Preamble

Quantitative Analysis of the Impact of Nuclide Bias Partitioning Strategies on Criticality Calculation Uncertainty Under Burnup Credit

Abstract: Burnup credit (BUC) enhances the realism of spent fuel criticality safety assessments by accounting for changes in isotopic composition during the depletion process. However, uncertainties arising from sample biases in isotopic concentrations can significantly impact the accuracy of criticality calculations. To quantify these uncertainties across different burnup levels and improve the precision of burnup credit analysis, this study establishes an isotopic bias partitioning strategy based on linear regression fitting and Mean Square Error (MSE) minimization. This strategy is designed to determine the optimal burnup interval divisions for target isotopes. Using the AFA-3G fuel assembly as a benchmark problem, the SCALE platform—integrated with the ORIGEN module—was employed to evaluate the propagation of isotopic bias uncertainties to criticality results via Monte Carlo sampling methods. The results demonstrate that this strategy effectively reduces the uncertainty in criticality calculations under low-burnup conditions, providing an optimized methodology for the criticality safety evaluation of spent fuel under the burnup credit framework.

1. Introduction

Burnup Credit (BUC) refers to the methodology of accounting for the reduction in reactivity of nuclear fuel due to its utilization in the reactor. Compared to the traditional "fresh fuel" assumption, BUC allows for higher storage densities and more efficient transport of spent nuclear fuel. However, implementing BUC requires rigorous quantification of uncertainties, particularly those arising from the prediction of isotopic concentrations during depletion.

A critical component of BUC analysis is the treatment of nuclide biases—the systematic differences between calculated and experimental nuclide concentrations. These biases are often handled through partitioning strategies, where the fuel assembly or the entire spent fuel pool is divided into zones based on burnup, enrichment, or spectral indices. The choice of partitioning strategy directly influences the calculated $k_{eff}$ and the required subcritical safety margin.

2. Methodology

2.1 Criticality Calculation Framework

The criticality calculations in this study are performed using high-fidelity Monte Carlo codes. The effective multiplication factor is defined as:

$$k_{eff} = \frac{\text{Production Rate}}{\text{Absorption Rate} + \text{Leakage Rate}}$$

To account for nuclide biases, the concentration of each important isotope $i$ is adjusted using a bias factor $\beta_i$ and an associated uncertainty $\sigma_i$. The adjusted concentration $N_i'$ is given by:

$$N_i' = N_{i,calc} \cdot (1 + \beta_i \pm k \cdot \sigma_i)$$

where $N_{i,calc}$ is the concentration predicted by the depletion code, and $k$ is the coverage factor.

2.2 Nuclide Bias Partitioning Strategies

We evaluate three primary partitioning strategies:

摘要

Abstract

Burnup credit (BUC) enhances the realism of spent fuel criticality safety assessments by accounting for changes in isotopic composition during the depletion process. However, uncertainties arising from sample biases in isotopic concentrations can significantly impact the accuracy of criticality calculations. To quantify these uncertainties across different burnup levels and improve the precision of burnup credit analysis, this study establishes an isotopic bias partitioning strategy based on linear regression fitting and Mean Square Error (MSE) minimization. This strategy is designed to determine the optimal burnup interval divisions for target isotopes. Using the AFA-3G fuel assembly as a benchmark problem, the SCALE platform—integrated with the ORIGEN module—was employed to evaluate the propagation of isotopic bias uncertainties to criticality results via Monte Carlo sampling methods. The results demonstrate that this strategy effectively reduces the uncertainty in criticality calculations under low-burnup conditions, providing an optimized methodology for the criticality safety evaluation of spent fuel under the burnup credit framework.

关键词

Burnup Credit, Monte Carlo Methods, and Criticality Safety Analysis

CLC Number: TL329
Document Code: A

Abstract

This paper explores the application of burnup credit (BUC) methodologies within the framework of criticality safety analysis, specifically utilizing Monte Carlo simulation techniques. Burnup credit refers to the accounting of the reduction in reactivity of nuclear fuel as it undergoes irradiation in a reactor. By incorporating the isotopic changes—specifically the depletion of fissile nuclides and the buildup of neutron-absorbing fission products and actinides—criticality safety assessments can be performed with greater accuracy and reduced conservatism compared to the traditional "fresh fuel" assumption.

1. Introduction

In the field of nuclear fuel cycle management, ensuring criticality safety during the storage, transportation, and reprocessing of spent nuclear fuel is of paramount importance. Traditionally, criticality safety analyses have relied on the "fresh fuel" assumption, which assumes that the fuel is at its maximum potential reactivity. While this approach provides a significant safety margin, it often leads to overly restrictive operational limits, such as reduced storage density or increased requirements for fixed neutron absorbers.

The concept of burnup credit (BUC) has emerged as a robust solution to these limitations. By accounting for the actual physical state of the spent fuel, BUC allows for a more realistic determination of the effective multiplication factor ($k_{eff}$). However, implementing BUC requires sophisticated computational tools capable of handling complex geometries and precise isotopic compositions. Monte Carlo methods have become the gold standard for these calculations due to their ability to model three-dimensional systems without the energy-group approximations inherent in deterministic methods.

2. Methodology

2.1 Burnup Credit Principles

The implementation of burnup credit involves two primary components: the depletion calculation and the criticality calculation. During the depletion phase, the evolution of the fuel composition is tracked over time, accounting for various reactor operating conditions such as power density, moderator temperature, and boron concentration. The resulting isotopic inventory typically includes major actinides (e.g., $\text{U}^{235}$, $\text{Pu}^{239}$) and, in higher-level BUC applications, stable fission products with significant neutron capture cross-sections (e.g., $\text{Sm}^{149}$, $\text{Rh}^{103}$, $\text{Gd}^{155}$).

