Acta Polytechnica CTU Proceedings doi:10.14311/APP.2018.14.0008 Acta Polytechnica CTU Proceedings 14:8–13, 2018 © Czech Technical University in Prague, 2018 available online at http://ojs.cvut.cz/ojs/index.php/app UNCERTAINTY PROPAGATION FOR LWR BURNUP BENCHMARK USING SAMPLING BASED CODE SCALE/SAMPLER Daniel Campolinaa, b, ∗, Jan Fryborta a Department of Nuclear Reactors, Czech Technical University in Prague, Prague, Czech Republic b Nuclear Technology Development Center, Belo Horizonte, Brazil ∗ corresponding author: campolina@cdtn.br Abstract. Sampling based method is adopted in many fields of engineering and it is currently used to propagate uncertainties from physical parameters and from nuclear data, to integral indicators of nuclear systems. The total uncertainty associated with a model simulation is of major importance for safety analysis and to guide vendors about acceptable tolerance limits for nuclear installations parts. This work presents some calculations to propagate uncertainties for a nuclear reactor fuel element modeled in SCALE/TRITON, using the sampling tool SCALE/SAMPLER. Results showed that the influence of input uncertainties on kinf is more pronounced in the fresh core other than the depleted core and the contribution from studied manufacturing uncertainties is smaller than the contribution of nuclear data uncertainties. Keywords: random sampling, uncertainty, nuclear fuel burnup, OECD UAM LWR. 1. Introduction In a global picture, uncertainty quantification (UQ) is the process of characterizing input uncertainties, forward propagating these uncertainties through a computational model, and performing statistical as- sessments on the resulting responses. In this forward propagation, illustrated in Fig.1, probabilistic or in- terval information on parametric inputs are mapped through the computational model to assess statistics or intervals on outputs. The computational model for the present application is the TRITON control mod- ule of SCALE, composed of cross-section processing codes, a neutron transport solver, and point depletion code [1]. For UQ, components of the input parameter are con- sidered to be uncertain as specified by probability dis- tributions (e.g., normal, beta). By stipulating specific distributional structure to the inputs, distributional structure for the outputs (i.e, response statistics) can be inferred [2]. Uncertainties and correlations among experimentally measured cross-sections constitute the so called nuclear data covariance libraries. The sam- pling based approach can be used to sample physical parameters like dimensions and densities and also joint probability density functions given in the nuclear data covariance libraries. The last produces a random sample for the nuclear cross-sections that are used in a transport calculation. One of the most important results of a nuclear reactor simulation is neutron multiplication factor (k) that measures the balance between production of neutrons and their absorption in the core and leakage out of the core. The multiplication factor is a key Figure 1. Realizations from specified input distribu- tions being run in a model simulation. indicator for the changes in the neutron flux that also causes changes in the power level of the reactor. If the multiplication factor is lower than 1, the system is subcritical and its power is decreasing. Loss of neutrons is greater than their production and it is impossible to maintain the fission chain reaction. A UQ analysis can provide the level of confidence in the multiplication factor predicted by simulations and identify through a sensitivity analysis the key pa- rameters whose uncertainties contribute most to the multiplication factor uncertainty. There are always safety margins to multiplication factor in criticality calculations. It reflects the uncertainty in all calcula- tions. When subcriticality of some system must be certified the calculations are usually aimed at maxi- mum multiplication factor by 5 % below the criticality. The UQ process can verify that such a margin is suffi- cient for any design systems. Since sampling based method only requires forward 8 http://dx.doi.org/10.14311/APP.2018.14.0008 http://ojs.cvut.cz/ojs/index.php/app vol. 14/2018 Uncertainty propagation for LWR burnup benchmark calculations, it