sink via the gas PCS and the secondary circuit. As for the standard procedure, 30 seconds after the IE, emergency electrical supplies are available to ensure a primary mass flow rate of 25% of its nominal value during the whole transient. 30 seconds after the IE, the secondary pumps are powered by backup systems and the secondary flow rate is maintained to 25% of its nominal value during the whole transient. The feasibility study and the sizing of the emergency supplies associated to the primary and secondary pumps were realised for the European Fast Reactor (EFR, 1999). Few seconds after the IE, a backup water flow rate is provided as well in the coolers to ensure the heat removal. Furthermore, the water flow rate regulates the gas inlet temperatures of the two compressors to the setpoint of 27 °C, in order to avoid damages on the compressors and keep an efficient heat sink. The power provided by emergency supply limits the maximal water flow rate to 40% of its nominal value. For this procedure, the PCS is operated in two different ways. First, it cools the reactor down to reach the cold shutdown state and then, it maintains this safe state. In the first step, the gas flow rate must be strong enough to remove the decay heat and the energy previously accumulated in the sodium; once the cold shutdown state is reached, only the decay heat must be removed to maintain the reactor in a safe state. For this purpose, a regulation of the TM rotation speed, by the BP1 valve, ensures a gas flow. The target of this regulation is described by the two parameters:

-A ramp-down period, called tramp, for which the target varies from 3000 rpm to a steady value; -The steady value of the rotation speed after the ramp-down period, called ωsteady.

Preliminary results, not mentioned in this paper, have shown that this kind of target for the TM rotation speed, induces a removed power through the SGHX higher than the decay heat. After a delay depending of the ωsteady value, a too much important cooling than needed occurs and could induce a sodium freezing. The selected solution, to adapt the removed power to the decay heat, is to regulate the sodium temperature, at the SGHX outlet, without changing the regulated TM rotation speed. A 180 °C set point is selected to maintain the reactor in the cold shutdown state and the regulation is actuated once the sodium temperature at the outlet of the SGHX becomes lower than 180 °C; the corresponding time is called tSGHX. A summary of this alternative procedure is provided in table 1. Then, to complete the definition of the alternative sequence, the actuator of the sodium temperature regulation and the controllers associated to each regulation must be defined.

Supporting studies for the alternative sequence

This part presents:

-The selection of the actuator to regulate the sodium temperature at the SGHX outlet; -The setting of the controller associated to the regulation of the TM rotation speed.

Sensitivity analysis to select the most influential actuator for regulation

The regulation of the sodium temperature at the SGHX outlet in the alternative sequence can be performed by different actuators available on the gas PCS:

-The valve located on the recuperator by-pass line; an opening of this one induces a temperature decrease of the sodium at the SGHX outlet; -The valve located on the SGHX by-pass line; an opening of this one induces a temperature increase of the sodium at the SGHX outlet; -The NSS removal line: a fully opening of the associated valve induces notably a decrease of the gas flow rate through the SGHX and hence a sodium temperature increase at the outlet of the SGHX.

In order to determine the most efficient to achieve the regulation, among these three actuators, a Global Sensitivity Analyses (GSA) is performed with advanced statistical tools. The target criterion (output of the sensitivity analysis) is the amplitude of the temperature variation induced by the three different actions (inputs of the sensitivity analysis). The more a specific action influences the temperature variation, the more the associated actuator is significant to control the sodium temperature.

