Multi-objective optimization of solid oxide fuel cell integrated system based on response surface and genetic algorithm

doi:10.11949/0438-1157.20250705

Abstract

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

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

CLC Number:

TM 911.42

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.

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.

Figures/Tables 13

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

Fig.2 Schematic structure of SOFC integrated system

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

Fig.4 Model validation for a single cell

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

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

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

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

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

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	—

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	—

Fig.5 Effect of key SOFC parameters on system efficiency

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

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

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

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

References 31

[1]	邹才能, 马锋, 潘松圻, 等. 世界能源转型革命与绿色智慧能源体系内涵及路径[J]. 石油勘探与开发, 2023, 50(3): 633-647.
	Zou C N, Ma F, Pan S Q, et al. Global energy transition revolution and the connotation and pathway of the green and intelligent energy system[J]. Petroleum Exploration and Development, 2023, 50(3): 633-647.
[2]	邹才能, 何东博, 贾成业, 等. 世界能源转型内涵、路径及其对碳中和的意义[J]. 石油学报, 2021, 42(2): 233-247.
	Zou C N, He D B, Jia C Y, et al. Connotation and pathway of world energy transition and its significance for carbon neutral[J]. Acta Petrolei Sinica, 2021, 42(2): 233-247.
[3]	Zhang X, Xu W T, Rauf A, et al. Transitioning from conventional energy to clean renewable energy in G7 countries: a signed network approach[J]. Energy, 2024, 307: 132655.
[4]	毕锐, 王孝淦, 袁华凯, 等. 考虑供需双侧响应和碳交易的氢能综合能源系统鲁棒调度[J]. 电力系统保护与控制, 2023, 51(12): 122-132.
	Bi R, Wang X G, Yuan H K, et al. Robust dispatch of a hydrogen integrated energy system considering double side response and carbon trading mechanism[J]. Power System Protection and Control, 2023, 51(12): 122-132.
[5]	Zhu Z G, Ning H L, Song C, et al. Effect of low plasma spraying power on anode microstructure and performance for metal-supported solid oxide fuel cells[J]. Journal of Thermal Spray Technology, 2024, 33(5): 1725-1735.
[6]	洪吉超, 马世琨, 梁峰伟, 等. 氢燃料电池数字孪生技术的系统集成与智能管理[J]. 工程科学学报, 2025, 47(11): 2296-2308.
	Hong J C, Ma S K, Liang F W, et al. System integration and intelligent management of hydrogen fuel cells based on digital twin technology[J]. Chinese Journal of Engineering, 2025, 47(11): 2296-2308.
[7]	Beyrami J, Chitsaz A, Parham K, et al. Optimum performance of a single effect desalination unit integrated with a SOFC system by multi-objective thermo-economic optimization based on genetic algorithm[J]. Energy, 2019, 186: 115811.
[8]	Wang J Y, Hua J, Pan Z H, et al. Novel SOFC system concept with anode off-gas dual recirculation: a pathway to zero carbon emission and high energy efficiency[J]. Applied Energy, 2024, 361: 122862.
[9]	Khani L, Mehr A S, Yari M, et al. Multi-objective optimization of an indirectly integrated solid oxide fuel cell-gas turbine cogeneration system[J]. International Journal of Hydrogen Energy, 2016, 41(46): 21470-21488.
[10]	Hosseinpour J, Sadeghi M, Chitsaz A, et al. Exergy assessment and optimization of a cogeneration system based on a solid oxide fuel cell integrated with a Stirling engine[J]. Energy Conversion and Management, 2017, 143: 448-458.
[11]	Huang Y, Turan A. Fuel sensitivity and parametric optimization of SOFC-GT hybrid system operational characteristics[J]. Thermal Science and Engineering Progress, 2019, 14: 100407.
[12]	Roy D, Samanta S, Ghosh S. Performance optimization through response surface methodology of an integrated biomass gasification based combined heat and power plant employing solid oxide fuel cell and externally fired gas turbine[J]. Energy Conversion and Management, 2020, 222: 113182.
