Prediction of Lithium Battery Health State Based on Temperature Rate of Change and Incremental Capacity Change

Abstract

With the use of Li-ion batteries, Li-ion batteries will experience unavoidable aging, which can cause battery safety issues, performance degradation, and inaccurate SOC estimation, so it is necessary to predict the state of health (SOH) of Li-ion batteries. Existing methods for Li-ion battery state of health assessment mainly focus on parameters such as constant voltage charging time, constant current charging time, and discharging time, with little consideration of the impact of changes in Li-ion battery temperature on the state of health of Li-ion batteries. In this paper, a new prediction method for Li-ion battery health state based on the surface difference temperature (DT), incremental capacity analysis (ICA), and differential voltage analysis (DVA) is proposed. Five health factors are extracted from each of the three curves as input features to the model, respectively, and the weights, thresholds, and number of hidden layers of the Elman neural network are optimized using the Whale of a Whale Algorithm (WOA), which results in an average decrease of 43%, 49%, and 46% in MAE, RMSE, and MAPE compared to the Elman neural network. For the problem where the three predictions depend on different sources, the features of the three curves are fused using the weighted average method and predicted using the WOA–Elman neural network, whose MAE, RMSE, and MAPE are 0.00054, 0.0007897, and 0.06547% on average. The results show that the proposed method has an overall error of less than 2% in SOH prediction, improves the accuracy and robustness of the overall SOH estimation, and reduces the computational burden to some extent.

1. Introduction

With the increasing severity of oil pollution, the development and utilization of new energy sources have become a hot topic of discussion [1]. Lithium batteries are widely used in the new energy vehicle industry because of their high energy density, long usable life, green energy saving, and environmental protection [2]. With the long-time use of lithium batteries, lithium batteries work during harsh environments, so lithium batteries have irreversible capacity degradation problems. For the lithium batteries used in electric vehicles, it is generally defined that when their capacity decreases to 80% of the rated capacity, the lithium batteries need to be replaced as soon as possible to avoid overheating, overcharging, over-discharging, and other safety problems caused by the decline in the performance of lithium batteries [3]. Therefore, research on battery health management systems becomes crucial [4]. Among them, Li-ion battery health status is the key factor of the Li-ion battery health management system.

Currently, the primary methods for lithium battery health prediction are categorized into two main groups: model-based methods and data-driven methods [5]. The model-based approach focuses mainly on analyzing the aging mechanism of lithium batteries by analyzing the states and variables inside the lithium batteries and establishing equivalent aging models using electronic components such as resistors, capacitors, and inductors [6,7]. However, since the lithium battery is a system integrated by a variety of complex physical and chemical reactions during operation, different lithium batteries are accompanied by different chemical reactions during operation due to the different electrode materials, diaphragms, and solutions. In addition, Li-ion batteries are subject to changes in temperature, charging and discharging multiplicity, and dynamic environment during operation, so it is difficult to determine the law of capacity degradation from the electrochemical reaction inside the Li-ion battery [8,9,10].

In recent years, the rapid development of the data-driven method offers another way of thinking about lithium battery health management systems. The data-driven method is used to collect a large amount of historical data on lithium-ion batteries, and through the analysis of a large amount of historical data, to establish the connection between the capacity degradation of lithium-ion batteries and the data characteristics only from the perspective of data. The advantage of the data-driven method is that it does not rely on physicochemical analysis and circuit model simulation, and it can be used without delving into the complex mechanistic changes within the lithium-ion battery [11,12,13,14]. Pang et al. [15] proposed a method combining the wavelet decomposition technique (WDT) and nonlinear autoregressive neural network (NARNN) to capture the capacity regeneration phenomenon of lithium batteries. Ng et al. [16] compared equivalent circuit models (ECM) and physically based models and outlined an exhaustive survey of the application of machine learning techniques in battery condition prediction. Chaoui et al. [17] proposed to estimate the SOC of Li-ion batteries using recurrent neural networks (RNNs), and experiments have shown that the RNN networks are robust to dynamic loads, nonlinear dynamic properties, aging, hysteresis, and parameter uncertainty. The RNN networks are also robust to the effects of dynamic loads, nonlinear dynamic properties, aging, hysteresis, and parametric uncertainty. Zhang et al. [18] used incremental capacity analysis (ICA) to characterize the relationship between the incremental capacity of Li-ion batteries and the SOH of Li-ion batteries. They demonstrated that the maximum peak height in the ICA curve is linearly and positively correlated with the SOH of the battery. Li et al. [19] extracted four health features highly correlated with the SOH decrease from lithium battery cyclic charge/discharge data as inputs to support vector regression (SVR) and optimized the SVR model parameters based on the improved ant optimization algorithm. Jia et al. [20] combine the Gaussian process regression (GPR) method with probabilistic prediction for short-term SOH prediction, in which the inputs to the model extract some features of the voltage, current, and temperature profiles of Li-ion batteries during charging and discharging and use gray correlation to analyze the correlation with the SOH of the Li-ion batteries.