2.2 Monte Carlo Simulation for Criticality Safety

Monte Carlo methods are employed to solve the neutron transport equation

1 引言

With the development of the nuclear power industry, the consumption of nuclear fuel in China's nuclear power plants has increased annually. It is estimated that by 2025, more than 1,000 tons of spent fuel will be generated, posing new challenges to the spent fuel reprocessing workflow \cite{1}. Compared to traditional criticality safety analysis methods, the Burnup Credit (BUC) approach accounts for the burnup effects of spent fuel during reactor operation to lower the estimated effective multiplication factor ($k_{eff}$) in criticality safety analyses. This reduces overly conservative safety margins and enhances both the economic efficiency and safety of spent fuel storage and transport \cite{2}. Burnup credit can significantly reduce the space required for spent fuel storage and transport as well as operational costs, demonstrating broad application prospects both domestically and internationally.

The use of burnup and criticality calculation codes in burnup credit analysis introduces various sources of uncertainty. In simulating the evolution of nuclide inventories in spent fuel, discrepancies between the nuclear databases and numerical models used in burnup codes and actual physical conditions introduce additional uncertainty. Furthermore, approximations in the treatment of geometric models and cross-sections in criticality calculation codes also contribute to uncertainties in $k_{eff}$ results. Applicability assessments of burnup codes reveal that calculation accuracy varies across different burnup depths; the limited accuracy of burnup models and cross-section libraries leads to varying degrees of bias between calculated nuclide concentrations and chemical experimental measurements. Significant progress has been made in this area through domestic and international research. Based on experimental measurements and simulations, the U.S. Nuclear Regulatory Commission (NRC) has compiled sample values of nuclide biases at different burnup depths, providing an important reference for correcting burnup calculations \cite{3}. Oak Ridge National Laboratory (ORNL) conducted burnup credit analyses for spent fuel storage facilities at various burnup levels, quantifying the impact of burnup depth on criticality results and their associated uncertainties \cite{4}. The China Institute of Atomic Energy (CIAE) performed theoretical analyses on nuclide inventories using experimental chemical analysis data from the TAKAHAMA-3 PWR burnup benchmark, determining the trends of calculated versus experimental values relative to burnup depth and correcting the experimental values \cite{5}. Previous studies have evaluated the degree of nuclide bias by partitioning burnup depth samples; however, the sample data used for such partitioning are inconsistent across studies, and the methodology for defining these intervals still lacks a rigorous scientific basis.

Building upon existing burnup credit analysis methods, this paper further evaluates the influence of selecting boundary points for burnup depth intervals on criticality calculation results and their uncertainties. A burnup partitioning selection strategy based on linear regression analysis is proposed. In this study, the burnup credit application level is set to APU-1. First, a piecewise function is established based on the mean nuclide bias samples within each sub-interval to evaluate the relationship between nuclide bias/uncertainty and sample burnup. Subsequently, the optimal burnup partitioning strategy is selected by evaluating the goodness-of-fit between the piecewise function and the sample points of the six actinide nuclide biases of interest under the APU-1 level. The study establishes a fuel assembly analysis model based on burnup and criticality calculation codes, utilizing a nuclide concentration sampling method to quantify the uncertainty propagated from burnup calculations to criticality calculations. By performing burnup credit analysis on benchmark cases under different partitioning strategies, this work provides theoretical support for the burnup partitioning of nuclide biases.

2.1 燃耗计算核素偏差

Burnup credit (BUC) is an approach used in the criticality safety analysis of spent nuclear fuel. By accounting for the actual isotopic composition of the fuel based on its irradiation history, BUC reduces the excessive conservatism inherent in traditional methods that assume the fuel is fresh (unirradiated). To maximize the economic efficiency of spent fuel storage and transport systems while ensuring safety, it is essential to evaluate the various uncertainties that affect the criticality calculation results within the framework of burnup credit analysis.

The uncertainty in criticality calculations primarily stems from several sources: statistical or convergence uncertainty inherent in the criticality calculation itself; uncertainties arising from manufacturing tolerances; uncertainties introduced by geometric or material approximations within the model; biases and uncertainties in criticality results caused by uncertainties in the isotopic composition derived from burnup calculations; uncertainties in the recorded burnup values; and uncertainties associated with the specific computational methods and nuclear cross-section libraries employed.

Regarding burnup calculations, burnup credit analysis typically focuses on isotopes that significantly impact criticality, such as major actinides and specific fission products. Different nuclear data libraries or cross-section libraries introduce varying degrees of uncertainty into both burnup and criticality calculations. Furthermore, commonly used burnup codes are subject to biases caused by modeling approximations and the precision of cross-section libraries when simulating fuel irradiation history and isotopic evolution. Consequently, it is necessary to introduce isotopic correction factors to account for these discrepancies. The isotopic bias is generally defined as the ratio of the measured isotopic concentration to the calculated isotopic concentration (i.e., $C/E$ or $E/C$ ratios).

M X C = (1)

Let $\delta_{i,j}$ represent a specific nuclide, where $j$ denotes a particular fuel sample. For a given nuclide $i$, the set of samples ${\delta_{i,j}}$ represents the deviations of that nuclide under different burnup conditions or across various measured samples. The sample mean and standard deviation of the nuclide deviation are defined as:

where $N$ is the number of evaluated fuel samples. $\bar{\delta}_i$ represents the average deviation of the nuclide within a specific sample set, while $s_i$ measures the sample dispersion for use in subsequent uncertainty analysis.

