Nuclear data uncertainty propagation: Perturbation vs. Monte Carlo

Two methods of nuclear data uncertainty propagation are compared, using the same nuclear data uncertainties and criticality-safety benchmarks. The first method, based on perturbation theory uses covariance files, covariance processing and the perturbation card of MCNP. The second method makes use of a large number of MCNP calculations, all alike, but using different random nuclear data libraries, consistent with the covariance files of the first method. The consistency of the nuclear data used by both methods is checked and results for 33 criticality-safety benchmarks are presented. Relatively good agreements are found, but depending on the benchmark cases, differences due to the elastic cross-section, ν-bar, angular and energy distributions are observed.     

Sensitivity and uncertainty analysis of the HCLL breeder blanket experiment in the frame of the EU fusion technology programme

The mock-up of the EU Test Blanket Module (TBM) based on the Helium-Cooled Lithium-Lead (HCLL) concept was irradiated at Frascatti, Italy, in order to assess the uncertainty in the tritium production rate (TPR) due to the uncertainty in the nuclear data and the computational methods. The benchmark should contribute also to the validation of the new nuclear cross-section and covariance data evaluations. This paper presents the final design of the benchmark and the analysis using the deterministic transport, sensitivity and uncertainty code system. The analysis includes the calculation of the tritium production rate (TPR) in LiPb layers and the neutron reaction rates, which were measured in the experimental set-up. The SUSD3D cross-section sensitivity and uncertainty code together with the 2D/3D deterministic transport code package DOORS is used for the analysis of the experiment. Based on the sensitivity analyses the most important nuclear reactions and energy ranges in the particular reaction rate measurements are identified, as well as the corresponding uncertainties.     

基于统计抽样的敏感性系数计算方法

分析核系统的不确定性和敏感性,对于减小核设计的设计余量、提高核系统的经济性具有重要意义。基于统计抽样的不确定度分析方法,由于算法简单、可考虑高阶效应且对响应量没有特殊要求等,越来越受到重视。但之前认为基于统计抽样法很难进行敏感性系数分析,其原因主要是响应量的变化是由多变量同时变化引起,很难把单独一个变量的变化导致的响应量的变化确定出来。本文首先推导了利用统计抽样法进行敏感性系数分析的理论公式,然后利用裸堆双群近似的临界公式和复杂的压水堆单栅元问题进行了验证,验证了统计抽样法的可行性。针对实际问题协方差矩阵求逆困难的问题,本文提出了两种替代解决方法,即采用简化协方差矩阵或统一微扰量的方法,利用²³⁵U裂变截面对上述方法进行了验证分析,证明了方法的可行性和正确性;同时分析了不同敏感性系数对不确定度计算的影响。     

Generalized formulation of extended cross-section adjustment method based on minimum variance unbiased linear estimation

ABSTRACT

By introducing a new assumption of linear estimation, we derive a new formulation of the extended cross-section adjustment (EA) method, which minimizes the variance of the design target core parameters. The new formulation is derived on the basis of minimum variance unbiased estimation with no use of the assumption of normal distribution. In this formulation, we found that EA has infinitely many solutions as the adjusted cross-section set. The new formulation of EA can represent all the possible solutions minimizing the variance of the design target core parameters and includes a special case identical to the classical Bayesian EA method, which was derived on the basis of the Bayes theorem under the assumption of normal distribution. Moreover, we prove that the special case minimizes not only the variance of the design target core parameters but also the variance of the nuclear data. Meanwhile, we show that the new assumption of linear estimation is consistent with the Kalman filter and demonstrate that we can formulate similarly the extended bias factor method, the conventional cross-section adjustment method, and the regressive cross-section adjustment method with no use of the assumption of normal distribution.     

