1. Introduction
Lithium-ion batteries are widely applied in many fields, which range from automobiles to ships and even satellites due to their high energy density, long lifespan, and other excellent characteristics [
1,
2]. The safety and reliability of lithium-ion batteries have always been of concern in their applications. The main risks are follows. Improper application can cause the battery temperature to rise or the battery to catch fire or even explode. Overcharge and overdischarge can cause changes in the material properties of the battery that result in irreversible capacity loss, reduced performance, and shortened lifetime. As the charging and discharging processes proceed, the performance between single cells in the battery pack gradually becomes unbalanced, which will shorten the lifetime of the battery pack [
3,
4]. Battery failures can result in failure of the electronic equipment.
As a discipline that is designed to assess battery life and thus to ensure the proper functioning of electronic devices or electronic systems, battery prognostic and health management (PHM) has recently been widely applied due to the prominence it has demonstrated in terms of various performance indicators [
5]. Considering the influences of the ambient temperature and the load conditions on battery performance, it is important to accurately assess the battery state of health (SOH), which, as a critical indicator of battery aging levels that is based on battery characterization parameters, can be used as a key indicator of the remaining useful life (RUL) of a battery and facilitates the accurate evaluation of the entire system [
6]. Nevertheless, due to the complex electrochemical reaction that occurs inside the battery, the SOH is not directly available.
In the past few years, various SOH estimation methods have been proposed. First, experimental data, such as the voltage, current, and impedance, can be extracted from the battery cycle processes. In addition, lithium batteries may differ in terms of their discharge methods according to their applications. However, the charging processes tend to be consistent. Therefore, we can utilize the charging processes of lithium batteries and various algorithms or models to realize SOH estimation. Therefore, considering various algorithms and models, the SOH estimation methods for lithium batteries can be roughly classified into three categories: empirical or semi-empirical methods, methods that are based on electrochemical/physical models, and data-driven methods [
7,
8,
9,
10,
11].
Various empirical or semi-empirical methods have been proposed: Zhong W et al. [
12] used the Thevenin model with battery polarization to simulate internal changes in the battery, but their simple model has substantial limitations; in [
13] the authors utilized a second-order equivalent circuit model (ECM) for estimation; in [
14] a standard equivalent circuit model was used, and then a genetic algorithm was employed to obtain its internal resistance. While the empirical models are easy to implement, the accuracy of methods of this type is limited by the robustness of the circuit models. The long-term trend prediction of the autoregressive synthesis (ARI) model was used as the capacity observation true value for the prediction stage of the SRCKF (square root cubature Kalman filter) algorithm to model the nonlinear degradation process of a lithium battery in [
10]. Electrochemical-based methods can accurately simulate the electrochemical reactions that occur inside the batteries, and researchers will use mathematical methods to simplify them. These methods mainly use electrochemical impedance spectroscopy during battery cycling to conduct SOH estimation [
15,
16]. These models can also lead to an increase in the computational costs while yielding accurate results. Therefore, they are not suitable for SOH estimation processes.
The data-driven methods rely mainly on the intrinsic analysis of historical data, where an in-depth understanding of the principles of electrochemistry is not required [
17]. These models typically use the extracted or processed features as inputs and output a battery capacity degradation curve. In the data-driven methods, many methods that are based on adaptive state estimation have been utilized for SOH prediction processes, such as the Kalman filter (KF) algorithm [
7,
8]; in [
9] the authors proposed an improved PF algorithm, which was used to map the capacity degradation to the SOH estimate, and proved the effectiveness of the method by using multiple testing results. However, these methods also have limitations: over time, particle degradation will occur. To improve the prediction accuracy of the model, it is necessary to modify the model and to process or enhance the experimental data. Among the data-driven methods, health indicators such as the capacity have been widely extracted and utilized for battery SOH or RUL prediction: Maitane et al. [
18] designed a new differential voltage curve for SOH prediction; Wu et al. [
19] employed a polynomial neural network (NN) which can be regarded as a black-box with parameters; the authors of [
20] utilized the fusion part incremental capacity and Gaussian process regression (GPR) to estimate the SOH; in [
21] the authors applied Dempster–Shafer theory and the Bayesian Monte Carlo method to develop a new method for SOH estimation; In [
22] relevance vector machine (RVM) and particle filtering were used to describe nonlinear relationships in dataset. Zheng C et al. [
23] used the long short-term memory network to estimate the SOH for electric vehicles.
