Elsevier

Measurement

Volume 187, January 2022, 110269
Measurement

An iterative model of the generalized Cauchy process for predicting the remaining useful life of lithium-ion batteries

https://doi.org/10.1016/j.measurement.2021.110269Get rights and content

Highlights

Abstract

The degradation process of lithium-ion batteries has memory, i.e. it has long-range dependence (LRD). In this paper, an iterative model of the generalized Cauchy (GC) process with LRD characteristics is proposed for the remaining useful life (RUL) prediction of lithium-ion batteries. The GC process uses two independent parameters, fractal dimension and Hurst exponent, to measure the LRD of the degradation process. The diffusion term of the GC iterative model is replaced by the increment of the GC time sequences, constructed via the autocorrelation function (ACF) to describe uncertainty and the LRD characteristics of the lithium-ion batteries capacity degradation. Linear and nonlinear drift terms are used to explain the degradation trend of the lithium-ion batteries capacity. A comparison is made with fractional Brownian motion (FBM) and long-short-term memory (LSTM) network models to show how the GC iterative model has the best performance in RUL prediction of lithium-ion batteries.

Introduction

Lithium-ion batteries gradually degrade and age by usage, eventually leading to functional failure [1], [2]. RUL prediction is an important task for lithium-ion batteries reliability. It can be based on the estimate of the failure time obtained based on state-of-health monitoring, and can guide predictive maintenance strategies for reducing accident risk [3]. The degradation of lithium-ion batteries is a long-term, slow process [4]. Generally, the capacity in the cycle of charging and discharging is considered as a suitable feature to reflect the degradation trend and, thus, it can be used as health indicator. Then, the RUL of lithium-ion batteries can be defined with respect to a minimum capacity threshold level, typically 20–30% of the rated capacity [5]. In this way, the RUL prediction problem is transformed into a capacity prediction problem with reference to the preset failure threshold (FT).

The existing RUL prediction methods for lithium-ion batteries can be mainly divided into three categories: filtering methods, artificial intelligence methods and stochastic process methods. Wang et al. [6] developed a degradation method based on a spherical particle filter to achieve RUL prediction of lithium-ion batteries. Zheng and Fang [7], [8] proposed a hybrid prediction model based on unscented Kalman filter to achieve RUL prediction of lithium-ion batteries. Other developments of filtering methods, such as particle filter and Kalman filter, can be found in [9], [10], [11]. Filtering methods have obvious advantages under certain application conditions [12], but there are also three main disadvantages that hinder the feasibility of their use in practical applications of lithium-ion batteries RUL prediction: in general, the prediction accuracy of filtering methods is susceptible to ambient temperature and time-varying current [13]; particle filtering methods suffer the problem of particle paucity due to resampling [14]; particle filtering methods may have difficulties in determining the key parameters to describe the process of lithium-ion battery capacity degradation [15]. Zhang et al. [2] developed a RUL prediction method based on LSTM recurrent neural network. Chen et al. [16] proposed a prediction model based on empirical mode decomposition and deep recurrent neural network to show the LSTM can give good prediction accuracy. Nuhic et al. [17] proposed a method based on support vector machines to predict the RUL of lithium-ion batteries. Li et al. [18] developed a least squares support vector machine model, based on data-driven and model fusion, to predict the RUL of lithium-ion batteries. Li et al. [19] proposed an Elman neural network model to predict the RUL of lithium-ion batteries. The disadvantages of artificial intelligence methods are that mainly the expected value of the RUL can be obtained, whereas the uncertainty in the prediction can be achieved only at the expense of large computational efforts [13]. In addition, the artificial intelligence methods require large-scale training data and overtraining can cause the problem of falling in a local optimum [20]. The stochastic process methods for RUL prediction have good characteristics, useful also for characterizing the uncertainty in the prediction of the degradation process of lithium-ion batteries. These methods perform the RUL prediction based on probability theory and the development of a stochastic model. The stochastic model can flexibly reflect the uncertainty of the degradation process [21]. For this reason, they are finding applications in lithium-ion batteries RUL prediction [22], e.g. based on Markov process and Wiener process [22], [23].

Since the degradation of lithium-ion batteries is a continuous and slow process, it has LRD characteristics [24]. However, modeling based on Wiener process relies on the Markov hypothesis [21], and the incremental independence of the Wiener process cannot reflect the LRD characteristics of the lithium-ion batteries degradation process. To overcome this limitation, in this paper we propose a GC iterative model with LRD characteristics to predict the RUL of lithium-ion batteries.

