A Novel SOH Estimation Method for Lithium-Ion Batteries Based on the PSO–GWO–LSSVM Prediction Model with Multi-Dimensional Health Features Extraction

Abstract

Accurate State of Health (SOH) estimation of lithium-ion batteries (LIBs) is critical for ensuring the safety of electric vehicles and improving the reliability of battery management systems (BMS). However, the use of individual health features (HFs) and the selection of hyperparameters can increase the data processing burden on the BMS and reduce the accuracy of data-driven models. To address the above issue, this paper proposes a novel SOH estimation method for lithium-ion batteries based on the PSO–GWO–LSSVM prediction model with multi-dimensional health feature extraction. To comprehensively capture the battery aging mechanisms, four categories of health features—time, energy, similarity, and second-order features—are extracted from the LIBs charging segments. The correlation between HFs and SOH is comprehensively evaluated through Pearson and Spearman correlation analyses, followed by Gaussian filtering and outlier detection to enhance feature quality. With strong generalization and robustness, least squares support vector machine (LSSVM) is widely applied to nonlinear computations and function approximation. To improve LSSVM model accuracy and efficiency, this paper develops a novel prediction model that uses particle swarm optimization (PSO) combined with grey wolf optimization (GWO) algorithms to optimize the LSSVM model. The generalization performance of the proposed method is validated through comparative experiments using a battery dataset provided by the Center for Advanced Life Cycle Engineering (CALCE) Research Center at the University of Maryland. Experimental results show that the coefficient of determination (R²) consistently exceeds 0.985, with the average absolute error in SOH prediction for four batteries remaining around 0.5%. The comparative experiments demonstrate that the proposed method has a certain degree of accuracy, robustness, and generalization capability.

1. Introduction

Lithium-ion batteries (LIBs) are increasingly replacing lead-acid and nickel-based batteries due to their superior environmental and performance characteristics, including high power density, low self-discharge rate, long cycle life, and broad temperature range [1,2,3,4]. With an increasing number of charge–discharge cycles, chemical degradation accelerates, contributing to the aging of the battery. This is caused by the continuous rise in internal resistance and the gradual reduction of the maximum usable capacity due to side reactions within the battery [5]. During real-world operation, harsh conditions and extreme environments can accelerate the aging process of LIBs, potentially leading to failure or thermal events. The SOH of LIBs is a critical parameter for battery management systems (BMS), as it provides an accurate assessment of the battery’s aging status. In practice, replacing batteries with a low SOH, based on predefined thresholds, is a key strategy to ensure the safe operation of battery energy storage systems [6]. The United States Advanced Battery Consortium (USABC) recommends an SOH threshold of 80% for battery packs. When the SOH drops below this threshold, the battery is no longer suitable for use in electric vehicles [7]. Therefore, accurate and effective SOH estimation is crucial for the safe operation of electric vehicles and plays a key role in the widespread adoption of LIBs.

SOH estimation methods for batteries are generally categorized into three main approaches: direct measurement, model-based, and data-driven methods. In the direct measurement approach, SOH is assessed by evaluating the battery’s capacity and internal resistance [8]. Although this method is relatively simple to implement, it requires specialized equipment to assess the battery’s internal resistance and capacity, imposing strict requirements on the testing environment. Therefore, it is only suitable for SOH estimation under controlled laboratory conditions and cannot be applied in practical settings [9]. Estimation methods based on battery models can be broadly classified into physical and mathematical models. The physical models include the equivalent circuit model (ECM) and the electrochemical model (EM) [10]. The internal chemical reactions in LIBs are complex, dynamic, and nonlinear, making them highly sensitive to external environmental factors and operating conditions. Due to the complexity of these reactions, identifying the parameters in the models is challenging. Additionally, anomalies during charging and discharging are difficult to explain, and the accuracy of these models’ predictions often depends on extensive prior knowledge and experience [11]. Although SOH estimation methods based on battery models under specific operating conditions offer high accuracy, they require significant computational resources and are difficult to generalize, which complicates their implementation in practical BMS.

