基于响应面与遗传算法的固体氧化物燃料电池集成系统多目标优化

doi:10.11949/0438-1157.20250705

摘要/Abstract

摘要：

固体氧化物燃料电池（solid oxide fuel cell， SOFC）具有高效、清洁的能源转换特性。SOFC集成系统的优化研究则通过多参数协同调控进一步提升系统效率与长期稳定性，对推动低碳能源技术的发展具有重要意义。本研究基于流道中设置矩形障碍物的SOFC模型，采用Plackett-Burman试验设计筛选关键工作参数，并利用响应面分析法分析参数交互作用，确定多目标优化的必要性。通过单因素试验确定参数范围，筛选出工作温度、压力等关键因素，结合辅助系统参数建立响应面模型。采用非支配排序遗传算法（non-dominated sorting genetic algorithm Ⅱ， NSGA-Ⅱ）和多目标粒子群优化遗传算法（multi-objective particle swarm optimization genetic algorithm， MOPSO-GA）对系统效率与衰退率进行多目标优化，结果表明：NSGA-Ⅱ优化后系统效率达87.19%（提升22.62%），衰退率为71.75%（增加4.64%）；MOPSO-GA优化后效率为83.89%（提升19.32%），衰退率为68.12%（增加1.01%）。NSGA-Ⅱ在效率提升方面更优，而MOPSO-GA更适用于长期稳定运行需求。

关键词: 燃料电池, 优化, 响应面分析法, NSGA-Ⅱ算法, MOPSO-GA算法, 数值模拟

Abstract:

Solid oxide fuel cell (SOFC) is of great significance in promoting the development of low-carbon energy technology due to its high efficiency and clean energy conversion characteristics, while the optimization study of SOFC integrated system further improves the system efficiency and long-term stability through the synergistic regulation of multi-parameter, which is of great significance in promoting the development of low-carbon energy technology. In this study, based on the SOFC model with rectangular obstacles set up in the cathode and anode flow paths, the Plackett-Burman (PB) experimental design was used to screen the key operating parameters, and the parameter interactions were analysed using the response surface analysis methodology to determine the necessity of multi-objective optimization. The parameter ranges were determined by a one-factor test, and key factors such as operating temperature, pressure and oxygen concentration were screened and combined with auxiliary system parameters to establish a response surface model. The NSGA-Ⅱ and MOPSO-GA algorithms were used to perform multi-objective optimization of system efficiency and recession rate, and the results showed that: after NSGA-Ⅱ optimization, the system efficiency reached 87.19% (an increase of 22.62%), and the recession rate was 71.75% (an increase of 4.64%); after MOPSO-GA optimization, the efficiency was 83.89% (an increase of 19.32%), and the recession rate was 68.12% (1.01% increase).NSGA-Ⅱ is superior in terms of efficiency improvement, while MOPSO-GA is more suitable for long-term stable operation requirements.

Key words: fuel cells, optimization, response surface analysis methodology, NSGA-Ⅱ algorithm, MOPSO-GA algorithm, numerical simulation

中图分类号:

TM 911.42

刘进一, 陈龙, 王巧, 付丽荣, 赵映. 基于响应面与遗传算法的固体氧化物燃料电池集成系统多目标优化[J]. 化工学报, 2025, 76(11): 5965-5979.

Jinyi LIU, Long CHEN, Qiao WANG, Lirong FU, Ying ZHAO. Multi-objective optimization of solid oxide fuel cell integrated system based on response surface and genetic algorithm[J]. CIESC Journal, 2025, 76(11): 5965-5979.

图/表 13

图1 SOFC几何结构示意图：(a)障碍物分布三维；(b)二维结构示意图；(c)障碍物分布二维图

Fig.1 Schematic of SOFC geometry: (a)3D schematic of obstacle distribution; (b)2D structure diagram; (c)2D schematic of obstacle distribution

图2 SOFC集成系统的结构示意图

Fig.2 Schematic structure of SOFC integrated system

图3 两种遗传算法的优化流程对比

Fig.3 Comparison of the optimization process of the two genetic algorithms

图4 单电池的模型验证

Fig.4 Model validation for a single cell

表1 PB试验显著性分析

Table1 Significance analysis of the PB experiment

参数	效应值	平方和	均方	F值	P值	显著性
A	0.00577	0.0001	0.0001	461.71	0.0296	显著
B	0.0439	0.0058	0.0058	28846.47	0.0037	极显著
C	0.0447	0.0060	0.0060	29724.87	0.0037	极显著
D	0.1968	0.1162	0.1162	578000.00	0.0008	极显著
E	0.2391	0.1715	0.1715	852000.00	0.0007	极显著

