Analysis of the APR1400 Benchmark Using High-Fidelity Pin-Wise Core Calculation Codes

Abstract

The KEPCO Nuclear Fuel Company (KNF) has undertaken considerable efforts to improve the KARMA/ASTRA nuclear design code system to meet the increasing demand for high-fidelity core analyses. Through years of effort, the KARMA lattice transport code based on the method of characteristics (MOC) has evolved into KARMA2, a direct whole core calculation code using a 3D calculation method based on planar (2D) MOC principles. Simultaneously, ASTRA2, designed as the successor to the ASTRA nodal diffusion code, has been developed. ASTRA2 exhibits enhanced capabilities in multigroup pin-by-pin core calculations, achieved by decoupling the 3D whole core problem into a series of planar and axial problems. The domain size of each axial problem can be adjusted, ranging from pin-cell to assembly scale, thereby optimizing efficiency. The verification and validation process of the KARMA2/ASTRA2 code system involves various benchmark problems and measured data from operational PWRs. In this study, the APR1400 benchmark analysis was performed to verify the neutronics calculation capabilities of both codes. The results underscore the reliability and accuracy of the KARMA2 solutions across various core conditions, exhibiting close agreement with the McCARD reference solutions. Similarly, the ASTRA2 results agree with the corresponding KARMA2 results. These successful results demonstrate the high-fidelity core calculation capabilities of KNF’s next-generation code system.

1. Introduction

Over the past decade, there has been a significant development of first-principle-based direct whole core (DWC) calculation codes, driven by the increasing demand for high-fidelity core analyses and remarkable advancements in computing technologies. Prominent DWC codes in the deterministic transport field include nTRACER [1] from Seoul National University (SNU), MPACT [2] from the University of Michigan, and STREAM [3] from UNIST. In the Monte Carlo field, notable codes include OpenMC [4] from MIT, Shift [5] from ORNL, and McCARD [6] and PRAGMA [7] from SNU. Despite their advantages, the widespread application of these codes is hindered by their considerable computational requirements. Even for small core problems such as the C5G7-TD benchmark [8], executing these codes may take tens of hours on high-performance computing nodes to simulate transient behavior for only a few seconds. Consequently, it is unlikely that DWC codes will replace the conventional two-step calculation (TSC) codes system shown in Figure 1 for commercial core design and analyses.

The conventional TSC method has a significant advantage in terms of lower computing costs. However, it may introduce inherent errors in highly heterogeneous problems owing to its reliance on assembly-wise homogenized few-group constants (few-GCs). To improve the accuracy of conventional TSC while managing computational overhead, advanced codes such as SCOPE2 [9] from Nuclear Fuel Industries, Ltd. (NFI) (Yokohama, Japan), Bamboo2.0 [10] from Xi’an Jiaotong University (XJTU), CORCA-3D [11] from the Nuclear Power Institute of China (NPIC), and SPHINCS [12] and VANGARD [13] from SNU have adopted the pin-by-pin calculation approach, employing pin-wise homogenized multigroup constants (MG-GCs). Utilizing pin-wise equivalence factors (EFs) to correct errors originating from various sources, coupled with pin-wise homogenization, has yielded results close to those obtained with direct whole core (DWC) calculations.

The strategy for the next generation of codes at KEPCO Nuclear Fuel (KNF) was developed by considering the pros and cons of various methodologies. Considering practical constraints such as computing resources and the necessity for high-fidelity reference solutions, the DWC code KARMA2 [14], which is a successor to the lattice transport code KARMA (Kernel Analyzer by Ray-tracing Method for fuel Assembly) [15], was developed for lattice calculations and to generate reference solutions. KARMA2 employs the widely used 2D/1D method [1], which combines planar (2D) transport calculations based on the method of characteristics (MOC) with axial (1D) nodal calculations based on diffusion, simplified P_N, or S_N theory.

