3.1. Burnup Credit System for the Swiss Reactors–Repository (BUCSS-R)
The BUCSS-R scheme applies BUC starting with the depletion calculation of a FA in the in-core burnup phase considering also shutdown cooling between cycles, followed by cooling during interim storage as well as cooling in the long term of final disposal. Based on detailed information about the irradiation conditions, this computational sequence allows for the determination of a realistic isotopic inventory in the FAs and its evolution over time. BUCSS-R is described in detail in [
5]. Nevertheless, the elements of the methodology shall be briefly outlined here in order to provide the background for this study, see also
Figure 2. Each step of the computational sequence is listed and outlined below, headed by the main code package(s) used for the current task:
The first task is a realistic estimation of operation histories, including in-core depletion and shutdown cooling between cycles. The calculations are based on reference CASMO-SIMULATE core models, developed and validated for all Swiss reactors and almost all operated cycles within the PSI core management system (CMSYS) platform [
14,
15,
16,
17].
- 2.
BOHR
The detailed irradiation history can be extracted from SIMULATE3 results by means of an in-house code called BOHR [
18]. State parameter values for every fuel assembly and axial elevation at numerous cycle instants are retrieved and corresponding CASMO5 [
19] input files are produced.
- 3.
CASMO5
New CASMO5 calculations are carried out based on BOHR output. Pin-wise and axially resolved isotopic compositions of the burnt fuel are determined. This step allows obtaining detailed spent nuclear fuel compositions, rather closely corresponding to the actual fuel assembly operating history.
- 4.
SERPENT2
The SNF isotopic composition change due to decay is calculated by means of SERPENT2 [
20]. In this way, the long-term nuclide content evolution inside the loaded disposal canister can be determined. The choice of the SERPENT decay module was stipulated by its good performance and also due to the possibility to easily use different sources of the decay data, facilitating the different data libraries assessment [
21].
- 5.
COMPLINK
The SERPENT2 output is processed with the in-house code COMPLINK [
22]. The isotopic composition data are automatically transferred into a pre-existing MCNP6-model template of the disposal canister loaded with FAs, producing MCNP6-input files ready to be executed [
3].
- 6.
MCNP6
The criticality calculation is finally performed with MCNP6 and the neutron multiplication factor keff is determined.
Nuclear data uncertainty is propagated through the execution of the calculation chain for n samples of perturbed nuclear data. The effect of ND uncertainties on criticality is then quantified by the standard deviation in the n-tuple of calculated keff values.
The BUCSS-R methodology has been used already for a preliminary criticality safety assessment of UO2 spent fuel assemblies from a Swiss pressurized water reactor (PWR) loaded into a disposal canister of NAGRA’s design [
11,
23]. The results of this assessment were presented in the form of so called loading curves, indicating what minimum fuel assembly burnup is required for a fuel assembly with a given initial fuel enrichment to ensure that the effective neutron multiplication factor of the flooded disposal canister would comply with the imposed nuclear criticality safety (NCS) criterion.
A distinctive feature of PSI’s CSE method in general has been outlined in [
23]: “Normally, the fuel depletion calculations for BUC evaluations are done with a set of bounding parameters in terms of power density, fuel and coolant temperatures, densities, etc., so that the reactivity of the fuel at discharge for such conservative assumptions will be higher than the reactivity obtained with any possible real irradiation history. This path, however, was not fully adopted for the joint PSI/NAGRA BUCSS-R […] project. The approach being utilised is different because real operational data could be employed (using available CASMO/SIMULATE3 validated core follow models at PSI) for all fuel assemblies operated in Switzerland. This allows estimating the loading curves on the basis of best-estimate assessments integrated with a conservative but rational treatment of uncertainties.” A thorough comparison of the present method with others in the field of criticality safety is indicated at this point, however, corresponding efforts would certainly go beyond the scope of this work.
