Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 8358025, 11 pages

https://doi.org/10.1155/2018/8358025

## Prognostics for State of Health of Lithium-Ion Batteries Based on Gaussian Process Regression

^{1}School of Electrical Engineering and Automation, Harbin Institute of Technology, Harbin, China^{2}School of Science, Harbin University of Science and Technology, Harbin 150080, China^{3}Shenzhen Academy of Metrology & Quality Inspection, Shenzhen, China

Correspondence should be addressed to Hongtao Yin; nc.ude.tih@thniy

Received 21 October 2017; Revised 27 January 2018; Accepted 6 February 2018; Published 1 April 2018

Academic Editor: Anna Vila

Copyright © 2018 Di Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Accurate estimation and prediction of the lithium-ion (Li-ion) batteries’ performance has important theoretical and practical significance to make better use of lithium-ion battery and to avoid unnecessary losses. State of health (SOH) estimation is used as a qualitative measure of the capability of a lithium-ion battery to store and deliver energy in a system. To evaluate and predict the SOH of batteries, the Gaussian process regression with neural network (GPRNN) as its variance function is proposed. Experimental results confirm that the proposed method can be effectively applied to Li-ion battery monitoring and prognostics by quantitative comparison with basic GPR, combination LGPFR, combination QGPFR, and the multiscale GPR (SMK-GPR, P-MGPR, and SE-MGPR). The criteria of RMSE and MAPE of the proposed three models are reduced significantly compared to those of other existing methods.

#### 1. Introduction

Lithium-ion battery has been widely applied to portable electronic devices, electric and hybrid vehicles, even military electronics, aerospace avionics, and other automotive vehicles because of the properties of lightness, high energy density, high output voltage, long lifetime, and low self-discharge [1, 2]. The safety and reliability of Li-ion battery in actual application are critical factors of the system performance whose failure would not only lead to the performance degradation, repair costs increase, and inconvenience, but also cause catastrophic accidents due to overheating and short circuiting [3]. Therefore, the effective monitoring and prognostics for Li-ion battery have been attracted much more attention and a reliable and effective BMS (battery management system) is indispensable [4, 5]. The precondition of BMS is monitoring and estimation of states which includes estimations of SOC (state of charge), SOH (state of health), and SOL (state of life). The above three states estimations are still the core content in the research of BMS [4, 6–8]. SOH is the ratio of the released capacity with a certain discharge from full charge state to cut-off standard voltage under standard condition and its corresponding initial capacity which is the response of the battery performance degradation and the accidents possibility. In general, SOC describes the short-term changes of the current parameters and SOH describes long-term changes. SOH does not require continuous measurements and only needs periodical measurements in most cases, and the measured periods depend on the applications. Based on the above advantages of SOH, this paper focuses on research of SOH to reflect the performance of Li-ion battery.

With the development of Li-ion battery, SOH estimation plays an important role as a quantitative description for the battery performance indicators to store and deliver energy in the system [9, 10]. Generally, there are two mainstream methods to estimate SOH [11]. One is experience-based approach such as cycle counting method, Ah Low and Ah weighted method, and event-oriented cumulative method of aging. The other method is performance-based approach which utilizes different forms of performance models and considers the aging process and stress factors. According to different sources of information used, the performance-based approach is divided into three categories which are mechanism-based method, characteristics-based method, and data-driven method. Due to the complex physical and chemical processes of the battery itself, a lot of the rules are difficult to describe directly through the mechanism. The method which describes the battery performance from the perspective of the test data is called data-driven approach. The acquired data tends to have a strong uncertainty and incompleteness; testing all the possible factors affecting the battery life in the practical application is unrealistic. Although the limitations exist, data-driven method is still an effective prediction method. It is used to excavate the rules of battery performance for SOH prediction through using battery performance test data. Data-driven prediction does not require mechanical knowledge of object systems. It excavates the implied information to predict SOH through a variety of data analysis learning methods. Therefore, it can avoid the difficulty of the model acquisition, so it is a more practical prediction method.

There are many common data-driven algorithms such as Particle Filtering (PF), Support Vector Machine (SVM), and Relevant Vector Machine (RVM). PF can combine available information from system measurements and analytic models for SOH prediction [10, 12]. Kalman filter and extended Kalman filter (EKF) are applied to SOH prediction of batteries by using estimation of equivalent circuit model [13, 14]. Unscented filtering and Bayesian filtering are classical methods for battery SOH or SOC estimation. These stochastic filtering methods for prognostics are easier to be obtained and have wider application scope. However, the methods are difficult in containing the environmental interference and loading dynamic characteristics factors and so on, and the dynamic accuracy and adaptability are still limited. Moreover, PF algorithm itself solves the problem of parameters initialization which is time-consuming in the case of more parameters. Some methods are mostly improvement and optimization of PF or KF such as sequential Monte Carlo method referring to PF which has attracted attention to SOH [15]. In recent years, SVM has been used in SOH prediction of Li-ion battery [16]. In addition to basic SVM algorithm, there are some improved SVM algorithms and fusion algorithms. SVM method still has some deficiencies: the kernel function must satisfy the Mercer condition, sparsity is limited, the number of support vectors is sensitive to error boundary, and the loss of function and the penalty factor are difficult to determine. SVM also lacks the expression of uncertainty management. To solve this problem, RVM approaches have been proposed. RVM algorithm has higher calculation accuracy and lower computational complexity compared with SVM and can output the uncertainty information of prediction results [17]. Kim obtained the relevant parameters of exponential model by using EIS test data based on RVM algorithm and gave the prediction outcomes [1]. Oai et al. developed a new technique to predict SOH using a dual-sliding-mode observer. The researchers presented a method, Li-ion battery SOH prediction concept, based on an appropriate SOH definition [18]. However, since the much sparse and dynamic fluctuation of capacity data will lead to forecasting the result of poor stability if RVM is directly used for prediction only, there are many others methods such as neural fuzzy logic [19], networks [20], regressions [21], and distributed active learning [22] that lack specific strategies in dealing with model uncertainties and would lead to prognostics performance deterioration when facing long-term changes in environmental and operating conditions.