presents some advantages if compared to adjoint-based perturbation theory approach used by e.g. TSUNAMI modules of SCALE. In problems with significant second order effects, the first order perturbation theory is not valid while the sampling method is fully applicable. The typical example of the second order effect is when the perturbation in the transport operator causes significant perturbation in the flux solution of the Boltzmann transport equation (see chapter 6 of [1]). SCALE/SAMPLER is one of the sampling tools available at Department of Nuclear Reactors of CVUT for uncertainty qualification. It is at first used in this study to assess the impact of covariance data on nuclear transport and depletion of a LWR Benchmark 2D model. Then effect of geometric uncertainties, also known as manufacturing uncertainties, are quantified for the same model. 2. Uncertainty analysis from nuclear data In this section, cross-section perturbation using SCALE/SAMPLER is performed for a benchmark proposed by the OECD expert group Uncertainty Analysis in Modeling (UAM) [3]. Resulted values of propagated uncertainties to kinf are compared against participant-averaged results for the UAM benchmark. 2.1. Methodology The UAM benchmark has several exercises. This anal- ysis is using fuel pin model of a typical PWR fuel rod in 15x15 assembly design from exercise I-b. This was proposed in scope of the UAM benchmark in order to address uncertainties in nuclear fuel depletion cal- culation due to the basic nuclear data as well as the impact of processing the nuclear and covariance data. Summary and discussion of neutronic cases performed by the participants is available in [4]. Participant- averaged results for exercise I-b are reproduced here for comparison purposes despite different input co- variance cross-sections were used by the participants. The UAM expert group selected 44-group SCALE 6.0 covariance library as a reference library for the benchmark exercise since it was the most complete and up-to-date compilation at the time of that review. Calculations in SCALE are conducted according to control modules and sequences. This study adopted T-DEPL sequence of TRITON that couples 2D multi- group deterministic transport in NEWT with ORI- GEN module depletion. The lattice physics calculations are based on the 56-group energy structure of the v7-56 SCALE library. This library is recommended for light-water reactor calculations. Trace amount of 231 nuclides was added in the inventory of burnable materials (addnux=3 card of TRITON) in order to reach more realistic fuel depletion from the very beginning. Infinite lattice of the elementary cell was simulated by reflective boundary condition adopted for all sides of the model. Parameter Value HFP conditions Fuel temperature (K) 900 Cladding temperature (K) 600 Moderator (coolant) temperature (K) 562 Moderator (coolant) density (g/cm3) 0.7484 Configuration Unit cell pitch (mm) 14.427 Fuel pellet diameter (mm) 9.391 Fuel pellet material UO2 Fuel density (g/cm3) 10.283 Fuel enrichment (w/o) 4.85 Cladding outside diameter (mm) 10.928 Cladding thickness (mm) 0.673 Cladding material Zircaloy-4 Cladding density (g/cm3) 6.55 Gap material He Moderator material H2O Operating history Operating cycle 1 Burn time (days) 1825 Final Burn-up (GWd/MTU) 61.28 Specific power (kW/kgU) 33.58 Table 1. Configuration and conditions of burnup pin-cell test problem. Materials: Pitch of unit cell Reflective boundary R e fl e ct iv e b o u n d a ry R e fl e ct iv e b o u n d a ry Figure 2. TRITON model model for Exercise I-1b Pin Cell. The SCALE/SAMPLER module was used to per- form uncertainty propagation/analysis of neutron mul- tiplication factor for the pin-cell shown in Fig. 2. Data library for cross-section perturbations is described in the following section. Cross section covariance library The new 56-group (56groupcov) SCALE 6.2 covari- ance library [1], recommended by SCALE 6.2.1 for 9 Daniel Campolina, Jan Frybort Acta Polytechnica CTU Proceedings kinf 0 GWd/MTU 60 GWd/MTU SAMPLER SAMPLER Ref.