Moreover, to enhance the safety and economics of advanced reactors such as the innovative Small Modular Reactor (iSMR) [16], the pin-by-pin core calculation code ASTRA2 [17] has been developed as a successor to the nodal diffusion code ASTRA (Advanced Static and Transient Reactor Analyzer) [18]. ASTRA2 solves the pin-wise diffusion equation with the finite difference method (FDM), and the super homogenization (SPH) factor [19] is applied to correct errors originating from the FDM mesh size. This approach effectively manages the inherent complexity of large 3D problems by decomposing them into a series of planar (2D) and axial (1D) problems. The domain size of each axial problem can be changed from a pin-cell to an assembly [20].

This study presents the results of the APR1400 benchmark [21] analysis performed to demonstrate the neutronics calculation capabilities of KARMA2/ASTRA2, KNF’s next-generation codes system, as shown in Figure 2. The benchmark comprises problems of various scales, ranging from a single fuel pin to the hot-full-power (HFP) core, denoted as APR01 to APR06, complete with detailed core specifications and McCARD Monte Carlo reference solutions provided. Among these, APR04, which is an all-rods-out (ARO) core problem set under various fuel/moderator temperatures and boron concentrations, and APR05, which is a control rod insertion problem set under fixed temperature and boron concentrations conditions, were selected for core calculations using KARMA2 and ASTRA2. The overall results are promising, with KARMA2 results aligning closely with the McCARD reference solutions and ASTRA2 results following those of KARMA2, irrespective of core conditions.

The subsequent section focuses on the core input modeling and calculation options for each code and the pin-wise MG-GC generation procedure. Section 3 elaborates on the accuracy of KARMA2/ASTRA2. Section 4 provides concluding remarks.

2. Numerical Methods

2.1. 2D/1D Decoupling Method

The fundamental ideas of the 2D/1D decoupling method include partitioning the 3D problem into 2D and 1D problems and leveraging parallelization to enhance calculation efficiency in contemporary multicore systems. Analogous to the concept of “separation of variables” in mathematics, the 3D neutron balance equation is separated into groups of 2D and 1D neutron balance equations, as depicted in Figure 3. Each 2D neutron balance equation, as described in Equation (1), can be independently solved, assuming that the axial leakage term is treated as a constant source term. Similar to the 2D problems, the radial leakage term of each pin in Equation (2) is considered a predetermined constant source term. This implies that each 2D planar and 1D pin problem is solved independently and concurrently. Subsequently, the leakage information for each segment is updated for the subsequent predetermined constant.

$${\sum }_{u=x,y}\frac{1}{h_u^m}\left(J_{gur}^m-J_{gul}^m\right)+{\Sigma }_{tg}^m{\varphi }_g^m=\frac{{\chi }_g^m}{k_{eff}}{\sum }_{g^{\prime }=1}^{G}v{\Sigma }_{f{g}^{\prime }}^m{\varphi }_{g^{\prime }}^m+{\sum }_{g^{\prime }=1}^G{\Sigma }_{g^{\prime }g}^m{\varphi }_{g^{\prime }}^m-\frac{1}{h_z^m}\left(J_{gzr}^m-J_{gzl}^m\right)$$

(1)

$$\frac{1}{h_z^m}\left(J_{gzr}^m-J_{gzl}^m\right)+{\Sigma }_{tg}^m{\overline{\varphi }}_g^m=\frac{{\chi }_g^m}{k_{eff}}{\sum }_{g^{\prime }=1}^{G}v{\Sigma }_{f{g}^{\prime }}^m{\overline{\varphi }}_{g^{\prime }}^m+{\sum }_{g^{\prime }=1}^G{\Sigma }_{g^{\prime }g}^m{\overline{\varphi }}_{g^{\prime }}^m-{\sum }_{u=x,y}\frac{1}{h_u^m}\left(J_{gur}^m-J_{gul}^m\right)$$

