Uncertainty Propagation in UAM Time-Dependent Neutronics PWR Studies

Abstract

A subset of pressurized water reactor (PWR) studies from Exercise II-2 of the Uncertainty Analysis in Modeling for Light Water Reactors (UAM-LWR) benchmark study has been performed to quantify the importance of both nuclear data uncertainties and manufacturing uncertainties in an assembly depletion calculation and a mini-core rod movement transient. The depletion study of the 15 × 15 PWR assembly using the SAMPLER and TRITON modules of the SCALE code system revealed a maximum uncertainty in k_eff of 0.49% for fresh fuel, decreasing to 0.38% at the midpoint of the fuel cycle and rising back to 0.48% at the end of the fuel cycle. Uncertainties and correlations of various homogenized cross sections and other group constant data, such as k_eff, have been determined, and the effect of randomly applied manufacturing uncertainties was found to be largely negligible relative to nuclear data uncertainties for bulk lattice parameters. However, for local parameters, such as the pin power factors, assembly discontinuity factors, and diffusion coefficients, the effects from manufacturing uncertainties were appreciable and sometimes dominant.

Nuclear data uncertainties were found to be the dominant contributors to uncertainty in the isotopic composition of the overall assembly, with the exception of very early in the fuel cycle, where manufacturing uncertainties such as perturbations to the fuel density and pin radius made nonnegligible contributions to total uncertainty. The contribution of manufacturing uncertainties to isotopic uncertainties was nonnegligible at a pin-by-pin level, but still smaller than the contributions from nuclear data uncertainty. Studies of the PWR mini-core rod movement transient using homogenized data from the SCALE models in the PARCS diffusion code showed little difference between the tested modeling approaches and demonstrated that nuclear data uncertainties dominated the manufacturing uncertainties in the global figures of merit considered, such as the equilibrium core boron concentration, the maximum core power factor, and the maximum reactivity insertion. For local effects, such as maximum pin power during the transient, the randomly applied manufacturing uncertainties were dominant. It was found in general that for global system properties, nuclear data uncertainties made significantly larger contributions to total uncertainty, whereas for local parameters the impact of manufacturing uncertainties was at least nonnegligible, and for some parameters, dominant.

Keywords:

• Uncertainty Analysis in Modeling for Light Water Reactors
• UAM-LWR
• uncertainty analysis
• nuclear data uncertainties
• manufacturing uncertainties

I. INTRODUCTION

The safety analysis and licensing process for nuclear power plants relies heavily on the use of simulation software to predict a plant’s response during both normal operation and under accident conditions. Of utmost importance in this analysis are not only the values for key figures of merit predicted by the simulation, such as peak pin powers or ramp rates, but also the degree of confidence with which each figure of merit can be predicted. Operational limits for various reactor parameters are determined using a combination of the predicted values and their margins of uncertainty to ensure safety limits are observed during any predicted accident within the design basis.

Due to limitations in computing power, early safety analyses used conservative assumptions to account for the uncertainty in modeling, code inputs, and operational plant states. While these conservative approaches allowed operators to define their safe operating envelope, past studies have demonstrated that conservative approaches can lead to simulations that deviate from actual plant behavior.¹ Recent approaches have taken advantage of significant improvements in computing power to perform detailed propagation of uncertainties through multistep Monte Carlo methods to generate more accurate predictions of the operating margins.²

In recognition of the international interest in quantifying these uncertainties, the Organisation for Economic Cooperation and Development’s Nuclear Energy Agency established a benchmark study on Uncertainty Analysis in Modeling for Light Water Reactors (UAM-LWR). The benchmark established a series of exercises split into three phrases to improve the quantification of uncertainties in the simulation of light water reactors (LWRs). The phases defined by the benchmark are³

1. Phase I (Neutronics):

   1. Exercise I-1 (Cell Physics): focused on the derivation of the multigroup microscopic cross-section libraries and their uncertainties

   2. Exercise I-2 (Lattice Physics): focused on the derivation of the few-group macroscopic cross-section libraries and their uncertainties

   3. Exercise I-3 (Core Physics): focused on the core steady-state standalone neutronics calculations and their uncertainties

2. Phase II (Core)

   1. Exercise II-1 (Fuel Physics): fuel thermal properties relevant to steady-state and transient performance

   2. Exercise II-2 (Time Dependent Neutronics): neutron kinetics and fuel depletion standalone performance

   3. Exercise II-3 (Bundle Thermal Hydraulics): thermal-hydraulic fuel bundle performance

3. Phase III (System)

   1. Exercise III-1 (Core Multi-Physics): coupled neutronics/thermal-hydraulic core performance

   2. Exercise III-2 (System Thermal Hydraulics): thermal-hydraulic system performance

   3. Exercise III-3 (Coupled Core-System): coupled neutronics/thermal-hydraulic performance

   4. Exercise III-4: Comparison of best estimate plus uncertainty versus conservative calculations.

In this work, a subset of the studies defined in Exercise II-2 have been performed. Specifically, case II-2a studying the propagation of uncertainties in a typical depletion calculation for a pressurized water reactor (PWR) fuel assembly and case II-2b, a neutronics study of the propagation of uncertainty in a rod movement transient simulation for a PWR mini core.⁴ Both exercises present a unique opportunity to elucidate the uncertainties in key reactor physics phenomena.