* ∆ Mean val 1.4023 0.9063 0.9030 0.4 % RSD (%) 0.55 0.58 0.89 - SD (pcm) 770 529 805 52 % Table 2. Initial condition and final burnup uncertainties in kinf for cases UAM* (using 44groupcov) and SAMPLER (using 56groupcov). all applications, was used by SAMPLER uncertainty module. The data correspond to relative uncertainties assembled from a variety of sources, including high- fidelity covariance evaluations from ENDF/B-VII.1 [5] as well as approximate uncertainties obtained from the collaborative projects involving BNL, LANL, and ORNL. The assumptions in generating the data, the library contents, and processing procedure for the SCALE 56-group covariance libraries can be found in [1]. Comparison between the two libraries adopted by SAMPLER and adopted by UAM participants results are provided in next section in terms of relative stan- dard deviation for some reactions, in order to clarify the difference and the probable impact on results. The number of samples/realizations in SAMPLER code was set to 1000, that is the maximum allowed for cross section sampling. Recent works show that 1000 calculations are enough to permit optimal convergence of the statistical moments of significance for any given response function [6–8]. It is important to emphasize that only cross-section (XS) perturbation is performed and it is the most im- portant contributor to the overall uncertainty. Neither fission yield perturbation nor decay data perturba- tions are performed. Some studies considering the same benchmark concluded that contribution of decay data uncertainty for burnup calculations is negligible [9]. On the other hand, other studies stated that fission yield and decay data uncertainties may be rele- vant for uncertainties of nuclide densities in depletion calculations [10]. 2.2. Results Propagation of uncertainties to initial and final burnup results performed using SAMPLER are presented in Tab. 2 together with the mean value of kinf calculated. Participant-averaged result from exercise I-b bench- mark (only available for the final burnup condition in [4]) is presented in the fifth column for comparison purposes. The symbol ∆ is the difference between the value calculated in this study by SAMPLER and the reference value. Note that SD stands for 1 standard deviation and RSD is the relative standard deviation. The difference between kinf mean value calculated by SAMPLER and the reference value was 0.4 %. It is typical difference that can be observed for criticality calculations of identical systems by different codes and XS libraries [11]. This multiplication factor value is the average for the all the 1000 perturbed cases. As the XS perturbations are based on normal distribution, Figure 3. Running kinf average and standard de- viation as function of sample size for simulation on SAMPLER. the mean value should stay almost the same as for a single case with average XS values. In order to provide information on sampling convergence, Fig. 3 presents the average kinf and the running standard deviation as function of the sample size. It is possible to see that after 800 samples, the average of calculated kinf mean values is stabilized. In contrast to the above results, the increase of prop- agated uncertainty in kinf for burnup 60 GWd/MTU was 52 %. SAMPLER estimated 529 pcm for the stan- dard deviation on kinf and the reference value is 805 pcm. The large difference (SCALE x UAM reference) can be explained by use of different versions of the covariance libraries. There are also SAMPLER results for the fresh fuel and the final burnup. The estimated standard de- viation on kinf was 770 pcm for the fresh fuel and it decreased to 529 pcm for the final burnup. The 31 % difference comes from change of fuel composition during depletion. Each nuclide has its own XS covari- ance data thus the total uncertainty changes during depletion. The reason for the uncertainty decrease is discussed below and the main contributors to this phenomenon are illustrated in Fig. 4 and 5. It is possible to select the main contributors to the overall kinf uncertainty. These examples can also serve as the explanation for differences between the current and the older version of the XS covariance library in SCALE; and they are also the reason for computed uncertainty decrease during fuel depletion. The usual main contributor to the calculation uncer- 10 vol. 14/2018 Uncertainty propagation for LWR burnup benchmark Variable Distribution Value Std.dev lower cutoff upper cutoff Fuel_radius normal 0.46955 2.1667e-4 0.4689 0.4702 Gap_thickness normal 0.00955 1.3333e-3 0.00555 0.01355 Clad_thickness normal 0.0673 1.3333e-3 0.0633 0.0713 All dimensions in cm. Table 3. Parameters for variable sampling in SAMPLER input file. Figure 4. 