The current estimation methods of SOH for Li-ion batteries mainly focus on the changes in voltage and current during the charging and discharging process of Li-ion batteries and seldom consider the changes in temperature during the decline of Li-ion batteries. Therefore, this paper proposes an SOH estimation method based on the change in surface temperature and incremental change in the capacity of Li-ion battery, which contributes as follows:

(1)

By analyzing the characteristics of the temperature difference (DT) curve, the capacity increment (IC) curve, and the differential voltage analysis (DVA) curve, five features were extracted from each of the three curves as the health factors, respectively, and the correlation with lithium battery SOH was analyzed using the Pearson correlation coefficient.

(2)

For the problem of the random initialization of Elman neural network weights and thresholds, the whale optimization algorithm (WOA) is used for optimization to find the best weights and thresholds. Meanwhile, for the problem that the number of hidden layers is set artificially, which leads to unsatisfactory training results, the same whale optimization algorithm is used to obtain the optimal number of hidden layers.

(3)

For the problem of machine learning models relying on different information sources, a weighted average method can be used to fuse the data, which improves the overall estimation accuracy and robustness, and effectively avoids the differences in prediction results due to different sources of data.

2. Feature Extraction

2.1. Oxford Dataset

This paper uses the Oxford University battery dataset, which contains aging data from eight small lithium batteries containing voltage, current, temperature, and time information. The battery cathode material consists of lithium cobalt oxide and lithium nickel cobalt oxide, and the anode material is graphite. These batteries were characterized by every one hundred charge/discharge cycles using the constant current and constant voltage charging mode at a room temperature of 40 °C.

SOH is an important characterization of the state of health of lithium batteries. SOH is commonly defined as:

$$SOH=\frac{C_i}{C_0}\times 100\%$$

(1)

$C_i$ is the current available capacity of the lithium battery and $C_0$ is the rated capacity of the lithium battery. The SOH variation curves of the Oxford University lithium battery dataset are shown in the figure below, where the SOH of the lithium batteries showed different degrees of decline after many cycles of charging and discharging processes [21,22]. Figure 1 shows the SOH for the Oxford dataset.

2.2. Temperature Difference Curve

The capacity degradation of Li-ion batteries will be accompanied by complex electrochemical mechanisms throughout their service life. According to battery aging theory, surface temperature has a strong relationship with lithium battery aging, hence the introduction of the DT curve.

According to reference [23] it is known that given the finite resolution of the temperature sensor, the direct calculation of DT is susceptible to measurement noise. To avoid this problem, the DT curve is approximated as a finite difference over L sample intervals, and a new DT curve is used with the following equation:

$$DT\left(t\right)\approx \frac{T\left(t\right)-T\left(t-L\right)}{V\left(t\right)-V\left(t-L\right)}$$

(2)

where: $T\left(t\right)$ is the temperature at time $t$, $L$ is the sampling interval, and $V\left(t\right)$ is the voltage at time $t$.

The principle is to solve the first-order derivative of a battery’s terminal voltage temperature (V − T) under constant current charging or discharging conditions to obtain the terminal voltage temperature variation (V − dTdV). V − dT/dV describes the magnitude of the temperature rise of the cell per unit voltage of the battery. Choosing a suitable sampling interval ($L$) is very important for the plotting of DT curves, and after simulation, this paper uses $L=120$. Gaussian filtering is used to filter the whole DT curve, and the results are obtained as shown in Figure 2.