This paper includes a summary of the description of each exercise performed, results obtained for the fuel assembly depletion case demonstrating the evolution of uncertainties throughout the expected life of the fuel, results of the uncertainty propagation on the mini-core rod movement transient, and some concluding remarks on this work and steps for future work.

II. MODEL DESCRIPTION

The UAM-LWR benchmark study has led to the development of numerous tools for the study of uncertainties in reactor physics. Among these tools is the SAMPLER super sequence of the SCALE code package, which is used in this work. SAMPLER controls the repeated execution of a SCALE sequence, in this case TRITON (Ref. 5). Within each sequence, perturbations can be applied to the nuclear data used in the calculation as well as the other input data provided by the user (such as the geometry of the assembly). Each execution of the TRITON code yields an output corresponding to the specific set of perturbations applied by SAMPLER. The major inputs requested by SAMPLER are the perturbed nuclear data sets included with SCALE that were based on the nuclear date covariance uncertainty matrices. The SCALE code package includes 1000 precalculated perturbed nuclear data libraries, allowing for a total of 1001 (the reference nuclear data plus each perturbed nuclear data set) simulations to be performed for a given model.

The SAMPLER/TRITON code sequence is sufficient on its own for the PWR fuel assembly depletion study. However, to complete the PWR mini-core transient study a core-level diffusion solver was required. In this work, PARCS (Ref. 6), a three-dimensional multigroup neutron diffusion solver released by the U.S. Nuclear Regulatory Commission was used. The output of each TRITON execution was converted to PARCS input data (in the form of PMAXS files) using the GenPMAXS tool.⁷ For each TRITON execution, the mini-core transient was simulated once. All perturbations were applied via the PMAXS homogenized cross-section data files (i.e., the PARCS input file was not altered between trials).

II.A. Exercise II-2a Description

Exercise II-2a models the typical depletion calculation of a PWR fuel assembly using a TMI-1 15 × 15 pin assembly as the test case. The fuel assembly, as defined by the benchmark specifications,^2,4 is shown in Fig. 1. The enrichment of the fuel pins is 4.12% ²³⁵U. The gadolinium poison pins (Gd) have the same enrichment as the fuel pins, but contain 2% Gd by weight. The depletion calculation is performed at nominal hot full-power (HFP) conditions, defined in Table I, to a final burnup of 60 megawatt days per kilogram of uranium (MWd/kg).

TABLE I HFP Conditions for the TMI-1 Fuel Assembly Depletion Calculation*

Parameter	HFP Value
Fuel temperature	900 K
Cladding temperature	600 K
Coolant temperature	562 K
Coolant density	748.4 kg/m^3
Power density	33.75 W/gU
Boron concentration	900 ppm

*Reference 4.

Fig. 1. The 15 × 15 TMI-1 fuel assembly modeled in this work. Cells containing g represent fuel pins with poison, cells containing G are guide tubes, and the cell containing I is the instrumentation tube. All other cells are standard fuel pins.⁴

In addition to propagating nuclear data uncertainties in the depletion calculation, a subset of manufacturing uncertainties defined in the UAM-LWR benchmark were considered (see Table II). The distribution of all manufacturing uncertainties was assumed to be normal.⁴

TABLE II Manufacturing Uncertainties Considered in the TMI-1 Fuel Assembly Depletion Study

Parameter	Uncertainty
Cladding inner diameter	±0.04 mm
Cladding thickness	±0.04 mm
Fuel pellet outer diameter	±0.013 mm
Fuel density	±0.91%

The TMI-1 fuel assembly was modeled in TRITON using the NEWT transport solver. The 15 × 15 assembly had a square pitch of 21.81 cm and was modeled with a 60 × 60 global mesh grid. Sensitivity analyses were performed on the mesh, and the grid size used provided sufficiently accurate results while still being computationally efficient. In this model, a single fuel material was used to track the composition of all fuel pins in the assembly simultaneously. Similarly, a single fuel-poison material was used to track the composition of all four poison pins. Additional work was performed to consider the impact of using pin-specific compositions, and the results are discussed in subsequent sections.

The PWR lattice considered in the benchmark study contained 225 pins (204 of which were fuel pins and an additional four which were fuel pins containing burnable poison). To accurately quantify the impact of manufacturing uncertainties on the lattice level calculation results, it was necessary to apply manufacturing uncertainties on a pin-by-pin basis. This approach is only possible if the pins are modeled within SCALE as separate entities, each requiring its own self-shielding and depletion calculations. To reduce the computational costs associated with this approach, a quarter-assembly model was considered with reflective boundary conditions on all sides. The quarter assembly was modeled using a 30 × 30 global grid, and mesh sensitivity studies demonstrated that the reduction in grid did not contribute significantly to the uncertainty calculations. The model had 49 complete pins and 15 partial pins modeled. Of these, 58 were regular fuel pins (including 14 of the partial pins), one was a fuel pin containing burnable poison, and the remaining pins were either guide tubes or the instrumentation tube. Manufacturing uncertainties were only applied to the fuel and fuel-burnable poison pins.