Due to the limited number of samples available for each nuclide, tolerance intervals must be employed to estimate the confidence range of the nuclide deviation. If the sample size is $N$, and assuming the data follows a normal distribution when the sample size is sufficient, it is necessary to ensure a confidence level of $\gamma$.

Accordingly, a two-sided tolerance limit factor $k$ can be introduced to adjust the mean and standard deviation, thereby obtaining more reliable upper and lower estimates for the nuclide deviation. The tolerance interval for the mean nuclide deviation is defined as:

The tolerance coefficient $k$ is calculated based on small-sample statistics, the required coverage $P$, and the confidence level $\gamma$. After accounting for the tolerance factor, the corrected standard deviation can be expressed as:

b g s = (5)

This uncertainty, referred to as the nuclide bias, is used as an input parameter when sampling nuclide concentrations during criticality calculations.

2.2 核素偏差随燃耗深度的变化关系

Establishing a nuclide bias sample set requires the support of existing spent fuel composition measurement data and burnup code calculation results. The resulting nuclide bias samples are widely distributed across various burnup levels. Notably, the nuclide bias and its associated uncertainty are closely related to the specific burnup of the sample. As burnup increases, the fuel undergoes longer neutron irradiation within the reactor, leading to more complex nuclear reactions such as fission and absorption. Furthermore, burnup calculation codes may neglect certain secondary burnup chains, which subsequently impacts the bias of different nuclide compositions.

According to the linear regression statistical plots of nuclide composition calculation bias, the calculated burnup values are generally higher than the experimental measurements under high burnup conditions. This observation is consistent with the conservative assumptions typically employed in criticality safety analysis. In the low burnup stage, both the calculation bias and the bias uncertainty of the nuclide composition are relatively small; however, both increase significantly in the high burnup stage. Consequently, a uniform calculation bias and uncertainty should not be applied across all burnup levels for evaluation purposes. Currently, the number of samples available in experimental benchmark databases, both domestically and internationally, is insufficient to support reliable statistical analysis of calculation bias and uncertainty for every individual burnup point. To address the discrepancies in bias and uncertainty across different burnup levels, this study introduces a partitioned processing method. By reasonably dividing burnup intervals, it is possible to accurately characterize samples at different burnup depths.

分析

Considering the variation patterns of nuclide bias and uncertainty during the burnup evolution process, and given the limited number of statistical samples, a segmented processing approach was implemented in this report to balance physical representativeness with statistical reliability. Specifically, the burnup depth is divided into three distinct intervals: low, medium, and high.

This partitioning method effectively reflects the statistical characteristics of nuclide bias across different burnup ranges while ensuring that each interval contains a sufficient number of samples. Based on these burnup depth ranges, the relationship between nuclide bias and burnup is modeled as a piecewise constant function within each respective interval.

ì £ ï ï ï ï = < £ í - ï ï ï > - ï î

$BU_{1}$ and $BU_{2}$ represent the boundary values for burnup depth used to divide the burnup range into three distinct stages: low, medium, and high. Here, $N$ denotes the total number of nuclide bias samples, $n_{1}$ represents the index of the sample with the highest burnup depth within the low burnup interval, and $n_{2}$ represents the index of the sample with the highest burnup depth within the medium burnup interval.

After partitioning the intervals, the mean and standard deviation of the nuclide composition bias are statistically calculated for each nuclide within its respective burnup interval based on Eq. (1). This process yields the distribution characteristics of the bias for various nuclides across different burnup stages. The burnup partitioning method determines the appropriate burnup interval for different nuclides within a given sample based on that sample's specific burnup depth. Subsequently, the mean and standard deviation of the nuclide bias corresponding to that interval are selected. These statistical parameters are then combined with the nuclide concentrations calculated by the burnup code to serve as inputs for Monte Carlo random sampling. This sampling generates nuclide concentration samples used in criticality calculations to quantify how the uncertainty of nuclide composition in criticality safety analysis propagates to the uncertainty of the final results across different burnup depths. Traditional burnup interval partitioning methods often rely on industry conventions, applying a uniform interval division to all nuclides of interest. Such approaches lack a theoretical basis and fail to provide statistical optimization tailored to the specific bias distributions of different nuclides, making it difficult to account for the inconsistent bias trends exhibited by various nuclides as burnup depth changes.

2.3 核素偏差分区

To address the characteristic variations in deviations for different nuclides across burnup depths, it is necessary to partition burnup intervals individually. This allows for a more precise quantification of their impact on the uncertainty of criticality calculations. Consequently, this study proposes a partitioning strategy based on the specificity of nuclide deviation samples. By analyzing the relationship between nuclide deviations and burnup depth, optimal boundary points are determined to implement a complete workflow—ranging from partitioned data processing to burnup credit calculations.

The optimal boundary points for the burnup partitions of each nuclide are determined by evaluating the Mean Square Error (MSE) between the nuclide deviation sample points and the fitting function at those boundaries. In this study, it is assumed that the deviation samples for each nuclide follow a piecewise constant distribution within the burnup depth intervals, as shown in Equation (6). The optimal boundary points are identified by minimizing the MSE between the sample deviations and the mean values within each respective interval.

$$
\text{MSE}(\tau, \dots) = \dots
$$

= Î = - åå (7)

The calculation of the Mean Square Error (MSE) is performed according to the formula shown in (7). First, the samples are divided into different partitions based on their burnup depth.

