近年來，全球趨勢和發展、政府支持與激勵措施等因素促使電動車的銷售和儲能設備的應用逐年增加，進而使鋰離子電池及其模組的壽命研究成為熱點。在本研究中，我們利用電池組的容量損失資料集來計算每個放電循環後的電池健康狀態(State of Health, SoH)，從而建立電池健康狀態的退化路徑。同時，我們考慮了實際電池組應用中分支電路上電池放電速率(C-rate)不一致性的問題。為更精確地描述電池組中各電池的退化路徑，我們提出了放電速率因子的概念，用以描述電池間每次放電週期中放電速率的差異對各自電池與電池組造成的影響，並結合隨機係數退化模型以掌握電池組之間的差異。相較於未考慮放電速率不一致性的退化模型，我們提出的方法能夠更準確地反映電池組內各電池的退化行為。此外，我們也提出了Cell-to-Pack估計方法來估計電池組壽命分佈，並使用Parametric Bootstrap來建構電池組壽命的累積分佈函數（CDF）的95%信賴區間。結果顯示，相較於傳統的可靠度方法，將放電速率因子納入模型中能夠更真實地描述電池的退化路徑，使用Cell-to-Pack估計方法可以利用單電池資料推估電池組壽命分佈，其95%信賴區間較窄且更符合真實電池組數據。綜上所述，本研究不僅提出了新的理論方法來提高電池壽命預測的準確性，而且在實踐中提供了可靠的數據支持，有助於提高鋰離子電池組的壽命與可靠性預測。

In recent years, global trends and developments, along with government support and incentives, have driven the annual increase in electric vehicle sales and the application of energy storage systems. Consequently, the study of the lifetime of lithium-ion batteries and their modules has become a focal point of research. In this study, we utilized capacity loss datasets from battery packs to calculate the State of Health (SoH) after each discharge cycle, thereby establishing degradation paths for battery health. Simultaneously, we addressed the issue of non-uniform discharge rates (C-rate) in the branched circuits of actual battery pack applications. To more accurately depict the degradation paths of individual batteries within a pack and of the pack itself, we introduced the concept of the discharge rate factor to describe the impact of variations in discharge rates during each discharge cycle. This factor was incorporated into a random coefficients degradation model. Compared to degradation models that do not consider discharge rate inconsistencies, our proposed method more accurately reflects the degradation behavior of individual batteries within a pack. Additionally, we proposed the Cell-to-Pack estimation method to estimate the lifetime distribution of battery packs by using the cell data. Using Parametric Bootstrap, we constructed the 95% confidence interval of the cumulative distribution function (CDF) for battery pack lifetime. The results indicate that, compared to traditional reliability methods such as the Reliability Block Diagram, incorporating the discharge rate factor into the model provides a more realistic description of battery degradation paths of individual cell in the pack. Furthermore, the Cell-to-Pack estimation method yields a more accurate battery pack lifetime distribution by using the cell data, with narrower 95% confidence intervals that align more closely with actual battery pack data.

[1] Bae, S. J., and Kvam, P. H. A nonlinear random-coefficients model for degradation testing. Technometrics 46, 4 (2004), 460–469.
[2] Catelani, M., Ciani, L., and Venzi, M. Rbd model-based approach for reliability assessment in complex systems. IEEE Systems Journal 13, 3 (2019), 2089–2097.
[3] Chen, L., An, J., Wang, H., Zhang, M., and Pan, H. Remaining useful life prediction for lithium-ion battery by combining an improved particle filter with sliding-window gray model. Energy Reports 6 (2020), 2086–2093.
[4] Chen, Z., Sun, M., Shu, X., Shen, J., and Xiao, R. On-board state of health estimation for lithium-ion batteries based on random forest. In 2018 IEEE International Conference on Industrial Technology (ICIT) (2018), IEEE, pp. 1754–1759.
[5] Chiu, T.-C. Remaining useful life prediction of lithiumion batteries based on piecewise random coefficient models. Master's Thesis of National Cheng Kung University (2020).
[6] Dong, G., Chen, Z., Wei, J., and Ling, Q. Battery health prognosis using brownian motion modeling and particle filtering. IEEE Transactions on Industrial Electronics 65, 11 (2018), 8646–8655.
[7] Fan, Y., Xiao, F., Li, C., Yang, G., and Tang, X. A novel deep learning framework for state of health estimation of lithium-ion battery. Journal of Energy Storage 32 (2020), 101741.
[8] Guo, Y., Huang, K., and Hu, X. A state-of-health estimation method of lithium-ion batteries based on multi-feature extracted from constant current charging curve. Journal of Energy Storage 36 (2021), 102372.
[9] Horiba, T. Lithium-ion battery systems. Proceedings of the IEEE 102, 6 (2014), 939–950.
[10] Jeng, S.-L., Tan, C. M., and Chen, P.-C. Statistical distribution of lithium-ion batteries useful life and its application for battery pack reliability. Journal of Energy Storage 51 (2022), 104399.
[11] Levitin, G. A universal generating function approach for the analysis of multi-state systems with dependent elements. Reliability Engineering & System Safety 84, 3 (2004), 285–292.
[12] Li, C., Chen, X., and Yi, X. Reliability analysis of primary battery packs based on the universal generating function method. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 223, 3 (2009), 251–257.
[13] Lingzhao, K., and Lee, R. A study on parameter variation of cells effects on battery groups with different topologies and load profiles. SAE International Journal of Advances and Current Practices in Mobility 3, 2021-01-0756 (2021), 2770–2781.