The assembly depletion studies were done with the full set of nuclear data perturbations available in SCALE (all 1000 perturbations for the cross sections, fission product yields, and radioactive decay constants) for both the full-assembly (with pin-averaged depletion) and the quarter-assembly model (with pin-specific depletion and self-shielding). The full-assembly model was simulated with the nuclear data uncertainties only since computational limits prevented the application of pin-by-pin depletion and manufacturing perturbations (i.e., only a single material was used to model the composition of the fuel pins, and thus depletion was tracked for the aggregate of the pins, rather than on a pin-by-pin basis). The quarter-assembly model was tested using both the nuclear data uncertainties alone and the combined nuclear data and manufacturing uncertainties. Additionally, the effect of using separate fuel materials for each pin (i.e., each pin was modeled with an individual, unique fuel material, which was depleted independent of the other fuel pins) was considered. The four key test cases are outlined Table III. The approximate computational cost of each study is also shown. The depletion calculations were performed on a Dell XPS desktop PC with a 10 core 3.3-GHz Intel Xeon W-2155 CPU and 251 Gb of usable DDR4 RAM. Comparison tests were performed to confirm the equivalency of the different models.

TABLE III Description of Cases of Interest Studied for the TMI Assembly Depletion Case

Case Identification	Assembly Geometry	Pin Modeling	Nuclear Data Perturbations	Manufacturing Perturbations	Number of Samples (Including Reference)	CPU Hours
FA-SP-ND	Full	Single material for all pins	Yes	No	1001	7 080
QA-IP-ALL	Quarter	Unique material tracking for individual pins	Yes	Yes	980^a	16 267
QA-IP-ND	Quarter	Unique material tracking for individual pins	Yes	No	1001	15 804
QA-SP-ND	Quarter	Single material for all pins	Yes	No	1001	2 102

^aA total of 1001 samples were performed but the application of manufacturing uncertainties caused 21 sample cases to fail to execute. Separate testing of these 21 cases with relaxed modeling parameters found the outputs produced by these failed cases were well dispersed throughout the distribution of results obtained from the successful cases (i.e., the failed cases do not represent outliers in the outputs considered in this work).

II.B. Exercise II-2b Description

Exercise II-2b describes a kinetics-specific transient within a mini PWR core. The mini core contained nine fuel assemblies (identical to the assembly modeled in Exercise II-2a) surrounded by reflector assemblies. All nine fuel assemblies were assumed to be fresh fuel. The layout of the PWR mini core is shown in Fig. 2. The center fuel assembly contained a control rod cluster of length 3657.6 mm (the same as the active height of the core). Reflectors at the top and bottom of core with heights of 21.84 cm were modeled. The mini-core transient was simulated using homogenized data from the FA-SP-ND model, the QA-IP-ND model, and the QA-IP-ALL model. Though the same models were used, separate simulations were required to prepare the homogenized data as Exercise II-2a was performed with HFP conditions and Exercise II-2b was performed with hot zero-power (HZP) conditions, as defined in Table IV. The initial power of the PWR mini core was defined in the benchmark as 0.1409 MW(thermal).

TABLE IV HZP Conditions for the PWR Mini-Core Rod Movement Transient Study*

Parameter	HZP Value
Fuel temperature	551 K
Cladding temperature	551 K
Coolant temperature	551 K
Coolant density	766.0 kg/m^3

*Reference 4.

Fig. 2. The layout of the mini PWR core considered for the kinetics-only rod movement transient. The unrodded and rodded assemblies are denoted by U and R, respectively.⁴

The transient studied in this case considered the movement of the central control assembly. The control rod, which extends the entire length of the active core, was fully inserted at t = 0. The rod was withdrawn at constant velocity over a period of 10 s by 438.912 mm, or 12% of the rod’s full length. The control rod was then fully reinserted at a constant velocity over a period of 20 s. An additional 20 s was simulated after the rod was fully reinserted.

The PWR mini core was modeled in PARCS using two and four calculation nodes per assembly (including reflector assemblies) in the x- and y-directions. It was found that transient pin power factors could not be calculated with more than two calculation nodes per assembly in the x- and y-directions, and thus the transient simulations were performed with both nodalizations to ensure the selected nodalization was sufficient. The z (axial) direction was modeled with 24 calculation nodes in the active height of the core, with an additional node for the top and bottom reflectors. Differences between the 2 × 2 calculation mesh per node and the 4 × 4 calculation mesh per node were found to be small.

III. ASSEMBLY DEPLETION STUDY RESULTS

Depletion studies were performed using each of the PWR assembly models from fresh conditions to the specified benchmark final burnup. These studies were performed using identical cycle data for each case. The following sections discuss key outputs of the lattice-level depletion calculations and how they were impacted by nuclear data perturbations, and in the case of the quarter-assembly model, manufacturing uncertainties.

III.A. k_eff

The average assembly neutron multiplication factor and its uncertainty are shown to agree closely across the sampled cases in Fig. 3. The similarity of the k_eff uncertainty curve for the quarter-assembly case with manufacturing uncertainties to the other cases implies the tested manufacturing uncertainties had little impact on the output uncertainty of k_eff. The relative uncertainty in k_eff was found to be about 0.49% for fresh fuel in all cases, decreasing to around 0.38% at the middle of the fuel’s irradiation, before increasing to around 0.48% at the highest observed burnup. The behavior of the uncertainty in k_eff with burnup was found to be similar to studies of other LWR assemblies, which provided context for possible drivers of this behavior⁸ (specifically, Fig. 14 in Ref. 8). The observed means and standard deviations were found to agree to within approximately 0.1% across all cases.