(2)

where

• $h_u^m$: mesh size of mesh m in u-direction;
• $J_{gur}^m$: g-th group average neutron current at the right-side surface of mesh m in u-direction;
• $J_{gul}^m$: g-th group average neutron current at the left-side surface of mesh m in u-direction;
• ${\varphi }_g^m$: g-th group average neutron flux of mesh m;
• ${\Sigma }_{tg}^m$: g-th group total XS of mesh m;
• $v{\Sigma }_{fg}^m$: g-th group nu-fission XS of mesh m;
• ${\Sigma }_{g^{\prime }g}^m$: scattering XS from group g’ to g of mesh m;
• ${\chi }_g^m$: g-th group fission spectrum of mesh m;
• $k_{eff}$: effective multiplication factor;
• $G$: number of neutron groups.

Figure 3. 2D/1D decoupling scheme.

Figure 3. 2D/1D decoupling scheme.

The simplest FDM was employed to solve 2D and 1D problems using the 2D pin-wise finite difference method (2D FDM) and a 1D finite difference approach with a fine mesh of ~1 cm (1D FDM). To address errors originating from the use of pin-cell size meshes, the SPH factor was employed for compensation. For a detailed application of the decoupling calculation, refer to previous research [17].

In contrast to an earlier study [17], the linear system was solved using the biconjugate gradient stabilized method (BiCGSTAB), with the system matrix preconditioned by the zero-fill-in incomplete LU decomposition method (iLU0). Given the dominant influence of pin-wise leakage effects, the previous method employing the conjugate gradient technique faced convergence issues in pin-wise analysis. Furthermore, the block incomplete LU (BiLU) preconditioning method used previously is not optimal for multigroup pin-wise core analysis because it requires additional inverse matrix calculations with increasing node count.

2.2. Modified 2D/1D Decoupling Method

The 2D/1D decoupling method has shown satisfactory accuracy and computational efficiency for practical application in commercial PWR nuclear design. Hereinafter, this method will be referred to simply as “2D/1D”. However, it is not fast enough and becomes burdensome when applied to reactor designs for commercial PWR-sized reactors, particularly for dynamic control rod worth measurement analyses and “N − 1” rod ejection analyses.

In the modified 2D/1D decoupling method, we focused on addressing the large number of 1D problems. For example, in the quarter and full core of APR1400, there were 17,664 and 61,696 1D fuel pin problems, respectively. To reduce the computational load associated with these 1D pin problems, an axial 1D pin-wise problem was replaced with an axial 1D box-wise problem (2 × 2 boxes in one FA), as illustrated in Figure 4. This approach was introduced in both steady-state and transient pin-wise calculations [12,20] to optimize computational efficiency while maintaining practical accuracy. This was based on the observation that neighboring pins exhibited a similar degree of axial leakage. Although dozens of pins in one box had similar axial leakage, two additional calculations were required to maintain box-wise balance. This entailed box-wise homogenization and box-wise surface leakage summation, as shown in Equations (3) and (4), respectively.

$${\overline{\varphi }}_g^{j,k}=\frac{{\sum }_{i\in j}{\varphi }_g^{i,k}{V}_{i,k}}{{\sum }_{i\in j}{V}_{i,k}},{\overline{\mathsf{\Sigma }}}_{xg}^{j,k}=\frac{{\sum }_{i\in j}{\mathsf{\Sigma }}_{xg}^{i,k}{\varphi }_g^{i,k}{V}_i}{{\sum }_{i\in j}{\varphi }_g^i{V}_i}$$

(3)

where

• $i$: radial pin index;
• $j$: radial node index;
• k: plane index;
• ${\mathsf{\Sigma }}_{xg}^i$: g-th group x-type XS of pin i;
• ${\varphi }_g^i$: g-th group average neutron flux of pin i;
• $V_i$: volume of pin i;
• ${\overline{\varphi }}_g^j$: g-th group average neutron flux of node j;
• ${\overline{\mathsf{\Sigma }}}_{xg}^j$: g-th group x-type homogenized XS of node j.

$$J_{g,z}^{kb}=\frac{1}{A_z^{kb}}{\sum }_{kp\in kb}{J}_{g,z}^{kp}{A}_z^{kp}$$

(4)

where

• $kp$: pin-wise axial surface index;
• $kb$: box-wise axial surface index;
• kp(l or r): pin-wise left of right node index in axial direction;
• kb(l or r): box-wise left of right node index in axial direction;
• $A_z^{kp(or kb)}$: area of axial pin surface kp or box surface kb.