The DT curve takes the voltage as the horizontal coordinate and the rate of change in temperature between two samples as the vertical coordinate, which can effectively respond to the health status of the lithium battery. The peaks and valleys of the DT curves can reflect the current positions of the phase transitions in the cathode and anode. The peak positions describe the peak potentials at which the lithium-ion de-embedding and embedding phases occur. The shift in peak position through degradation can characterize the impedance rise and stoichiometric drift of the cell. The peak height indicates the point at which the maximum heat production rate of those phases occurs. The peak width describes the potential window of the combined phase in both electrodes. The peak area gives information about the heat produced during the lithium-ion embedding and de-embedding phases. The peaks and valleys of the DT curves correspond to cell aging and the combined phase transition of the cathode and anode materials.

DT Curve Feature Extraction

As shown in Figure 2, the temperature change in the Li-ion battery takes the lead in monotonically decreasing as the voltage increases, monotonically increasing when it goes down to the minimum value, and monotonically decreasing again when it goes up to the maximum value. Thus, the temperature profile shows a trend of peaks and valleys. As the SOH of the lithium battery decreases, so does the peak temperature rate of change. Therefore, in this paper, the peak value of the DT curve (HF1), the value of the valley (HF2), the area of the peak (HF3), the area of the valley (HF4), and the difference in the height of the peaks and valleys (HF5) are selected as features. The Pearson correlation coefficients for these features are shown in

Table 1. Pearson correlation coefficient analysis of health factors and SOH for DT characteristics.

Battery	HF1	HF2	HF3	HF4	HF5
Cell 1	0.7812	−0.7768	−0.8952	0.8792	0.8397
Cell 2	0.8875	−0.8436	−0.9696	0.9531	0.9624
Cell 3	0.8811	−0.8080	−0.9647	0.9306	0.9480
Cell 4	0.9441	−0.7911	−0.9776	0.9415	0.9609
Cell 5	0.7046	−0.8104	−0.9064	0.9159	0.91552
Cell 6	0.8457	−0.8247	−0.9666	0.9450	0.9779
Cell 7	0.8681	−0.7692	−0.9700	0.9248	0.9347
Cell 8	0.8983	−0.9174	−0.9774	0.9724	0.9849

. The closer the absolute value of the Pearson correlation coefficient is to 1, the stronger the correlation between the two.

2.3. Incremental Capacity Analysis

During the charging and discharging process of the Li-ion battery, its internal relative equilibrium state is manifested as the voltage plateau period on the external characteristics, and the change in the voltage plateau period has a strong relationship with the decline of the Li-ion battery, so the incremental capacity (IC) is introduced. The principle is to take the first-order derivative of a battery’s terminal voltage capacity (V − Q) under constant current charging or discharging conditions to obtain the terminal voltage capacity change rate (V − dQ/dV). V − dQ/dV describes the power charged or discharged by the battery per unit voltage [24]. Mathematically, IC is expressed as the amount of battery power added over a continuous voltage increment, and in constant current charging mode, IC is calculated as:

$$ICA=\frac{\mathrm{d}Q}{\mathrm{d}V}\approx \frac{\mathsf{\Delta }Q}{\mathsf{\Delta }V}=\frac{Q_t-Q_{t-1}}{V_t-V_{t-1}}$$

(3)

where: Q is the capacity, and V is the voltage. The ΔV selected varies from cell to cell, depending on the severity of the measurement noise contained in the IC curve. Unlike standard battery aging measurements (e.g., internal resistance and capacity degradation assessment) methods, IC analysis investigates the aging mechanism of the battery in terms of the electrode level, and one of its significant advantages is the ability to convert the battery voltage plateaus into clearly identifiable peaks on the IC curves, respectively, which characterizes the phase transition of the battery during lithium-ion active material intercalation and decalcification processes [25].

Before calculating dQ/dV, the charging or discharging capacity of the battery should be calculated so it can be obtained using the ampere–time integration method:

$$Q=\int I\times dt$$

(4)

The relationship between the capacity and terminal voltage of a lithium battery is composed of Equations (5) and (6):

$$V=f\left(Q\right)$$

(5)

$$Q=f^{-1}\left(V\right)$$

(6)

IC Curve Feature Extraction

As can be seen in Figure 3, with the increase in the number of cycles, the change in the IC curve tends to be more and more smooth, and the peak of the curve is also gradually reduced. The IC curve peaks’ height, position, and shape reflect SOH’s decline during the charging and discharging process of Li-ion batteries. The first peak gradually disappears with the increase in cycles, and the second peak gradually decreases. At the same time, the position is shifted back due to the loss of active materials and lithium ions inside the battery, making the chemical reaction change, leading to an increase in the internal resistance and causing the voltage plateau to be shifted. The height of the central peak (HF1), the coordinates of the principal peak (HF2), the area of the central peak (HF3), the left slope of the central peak (HF4), and the proper slope of the central peak (HF5) were selected as features. The correlation of their characteristics with the SOH of lithium batteries is shown in

Table 2. Pearson correlation coefficient analysis of health factors and SOH for IC characteristics.