The sample mean within each interval and the difference between each sample and its corresponding interval mean are calculated. The squares of these differences are then accumulated and divided by the total number of samples to obtain the MSE for the entire sample set under a three-segment partitioning strategy. An automated program was developed to independently optimize the partitioning strategy for different nuclides based on existing nuclide bias sample data. The program discretizes the burnup depth using a step size of 1 GWd/tU. Under the prerequisite of ensuring a sufficient number of samples within each interval, the program traverses all possible combinations of boundary points to calculate the corresponding MSE, selecting the combination with the minimum MSE as the optimal boundary points. After obtaining the nuclide bias and uncertainty under the optimal boundary points, these values are used as parameters for determining nuclide concentrations via Monte Carlo sampling under the burnup credit system. Criticality calculations are then performed on the generated nuclide composition samples. The specific operational workflow of the automated program is illustrated in [FIGURE:1].

3.1 基准例题与计算模型选取

Based on the above, this study aims to demonstrate the optimized criticality calculation results of the burnup credit system using a nuclide bias partitioning strategy. The analysis focuses on the effectiveness of this strategy in reducing uncertainty and improving computational accuracy. This paper selects the AFA-3G fuel assembly as a benchmark case for burnup credit analysis. In the criticality calculation model, the spent fuel storage pool utilizes storage racks to house this specific assembly model. Detailed parameters for the fuel assembly and its fuel rods are provided in [TABLE:1].

The assembly parameters include the pitch between cell centers, guide tube material, and the inner and outer diameters of the guide tubes, as well as the number, material, and dimensions of the instrument tubes. Fuel rod parameters specify the active zone height, fuel pellet diameter, and the composition of the gap filler gas. The burnup and criticality calculations are based on the SCALE code package, utilizing two primary functional modules. The ORIGEN module is employed to simulate the burnup history of the fuel within the reactor, calculating the evolution of nuclide concentrations at various burnup depths. For the criticality calculation, the KENO module, based on the Monte Carlo method, is used to evaluate the criticality of the fuel assemblies. This module further facilitates the analysis of uncertainties introduced by deviations in nuclide concentrations during the calculation process.

The fuel rod arrangement for the AFA-3G benchmark case is shown in [FIGURE:1].

[FIGURE:1]: Fuel Rod Arrangement. To account for the impact of the axial power distribution of the AFA-3G fuel assembly, a three-dimensional criticality model was constructed using the STARBUCS sequence within the SCALE burnup credit framework. This model references the axial burnup envelope distributions determined by Parish from the PWR spent fuel distribution curve database. This envelope distribution divides the active zone into segments along the axial direction and provides corresponding envelope distributions tailored to different burnup ranges.

30GWd/t 18-30GWd/t <18GWd/t

In the nuclide bias sample set used in this study, experimental data were sourced from the international spent fuel composition database, SFCOMPO 2.0. The corresponding calculated results were obtained using Bamboo-SFuel, a domestic, independently developed nuclide composition calculation code. Both Bamboo-SFuel and ORIGEN utilize the same ENDF/B-VII.0 nuclear data cross-section library for burnup calculations. To verify the consistency between the two codes, benchmark problems were used to perform comparative burnup calculations within the ORIGEN module. The results indicate that the maximum relative deviation in the calculated concentrations of the six primary actinide categories remains within acceptable limits. Consequently, the nuclide composition biases quantified based on Bamboo-SFuel calculations can be applied to burnup credit analyses performed with the SCALE program to correct its calculated values. The calculation benchmarks for the pressurized water reactor (PWR) spent fuel assemblies selected for the nuclide bias sample set are shown in [TABLE:1]. The initial fuel enrichment ranges from 2.453 wt% to 4.66 wt%, and the burnup depth ranges from 6.9 GWd/tU to 75 GWd/tU. These parameters cover the initial enrichment and burnup levels of most spent fuel assemblies at the time of discharge.

[TABLE:1] (Summary of Reactor Data: Mihama-3, etc., with enrichments such as 3.203 wt%, 3.21 wt%, 3.00 wt%, 3.13 wt%, 2.719 wt%, 3.897 wt%, 2.453 wt%, 3.038 wt%, and 2.72 wt%; Burnup range: 6.9~8.3 GWd/tU and others).

For the Neckarwestheim-2 benchmark (3.8 wt%), this paper employs a Monte Carlo sampling method to obtain the nuclide composition inputs for criticality calculations, thereby quantifying the uncertainty propagation of nuclide bias to the criticality results. The specific steps are as follows: 1) Determine the mean and corrected standard deviation for each target nuclide based on the sample data of nuclide biases at different burnup depths; 2) Assume that the bias of each target nuclide follows a normal distribution based on the aforementioned statistical parameters, and perform random Monte Carlo sampling to generate a large number of nuclide concentration samples; 3) Use the sampled nuclide concentration distributions as input parameters for Monte Carlo-based criticality simulations to evaluate the distribution characteristics of $k_{eff}$. This process allows for the assessment of the uncertainty in criticality calculation results under the burnup credit framework.

3.2 不确定度结果量化

Under the application level of burnup credit for APU-1, this study considers the impact of the net reduction of fissile isotopes and the neutron absorption of actinide nuclides on the results of criticality safety analysis. To verify the impact of the proposed partitioning strategy screening method on burnup credit analysis, an automated program was used to evaluate the partitioning strategies for each nuclide of interest. Given the limited number of nuclide bias samples available in this study, and referring to established statistical methods for handling sample biases across different burnup ranges in previous research, several constraints were implemented to ensure the reliability of the partitioning. Specifically, the process ensured that each burnup interval after partitioning covered a sufficient number of nuclide bias samples (at least 10) and a diverse range of benchmark case types (at least 3). Furthermore, to ensure the burnup intervals were sufficiently wide (at least 5 GWd/tU), the first boundary point $B_1$ was traversed within the range of 5 to 55 GWd/tU, while the second boundary point $B_2$ was traversed within the range of 10 to 55 GWd/tU. Combinations of boundary points that did not meet these requirements were excluded to avoid introducing additional errors due to improper interval selection. The results of the nuclide partitioning strategy screening are shown in [TABLE:4].