This adjustment significantly decreased the number of 1D problems. For example, in the APR1400 quarter and full cores, the number of 1D fuel pin problems was reduced from 17,664 and 61,696 to 276 and 964 1D box problems, respectively. Hereinafter, this modified 2D/1D decoupling method will be referred to as “2D1DBOX”. Further insights into the computational performance of both methods are provided in Section 3.

Figure 4. Pin-wise 1D FDM (2D1D) vs. box-wise 1D FDM (2D1DBOX).

Figure 4. Pin-wise 1D FDM (2D1D) vs. box-wise 1D FDM (2D1DBOX).

3. Numerical Results

3.1. Modeling for APR1400 Benchmark Problems

The Korea Atomic Energy Research Institute has released an APR1400 benchmark problem book [21]. This benchmark problem loads the 241 CE (combustion engineering)-type 16 × 16 fuel assemblies and has a total thermal power output of 3983 MW. These assemblies are categorized into A, B, and C types, with the B and C types further divided into four variants (indexed 0–3) based on radial enrichment zoning patterns and the number of Gadolinia burnable absorber pins. Figure 5 shows the loading pattern and the locations of control rods, including two shutdown banks (A and B) and five regulating banks (R1 to R5). The shutdown banks and some regulating banks are equipped with 12-finger type control rods, while others have 4-finger type control rods.

The benchmark sets consist of six categories: (1) single fuel pin problems (APR01), (2) 2D fuel assembly problems (APR02), (3) 2D core problems (APR03), (4) 3D core problems (APR04), (5) control rod worth problems (APR05), and (6) 3D core depletion problem (APR06).

Table 1. Nine conditions of the APR1400 benchmark [21] (APR02–APR04).

Soluble Boron Concentration	CZP *	HZP **	HFP ***
0 ppm	V01	V02	V03
1000 ppm	V04	V05	V06
2000 ppm	V07	V08	V09

CZP * (cold zero power) = 300 K (moderator temperature), 300 K (fuel temperature) HZP ** (hot zero power) = 600 K (moderator temperature), 600 K (fuel temperature), HFP *** (hot full power) = 600 K (moderator temperature), 900 K (fuel temperature).

summarizes the nine conditions of the APR02 to APR04 benchmark problems, while

Table 2. Seven control rod conditions of APR1400 benchmark [21] (APR05).

Variants	Inserted CEA Group
APR04V02	ARO
APR05V01	R5
APR05V02	R5 + R4
APR05V03	R5 + R4 + R3
APR05V04	R5 + R4 + R3 + R2
APR05V05	R5 + R4 + R3 + R2 + R1
APR05V06	R5 + R4 + R3 + R2 + R1 + B
APR05V07	R5 + R4 + R3 + R2 + R1 + B + A

details the control rod conditions for APR05. Unfortunately, the benchmark book does not include depletion results for APR06.

In this study, we focused on evaluating the computational efficiency and accuracy of the proposed modified 2D/1D decoupling method (2D1DBOX) through analysis of the 3D core problem (APR04) and the control rod worth problem (APR05). Four-group pin-wise homogenized group constants (GCs) were generated via conventional 2D lattice calculations performed by KARMA, supplemented by corresponding SPH iterations. For the reflector regions, pin-wise homogenized GCs were determined following the methodology outlined in previous work [17].