Fig. 3. Average k_eff and uncertainty in k_eff throughout the fuel cycle for each case.

Tests for normality were performed using the Anderson-Darling test. The null hypothesis of normality was not rejected at the 5% significance level for the k_eff distributions of the beginning of cycle (BOC) and end of cycle (EOC) fuel and at most calculation points during the fuel cycle. However, it was observed that for burnups between 13.7 and 30 MWd/kg (a total of six calculation steps), the null hypothesis was rejected at the 5% significance level in the nuclear data only cases. Similarly, for the quarter-assembly case with nuclear data and manufacturing uncertainties, the null hypothesis was rejected for burnups between 12.39 and 30 MWd/kg (this represented only one additional calculation step where the null hypothesis was rejected relative to the other cases). This indicates that as the uncertainty transitions toward the minimum, there is some deviation from normality in the output uncertainty distributions.

To ensure a sufficient number of samples were used, the convergence of the average k_eff value and its uncertainty was checked at BOC and EOC for the full-assembly case with nuclear data uncertainties and the quarter-assembly case, which included manufacturing uncertainties. The convergence of k_eff and its uncertainty for the full-assembly model is shown in Fig. 4. In all cases, it was found the absolute standard deviation of k_eff was lower for EOC fuel. Additionally, the mean k_eff value was found to vary less as a function of the number of samples for the EOC fuel than for the BOC fuel. In all four convergence tests, the results appear to be well converged after approximately 600 samples.

Fig. 4. k_eff convergence at BOC and EOC for the full-assembly nuclear data only case.

III.B. Homogenized Cross Sections

Homogenized cross sections are a critical output of lattice calculations as they are typically used as input for core behavior modeling in a diffusion code. The output uncertainties of these homogenized cross sections are thus essentially input uncertainties to the next stage of the reactor modeling process.

The mean of the fast absorption cross section and its uncertainty are shown in Fig. 5. Differences of up to 0.02% were observed between the mean values obtained using the full-assembly geometry compared to the quarter-assembly model. The application of manufacturing uncertainties was found to slightly increase the uncertainty in ∑_A1 compared to nuclear data uncertainties alone, with the maximum relative difference between standard deviations found to be 2.8% between the quarter-assembly cases with and without manufacturing uncertainties (QA-IP-ALL and QA-IP-ND, respectively) at the second calculation step. The application of manufacturing uncertainties was also found to cause the ∑_A1 distributions to reject normality at some calculation steps. For the case including manufacturing uncertainties, normality of the ∑_A1 distribution was rejected at the 5% significance level at the calculation step corresponding to a burnup of 0.65 MWd/kg and for burnups at or above 37.6 MWd/kg (seven calculation steps).

Fig. 5. Mean and standard deviation in the fast group absorption homogenized cross section throughout fuel life.

The thermal absorption cross section ∑_A2 mean and uncertainty are shown in Fig. 6. The maximum variation in the mean between cases was 0.13% and occurred at 45 MWd/kg. The maximum relative difference in the standard deviations between cases was found to be about 9.0% and occurred at the first calculation step between the quarter-assembly case with manufacturing uncertainties (QA-IP-ALL) and the quarter-assembly case with a single fuel material (QA-SP-ND). It was found that the null hypothesis of normality was rejected at the 5% significance level for the first calculation step in the nuclear data cases. The null hypothesis was not rejected at any subsequent calculation steps, nor was it rejected for the first calculation step in the manufacturing uncertainty case.

Fig. 6. Mean and standard deviation in thermal group absorption homogenized cross section throughout the fuel life.

The average ν∑_f1 cross sections are shown in Fig. 7 and were found to agree closely across all cases (within 0.1%). The inclusion of manufacturing uncertainties was found to slightly increase the uncertainty in ν∑_f1 throughout the fuel cycle, with a maximum relative difference of 13.6% observed between the standard deviations of the quarter-assembly cases with and without manufacturing uncertainties (QA-IP-ALL and QA-IP-ND) at approximately 10 MWd/kg, with similar magnitude discrepancies observed between the case with manufacturing uncertainties and the remaining cases. The null hypothesis of normality was not rejected for any calculation step across the four test cases.

Fig. 7. Mean and standard deviation in fast group ν∑_F throughout the fuel cycle.

The mean and standard deviation of the thermal fission cross section, multiplied by ν, is shown in Fig. 8. The largest observed difference in the ν∑_f2 cross section between cases was 0.21% at 45 MWd/kg. The largest relative difference in standard deviations was 2.66% at 8.48 MWd/kg between the quarter-assembly cases with and without manufacturing uncertainties (QA-IP-ALL and QA-IP-ND). The null hypothesis of normality was not rejected for any calculation step across the four test cases.

Fig. 8. Mean and standard deviation in thermal group ν∑_F throughout the fuel cycle.

The mean and relative uncertainty in the fast to thermal downscatter cross section is shown in Fig. 9. The mean value agrees closely across all cases at BOC. The presence of manufacturing uncertainties was found to increase the uncertainty in the downscatter cross section, with a maximum relative difference in the standard deviation of about 40.1% observed early in the fuel cycle at around 1.95 MWd/kg. The null hypothesis of normality was rejected at the 5% significance level for burnups between 4.56 and 15 MWd/kg in the quarter-assembly case with the individual pin materials (QA-IP-ND) case and between 5.87 and 15 MWd/kg (one fewer calculation step) for the single material cases (FA-SP-ND and QA-SP-ND). The null hypothesis of normality was not rejected at the 5% significance level for the case including manufacturing uncertainties.