However, most feature-based data-driven methods simply use signal processing methods to process single health indicators without considering the possible coupling relationships between the various health indicators. In addition, modification of the model affects the accuracy of the prediction. Moreover, most available optimization programs often fall into locally optimal solutions due to the prevailing difficulty of adjusting the hyperparameters in a data-driven method. Hence, a differential evolution (DE) algorithm is used here for parameter optimization and a hybrid model for SOH estimation is proposed that is based on the DE algorithm and an improved support vector regression (SVR) algorithm. The main contributions of this paper are as follows:
- (1)
Dynamic health indicators are extracted from the charging curves. The SOH prediction accuracy is regarded as the fitness of the evolution algorithm. Differential evolution algorithms are used to obtain the weights of the health indicators, and health indicators and their weights are combined to form an enhanced health indicator for the estimation of battery SOH.
- (2)
A DE algorithm is used to form an enhanced health indicator (HI) and hyperparameters in an improved SVR method. The article also analyzes the improvement in the prediction accuracy of the enhanced health indicator compared to the original health indicators.
The remainder of this article is organized as follows:
Section 2 presents the background information;
Section 3 extracts the health indicators from charging and discharging curves;
Section 4 presents the hybrid model framework, the improved SVR method, and the SOH estimation process;
Section 5 compares estimation performances of various methods; and the final section presents the conclusions of this study.
3. New Health Indicator for SOH Estimation
In this section, the internal electrochemical reaction of the lithium battery is analyzed so that effective and suitable health indicators can be selected for the accurate modeling of the degradation phenomenon of batteries, and the accessibility of the selected features in tests will be considered. Three datasets were obtained from the NASA database for research, based on which the following four subsections will focus on the introduction of SOH, experimental data analysis, feature extraction from the charging curves, and feature analysis that is based on grey correlation analysis (GCA).
3.1. Definition of SOH
SOH, which is an indicator that quantitatively describes the battery state of health, characterizes the ratio of a performance parameter to a nominal parameter after a period of use of the battery. However, no uniform definition has been proposed except for indicators and concepts that were established in [
30] for describing it. In this paper, the ratio of the current capacity to the nominal capacity is used to represent SOH, as expressed in Equation (8).
where
represents the SOH value at the
ith cycle,
represents the capacity at the
ith cycle, and
represents the initial capacity. As charging and discharging progress, the capacity curves show an overall degradation trend.
3.2. Experiment Data Analysis
To observe the declining trend of SOH under various conditions, it is critical to conduct cyclic charge-discharge experiments. The dataset was provided by the Prognostics Center of Excellence at NASA Ames [
31]. The three 18,650 LIBs numbered 05, 06, and 07 were produced by Idaho National Laboratory. The rated capacity and voltage of each battery are 2.2 Ah and 3.7 V [
32], respectively. Three batteries were run through three operations (charging, discharging, and measuring impedance) at 24 °C: first, the batteries were charged at a constant current of 1.5 A until the voltage reached 4.2 V. Then, they were charged at a constant voltage until the current dropped to 20 mA. Next, the batteries were discharged at a constant current of 2 A until the voltage reached the discharge cut-off voltage. The conditions are enumerated in the
Table 1, and their discharge cut-off voltages are inconsistent.
Figure 1 shows the capacity aging curves of these three batteries.
The byproduct may dissipate after the battery completes a charge-discharge cycle. Therefore, the capacity curves fluctuate in some cycles. Continuous switching between charging and discharging cycles may cause a momentary increase in battery capacity [
33]. Hence, the battery capacity will increase in the next cycle compared with that in the previous cycle. This phenomenon is called capacity regeneration [
34]. It will affect the precision and accuracy of SOH estimation [
22]. To solve the problem and reduce the estimation error, we set one parameter of the kernel function to represent the capacity regeneration in
Section 4.2.