Extended cross-section adjustment method to improve the prediction accuracy of core parameters

An extended cross-section adjustment method has been developed to improve the prediction accuracy of target core parameters. The present method is on the basis of a cross-section adjustment method which minimizes the uncertainties of target core parameters under the conditions that integral experimental data are given. The present method enables us to enhance the prediction accuracy better than the conventional cross-section adjustment method by taking into account the target core parameters, as well as the extended bias factor method. In addition, it is proved that the present method is equivalent to the extended bias factor method when only one target core parameter is taken into account. The present method is implemented in an existing cross-section adjustment solver. Numerical calculations verify the derived formulation and demonstrate an applicability of an adjusted cross-section set which is specialized for the target core parameters.     

面向协方差矩阵抽样的快堆不确定性分析方法研究

基于传统统计学抽样的不确定性分析方法由于算法简单、程序容易实现及同时考虑高阶效应受到国内外广泛关注,但上述方法通常需要大量样本才能保证响应量计算精度。研究发现,产生以上现象的原因是抽样样本质量不高。通过改进抽样方法,面向协方差矩阵抽样时小样本量可以保证较高的计算精度。文中首先从理论上证明了面向协方差矩阵抽样方法的可行性,用简单测试题对其进行验证。在此基础上,使用自主开发的快能谱反应堆敏感性和不确定性分析程序SUFR,选取国际快堆基准装置ZPR-6/7,计算多个核素不同反应类型的核截面引起的有效增殖因子（k_eff）的不确定度,并与使用确定论方法计算的不确定度进行对比。结果表明,使用面向协方差矩阵抽样的情况下,样本量为50时,2种方法计算的不确定度偏差均低于1.3%。由此说明,面向协方差矩阵抽样方法可以很好地解决传统抽样方法计算不确定度时存在的问题,且SUFR程序面向协方差矩阵抽样功能的开发是正确的,该方法是对传统抽样方法的进一步发展。      

高精度相关变量随机数序列产生方法

当对反应堆物理计算结果进行不确定性分析时,需产生多维相关变量随机数序列。为产生高质量的相关变量随机数序列以减少样本数量,本文首先从理论上分析给出了之前的多维相关变量随机数序列的协方差矩阵与真实的协方差矩阵有差别的原因,据此提出了解决方法,并采用数值计算对解决方法进行了验证。验证结果表明,对于3个变量的抽样序列,高精度相关变量抽样方法采用20个样本便得到与原相关系数矩阵一致的矩阵,抽样样本数量较之前的方法减少了5个量级;而对于33群的²³⁸U辐射俘获反应道,即使抽样样本数为34,最大相对误差亦仅0061%,由此证明了方法的有效性。最后,利用不同方法对铅基快堆LFR进行了分析,传统正态分布抽样样本总数较高精度相关变量抽样方法的样本总数高1倍,其最大相对误差为12.5%,而高精度相关变量抽样方法的最大相对误差仅1.7%,计算精度有明显提高。结果表明该方法具有工程应用前景。     

Cross-section adjustment methods based on minimum variance unbiased estimation

On the basis of the minimum variance approach, the unified formulation for three types of the cross-section adjustment methods has been derived in a straightforward way without assuming the normal distribution. These methods are intended to minimize the variances of the predicted target core parameters, the adjusted cross-section set, and the calculated integral experimental values. The first and the second methods are found to be slightly different from the extended and the conventional cross-section adjustment methods based on the Bayesian approach with the normal distribution assumption, respectively. However, they become equivalent in some cases and results. The third method is a new method, which is necessary from the viewpoint of the symmetry of the formulation. In addition, it is verified by numerical calculations that the derived formulation gives the minimized variances as intended. The derivation procedure proposed in the present paper is potentially applicable to developing more sophisticated cross-section adjustment methods because of the less assumptions on the probability density function.     