Battery	HF1	HF2	HF3	HF4	HF5
Cell 1	0.9633	−0.9625	0.9794	0.9524	0.9711
Cell 2	0.9451	−0.9682	0.9722	0.9156	0.9652
Cell 3	0.9671	−0.9707	0.9806	0.9606	0.9717
Cell 4	0.9771	−0.9731	0.9839	0.9756	0.9775
Cell 5	0.9048	−0.9917	0.9448	0.8724	0.9303
Cell 6	0.9741	−0.9832	0.9816	0.9733	0.9736
Cell 7	0.9704	−0.9728	0.9836	0.9665	0.9724
Cell 8	0.9656	−0.9698	0.9811	0.9547	0.9736

. The capacity increment curve of the Cell 1 battery is shown in Figure 3.

2.4. Differential Voltage Analysis

Differential voltage analysis (DVA) is an essential method for battery health state estimation. The differential voltage on the expression of the formula is the reciprocal of the capacity increment, and its relationship with the charge capacity is called the DVA curve, as shown in Figure 4. The distance between the peaks of the curves can characterize the amount of electricity transferred during the phase transition [26].

$$DVA=\frac{\mathrm{d}V}{\mathrm{d}Q}\approx \frac{\mathsf{\Delta }V}{\mathsf{\Delta }Q}=\frac{V_t-V_{t-1}}{Q_t-Q_{t-1}}$$

(7)

In this case, the lithium battery charging capacity is calculated similarly when calculating the IC module, i.e., using the ampere–time integration method. One of the DVA curves for the Oxford dataset is shown in Figure 4.

Q is the lithium battery’s charging capacity, and V is the terminal voltage of the lithium battery. The DVA curve takes the charging capacitance as the horizontal coordinate and the rate of change in the capacity between the two samples as the vertical coordinate. The DVA curve, as the inverse of the IC curve, shows an opposite trend to the IC curve in terms of the curve change.

DVA Curve Feature Extraction

It was shown that the voltage interval between the two peaks of the DV curve is linearly correlated with the battery capacity and can be used for SOH estimation. The height of the peaks (HF1), the coordinates of the peaks (HF2), the area of the peaks (HF3), the left slope of the peaks (HF4), and the right slope of the peaks (HF5) were chosen as the characteristics. The correlation between their properties and the SOH of lithium batteries is shown in

Table 3. Pearson correlation coefficient analysis of health factors and SOH for DVA characteristics.

Battery	HF1	HF2	HF3	HF4	HF5
Cell 1	−0.9863	0.2285	−0.9878	−0.2221	−0.9867
Cell 2	−0.9708	0.7995	−0.9755	−0.8056	−0.9588
Cell 3	−0.9874	0.3447	−0.9785	−0.3081	−0.9843
Cell 4	−0.9863	0.2873	−0.9895	−0.2430	−0.9841
Cell 5	0.9048	0.6261	−0.9673	−0.6233	−0.9521
Cell 6	−0.9829	0.1944	−0.9892	−0.2928	−0.9819
Cell 7	−0.9837	0.6072	−0.9887	−0.2071	−0.9833
Cell 8	−0.9849	0.3157	−0.9878	−0.2140	−0.9842

.