Fig. 9. Mean and standard deviation in downscatter cross section throughout the fuel cycle.

The value and uncertainty of D₁ and D₂ were not found to vary significantly with burnup. The mean values of D₁ were found to agree to within 0.03% throughout the fuel cycle, while the presence of manufacturing uncertainties led to a maximum relative increase in the standard deviation of 32.4% at 7.17 MWd/kg. Normality was rejected at the 5% significance level for burnups between 30 and 60 MWd/kg in the nuclear data cases.

For the thermal diffusion coefficient, the mean values agreed to within 0.03% throughout the fuel cycle, and the incorporation of manufacturing uncertainties led to a maximum relative increase in the standard deviation of 3.65% at 9.78 MWd/kg. The null hypothesis of normality was not rejected at the 5% significance level for D₂.

Last, the assembly discontinuity factors (ADFs) are considered in Figs. 10 and 11. The mean value was found to be affected by the pin modeling (single or individual) used. A maximum difference in the ADF₁ mean of 0.52% was observed at EOC. Large differences in the uncertainty were noted, primarily due to the inclusion of manufacturing uncertainties. This was expected since the ADFs are a local phenomenon primarily dependent on the adjacent pins. So, while global phenomena (e.g., k_eff) have the effect of manufacturing uncertainties canceled out by randomly generated perturbations offsetting each other when a large number of perturbations are applied, for local parameters such as ADFs or pin powers, the uncertainty is dependent on the specific properties of the pins in each local region of the assembly, and the number of pins is small so the variations do not completely cancel out. The standard deviation for the case including manufacturing uncertainties (QA-IP-ALL) was up to 13 times larger than the other cases, but it was noted the largest observed relative uncertainty across all cases and burnup steps was 0.05%, suggesting the large variation in standard deviations likely has little practical importance. Normality was not rejected at the 5% significance level for ADF₁.

Fig. 10. Mean and standard deviation in fast group ADF throughout the fuel cycle.

Fig. 11. Mean and standard deviation in thermal group ADF throughout the fuel cycle.

The impact of the pin modeling selections was once again apparent in ADF₂. The maximum difference in mean values between the different cases was 1.6% and was observed at EOC. Like ADF₁, a large relative difference in standard deviations was observed (up to nearly a factor of 12 difference in absolute standard deviations), but the maximum relative uncertainty in ADF₂ was found to be below 0.4%, again suggesting the large relative change in standard deviations has only minor practical effects. The null hypothesis of normality was not rejected for any burnup step across the four cases.

III.B.1. Homogenized Cross-Section Correlations

For each calculation point, Pearson correlation coefficients were calculated for the homogenized data using MATLAB. The correlation coefficients at each burnup step were calculated independently of the correlation coefficients at all other burnup points. The correlation coefficients at select burnup steps are shown for select cases in Figs. 12 and 13. The single pin material cases were found to agree closely, and thus only the full-assembly case is shown. The tracking of individual pin compositions mainly affected the correlations between ADFs and the remaining homogenized data. The BOC correlation coefficients were found to agree well between the nuclear data cases. The inclusion of manufacturing uncertainties significantly impacted the correlation between ADFs in the QA-IP-ALL case. The ADFs in the manufacturing uncertainty case at BOC were strongly negatively correlated, while in the remaining cases they were strongly positively correlated at BOC. The manufacturing uncertainties were found to have the most significant impact on correlations involving ADFs or diffusion coefficients.

Fig. 12. Correlation coefficients for the homogenized properties from the full-assembly case at select burnups.

Fig. 13. Correlation coefficients for the homogenized properties from the quarter-assembly case with manufacturing uncertainties at select burnups.

It was found that some correlation coefficients changed with burnup. Changes with burnup were typically observed for pairs of homogenized data that had relatively weak relationships at BOC. The correlations (or anticorrelations) were generally observed to grow stronger as burnup increased, although the rate of change was typically higher in the first half of the fuel cycle. Most correlation coefficients were observed to maintain the same sign throughout the fuel cycle, with the strength of the correlation (or anticorrelation) growing. However, some correlations, for example, the ADF₁-∑_A2, ADF₁-ADF₂, and ∑_A1-ν∑_f1 correlations, were found to change signs over the fuel cycle. The most significant changes were observed in the correlation between ADF₁ and ADF₂, shown in Fig. 14. Also shown is the ADF₁-D₂ correlation, which underwent a less significant change with burnup when manufacturing uncertainties were included. The ADF₁-ADF₂ correlation was found to be affected by both the use of individual pin material tracking and the inclusion of manufacturing uncertainties, but still underwent significant changes throughout the fuel cycle in the full-assembly model with a single pin material.

Fig. 14. Correlation coefficient between the ADFs is shown over the fuel cycle to highlight the differences between models at BOC and the significant change the correlation coefficient undergoes.