Covariances of Evaluated Nuclear Cross Section Data for Th, W and Mn

The EMPIRE code system is a versatile package for nuclear model calculations that is often used for nuclear data evaluation. Its capabilities include random sampling of model parameters, which can be utilised to generate a full covariance matrix of all scattering cross sections, including cross-reaction correlations. The EMPIRE system was used to prepare the prior covariance matrices of reaction cross sections of ²³²Th, ^{180,182,183,184,186}W and ⁵⁵Mn nuclei for incident neutron energies up to 60 MeV. The obtained modelling prior was fed to the GANDR system, which is a package for a global assessment of nuclear data, based on the Generalised Least-Squares method. By introducing experimental data from the EXFOR database into GANDR, the constrained covariance matrices and cross section adjustment functions were obtained. Applying the correction functions on the cross sections and formatting the covariance matrices, the final evaluations in ENDF-6 format including covariances were derived. In the resonance energy range, separate analyses were performed to determine the resonance parameters with their respective covariances. The data files thus obtained were then subjected to detailed testing and validation. Described evaluations with covariances of ²³²Th, ^{180,182,183,184,186}W and ⁵⁵Mn nuclei are included into the ENDF/B-VII.1 library release.     

Nuclear data sensitivity,uncertainty and target accuracy assessment for future nuclear systems

A sensitivity and uncertainty study has been performed to evaluate the impact of neutron cross-section uncertainty on the most significant integral parameters related to the core and fuel cycle. This work is a contribution to the feasibility assessment of innovative reactor and fuel cycle systems, proposed within the Generation IV initiative. Results of an extensive analysis indicate the most relevant parameters and show any potential significant problems arising from the quality of existing nuclear data, in the assessment of the systems considered. In order to perform these studies, uncertainty covariance data have been produced, mostly based on selected, high accuracy integral experiments. A target accuracy assessment has been also performed in order to evaluate nuclear data improvement requirements. The results of the assessment allows to give guidelines in order to define the most appropriate and effective strategy for data uncertainty reduction.     

Uncertainty quantification in (α,n) neutron source calculations for an oxide matrix

We present a methodology to propagate nuclear data covariance information in neutron source calculations from (α,n) reactions. The approach is applied to estimate the uncertainty in the neutron generation rates for uranium oxide fuel types due to uncertainties on 1) ^17,18 O(α,n) reaction cross-sections and 2) uranium and oxygen stopping power cross sections.The procedure to generate reaction cross section covariance information is based on the Bayesian fitting method implemented in the R-matrix SAMMY code. The evaluation methodology uses the Reich-Moore approximation to fit the ^17,18 O(α,n) reaction cross-sections in order to derive a set of resonance parameters and a related covariance matrix that is then used to calculate the energy-dependent cross section covariance matrix. The stopping power cross sections and related covariance information for uranium and oxygen were obtained by the fit of stopping power data in the α-energy range of 1 keV up to 12 MeV.Cross section perturbation factors based on the covariance information relative to the evaluated ^17,18 O(α,n) reaction cross sections, as well as uranium and oxygen stopping power cross sections, were used to generate a varied set of nuclear data libraries used in SOURCES4C and ORIGEN for inventory and source term calculations. The set of randomly perturbed output (α,n) source responses, provide the mean values and standard deviations of the calculated responses reflecting the uncertainties in nuclear data used in the calculations. The results and related uncertainties are compared with experiment thick target (α,n) yields for uranium oxide.     

基准模型keff对核数据的灵敏度分析及不确定度量化

为研究有效增殖因数（k_eff）对核反应数据的灵敏度,以科学量化核数据导致k_eff计算的不确定度,编制了输运计算积分量灵敏度及不确定度分析程序SURE。该程序采用多群S_N输运计算方法计算k_eff、角通量和伴随角通量,基于微扰理论确定k_eff对核数据的灵敏度,利用协方差数据量化评估k_eff计算的不确定度。利用ENDF/B-Ⅶ.1评价中子核数据库,制作了输运计算所需的多群核数据、灵敏度分析所需的各反应道多群截面和中子群转移矩阵、不确定度分析所需的多群协方差数据。采用上述数据,利用SURE分析了基准模型Godiva和Jezebel的k_eff计算值对核数据的灵敏度,以及核数据导致的模拟计算的不确定度。SURE的灵敏度计算结果与MCNP程序及FORSS程序计算结果符合较好。     