表2 系统衰退率模型BBD方差分析

Table 2 Variance analysis for the BBD of the system recession rate model

方差来源	平方和		自由度	均方		F值	P值		显著性
R² 0.9673		修正R² 0.9232			信噪比显著 16.8110			变异系数 4.40
模型	6967.46		35	199.07		21.96	<0.0001		极显著
F: p_air	38.32		1	38.32		4.23	0.0421		显著
G: p_fuel	40.04		1	40.04		4.42	0.03967		显著
H: p_pump	60.98		1	60.98		6.72	0.01159		显著
I:η_HE	69.68		1	69.68		7.68	0.0072		显著
J: T_SOFC	1284.79		1	1284.79		141.76	<0.0001		极显著
K: V	32.17		1	32.17		3.55	0.0708		不显著
L: $m O 2$	1550.40		1	1550.40		171.07	<0.0001		极显著
FG	42.53		1	42.53		4.69	0.03391		显著
FH	4.35		1	4.35		0.48	0.6569		不显著
FI	26.82		1	26.82		2.96	0.8989		不显著
FJ	6.25		1	6.25		0.69	0.5330		不显著
FK	1.54		1	1.54		0.17	0.1226		不显著
FL	2.08		1	2.08		0.23	0.1690		不显著
GH	3.90		1	3.90		0.43	0.3257		不显著
GI	1.36		1	1.36		0.15	0.0997		不显著
GJ	16.67		1	16.67		1.84	0.8001		不显著
GK	1.18		1	1.18		0.13	0.0764		不显著
GL	1.09		1	1.09		0.12	0.0669		不显著
HI	50.10		1	50.10		5.53	0.0210		显著
HJ	13.05		1	13.05		1.44	0.2139		不显著
HK	5.44		1	5.44		0.60	0.5770		不显著
HL	15.86		1	15.86		1.75	0.1863		不显著
IJ	7.43		1	7.43		0.82	0.4454		不显著
IK	6.98		1	6.98		0.77	0.4721		不显著
IL	4.80		1	4.80		0.53	0.6301		不显著
JK	1.21		1	1.21		0.1332	0.7181		不显著
JL	41.55		1	41.55		4.58	0.0418		显著
KL	157.97		1	157.97		17.43	0.0003		显著
F²	12.01		1	12.01		1.33	0.2601		不显著
G²	0.9242		1	0.9242		0.1020	0.7520		不显著
H²	20.91		1	20.91		2.31	0.1409		不显著
I²	19.60		1	19.60		2.16	0.1534		不显著
J²	2602.41		1	2602.41		287.14	<0.0001		极显著
K²	2.15		1	2.15		0.2377	0.6299		不显著
L²	1339.39		1	1339.39		147.78	<0.0001		极显著
残差	235.64		26	9.06