Metamodel-based methodology for sensitivity analysis

GSA is an invaluable tool to identify the most influential uncertain parameters, their possible significant interactions, and the non-influential parameters [START_REF] Saltelli | Sensitivity analysis for importance assessment[END_REF]. A widely used GSA approach is the one based on variance decomposition, leading to the definition of Sobol' indices [START_REF] Sobol | Sensitivity analysis for non-linear mathematical models[END_REF]. These sensitivity measures determine how to share the output variance resulting from each input variable alone (first order Sobol' indices) or from interaction between variables (higher order indices). Higher order indices can also be defined but are not computed in practice. All the Sobol' indices are all included in the interval [0 ; 1], their sum is one in the case of independent input variables and their interpretation is natural: the larger the index value, the more influential the input related to this index. In practice, first order are only computed. Several statistical Monte-Carlo estimators [START_REF] Saltelli | Sensitivity analysis for importance assessment[END_REF][START_REF] Jansen | Analysis of variance designs for model output[END_REF] have been proposed to estimate these indices. However, such estimators require several thousand simulations and the loop transient simulated with CATHARE2 is here too time expensive to be directly used to compute efficiently the Sobol' indices. One solution is then to approximate the CATHARE2 outputs by a CPU inexpensive mathematical function, called surrogate model or metamodel [START_REF] Fang | Design and modeling for computer experiments[END_REF][START_REF] Gratiet | Metamodel-Based Sensitivity Analysis: Polynomial Chaos Expansions and Gaussian Processes[END_REF]). This metamodel is built from a set of simulations and must be as representative as possible of the simulator in the variation domain of inputs. Among the metamodels classically used in computer experiments, the Gaussian Process (GP) model has shown strong advantages and capabilities [START_REF] Rasmussen | Gaussian processes for machine learning[END_REF]Williams, 2006 or Marrel et al., 2008).

To build this metamodel, a set of code simulations, namely the learning sample, is needed. For this, the use of space-filling designs, which ensure a full coverage of the input space, is recommended. Among classical space-filling designs for computer experiments, we use here optimized Latin Hypercube Samples (LHS, see [START_REF] Fang | Wrap-around L2-discrepancy of random sampling, Latin hypercube and uniform designs[END_REF][START_REF] Damblin | Numerical studies of space-filling designs: Optimization of Latin Hypercube Samples and subprojection properties[END_REF] for details).

Once the metamodel has been built on the learning sample, its accuracy must be assessed by comparing the observations with the predictions of metamodel, either on an independent test sample or by cross-validation on the learning sample [START_REF] Hastie | The Elements of Statistical Learning: Data Mining, Inference, and Prediction[END_REF]. Finally, the validated metamodel is used for GSA by computing Sobol' indices. The Figure 4 details the methodology.

Application and results for the temperature of sodium

The above-described methodology is applied to the variation of the sodium temperature at the outlet of the SGHX, computed by CATHARE2 code, for a transient induced by the simultaneous using of the three different actuators. These actuators are tested for a nominal power and their explored ranges of variation are summarised in the table 2. Note that these ranges are chosen to be of the same magnitude order as in normal operation.

Table 2. Actuators and associated variation ranges for the sensitivity analysis Actions Lower limit Upper limit Time for which gas is removed with the NSS (s) 0 200 Opening of the recuperator by-pass valve (%) 0 25 Opening of the SGHX by-pass valve (%) 0 25

An optimized LHS2 of N = 600 experiments with three inputs is generated with the R package "DiceDesign" (Dupuy et al., 2005). Each experiment is defined by a set of values for the three above actions; the CATHARE2 code simulates the transient induced by these three simultaneous actions. The CATHARE2 output of interest is the difference between the 345 °C for a nominal power state and the achieved temperature for a new steady state. A GP metamodel is fitted on the learning sample to emulate the output of interest. It has a very high predictivity (99% of output variance correctly predicted by the metamodel). Finally, from the metamodel, Sobol' indices are computed with the Jansen's estimator implemented in the R package "Sensitivity" [START_REF] Pujol | Sensitivity: Global Sensitivity Analysis of Model Outputs[END_REF]. The results obtained for first Sobol' indices are given in table 3. The NSS with a Sobol index of 0.59 (59% of output variance explained) is the most significant actuator. Even if the delay to affect the sodium temperature is higher for the NSS than for the two by-pass valves, its capacities are considered adequate to regulate the sodium temperature. Note that higher order indices are null (sum of first orders equal to one): this means that there is no significant interactions between the inputs.

Corrector setting for the rotation speed regulation of the turbomachinery

The use of the gas PCS as a DHR system required the following regulations:

-The TM rotation speed; -The sodium temperature at the SGHX outlet; -The gas temperature at the compressor inlets.

Their settings are presented in the following.

The control line of the TM rotation speed regulation is designed as the Figure 5. Preliminary studies have revealed that the set of values (𝐾 𝑟 ; 𝜏 𝑖 ; 𝜏 𝑑 ) impacts significantly the quality of the rotation speed regulation. Hence, we fixed this set in order to ensure a high quality for the regulation.

