The dataset used in this article is a prismatic battery with a nominal capacity of 1100 mAh, designated as CS2-36, publicly available at the University of Maryland. Due to the fact that a small rate of charging and discharging current can better extract features that characterize the changes in battery SOH, and batteries with a capacity lower than 80% of the initial capacity are considered failed batteries. Therefore, a small magnification charging dataset with a remaining capacity of 100%–80% of the initial capacity was selected. Battery was tested using the Arbin Battery Tester. The specific operation process is as follows. (1) Charge the battery to 4.2 V using a constant current charging method of 0.5 C, (2) Charge the battery current to below 0.05 A using a constant voltage charging method of 4.2 V, (3) Discharge at a rate of 1 C to the battery cut-off voltage of 2.7 V, (4) Cycle steps (1), (2) and (3) until the battery capacity drops to around 80%. Fig. 1 shows the remaining capacity variation curve of the battery after 463 cycles.

From the capacity change chart, it can be seen that after many cycles, the capacity increases compared to the previous one, which is a phenomenon of capacity increase. That is, if the battery is left idle for a period of time after several cycles, the next discharge capacity will increase, which is a normal phenomenon. Some cycles show a significant drop in capacity after completion, and the capacity returns to normal values in the next cycle, which is obviously abnormal. The solution is to set a capacity decrease threshold. If it exceeds this threshold, it is considered abnormal data and needs to be removed. Fig. 2 shows the capacity decline curve after removing abnormal data. To obtain the capacity change curve, it is necessary to convert it into battery SOH. Take the discharge capacity of the first cycle as the initial capacity, and calculate the ratio of each discharge capacity to the initial battery capacity to obtain the battery SOH for the current cycle. Fig. 3 shows the variation curve of battery SOH with the number of battery discharge cycles.

Before obtaining the IC curve, it is necessary to obtain the Q-V curve of constant current charging, that is, during the offline training stage, extract the charging process from the battery degradation dataset and analyze the features of battery capacity degradation [11]. Fig. 4 shows the relationship between capacity and voltage changes during a certain cycle. It can be seen from the graph that the Q-V curve does not increase linearly. In the initial stage of charging, the voltage rises rapidly. As the charging current flows in, the voltage and capacity also increase rapidly, but the rate of increase is different and constantly changing. Therefore, it is necessary to convert this subtle rate change into intuitive rate data.

The ICA curve method organizes the initial charging data to obtain the dQ/dV data during the charging or discharging process, and then obtains the capacity change rate-terminal voltage (dQ/dV‐V) curve, as shown in the figure. The principle of IC curve is to calculate the first derivative of the Q-V curve of the battery under constant current charging or discharging conditions to obtain the dQ/dV‐Vcurve. To solve the first-order derivative of the IC curve, it is necessary to first perform function fitting on the Q‐V curve, which inevitably introduces fitting errors. Therefore, this article adopts a simple and practical method to draw the IC curve, ΔV replacing it with a fixed voltage interval ΔQ to represent the ΔV capacity change in the corresponding interval. When it ΔV approaches 0, it is approximately dQdV≈ΔQΔV. Fig. 5 shows the IC curve derived from the Q-V curve variation.

Wavelet transform can perform finer processing on signals and better express certain features of the signal [12]. From the perspective of the capacity increment curve, the curve has high noise and is not smooth enough, which is not conducive to collecting the required sample points. Therefore, it is necessary to denoise the IC curve. There are many methods that can be used to handle the noise of capacity increment curves, such as sliding average filtering, first-order low-pass filtering, wavelet packet transform, Fourier transform, etc. In this article, considering the simplicity of operation and good filtering effect, wavelet threshold filtering is chosen. By selecting appropriate decomposition levels, threshold rules, and processing methods, noise in the signal can be effectively removed while retaining useful information, achieving wavelet denoising function. The specific parameter settings are as follows: the threshold rule is heursure, the denoising method is soft thresholding, the reconstruction mode is one stage reconstruction, the decomposition level is 5, and the wavelet type is db4. Fig. 6 shows the noise reduction results of the IC curve.

