Prediction of remaining useful life of lithium batteries based on grey wolf optimization and combined kernel function GPR model

doi:10.3969/j.issn.2097-0706.2025.12.003

Abstract

Abstract:

Accurately predicting the remaining useful life （RUL） of lithium-ion batteries can help users formulate reasonable maintenance strategies. As a data-driven method， Gaussian process regression （GPR） algorithm can effectively capture the nonlinear relationship between features and variables， and thus is widely used in the RUL estimation of lithium batteries. To address the deficiency of traditional single-kernel GPR in feature capture， a combined kernel GPR model based on the grey wolf optimization （GWO） algorithm was proposed. By combining kernel functions， the model's ability to capture nonlinear features was enhanced， and the GWO algorithm was utilized to overcome the difficulty in optimizing the hyperparameters of the combined kernel function. The NASA lithium battery cycle aging dataset was adopted to verify this model， and the single-kernel GPR model based on particle swarm（PSO） optimization was selected for comparison. The experimental results showed that the GWO-optimized combined kernel GPR model achieved a 37.89% reduction in root mean square error （RMSE） and a 70.42% reduction in mean absolute error （MAE） compared with the PSO-optimized single-kernel GPR model， demonstrating a stronger ability to capture capacity degradation. The results indicate that compared with the traditional GPR model， the GWO-optimized combined kernel GPR model has higher accuracy for the RUL prediction of lithium batteries.

Key words: lithium-ion battery, remaining useful life, Gaussian process regression, grey wolf optimization algorithm, battery aging, capacity degradation, prediction error, particle swarm optimization

CLC Number:

TM912

HU Linjing, LI Zaiwei. Prediction of remaining useful life of lithium batteries based on grey wolf optimization and combined kernel function GPR model[J]. Integrated Intelligent Energy, 2025, 47(12): 25-33.

Figures/Tables 15

Fig.1

Fig.2

Fig.3

Fig.4

Table 1

Pearson correlation between characteristics and capacity

电池编号	与容量的Pearson相关性
电池编号	$H 1$	$H 2$	$H 3$	$H 4$
B0005	0.908	0.982	0.988	-0.972
B0006	0.918	0.971	0.983	-0.970
B0007	0.785	0.978	0.979	-0.959

Table 1

Table 2

Fig.5

Table 3

Common simple kernel functions

函数	数学表达式
高斯核	$k (x, x') = σ 2 e x p (- x - x' 2 2 l 2)$
马顿核	$k (x, x') = σ 2 (1 + 3 x - x' l) e x p (- 3 x - x' l)$
线性核	$k (x, x') = σ 02 + x x'$
周期核	$k (x, x') = σ 2 e x p (- 2 s i n 2 (π x - x' / p) l 2)$
指数核	$k (x, x') = σ 2 e x p (- x - x' l)$
拉普拉斯核	$k (x, x') = e x p (- x - x' l)$

Table 3

Fig.6

Fig.7

Fig.8

Fig.9

Table 4

Fig.10

Table 5

References 27

[1]	喻子逸, 黄晓凡, 李佳瑞, 等. 梯次利用动力电池储能系统综合效益分析[J]. 综合智慧能源, 2024, 46(7): 63-73. doi: 10.3969/j.issn.2097-0706.2024.07.008
	HUANG Xiaofan, LI Jiarui, LIU Hui, et al. Comprehensive benefit analysis on the cascade utilization of a power battery system[J]. Integrated Intelligent Energy, 2024, 46(7): 63-73. doi: 10.3969/j.issn.2097-0706.2024.07.008
[2]	刘月峰, 张公, 张晨荣, 等. 锂离子电池RUL预测方法综述计算机工程[J]. 2020, 46 (4): 11-18.
	LIU Yuefeng, ZHANG Gong, ZHANG Chenrong, et al. Review of RUL prediction method for lithium-ion batteries[J]. 2020, 46 (4): 11-18.
[3]	张若可, 郭永芳, 余湘媛, 等. 基于数据驱动的锂离子电池RUL预测综述[J]. 电源学报, 2023, 21(5): 182-190. doi: 10.13234/j.issn.2095-2805.2023.5.182
	ZHANG Ruoke, GUO Yongfang, YU Xiangyuan, et al. Summary of data-driven prediction of RUL for lithium-ion batteries[J]. Journal of Power Supply, 2023, 21(5): 182-190. doi: 10.13234/j.issn.2095-2805.2023.5.182
[4]	刘大同, 宋宇晨, 武巍, 等. 锂离子电池组健康状态估计综述[J]. 仪器仪表学报, 2020, 41(11): 1-18.
	LIU Datong, SONG Yuchen, WU Wei, et al. Review of state of health estimation for lithium-ion battery pack[J]. Chinese Journal of Scientific Instrument, 2020, 41(11): 1-18.
[5]	LI L, SONG Y C, PENG Y, et al. Lithium-ion battery remaining useful life prognostics using data-driven deep learning algorithm[C]// Prognostics and System Health Management Conference, 2018.
[6]	熊庆, 邸振国, 汲胜昌. 锂离子电池健康状态估计及寿命预测研究进展综述[J]. 高电压技术, 2024, 50(3): 1182-1195.
	XIONG Qing, DI Zhenguo, JI Shengchang. Review on health state estimation and life prediction of lithium-ion batteries[J]. High Voltage Engineering, 2024, 50(3): 1182-1195.
[7]	胡晓亚, 郭永芳, 张若可. 锂离子电池健康状态估计方法研究综述[J]. 电源学报, 2022, 20(1): 126-133.
	HU Xiaoya, GUO Yongfang, ZHANG Ruoke. Review of state-of-health estimation methods for lithium-ion battery[J]. Journal of Power Supply, 2022, 20(1): 126-133. doi: 10.13234/j.issn.2095-2805.2022.1.126
[8]	韦海燕, 安晶晶, 陈静, 等. 基于改进粒子滤波算法实现锂离子电池RUL预测[J]. 汽车工程, 2019, 41(12): 1377-1383. doi: 10.19562/j.chinasae.qcgc.2019.012.005
	WEI Haiyan, AN Jingjing, CHEN Jing, et al. RUL prediction of lithium-ion battery based on improved particle filtering algorithm[J]. Automotive Engineering, 2019, 41(12): 1377-1383.
[9]	FANG P Y, SUI X X, ZHANG A H, et al. Fusion model based RUL prediction method of lithium-ion battery under working conditions[J]. Maintenance and Reliability, 2024, 26(3):186537
[10]	PANG X Q, HUANG R, WEN J, et al. A lithium-ion battery RUL prediction method considering the capacity regeneration phenomenon[J]. Energies, 2019, 12(12). :12122247.
[11]	冯娜娜, 杨明, 惠周利, 等. 基于蚁狮优化高斯过程回归的锂电池剩余使用寿命预测[J]. 储能科学与技术, 2024, 13(5): 1643-1652. doi: 10.19799/j.cnki.2095-4239.2023.0865
	FENG Nana, YANG Ming, HUI Zhouli, et al. Prediction of the remaining useful life of lithium batteries based on antlion optimization Gaussian process regression[J]. Energy Storage Science and Technology, 2024, 13(5): 1643-1652. doi: 10.19799/j.cnki.2095-4239.2023.0865
[12]	刘迎迎, 张孝远, 刘梦楠, 等. 基于自适应最优组合核函数高斯过程回归的锂电池健康状态区间估计[J]. 储能科学与技术, 2025, 14(1): 346-357. doi: 10.19799/j.cnki.2095-4239.2024.0473
	LIU Yingying, ZHANG Xiaoyuan, LIU Mengnan, et al. State of health interval estimation for lithium battery via Gaussian process regression with adaptive optimal combination kernel function[J]. Energy Storage Science and Technology, 2025, 14(1): 346-357. doi: 10.19799/j.cnki.2095-4239.2024.0473
[13]	周洪宇, 肖佳鹏. 基于高斯过程回归组合核函数的磷酸铁锂电池荷电状态估算[J]. 农业装备与车辆工程, 2020, 58(8): 87-91, 111.
	ZHOU Hongyu, XIAO Jiapeng. Estimation for state of charge of lithium iron phosphate battery based on Gaussian process regression with composite kernel function[J]. Agricultural Equipment & Vehicle Engineering, 2020, 58(8): 87-91, 111.
[14]	GUO Y, YU X, WANG Y, et al. Health prognostics of lithium-ion batteries based on universal voltage range features mining and adaptive multi-Gaussian process regression with harris hawks optimization algorithm[J]. Reliability Engineering & System Safety, 2024, 244: 109913. doi: 10.1016/j.ress.2023.109913
[15]	王琛, 闵永军. 基于容量增量曲线与GWO-GPR的锂离子电池SOH估计[J]. 储能科学与技术, 2023, 12(11): 3508-3518. doi: 10.19799/j.cnki.2095-4239.2023.0458
	WANG Chen, MIN Yongjun. SOH estimation of lithium-ion batteries based on capacity increment curve and GWO-GPR[J]. Energy Storage Science and Technology, 2023, 12(11): 3508-3518. doi: 10.19799/j.cnki.2095-4239.2023.0458
[16]	庞景月, 马云彤, 刘大同, 等. 锂离子电池剩余寿命间接预测方法[J]. 中国科技论文, 2014, 9(1): 28-36.
	PANG Jingyue, MA Yuntong, LIU Datong, et al. Indirect remaining useful life prognostics for lithium-ion battery[J]. China Science Paper, 2014, 9(1): 28-36.
[17]	陈毅, 黄妙华, 王树坤. 基于数据驱动的锂电池剩余容量估计[J]. 自动化与仪表, 2017, 32(8): 69-73.
	CHEN Yi, HUANG Miaohua, WANG Shukun. Lithium battery residual capacity estimation based on the data-driven[J]. Automation & Instrumentation, 2017, 32(8): 69-73.
[18]	ZHOU Y, HUANG M. On-board capacity estimation of lithiumion batteries based on charge phase[J]. Journal of Electrical Engineering and Technology, 2018, 13(2):733-741.
[19]	朱振威, 苗嘉伟, 祝夏雨, 等. 基于机器学习方法的锂电池剩余寿命预测研究进展[J]. 储能科学与技术, 2024(9): 3134-3149. doi: 10.19799/j.cnki.2095-4239.2024.0713
	ZHU Zhenwei, MIAO Jiawei, ZHU Xiayu, et al. Research progress of residual life prediction of lithium batteries based on machine learning method[J]. Energy Storage Science and Technology, 2024(9): 3134-3149. doi: 10.19799/j.cnki.2095-4239.2024.0713
[20]	陈璐, 于仲安, 熊莹燕. 不同温度下基于PSO-LSSVM的锂电池SOH估计与RUL预测[J]. 传感器与微系统, 2023, 42(6): 141-145.
	CHEN Lu, YU Zhongan, XIONG Yingyan. SOH estimation and RUL prediction of Li battery based on PSO-LSSVM at different temperatures[J]. Transducer and Microsystem Technologies, 2023, 42(6): 141-145.
[21]	赵靖华, 闻龙, 汪守丰, 等. 基于改进组合核函数高斯过程回归的车速预测[J]. 吉林大学学报(理学版), 2025, 63(2): 454-464.
	ZHAO Jinghua, WEN Long, WANG Shoufeng, et al. Vehicle speed prediction based on Gaussian process regression with improved combination kernel function[J]. Journal of Jilin University (Science Edition), 2025, 63(2): 454-464.
[22]	PAN Y, ZENG X K, XU H X, et al. Evaluation of Gaussian process regression kernel functions for improving groundwater prediction[J]. Journal of Hydrology, 2021, 603:126960. doi: 10.1016/j.jhydrol.2021.126960
[23]	DEND T Q, YE D S, MA R, et al. Low-rank local tangent space embedding for subspace clustering[J]. Information Sciences, 2020, 508:1-21. doi: 10.1016/j.ins.2019.08.060
[24]	YANG X H, JIANG X Y, TIAN C X, et al. Inverse projection group sparse representation for tumor classification: A low rank variation dictionary approach[J]. Knowledge-Based Systems, 2020, 196(4):105768. doi: 10.1016/j.knosys.2020.105768
[25]	SI S, HSIEH C J, DHILLON I S. Memory efficient kernel approximation[J]. Journal of Machine Learning Research, 2017, 18(20):1-32.
[26]	蒋剑, 杜董生, 苏林. 基于改进HHO-LSTM-Self-Attention的质子交换膜燃料电池剩余使用寿命预测[J]. 综合智慧能源, 2025, 47(6): 47-56. doi: 10.3969/j.issn.2097-0706.2025.06.006
	JIANG Jian, DU Dongsheng, SU Lin. Remaining useful life prediction of proton exchange membrane fuel cells based on improved HHO-LSTM-Self-Attention[J]. Integrated Intelligent Energy, 2025, 47(6): 47-56. doi: 10.3969/j.issn.2097-0706.2025.06.006
[27]	李云, 周世杰, 胡哲千, 等. 基于NSGA-Ⅱ-WPA的综合能源系统多目标优化调度[J]. 综合智慧能源, 2024, 46(4): 1-9. doi: 10.3969/j.issn.2097-0706.2024.04.001
	LI Yun, ZHOU Shijie, HU Zheqian, et al. Optimal scheduling of integrated energy systems based on NSGA-Ⅱ-WPA[J]. Integrated Intelligent Energy, 2024, 46(4): 1-9. doi: 10.3969/j.issn.2097-0706.2024.04.001