3.2. Uncertainty Propagation and Quantification
Following the BEPU concept, the fuel depletion and canister criticality calculations with BUCSS-R are furthermore complemented by UQs. CSEs in preparation of long-term geological disposal are in general affected by the following uncertainty sources:
Nuclear data;
Operating conditions of the FA in the reactor core;
Design and technological parameters of the FAs and the disposal canister;
Radiation-induced changes in the FA geometry.
A comprehensive, although still preliminary, quantification of the uncertainties from all of these sources was summarised in [
11]. ND-UQ results therein have been adopted from another study, also conducted at PSI [
4]. Since ND are, in one format or another, essential input required for steps 3, 4 and 6 of the BUCSS-R scheme, several tools were developed and eventually used to quantify ND-related uncertainties stemming from each of those steps, see red frame in
Figure 2. Using these tools, ND uncertainties can be propagated through the calculation chain by repeating the calculations for a large number of times, each time with a different set of ND, randomly sampled following information on the ND uncertainties provided in the selected ND library. The methods of the tools are based on stochastic sampling (SS), which has been explored extensively in the context of ND-uncertainty assessments for reactor physics applications and validation studies [
24]. The UQ finally consists in the statistical analysis of the outcome of all the simulations. The following tools can be employed independently from each other:
ND (i.e., cross sections, fission neutron yields
and fission spectra) can be perturbed with the SHARK-X [
16,
25,
26,
27] tool, which also takes care of the correct formatting for usage in CASMO5. First, ENDF covariance files are processed with NJOY into a number of energy groups from thermal energy up to 20 MeV. These matrices are then used to produce random realizations of ND by means of SS from joint probability distributions defined with normal distributions. SHARK-X has been used for a previous study [
28] assessing uncertainties for Swiss LWR spent nuclear fuels due to ND, however, with a focus on other quantities of interest: assembly burnup, decay heat, neutron and gamma sources, as well as isotope inventory for realistic fuel histories.
- 2.
Fission yields sampling
The UQ due to fission yields is similar to the case of the above ND. A stochastic sampling of evaluated fission yields is performed, based on fission yield uncertainties (as provided in the evaluated libraries), and on an in-house correlation matrix (as no fission yield correlations are provided in such evaluated libraries). Details on the correlation matrix can be found in [
29].
- 3.
ENDF2C
The impact of decay data uncertainties on keff results can be investigated with a modified version of the ENDF2C tool [
30]. This routine uses stochastic sampling for a perturbation of the decay constants and branching ratios, whose uncertainties are contained and read from original ENDF decay data files. Furthermore, the physical relationship between the branching ratios is respected. The perturbed decay data are formatted for use in SERPENT2.
A previous study with ENDF2C [
4] showed that the effect of decay data uncertainties on system criticality in the present context is negligible when compared with the effects of uncertainties in cross section or fission yield (FY) data. This finding is in line with other publications addressing decay data uncertainties in this respect [
31,
32,
33]. An assessment of the effects of decay data uncertainties is, hence, not part of this work.
- 4.
NUSS
The Nuclear data Uncertainty Stochastic Sampling method (NUSS) [
5,
25,
34,
35] applies perturbations in groups to pointwise nuclear data given in the specific ACE format which is used for criticality calculations with MCNP6. NUSS hereby follows the descriptions of ND sampling in [
25]. The pointwise ND are divided into
n segments with
n being the specified number of energy-groups. Perturbation factors, i.e., the ratio between sampled and nominal ND, are then applied to each of these ND segments varying the data uniformly within each energy group. In order to ensure the consistency of the ACE file, total cross sections and the reaction cross sections are adjusted accordingly. With NUSS, perturbations of (
n,
n), (
n,
n′), (
n,2
n), (
n,
f), (
n,
γ) cross sections and of
,
χ data can be applied.
An example of the functionality of NUSS and a comparison of SS with other UQ methods, e.g., the Total Monte Carlo (TMC) method and sensitivity based calculations, can be found in [
5].