After screening the boundary points for different nuclides, the partitioning strategy with the minimum mean square error—representing the best fit with the nuclide bias samples—was obtained. [TABLE:N] presents the relevant parameters of the nuclide sample bias after the optimization of the partitioning strategy.

Low burnup interval

241 Pu

GWd/tU

242 Pu

GWd/tU

235 U

GWd/tU

238 U

GWd/tU

239 Pu

In the context of nuclear engineering and fuel management, the term "burnup range in GWd/tU" refers to the specific interval of energy extracted from a given mass of uranium fuel.

Specifically, GWd/tU (Gigawatt-days per metric ton of initial heavy metal uranium) is the standard unit used to measure fuel burnup, representing the total thermal energy produced per unit mass of the fuel. The burnup range typically defines the operational stages or discharge limits of the fuel, such as:

• Low burnup: Often associated with research reactors or early-stage cycles.
• Standard/High burnup: Typically ranging from 33 to 45 GWd/tU for conventional Light Water Reactors (LWRs).
• Ultra-high burnup: Extending beyond 60 GWd/tU, which is a focus of current research into Accident Tolerant Fuels (ATF) and advanced fuel cycles to improve economic efficiency and reduce waste volume.

240 Pu

GWd/tU

241 Pu

GWd/tU

242 Pu

GWd/tU

235 U

GWd/tU

238 U

GWd/tU

239 Pu

GWd/tU

240 Pu

GWd/tU high burnup range

241 Pu

GWd/tU

242 Pu

Compared to analyzing nuclide bias across the entire burnup range without partitioning, the burnup partitioning strategy fully accounts for the variations in interval division for different nuclide bias samples, as well as the relationship between nuclide bias, its uncertainty, and the change in burnup depth. Taking a specific nuclide as an example, the nuclide bias and its uncertainty under the conditions of no partitioning versus the burnup partitioning strategy are shown in [FIGURE:1].

Statistical plot of nuclide composition uncertainty. Statistical plot of partitioned nuclide composition uncertainty.

Statistical plot of partitioned nuclide composition uncertainty. Statistical plot of nuclide composition uncertainty.

Statistical plot of partitioned nuclide composition uncertainty. Statistical plot of partitioned nuclide composition uncertainty. Based on the mean nuclide bias and its uncertainty for each nuclide across the three burnup intervals, this study employs the Monte Carlo sampling method to obtain the concentrations of various nuclides at different burnup points within the range of 0 to 60 GWd/tU for the benchmark case. During the sampling process, the optimal burnup partition boundary corresponding to each nuclide is first used to determine its position at each burnup point.

Depending on whether the point falls within the low, medium, or high burnup segment, the mean nuclide bias and standard deviation obtained through the burnup partitioning strategy are used as random sampling parameters. Monte Carlo sampling is then applied to generate corrected nuclide concentration samples, as shown in Eq. (1):
$$ C_{i,j}^{corr} = C_{i,j}^{calc} \cdot (1 + \mu_{i,k} + \sigma_{i,k} \cdot \epsilon_{j}) $$
where $C_{i,j}^{corr}$ is the corrected nuclide concentration value based on Monte Carlo sampling; $C_{i,j}^{calc}$ is the calculated value of the $i$-th nuclide at burnup point $j$; $\epsilon_{j}$ is a random value sampled from a standard normal distribution; and $\mu_{i,k}$ and $\sigma_{i,k}$ represent the bias and standard deviation of the $i$-th nuclide component, respectively. These parameters are determined by the statistical values within the burnup interval $k$ to which the nuclide belongs at burnup point $j$ after processing via the partitioning strategy.

Each nuclide is sampled $N$ times, and these samples are substituted as input parameters into the SCALE/KENO module input file to obtain the sample mean $\bar{k}_{eff}$ and sample standard deviation $s$ of the criticality calculation results. A sample size is selected to cover a one-sided tolerance factor of 95%/95%. The impact of nuclide composition uncertainty on the criticality calculations is presented in [TABLE:1], which details the burnup partition and burnup depth (GWd/tU) for each nuclide.

70 高

After applying the partitioning strategy:

Based on the criticality calculation results, the $k_{eff}$ values before and after optimization are essentially consistent across most burnup points, with the exception of a discrepancy of approximately $\Delta k = 0.001$ at a burnup of $30$ GWd/tU. Analysis of the partitioning strategy results reveals that for the $30$ GWd/tU sample, the nuclides making the largest contribution to the criticality reactivity and uncertainty utilize statistical parameters from the medium-burnup interval. In this interval, the mean deviation ($\mu$) is the largest among the three partitions and is greater than the value used in the non-partitioned approach. This introduces additional reactivity, causing the $k_{eff}$ results to increase compared to the non-partitioned case. At a burnup of $45$ GWd/tU, while certain nuclides also fall within this medium-burnup interval, the fissile nuclides are situated in the interval with the smallest deviation; consequently, the increase in $k_{eff}$ is less significant than that observed at $30$ GWd/tU. At low burnup depths, the uncertainty values before applying the partitioning strategy were $0.0045$ and $0.0052$, respectively. After processing with the partitioning strategy, these values decreased to $0.0028$ and $0.0031$. This significant reduction indicates that the partitioning strategy more accurately reflects nuclide deviations and their associated uncertainties at low burnup, thereby facilitating a more economical criticality safety assessment. Within the higher burnup range, the criticality calculation uncertainty fluctuates slightly after partitioning, but remains within a margin of $\pm 0.0005$. Across the entire burnup range, the optimized criticality calculation uncertainty exhibits an upward trend as burnup increases, which is more consistent with both theoretical expectations and empirical observations.