表2 系统衰退率模型BBD方差分析

Table 2 Variance analysis for the BBD of the system recession rate model

方差来源	平方和		自由度	均方		F值	P值		显著性
R² 0.9673		修正R² 0.9232			信噪比显著 16.8110			变异系数 4.40
模型	6967.46		35	199.07		21.96	<0.0001		极显著
F: p_air	38.32		1	38.32		4.23	0.0421		显著
G: p_fuel	40.04		1	40.04		4.42	0.03967		显著
H: p_pump	60.98		1	60.98		6.72	0.01159		显著
I:η_HE	69.68		1	69.68		7.68	0.0072		显著
J: T_SOFC	1284.79		1	1284.79		141.76	<0.0001		极显著
K: V	32.17		1	32.17		3.55	0.0708		不显著
L: $m O 2$	1550.40		1	1550.40		171.07	<0.0001		极显著
FG	42.53		1	42.53		4.69	0.03391		显著
FH	4.35		1	4.35		0.48	0.6569		不显著
FI	26.82		1	26.82		2.96	0.8989		不显著
FJ	6.25		1	6.25		0.69	0.5330		不显著
FK	1.54		1	1.54		0.17	0.1226		不显著
FL	2.08		1	2.08		0.23	0.1690		不显著
GH	3.90		1	3.90		0.43	0.3257		不显著
GI	1.36		1	1.36		0.15	0.0997		不显著
GJ	16.67		1	16.67		1.84	0.8001		不显著
GK	1.18		1	1.18		0.13	0.0764		不显著
GL	1.09		1	1.09		0.12	0.0669		不显著
HI	50.10		1	50.10		5.53	0.0210		显著
HJ	13.05		1	13.05		1.44	0.2139		不显著
HK	5.44		1	5.44		0.60	0.5770		不显著
HL	15.86		1	15.86		1.75	0.1863		不显著
IJ	7.43		1	7.43		0.82	0.4454		不显著
IK	6.98		1	6.98		0.77	0.4721		不显著
IL	4.80		1	4.80		0.53	0.6301		不显著
JK	1.21		1	1.21		0.1332	0.7181		不显著
JL	41.55		1	41.55		4.58	0.0418		显著
KL	157.97		1	157.97		17.43	0.0003		显著
F²	12.01		1	12.01		1.33	0.2601		不显著
G²	0.9242		1	0.9242		0.1020	0.7520		不显著
H²	20.91		1	20.91		2.31	0.1409		不显著
I²	19.60		1	19.60		2.16	0.1534		不显著
J²	2602.41		1	2602.41		287.14	<0.0001		极显著
K²	2.15		1	2.15		0.2377	0.6299		不显著
L²	1339.39		1	1339.39		147.78	<0.0001		极显著
残差	235.64		26	9.06

表3 系统效率模型的BBD方差分析

Table 3 Variance analysis for the BBD of the system efficiency model

方差来源	平方和	自由度	均方	F值	P值	显著性
R² 0.9988		修正R² 0.9972		信噪比 109.8268	变异系数 0.5896
模型	3213.0200	35	91.8000	613.83	<0.0001	极显著
F: p_air	0.6930	1	0.6930	6.03	0.0201	显著
G: p_fuel	0.5400	1	0.5400	4.69	0.0339	显著
H: p_pump	0.5750	1	0.5750	5.00	0.0311	显著
I: η_HE	0.7240	1	0.7240	6.30	0.0159	显著
J: T_SOFC	1433.1700	1	1433.1700	9582.95	<0.0001	极显著
K: V	3.8200	1	3.8200	25.53	<0.0001	极显著
L: $m O 2$	1715.4700	1	1715.4700	11470.53	<0.0001	极显著
FG	0.4860	1	0.4860	4.23	0.0477	显著
FH	0.0437	1	0.0437	0.38	0.7101	不显著
FI	0.0310	1	0.0310	0.27	0.7990	不显著
FJ	0.1391	1	0.1391	1.21	0.2661	不显著
FK	0.0999	1	0.0999	0.87	0.3774	不显著
FL	0.0632	1	0.0632	0.55	0.5866	不显著
GH	0.2391	1	0.2391	2.08	0.1559	不显著
GI	0.0563	1	0.0563	0.49	0.6334	不显著
GJ	0.3196	1	0.3196	2.78	0.1197	不显著
GK	0.0816	1	0.0816	0.71	0.4770	不显著
GL	0.1127	1	0.1127	0.98	0.3391	不显著
HI	0.8050	1	0.8050	7.00	0.0128	显著
HJ	0.1586	1	0.1586	1.38	0.2408	不显著
HK	0.6103	1	0.6103	4.08	0.5038	不显著
HL	0.3572	1	0.3572	2.39	0.1343	不显著
IJ	0.1333	1	0.1333	1.16	0.3018	不显著
IK	0.1357	1	0.1357	1.18	0.2933	不显著
IL	0.2747	1	0.2747	2.39	0.1107	不显著
JK	0.1688	1	0.1688	1.13	0.2978	不显著
JL	0.5200	1	0.5200	4.53	0.0329	显著
KL	0.4400	1	0.4400	3.83	0.0418	显著
F²	0.0629	1	0.0629	0.42	0.5224	不显著
G²	0.0011	1	0.0011	<0.01	0.9336	不显著
H²	0.1168	1	0.1168	0.78	0.3849	不显著
I²	0.0476	1	0.0476	0.32	0.5774	不显著
J²	30.7700	1	30.7700	205.78	<0.0001	极显著
K²	16.6400	1	16.6400	111.29	<0.0001	极显著
L²	6.2200	1	6.2200	41.58	<0.0001	极显著
残差	3.8900	26	0.1149

