基于并行EGO和代理模型辅助的多参数优化方法研究

doi:10.11949/0438-1157.20221345

摘要/Abstract

摘要：

化工流程模拟优化问题常常具有高维、非线性的特点，使得仿真计算难以收敛。过长的求解时间是调度优化和运行优化的主要瓶颈之一。采用代理模型对机理模型进行替代是降低计算复杂度、保证结果准确性的有效途径。Kriging代理模型具有较强的非线性近似性，但处理高维问题依然较为困难。因此，本文研究并行EGO(efficient global optimization)算法与代理模型集成，并将模型应用于化工过程。并行EGO算法以Kriging代理模型的预测函数和误差函数为基础，先推导出样本分布概率密度函数与累积分布函数相结合的解析表达式；然后通过PEI(pseudo expected improvement)准则得到新的样本点以更新代理模型；最后结合改进的差分进化算法对优化参数进行全局搜索。在保证结果准确性的前提下，将本文算法与其他优化算法进行比较。8个多峰测试函数的测试结果表明，该算法的收敛速度提高了85%。然后将其应用于双级氨吸收制冷过程的模拟，结果表明该方法的模拟误差小于0.01%，优化时间从9846 s缩短至3705 s。

关键词: 代理模型, 并行EGO算法, 多参数优化, 流程模拟

Abstract:

Process simulation and optimization have the characteristics of high-dimensional and non-linear, which makes the simulation calculation difficult to converge. Excessive solution time is one of the bottlenecks during scheduling and operation optimization. Using surrogate models to replace mechanistic models is an effective way to reduce computational complexity and ensure the accuracy of results. Though the Kriging surrogate model has a stronger nonlinear approximation, it is still difficult to deal with high-dimensional problems. Therefore, this paper investigates the parallel EGO (efficient global optimization) algorithm integrated with the surrogate model and applies the model to typical chemical process. The parallel EGO algorithm is based on the prediction function and error function of the Kriging surrogate model. First, an analytical expression is derived from data sample combined by the probability density function and cumulative distribution function. Then the surrogate model is updated by new sample points obtained by PEI (pseudo expected improvement) criterion. Finally, combined with the improved differential evolution algorithm, the optimization parameters are searched globally. Under the premise of ensuring the accuracy of the results, the algorithm in this paper is compared with other optimization algorithms. Taking eight multi-peaked test functions as a test case, it is found that the convergence speed of the algorithm is improved by 85%. Then, it is applied to the study case of the two-stage ammonia absorption refrigeration process. The results show that the simulation error is less than 0.01%, and the optimization time is reduced from 9846 s to 3705 s.

Key words: surrogate model, parallel efficient global optimization algorithm, multi-parameter optimization, process simulation

中图分类号:

TQ 062.2

顾学荣, 刘硕士, 杨思宇. 基于并行EGO和代理模型辅助的多参数优化方法研究[J]. 化工学报, 2023, 74(3): 1205-1215.

Xuerong GU, Shuoshi LIU, Siyu YANG. Research on multi-parameter optimization method based on parallel EGO and surrogate-assisted model[J]. CIESC Journal, 2023, 74(3): 1205-1215.