To find a good set of values, we first define pertinent upper and lower bounds of the inputs (given in table 4), according to expert judgement. Then, we explore the domain of variation of these inputs thanks to a space-filling design. For this, an optimized LHS of N = 250 is used (cf. Section 5.1.1). For each set of (𝐾 𝑟 ; 𝜏 𝑖 ; 𝜏 𝑑 ), CATHARE2 code simulates the LOOP accident as defined in the 4.3 part. To assess the quality of each regulations, the difference between the regulated TM rotation speed and its target is considered, via the following Integral Time Square Error (ITSE) criterion:

𝐼𝑇𝑆𝐸 = ∫ 𝑡 * (𝜔(𝑡) -𝜔 𝑡 (𝑡)) 2 (𝑡)𝑑𝑡 t 5% 0 (4)
With:

-𝜔(𝑡): TM rotation speed (rpm); -𝜔 𝑡 (𝑡): TM rotation speed target (rpm); -t 5% : Time from which the rotation speed is 5% close of the target (s).

The selected PID corrector is the one that minimizes the ITSE criteria for the 250 experiments, thus defining an efficient regulation:

-𝐾 𝑟 𝑜𝑝𝑡 = 1.3.10 -3 rpm -1 ; -𝜏 𝑖 𝑜𝑝𝑡 = 10.7 s; -𝜏 𝑑 𝑜𝑝𝑡 = 0.13 s.

The two other regulations used for the alternative procedure preliminary rely on proportional correctors (Table 5).

Table 5. Gain of the proportional correctors used for the standard procedure Regulation Gain of the proportional corrector Temperature of the gas at the inlet of the compressors 𝐾 𝑝 𝑐𝑜𝑜𝑙𝑒𝑟𝑠 = 160 kg. s -1 . K -1

Temperature of the sodium at the outlet of the SGHX 𝐾 𝑝 𝑁𝑆𝑆 = 500 kg. °K-1

The three regulations associated to the gas PCS and the required actions to keep an efficient heat sink and an available secondary circuit are now clearly defined. Hence, the gas PCS could be used to remove the decay heat.

Optimisation of the alternative sequence and comparison to the standard sequence

In this part, the sequence using the PCS is optimised to find a compromise between fast cooling and low thermal stresses. Then, a comparison of optimal alternative procedures to the standard procedures is performed for the following points:

-Their capability to cool down the reactor; -Thermal stresses induced on the main vessel during the transient.

For the whole following figures, the LOOP IE occurs at 100 seconds.

Studied variables

The capability of a procedure to cool down the reactor is assessed with the delay to reach the cold shutdown state. In the present study, this time (called 𝑡 𝐶𝑆𝐷 ) is reached when the temperature of the sodium at the inlet of the core become lower to 210 °C. Furthermore, the main vessel has to resist to all stresses during its lifetime and avoid creeping conditions. On the one hand, its temperature has to stay below a threshold for which creep is not significant. The main vessel is made of austenitic steel (316L (N)), for this material creeping condition occurred for temperatures higher than 450 °C (Afcen, 2010). A verification of this threshold excess will be made. On the other hand, thermal stresses induced on the main vessel by thermal gradients have to be as low as possible. These stresses are assessed during a transient by the absolute value of the thermal gradient through the main vessel. With:

𝑔𝑟𝑎𝑑(𝑇) = 𝑇 𝑖 -𝑇 𝑜 𝑒 (7)
-𝑇 𝑖 : Inside temperature of the vessel (°C); -𝑇 𝑜 : Outside temperature of the vessel (°C); -𝑒: main vessel width (4cm).

The thermal gradient is studied at the higher part of the main vessel, where the highest thermal stresses are observed, because of the proximity of the hot pool, which induces a heat-up of the main vessel. Cold sodium flow from the core inlet to an annular space inside and against the main vessel is hence derived to cool down the main vessel. This annular space is called the immersed weir and ensures creep and fatigue resistance of the main vessel. It is at this cooling system that the most significant thermal gradients are observed and in this paper, the two needed temperatures to assess the thermal gradient through the vessel are located at the inlet of the immersed weir. In the following, the maximum absolute value of the thermal gradient is the criterion to assess the thermal stresses induced on the main vessel. We defined it as MTG (°C/cm) (equation 8) depending on the thermal gradient given by the equation 7:

𝑀𝑇𝐺 = max (|grad(T)|) (8)

Optimisation of the alternative procedure

For the alternative procedure, the definition of the target for the regulation of the TM rotation speed is defined as a solution of the Multiobjective Optimisation Problem (MOP), for which the two variables (𝑡 𝐶𝑆𝐷 and 𝑀𝑇𝐺) have to be minimized simultaneously. The inputs of the MOP are the two descriptive parameters of the TM rotation speed target: 𝑡 𝑟𝑎𝑚𝑝 and 𝜔 𝑠𝑡𝑒𝑎𝑑𝑦 .