3. Elman Neural Network

After determining the input features, it is necessary to establish the relationship between the input features and the SOH, so an efficient machine learning algorithm is used. The Elman neural network adds a connection layer to the BP neural network. The connection layer is equivalent to a delayed output unit that stores the output of the previous moment of the hidden layer and improves the input for the next moment of the hidden layer. As a well-known regression machine learning algorithm, Elman has been successfully used for SOH estimation [27]. The structure of the Elman neural network is shown in Figure 5. The goal of Elman is to establish the following mapping between input features and SOH:

The signaling expression for the Elman neural network is:

$$\begin{array}{c}{y}_i\left(t\right)=g\left[{\omega }_2{y}^1\left(t\right)+B_2\right]\\ y^1\left(t\right)=f\left[{\omega }_3{y}_1\left(t\right)+{\omega }_1{x}_i\left(t-1\right)+B_1\right]\\ y_1^{*}\left(t\right)=y^1\left(t+1\right)\end{array}$$

(8)

where: t is the time point and $x_i\left(t-1\right)$ is the ith input of the input layer at time t−1. $y_i\left(t\right)$ is the jth output of the output layer at moment t. $y^1\left(t\right)$ is the 1st output of the implicit layer at moment t; $y_1^{*}\left(t\right)$ is the 1st output of the connection layer at moment t; ${\omega }_1$ is the input layer and implicit layer connection weights; ${\omega }_2$ is the implicit layer and output layer connection weights; ${\omega }_3$ is the connectivity layer and implicit layer connection weights; $B_1$ is the implicit layer threshold; $B_2$ is the output layer threshold. The transfer function for the input layer is $g\left[{\omega }_2{y}^1\left(t\right)+B_2\right]$. The transfer function of the implicit layer is $f\left[{\omega }_3{y}_1\left(t\right)+{\omega }_1{x}_i\left(t-1\right)+B_1\right]$.

Whale Optimization Algorithm (WOA)

When using Elman neural networks for predictive modeling, the initial weights and thresholds are generally initialized by pseudo-random numbers, which makes the performance of models trained on training data unstable. For the fact that the prediction accuracy of the Elman neural network is greatly affected by the weight values, thresholds, and the number of hidden layers, the parameters are optimized using the whale optimization algorithm in order to improve the prediction accuracy of the Elman neural network.

The whale optimization algorithm simulates the hunting behavior of humpback whales, which use a spiral path to encircle their prey and generate a large number of bubbles in their path during the hunting process. Therefore, the algorithm includes three phases: encirclement, bubble generation, and search [28].

Whales search in a global solution space and must locate their prey first to encircle. Since the search space of the optimal solution is unknown, the WOA algorithm assumes that the current best candidate solution is the target prey or close to the optimal solution. After defining the optimal search agent, other search agents will try to update their positions to the optimal search agent [29,30]. Its formula is:

$$D=\mid C\cdot {\mathrm{x}}^{*}\left(t\right)-\mathrm{x}\left(t\right)\mid$$

(9)

$$x\left(t+1\right)=X^{*}\left(t\right)-A\cdot D$$

(10)

where: t is the current number of iterations; $A$ and $C$ are vector coefficients, which are used to control the movement mode of the whale; $\cdot$ for element-by-element multiplication; ${\mathrm{x}}^{*}\left(t\right)$ is the current location of the best solution obtained; $\mathrm{x}\left(t\right)$ is the position vector of the individual whale; $D$ is the distance between the individual whale and its prey. where $A$ and $C$ are calculated as shown below [31]:

$$A=2a\times r_1-a$$

(11)

$$C=2\times r_2$$

(12)

$$a=2-\frac{2t}{T_{max}}$$

(13)

$a$ is a control parameter that decreases from bilinear to zero throughout the iteration, $r_1$ and $r_2$ are random variables in [0, 1]. $T_{max}$ is the maximum number of iterations.

Humpback whales feed by two main mechanisms: encircling predation and bubble net predation. The positional update between the humpback whale and its prey when feeding with bubble nets is expressed by the logarithmic spiral equation as in Equation:

$$x\left(t+1\right)=D^{\prime }\times e^{bi}\times \cos \left(2\pi l\right)+X^{*}\left(t\right)$$

(14)

$$D^{\prime }=\mid x^{*}\left(t\right)-x\left(t\right)\mid$$

(15)

where: $D^{\prime }$ is the distance to the current individual optimal solution, $b$ is the spiral shape parameter, $i$ is a random number whose value domain is evenly distributed in [−1, 1].