电池编号	主成分1	主成分2	主成分3	主成分4
B0005	0.804 1	0.169 0	0.014 4	0.012 5
B0006	0.900 0	0.090 1	0.007 9	0.002 1
B0007	0.692 3	0.246 2	0.033 2	0.028 2

项目		RMSE			MAE
项目		以80为起点	以100为起点	以120为起点	以80为起点	以100为起点	以120为起点
LIN+RBF 核模型	B0005	0.006 1	0.005 1	0.005 9	0.003 7	0.001 8	0.002 1
	B0006	0.005 9	0.004 3	0.004 1	0.004 6	0.003 2	0.002 8
	B0007	0.027 3	0.016 2	0.012 4	0.023 2	0.014 0	0.011 1
LIN核模型	B0005	0.070 9	0.039 6	0.025 0	0.058 4	0.031 9	0.019 6
	B0006	0.059 5	0.022 8	0.052 6	0.051 4	0.018 7	0.051 0
	B0007	0.052 6	0.035 2	0.028 6	0.041 9	0.028 9	0.024 6
RBF核模型	B0005	0.019 4	0.013 2	0.015 0	0.017 5	0.011 0	0.012 6
	B0006	0.011 6	0.008 0	0.010 2	0.008 2	0.005 9	0.008 1
	B0007	0.028 6	0.018 9	0.017 0	0.025 4	0.016 7	0.015 5
Matern核模型	B0005	0.058 7	0.029 9	0.009 1	0.048 6	0.026 3	0.008 0
	B0006	0.053 5	0.035 4	0.021 4	0.046 6	0.030 7	0.018 6
	B0007	0.026 5	0.016 2	0.008 3	0.020 0	0.013 1	0.007 2

起始循环次数	RMSE			MAE
起始循环次数	GWO	无优化	PSO	GWO	无优化	PSO
80	0.006 1	0.062 6	0.016 7	0.003 7	0.052 3	0.013 9
100	0.005 1	0.028 0	0.010 9	0.001 8	0.012 5	0.008 6
120	0.005 9	0.018 0	0.009 5	0.002 1	0.013 0	0.007 1