The retained problem-solving method is to perform, with the code CATHARE2, several alternative procedures applied for the LOOP accident, for which the TM rotation speed target is modified, and then to select a good candidate to solve the MOP. A LHS (cf. section 5.1.1) including 250 calculations built the database. Its inputs are the two parameters of the TM rotation speed target which variation ranges are set in the Table 6. The MOP is characterized by two objectives whose variations are conflicting. On the one hand, the 𝑡 𝐶𝑆𝐷 objective decreases significantly with an increase of the 𝜔 𝑠𝑡𝑒𝑎𝑑𝑦 parameter. On the other hand, the MTG increases with an increase of the tramp parameter. These qualitative variations are observed in the Figure 6, in which each objectives of optimisation of the database are plotted in function of the two descriptive parameters of the TM rotation speed target. However, as shown in the Figure 6, the relation between MTG and 𝜔 𝑠𝑡𝑒𝑎𝑑𝑦 and the relation between tCSD and tramp are not very clear according to this scatterplot analysis. Nevertheless, it can be expected according to engineering judgment that for a high 𝜔 𝑠𝑡𝑒𝑎𝑑𝑦 the MTG cannot be minimal because a large TM rotation speed induces a large PCS flow rate that induces a stronger cooling of the structures. In the same way, a small tramp does not able to have a short tCSD. In this case, the cooling is not efficient enough because the TM rotation speed and so the gas flow rate are reduced very fast.

Finally, to conclude it can be kept in mind that the peak of thermal stress being reached early in the sequence, this value is very impacted by the decrease phase of the TM rotation speed up to its asymptotic value. Conversely, the time to reach the cold shutdown state is largely impacted by the final value of the TM rotation speed because the variable tCSD is about several hours and therefore is only slightly impacted by the TM rotation speed decrease occurring at the beginning of the sequence. 𝑀𝑇𝐺 (°C/cm) 𝑀𝑇𝐺 (°C/cm) Based on the LHS above-mentioned, the solution of this MOP is the following Pareto front (Figure 7) associated to the Pareto set (Figure 8) constructed with the R package "rPref" (Roocks, 2016). The Pareto front is included between two extremes (Experiments A and B in the Figure 7), presented in the table 7, which define two large ranges of variation for the two objectives of optimization. The alternative procedure allows the reactor to reach the cold shutdown state between 30 minutes and 4 hours. A short delay to reach the safety state induces a MTG through the main vessel close to 5 °C/cm ("B" experiment), whereas a long delay to reach the safety state induces a MTG through the main vessel close to 1 °C/cm ("A" experiment).

It should be noticed that by considering the set of calculations in a graph representing their input parameters in axis 𝑡 𝑟𝑎𝑚𝑝 as a function of 𝜔 𝑠𝑡𝑒𝑎𝑑𝑦 a space defined by a 𝑡 𝑟𝑎𝑚𝑝 between 500 s and 2000 s and a 𝜔 𝑠𝑡𝑒𝑎𝑑𝑦 between 1000 rpm and 1875 rpm (delimitated by a red rectangle in the Figure 8) is not included in the Pareto set. To solve the MOP, this input space area must not be exploited to define the TM rotation speed target. Thanks to the TM rotation speed target it is possible to choose the objective that has to be minimized. The following part compares the standard procedure to optimised alternative procedure and highlights the relations between the outputs and the inputs.