Experimental analyses of bismuth sample reactivity worth at Kyoto University Critical Assembly

Sample reactivity worth experiments are carried out by substituting aluminum (Al) plates for bismuth (Bi) ones at the Kyoto University Critical Assembly. At the beginning, uncertainty quantification of bismuth isotope is conducted by deterministic calculations with nuclear data library JENDL-4.0, with the use of experimental results of sample reactivity worth. Then, with the combined use of current (Bi) and previous (Pb) experimental results that demonstrate the comparative difference in the sensitivity and uncertainty of Bi and Pb isotopes, experimental results of cross-section uncertainties of Bi isotope are available for examination of neutron characteristics of Pb–Bi coolant material in the accelerator-driven system. From the experimental analyses, further uncertainty analyses by neutron transport calculations are needed for several reactions of Bi isotope, especially with the use of the covariance data of capture, elastic scattering and inelastic scattering reactions in another nuclear data library.     

Improved cross-section modeling methodology for coupled three-dimensional transient simulations

The purpose of this paper is to introduce a more accurate and sophisticated methodology for use in three-dimensional coupled neutronic/thermal hydraulic analysis. The approach described in this paper is an original method of modeling cross-section variations for off-nominal core conditions, which is becoming an important issue with the increased use of coupled three-dimensional neutronic/thermal hydraulic simulations. This proposed method improves the accuracy of the cross-section modeling for transient applications and it is called the adaptive high-order table lookup method (AHTLM). During nuclear power plant (NPP) transient and accident simulations AHTLM interpolates into multi-dimensional cross-sections tables, which form a box envelope bounding the expected range of change of both nominal and off-nominal NPP conditions. This paper further addresses the methodologies for the development of the cross-section libraries and issues that affect the proper formulation of accurate data. The automated generation procedure outlined in this paper gives the user the tools and the ability of generating accurate cross-sections that cover a large range of thermal hydraulic parameters. Further improvements and expansions for future applications are also discussed.     

Cross-section adjustment in the analysis of benchmark experiments

An iteration procedure is given for the solution of the exact non-linear least-squares equation for cross-section adjustment. The solution is made feasible by properly weighting all detectors used in the Benchmark experiment and combining them to one ‘global’ detector. The adjoint fluxes and sensitivity coefficients must be calculated only for this one global detector so that in each iteration cycle the code ANISN and SWANLAKE have to be run only once.Simultaneously an expression was found for the error-covariance matrix of the adjusted cross-section set.     

多群核数据不确定性对堆芯物理计算的影响

核数据不确定性是造成反应堆物理计算结果不确定性的重要因素之一。基于所需抽样核数据的协方差矩阵开发了随机抽样模块(Stochastic Sampling,SAMP),在此基础上利用SCALE(Standardized Computer Analyses for Licensing Evaluation)软件包实现了混合法和随机抽样法两种不确定性分析方法,以研究多群核数据不确定性对堆芯物理计算的影响。以3×3假想堆芯为对象,对两种方法进行了验证,然后应用于国际原子能机构(International Atomic Energy Agency,IAEA)燃料管理基准题中的Almaraz核电厂首循环堆芯。分析结果表明,两种方法结果符合良好,Almaraz核电厂堆芯keff不确定性约为0.5%,堆芯径向和轴向功率的最大不确定性分别为1.9%和0.45%。     

连续能量核数据敏感性分析及其在AP1000中的应用

AP1000是典型的第三代核电技术,对AP1000反应堆进行核数据的敏感性分析是不确定度量化分析的基础,对AP1000后续的安全分析有重要作用。本文基于反复裂变几率方法在蒙特卡罗前向计算中求解共轭通量,并根据一阶微扰理论得到keff对核数据的灵敏度系数。针对反复裂变几率方法普遍存在占用内存大的问题,采用稀疏矩阵的存储方式降低内存。针对计数效率低、统计涨落大的问题,采用重叠块法提高计数效率。通过在蒙特卡罗程序NECP-MCX中开发连续能量核数据敏感性分析功能模块,并对AP1000进行连续能量核数据灵敏度系数的计算,得到了对keff的灵敏度系数影响较大的核数据,同时将计算结果与MCNP6进行了比较。结果表明,NECP-MCX和MCNP6的计算结果吻合较好。     