图/表 11

表1 Hartman3和Hartman6相关参数

Table 1 Parameters related to Hartman3 and Hartman6

函数	参数
Hartman3	$α_{i j} = (\begin{matrix} 3.0 & 10.0 & 30.0 \\ 0.1 & 10.0 & 35.0 \\ 3.0 & 10.0 & 30.0 \\ 0.1 & 10.0 & 35.0 \end{matrix})$
Hartman3	$p_{i j} = (\begin{matrix} 0.3689 & 0.1170 & 0.2673 \\ 0.4699 & 0.4378 & 0.7470 \\ 0.1091 & 0.8732 & 0.5547 \\ 0.03815 & 0.5743 & 0.8828 \end{matrix})$
Hartman6	$α_{i j} = (\begin{matrix} 10.00 & 3.00 & 17.00 & 3.50 & 1.70 & 8.00 \\ 0.05 & 10.00 & 17.00 & 0.10 & 8.00 & 14.00 \\ 3.00 & 3.50 & 1.70 & 10.00 & 17.00 & 8.00 \\ 17.00 & 8.00 & 0.05 & 10.00 & 0.10 & 14.00 \end{matrix})$
Hartman6	$p_{i j} = (\begin{matrix} 0.1312 & 0.1696 & 0.5569 & 0.0124 & 0.8283 & 0.5886 \\ 0.2329 & 0.4135 & 0.8307 & 0.3736 & 0.1004 & 0.9991 \\ 0.2348 & 0.1451 & 0.3522 & 0.2883 & 0.3047 & 0.6650 \\ 0.4047 & 0.8828 & 0.8732 & 0.5743 & 0.1091 & 0.0381 \end{matrix})$

表1 Hartman3和Hartman6相关参数

Table 1 Parameters related to Hartman3 and Hartman6

函数	参数
Hartman3	$α_{i j} = (\begin{matrix} 3.0 & 10.0 & 30.0 \\ 0.1 & 10.0 & 35.0 \\ 3.0 & 10.0 & 30.0 \\ 0.1 & 10.0 & 35.0 \end{matrix})$
Hartman3	$p_{i j} = (\begin{matrix} 0.3689 & 0.1170 & 0.2673 \\ 0.4699 & 0.4378 & 0.7470 \\ 0.1091 & 0.8732 & 0.5547 \\ 0.03815 & 0.5743 & 0.8828 \end{matrix})$
Hartman6	$α_{i j} = (\begin{matrix} 10.00 & 3.00 & 17.00 & 3.50 & 1.70 & 8.00 \\ 0.05 & 10.00 & 17.00 & 0.10 & 8.00 & 14.00 \\ 3.00 & 3.50 & 1.70 & 10.00 & 17.00 & 8.00 \\ 17.00 & 8.00 & 0.05 & 10.00 & 0.10 & 14.00 \end{matrix})$
Hartman6	$p_{i j} = (\begin{matrix} 0.1312 & 0.1696 & 0.5569 & 0.0124 & 0.8283 & 0.5886 \\ 0.2329 & 0.4135 & 0.8307 & 0.3736 & 0.1004 & 0.9991 \\ 0.2348 & 0.1451 & 0.3522 & 0.2883 & 0.3047 & 0.6650 \\ 0.4047 & 0.8828 & 0.8732 & 0.5743 & 0.1091 & 0.0381 \end{matrix})$

表2 测试函数

Table 2 Test functions

测试函数	变量维度	初始样本点数	最大迭代次数
Six-hump	2	20	400
Branin	2	20	400
GoldPrice	2	20	400
Hartman3	3	30	400
Shekekl5	4	40	400
Shekekl7	4	40	400
Shekekl10	4	40	400
Hartman6	6	60	400

图1 不同优化算法在测试函数上的迭代历史曲线

Fig.1 Iteration history curves of different optimization algorithms on test functions

表3 收敛到最优解所需的迭代次数

Table 3 The number of iterations required to converge to the optimal solution

测试函数	变量维度	EGO算法	并行EGO算法
测试函数	变量维度	q=1	q=2	q=4	q=6	q=8	q=10
Six-hump	2	29.4	16.2	9.0	6.9	6.1	4.4
Branin	2	31.9	14.5	10.8	7.5	6.2	5.3
GoldPrice	2	97.8	47.3	24.3	21.2	15.6	11.6
Hartman3	3	16.0	14.0	9.0	7.0	6.0	4.0
Shekekl5	4	67.9	37.9	17.7	13.7	10.4	9.6
Shekekl7	4	165.0	54.1	33.5	19.8	17.2	13.0
Shekekl10	4	84.6	43.3	26.5	17.0	12.6	10.9
Hartman6	6	28.7	12.6	8.3	6.9	5.7	4.9