Analyse of the optimised alternative procedure and comparison to the standard procedure

The following part provides a comparison of the three optimal procedures defined in the table 7 and the standard procedure. The thermal gradient (Eq. 7) through the main vessel is illustrated for the four compared procedures in the Figure 9. Thermal gradients through the main vessel vary differently for the standard and the alternative procedures. The significant cooling induced by the PCS, leads to an inside temperature of the main vessel lower than its outside temperature. In such situation, the main vessel inside is compressed and the outside is under traction. For the procedure with the RRB the situation is inverted, the main vessel inside is under traction and the outside is compressed. For the RRB procedure the MTG is equal to 3.3 °C/cm. In spite of these different behaviours, the maximum absolute values of the thermal gradients are of the same order of magnitude for the two procedures. Furthermore, for few optimised alternative procedures, it is possible to reduce these thermal stresses. The inlet temperature of the core are plotted for the three procedures defined in the table 7 and for the standard procedure (Figure 10). For the standard procedure, the sodium temperature at the inlet of the core exceeds 450 °C, to reach the temperature threshold of 470 °C which turns on the RRB, and hence induces a significant creep on the main vessel. The passive RRB systems require a delay of 560 seconds to be turned on. Then, 110 seconds after the actuation of the RRB, a decrease of the sodium temperature at the inlet of the core is observed. On the other hand, the procedures with the gas PCS are directly efficient to decrease the sodium temperature at the core inlet. It presents the real advantage to not exceed the threshold of 450 °C and hence to avoid creeping condition. Furthermore, the passive RRB systems do not allow the reactor to reach the cold shutdown state after a period of 24 hours. For the alternative procedure, the operating of the gas PCS leads the reactor to the cold shutdown state with a delay depending of the TM rotation speed target (Figure 10). To make a choice through the optimal targets a cobweb plot is a useful tool. This one associated to the Pareto front is built with the R package "plotly" (Carson, 2018) (Figure 12). The Pareto front can be arbitrarily separated in three parts.

-The first part is compounded of elements which favour the MTG: the 𝑡 𝐶𝑆𝐷 is higher than 5000 s and the MTG lower than 1.6 °C/cm (green lines in the Figure 12). These transients required a low 𝑡 𝑟𝑎𝑚𝑝 combined to an ω steady inferior to 1600 rpm, in the aim to reduce as low as possible the 𝑡 𝐶𝑆𝐷 without increase too much the MTG. The point A and C of the table 7 are included in this part. -The second part is compounded of elements which favour the 𝑡 𝐶𝑆𝐷 : this one is lower than 3000 s and the MTG higher than 2.8 °C/cm (red lines in the Figure 12). These transients required an 𝜔 𝑠𝑡𝑒𝑎𝑑𝑦 superior to 1930 rpm necessarily combined to a 𝑡 𝑟𝑎𝑚𝑝 superior to 500 s, to reduce as low as possible the MTG without increasing significantly the 𝑡 𝐶𝑆𝐷 . The point B of the figure 7 is included in this part. -The third part includes optimal compromises for which the 𝑡 𝐶𝑆𝐷 value is included between 3000 s and 5000s, and the MTG value is included between 1.6 °C/cm and 2.8 °C/cm (blue lines in the Figure 12). The two first parts can be qualified as compromises privileging one of the two variables of optimisation.

Conclusion and prospects

This paper presents an alternative sequence, in case of the LOOP accident, which removes the decay heat with the gas PCS instead of exploiting the RRB system. The three regulations required to operate the cooling of the reactor are:

-The regulation of the TM rotation speed to keep an efficient gas flow; -The regulation of the sodium temperature at the outlet of the SGHX, to adapt the heat removal to the decay heat, once the cold shutdown state reached; -The regulation of the gas temperature at the inlet of the compressors, in the PCS, to keep an efficient heat sink.

The choice of the NSS to regulate the sodium temperature at the outlet of the SGHX is supported by Sobol indices calculation. The NSS is the tested actuator that induces the highest variations of this temperature and is hence selected. The setting of the PID corrector associated to the TM rotation speed is performed by a comparison of different settings and the one with the lowest difference between the target and the rotation speed is held.

