# Exploring Stochastic Sampling in Nuclear Data Uncertainties Assessment for Reactor Physics Applications and Validation Studies

^{*}

## Abstract

**:**

## 1. Introduction

_{eff}. The k

_{eff}sensitivity vectors required by the TSUNAMI methodology originally were computed within the SCALE system with the help of the multigroup Monte Carlo (MC) particle transport code KENO [9]. Recently significant progress was also achieved in the continuous-energy sensitivity calculations, see e.g., [10,11].

^{®}codes MCNP

^{®}or MCNPX (hereafter MCNP(X) stands for both the MCNP6 and MCNPX versions of the codes owned by USA Los Alamos National Security, LLC, manager and operator of Los Alamos National Laboratory, being in use at Paul Scherer Institute (PSI)) [12,13], however the realization of this option in the standard versions of MCNP(X) is known to have deficiencies in the case of eigenmode calculations [11]. The generalized perturbation theory framework exists for both the deterministic [14] (see the link “Bibliography relative to Sensitivity and Uncertainty (S/U) Analysis” for a list of relevant references) and the MC based neutron transport solution options. For instance the continuous-energy MC calculation capability is already available [15], though such studies are still occasional in practice.

_{eff}responses.

_{eff}value analysis, as e.g., applied in [50,51]. A prospective use of the stochastic sampling including the correlation assessment and enclosed with Bayesian updating algorithms [21] is pushing forward by the developers of Nuclear Data Uncertainty Analysis (NUDUNA) [34,51] and MOCABA (MOCABA is a combination of Monte Carlo sampling and Bayesian updating algorithms) [52] tools. The use of the Bayesian updating procedures has been already in place at PSI with the Total Monte Carlo (TMC) method and associated tools [53]. Thus, further adaptation of such techniques is foreseen at PSI and the given study is a step in this direction.

## 2. Subject of the Given Study

_{eff}/criticality safety assessments).

## 3. Description of the Calculation Models and the Calculation Methodologies

_{eff}parameter) with a self-consistent fission source distribution [13]. Thus, the role of ND is not fully the same in the fixed-source and in the eigenmode calculations. In the first approach the uncertainties of the fuel isotopes which are related to the neutrons generation at the fission process can be accounted only in the given fission source specifications, but they cannot be propagated through the fixed-source simulations. However the fuel isotopes continue to affect the neutron transport through other than fission interactions with neutrons (and actually the fission cross-section remains to be a part of the absorption cross-section) and thus they still contribute (partly) to the calculated outcome uncertainties. In this sense only the impacts of the neutron multiplicity and the fission neutron spectra are missed. This said, one should notice that in the fixed-source model the source amplitude is always normalized to the same reference value (corresponding to the total reactor power) and therefore the neutron multiplicity perturbation is not much relevant for consideration in the neutron transport simulation. Thus, in the given study when using the fixed-source modeling option we consider the ND uncertainties effects only with respect to the neutron transport and we ignore the uncertainties in the fission source spatial-energy formation. Note that normally in the PSI methodology the neutron source for MC calculations with MCNP(X) is prepared based on the detailed and validated reactor core-follow calculations with deterministic codes CASMO/SIMULATE (see for details [67]). A special methodology was also developed at PSI for assessment of the ND-related uncertainties in the core-follow calculations [39,41,42]. Thus, in principle, the uncertainty of the neutron source specifications used in the MCNP(X) fixed-source calculations can be taken into account as well and such studies are planned.

_{p}) and the prompt fission neutron spectra (χ

_{p}) for actinides (see for more details [54]). Thus, the dimension of the varied inputs is about ~50 isotopes times ~5 reactions times ~187 energy bins of covariance matrices (as applied for the calculations of Section 4, in consistency with one of the default group structures used in the NJOY nuclear data files processing code, applicable for light water reactor (LWR) lattices, see also for illustration [57]), i.e., about ~50,000 of “different” input parameters.