[13]	Khoshgoftar Manesh M H, Ghorbani S, Blanco-Marigorta A M. Optimal design and analysis of a combined freshwater-power generation system based on integrated solid oxide fuel cell-gas turbine-organic Rankine cycle-multi effect distillation system[J]. Applied Thermal Engineering, 2022, 211: 118438.
[14]	Thanganadar D, Asfand F, Patchigolla K, et al. Techno-economic analysis of supercritical carbon dioxide cycle integrated with coal-fired power plant[J]. Energy Conversion and Management, 2021, 242: 114294.
[15]	Xu G P, Yu Z T, Xia L, et al. Performance improvement of solid oxide fuel cells by combining three-dimensional CFD modeling, artificial neural network and genetic algorithm[J]. Energy Conversion and Management, 2022, 268: 116026.
[16]	Tang Z G, Xiang Y, Li M, et al. Multi-objective optimization of liquid-cooled battery thermal management system with biomimetic fractal channels using artificial neural networks and response surface methodology[J]. International Journal of Thermal Sciences, 2024, 206: 109304.
[17]	Su H R, Hu Y H. Progress in low-temperature solid oxide fuel cells with hydrocarbon fuels[J]. Chemical Engineering Journal, 2020, 402: 126235.
[18]	Hagen A, Wulff A C, Zielke P, et al. SOFC stacks for mobile applications with excellent robustness towards thermal stresses[J]. International Journal of Hydrogen Energy, 2020, 45(53): 29201-29211.
[19]	Salvador C A F, Rego J S, Lopes T, et al. Materials selection of non-metallic glasses for planar solid-oxide fuel cell sealants[J]. Ceramics International, 2025, 51(6): 6867-6878.
[20]	Hesami H, Borji M, Rezapour J. Three-dimensional numerical investigation on the effect of interconnect design on the performance of internal reforming planar solid oxide fuel cell[J]. Korean Journal of Chemical Engineering, 2021, 38(12): 2423-2435.
[21]	Sayadian S, Ghassemi M, Ahmadi S, et al. Numerical analysis of transport phenomena in solid oxide fuel cell gas channels[J]. Fuel, 2022, 311: 122557.
[22]	Sayadian S, Ghassemi M, Robinson A J. Multi-physics simulation of transport phenomena in planar proton-conducting solid oxide fuel cell[J]. Journal of Power Sources, 2021, 481: 228997.
[23]	Milewski J, Swirski K, Santarelli M, et al. Advanced Methods of Solid Oxide Fuel Cell Modeling [M]. Berlin: Springer Science & Business Media, 2011.
[24]	Patcharavorachot Y, Arpornwichanop A, Chuachuensuk A. Electrochemical study of a planar solid oxide fuel cell: role of support structures[J]. Journal of Power Sources, 2008, 177(2): 254-261.
[25]	Wu Z, Zhu P F, Yao J, et al. Thermo-economic modeling and analysis of an NG-fueled SOFC-WGS-TSA-PEMFC hybrid energy conversion system for stationary electricity power generation[J]. Energy, 2020, 192: 116613.
[26]	Yahya A, Ferrero D, Dhahri H, et al. Electrochemical performance of solid oxide fuel cell: experimental study and calibrated model[J]. Energy, 2018, 142: 932-943.
[27]	Chang J S, Jiao M H, Zhang P P, et al. Analyzing the degradation mechanism of solid oxide fuel cell during different time periods[J]. Electrochimica Acta, 2024, 498: 144615.
[28]	Zhu P F, Wu Z, Wang H, et al. Ni coarsening and performance attenuation prediction of biomass syngas fueled SOFC by combining multi-physics field modeling and artificial neural network[J]. Applied Energy, 2022, 322: 119508.
[29]	He J, Wang X X, Jin F J, et al. System simulation and parametric analysis of low-concentration coal mine gas fed SOFC-CHP with deoxygenation and enrichment pretreatment[J]. Fuel, 2024, 373: 132300.
[30]	Petrone R, Hissel D, Péra M C, et al. Accelerated stress test procedures for PEM fuel cells under actual load constraints: state-of-art and proposals[J]. International Journal of Hydrogen Energy, 2015, 40(36): 12489-12505.
[31]	Shao Y Q, Shen J J, Ren H, et al. The influence of microstructural evolution on performance degradation in solid oxide fuel cells[J]. Materials Science and Engineering: B, 2025, 317: 118187.