4 总结

This study proposes a nuclide bias partitioning strategy based on the minimization of Mean Square Error (MSE). This strategy addresses the variations in nuclide sample bias and its associated uncertainty across different burnup levels by dividing the burnup range into three distinct intervals: low, medium, and high. Through an automated procedure, the partitioning boundaries are independently optimized for each nuclide. Furthermore, Monte Carlo sampling is employed to propagate the resulting nuclide concentration uncertainties into criticality calculations, achieving a refined treatment of nuclide composition correction within the framework of burnup credit. The burnup partitioning strategy ensures that the mean bias and uncertainty parameters used for nuclide correction at various burnup depths more accurately reflect the statistical characteristics of their respective intervals. This approach prevents samples from disparate burnup ranges from improperly influencing the calculation results at specific burnup points.

This method primarily addresses the issue in traditional burnup credit analysis where the inclusion of high-burnup bias samples during the low-burnup stage leads to excessive uncertainty. By isolating these samples, the strategy enhances the safety margin for burnup credit analysis of low-burnup fuel. For higher burnup stages, although the exclusion of low-burnup nuclide bias samples may lead to a slight increase in criticality calculation uncertainty for some samples, this additional uncertainty is limited compared to non-partitioned processing. Crucially, the uncertainty results obtained through partitioning maintain the correct statistical trend as a function of burnup depth. The partitioning method proposed in this paper can further compress the safety margins at low burnup depths while ensuring criticality safety, thereby improving the efficiency and economy of spent fuel storage and transportation systems.

References

Hong, Zhao, Shanguai; Zhang, Chunlong;

Analysis

demand away-from-reactor storage spent China[J].Nuclear Science Engineering, 2016,36(3):411-418(in Chinese).

Analysis of Away-from-Reactor Spent Fuel Storage Requirements in China

Abstract

As the scale of nuclear power in China continues to expand, the accumulation of spent fuel has become an increasingly pressing issue. This paper analyzes the current status and future trends of spent fuel generation from commercial nuclear power plants in China. By evaluating the existing storage capacities at reactor pools and centralized wet storage facilities, we project the timeline and scale of the demand for away-from-reactor (AFR) storage. Furthermore, the paper discusses the technical requirements and safety considerations for spent fuel storage, with a particular focus on the application of burnup credit in criticality safety analysis.

1. Introduction

The sustainable development of nuclear energy is a critical component of China's energy strategy. However, the "back-end" of the nuclear fuel cycle, specifically the management and disposal of spent fuel, poses significant challenges. Spent fuel discharged from reactors is initially stored in at-reactor (AR) pools to allow for thermal cooling and radioactive decay. As these pools approach their design capacity, there is an urgent need to transfer the fuel to away-from-reactor (AFR) storage facilities or reprocessing plants to ensure the continuous and safe operation of nuclear power units.

2. Current Status of Spent Fuel Storage in China

At present, the majority of spent fuel in China is stored in pools adjacent to the reactor cores. While some plants have expanded their storage capacity through the use of high-density racks, these measures only delay the saturation of storage space.

[TABLE:1]

As shown in [TABLE:1], the storage pressure varies across different nuclear power bases. Older units, such as those at the Qinshan and Daya Bay sites, are closer to reaching their storage limits compared to newly commissioned units. The centralized wet storage facility at the Lanzhou Nuclear Fuel Complex provides some relief, but its capacity is limited relative to the projected national discharge rates.

3. Demand Projection for Away-from-Reactor Storage

Based on the current nuclear power development plan, we estimate the cumulative volume of spent fuel through 2030. The projection accounts for the installed capacity, average burnup rates, and the refueling cycles of various reactor types, including CPR1000, AP1000, and HPR1000 units.

[FIGURE:1]

[FIGURE:1] illustrates the projected gap between total spent fuel generation and available AR storage capacity. The analysis indicates that

Nuclear Power Engineering, 2010, 31(02):24-28(in Chinese).

Criticality Calculation of Spent Fuel Storage Pools in Nuclear Power Plants Based on Burnup Credit

Abstract

This study investigates the application of burnup credit (BUC) in the criticality safety analysis of spent fuel storage pools. By accounting for the reduction in reactivity that occurs as fuel is depleted during reactor operation, burnup credit allows for higher storage densities and more efficient management of spent fuel. Using advanced computational codes, we perform criticality calculations for a typical pressurized water reactor (PWR) spent fuel pool. The results demonstrate that incorporating burnup credit significantly increases the safety margin compared to the traditional "fresh fuel" assumption, while maintaining strict adherence to regulatory subcriticality requirements.

1. Introduction

The management of spent nuclear fuel is a critical aspect of the nuclear fuel cycle. Traditionally, criticality safety analyses for spent fuel storage facilities have relied on the "fresh fuel" assumption, which assumes that the fuel assemblies are at their maximum possible reactivity (i.e., unirradiated). While this approach provides a robust safety margin, it is increasingly viewed as overly conservative, leading to underutilization of storage capacity.

Burnup credit (BUC) is a methodology that accounts for the actual composition of spent fuel, including the depletion of fissile isotopes and the buildup of neutron-absorbing fission products and actinides. Implementing BUC requires sophisticated calculation tools and rigorous validation to ensure that the effective multiplication factor ($k_{eff}$) remains below the regulatory limit of 0.95 under all normal and credible accident conditions.

2. Methodology and Computational Models

2.1 Burnup Credit Levels

The application of BUC is generally categorized into different levels based on the isotopes considered in the analysis:
- Actinide-Only BUC: Accounts only for the change in uranium and plutonium isotopes.
- Full BUC: Accounts for both actinides and major fission product poisons (such as $^{149}Sm$, $^{155}Gd$, and $^{143}Nd$).