The charging process of lithium batteries consists of a constant current section (CC) and a constant voltage section (CV). The battery is first charged at a constant current to the charging cut-off voltage, and then charged at a constant cut-off voltage to the cut-off current. The time required to complete the constant current and constant voltage sections varies in different battery health states. However, due to the large proportion of time spent in the constant current section of a typical battery to the entire charging time, the constant current section charging time is chosen as a parameter for battery SOH estimation. In the MATLAB environment, record the charging time t1 at the starting point of the constant current section, and then record the charging time t2 at the ending point of the constant current section. t2−t1 is the charging time of the constant current section. Fig. 7 shows the charging curve of the constant current section as a function of the battery discharge cycle. Fig. 8 shows the relationship between the extracted charging time and the number of battery cycles. Take the battery charging time as the battery health feature 1, denoted as H₁.

With the intensification of battery aging, the charging and discharging performance of batteries is also deteriorating. In order to accurately reflect the degree of battery aging and estimate the battery SOH, it is necessary to select accurate battery health features. To draw an IC curve, it is necessary to convert it. In this article, an IC curve is drawn every 50 charge and discharge cycles until the battery SOH value is discharged to around 80%, in order to identify the health features that can describe battery aging, as shown in Fig. 9. From the graph, it can be seen that as the number of battery cycles increases, the overall IC curve moves towards the lower right corner of the coordinate axis, with the two IC peaks particularly prominent moving towards the lower right corner of the coordinate axis. Therefore, this article selects the horizontal axis (H₂) of the first IC peak, the vertical axis (H₃) of the first IC peak, the horizontal axis (H₄) of the second IC peak, and the vertical axis (H₅) of the second IC peak as the second, third, fourth, and fifth health features of the battery, respectively.

In the MATLAB environment, extract the horizontal and vertical coordinates of the two peaks of each IC curve as health factors. The specific operation is to divide the IC curve into two parts, left and right, with 3.86 V as the boundary, and determine the peak value of each part in bubble sorting. The variation pattern of the four health features with the number of battery cycles is shown in Figs. 10–13. From the four graphs, it can be seen that the health features H₂, H₃, and H₅ show a significant upward/downward trend with the number of cycles, while the fluctuation of H₄ is significant. Therefore, quantitative analysis of health features is necessary.

In order to quantitatively analyze the relationship between five health features and battery SOH, this article uses the Pearson correlation coefficient method for correlation analysis. The principle is the ratio of the covariance of variables X and Y to the standard deviation of X and Y. The formula is: r=∑i=1N(Xi−X¯)(Yi−Y¯)∑i=1N(Xi−X¯)2∑i=1N(Yi−Y¯)2, where N represents the number of samples, Xi represents the inth observation value of the variable X, X¯ represents the average value of the variable X, Yi represents the inth observation value of the variable Y, and Y¯ represents the average value of the variable Y.

The range of Pearson correlation coefficient is between [−1, 1]. The correlation coefficient shows directionality: if the correlation coefficient is close to 1, it indicates a high positive correlation between the two variables; If the correlation coefficient is close to −1, it indicates a high negative correlation between the two variables; If the correlation coefficient is close to 0, it indicates that the two variables are independent of each other and have no correlation. Specifically, if the absolute value of the Pearson coefficient is between 0.8–1.0, it indicates extreme correlation between the two variables. If it is between 0.6–0.8, it indicates strong correlation between the two variables. If it is between 0.4–0.6, it indicates moderate correlation. If it is between 0.2–0.4, it indicates weak correlation between the two variables. If it is between 0–0.2, it indicates extremely weak or no correlation between the two variables. From the table below, it can be seen that features 1, 2, 3, and 5 have a strong correlation with battery SOH, while feature 4 has a strong correlation with battery SOH. Therefore, all five health features can be used as indicators for battery SOH estimation. Table 1 shows the Pearson coefficients between health factors and battery capacity.

WOA is an algorithm that simulates whales hunting prey, covering three types of predation processes: encirclement, predation, and search [13]. The specific predation process is as follows:

(1) Surrounding

In the WOA algorithm, the position of prey is treated as the optimal solution, and whales can find the position of prey and surround them [14]. The iterative process of the WOA algorithm is the process of whales surrounding their prey. Each iteration calculates the fitness of the whale population according to the objective function, and the whale position with the highest fitness value is used as the optimal solution for this iteration, while other whales surround towards the direction of the optimal solution. The specific process is as follows: (1)D→=|C→⋅X→b(t)−X→(t)| (2)X→(t+1)=|X→b(t)−A→⋅D→|

In the formula, in the tnth iteration process, X→(t) represents the current whale position and X→b(t) represents the current optimal position. D→ represents the enclosing step size, where have the A→ and C→ following specific meanings: (3)A→=2a→⋅r→−a→ (4)C→=2⋅r→

In the formula, a→ represents a gradual linear decrease from 2 to 0; r→ is a random number within the interval of [0, 1].

