Intelligent Optimization of Lead-bismuth Reactor Core Based on Radial Basis Function Surrogate Model and Niche Genetic Algorithm
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摘要: 为解决铅铋反应堆多因素耦合影响下的复杂非线性多维优化问题，构建了基于径向基（RBF）代理模型预测、正交拉丁超立方抽样（OLHS）和小生境遗传算法（NGA）寻优的堆芯智能优化方法，开发了包含抽样、蒙卡程序耦合处理、堆芯参数预测寻优等功能的铅铋反应堆设计优化平台，并以堆芯最小燃料装载量为优化目标进行方案寻优验证。研究结果表明：RBF代理模型可准确快速地预测铅铋反应堆堆芯特性参数，与蒙卡程序计算值比较，其预测的堆芯有效增殖因子（keff）相对误差在±0.1%以内；该智能优化方法应用于铅铋反应堆堆芯优化是可行的，能找到多因素共同变化约束下的最优目标方案，且极大缩减了设计方案的搜索计算时间。本研究建立的堆芯智能优化方法可为铅铋反应堆多物理、多变量、多约束耦合影响的优化设计提供思路。
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关键词:
- 铅铋反应堆 /
- 堆芯设计 /
- 径向基（RBF）代理模型 /
- 小生境遗传算法（NGA） /
- 正交拉丁超立方抽样（OLHS） /
- 智能优化
Abstract: In order to solve the complex nonlinear multi-dimensional optimization problem under the influence of multi-factor coupling of lead-bismuth reactor, an intelligent optimization method for reactor core was constructed based on radial basis function (RBF) surrogate model prediction, orthogonal Latin hypercube sampling (OLHS) and niche genetic algorithm optimization. A design optimization platform for lead-bismuth reactor was developed, which included the functions of sampling, Monte Carlo program coupling treatment, and core parameter prediction and optimization. The scheme optimization verification was carried out with the minimum fuel loading of the core as the optimization objective. The results show that the RBF surrogate model can accurately and quickly predict the core characteristic parameters of the lead-bismuth reactor. Compared with the calculated values of the Monte Carlo program, the relative error of the predicted core effective multiplication factor k eff is within ± 0.1%. This intelligent optimization method is feasible for lead-bismuth reactor core optimization, which can find the optimal target scheme under the constraint of multi-factor co-variation, and greatly reduce the search calculation time of the design scheme. Therefore, the intelligent optimization method established in this study can provide new ideas for the optimization design of multi-physics, multi-variable and multi-constraint coupling effects of lead-bismuth reactor. -
图 1 标准K折交叉验证法（K=10）流程图
Figure 1. Flow Chart of Standard K Fold Cross Validation Method ( K=10 )
图 3 基于RBF代理模型、OLHS和NGA的堆芯智能优化方法
Figure 3. Core Intelligent Optimization Method Based on RBF Surrogate Mmodel, OLHS and NGA
图 8 3种核函数构建的代理模型拟合相对误差随散布常数的变化情况
Figure 8. Variation of Fitting Relative Error with Dispersion Constant of Surrogate Models Constructed by Three Kernel Functions
图 9 散布常数取值范围内RMSE变化情况
Figure 9. RMSE Variation within the Value Range of Dispersion Constant
表 1 常用RBF核函数
Table 1. Commonly Used RBF Kernel Functions
核函数类型 表达式 线性函数 $\varphi \left(x,{x}_{_{i}}\right)={\sigma }_{{\rm{r}}}\left\|x-{x}_{_{i}}\right\|$ 高斯函数 $\varphi \left(x,{x}_{_{i}}\right)={\rm{exp} }(-{\left\|x-{x}_{_{i}}\right\|^{2}}/2{ {\sigma }_{{\rm{r}}} ^{2}})$ 多项式函数 $\varphi \left(x,{x}_{_{i}}\right)={\left({\sigma }_{{\rm{r}}}+\left\|x-{x}_{_{i}}\right\|\right)}^{n}$ 样条函数 $\varphi \left(x,{x}_{_{i}}\right)={\left\|x-{x}_{_{i} }\right\|^{2}}\mathrm{log}\left({\sigma }_{ {\rm{r} } }{\left\|x-{x}_{_{i} }\right\|^{2}}\right)$ 多二次函数 $\varphi \left(x,{x}_{\boldsymbol{{i}}}\right)=\sqrt{\left({ {\sigma }_{{\rm{r}}} ^{2}}+{\left\|x-{x}_{_{i}}\right\|^{2}}\right)}$ 逆多二次函数 $\varphi \left(x,{x}_{_{i}}\right)={\left({ {\sigma }_{{\rm{r}}} ^{2}}+{\left\|x-{x}_{_{i}}\right\|^{2}}\right)}^{-1/2}$ 
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表 2 SPALLER-4设计参数
Table 2. Design Parameters of SPALLER-4
设计参数 参数值 热功率/MW 4 换料周期/EFPY 10 燃料装载量/kg 577.89 活性区高度/cm 80 燃料中Pu的质量分数/% 31/48（内/外） 固体慢化剂厚度/cm 3.5 栅径比 1.2 燃料棒内径/cm 0.6 燃料材料 PuN-ThN 冷却剂材料 208Pb-Bi 反射层材料 208Pb-Bi 屏蔽层材料 B4C 固体慢化剂材料 BeO 燃料棒包壳材料 HT-9 燃料棒气隙填充物 He
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表 3 SPALLER-4优化变量取值区间
Table 3. SPALLER-4 Optimization Variable Value Interval
设计参数 优化后取值区间 活性区高度/cm [30, 150] 燃料中Pu的质量分数/% [25, 50] 固体慢化剂厚度/cm [0, 20] 栅径比 [1.01, 1.5] 燃料棒内径/cm [0.2, 0.6]
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表 4 RBF代理模型预测结果与蒙卡程序RMC计算值对比
Table 4. Comparison between Predicted Results of RBF Surrogate Model and RMC Calculated Values of Monte Carlo Program
参数名 参数值 第1组 第2组 第3组 第4组 第5组 固体慢化剂厚度/cm 4.4953 4.8964 4.7991 4.8642 5.2025 Pu的质量分数/% 49.4608 48.2705 47.7474 49.8008 47.7586 燃料芯块半径/cm 0.2072 0.2084 0.2394 0.2228 0.2370 堆芯活性区高度/cm 96.0945 95.2556 92.2671 82.7424 97.9885 栅径比 1.1167 1.2655 1.2864 1.2199 1.1223 第三年keff RBF预测值 0.9836 0.9835 0.9975 0.9753 0.9957 RMC计算值 0.9831 0.9830 0.9977 0.9756 0.9952 相对误差/% 0.0509 0.0509 −0.0200 −0.0308 0.0502
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表 5 SPALLER-4优化方案
Table 5. Optimization Scheme of SPALLER-4
设计方案 优化参数 固体慢化剂厚度/cm 4.8256 燃料中Pu的质量分数/% 49.9995 燃料芯块半径/cm 0.2083 堆芯活性区高度/cm 97.8064 栅径比 1.0753 堆芯初始keff 1.0338 第三年的keff RBF预测值 1.0010 RMC计算值 1.0005 相对误差/% 0.0475 换料周期/EFPY 3 燃料装载量/kg 86.0881 燃料包壳最大温度/℃ 576.7063 燃料芯块中心温度/℃ 923.3051
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