To consider modeling flexibility and uncertainty representations, Gaussian process regression (GPR) has been investigated for Li-ion battery prognostics, where the degradation trends are learnt from battery datasets with the combination of Gaussian process functions [23]. GPR could provide a confidence measure of output and has been used for SOH predictions as a kind of new machine learning method based on Bayesian theory and statistical learning theory. Meanwhile, it is a nonparametric method which is suitable for high dimension nonlinear regression problems and can give the uncertainty representation of SOH prediction. An improved GPR method has been proposed to capture regeneration phenomena [24]. Li and Xu proposed a new prognostics method for state of health estimation of Li-ion batteries based on a mixture of Gaussian process models and particle filter proposed [25]. However, forecasting accuracy and uncertainty of the above two methods do not meet the requirements of practical application.

In this paper, an approach based on GPR with neural network kernel function is presented to predict the SOH of Li-ion batteries. Neural network is ideally suited to the SOH data, since it has smaller misspecification than other stationary covariance functions and draws saturation at different values in the positive and negative directions of variable [23]. Meanwhile, the log marginal likelihood is much larger than other covariance functions. The proposed method consists of three models: the neural network itself, the sum of the neural network and the Maternard covariance function, and the product of the neural network and the periodic covariance function. Finally, experiments based on the NASA battery datasets are provided to demonstrate the performance of the proposed three prognostics models. The experimental results show that the predictions are near perfect.

The remainder of this paper is organized as follows. In Section 2, the background knowledge about the SOH of Li-ion batteries is described. Then, the proposed methodology including GPR and its parameters is provided in Section 3. Experiments and analysis are given in Section 4 to demonstrate the performance of proposed prognostics algorithm. Finally, conclusions are drawn in Section 5.

#### 2. The SOH of Lithium-Ion Batteries

There are four definitions of SOH according to the battery characteristics which are shown as follows [26].(1)From the perspective of remaining power of battery to determine the battery SOH, its definition can be given by where is the current biggest power of battery and is the initial power.(2)From the perspective of starting power to define the battery SOH, the expression is given by where is the real-time starting power and is the predicted starting power released by battery when the SOC is 100% and is the minimum starting power needed.(3)From the perspective of impedance measurement to define the battery SOH, the definition is depicted as follows:where is th impedance measurement varying with the cycles of charging and discharging and is the initial impedance.(4)From the perspective of battery capacity power to define the battery SOH, this method can be expressed aswhere is the th capacitance value degenerated with cycles and is the initial capacity.

In this paper, this definition of SOH using the capacity power is utilized to study the performance of battery.

#### 3. Methodology

##### 3.1. Gaussian Process Regression

Gaussian process (GP) is used to describe random variables which are scalars or vectors. GP is a stochastic process which also governs the properties of functions and gets Bayesian inferences with the function-space view. GP is a set where any finite random variable has joint Gaussian distribution, and the characters of GP are completely determined by the mean function and the covariance function [28].

Consider the simple regression modelThe training set is , where , are the input values, is a function value, and , , are the output values. represents noise which follows an independent, identically distributed Gaussian distribution with zero mean and variance ; that is,Then the prior distribution of the observation is obtained:It can be used to estimate the predictive values. The joint distribution of the observation and the predictive values is derived as follows:where represents the covariance of the training data points and predictive values and is the covariance of the predictive values.

Computed by Bayes’ rule, the posterior distribution over the weights can be gotten. The posterior distribution to inference in the Bayesian linear model can be shown as

##### 3.2. Choice of Parameters

GP is usually parameterized in terms of their covariance functions. There are some most popular covariance functions [23, 29]: the squared exponential covariance function,the constant covariance function,and the periodic covariance function,

These covariance functions are stationary and are usually used in battery health prognostic with good results. Here in this paper a particular type of neural network is applied in battery health prognostic, which is called neural network covariance function; see (7). Neural network is ideally suited to the SOH data, since it allows saturation at different values in the positive and negative directions of [23]*.* For comparison, the predictive distributions for three covariance functions are shown in Figure 1. The 64 data points are produced by a step function with Gaussian noise with standard deviation which is totally the same as the example taken in [23]. Figure 1 shows the means and 95% confidence intervals for the noisy signal in grey. Particularly, Figure 1(a) expresses the single squared exponential covariance and Figure 1(b) illustrates the sum of two squared exponential covariances. Obviously, this covariance function is more flexible than a single squared exponential, since it has two magnitudes and two length-scale parameters. The predictive distribution looks a little bit better, but they both are not ideal for this scenario. Figure 1(c) shows the neural network covariance function which is ideally suited to this case, since it allows saturation at different values in the positive and negative directions of . As shown in Figure 1, the predictions are also near perfect.