III.C. Isotope Concentrations

The concentration and uncertainty of isotopes within the fuel were tracked throughout the depletion cycle. For demonstration, the assemblywise concentrations of ²³⁵U and ²³⁹Pu are shown in Figs. 15 and 16. In general, it was observed that the inclusion of manufacturing uncertainties did not significantly affect the uncertainty of the assembly isotopic compositions (again, the global effect of manufacturing uncertainties was being reduced by individual pin perturbations canceling each other out). However, some effect was observed in the uncertainties of the composition of individual pins, as shown in Figs. 17 and 18. As expected, for isotopes in the specified composition of the fuel (such as ²³⁵U and ²³⁸U) there is no uncertainty at BOC except in the quarter-assembly case with manufacturing uncertainties. However, the uncertainty of ²³⁵U at the assembly level in the case including manufacturing uncertainties was found to converge toward the uncertainty of the remaining cases over the life of the fuel. The uncertainty in ²³⁸U in the manufacturing uncertainty case was found to be significantly higher than the remaining cases throughout the fuel cycle, but the overall magnitude of the uncertainty was small (reaching a maximum below 0.13% by EOC), and thus the increased uncertainty due to manufacturing uncertainties likely has few practical impacts. The concentration of ¹⁵⁵Gd (from the burnable poison pins) was found to decrease to negligible levels around approximately 10 MWd/kg, with its uncertainty generally increasing with burnup to that point.

Fig. 15. The ²³⁵U concentration and uncertainty for the entire assembly.

Fig. 16. The ²³⁹Pu concentration and uncertainty for the entire assembly.

Fig. 17. The ²³⁵U concentration and uncertainty for the inner and outer assembly pins.

Fig. 18. The ²³⁹Pu concentration and uncertainty for the inner and outer assembly pins.

III.D. Pin Power Factors

Assembly pin power factors were observed throughout the depletion of the fuel. For ease of comparison, the quarter-assembly model pin powers were reflected to generate 15 × 15 pin power maps, consistent with the full-assembly geometry. Mean pin power factors for select cases at BOC are shown in Fig. 19. No notable variations were observed between the four cases. The relative standard deviation (RSD) of the pin power factors at BOC is shown in Fig. 20. It was found that the introduction of manufacturing uncertainties significantly impacted the uncertainty in the pin power factors, as uncertainties on the order of 0.75% were observed in the fuel pins of the quarter-assembly case with manufacturing uncertainties compared to negligible uncertainty in the remaining cases. Similarly, the introduction of manufacturing uncertainties increased the uncertainty of the pin power factors in the burnable poison pins, rising from around 0.77% to around 1%.

Fig. 19. Mean pin power factors at BOC for select cases.

Fig. 20. Pin power RSDs at BOC for select cases.

For EOC pin powers, slight differences were observed between the models. Small variations between the case with individual pin materials and a single pin material demonstrated the impact of tracking pin compositions individually rather than as a lumped material, but the impact of manufacturing uncertainties was found to be much more significant than that of individual pin materials. RSDs for the EOC pin power factors are shown in Fig. 21. In all cases, the uncertainty of the pin power factor in the pins containing burnable poison decreased to a level consistent with the fuel pins (as expected, as very little burnable poison remains in the pins).

Fig. 21. Pin power RSDs at EOC for select cases.

IV. PWR MINI-CORE TRANSIENT RESULTS

The PWR mini-core transient from Exercise II-2b of the UAM-LWR benchmark study was completed with three sets of homogenized cross-section data corresponding to the full-assembly nuclear data case (FA-SP-ND), the quarter-assembly case with nuclear data uncertainties (QA-IP-ND), and the quarter-assembly case with manufacturing uncertainties (QA-IP-ALL). Note that while the geometry model, pin material modeling, and uncertainties selected were the same, the conditions of Exercise II-2b are at HZP, rather than the HFP conditions used in Exercise II-2a. The reference homogenized cross sections were computed for the unrodded assembly at HZP thermal-hydraulic conditions and a boron level of 900 ppm. Three additional branches were calculated, each with only a single change relative to the reference conditions: a rodded assembly, an assembly with a low boron level (5 ppm), and an assembly with a high boron level (1800 ppm).

The following sections summarize the key results of the mini-core rod movement transient and the effect of the uncertainties studied. It should be noted that while some uncertainty was observed in the kinetic parameters critical to the evolution of the transient (as a result of differences in the nuclear data), a more complete accounting of the kinetic parameter uncertainty, as described in other works,⁹ was not able to be performed with the available tools, and thus future improvements to the results presented here may be required.^a The results presented in this section were obtained with the 4 × 4 radial nodalization, with the exception of the pin power factors, which were obtained with the 2 × 2 radial nodalization.

IV.A. Initial Core State

The application of nuclear data and manufacturing uncertainties caused variations in the initial (t = 0) core state. Before the rod withdrawal began, a critical boron search was performed for each trial to ensure each transient started from a critical state. The distribution of critical boron levels for the two test cases is shown in Fig. 22. The application of manufacturing uncertainties was not found to significantly change the observed results. The null hypothesis of normality was not rejected at the 5% significance level for either case. The mean and standard deviations of the equilibrium boron levels are shown in Table V.

TABLE V Equilibrium Boron Levels for Each Case

Case	Mean Equilibrium Boron (ppm)	Equilibrium Boron Standard Deviation (ppm)
FA-SP-ND	1134.1	58.40
QA-IP-ALL	1134.0	58.48
QA-IP-ND	1134.2	58.40

Fig. 22. Equilibrium boron concentration distribution for the PWR mini-core models.

The initial mean radial assembly power distributions and RSDs for the 3 × 3 mini core are shown in Fig. 23 for the full-assembly case. Differences between the nuclear data and manufacturing uncertainty cases were negligible. The initial axial power distribution and its uncertainty are shown for both cases in Fig. 24. Pin power factors and uncertainties for the corner and center assemblies are shown in Figs. 25 and 26. The introduction of manufacturing uncertainties was not found to significantly affect the initial power distributions or uncertainties for the PWR mini core, but the increase in pin power uncertainties was found to be nonnegligible.

Fig. 23. Initial mean channel powers and RSDs for the PWR mini core modeled with homogenized data from the full-assembly model.

Fig. 24. Initial axial power distribution for both cases.

Fig. 25. Mean pin power factors and uncertainty for corner and center assemblies with only nuclear data uncertainty at t = 0.

Fig. 26. Mean pin power factors and uncertainty for corner and center assemblies with nuclear data and manufacturing uncertainties at t = 0.

IV.B. Transient Evolution

The PWR mini-core transient experienced an initial increase in reactivity as the center assembly control rod was withdrawn over the first 10 s, followed by a decrease in reactivity back to criticality as the control rod was reinserted over a period of 20 s. An additional 20 s were simulated following the complete reinsertion of the rod. The transient was simulated using time steps of 0.05 s. No thermal-hydraulic modeling was performed. The peak core power was observed at either 10.6 or 10.65 s for all samples across both models, with the majority of peak powers occurring at 10.65 s.

For the full-assembly case, a total of 893 of the 1001 samples reached the peak core power at 10.65 s. For the quarter-assembly case with manufacturing uncertainties, a total of 876 out of 980 samples reached the peak core power at 10.65 s. Testing of the mini-core transient with homogenized data from the quarter-assembly case with nuclear data only found 895 out of 1001 samples reached the peak core power at 10.65 s. The similarities between cases with and without manufacturing uncertainties was reasonable based on other works, which identified D₁ and ∑_A1 as the most significant contributors to the variance in results for similar transients.^10,11 As discussed earlier in this work, the variability in D₁ is small in all cases (incorporating manufacturing uncertainties had a minimal impact on the variability of D₁) and the uncertainty in ∑_A1 for fresh fuel was driven mostly by nuclear data uncertainties, which are the same between cases, rather than manufacturing uncertainties.

The mean and standard deviations of the mini-core power (normalized to the initial power) and reactivity (in dollars) are shown alongside the results for the reference case in Figs. 27 and 28, respectively. The maximum uncertainty in the core power factor was found to be 3.4% and was observed at the time for which peak power was most common, 10.65 s. The maximum absolute uncertainty in reactivity was approximately 0.013 $ and was observed at 10 s in both cases. The average maximum core power factor and reactivity insertion for each case are shown in Table VI.

TABLE VI Average and RSD of Maximum Core Power Factor and Reactivity Insertion During the Rod Movement Transient

Parameter	FA-SP-ND		QA-IP-ALL		QA-IP-ND
	Mean	RSD (%)	Mean	RSD (%)	Mean	RSD (%)
Maximum core power factor	2.1668	3.41	2.1668	3.40	2.1668	3.41
Maximum reactivity insertion ($)	0.3559	3.76	0.3559	3.75	0.3559	3.76

Fig. 27. Core power factor over the evolution of the transient.

Fig. 28. Core reactivity over the evolution of the transient.

The distributions of maximum core power factor and maximum reactivity were found to reject the null hypothesis of normality at the 5% significance level for both cases.

Radial power distributions at the most common peak power time step, 10.65 s, are shown for both cases in Figs. 29 and 30. The effect of manufacturing uncertainties was found to be essentially nonexistent. The axial power distribution at this point is shown in Fig. 31 for both cases, where again the impact of manufacturing uncertainties on the power distribution at the transient peak was found to be negligible compared to the impact of nuclear data uncertainties. Pin power factors for the corner and center assemblies are shown for the manufacturing uncertainty case in Fig. 32, where the effect of manufacturing uncertainties on the local pin power factors was again nonnegligible.

Fig. 29. Mean assembly powers and RSDs for the FA-SP-ND case at the time of maximum core power.

Fig. 30. Mean assembly powers and RSDs for the QA-IP-ALL case at the time of maximum core power.

Fig. 31. Axial power distribution and uncertainty at peak core power for both cases.

Fig. 32. Mean pin power factors and uncertainty for corner and center assemblies with nuclear data and manufacturing uncertainties at t = 10.5 (the closest time step to the peak power for which pin power factors were reported).

IV.C. Final Core State

The state of the PWR mini core was examined at 50 s, 20 s after the completion of control rod movements. The mean core power factor at the end of the simulated period was found to be approximately 1.36, with a RSD of about 1.6% in both cases. In both cases, the null hypothesis of normality was rejected at the 5% significance level.