Simulation of robust resonance parameters using information theory

Due to complex nature of resonance region interactions, significant effort has been devoted to quantify the resonance parameter uncertainty information through covariance matrices. Statistical uncertainties arising from measurements contribute only to the diagonal elements of the covariance matrix, but the off-diagonal contributions arise from multiple sources like systematic errors in cross-section measurement, correlation due to nuclear reaction formalism, etc. All the efforts have so far been devoted to minimize the statistical uncertainty by repeated measurements but systematic uncertainty cannot be reduced by mere repetition. The computer codes like SAMMY and KALMAN so far developed to generate resonance parameter covariance have no provision to improve upon the highly correlated experimental data and hence reduce the systematic uncertainty. We propose a new approach called entropy based information theory to reduce the systematic uncertainty in the covariance matrix element wise so that resonance parameters with minimum systematic uncertainty can be simulated. Our simulation approach will aid both the experimentalists and the evaluators to design the experimental facility with minimum systematic uncertainty and thus improve the quality of measurement and the associated instrumentation. We demonstrate, the utility of our approach in simulating the resonance parameters of Uranium-235 and Plutonium-239 with reduced systematic uncertainty.     

Modeling of distribution parameter,void fraction covariance and relative velocity covariance for upward steam-water boiling flow in vertical rod bundle

In order to improve the prediction accuracy of one-dimensional interfacial force formulated by ‘Andersen’ approach, the distribution parameter in a drift–flux correlation, void fraction covariance, and relative velocity covariance has been modeled for dispersed boiling two-phase flow in a vertical rod bundle. The distribution parameter has been derived by a bubble-layer thickness model. The correlations of void fraction covariance and relative velocity covariance have been developed based on prototypic 8 × 8 rod bundle data. The correlation of void fraction covariance agrees with the bundle data with the mean absolute error, standard deviation, mean relative deviation, and mean absolute relative deviation being 0.00120, 0.0415, ?0.173%, and 1.80%, respectively. The correlation of relative velocity covariance agrees with the bundle data with the mean absolute error, standard deviation, mean relative deviation, and mean absolute relative deviation being ?0.00241, 0.0452, ?0.0316%, and 2.52%, respectively. In view of the great importance of void fraction covariance and relative velocity covariance on the one-dimensional interfacial drag force formulation, it is highly recommended to include the void fraction covariance and relative velocity covariance in the one-dimensional formulation of interfacial drag force used in nuclear thermal-hydraulic system analysis codes.     

Test of235U Covariance Data with Nuclear Data Adjustment BenchmarkWU Haicheng

Nuclear data covariances are important input data for quantificational assessment of nuclear facility design uncertainty and nuclear data adjustment (NDA), which give direct impact on estimated uncertainties and results of NDA. To test the rationality of the newly evaluated nuclear reaction cross sections covariance in smooth region for²³⁵U which is generated based on the analysis of source of experiments uncertainties and linear least-square method, the data were tested with the NDA benchmark exercises recommended by the OECD/NEA WPEC/SG33. The input data of the NDA generated from JENDL-4.0 library were updated with the cross sections and covariances of ²³⁵U fission, capture and inelastic scattering reactions from²³⁵U cov, and used in NDA calculation. The new results were compared with the original JENDL-4.0 ones. The test results show that the covariance data from²³⁵U cov are unreasonable. Too large uncertainties around threshold energy of inelastic scattering reaction were found. The uncertainties of fission and capture cross sections in smooth region are too small to be supported by NDA results.