While the APR1400 benchmark book provides reference solutions obtained using McCARD [21], it lacks pin-wise information. Therefore, in this study, solutions derived from 3D calculations using KARMA were considered as the reference solutions. The model adhered to the specifications provided in the benchmark, with each pin-cell modeled to include all components, such as the pellet, air gap, cladding, and moderator. The fuel pellet was subdivided into 40 flat source regions (FSRs) by 5 annular rings and 8 azimuthal sections, and the number of annular rings was increased to 8 for the burnable poison pellet. Similarly, the moderator region was subdivided into 32 FSRs by 4 annular rings and 8 azimuthal sections. This FSR subdivision ensured accurate consideration of sub-pin level flux variations.

Each fuel assembly comprised 3 bottom reflector planes, 30 active fuel planes, and 3 top reflector planes. The active fuel planes were loaded with detailed models for fuel, burnable poison, and guide and instrument tube pin-cells. Additionally, the water gap was explicitly modeled, while the nine spacer grids were semi-explicitly modeled by smearing ZIRLO in the moderator. In the axial reflector planes, the fuel bottom and top ends and the guide and instrument tubes were explicitly modeled, while other structures, such as the core plate, bottom and top nozzles, etc., were smeared in the moderator.

The core loaded with 241 fuel assemblies was surrounded by an assembly-thick water reflector with an explicitly modeled stainless steel shroud. However, other structures, such as the core vessel and neutron pads, were not included in the model.

For a fully consistent code-to-code comparison, the KARMA2 core model was used to generate pin-homogenized four-group cross sections (XSs). Notably, the lower energy boundaries of the four-group were established at 9.119 × 10³, 3.928, 6.251 × 10⁻¹, and 1.000 × 10⁻⁴ eV, determined as optimal in previous studies [12,17]. KARMA2 calculations were performed using a 47-group (47G) transport-corrected P0 (TCP0) XS library based on ENDF/B-VII.1. The ray parameter set featured 0.02 cm ray spacing with 16 azimuthal and 3 polar angles per octant sphere and was used for the 3D core calculations.

The pin-wise XSs and 4G fluxes for fuel assemblies were obtained from 2D single assemblies with all reflective boundary conditions. Meanwhile, the XSs and fluxes for radial and axial reflectors were derived from fuel-reflector local problems. Pin-wise SPH factors were determined using the standard procedure, using the corresponding coarse group XSs and fluxes.

3.2. Computing Performance of 2D/1D Decoupling Methods

Table 3. Time required for each calculation module for both decoupling methods (single thread).

Item	2D1D		2D1DBOX
	Computing Time (s)	Portion (%)	Computing Time (s)	Portion (%)
1D FDM	43.73	64.89	1.07	4.03
2D FDM	23.43	34.77	25.23	94.92
3D CMFD	0.23	0.34	0.28	1.05
Total	67.39	100.00	26.58	100.00

outlines the required calculation time for each module of the method. For the test problem, an un-rodded APR1400 core (APR04V05) was selected. Notably, the 2D1DBOX method significantly reduced computation time, particularly in the 1D FDM module, which constituted the largest segment (approximately 64%) of the 2D1D method. Despite the additional iterations required in the 2D FDM and 3D CMFD modules, the increase in computation time was negligible compared to the significant time reduction achieved in the 1D FDM module.

To evaluate the parallel performance of both methods, the APR04V05 problem was solved using various numbers of threads. The calculations were parallelized using the OpenMP platform and executed on a Linux system featuring an Intel^® Xeon^® Platinum 9242 processor (Intel, Santa Clara, CA, USA). Figure 6 shows the total execution time and speed-up factor against the number of threads. Speedup is defined as ratio of single-thread calculation time to multi-thread calculation time to represent the performance of multi-threading. With a single thread, the total execution time of the 2D1D method was approximately 86.2 s, reduced to 6.6 s with 32 threads. It is worth noting that the execution time of the current 2D1D-based pin-by-pin core calculation was significantly shorter than in previous studies [17]. Furthermore, the execution time was further reduced with the proposed 2D1DBOX method, requiring approximately 39.8 s and 3.6 s with a single thread and 32 threads, respectively. With 32 threads, the 2D1DBOX method saved approximately 45% of computing time. However, the speed-up factor of the 2D1DBOX method was lower than that of the 2D1D method owing to the smaller number of 1D FDM problems.

As mentioned in the previous section, the proposed 2D1DBOX method was based on the observation of box (or node)-wise similarity in axial leakage. Therefore, it was imperative to validate the accuracy of the 2D1DBOX method when applied to axially heterogeneous problems, such as rodded cases.

Table 4. Discrepancies in major design parameters in APR04V05.

Item	2D1D	2D1DBOX	Error (%)
k-eff	1.01681	1.01683	0.002
Fr	1.4594	1.4589	−0.034
Fxy	1.5089	1.5082	−0.046
Fq	2.2099	2.2075	−0.109
Max. FA power	1.2227	1.2224	−0.025
RMS value of relative pin-power error (%) *			0.029

Where RMS * is defined as follows: $RMS=\sqrt{\frac{\sum {\left(Er{r}_{pow}\right)}^2}{N}}$, $Er{r}_{pow}=\frac{po{w}_{2D1DBOX}-po{w}_{2D1D}}{po{w}_{2D1D}}\times 100$.

and

Table 5. Discrepancies in major design parameters in APR05V05.

Item	2D1D	2D1DBOX	Error (%]
k-eff	1.08088	1.08092	0.004
Fr	1.8364	1.8369	0.027
Fxy	1.8473	1.8481	0.043
Fq	2.7846	2.7828	−0.064
Max. FA power	1.4747	1.4752	0.034
RMS value of relative pin-power error (%) *			0.037

Where RMS * is defined as follows: $RMS=\sqrt{\frac{\sum {\left(Er{r}_{pow}\right)}^2}{N}}$, $Er{r}_{pow}=\frac{po{w}_{2D1DBOX}-po{w}_{2D1D}}{po{w}_{2D1D}}\times 100$.

present the discrepancies in the major design parameters between the two methods for the rodded problem of APR05V05. Fr and Fq are an axially integrated radial peaking factor and a 3D peaking factor, respectively. Fxy is maximum value of planar radial peaking factor, Fxy(z). It is worth noting that the difference between the two methods was negligible in terms of nuclear design accuracy.

3.3. Accuracy of Pin-by-Pin Core Calculation

Table 6. Numerical results for APR04 variants.

Condition	$\mathbf{\Delta }$keff (pcm)	FA Power		Pin Power
		Max. (%]	RMS (%]	Max. (%)	RMS (%)
V01	−37	−2.02	0.94	−7.52	1.27
V02	−26	1.48	0.74	−2.43	0.84
V03	−25	1.49	0.75	−2.40	0.84
V04	−34	−2.92	1.19	−8.08	1.51
V05	2	−1.50	0.77	−3.16	0.86
V06	2	−1.51	0.78	−3.12	0.86
V07	−42	−3.15	1.38	−8.29	1.67
V08	12	−1.44	0.72	−3.76	0.83
V09	12	−1.47	0.73	−3.77	0.83

presents the pin-wise calculation results of the APR04 benchmark problems compared to the reference solution obtained with KARMA. Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12, Figure 13, Figure 14 and Figure 15 show each variant’s reference normalized 2D pin-power distribution and the relative pin-power error distribution. The normalized 2D pin-power distribution was axially integrated pin-power distribution and normalized by a factor which made the average ‘one’. The eigenvalue differences across all variants were below 50 pcm, and the error in fuel assembly (FA) power was smaller than the typical value obtained through conventional nodal analysis. Regarding the relative pin-power error, maximum values were below 4%, and the root mean square (RMS) values were below 1%, except for the cold zero power (CZP) conditions (V01, V04, V07). The maximum values of all variants were located at pins which had relatively small pin power, around 0.2~0.3. Therefore, the actual effect of maximum pin power error was not much.