Since there are two predatory behaviors during the proximity to the prey, the WOA chooses either bubble net predation or contraction encirclement based on the probability p. The position update formula is shown in:

$$x\left(t+1\right)=\{\begin{array}{cccc}{X}^{*}\left(t\right)-A\cdot D& & p\le 0.5& \\ D^{*}\times e^{bi}\times \cos \left(2\pi l\right)+X^{*}\left(t\right)& & p\ge 0.5& \end{array}$$

(16)

where: $p$ is the probability of predation mechanism, a random number with value domain [0, 1]. As the number of iterations t increases, the parameter $A$ and the convergence factor gradually decrease, and if |A| < 1, each whale gradually surrounds the current optimal solution, which belongs to the local optimization stage in WOA.

To ensure that all whales are adequately searched in the solution space, the WOA updates positions based on the distance between whales to allow for randomized searching. Thus, when |A| ≥ 1, the searching individual swims to a random whale individual. As in style:

$$D^{″}=\mid C\cdot {\mathrm{X}}_{\mathrm{rand}}\left(t\right)-\mathrm{X}\left(t\right)\mid$$

(17)

$$x\left(t+1\right)={\mathrm{X}}_{\mathrm{rand}}\left(t\right)-A\cdot D$$

(18)

where: $D^{″}$ denotes the distance between a random whale individual and its prey, and ${\mathrm{X}}_{\mathrm{rand}}$ denotes the position vector of a randomly selected whale individual from the whale population [32,33].

The optimal weights, thresholds, and the number of layers of hidden layers in the Elman neural network are determined by the whale optimization algorithm.

4. Evaluation Indicators

The extracted health factors are input into the model as features, and the data set is divided into two groups using the leave-one-out method; one group is the training set, and one group is the test set. The training set consists of seven samples; the remaining group of samples is used as the test set, and the eight groups of samples are used as the test set in turn.

In order to assess the validity and accuracy of the model comprehensively, the root mean square error (MSE), mean absolute percentage error (MAPE), and mean absolute error (MAE) were selected to assess the performance of the model with the following formulas:

$$\mathrm{MSE}=\sqrt{\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}$$

(19)

$$MAPE=\frac{1}{n}{{\displaystyle \sum }}_{i=1}^n\frac{\mid y_i{}^{\prime }-y_i\mid }{y_i}$$

(20)

$$\mathrm{MAE}=\frac{{{\displaystyle \sum }}_{i=1}^n\mid y_i-x_i\mid }{n}$$

(21)

5. Projected Results

5.1. DT Forecast Results

The five features extracted from the DT curves were input into the WOA–Elman neural network and the unoptimized Elman neural network to predict the SOH of Li-ion batteries. The prediction curves are shown in Figure 6, and the RMSE, MAE, and MAPE of the estimation results are shown in

Table 4. WOA–Elman error for DT features.

Test Set	MAE	RMSE	MAPE
Cell 1	0.0010254	0.0012192	0.12462%
Cell 2	0.00025247	0.0003664	0.030107%
Cell 3	0.0015656	0.0021123	0.18437%
Cell 4	0.00023486	0.00035171	0.027124%
Cell 5	0.00064367	0.00091341	0.073228%
Cell 6	0.00059183	0.00076863	0.067114%
Cell 7	0.00033889	0.00043897	0.039798%
Cell 8	0.00039627	0.00057397	0.047694%

and

Table 5. Elman error for DT features.

Test Set	MAE	RMSE	MAPE
Cell 1	0.0039592	0.0044058	0.45667%
Cell 2	0.0016823	0.0020586	0.21423%
Cell 3	0.0090102	0.010912	1.1042%
Cell 4	0.0015892	0.0018576	0.18737%
Cell 5	0.0017174	0.0019943	0.20165%
Cell 6	0.001516	0.001828	0.17711%
Cell 7	0.0012513	0.0015811	0.1518%
Cell 8	0.002599	0.0029838	0.32145%

.

The prediction results of the Elman neural network not optimized by the whale algorithm showed different degrees of fluctuation, especially for Cell 3 cells, whose MAPE reached 1.1042%. Thus, there was a large error between the prediction curve and the true value. The Elman neural network after optimization by the whale optimization algorithm (WOA–Elman), on the other hand, has a substantial increase in accuracy, with maximum reductions in MAE, RMSE, and MAPE of 84%, 82%, and 85%, and average reductions of 66%, 72%, and 76%. It is shown that the whale optimization algorithm (WOA) for optimization of the Elman neural network in terms of weights, thresholds, and number of hidden layers can predict the SOH of lithium batteries more accurately.