235U nubar relative standard deviation by energy for covariance libraries 44groupcov and 56groupcov. tainty is average fission neutron yield (nubar) of the major fissile nuclides, i.e. 235U and 239Pu. Fig. 4 and Fig. 5 shows the relative standard deviation of this fission neutron yield in 44-groupcov and 56-groupcov for 235U and 239Pu, respectively. The relative standard deviation of 235U nubar below around 10 eV is approximately by 24 % higher for 56-groupcov. Much larger difference can be observed for 239P u nubar. Its RSD is about three times larger in 44-groupcov when compared to 56-groupcov. It should be also noted that its uncertainty value is lower when compared to 235U nubar. It is the reason for multiplication factor calculation uncertainty decrease during fuel depletion when adopting 56-groupcov. In the final Figure 6, this comparison shows that the relative standard deviation of absorption in 238U is comparable for these two covariance libraries and it is below 5 % for all neutrons except fast neutrons. In addition to the different covariance data used for the two methodologies compared, uncertainties may be introduced into the broad-group cross-sections due to approximations in the grouping procedure. In [12], it is showed that the dominant uncertainty is generally with regard to the energy weighting function used to average the point-wise data within a broader groups. Figure 5. 239Pu nubar relative standard deviation by energy for reproduced from covariance libraries 44groupcov and 56groupcov. Figure 6. 238U(n,γ) relative standard deviation by energy for covariance libraries 44groupcov and 56groupcov. 3. Uncertainty analysis from geometry perturbation The UAM benchmark model of PWR fuel rod pre- sented in section 2.1 had its geometrical uncertain parameters of fuel radius, gap thickness, and clad thickness modeled in SAMPLER. The methodology adopted and resulting propagated uncertainty to the neutron multiplication factor are presented. 11 Daniel Campolina, Jan Frybort Acta Polytechnica CTU Proceedings 0 GWd/MTU 60 GWd/MTU Name variable # samples kinf RSD(%) SD(pcm) kinf RSD(%) SD(pcm) Case1 Fuel_radius 300 1.40749 0.010 14 0.90494 0.002 2 Case2 Gap_thickness 300 1.40749 0.004 6 0.90494 0.000 0 Case3 Clad_thickness 300 1.40749 0.005 8 0.90494 0.001 1 Case4 All together 300 1.40749 0.012 17 0.90494 0.003 3 Table 4. Variables sampled for each study case and results of UQ with the contributors to uncertainty in kinf. 3.1. Uncertain variables Normal distribution of the geometrical parameters was assumed. The UAM benchmark specifies values of standard deviations and there is also requirement to truncate the distribution at 3 standard deviations as lower cutoff and upper cutoff. Table 3 presents the parameters considered in the sampling of variables. UQ for each variable was calculated in order to perform the sensitivity study of the geometric parame- ters. The values of fuel radius (fuelr), gap outer radius (gapr) and cladding outer radius (cladr) as function of the sampled variables (Tab. 3) are the following: • Case1 (variable sampled: Fuel_radius ) fuelr = Fuel_radius gapr = Fuel_radius+0.00955 cladr = Fuel_radius + 0.00955 + 0.0673 • Case2 (variable sampled: Gap_thickness) fuelr = 0.46955 gapr = 0.46955 + Gap_thickness cladr = 0.46955 + Gap_thickness+0.0673 • Case3 (variable sampled: Clad_thickness) fuelr = 0.46955 gapr = 0.4791 cladr = 0.4791 + Clad_thickness • Case4 (variable sampled: All together) fuelr = Fuel_radius gapr = Fuel_radius + Gap_thickness cladr = Fuel_radius + Gap_thickness + Clad_thickness Since the cladding outer radius (cladr) depends on uncertain variables values, the moderator/fuel volume ratio and the cladding density vary according to the sampled values. However, for being a neutron non- absorbing material (Zircaloy-4), the density change was disregarded. The fuel meat density (D), dependent on fuel ra- dius for Case 1 and Case 4, is updated according to equation 1. This allows to keep the value of fuel mass constant when fuel radius sampled value is updated in the model geometry. The nominal density is worth 10.283 g/cm3 according to Tab. 1. Dnew[g/cm3] = 10.283 × ( 0.46955 F uel_radius )2 (1) 3.2. Results The cases considered are presented in Tab. 4, where the variable under study is given in the second column. The contribution to uncertainty in kinf are presented for 0 GWd/MTU and 60 GWd/MTU burnups. Un- certainty in the fuel radius (Case 1) is the biggest contributor to the overall calculation uncertainty due to the fact that its value directly influences fuel volume. The second in rank is the clad thickness represented by Case 3. It was noticed that the effect of uncertain- ties in kinf is much more pronounced for the fresh fuel rather than for the final burnup condition. 4. Conclusions Uncertainty propagation analysis was performed for the burnup pin-cell exercise I-b of the UAM bench- marks. The SAMPLER code using SCALE 6.2 56- groupcov covariance library had the results compared against participant-averaged result from the bench- mark exercise I-b. The difference between propagated uncertainty (SD) in kinf calculated by SAMPLER and the reference case (UAM benchmark) was 52 %. Such difference is expected since the codes used different versions of covariance libraries with different group structure. Comparison of results for the fresh and spent fuel for the same covariance library was done for the SAM- PLER results alone, as there are no benchmark refer- ence results. The observed 31 % difference in standard deviations is attributed to the fact that propagated uncertainty is mainly affected by nuclide composition that changes along the burnup. Future research must include other nuclear libraries for comparison. When propagating geometrical uncertainties, the effect of uncertainty in fuel radius, clad and gap thick- ness was calculated. It was noted that the influence of geometric input uncertainties on kinf is more pro- nounced for the fresh than for the depleted fuel. This can be explained by the fact that fresh fuel contains higher density of fissile material and influences more the neutron population of the reactor. The sensitivity study also confirmed that uncertainty of fuel radius (Case 1) is the biggest contributor to the overall calcu- lation uncertainty since it influences directly fuel vol- ume even though the total fuel mass is preserved. This study also confirmed that the contribution of manu- facturing geometrical uncertainties to the prediction 12 vol. 14/2018 Uncertainty propagation for LWR burnup benchmark of kinf is smaller than the contribution of cross-section uncertainties. Acknowledgements The mobility of the Brazilian co-author to Department of Nuclear Reactors, Czech Technical University in Prague has been possible with the Erasmus Mundus SMART2 sup- port (Project Reference: 552042-EM-1-2014-1-FR-ERA MUNDUS-EMA2) coordinated by Centrale Supelec. References [1] E. B.T. Rearden, M.A. Jessee. SCALE Code System Version 6.2.1. Tech. Rep. ORNL/TM-2005/39, Oak Ridge National Laboratory, Oak Ridge, Tennessee, 2016. [2] A. Haldar, S. Mahadevan. Probability, reliability, and statistical methods in engineering design. John Wiley, 2000. [3] K. Ivanov, M. Avramova, S. Kamerow, et al. Benchmark for uncertainty analysis in modeling (UAM) for design, operation and safety analysis of LWRs, vol. 1. Citeseer, 2013. [4] R. N. Bratton, M. Avramova, K. Ivanov. OECD/NEA benchmark for uncertainty analysis in modeling (UAM) for LWRs–summary and discussion of neutronics cases (phase I). 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Ph.D. thesis, École Polytechnique Féeérale de Lausanne, 2015. doi:10.5075/epfl-thesis-6598. 13 http://dx.doi.org/https://doi.org/10.5516/NET.01.2014.710 http://dx.doi.org/https://doi.org/10.1016/j.nds.2011.11.002 http://dx.doi.org/https://doi.org/10.1016/j.anucene.2015.03.046 http://dx.doi.org/http://dx.doi.org/10.1155/2013/549793 http://dx.doi.org/http://dx.doi.org/10.1155/2012/767096 http://dx.doi.org/http://dx.doi.org/10.1155/2013/790206 http://dx.doi.org/https://doi.org/10.1016/j.anucene.2016.11.025 http://dx.doi.org/https://doi.org/10.1016/j.anucene.2011.01.032 http://dx.doi.org/10.5075/epfl-thesis-6598 Acta Polytechnica CTU Proceedings 14:36–41, 2018 1 Introduction 2 Uncertainty analysis from nuclear data 2.1 Methodology 2.2 Results 3 Uncertainty analysis from geometry perturbation 3.1 Uncertain variables 3.2 Results 4 Conclusions Acknowledgements References