Table 7. Detailed pin information for APR04 variants.

Condition	Radial Peaking Factor		Effective Max. Pin-Power Error (%)
	Value	Error (%)
V01	1.700	−1.17	−4.66
V02	2.023	−0.94	−1.51
V03	2.019	−0.94	−1.51
V04	2.573	−2.07	−6.12
V05	1.459	−1.16	−1.87
V06	1.454	−1.17	−1.89
V07	2.852	−2.25	−6.33
V08	1.571	−0.33	−1.99
V09	1.577	−0.33	−1.99

shows additional analyses of the pin-power distribution. First, the pin-power error of the radial peak pin was acceptably small. Second, the maximum pin-power error was evaluated again using only pin-power values greater than average value (1.0). This was to obtain more meaningful pin-power error information by excluding practically meaningless data. Hereinafter, this pin-power error will be referred to simply as “effective max. pin-power error”. As expected, the effective max. pin-power errors were below 2%, except for the CZP conditions.

Based on the results presented in Table 6 and Table 7, discrepancies in the CZP conditions slightly surpassed those in hot zero power (HZP) and HFP conditions. This discrepancy can be attributed to the “outward” power distribution observed in the CZP conditions, as shown in Figure 7, Figure 10 and Figure 13.

Similar to conventional two-step procedures, the pin-wise core analysis in this study also exhibited an inherent limitation, leading to discrepancies when the pin-wise GCs were not applied in areas where they were originally generated. Disagreements arose in fuel assemblies that faced the baffle-reflector or a different type of FA because the pin-wise GCs were generated in a FA problem with all-reflective boundary conditions. Additionally, in the CZP condition, the peripheral fuel assemblies had relatively high power (“outward” power distribution) and steeper gradients in the pin-power distribution compared to other conditions, as illustrated in Figure 16. Despite resulting in relatively high effective max. pin-power errors (4–6%) in the peripheral fuel regions, they were not important owing to significantly small pin powers in the cold “zero power” condition.

Figure 17 shows the axial power distribution and relative axial power error distribution. This axial power distribution was determined by radially integrating power distribution of each layer. The pin-by-pin solution demonstrated good agreement with the reference KARMA solution, with most axial power errors below 0.8%.

Table 8. Numerical results for APR05 variants.

Condition	$\mathbf{\Delta }$keff (pcm)	FA Power		Pin Power
		Max. (%]	RMS (%]	Max. (%)	RMS (%)
V01	−25	1.66	0.83	2.91	0.92
V02	−31	1.46	0.66	3.10	0.78
V03	−25	−2.12	1.16	3.43	1.20
V04	−19	−2.70	0.86	3.16	0.93
V05	9	−3.52	1.31	−3.96	1.30
V06	−10	−1.06	0.54	−3.07	0.77
V07	7	−2.11	0.94	−3.98	1.05

presents the pin-wise calculation results of the rodded benchmark problems. The eigenvalue differences across all variants were below 31 pcm. Compared to conventional nodal analysis, the pin-by-pin calculation showed better agreement in FA power error, with errors smaller than typical values obtained through conventional nodal analysis. The maximum and RMS values of the relative pin-power error were below 4% and 1.3%, respectively. Upon inserting all control rods, including the two shutdown banks, the pin-power errors marginally increased compared to the un-rodded problem (APR04V02). The maximum value shifted from −2.43% to −3.98%, and the RMS value changed from 0.84% to 1.05%. The maximum value of V07, −3.98%, was located at a fuel pin just next to the R3 control bank. It is noted that the corresponding fuel pin was in the B0 type FA, which was loaded in a peripheral region of the APR1400 core, and the normalized pin-power of corresponding fuel pin was around 0.36.

Table 9. Detailed pin information for APR05 variants.