## 4. Obtained Uncertainties Results

#### 4.1. General Description and Assessment

_{eff}ND-related uncertainties, typical for the LWR-type of fresh fuel critical benchmark configurations can be recalled. The average value obtained with the ENDF/B-VII.1 CM data for PSI CSE validation benchmark suite subset with UO

_{2}fuel was: ${\overline{\mathsf{\sigma}}}_{ND}$ = 598 pcm (pcm: per cent mille, 1 × 10

^{−5}), there the SCALE 44-group structure was used for CM processing with NJOY [54]. For comparison, in work [71], presenting results of TSUNAMI/SCALE calculations, the ND-related uncertainties are close to the ones mentioned above. The ${\overline{\mathsf{\sigma}}}_{ND}$ values obtained for the UO

_{2}cases (with not exactly the same set of benchmarks, comparing to [54]) with ENDF/B-VII.0 and SCALE 6.2 ND libraries were 583 pcm (see also [29] for details) and 597 pcm respectively. For the full set of 149 benchmarks from PSI validation suite, which includes 27 mixed oxide fuel (MOX) cases, the ${\overline{\mathsf{\sigma}}}_{ND}$ value was 614 pcm.

^{93}Nb fluence monitor the average (C/E-1) values obtained with the PWR-GP, PWR-ST and BWR-D validation models (with ENDF/B-VII.0 in [62] and ENDF/B-VII.1 in [68] as the base ND and CM library) were respectively ~5%, ~5% and ~18%. For the same models and the

^{54}Fe fluence monitors the (C/E-1) values were ~−1%, ~3% and ~9%. It should be noted that the given values are still preliminary since some improvements of the calculation models and the methodology are still on-going at PSI [73]. As it is seen from Table 1, the ND related uncertainties can have a significant contribution to the overall uncertainty of the calculations.

#### 4.2. Observations and Trends Analysis

^{1}H is usually one of the major contributors to the uncertainty in the fast neutron flux calculations. As it was mentioned above, for the model analyzed in [64], which was very similar to the case PWR-CRs, the

^{1}H related uncertainty was about ~1%, while the total uncertainty was about 4%–5% if compared with the same responses as considered here (note that the uncertainties shall be summed up as variances). The principle contributor for the in-core neutron transport was however the

^{238}U inelastic scattering cross-section. Next, for the case PWR-GP, the impact of

^{1}H was found about 2.5%, which gives approximately 12% to the total uncertainty for the fast flux, which was tentatively assessed about 5.5%. The major contributor was still the

^{238}U inelastic scattering cross-section. Now, for the given study it was interesting to see the value of the

^{1}H contribution for the case BWR-D, since it has the highest uncertainty together with the highest thickness of the water zones between the neutron source and the response location. To assess the effect of

^{1}H impact, the NUSS/MCNP(X) calculations were repeated with perturbation of only the

^{1}H cross-sections (200 sample calculations were done this time). The resulting sample standard deviation for the fast flux outputs was about ~13%. Therefore for the BWR dosimetry and from the point of view of only the neutron transport (without neutron source uncertainties considerations), the contribution of the single isotope

^{1}H is dominant for the total uncertainty. This can be explained by the fact that

^{1}H is a very strong moderator of neutrons and changing its elastic scattering cross-section significantly affect how many neutrons will be slowed down below the energy cut-off for the fast neutron flux of 1 MeV (note that the uncertainties on angular distributions are not considered yet).

## 5. Further Profits from the Stochastic Sampling Results

#### 5.1. Evaluation and Preliminary Assessment of Correlations

_{eff}response to the perturbations of the input parameters, viz. nuclear data. Such analysis can be efficiently performed using the deterministic approaches (see e.g., [29]). However, in the case of consideration of arbitrary system responses, the way of computing the sensitivity coefficients in a production manner still needs further developments [15]. Alternatively, when, for instance, nuclear data stochastic sampling is performed with a tool like NUSS, it is straight forward to assess the Pearson correlation coefficient, r, for a pair of calculation sets, e.g., with different models and/or outputs [47,48,50,51,75] (note also that r

^{2}is equivalent to the “coefficient of determination” [76], which can be used to assess linear relation between the output and input values, as demonstrated in [31,75], but here the discussion is limited to only different systems output correlations in which case r

^{2}can be used to assess the proportion of variance in common between two outputs [76]; actually the NUSS tool has been further extended to the version NUSS-RF (where -RF stands for “Random balance design and Fourier amplitude sensitivity testing”) to allow performance of Global Sensitivity Analysis, which concerns the first order global sensitivity indices instead of the local (one at a time) linear sensitivity coefficients [21,46,48,49], but for the context of the given paper we limit the discussion by using only the regular NUSS tool.