(2) Predation

After completing the encirclement behavior, whales will engage in contraction encirclement and spiral approach to launch an attack. Control A→ by linearly reducing a→ in Eq. (3), this is the contraction and enclosure behavior of whales; Whales use spiral swimming to shrink their encirclement and gradually approach their prey, which is their spiral approach behavior. The expression of this spiral approximation equation [15] is shown in Eq. (5). (5)X→(t+1)=D∗→⋅e(nl)⋅cos⁡(2πl)+X→b(t)

In the formula, l is a random number within the interval [−1, 1], and n is the helix shape constant. D∗→ is the distance between the current prey and the current whale position. In the predation stage, contraction and spiral approximation cannot occur simultaneously. Therefore, a probability function is determined to distinguish between the two, and a probability threshold of 0.5 is set, which means that the probability P is randomly taken within the interval [0, 1]. The expression is shown in Eq. (6). (6)X→(t+1)={X→b(t)−A→⋅D→,P≤0.5D∗→⋅e(nl)⋅cos⁡(2πl)+X→b(t),P≥0.5

(3) Search

The WOA model updates positions through random whale individuals [16]. During this process, whales can either surround themselves towards the optimal whale position or randomly move towards a certain whale position, making the algorithm more capable of global search. Its mathematical model is represented by Eqs. (7) and (8). (7)D→=|C→⋅X→xrand(t)−X→(t)| (8)X→(t+1)=X→rand(t)−A→⋅D→

In the formula: X→rand(t) is the randomly selected whale position. This article chooses to optimize the radial basis function extension velocity parameters in radial basis function neural networks using the whale algorithm, and compares it with traditional particle swarm optimization algorithms.

RBF neural network is a single hidden layer feedforward neural network that can effectively process nonlinear data [17]. Radial basis function neural networks, like traditional neural networks, are generally divided into three layers: input layer, hidden layer, and output layer, as shown in Fig. 14.

The input layer is the first step in RBF prediction [18] and is generally selected as the parameter that the neural network needs to train. The hidden layer is the second step of RBF, and a radial basis function is required from the input layer to the hidden layer. The radial basis function is a function of the Euclidean distance from a point in space to a center, which can be a Gaussian kernel function, a quadratic function, or an abnormal S-type function. Due to the better adaptability of Gaussian kernel function to different types of datasets, this article chooses Gaussian kernel function as the radial basis function. Its expression is: (9)Zi(x)=exp⁡(−12σi2||x−Ci||2),i=1,2,…,p

In the formula: x is the input matrix; Ci is the center of the inth basis function; σi is the variance of the ith Gaussian function; x−Ci is the Euclidean distance between x and Ci, which is a nonlinear transformation. The output layer is the third step of RBF prediction, which assigns different weights to the output values of the hidden layer. Therefore, this process is a linear transformation. The output result of the RBF neural network is obtained through the non-linear transformation from the input layer to the hidden layer and the linear transformation from the hidden layer to the output layer.

This article combines the advantages of good optimization performance of WOA algorithm and fewer adjustable parameters of RBF algorithm as an estimation method for battery SOH. Specifically, the extracted five health features and the actual capacity will be used as inputs and outputs for the algorithm, and the dataset will be divided. Using the RBF algorithm as the estimation subject, optimize the extension speed of the radial basis function in the RBF algorithm using the WOA algorithm, and use the root mean square error as the fitness function of the WOA algorithm. The parameters of the WOA algorithm are set to a population of 10, a maximum number of iterations of 20, and a dimension of 1. The radial basis function expansion speed in RBF is optimized by the WOA algorithm. Fig. 15 shows the flowchart of the WOA-RBF algorithm. Table 2 shows the parameter settings for the WOA-RBF algorithm.