Condition	Radial Peaking Factor		Effective Max. Pin-Power Error (%)
	Value	Error (%)
V01	1.860	−1.14	1.96
V02	2.096	−0.86	−1.42
V03	1.643	−1.21	−2.61
V04	1.892	−0.94	−1.89
V05	1.837	1.13	−2.44
V06	3.599	−0.67	2.27
V07	3.545	0.11	3.08

shows that the pin-power errors of the radial peak pin ranged from 1% to 2% or below. The effective max. pin-power errors were around 3% or below. This accuracy is much better than that of the conventional nodal code.

Table 10. Control rod worth in APR05 benchmark problems.

Condition	Rod Worth (KARMA)		Rod Worth Error (%)
	Cumulative	Group	Cumulative	Group
V01	365.35	365.35	−0.17	−0.17
V02	685.98	320.62	0.62	1.51
V03	1685.20	999.23	0.00	−0.43
V04	2724.46	1039.26	−0.16	−0.43
V05	4723.63	1999.17	−0.59	−1.16
V06	8842.91	4119.28	−0.12	0.41
V07	15964.57	7121.66	−0.17	−0.24

presents the control rod worth for both cumulative and group-wise banks. The cumulative control rod worth represents the total control rod worth of the rodded banks. In contrast, the group-wise control rod worth is the individual group control rod worth under the control rod insertion sequence detailed in Table 2. For the pin-by-pin calculations, the error range of the cumulative and group control rod worth was between −0.59% and 0.62% and between −1.16% and 1.51%, respectively.

Figure 18 illustrates the axial power distribution and axial power error distribution of a representative rodded benchmark problem. Similar to the un-rodded case, the pin-by-pin solution demonstrated excellent agreement, with errors below 1.0%.

4. Conclusions

In this paper, we introduce the modified 2D/1D decoupling method to enhance the performance of pin-by-pin core analysis. In the 2D1DBOX method, the number of 1D FDM problems, known as the most time-consuming calculation module, was significantly reduced by replacing dozens of pin-wise 1D FDM problems with a single box-wise 1D FDM problem. Although additional calculations, such as homogenization and summation of determining the box representation values, were necessary to maintain neutron balance, these tasks were negligible compared to the resulting time savings. Additionally, we optimized the linear solver and preconditioning method (BiCGSTAB with iLU0) to enhance stability.

In terms of computing performance, the 2D1DBOX method required 39.8 and 3.6 s for single-threaded and 32-threaded computations, respectively, saving approximately 60% and 45% of the computing time compared to the 2D1D method. In contrast, the 2D1D method required 86.2 and 6.6 s for single-threaded and 32-threaded computations, respectively. Regarding solution accuracy, the 2D1DBOX method demonstrated no noticeable discrepancies compared to the original 2D1D method for both all-rods-out (ARO) and all-rods-in (ARI) cases.

We solved the 3D un-rodded APR1400 (APR04) and rodded APR1400 (APR05) benchmark problems, comparing the results with those derived from DWC transport calculations. The discrepancies in eigenvalues were less than 50 pcm for both rodded and un-rodded problems, while conventional nodal calculations had differences of approximately 100 pcm. Across all benchmark problems, except under control-rod-withdrawn CZP conditions, the maximum and RMS pin-power errors were less than 4.0% and 1.7%, respectively. The maximum pin-power error usually occurred at the core peripheral fuel regions or fuel pin near the control rods. These pins have relatively low pin-power around 0.1 to 0.3 and the actual effect is negligible. The effective max. pin-power errors ranged from 3% (for ARI case) or below.

In the CZP condition, the relatively high pin-power error (~8%) was caused by composite conditions, such as outward power distribution, and the inherent limitations of the two-step procedure. However, the corresponding pin powers were minimal, rendering their actual effects negligible in terms of reactor design.

Regarding control rod worth, the maximum errors for cumulative and group-wise control rods were less than 1.5% and 0.6%, respectively. Therefore, we are confident that the performance of the suggested 2D1DBOX method has been demonstrated in terms of both accuracy and computing efficiency.