_{2}(“Low-enriched uranium Compound Thermal systems”—LCT cases category in [3]) and with mixed oxide (MOX, contains plutonium fraction) fuels (MOX cases correspond to “Mixed plutonium-uranium Compound Thermal systems”—MCT category in [3]). Presence of these two types of fuels in the validation suite is stipulated by the fact of use of both of them in Swiss reactors during their exploitation history. Figure 2 shows the complete correlation matrix obtained after 300 sample calculations for all the 149 benchmarks from the PSI validation suite. For clarity, the first 122 sequential benchmarks correspond to the LCT category and the next benchmarks from number 123 to 149 correspond respectively to the MCT category.

_{eff}values obtained for individual benchmarks on the resulting values of the benchmarks’ correlation coefficients was reported recently [79]. It indicates the need for further verification of the currently obtained results on the ND-related benchmark correlations, with respect to the MCNP(X) statistical uncertainty. It is also worth noting that in work [80] the correlations due to nuclear data uncertainties were analyzed as well for the set of benchmarks from the UACSA Phase-IV exercise, with the help of SCALE/TSUNAMI calculation sequences. For the verification purpose, NUSS methodology with the ENDF/B-VII.1 library has been also recently applied for such study [81] and preliminary results were found in a very reasonable agreement with those reported in work [80], noting that in the latter the ENDF/B-VII.0 library was used. That preliminary comparison can be seen as an additional justification of the NUSS workability for similarity assessments.

#### 5.2. Results Examination and Verification

_{eff}is completely outstanding here and it is included just on the purpose to verify, if the correlations between the outcomes of interest (fast flux) and an outcome which is by definition not expected to be correlated with the major outcomes of interest will be indeed found weak, as it should be. This exercise can serve as a “stress-test” for the computation procedures employed here on the basis of the NUSS tool. Finally, even if a better statistical precision of MC calculations and larger sample sizes are needed for more confident statements, there is an indication that the correlations between the cases PWR-CRs and the cases PWR-GP, PWR-ST and BWR-D are systematically higher than the correlations of the same cases and PWR-CRk model. This also looks logical since all the cases except PWR-CRk correspond to the fixed source option and thus shall demonstrate higher correlations compared to the eigenmode case PWR-CRk.

_{eff}results are weak, while the correlations among BWR-D, PWR-ST and PWR-GP are strong.

## 6. Discussion

## 7. Materials and Methods

## 8. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**MCNP(X) schematic models of: (

**a**) pressurized water reactor (PWR); and (

**b**) boiling water reactor (BWR).

**Figure 2.**Matrix of Pearson correlation coefficients obtained for criticality validation benchmarks with 300 randomly sampled libraries.

**Figure 3.**Pearson coefficient behavior vs. sample size for the selected couple of model calculations.

**Figure 4.**Upper triangular matrix of Pearson correlation coefficients obtained with 300 randomly sampled ND libraries.

**Table 1.**Fast neutron flux (E > 1 MeV) calculation uncertainties (one relative (rel.) sample standard deviation (STD)) (%). CR: control rods; and NUSS: Nuclear data Uncertainty Stochastic Sampling.

Model | PWR-CRs | PWR-CRk | PWR-GP | PWR-ST | BWR-D | PWR-CRs-t | PWR-CRk-t |
---|---|---|---|---|---|---|---|

NUSS Rel. STD | 4.0 | 5.1 | 7.6 | 10.5 | 14.5 | 1.0 | 1.8 |

MC Rel. STD | 0.5 | 0.6 | 0.4 | 0.2 | 0.6 | 1.1 | 1.2 |

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**MDPI and ACS Style**

Vasiliev, A.; Rochman, D.; Pecchia, M.; Ferroukhi, H.
Exploring Stochastic Sampling in Nuclear Data Uncertainties Assessment for Reactor Physics Applications and Validation Studies. *Energies* **2016**, *9*, 1039.
https://doi.org/10.3390/en9121039

**AMA Style**

Vasiliev A, Rochman D, Pecchia M, Ferroukhi H.
Exploring Stochastic Sampling in Nuclear Data Uncertainties Assessment for Reactor Physics Applications and Validation Studies. *Energies*. 2016; 9(12):1039.
https://doi.org/10.3390/en9121039

**Chicago/Turabian Style**

Vasiliev, Alexander, Dimitri Rochman, Marco Pecchia, and Hakim Ferroukhi.
2016. "Exploring Stochastic Sampling in Nuclear Data Uncertainties Assessment for Reactor Physics Applications and Validation Studies" *Energies* 9, no. 12: 1039.
https://doi.org/10.3390/en9121039