3.2.1. Consistent Sampling Approach Realisation
For each of steps 3, 4 and 6 of the BUCSS-R scheme, it is thus possible to introduce ND uncertainties into the calculation chain and to propagate them through it to evaluate their effect on the criticality of the system. In this way, ND-UQs have been carried out in an independent manner for each step, with BUCSS-R executions using perturbed ND sets for only one step at a time [
4]. Different sets of sampled perturbation factors were applied in separate batches of BUCSS-R computations addressing either the depletion, decay or the criticality aspect. The present work, however, allowed for an all-encompassing ND-UQ featuring a simultaneous perturbation of ND input for step 3 (depletion) and step 6 (criticality), ignoring the negligible uncertainties in step 4 (decay). The simultaneous perturbation could be achieved by linking SHARK-X and NUSS to produce a consistent set of perturbed ND, formatted for usage in both, CASMO5 and MCNP6, respectively. In the following, this new approach is referred to as ‘consistent ND-UQ’. Both, the results obtained with consistent ND-UQ and those of an independent ND-UQ are subject of a more detailed comparison and discussion in
Section 4.
3.2.2. Consistent Sampling Approach Verification
Prior to this work, the capability of producing and using consistent sets of perturbed ND for both depletion (with Shark-X/CASMO5) and criticality (with NUSS/Serpent) calculations has been demonstrated. The Peach Bottom 2 pin-cell model of UAM in Hot Zero Power (HZP) conditions served as a test case [
25]. The same Shark-X model was used while a consistent Serpent model was established based on the MCNP model used in [
25]. The ENDF/B-VII.0 nuclear data library is used in both codes to calculate kinf of the pincell.
Various sources of covariance information were considered for testing the consistent ND sampling, namely ENDF/B-VII.1, ENDF/B-VIII.0 and JEFF-3.3. 300 perturbed ACE formatted files were generated with the NUSS scripts for each of the covariance libraries. Covariance matrices with a 19 energy group structure were applied in the case of ENDF/B-VIII.0 and JEFF-3.3 while for ENDF/B-VII.1 they had a 44 energy group structure. Only 235U, 238U and 1H have been considered for this test albeit more nuclides are available. The following reaction channels were perturbed, when available: (n,n), (n,n′), (n,2n), (n,f), (n,γ), total and χ.
In order to provide the consistency of the ND sets employed in the CASMO5 and Serpent calculations, the perturbation factors produced by NUSS were used to generate the corresponding ND perturbations in Shark-X.
All Serpent calculations were performed with 100,000 histories per cycle, 200 active cycles and 20 inactive cycles. This leads to a kinf uncertainty of about 15 pcm, e.g., much less than the ND uncertainty. Furthermore, the seed number of each perturbed calculation is set to that of the nominal calculation.
Uncertainties in the kinf values, calculated with CASMO5 and Serpent on the basis of 300 sampled ND sets for each library, ENDF/B-VII.1, ENDF/B-VIII.0 and JEFF-3.3, are reported in
Table 1. The kinf relative uncertainties are consistent, even though small discrepancies are observed, most likely due to the different mean values for kinf. The consistency is further demonstrated by ordering the Serpent ratio of the perturbed kinf with respect to the nominal value in increasing order and plotting it against the same quantity produced by Shark-X. This is shown in
Figure 3a for ENDF/B-VII.1.
As expected, a slope of 1 is observed. Fluctuations around the unity slope are visible in
Figure 3a. Further analysis on these fluctuations is carried by computing the “distance” between the Shark-X and Serpent perturbed kinf for a consistent perturbation; this distance is expressed as the ratio defined by Equation (1):
The standard deviation of the distances is in the 50 pcm range and appears normally distributed, see
Figure 3b. However, such fluctuations are somewhat larger than the inherent statistical spread expected from the ratio of two independent Serpent calculations, each calculation having a 15 pcm standard deviation (the resulting standard deviation, assuming that the correlation between both Serpent
k-
eff values in Equation (1) is zero, should then be
≈ 20 pcm). The larger fluctuations are most likely caused by the different neutron transport approaches of Serpent and CASMO5, MC and deterministic method of characteristics (MoC).