3.3. Electrochemical Reaction Analysis and Features Extraction
Electrochemical reaction analysis shows that lithium ions, while charging, are forced by the external current to move from the cathode to the anode, thereby resulting in a negative concentration gradient in both electrodes, which increases in the direction of the current, peaks in the final stage of the constant current (CC) mode, and subsequently decreases to a minimum with a current drop at the constant voltage (CV) stage. As the battery ages, the charge quantity that is available on the cathode material will gradually decrease. Moreover, the growth of the solid electrolyte interface (SEI) layer will cause the internal resistance to continue to increase as the battery ages, thus, will reduce the duration of the CC charge mode. This analysis also shows that when discharging, lithium ions naturally move, in reverse, from the anode to the cathode to generate current, and the reduction of the voltage platform is mainly affected by ohmic resistance and polarization resistance. Therefore, battery aging leads to the increase of the polarization phenomenon and the decrease of the discharge time of the battery, where a relationship can be identified between the discharge time and the number of cycles.
Consider battery No. 06 as an example. Due to the inconsistency of the discharging modes under practical operating conditions, we focused on extracting features from the charging curves that reflect the battery degradation. In a later subsection of this article, the higher effectiveness of this approach compared to using the numbers of cycles as model inputs is discussed. According to
Figure 2, during the charging phase, the time that the battery spends in CC mode decreases as the number of cycles increase, and the curve becomes steep.
According to the above analysis, the internal electrochemical reactions of a lithium battery are closely associated with the battery charging mode. Therefore, to improve the accuracy of SOH estimation, three features (the constant current charging time (CCCT), the constant voltage charging time (CVCT), and the time duration of the fixed segment (FST) that reflect these dynamic changes are extracted as follows:
- (1)
CCCT: This is the time duration of the CC mode, namely, the amount of time for which the battery is polarized. According to the charging curves of batteries, the CCCT decreases as the cycle life increases.
- (2)
CVCT: CV mode plays a role in eliminating polarization during the battery charging process. According to
Figure 3b, the jitter of the CVCT curve of battery 06 is severe. According to the experimental results, its root means square error (RMSE) is indeed larger than those of the other two batteries, but it is smaller than those of the other compared methods.
- (3)
FST: This is the time that it takes for a battery to experience a fixed voltage interval during charging process. We focus on the constant current charging process. According to the experimental results, if only the slope of a point on the curve is considered and used as the input of the model, the obtained results will be affected by various factors (current fluctuations, ambient temperature, testing errors, etc.). The selection of this point such that high accuracy of the model is realized will be difficult. In the constant current charging process, the energy that was charged into the battery is proportional to FST. Previous research indicated that FST showed a downward trend as the battery was ageing.
Graphical representations of the selected features over many battery cycles are shown in
Figure 3.
3.4. Feature Analysis Based on GCA and Proposal of the Enhanced Health Indicator
To evaluate the performance of the extracted health indicators, it is necessary to determine the correlations between the original health indicators and the battery SOH. Therefore, this paper used the grey correlation analysis (GCA) method [
35] to assess the relational grades. The detailed procedure of the GCA algorithm is as follows:
For a specified dataset, first, set the reference sequence as
. Then, determine comparative sequences
, where
i represents the sequence number and
. In this case,
denotes the SOH sequence, and
represents the extracted features sequence. Then, calculate the relational coefficients:
where
denotes the identification coefficient. To make the final result accord with people’s habits, namely, to make the length of distribution interval of
not less than 0.5,
should be between 0 and 1. In this paper, we chose
= 0.5. After that, we calculated the relational grades
.
According to the above GCA calculation, as presented in
Table 2, a high relational grade is observed among the three features, whereas strong inconsistency of the relational grades is observed between the characteristics of the same battery and SOH. By comparison, as presented in
Table 2, the degree of correlation between the cycle numbers and SOH is the lowest among these features. It is inferred that more accurate SOH estimation results will be obtained if the extracted features are used as model inputs.
In addition, we calculated the relational grades between each two of the three features, which we used to describe the coupling relationships of the three features.
According to
Table 3, relational grade between any two features exceeds 0.6, and most of them exceed 0.7, thereby indicating that there are indeed coupling relationships of the three features. Therefore, to take advantage of this trait, the new concept of an enhanced health indicator is proposed, as expressed in Equation (10).
where
a,
b, and c are the corresponding features parameters, which will be optimized together with the parameters of the kernel function. Subscript
N indicates that the variable is normalized. The CVCT curve exhibits an upward trend. To increase the accuracy, we subtract 1 from the normalized CVCT so that the processed feature curve shows a similar downward trend to the other feature curves. In several references [
34,
35,
36], HIs were used directly to estimate SOH. Tests are conducted in
Section 5 on which the estimated values that were obtained using original HIs and the enhanced HI are compared in terms of accuracy.