In this study, we utilize a comprehensive approach that incorporates the most significant isotopes contributing to negative reactivity.

2.2 Calculation Codes

The criticality calculations were performed using the SCALE (Standardized Computer Analyses for Licensing Evaluation) modular code system. Specifically, the TRITON sequence was used for depletion calculations to determine the isotopic concentrations as a function of burnup, and the KENO V.a Monte Carlo code was employed for the 3D criticality safety analysis of the storage pool configuration

methodology

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Research on Benchmarking Methods for Burnup Databases

Authors: Zhaodong, Zhou Xiaoping, Xiaobo, Zheng Jiye.
Journal: Atomic Energy Science and Technology, 2022, 56(05).

Abstract

The accuracy of burnup databases is fundamental to the safety and efficiency of nuclear reactor analysis. This study investigates methodologies for the benchmarking and validation of burnup databases, focusing on the systematic comparison between experimental measurements and computational predictions. By analyzing isotopic composition changes and decay heat characteristics, this research establishes a robust framework for evaluating the reliability of nuclear data libraries used in burnup calculations. The results provide a technical basis for the optimization of reactor core design and spent fuel management.

Introduction

In the field of nuclear engineering, the precise prediction of isotopic inventory in spent fuel is critical for reactor physics design, radiation shielding, and nuclear waste management. Burnup calculations rely heavily on comprehensive burnup databases, which include cross-sections, decay constants, and fission product yields. However, uncertainties in these fundamental nuclear data can propagate through complex depletion chains, leading to significant discrepancies in macroscopic reactor parameters.

To ensure the reliability of these calculations, it is essential to perform rigorous benchmarking against experimental data. This paper explores systematic benchmarking methods, utilizing high-fidelity experimental assays to validate the performance of various burnup databases and simulation codes.

Methodology

1. Selection of Experimental Benchmarks

The foundation of any benchmarking study is the selection of high-quality experimental data. This research utilizes spent fuel radiochemical assay data from international databases, such as the SFCOMPO (Spent Fuel Composition) database. These benchmarks provide measured concentrations of actinides and fission products for various reactor types, including Pressurized Water Reactors (PWRs) and Boiling Water Reactors (BWRs).

2. Computational Modeling

The burnup simulations are conducted using state-of-the-art transport and depletion codes. The modeling process involves:
- Defining the precise geometry and material composition of the fuel assembly.
- Implementing detailed power history and operating conditions (e.g., moderator temperature, boron concentration).
- Utilizing different nuclear data libraries (e.g., ENDF/B, JEFF, JENDL) to assess the impact of data variations.

3. Comparison Metrics

The performance of the burnup database is evaluated using the Calculation-to-Experiment ($C/E$) ratio. For a given isotope $i$, the ratio is

Calculation Study Spent Using Burnup Credit[J]. Atomic Energy Science Technology, 2013, 47(11): 2102(in Chinese).

Computational Study on the Application of Burnup Credit in the Spent Fuel Pool of Tianwan Nuclear Power Plant

Introduction

The storage and management of spent fuel are critical components of the nuclear fuel cycle. As nuclear power plants operate over extended periods, the density of spent fuel storage must be optimized to accommodate increasing inventories. Traditionally, criticality safety analysis for spent fuel storage pools has relied on the "fresh fuel assumption," which treats all fuel assemblies as having their maximum initial enrichment without accounting for the depletion of fissile isotopes and the buildup of neutron absorbers during reactor operation.

However, the application of Burnup Credit (BUC) allows for a more realistic assessment by accounting for the reduction in reactivity that occurs during irradiation. This study investigates the implementation of BUC for the spent fuel pool at the Tianwan Nuclear Power Plant (TNPP), aligning with international safety standards and the domestic regulatory framework, specifically GB 15146.12-2017, Nuclear Criticality Safety for Fissile Materials Outside Reactors—Part 12: Burnup Credit for Light Water Reactor Fuels.

Methodology and Computational Models

To evaluate the criticality safety of the spent fuel storage racks, high-fidelity computational tools were employed to simulate fuel depletion and subsequent neutron multiplication factors ($k_{eff}$). The analysis accounts for the complex isotopic evolution within the fuel assemblies, including the depletion of $\text{ }^{235}\text{U}$ and the production of plutonium isotopes and various fission product poisons.

The computational framework integrates depletion codes with Monte Carlo N-Particle (MCNP) simulations to determine the reactivity of the storage configurations. The geometric model of the spent fuel pool includes the storage cell pitch, the thickness of the stainless steel cladding, and the presence of neutron-absorbing materials such as boron in the pool water.

[FIGURE:1]

Burnup Credit Analysis

The transition from the fresh fuel assumption to BUC requires rigorous validation to ensure safety margins are maintained. In this study, we consider "actinide-only" BUC as well as "principal fission product" BUC. The former accounts only for the changes in uranium and plutonium isotopes, while the latter includes the negative reactivity contribution from stable fission products with high neutron capture cross-sections.

Key parameters influencing the $k_{eff}$ include:
1. Initial Enrichment: The starting concentration of $\text{ }^{235}\text{U}$ in the fuel.
2. **Burnup Level

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Atomic Energy Science Technology, 2022, 56(09): 1915-1923(in Chinese).

Research on Deviations and Uncertainties in Burnup Calculations for Criticality Safety under Burnup Credit

Atomic Energy Science and Technology, 2022, 56(09).