图2 不同多峰测试函数的运行时间

Fig.2 Running time of different multi-peak test functions

图3 双级氨吸收制冷系统流程简图

Fig.3 Schematic diagram of two-stage ammonia absorption refrigeration process

表4 氨吸收制冷流程的优化变量及约束范围

Table 4 Optimization variables and their constraints on ammonia absorption refrigeration

符号	物理意义	下限	上限
x₁	氨气( $N H_{3}$ )进料流量/（kmol/h）	5000	7000
x₂	闪蒸器气相分数	0.05	0.35
x₃	闪蒸器操作压力/kPa	1.0	1.4
x₄	换热器1换热温度/℃	40	50
x₅	泵压(泵1)/kPa	1450	1550
x₆	预热器换热温度/℃	10200	10700
x₇	精馏塔回流比	0.4	0.6
x₈	精馏塔塔顶采出率	0.45	0.65
x₉	过冷器换热温度/℃	10	15
x₁₀	压力调节阀/kPa	70	80
x₁₁	压缩机排气压力/kPa	100	120
x₁₂	换热器2换热温度/℃	35	45
x₁₃	吸收器换热温度/℃	40	50
x₁₄	泵压(泵2)/kPa	700	900
x₁₅	出口氨溶液(S)分流率	0.01	0.05

表4 氨吸收制冷流程的优化变量及约束范围

Table 4 Optimization variables and their constraints on ammonia absorption refrigeration

符号	物理意义	下限	上限
x₁	氨气( $N H_{3}$ )进料流量/（kmol/h）	5000	7000
x₂	闪蒸器气相分数	0.05	0.35
x₃	闪蒸器操作压力/kPa	1.0	1.4
x₄	换热器1换热温度/℃	40	50
x₅	泵压(泵1)/kPa	1450	1550
x₆	预热器换热温度/℃	10200	10700
x₇	精馏塔回流比	0.4	0.6
x₈	精馏塔塔顶采出率	0.45	0.65
x₉	过冷器换热温度/℃	10	15
x₁₀	压力调节阀/kPa	70	80
x₁₁	压缩机排气压力/kPa	100	120
x₁₂	换热器2换热温度/℃	35	45
x₁₃	吸收器换热温度/℃	40	50
x₁₄	泵压(泵2)/kPa	700	900
x₁₅	出口氨溶液(S)分流率	0.01	0.05

图4 Kriging模型预测值与实际值对比

Fig.4 Comparison between predicted and actual values of Kriging model

图5 不同优化算法寻优迭代曲线

Fig.5 Optimization iteration curves for different optimization algorithms

表5 参数优化结果

Table 5 The results of parameter optimization

变量	q=1	q=2	q=4	q=6	q=8	q=10
x₁	7000.00	5954.92	5959.70	5950.70	5952.17	5953.89
x₂	0.16	0.20	0.24	0.14	0.13	0.17
x₃	800.67	807.86	806.46	804.51	804.41	803.06
x₄	42.79	46.63	48.87	46.58	47.45	48.16
x₅	1550.00	1497.80	1507.33	1506.61	1500.00	1497.38
x₆	10.00	5.92	7.63	6.34	5.95	5.71
x₇	0.47	0.48	0.45	0.51	0.46	0.52
x₈	0.65	0.53	0.61	0.57	0.57	0.56
x₉	10.00	11.95	12.61	10.00	10.00	10.92
x₁₀	70.00	74.51	70.67	76.00	71.21	71.00
x₁₁	100.24	110.65	100.00	107.04	106.37	108.15
x₁₂	43.83	39.79	42.72	41.98	45.00	41.01
x₁₃	50.00	42.41	40.00	45.80	44.00	40.42
x₁₄	800.00	819.71	813.15	816.92	806.09	808.41
x₁₅	0.04	0.04	0.05	0.04	0.03	0.04
绝对误差	0.1566	0.0041	0.0010	0.00058	0.00044	0.00016

图6 并行EGO优化算法的运行时间

Fig.6 Running time of parallel EGO optimization algorithms

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