The radial power distributions for the full-assembly nuclear data case at the end of the simulated period is shown in Fig. 33, with no significant differences observed for the manufacturing uncertainty case. The axial power distribution and its uncertainty are shown for both cases in Fig. 34. Pin power factors for the corner and center assemblies are shown in Fig. 35 for the manufacturing uncertainty case. The impact of manufacturing uncertainties relative to nuclear data uncertainties was found to be small at the conclusion of the transient for global parameters, while the local pin power factors were found to be affected by the manufacturing uncertainties in a similar manner to the initial and peak points of the transient.

Fig. 33. Mean assembly powers and RSDs for the full-assembly case at the end of the transient.

Fig. 34. Final axial power distribution and uncertainty for both cases.

Fig. 35. Mean pin power factors and uncertainty for corner and center assemblies with nuclear data and manufacturing uncertainties at the end of the transient.

V. CONCLUSIONS

This work presented results for a subset of the PWR time-dependent neutronics exercises of the UAM-LWR benchmark study. The SCALE code system was used to perform lattice-level calculations of the benchmark PWR fuel assembly in Exercise II-2a, and the PARCS diffusion code was used to study the mini PWR core rod movement transient from Exercise II-2b. For both studies, the importance of both nuclear data uncertainties and manufacturing uncertainties was considered and different modeling approaches were compared.

In the depletion studies of the PWR lattice, it was found that global parameters, such as k_eff, homogenized cross sections, and the assembly isotopic composition, were more significantly affected by nuclear data uncertainties. Some local parameters, such as the ADFs, diffusion coefficients, and individual pin compositions, were affected by both nuclear data and manufacturing uncertainties. Finally, some local parameters, such as pin power factors, were predominantly affected by manufacturing uncertainties. The effect of tracking isotopic compositions of each pin individually was typically found to be small, with minor effects noted as burnup increased in the ADFs and pin powers.

Correlation coefficients for the homogenized group data were studied over the length of the fuel cycle for each of the cases considered. The most significant effects of manufacturing uncertainties on the correlation coefficients were observed for parameters involving either ADFs or diffusion coefficients. Since the majority of correlation coefficients considered were between global lattice parameters, it was expected that the effect of manufacturing uncertainties would be less prevalent than the effect of nuclear data uncertainties. It was also noted that some correlation coefficients vary significantly with burnup, typically going from having a very weak relationship at BOC to a moderately strong correlation (or anticorrelation) at EOC. Changes in the correlation coefficients were found to occur nonlinearly, with most of the change typically occurring over the first half of the fuel cycle before mostly leveling off over the second half.

Similar to the lattice-level studies, it was found in the mini-core transient studies that global parameters, such as core power, core reactivity, and the radial/axial power distributions, were primarily affected by nuclear data uncertainties. Local parameters, such as the corner and center assembly pin power factors, were found to be more significantly affected by manufacturing uncertainties. Only minor differences in assembly powers, core power, axial power distribution, and reactivity were observed between models at the initial, peak, and final points of the transient. Future work on the PWR transient is required to better account for uncertainty effects in the kinetics factors, such as the delayed neutron fractions and decay constants. Additionally, future work to incorporate the effects of thermal-hydraulic feedback on the PWR transient and to determine the significance of thermal-hydraulic uncertainties relative to nuclear data and manufacturing uncertainties is required.

The results presented in this work demonstrate that while both nuclear data and manufacturing uncertainties have measurable effects on uncertainty in PWR studies, the relative importance of the two depends on whether the parameter of interest is a global or local phenomenon. For most global parameters, nuclear data uncertainties are typically much more significant, while manufacturing uncertainties make only a negligible contribution. For local parameters, manufacturing uncertainties, applied on a pin-by-pin basis, were found to be significant contributors to uncertainty. The additional computational effort required to completely simulate the effect of manufacturing uncertainties was found to be significant, and thus future work should weigh the importance of manufacturing uncertainties on the figures of merit being considered when deciding how to proceed.

Nomenclature

ADF1	=	= fast group ADF
ADF2	=	= thermal group ADF
D1	=	= fast group diffusion coefficient
D2	=	= thermal group diffusion coefficient
keff	=	= neutron multiplication factor

Greek

∑A1	=	= fast group absorption cross section
∑A2	=	= thermal group absorption cross section
ν∑f1	=	= fast group fission cross section multiplied by average number of neutrons per fission
ν∑f2	=	= fast group fission cross section multiplied by average number of neutrons per fission
∑S 1->2	=	= fast to thermal energy group scatter cross section (i.e., downscatter)

Acronyms

ADF	=	= assembly discontinuity factor
BOC	=	= beginning of cycle
EOC	=	= end of cycle
HFP	=	= hot full power
HZP	=	= hot zero power
LWR	=	= light water reactor
MWd/kg	=	= megawatt-days per kilogram of uranium
PWR	=	= pressurized water reactor
ppm	=	= parts per million
RSD	=	= relative standard deviation
UAM-LWR	=	= Uncertainty Analysis in Modelling for Light Water Reactors benchmark study

Acknowledgments

This work was funded by the Natural Sciences and Engineering Research Council of Canada and the University Network of Excellence in Nuclear Engineering.

Disclosure Statement

No potential conflict of interest was reported by the authors.

Notes

^a While the work performed in this paper includes some uncertainty in kinetics parameters, which results from differences in nuclear data between trials, the inclusion of uncertainties in the delayed neutron group fractions and precursor decay constants was not available in the official SCALE release at the time of this work.