Introduction

Burnup credit (BUC) refers to the methodology of accounting for the reduction in reactivity of spent nuclear fuel caused by the change in isotopic composition during irradiation. In criticality safety analyses of spent fuel storage and transportation systems, traditional methods often adopt the "fresh fuel assumption," which ignores the depletion of fissile isotopes and the buildup of neutron-absorbing fission products. While this assumption provides a significant safety margin, it leads to overly conservative designs that limit the capacity of storage pools and transport casks. Implementing burnup credit allows for higher loading densities and improved economic efficiency while maintaining rigorous safety standards.

A critical component of burnup credit is the accurate prediction of the isotopic inventory in spent fuel. The validation of burnup codes typically involves comparing calculated concentrations with experimental data from Radiochemical Analysis (RCA). However, the uncertainties inherent in these depletion calculations—arising from nuclear data, modeling approximations, and operational history—propagate into the subsequent criticality calculations ($k_{eff}$). This study investigates the impact of burnup calculation deviations and their associated uncertainties on criticality safety margins, following the framework established in the Radulescu and Gauld approach for validating actinide and fission product burnup credit.

Methodology for Validating Isotopic Predictions

The validation process for burnup credit criticality safety analyses generally follows a two-step procedure. First, the isotopic composition of the spent fuel is predicted using a depletion code. Second, these compositions are used as input for a Monte Carlo or deterministic transport code to calculate the effective multiplication factor ($k_{eff}$).

Isotopic Composition Prediction

The accuracy of isotopic predictions is evaluated by comparing the calculated values ($C$) with experimental values ($E$). The $C/E$ ratio serves as a primary metric for identifying systematic biases in the depletion software and underlying library data. This study considers both major actinides (e.g., $^{235}\text{U}$, $^{238}\text{U}$, $^{239}\text{Pu}$) and principal fission products (e.g., $^{149}\text{Sm}$, $^{143}\text{Nd}$, $^{155}\text{Gd}$) that contribute significantly to neutron absorption.

Uncertainty Propagation

The uncertainty in the predicted $k_{eff}$ due to isotopic depletion is not

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Analysis of Nuclide Composition in Spent Fuel Assemblies of Commercial PWRs Using the NECP-Bamboo Code

Atomic Energy Science and Technology, 2023, 57(03): 545-554.

Abstract

Accurate prediction of the nuclide composition in spent fuel is essential for nuclear fuel cycle analysis, including radioactive waste management and reprocessing. This study utilizes the NECP-Bamboo code, developed by Xi'an Jiaotong University, to perform high-fidelity depletion calculations for commercial Pressurized Water Reactor (PWR) spent fuel assemblies. By comparing the numerical results with experimental data from international benchmarks, the accuracy of the code in predicting key actinides and fission products is validated. The results demonstrate that NECP-Bamboo provides reliable estimates of isotopic concentrations across a wide range of burnup levels, meeting the requirements for engineering applications in the nuclear industry.

Introduction

The accurate determination of nuclide inventory in spent nuclear fuel is a fundamental requirement for the safety and efficiency of the back-end of the nuclear fuel cycle. Precise data on nuclide concentrations are necessary for calculating decay heat, radiation source terms, and criticality safety during the storage, transportation, and reprocessing of spent fuel. As commercial Pressurized Water Reactors (PWRs) move toward higher burnup and the use of advanced fuel designs, the demand for high-precision depletion analysis codes has increased significantly.

The NECP-Bamboo code system is a comprehensive reactor physics analysis tool developed by the Nuclear Engineering Computational Physics (NECP) laboratory at Xi'an Jiaotong University. It is designed to perform lattice physics calculations, core simulations, and fuel management analysis. A critical component of this system is its depletion module, which solves the Bateman equations to track the evolution of nuclide concentrations over time. This paper aims to validate the performance of NECP-Bamboo specifically for commercial PWR spent fuel by comparing its predictions against measured isotopic data from destructive chemical assays.

Methodology

1.1 The NECP-Bamboo Code System

NECP-Bamboo utilizes a modular architecture to handle cross-section generation, resonance calculations, and transport solutions. For depletion analysis, the code employs a high-order Taylor series expansion method or the CBRM (Chebyshev Rational Approximation Method) to solve the depletion equations. The system relies on a multi-group cross-section library processed from the latest evaluated nuclear data files (e.g., ENDF/B-VII.1).

1.2 Physical Modeling and Dep

Quantitative

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Quantitative

Analysis

Impact Nuclide-Bias Partitioning Strategies Criticality Calculation Uncertainty under Burnup Credit School Nuclear Science Technology University South China, Hunan Hengyang, 421001, China;

2. Key

Advanced Nuclear Energy Design Safety, Ministry Education, Hunan Hengyang, 421001, China

Abstract

Burnup credit accounts changes nuclide composition during irradiation reduces estimated effective multiplication factor criticality-safety assessments spent nuclear fuel.

However, uncertainties arising sample-to-calculation deviations nuclide inventories degrade accuracy criticality calculations. quantify those uncertainties across different burnup ranges improve precision burnup-credit analyses, develops nuclide-bias partitioning strategy based linear-regression fitting mean-squared-error minimization determine optimal burnup interval boundaries target nuclides.

Using AFA-3G assembly benchmark, depletion criticality calculations carried SCALE ORIGEN modules, Monte-Carlo sampling employed propagate nuclide-bias uncertainty

Results

indicate proposed partitioning strategy reduces criticality uncertainty approximately under low-burnup conditions, offering optimized approach burnup-credit evaluations spent-fuel criticality safety assessments. words: burnup credit; burnup calculation; Monte-Carlo sampling; criticality safety

analysis

Foundation item: National Natural Science Foundation China (12475174, U2267207) Project Yuelu Mountain Industrial Innovation Center (2024YCII0108) author:

Zhenping E-mail: