Verification of MPACT for the APR1400 Benchmark

This paper describes benchmark calculations for the APR1400 nuclear reactor performed using the high-fidelity deterministic whole-core simulator MPACT compared to reference solutions generated by the Monte Carlo code McCARD. The methodology presented in this paper is a common approach in the field of nuclear reactor analysis, when measured data are not available for comparison, and may be more broadly applied in other simulation applications of energy systems. The benchmark consists of several problems that span the complexity of single pins to a hot full power cycle depletion. Overall, MPACT shows excellent agreement compared to the reference solutions. MPACT effectively predicts the reactivity for different geometries and several temperature and boron conditions. The largest deviation from McCARD occurs for cold zero conditions in which the fuel, moderator, and cladding are all 300 K. Possible reasons for this are discussed. Excluding these cases, the ρ reactivity difference from McCARD is consistently below 100 pcm. For single fuel pin problems, the highest error of 151 pcm occurs for the lowest fuel enrichment of 1.71 wt.% UO2, indicating possible, albeit small, enrichment bias in MPACT’s cross-section library. Furthermore, MOC and spatial mesh parametric studies indicate that default meshing parameters and options yield results comparable to finely meshed cases. Additionally, there is very good agreement of the radial and axial power distributions. RMS radial pin and assembly power differences for all cases are at or below 0.75%, and all RMS axial power differences are below 1.65%. These results are comparable to previous results from the VERA progression problems benchmark and meet generally accepted accuracy criteria for whole-core transport codes.


INTRODUCTION
To address pressing challenges to nuclear power, the United States Office of Nuclear Energy supports projects that reduce costs, improve safety, and limit proliferation risk [1]. International collaborations are essential to this work, leading to the formation of programs such as the International Nuclear Energy Research Initiative (I-NERI), which is a joint venture between the United States and the Republic of Korea. A current I-NERI program specifically aims to improve high-fidelity, multiphysics simulation codes for advanced nuclear reactors. Under this collaboration, the APR1400 benchmark has been developed to facilitate code-to-code verification [2]. The benchmark reference solution was generated by the Monte Carlo code McCARD [3]. Hence, the purpose of this report is to describe and present the results of a series of benchmark calculations performed using the MPACT code [4] and discuss the extent of agreement with the McCARD reference solution.
The benchmark problems are based on the core design of the Korea Electric Power Corporation APR1400. The APR1400 core consists of 241 fuel assemblies in a rectangular lattice. Each fuel assembly is composed of 236 fuel or burnable absorber rods, 4 guide tubes, and 1 central tube arranged in a rectangular lattice. The guide tubes and central tubes are the size of four regular pin cells. Nine assembly types are specified utilizing different configurations of 1.71 wt% UO 2 , 2.00 wt% UO 2 , 2.64 wt% UO 2 , 3.14 wt% UO 2 , 3.64 wt% UO 2 , and gadolinia burnable absorbers. The reactor is controlled using seven control rod banks: two shutdown banks labeled A and B and five regular banks labeled 1, 2, 3, 4, and 5. Banks A, B, 1, and some of bank 2 have 12-fingered control rod assemblies, and some of bank 2 as well as all of banks 3, 4, and 5 have 4-fingered control rod assemblies. More details summarizing the benchmark problems are included in the following section, and the complete core specifications can be found in reference [2].
The analysis of these benchmark problems have been completed by researchers at the Seoul National University Reactor Physics Laboratory using nTRACER and at the Korea Atomic Energy Research Institute (KAERI) using DeCART. DeCART is a "whole core neutron transport code capable of direct subpin level flux calculation at power generating conditions" [5]. nTRACER is a "high-fidelity multi-physics simulation code" for commercial PWRs and fast reactors that utilizes a "direct whole core calculation module and a sub-channel thermal/hydraulic (T/H) solver" [6]. Both DeCART and nTRACER accurately predict reactivity in various geometries and conditions, with reactivity differences from McCARD below 100 pcm in almost every case and control rod worth differences below 1.0%. For both codes, the greatest reactivity differences occur for cold zero conditions in which the fuel, moderator, and cladding are 300 K. These results are similar to those obtained using MPACT.
Regarding the power distribution, nTRACER results initially exhibited a radial power tilt, and this was resolved by correcting the reflector cross sections for the large spatial dependency of the multigroup cross-sections for stainless steel. Once this was done, RMS radial power distribution errors between nTRACER and McCARD were all below 1.0% and RMS axial power distribution errors were below 1.7%. Radial power distribution errors were higher for DeCART; most RMS radial power distribution errors are below 1%, but there are several outliers, most of which are cases with cold zero power conditions. All RMS radial power distribution errors are below 2.27%. All RMS axial power differences from McCARD are below 2.25%. Excluding cases with cold zero conditions, all axial power differences are below 1.52% [7]. These results indicate that both DeCART and nTRACER effectively predict reactivity and power distributions for the APR1400, but errors are most significant for cases using cold zero conditions.
The remainder of the report is outlined as follows. In Section 2, the reactor geometry, benchmark problems, and conditions studied are described in detail. In Section 3, modeling parameters used in the MPACT models as well as equations relevant for the analysis are included. In Section 4, all benchmark problem results as well as the results of spatial and MOC parametric mesh studies on single pin and 2-D single assembly cases are presented. Finally, sections 5 and 6 contain conclusions and future work.

BENCHMARK PROBLEMS
The benchmark involves six problem types: single fuel pin, single 2-D fuel assembly, 2-D core, 3-D core, control rod worth calculations, and a 3-D core depletion. Additionally, mesh sensitivity studies were performed on the single fuel pin and single 2-D fuel assembly problems.
For each benchmark problem, several operating conditions are specified. For this report, cold zero (CZ) power refers to a moderator, cladding, and fuel temperature of 300 K. Hot zero (HZ) power indicates a moderator, cladding, and fuel temperature of 600 K. Hot full (HF) power refers to a moderator and cladding temperature of 600 K and a fuel temperature of 900 K. The specified boron concentrations are 0 ppm, 1000 ppm, and 2000 ppm. The naming convention adopted for describing benchmark results in this report references cases using two letters, that refer to the temperature condition, followed by a number, that indicates the boron concentration. A boron concentration of 0 ppm is indicated by a 0, 1000 ppm is indicated by a 1, and 2000 ppm is indicated by a 2. For example, when the moderator, fuel, and cladding temperature are all 300 K and the boron concentration is 1000 ppm, the case is referred to as CZ1. Table 1 summarizes the naming conventions for all temperature and boron conditions that are analyzed.  Fig. 1 -Fig. 7 show key elements of the APR1400 core configuration.
NURAM-2020-004-00   All guide tubes and central tubes are the same size as four pin cells, which can be seen in Fig. 4, which depicts the radial configuration of a fuel assembly. Each assembly has nine spacer grids with the axial layout shown in Fig. 5.

Modeling Parameters
Results were generated using MPACT's 2-D transport solver. All cases used the mpact51g_71_4.3m2_03262018 51-group cross-section library and default meshing parameters. For pin cells, the default Method oc Characteristics (MOC) flat source discretization was used. In fuel cells, the default flat source discretization creates 3 equal-area radial subdivisions in the fuel and one ring each in the fuel-clad gap, zircaloy cladding, and moderator. The guide tube pin cell has 3 radial subdivisions in the interior moderator and radial subdivision each in the cladding and external moderator. In the gadolinia burnable absorbers, there are 10 radial subdivisions in the fuel and 1 radial subdivision in all other regions. Each radial subdivision in all cell types has 8 azimuthal divisions. The flat source characteristics solver was used instead of the linear source characteristics solver because the linear source solver is still undergoing validation. For the MOC discretization, the Chebyshev-Yamamoto quadrature type was used with a ray spacing of 0.05 cm, 16 azimuthal angles per octant, and 2 polar angles per octant. The P 2 scattering method was used for all problems.
All materials were defined based on the isotope number densities provided in the benchmark specifications [2]. However, there were some isotopes missing from MPACT's cross section library for which substitutions were necessary. Specifically, for silicon, carbon, and molybdenum, the benchmark specified number densities for individual stable isotopes of these elements, but MPACT's cross-section library does not have entries for the individual isotopes. Rather, the cross sections for natural silicon, natural carbon, and natural molybdenum were used instead because the number densities of various isotopes in the natural elements are the same as the individual isotopes specified in the benchmark.

Reactivity differences
All reactivity differences are reported in terms of Δ , where Δ is defined in Eq. (1).
When results from multiple cases are combined, the arithmetic mean of Δ is used.

Pin and assembly power comparisons
Pin and assembly power comparisons are usually reported in terms of the relative Root Mean Square (%RMS) difference. The relative difference, , between MPACT and McCARD is defined in Eq. (2).
Using this result, the %RMS different is defined in Eq. (3).
where i is the pin or assembly index, and n is the total number of pins or assemblies considered. For maximum pin and assembly power differences, the relative maximum difference of all pins or assemblies considered, calculated using the above relative difference formula, is used.

Benchmark problem results
For the single pin problems, each of the five enrichments (1.71 wt% UO 2 , 2.00 wt% UO 2 , 2.64 wt% UO 2 , 3.14 wt% UO 2 , and 3.64 wt% UO 2 ) was modeled for each of the nine temperature and boron conditions, for a total of 45 cases studied. The complete isotopic compositions of the fuel are specified in the benchmark and were manually defined in the inputs. Fig. 19 is a histogram depicting the number of cases that fall in each reactivity difference range defined on the horizontal axis. MPACT and McCARD generally agree very well; the average difference in kinf between the two solutions was 63 pcm ± 44 pcm, which can be attributed to MPACT's usage of a multigroup approximation as opposed to McCARD's continuous energy cross section representation. In 34 of the 45 cases studied, MPACT had a lower k inf than McCARD, demonstrating a possible bias in the cross-section libraries. The maximum difference in k inf was 151 pcm, and the minimum difference was 3 pcm. Table 2 contains the average reactivity difference, standard deviation, and maximum reactivity difference for various categories of cases to better identify specific trends in the results. It was determined that the largest relative differences were observed for cases with an enrichment of 1.71 wt% UO 2 , suggesting a possible, slight enrichment bias of approximately -50 pcm, although the exact value depends on other conditions, e.g. temperature, moderator density, boron concentration, in MPACT's cross section library. Additionally, boron concentration had a moderate impact on accuracy, as observed in Table 2, which shows agreement with McCARD generally improving as boron concentration increases. Reference [8] outlines several accuracy goals, including that reactivity differences should be below 200 pcm. Given that the maximum reactivity difference from McCARD for pin cell cases is 151 pcm, these results are acceptable.  In Fig. 20 Table 2, there is a slight improvement in agreement with McCARD when using parameters finer than the default.

MOC parametric studies
One of the parametric studies performed combined all the finest parameters and was called the "fine" solution. The fine solution had a ray spacing of 0.005 cm, 32 azimuthal angles per octant, and 3 polar angles per octant. The error from the McCARD k inf for the fine solution is only 38 pcm, which is 10 pcm lower than the error when using default parameters. Fig. 21 and Fig. 22 are histograms the number of cases that fall in each reactivity difference range defined on the horizontal axis for the fine solution (Fig. 21) and when default parameters are used (Fig. 22). The cases shown all have 3.64 wt% UO 2 enrichment.  In comparing the two figures, it is clear that most reactivities predicted by MPACT when using the fine parameters are lower than the reactivities reported by McCARD. Additionally, when using fine parameters, the reactivities are generally closer to the McCARD reference than when default parameters are used. However, the fine solution requires significantly more computational resources to compute; it takes three times as long and uses almost four times as much memory. Given that the average reactivity agreement only improves by 10 pcm when using the finest parameters instead of default values and the tradeoff between accuracy and computational resources, the default parameters obtain suitable k inf values.

Spatial mesh parametric studies
For the spatial mesh studies, the number of radial subdivisions in the innermost region of the fuel cells was changed to be 1, 2, 4, or 5. For reference, the default solution has 3 subdivisions. The cases examined all had 3.64 wt% UO 2 . When changing the mesh, ray spacing was set as 0.01 cm, and the azimuthal and polar angles per octant were left at the default values of 16 and 2, respectively. The average differences and standard deviation between the k inf generated using 1, 2, 4, and 5 rings and the k inf calculated by McCARD are shown in 23. In Fig. 23, blue bars indicate parameters that are less fine than the default values, and maize bars indicate parameters that are finer than the default. Using coarser spatial meshes than the default results in reactivity differences from McCARD of 56 pcm for two fuel rings to 76 pcm for one fuel ring. A finer mesh results in reactivity differences from 47 pcm for four fuel rings to 44 pcm for five fuel rings. The reactivity differences for the finer meshes are only 1 pcm to 4 pcm below the reactivity difference of 48 pcm for the default mesh values. Considering that the finer cases take about 1.6 times longer and require about 1.4 times as much memory, and the minimum reactivity difference with McCARD is only 4 pcm lower than when default values are used, the default values are appropriate. Furthermore, altering the MOC parameters has a more significant impact on accuracy than altering the mesh of the problem.

Moderator mesh parameteric studies
For the moderator mesh studies, the number of radial subdivisions in the moderator was changed to be 2, 3, 4, or 5. For reference, the default solution has 1 subdivision. The cases examined all had 3.64 wt% UO 2 . When changing the mesh, ray spacing was set to a fine spacing of 0.005 cm, and the azimuthal and polar angles per octant were set at 32 and 3, respectively. Additionally, the number of fuel rings was increased to 5 from the default of 3. The average differences and standard deviation between the k inf generated using 2, 3, 4, and 5 rings and the k inf calculated by McCARD are shown in Fig. 24. When using the default value of one moderator ring, the average reactivity difference between MPACT and McCARD for 3.64 wt% pins is 48 pcm. As Fig. 24 shows, making the mesh finer causes agreement with McCARD to worsen in every case except for four moderator rings. Moreover, the finest moderator mesh of five rings has an average reactivity difference from McCARD of 53 pcm, which is 5 pcm higher than when using default values. Since, as noted earlier, when using the fine mesh, computation time is increased threefold and memory usage is increased fourfold, it was determined that the default value of one moderator ring is most appropriate for use in generating benchmark results.

Benchmark problem results
Each of the nine assembly types (A0, B0, B1, B2, B3, C0, C1, C2, C3) was run using all nine temperature and boron conditions. Fig. 25 is a histogram that shows the number of cases falling each reactivity difference range defined on the horizontal axis.   Fig. 26 indicates that greatest deviation from the reference solutions was observed in all CZ cases, no matter the boron concentration. These cases are marked in red, and the pin power differences are significantly higher than all other cases for that assembly. Table 3 summarizes the data in Fig. 25 and Fig. 26 and shows the average k inf difference as well as the average %RMS pin power difference from McCARD for each assembly type. As mentioned in the previous section, reactivity differences are ideally below 200 pcm. For the 2-D assembly problems, eight cases do not meet this goal, and all but two of these have CZ conditions. As such, MPACT shows good agreement with McCARD. Reference [8] also outlines accuracy goals of less than 1.0% RMS difference and less than 1.5% maximum difference for 2-D assembly pin power distributions. Since the highest RMS pin power difference between MPACT and McCARD is 0.57% and the maximum pin power difference is 1.15%, the pin powers calculated by MPACT show excellent agreement with McCARD.

MOC parametric studies
Nine cases were considered for the 2-D assembly MOC studies: for each of the nine assemblies, MOC parameters were independently changed for the CZ0 case. The results were compared to the McCARD k inf . Just as for the single pin cases, variations included using 4, 8, and 32 azimuthal angles per octant instead of the default 16; 1 and 3 polar angles per octant instead of the default 2; and 0.025 cm, 0.01 cm, and 0.005 cm ray spacing instead of the default 0.05 cm. Fig. 27 summarizes the average differences between the k inf generated using each of these parameters and the k inf generated by McCARD as well as the standard deviation. In Fig. 27 However, when MOC parameters are made finer than the default, the average reactivity difference with McCARD is worse than reactivity difference of 133 pcm when using default parameters in every case; the average difference ranges from 143 pcm to 179 pcm. Even compared to a "fine" solution that had a ray spacing of 0.005 cm, 32 azimuthal angles per octant, and 3 polar angles per octant, the average reactivity difference is 146 pcm, which is 13 pcm higher than the difference when default values are used. A possible explanation is that refining the MOC mesh could be reducing some error cancellation. Specifically, the MPACT cross section library uses super homogenization (SPH) factors to obtain the best possible agreement, even when different calculation methods are used. The SPH factors are determined by comparing pin cell solutions generated using transport corrected P 0 (TCP 0 ) scattering with solutions generated using continuous energy Monte Carlo methods. The multigroup resonance integral data is multiplied by these SPH factors. When solving problems using P 2 scattering instead of TCP 0 scattering, as with these benchmark results, the SPH factors become inconsistent, thereby removing some error cancellation and contributing to the significant disagreement observed in Fig. 27 above. Hence, given that every alteration to MOC parameters causes worse agreement with McCARD than for default values, default MOC parameters are most appropriate for use in generating benchmark results.

Spatial mesh parametric studies
For the 2-D assembly mesh study, the number of radial subdivisions in the innermost region of the fuel cells was changed from the default of three rings to be one, two, four, or five rings. The gadolinia mesh was also altered from the default value of ten rings to be one, five, or 15 rings. When varying the mesh, ray spacing was 0.01 cm, and the azimuthal and polar angles per octant were left at the default values. The mesh azimuthal angles were left at the default of 8 per octant. The CZ0 case for each assembly type, the same as was used for the MOC studies, was examined for the spatial mesh studies. The average differences between the k inf generated using the various spatial meshes and the k inf calculated by McCARD as well as the standard deviation are shown in Fig. 28.

Figure 28. Average difference from MPACT default k inf for 2-D assembly spatial mesh studies.
In Fig. 28, blue bars indicate parameters that are less fine than the default values, and maize bars indicate parameters that are finer than the default. Using coarser spatial meshes than the default results in average reactivity differences from McCARD ranging from 190 pcm to 264 pcm, and using finer spatial meshes results in reactivity differences from 171 pcm to 241 pcm. The maximum absolute difference from McCARD of all of the cases was 331 pcm. Like the MOC parametric studies, every alteration of the spatial mesh, including making it finer, has worse average agreement with McCARD than the default mesh, which has an average reactivity difference from McCARD of 133 pcm. This is again likely due to inconsistencies in the SPH factors due to the use of the P 2 scattering method that are accentuated when the mesh is changed.
Every alteration of the gadolinia mesh results in substantially worse agreement with McCARD than is observed for changing the fuel mesh; the finest gadolinia mesh of 15 rings has an average reactivity difference of 241 pcm with McCARD, which is 30 pcm worse than the average reactivity difference for the coarsest fuel spatial mesh of one ring.
As with the MOC parametric study, given that every alteration to the fuel and gad meshes causes agreement with McCARD to worsen, it is most optimal to use the default values of three fuel rings and ten gadolinia rings when generating benchmark results.

Comparison of 2-D assembly MPACT "fine" solution to McCARD
To further examine if default parameters generate appropriate benchmark solutions, 2-D assembly results obtained using the "fine" solution, which has a ray spacing of 0.005 cm, 32 azimuthal angles per octant, and 3 polar angles per octant, were compared to McCARD. These solutions were then compared to how close the default solution was to McCARD. All cases considered were CZ0 cases. Table 4 shows the k inf difference and %RMS pin power difference from McCARD for both the solution generated using MPACT's default parameters and the "fine" solution. As seen in Table 4, the average k inf relative difference from McCARD and average RMS pin power differences are both higher for the "fine" solution than using the default parameters. Again, this is likely due to the super homogenization (SPH) factors used by MPACT's cross section library that are determined using TCP 0 scattering. So, when solving problems using P 2 scattering instead of TCP 0 scattering, as with these benchmark results, the SPH factors become inconsistent, thereby removing some error cancellation and contributing to increased disagreement from a more refined mesh. Thus, the default parameters are sufficiently optimized and are appropriate to use to generate benchmark problem results.

In-out tilt with TCP 0 scattering
MPACT's default scattering method is TCP 0 . However, the 2-D core radial power distribution results exhibited significant in-out tilt when the default TCP 0 scattering was used. Fig. 30 shows the radial assembly powers for the HZ1 case when TCP 0 scattering was used. In the figure, maize indicates over-estimation of assembly powers, and blue represents underestimation. The greatest deviations from McCARD are outlined in red. There are clearly defined regions of over-and under-estimation of assembly powers; maize is concentrated in the center regions, with over-estimation ranging from 0.08% to 1.03%. The periphery is dominated by blue, with under-estimation from -0.16% to -0.63%. For the HF1 case, the RMS assembly power difference was 0.45%, and the maximum difference was 1.03% To address this in-out tilt, P 2 scattering was used instead. Fig. 31 shows the radial power distribution for the HZ1 case when the P 2 scattering method is used. Again, maize represents over-estimation of assembly powers while blue indicates under-estimation. The same color scale is used for Fig. 30 and Fig. 31 to better compare the two. The power distribution is now much more even, and the regions of over-and under-estimation are not so clear. Additionally, the radial assembly powers show much greater agreement with McCARD; The RMS assembly power difference decreased from 0.45% to 0.23%, and the maximum relative power difference decreased from 1.03% to 0.49%.
The HZ1 case is not unique; every case demonstrated substantially improved agreement when P 2 scattering was used. The specific changes to RMS and maximum assembly power differences when changing from TCP 0 to P 2 scattering are outlined in Table 5. As shown in Table 5, the average RMS assembly power difference decreased from 1.01% to 0.36% and the average maximum assembly power difference decreased from 2.24% to 0.80% when P 2 scattering was used instead of TCP 0 scattering, clearly indicating the P 2 scattering must be used to have acceptable agreement with McCARD. Thus, for all benchmark problems, P 2 scattering was used. P 2 scattering was also tested for the single pin and single 2-D assembly problems, and results were significantly different, so those problems were redone using P 2 scattering. The original, TCP 0 results are included in Appendix 1 for reference.

Results generated using P 2 scattering
As outlined above, when P 2 scattering is used, the 2-D core results generated by MPACT demonstrate strong agreement with the McCARD reference solution. Fig. 18 summarizes and compares the 2-D core results generated by McCARD and MPACT for each condition. The rho difference for each condition is below 200 pcm for every case, and the overall average rho difference is 83 pcm ± 61 pcm. The greatest reactivity difference is for the CZ2 case, which overpredicts k-eff by 182 pcm. This maximum difference is below the accuracy goal of 200 pcm difference outlined in [8], indicating good reactivity agreement between MPACT and McCARD.
Regarding the assembly comparisons, MPACT's results agree very well with McCARD; the average RMS assembly power difference is only 0.34%, and the average maximum difference is 0.73%. Reference [8] presents accuracy goals of below 1.5% RMS assembly power differences and below 2.5% maximum assembly power differences. The greatest RMS assembly power difference is 0.47% and the maximum assembly power difference is 1.02%, indicating that MPACT shows good agreement with the power distributions calculated by McCARD.
Generally, the CZ cases have worse reactivity agreement with McCARD than other case conditions. There are several possible explanations for this. Specifically, it is possible that the hydrogen scattering matrix used by either McCARD or MPACT is incorrect; temperature dependence is not accounted for in the scattering kernel for hydrogen, which results in inaccuracies in the scattering matrix. Since hydrogen plays such an important role in light water reactors, small changes in the scattering kernel can substantially impact results [9]. To determine if the scattering matrix used by McCARD caused these issues, single fuel pin and 2-D assembly results generated by MPACT using CZ conditions were compared to those generated by the Monte Carlo code Serpent [10].
The disagreement in pin power and reactivity between MPACT and Serpent was comparable to the disagreement between MPACT and McCARD, indicating that if the cold zero bias is caused by an incorrect scattering matrix, it is the scattering matrix used by MPACT and not McCARD. It was also suggested that there may be a spatial discretization error in which there are too few rings in the mesh for the moderator. However, this was investigated, and refining the moderator mesh did not resolve the cold zero bias. Table 7 summarizes and compares the 3-D core results from McCARD and MPACT for each condition. The standard deviation for all Monte Carlo cases is 4 pcm. Additionally, all RMS assembly power differences are below 0.56% with the exception of the HF0 case, which had an RMS assembly power difference 1.17%, respectively. Axial power differences are slightly larger; the largest RMS axial power difference is 1.65%, and the maximum axial power difference is 3.04%. Reference [8] gives an accuracy goal of 2.0% RMS difference and 3.0% maximum difference for both radial and axial power distributions. The radial RMS and maximum assembly power differences are all below these goals, indicating good agreement of radial power distributions between MPACT and McCARD. Regarding axial power differences, the RMS differences are all below 2.0%, but the maximum difference of 3.04% exceeds the accuracy goal of 3.0%, albeit slightly. However, all other maximum axial power differences are below 3.0%, with most falling below 2.0%, indicating acceptable agreement between the axial powers calculated by MPACT and those found by McCARD.

3-D core results
For each boron concentration, the temperature condition with the greatest overall deviation from McCARD is the CZ case, which is consistent with the results from previous problems. As noted above, this may be caused by an incorrect hydrogen scattering matrix being used by MPACT. Fig. 32 depicts the RMS axial power differences for each case.      Fig. 35 shows the control rod assembly configuration for the APR1400. In total, there are seven control rod banks. Five banks are regulating groups, and are labeled 1, 2, 3, 4, and 5, and two banks are shutdown groups labeled A and B. Banks A, B, 1, and some of bank 2 are 12-fingers All cases used HZ0 conditions. There were seven cases studied for the control rod worth problems, plus the all rods out (ARO) case, which is the HZ0 case from the 3-D core problems. For each case, all seven banks were inserted one at a time, following the order 5 -4 -3 -2 -1 -B -A. There was no withdrawal of previously considered banks.

Worth equation
The values used to compare results generated by MPACT and McCARD were the accumulated worth and the group worth.
Worth was calculated using Eq. (4) where is the case index. Since the insertion order follows 5 -4 -3 -2 -1 -B -A, = 0 corresponds to the ARO case, which has a worth of 0 pcm, = 1 corresponds to just bank 5 inserted, = 2 corresponds to banks 5 and 4 inserted, etc.

Accumulated worth
Accumulated worth is the sum of the worth for all banks inserted thus far and was calculated using the relationship in equation Eq. (5).
where is the number of inserted banks.

Group worth
Group worth refers to the difference in accumulated worth due to inserting a specific control rod bank and was determined using the formula in Eq. (6).
For example, Table 8 summarizes the control rod worth results and compares the results from MPACT and McCARD.  Again, there is substantial agreement between MPACT and McCARD. All group and accumulated differences are at or below 0.6%. Reference [11] outlines an accuracy goal of below 5% difference in group and accumulated rod bank worths. Since the maximum difference in group and accumulated worths is only 0.6%, MPACT's control rod worths show excellent agreement with McCARD. The greatest deviations from the group and accumulated control rod worths generated by McCARD occur when banks 5 and B are inserted.

Control rod worth results
Additionally, there is excellent agreement in the assembly powers; all RMS assembly power differences are below 0.71%, and the maximum difference is 1.86%. Assembly power agreement is significantly worse than for all other cases when banks 1 and B are inserted. Similar results were noted for DeCART, and assembly power differences between nTRACER and McCARD were not reported. When examining the radial power distribution, it appears that the greatest differences are in the center of the core, with agreement much stronger on the periphery. Since more absorbers have been inserted, power peaks are pushed away and the power shape becomes very complicated, resulting in errors in calculations when using transport codes as opposed to Monte Carlo codes. When these cases are excluded, the RMS assembly power differences are at or below 0.37%, and the maximum difference is only 0.90%.

3-D core depletion results
The depletion was done using a 3-D core model with 100% rated power, a coolant inlet temperature of 563.75 K, and critical boron concentration search at each burnup step. In the input, the axial geometry was adjusted slightly; the height of the upper axial reflector was decreased from 25 cm to 5 cm.
No McCARD reference solution was provided for the 3-D core depletion problem, so MPACT results were compared to results generated by other benchmark participants using DeCART and nTRACER [12]. The result from all three codes are shown in Fig. 36.   Table 9 provides more detail about the different boron concentrations by presenting the boron concentrations calculated by MPACT, DeCART, and nTRACER for different burnups and the difference of DeCART and nTRACER from MPACT. The average boron concentration difference from MPACT is only 15 ppm for nTRACER and 32 pcm for , indicating that MPACT's calculations are slightly closer to nTRACER's, as noted above. The largest differences from MPACT for occur at 8 MWD/kgHM, this appears to be around the time that the gadolinia is burning out. The prediction of the burnout of gadolinium is well understood to be challenging, so the peak difference being here is not surprising. Additional investigations should be performed to try to identify and confirm the root cause of the peak difference here. For MPACT and nTRACER, the largest difference occurs near the end of cycle. When comparing the two critical boron concentration curves in Fig. 36, it appears that the rate of fuel consumption is different between the two codes, as indicated by the differing slopes after 10 MWD/kgHM. This suggests the values, or energy release per fission used for flux normalization in depletion, are different. A suggested follow up study would be to have the 3 codes perform the cycle depletion calculation with the same .

Burnup interval sensitivity for Gadolinia
The burnup interval used for the depletion was 14 EFPD, which equals approximately 0.54 MWD/kgHM. There were concerns that this interval may be too large because gadolinia has a large absorption cross section, so the reaction rate of gadolinia changes dramatically over time. As such, a smaller interval may be necessary to obtain accurate results. To test this, burnup intervals of 0.5 MWD/kgHM and 0.25 MWD/kgHM burnup interval for 2D assembly depletion were tested, and the calculated k inf values were compared. The results are shown in Fig. 37. In Fig. 37, the k inf calculated using the different burnup steps is nearly indistinguishable for every level of burnup. The average rho difference is 21 pcm, and the maximum rho difference is only 63 pcm. Given the strong agreement between the k inf values calculated using both burnup intervals, it was determined that the burnup interval of 0.54 MWD/kgHM, or 14 EFPD, used for the 3-D core depletion problem was acceptable.