The alternative procedure in case of the LOOP accident is efficient to reach the cold shutdown state and to maintain it, whereas it is not possible with the RRB for the same scale of time. The PCS, thanks to the regulation of the TM rotation speed with the BP1 valve, is an adaptable system to optimise simultaneously the two conflicting objectives:

-The delay to reach the cold shutdown state; -The thermal stresses on the main vessel. The Pareto front, associated to this MOP, has been determined and it groups all the optimum compromises. It is up to designers to choose among the Pareto front which transients are the more interesting for them. On the one hand, it is possible to decrease the thermal gradient through the main vessel up to 0.7 °C/cm by accepting the delay to reach the cold shutdown state increases to about 4 hours. On the other hand, it is possible to decrease the delay to reach the cold shutdown state up to about 30 minutes by accepting the thermal gradient through the main vessel increases up to 4.7 °C/cm. With a thermal gradient of 3.3 °C/cm, for the procedure with RRB, it is highlighted that the alternative procedure can reduce the stresses on the main vessel. Another interesting objective to optimise would be the period for maintaining the reactor in the safe state. The longer this period is the latter the RRB system would be actuated. This possibility is interesting for the probabilistic safety approach by keeping for a long time a diversification of the systems to perform the DHR function. This study is a first step towards a methodology to optimise the reactor operation in case of various transients. To finalize the methodology, other tools have to be chained to the building of LHS and Pareto front. The latter would be able to define, for a specific transient and a MOP, the procedures to obtain the best compromises.

Figure 3 .

 3 Figure 3. Sketch of the SFR with a focus on the Gas-PCS

Figure 4 .

 4 Figure 4. Metamodel-based methodology for sensitivity analysis

Figure 5 .

 5 Figure 5. Control line of the TM rotation speed layout

Figure 6 .

 6 Figure 6. Scatterplot of the two objectives of optimisation as functions of tramp and ωsteady

Figure 7 .Figure 8 .

 78 Figure 7. Experiments of the database and Pareto front: tCSD=f(MTG)

Figure 9 .

 9 Figure 9. Thermal gradient through the main vessel

Figure 10 .

 10 Figure 10. Inlet temperature of the core

Figure 11 .

 11 Figure 11. TM rotation speed associated to the three experiences defined in the table 7

Figure 12 .

 12 Figure 12. Cobweb plots of the Pareto front

Table 1 .

 1 Chronological summary for the standard LOOP and the alternative procedure

	Time	LOOP with RRB	LOOP with PCS
			IE: Generator disconnection from the grid
	t = 0s		Beginning of the primary and secondary flow rates decrease
			Stopping of the coolers water-feeding
	t = 1s		Turbomachinery protection: full opening of the by-pass 1 valve
	t ~ 2s		Reactor scram
	t = 2s	x	Regulation of the TM rotation speed
	t = 3s	x	Water feeding of the coolers thanks to backup systems and regulation of the gas temperature at the compressor inlets with water flow rate
		Primary flow rate fixed to 25% of its nominal value thanks to emergency electrical
	t =		supplies
	30s	x	Secondary flow rate fixed to 25% of its nominal value thanks to emergency electrical supplies
	t = tRRB	RRB switch on	x
	tSGHX	x	Regulation of the sodium temperature at the SGHX outlet

Table 3 . First Sobol' indices for the sodium temperature variation at the outlet of SGHX Actuators (Inputs)

 3

		Time of gas	Opening of the	Opening of the
		removal with the	recuperator by-pass	SGHX by-pass
		NSS	valve	valve
	First order Sobol' indices of
	the sodium temperature	0.59	0.25	0.16
	variation

Table 4 .

 4 Explored ranges to set the PID corrector associated to the BP1 valve

		Lower value	Upper value
	𝐾 𝑟 (rpm -1 )	9.10 -6	9.10 -2
	𝜏 𝑖 (s)	0	3.10 5
	𝜏 𝑑 (s)	0	2.10 4

Table 6 .

 6 Variation ranges to the exploration of different TM rotation speed regulation target

		Lower limit	Upper limit
	tramp (s)	0	2000
	ωsteady (rpm)	1000	2000

Table 7 .

 7 Points of interest of the Pareto front

		𝑡 𝑟𝑎𝑚𝑝 (𝑠)	𝜔 𝑠𝑡𝑒𝑎𝑑𝑦 (rpm)	𝑡 𝐶𝑆𝐷 (s)	𝑀𝑇𝐺 (°C/cm)
	A	145	1141	13915	0.7
	B	1340	1903	2157	4.7
	C	302	1508	6078	1.4

CFV stands for « Coeur à Faible effet de Vidange » in French that means core with low sodium void worth effect.

Optimized by discrepancy criterion[START_REF] Damblin | Numerical studies of space-filling designs: Optimization of Latin Hypercube Samples and subprojection properties[END_REF].