表3 系统效率模型的BBD方差分析

Table 3 Variance analysis for the BBD of the system efficiency model

方差来源	平方和	自由度	均方	F值	P值	显著性
R² 0.9988		修正R² 0.9972		信噪比 109.8268	变异系数 0.5896
模型	3213.0200	35	91.8000	613.83	<0.0001	极显著
F: p_air	0.6930	1	0.6930	6.03	0.0201	显著
G: p_fuel	0.5400	1	0.5400	4.69	0.0339	显著
H: p_pump	0.5750	1	0.5750	5.00	0.0311	显著
I: η_HE	0.7240	1	0.7240	6.30	0.0159	显著
J: T_SOFC	1433.1700	1	1433.1700	9582.95	<0.0001	极显著
K: V	3.8200	1	3.8200	25.53	<0.0001	极显著
L: $m O 2$	1715.4700	1	1715.4700	11470.53	<0.0001	极显著
FG	0.4860	1	0.4860	4.23	0.0477	显著
FH	0.0437	1	0.0437	0.38	0.7101	不显著
FI	0.0310	1	0.0310	0.27	0.7990	不显著
FJ	0.1391	1	0.1391	1.21	0.2661	不显著
FK	0.0999	1	0.0999	0.87	0.3774	不显著
FL	0.0632	1	0.0632	0.55	0.5866	不显著
GH	0.2391	1	0.2391	2.08	0.1559	不显著
GI	0.0563	1	0.0563	0.49	0.6334	不显著
GJ	0.3196	1	0.3196	2.78	0.1197	不显著
GK	0.0816	1	0.0816	0.71	0.4770	不显著
GL	0.1127	1	0.1127	0.98	0.3391	不显著
HI	0.8050	1	0.8050	7.00	0.0128	显著
HJ	0.1586	1	0.1586	1.38	0.2408	不显著
HK	0.6103	1	0.6103	4.08	0.5038	不显著
HL	0.3572	1	0.3572	2.39	0.1343	不显著
IJ	0.1333	1	0.1333	1.16	0.3018	不显著
IK	0.1357	1	0.1357	1.18	0.2933	不显著
IL	0.2747	1	0.2747	2.39	0.1107	不显著
JK	0.1688	1	0.1688	1.13	0.2978	不显著
JL	0.5200	1	0.5200	4.53	0.0329	显著
KL	0.4400	1	0.4400	3.83	0.0418	显著
F²	0.0629	1	0.0629	0.42	0.5224	不显著
G²	0.0011	1	0.0011	<0.01	0.9336	不显著
H²	0.1168	1	0.1168	0.78	0.3849	不显著
I²	0.0476	1	0.0476	0.32	0.5774	不显著
J²	30.7700	1	30.7700	205.78	<0.0001	极显著
K²	16.6400	1	16.6400	111.29	<0.0001	极显著
L²	6.2200	1	6.2200	41.58	<0.0001	极显著
残差	3.8900	26	0.1149

表4 决策参数的约束条件

Table 4 Constraints on decision parameters

决策变量	符号	上限	下限	单位
空气压缩机工作压力	p_air	10	1	atm
燃料压缩机工作压力	p_fuel	10	1	atm
水泵工作压力	p_pump	5	0.5	atm
热交换效率	η_HE	0.9	0.5	—
工作温度	T_SOFC	900	600	℃
工作电压	V	0.95	0.1	V
氧气浓度	$m O 2$	0.448	0.21	—

表4 决策参数的约束条件

Table 4 Constraints on decision parameters

决策变量	符号	上限	下限	单位
空气压缩机工作压力	p_air	10	1	atm
燃料压缩机工作压力	p_fuel	10	1	atm
水泵工作压力	p_pump	5	0.5	atm
热交换效率	η_HE	0.9	0.5	—
工作温度	T_SOFC	900	600	℃
工作电压	V	0.95	0.1	V
氧气浓度	$m O 2$	0.448	0.21	—

图5 SOFC关键参数对系统效率的影响

Fig.5 Effect of key SOFC parameters on system efficiency

图6 SOFC关键参数对系统衰退率的影响

Fig.6 Effect of key SOFC parameters on system recession rate

图7 辅助元件关键参数对系统效率的影响

Fig.7 Effect of key parameters of auxiliary components on system efficiency

图8 辅助元件关键参数对系统衰退率的影响

Fig.8 Effect of key parameters of auxiliary components on system recession rate

图9 Pareto前沿与最优解和优化结果对比

Fig.9 Pareto frontier and optimal solution and comparison of optimization results

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