5.2. IC Forecast Results

The five features extracted from the IC curves were input into the WOA–Elman neural network and the unoptimized Elman neural network to predict the SOH of Li-ion batteries; the prediction curves are shown in Figure 7, and the RMSE, MAE, and MAPE of the estimation results are shown in

Table 6. WOA–Elman error for IC features.

Test Set	MAE	RMSE	MAPE
Cell 1	0.0015355	0.0019628	0.186%
Cell 2	0.0036111	0.0053521	0.48069%
Cell 3	0.0017421	0.0020672	0.20905%
Cell 4	0.0012496	0.0015615	0.15047%
Cell 5	0.0027543	0.0029526	0.32258%
Cell 6	0.0015464	0.0020215	0.1853%
Cell 7	0.0024968	0.0028942	0.30831%
Cell 8	0.0017175	0.0022618	0.22121%

and

Table 7. Elman error for IC features.

Test Set	MAE	RMSE	MAPE
Cell 1	0.0020832	0.0027034	0.25851%
Cell 2	0.0044156	0.0062386	0.58832%
Cell 3	0.0028335	0.0031952	0.33774%
Cell 4	0.0029127	0.0036651	0.3591%
Cell 5	0.0046339	0.0077523	0.58816%
Cell 6	0.0019521	0.002523	0.23102%
Cell 7	0.0041325	0.0048961	0.51481%
Cell 8	0.0027281	0.0037465	0.33547%

.

Based on the prediction of the SOH of lithium batteries by IC features, both the WOA–Elman neural network and the Elman neural network showed high stability and accuracy, which demonstrated the strong correlation between ICA features and the SOH of lithium batteries. The Elman optimized by the whale optimization algorithm has its MAE, RMSE, and MAPE reduced by a maximum of 57%, 61%, and 58% and an average of 34%, 37%, and 36%.

5.3. DVA Forecast Results

The five features extracted from the IC curves were input into the WOA–Elman neural network and the unoptimized Elman neural network to predict the SOH of Li-ion batteries, the prediction curves are shown in Figure 8, and the RMSE, MAE, and MAPE of the estimation results are shown in

Table 8. WOA–Elman error for DVA features.

Test Set	MAE	RMSE	MAPE
Cell 1	0.0022685	0.0029795	0.28117%
Cell 2	0.0048903	0.0079513	0.66421%
Cell 3	0.0024732	0.0037361	0.30522%
Cell 4	0.0027153	0.0032519	0.32872%
Cell 5	0.0036418	0.0042333	0.42154%
Cell 6	0.0016742	0.0024906	0.19799%
Cell 7	0.001477	0.0017831	0.17052%
Cell 8	0.001571	0.001917	0.1962%

and

Table 9. Elman error for DVA features.

Test Set	MAE	RMSE	MAPE
Cell 1	0.0021908	0.0036268	0.27287%
Cell 2	0.0084709	0.013501	1.1566%
Cell 3	0.0038561	0.0058704	0.41307%
Cell 4	0.0037689	0.0069539	0.46528%
Cell 5	0.0032137	0.0080635	0.43145%
Cell 6	0.002645	0.0032587	0.3141%
Cell 7	0.0028182	0.0034156	0.33942%
Cell 8	0.0023898	0.0041675	0.30402%

.

In the SOH prediction of Li-ion batteries based on DVA features, the prediction results of the WOA–Elman neural network perform weaker than the Elman neural network in some aspects. For example, in the results of Cell 1 and Cell 5 batteries, the Elman neural network optimized by the whale optimization algorithm improves the MAE by 3.4% and 17%, i.e., there is a negative optimization result, which is due to the fact that the whale optimization algorithm is affected by the initialization of the pseudo-random numbers to a certain extent. However, overall, the WOA–Elman neural network outperforms the Elman neural network with a maximum reduction of 47%, 54%, and 49% in MAE, RMSE, and MAPE, and an average reduction of 25%, 39%, and 26%.

