Battery prognostics at different operating conditions Dong Wang a , Jin-zhen Kong a , Fangfang Yang b , Yang Zhao b,⇑ , Kwok-Leung Tsui b a The State Key Laboratory of Mechanical Systems and Vibration, Shanghai Jiao Tong University, Shanghai 200240, PR China b School of Data Science, City University of Hong Kong, Hong Kong, China article info Article history: Received 29 March 2019 Received in revised form 2 September 2019 Accepted 19 October 2019 Available online xxxx Keywords: Battery management systems Bayesian methods Lithium batteries Prognostics and health management Remaining life assessment abstract Rechargeable batteries become one of the most popular energy storage devices. For battery state of health prediction, discharge rate and temperature are two crucial factors that significantly affect battery dis- charge capacity fade. Battery discharge capacity fade modeling at different operating conditions is still an ongoing research direction. In this paper, two new battery discharge capacity fade models are pro- posed. In the first model, a relationship between capacity fading and discharge rate is formulated. The second model derives a relation between capacity fading and temperature. Then, two Bayesian updating procedures are respectively designed to model a unit-to-unit capacity fade variance and incorporate on-line data of a single operating battery into the two battery fade models. At last, two case studies are provided to illustrate how the proposed two discharge capacity fade models work. Results show that the proposed two new models can accurately predict battery state of health at different discharge rates and different temperatures. Ó 2019 Elsevier Ltd. All rights reserved. 1. Introduction Lithium-ion batteries become one of the most useful energy storage devices and they have been widely used in hybrid electric vehicles, portable electronic devices, etc [1]. When batteries are charged and discharged over time, their discharge capacity fades. To provide continuous and reliable electric power sources for the aforementioned systems and devices, battery prognostics and health management [2] is required. Rich literature [3] can be found to characterize battery health status including state of charge and state of health. State of charge is a short-term parameter to indi- cate remaining energy at a given charge–discharge cycle. State of health is a long-term parameter to indicate remaining charge–dis- charge cycles, namely remaining useful life (RUL) [4,5]. Because these two basic battery parameters can not be directly measured, they are estimated and predicted by using other collected mea- surements, such as current, voltage, temperature, etc. In the past years, many state of charge estimation models [6–8] have been proposed and they can work at different operating con- ditions, such as different discharge rates and different tempera- tures. Wei et al. [9] proposed a state of charge estimation method based on a recursive total least squares-based observer, which is able to accurately estimate battery state of charge. Ali et al. [10] used the Lagrange multiplier method to adaptively esti- mate battery state of charge and achieved high prediction accura- cies. Liu et al. [11] proposed a partial adaptive forgetting factor least square method to estimate deep-discharging lithium-ion bat- tery state of charge. However, most battery state of health predic- tion models [12] solely consider constant operating conditions, such as a fixed discharge rate and a room temperature. Saha et al. [13,14] proposed using an equivalent electric circuit model with an exponential form to model battery capacity fade and then employed the combination of relevance vector machines and sev- eral particle filters to predict battery RUL. Following this pioneer work, many battery prognostic methods have been reported. He et al. [15] proposed using an bi-exponential fade model instead of an exponential fade model to achieve a better regression ability for fitting battery capacity fade and then employed a particle filter to estimate the posterior parameters of the bi-exponential model and extrapolate the determined model to a failure threshold, such as 80 percent of an initial capacity, for battery RUL prediction. Xing et al. [16] proposed the integration of an exponential model and a polynomial model with an order of 2 to improve the local fitting ability of the bi-exponential model and then used a particle filter to posteriorly update model parameters for on-line battery RUL prediction. Yang et al. [17] developed a two-term logarithmic model to capture two-phase battery degradation and predict bat- tery RUL using a particle filter. Yu et al. [18] proposed a probabilis- tic health indicator based state space model and used a particle https://doi.org/10.1016/j.measurement.2019.107182 0263-2241/Ó 2019 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail addresses: dongwang4-c@sjtu.edu.cn (D. Wang), kongjinzhen@sjtu.edu. cn (J.-z. Kong), fangfyang2-c@my.cityu.edu.hk (F. Yang), yang.zhao@my.cityu.edu. hk (Y. Zhao), kltsui@cityu.edu.hk (K.-L. Tsui). Measurement xxx (xxxx) xxx Contents lists available at ScienceDirect Measurement journal homepage: www.elsevier.com/locate/measurement Please cite this article as: D. Wang, J. z. Kong, F. Yang et al., Battery prognostics at different operating conditions, Measurement, https://doi.org/10.1016/j. measurement.2019.107182