[14] Liu, Z., Tan, C., and Leng, F. A reliability-based design concept for lithium-ion battery pack in electric vehicles. Reliability Engineering & System Safety 134 (2015), 169–177.
[15] Lu, C. J., and Meeker, W. O. Using degradation measures to estimate a time-to-failure distribution. Technometrics 35, 2 (1993), 161–174.
[16] Lu, J.-C., Park, J., and Yang, Q. Statistical inference of a time-to-failure distribution derived from linear degradation data. Technometrics 39, 4 (1997), 391–400.
[17] Mawonou, K. S., Eddahech, A., Dumur, D., Beauvois, D., and Godoy, E. State-ofhealth estimators coupled to a random forest approach for lithium-ion battery aging factor ranking. Journal of Power Sources 484 (2021), 229154.
[18] Meeker, W. Q., and Escobar, L. A. Statistical methods for reliability data. John Wiley & Sons, 1998.
[19] Meeker, W. Q., Escobar, L. A., and Pascual, F. G. Statistical methods for reliability data. John Wiley & Sons, 2022.
[20] Naguib, M., Kollmeyer, P., and Emadi, A. Lithium-ion battery pack robust state of charge estimation, cell inconsistency, and balancing: Review. IEEE Access 9 (2021), 50570–50582.
[21] Peng, C.-Y. Inverse gaussian processes with random effects and explanatory variables for degradation data. Technometrics 57, 1 (2015), 100–111.
[22] Peng, C.-Y., and Tseng, S.-T. Mis-specification analysis of linear degradation models. IEEE Transactions on Reliability 58, 3 (2009), 444–455.
[23] Qu, J., Liu, F., Ma, Y., and Fan, J. A neural-network-based method for rul prediction and soh monitoring of lithium-ion battery. IEEE Access 7 (2019), 87178–87191.
[24] Shen, X., Luo, Z., Xiong, J., Liu, H., Lv, X., Tan, T., Zhang, J., Wang, Y., and Dai, Y. Optimal hybrid energy storage system planning of community multi-energy system based on two-stage stochastic programming. IEEE Access 9 (2021), 61035–61047.
[25] Singh, P., Chen, C., Tan, C. M., and Huang, S.-C. Semi-empirical capacity fading model for soh estimation of li-ion batteries. Applied Sciences 9, 15 (2019).
[26] Tan, C. M., Singh, P., and Chen, C. Accurate real time on-line estimation of state-of-health and remaining useful life of li ion batteries. Applied Sciences 10, 21 (2020).
[27] Tseng, S.-T., and Lee, I.-C. Optimum allocation rule for accelerated degradation tests with a class of exponential-dispersion degradation models. Technometrics 58, 2 (2016), 244–254.
[28] Wang, L., Sun, Y., Wang, X., Wang, Z., and Zhao, X. Reliability modeling method for lithium-ion battery packs considering the dependency of cell degradations based on a regression model and copulas. Materials 12, 7 (2019).
[29] Wang, L., Sun, Y., Wang, X., Wang, Z., and Zhao, X. Reliability modeling method for lithium-ion battery packs considering the dependency of cell degradations based on a regression model and copulas. Materials 12, 7 (2019), 1054.
[30] Wang, X., Fang, Q., Dai, H., Chen, Q., and Wei, X. Investigation on cell performance and inconsistency evolution of series and parallel lithium-ion battery modules. Energy Technology 9, 7 (2021), 2100072.
[31] Wang, X., Wang, Z., Wang, L., Wang, Z., and Guo, H. Dependency analysis and degradation process-dependent modeling of lithium-ion battery packs. Journal of Power Sources 414 (2019), 318–326.
[32] Wang, X., Wang, Z., Wang, L., Wang, Z., and Guo, H. Dependency analysis and degradation process-dependent modeling of lithium-ion battery packs. Journal of Power Sources 414 (2019), 318–326.
[33] Wang, Y., Zhao, Y., Zhou, S., Yan, Q., Zhan, H., Cheng, Y., and Yin, W. Impact of individual cell parameter difference on the performance of series–parallel battery packs. ACS omega 8, 11 (2023), 10512–10524.
[34] Wang, Y., Zhao, Y., Zhou, S., Yan, Q., Zhan, H., Cheng, Y., and Yin, W. Impact of individual cell parameter difference on the performance of series–parallel battery packs. ACS Omega 8, 11 (2023), 10512–10524.
[35] Wei, J., Dong, G., and Chen, Z. Remaining useful life prediction and state of health diagnosis for lithium-ion batteries using particle filter and support vector regression. IEEE Transactions on Industrial Electronics 65, 7 (2018), 5634–5643.
[36] Xia, Q., Wang, Z., Ren, Y., Sun, B., Yang, D., and Feng, Q. A reliability design method for a lithium-ion battery pack considering the thermal disequilibrium in electric vehicles. Journal of Power Sources 386 (2018), 10–20.
[37] Xue, Z., Zhang, Y., Cheng, C., and Ma, G. Remaining useful life prediction of lithiumion batteries with adaptive unscented kalman filter and optimized support vector regression. Neurocomputing 376 (2020), 95–102.
[38] Yang, Y., Wang, R., Shen, Z., Yu, Q., Xiong, R., and Shen, W. Towards a safer lithiumion batteries: A critical review on cause, characteristics, warning and disposal strategy for thermal runaway. Advances in Applied Energy 11 (2023), 100146.
[39] Zhang, Y., Zheng, J., Lin, S., Bai, F., Tanveer, W. H., Cha, S., Wu, X., and Feng, W. Nonuniform current distribution within parallel-connected batteries. International Journal of Energy Research 42, 8 (2018), 2835–2844.