6. Weighted Average Method

The prediction results of the three methods show strong robustness and stability for batteries with different capacities, and the errors of the prediction results are controlled within 1%. For Cell 2, the SOH drop after 40 cycles and the SOH regeneration after 70 cycles are accurately predicted by the three methods. For Cell 4, Cell 5, and Cell 6, the charge/discharge cycle data are less, but the three methods are still able to estimate the SOH over the whole battery life with a maximum error of less than 1%. However, the results of the three methods for the same battery can show a large difference in the predicted data, such as the Cell 2 battery. The MAE errors of the three methods for their predictions are 0.00025247, 0.0036111, and 0.0048903, respectively, and this discrepancy implies that the three methods rely on different sources of information and that combining the three methods has the potential to obtain more accurate SOH estimation results. Therefore, the weighted average method was introduced.

In this paper, the weight of DT curve features is 0.7, the weight of ICA curve features is 0.2, and the weight of DVA curve features is 0.1. The Pearson correlation coefficients of the characteristics of the data after the weighted average method are shown in

Table 10. Pearson’s correlation coefficient between the weighted average processed features and SOH.

Battery	HF1	HF2	HF3	HF4	HF5
Cell 1	0.8848	−0.7813	−0.8953	0.8754	0.9740
Cell 2	0.9827	−0.8479	−0.9697	0.9515	0.9734
Cell 3	0.9460	−0.8132	−0.9649	0.9274	0.9759
Cell 4	0.9801	−0.7945	−0.9777	0.9392	0.9821
Cell 5	0.8851	−0.8139	−0.9066	0.9139	0.9387
Cell 6	0.9403	−0.8273	−0.9667	0.9431	0.9800
Cell 7	0.9414	−0.7745	−0.9701	0.9224	0.9751
Cell 8	0.9457	−0.9194	−0.9775	0.97714	0.9794

.

The weighted average processed features are imported into the WOA–Elman neural network and the results are shown in Figure 9 and

Table 11. Prediction error of WOA–Elman neural network after weighted average processing.

Test Set	MAE	RMSE	MAPE
Cell 1	0.00055622	0.00083431	0.06454%
Cell 2	0.00044995	0.00058576	0.058233%
Cell 3	0.00121419	0.0019488	0.15269%
Cell 4	0.00026275	0.00039682	0.030537%
Cell 5	0.00082164	0.0011593	0.093373%
Cell 6	0.00034954	0.00047364	0.039561%
Cell 7	0.0003449	0.00042638	0.041071%
Cell 8	0.00036636	0.00049316	0.043783%

.

As seen from Figure 9 and Table 11, the SOH predicted by the weighted average method is very close to the actual value. Among them, the MAE of the Cell 6 cell reaches the lowest value, and the accuracy of the Cell 2 cell is greatly improved. The maximum MAE is 0.0012419, the maximum RMSE is 0.0011593, and the maximum MAPE is 0.15269%, much lower than the maximum of the previous three characteristics. After data fusion by the weighted average method, the error of SOH prediction of a lithium battery by machine learning is reduced, the robustness is significantly improved, and the prediction curve does not fluctuate much, indicating that the dependence on different information sources is reduced.

7. Conclusions

In this paper, three SOH estimation methods based on lithium battery surface temperature change, capacity increment analysis, and differential voltage analysis are proposed, and five health factors are extracted from the three curves, respectively, which are found to have a strong relationship with lithium battery SOH through the Pearson correlation coefficient. For the problems of random initialization of Elman neural network weights and thresholds and artificial setting of the number of hidden layers, the whale optimization algorithm is introduced to optimize the weights, thresholds, and the number of hidden layers of the Elman neural network, and the MAE, RMSE, and MAPE are reduced by 43%, 49%, and 46% on average, and the maximum reduction is 84%, 82%, and 85%. However, negative optimization also occurs, such as in the Cell 2 and Cell 5 cells of the results predicted by DVA, and the MAE of WOA–Elman is improved by 3.4% and 17% compared to Elman, which is because the whale optimization algorithm also belongs to a kind of machine learning algorithm, which is affected by the initialization of pseudo-random numbers to a certain extent. On the whole, the WOA–Elman neural network is better than the Elman neural network in predicting the SOH of lithium batteries, and the prediction error is less than 2% on average. For the problem that the three methods depend on different sources of information, a weighted average method was introduced for data fusion and prediction using the WOA–Elman neural network, with an average MAE, RMSE, and MAPE of 0.00054, 0.0007897, and 0.06547%. The validity of the method was verified with the data of eight batteries from the Oxford dataset, which showed strong stability and accuracy for lithium batteries with different capacities.