CONCLUSIONS
Overall, MPACT shows excellent agreement compared to the Monte Carlo reference solution generated by McCARD. MOC and spatial mesh parametric studies indicate that default meshing parameters and options yield results comparable to finely meshed cases, so default parameters are appropriate for use in generating results for benchmark problems. MPACT effectively predicts reactivity for several problem types and conditions. The highest errors exist for CZ conditions, but excluding these cases, the rho reactivity difference is consistently below 100 pcm. Additionally, for the single fuel pin problems, the greatest disagreement existed for the lowest fuel enrichment of 1.71 wt% UO 2 , indicating possible enrichment bias in MPACT's cross section library. Moreover, there is strong agreement of the radial and axial power distributions. The use of P 2 scattering corrected the in-out radial power tilt caused by using the default TCP 0 scattering method. With P 2 scattering, all RMS pin and assembly power are differences below 1%, and all RMS axial power differences are below 1.65%. These results are comparable to previous results from the VERA progression problems benchmark [13] [14] and below the accuracy goals outlined in [8] and [11].
Regarding the hot full power 3-D core depletion, there was some variation in the critical boron concentration calculated by MPACT compared to nTracer and . MPACT's boron concentration is closer to nTRACER's concentration than 's for each burnup level.

FUTURE WORK
The results of this research generate questions that will be investigated in future work. Specifically, for the single fuel pin studies, there is greater agreement for pin cells with higher fuel enrichments, so this enrichment bias, which may be due to bias in MPACT's cross section library, will be investigated. Also, for each problem, the highest errors existed for cases with cold zero power conditions. This can possibly be explained by an incorrect hydrogen scattering matrix used by MPACT. This possibility will be investigated further. Finally, there were significant differences between MPACT, nTRACER, and in the hot full power 3-D depletion problem. The causes of these differences will be investigated.

Single Fuel Pin Benchmark Problem Results
For the single pin problems, each of the five enrichments (1.71 wt% UO 2 , 2.00 wt% UO 2 , 2.64 wt% UO 2 , 3.14 wt% UO 2 , and 3.64 wt% UO 2 ) was modeled for each of the nine temperature and boron conditions, for a total of 45 cases studied. Fig. 38 is a histogram depicting the number of cases that fall in each reactivity difference range defined on the horizontal axis when TCP 0 scattering is used. MPACT and McCARD generally agree very well when TCP 0 scattering is used; the average difference in k inf between the two solutions was 54 pcm ± 46 pcm, which is lower than the average reactivity difference of 63 pcm ± 44 pcm when P 2 scattering is used. As before, this difference can be attributed to MPACT's usage of a multigroup approximation as opposed to McCARD's continuous energy cross section representation. Similarly to when P 2 scattering is used, in 31 of the 45 cases studied, MPACT had a lower k inf than McCARD, demonstrating a possible bias in the cross-section libraries. The maximum difference in k inf was 151 pcm, and the minimum difference was 2 pcm. Table 10 contains the average reactivity difference, standard deviation, and maximum reactivity difference for various categories of cases to better identify specific trends in the results. The largest relative differences were observed for cases with CZ conditions, possibly due to the aforementioned incorrect hydrogen scattering matrix used by MPACT. Additionally, as observed when P 2 scattering was used, boron concentration had a substantial impact on accuracy, as observed in Table 10, which shows agreement with McCARD improving significantly as boron concentration increases. Every group of single pin problems has an equal or lower reactivity difference or maximum difference when TCP 0 scattering is used compared to when P 2 scattering is used.

Single 2-D Assembly Benchmark Problem Results
Each of the nine assembly types (A0, B0, B1, B2, B3, C0, C1, C2, C3) was run using all nine temperature and boron conditions as well as TCP 0 scattering. Fig. 39 is a histogram that shows the number of cases falling each reactivity difference range defined on the horizontal axis when TCP 0 scattering is used. The average difference from the McCARD k inf values was 106 pcm ± 86 pcm, which is higher than the average difference of 99 pcm ± 62 pcm when P 2 scattering is used.
The pin powers within the assemblies generated by MPACT using TCP 0 scattering are very similar to the McCARD reference solutions, with an overall average RMS pin power difference of 0.21%, which is the same as when P 2 scattering is used. Thus, the scattering method impacts reactivity agreement much more than pin power agreement. The %RMS pin power differences for each case are depicted in Fig. 40 alongside a list of the ordering of the case conditions within each assembly.  Fig. 40 indicates that greatest deviation from the reference solutions was observed in all CZ cases, no matter the boron concentration, which is the same as when P 2 scattering is used. These cases are marked in red, and the pin power differences are significantly higher than all other cases for that assembly. Table 11 summarizes the data in Fig. 39 and Fig. 40 and shows the average k inf difference as well as the average %RMS pin power difference from McCARD for each assembly type. Overall, compared to when P 2 scattering is used, the results when TCP 0 scattering is used have greater reactivity differences and the same or 0.1% lower RMS pin power differences.   NURAM-2020-004-00    NURAM-2020-004-00