Li defined a GC process with LRD characteristics and associated uncertainty to describe fractal time sequences [25]. The LRD characteristic of the GC process is evaluated by its ACF, which is composed of the fractal dimension D and the Hurst exponent H [26]. If the ACF integral -+R(τ)dτ= of the GC process diverges, the GC process is considered to have LRD characteristics [27]. The GC process satisfies LRD, which means that its ACF slowly decays to the extent of integral divergence, so that the correlation between two different points that are far apart in time cannot be ignored. Based on the integration of the above-mentioned ACF, Lim and Li obtained that the GC process satisfies the LRD characteristic when 0<(4-2D)(2-2H)1 [27]. Therefore, the LRD characteristics are jointly described by the Hurst exponent H and the fractal dimension D, and the GC time sequences are randomly generated considering the ACF of the GC, which captures the uncertainty in the process [25]. Therefore, the GC iterative model not only overcomes the problem of accounting for the LRD characteristics of the lithium-ion batteries degradation process, which the Markov process and the Wiener process cannot, but also overcomes the problem of representing the uncertainty in the prediction, which cannot be inexpensively achieved by the artificial intelligence methods. The FBM model is another model with LRD characteristics, which can be used to predict the RUL of lithium-ion batteries [24]. However, there is a linear relationship H+D=2 between H and D, so that only one parameter can be used to describe the LRD characteristics. When 0.5<H<1, the FBM has LRD characteristics [24]. Compared with the FBM model, the superiority of the GC iterative model lies in the availability of the two parameters H and D, which are independent of each other: this makes the GC iterative model more flexible than FBM to describe the LRD characteristics.

In this paper, linear and nonlinear drift terms (such as a power function drift term) are used to describe the degradation trend of lithium-ion batteries. In addition, the diffusion term of the GC iteration model is replaced by the increment of the GC time sequences to solve the problem of the uncertainty and LRD characteristic prediction of the lithium-ion batteries degradation process. Then, the parameters of the GC iterative model are obtained by the maximum likelihood estimation (MLE) method. The first arrival time of the lithium-ion batteries RUL can be obtained by evaluating when capacity exceeds the preset FT. Furthermore, the probability density function (PDF) of the RUL can be obtained by Monte Carlo simulation [28], [29]. Finally, real data of lithium-ion battery capacities are used to verify the feasibility of the GC iterative model, and a comparison is made with FBM and LSTM.

The organization of this paper is as follows. The LRD characteristics of the GC process are briefly discussed in Section 2. In Section 3, we describe the iterative model based on the GC process and the procedure for estimating the parameters of the iterative model. In Section 4, the lithium-ion batteries RUL is predicted by the GC iterative model, and a comparison is made with the FBM and LSTM models. Finally, the conclusion of this paper is given in Section 5.

Section snippets

The LRD characteristics of the generalized Cauchy process

To analyze the LRD characteristics of the GC process, it is essential to introduce the ACF. Li et al. [25] gave the definition of ACF for the GC process:ryτ=Eyt+τyt=(1+τα)-β/αwhere τ>0,β>0, 0<α2, and yt is GC process. When α=2 and β=2, the GC process degenerates into a classical Cauchy process.

In the GC process, the LRD characteristics are described by parameters D and H, which reflect the local and global properties, respectively. The parameter α can be described by the fractal dimension D,

Iterative model based on generalized Cauchy process

Since the degradation of lithium-ion batteries is a continuous and slow process, it has LRD characteristics. In this paper, we use the GC-driven diffusion term to describe the uncertainty and LRD characteristics of the degradation process of lithium-ion batteries. Similarly to the stochastic differential equation of Wiener process and FBM, the stochastic differential equation of the GC iterative model is written as [31], [32]:dXt=λφtdt+σtdGC(t)where Xt is the degradation process of lithium-ion

Case study

We consider the lithium-ion batteries capacity data from NASA Ames database to verify the feasibility of the GC iterative model [36]. The ambient temperature is 24 °C. Repeated charging and discharging cycles are induced to cause accelerated lithium-ion batteries aging. The capacity generated by the charge and discharge cycles is taken as a suitable health indicator to describe the degradation process of lithium-ion batteries. The experiment is terminated when the lithium-ion batteries drop

Conclusion

This paper proposes an iterative model with LRD characteristics and applies it to predict lithium-ion batteries RUL. The specific conclusions are as follows.

  • (1)

    Accurate prediction of lithium-ion batteries RUL is of great significance to equipment safety and maintenance. In this paper, a method for predicting the lithium-ion batteries RUL using a GC iterative model is introduced. Firstly, this paper has introduced that the LRD characteristics of the GC iterative model are simultaneously described

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The work was supported by major project of Ministry of Science and Technology of the People’s Republic of China, Grand number: 2020AAA0109301.

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