Compared to the model-based approach, the data-driven approach does not require a detailed understanding of the internal mechanisms of the battery. Instead, it learns the variation in input data through iterative cycles, offering greater flexibility [12]. Data-driven methods have proven practical and are increasingly used for battery state estimation. In this approach, only the relevant health factors, such as voltage, current, and temperature, are extracted from the battery data. Machine learning or deep learning algorithms are then used to model the nonlinear relationship between these factors and the battery’s SOH [13]. The accuracy of these methods depends on the BMS continuously monitoring and processing large amounts of real-time battery data [13]. Reference [14] highlights that, in data-driven approaches, the selection, processing, and modeling of prediction models play a key role in enhancing the accuracy of SOH estimation. Furthermore, reference [15] emphasizes that high-quality health features (HFs) such as voltage HFs, current HFs, internal resistance HFs, and energy HFs during the charging process are crucial for accurately reflecting the degradation trends of the battery’s SOH. Machine learning techniques, such as (support vector machines, SVM), (gated recurrent units, GRU), and (long short-term memory, LSTM), have been increasingly applied in SOH prediction [16]. For example, Reference [17] shows SVM to estimate the battery’s SOH, by utilizing the constant voltage charging and discharging times of LIBs as input variables and SOH as the output; an accurate SOH estimation was achieved through the training of an SVM model. Reference [18] shows a transfer learning-based (Gaussian process regression, GPR) model was proposed to achieve unified SOH prediction without causing prediction delays. Reference [19] shows a method based on the (time pattern attention, TPA) mechanism and (convolutional neural network—long short-term memory network, CNN–LSTM) model to improve the accuracy of SOH estimation. In model design, when the structure and parameters of data-driven models are not flexibly adjusted to the battery’s characteristic data, it often results in poor generalization performance. However, the design of hyperparameters such as learning rate, number of layers, and hidden units, along with the complex architecture of LSTM and GRU models, makes training more challenging. This complexity may lead to overfitting, particularly with small sample sizes, thereby affecting the stability and accuracy of SOH prediction [20]. On the other hand, SVM, which optimizes by maximizing the margin between classes, exhibits strong nonlinear fitting capabilities, but it has limitations in handling large-scale data, hyperparameter tuning, noise tolerance, and computational efficiency. To address this issue, the least squares support vector machine (LSSVM) model is introduced to establish a prediction model that captures the nonlinear relationship between SOH and feature data [21,22]. The performance of LSSVM models is strongly influenced by the optimal selection of hyperparameters, particularly the regularization factor and the kernel function type. Heuristic optimization techniques, such as particle swarm optimization (PSO), grey wolf optimization (GWO), and so on, are commonly used to optimize these parameters and enhance model accuracy [23]. Although these methods can improve prediction accuracy to some extent, models based solely on a single algorithm often face challenges such as slow convergence and a tendency to become trapped in local minima, limiting their applicability in real-world scenarios [24]. Additionally, outliers can negatively affect the stability and robustness of the LSSVM model, significantly reducing its performance when handling noisy data.

To enhance the accuracy of SOH prediction and achieve precise estimation of the SOH of LIBs, this paper introduces a novel approach that combines the PSO–GWO–LSSVM model with the extraction of multi-dimensional HFs. The main innovations of this method include: (1) to effectively capture the LIBs aging process, eight HFs were extracted from battery charging segments and categorized into four main types: time-based, energy-based, similarity-based, and second-order HFs; (2) correlation analysis of the extracted HFs identified six highly correlated parameters; (3) to optimally determine the regularization and other parameters of the LSSVM, the PSO–GWO optimization algorithm is introduced. This method leverages the strengths of both algorithms, balancing global and local search capabilities to address issues such as overfitting, underfitting, and convergence to local optima; (4) The PSO–GWO–LSSVM model is subsequently developed for SOH prediction, providing high accuracy. The paper is structured as follows: Section 2 covers HFs extraction and dataset processing; Section 3 introduces the PSO–GWO–LSSVM-based SOH prediction model; Section 4 compares the performance of the developed method with other methods in terms of accuracy and effectiveness; and Section 5 offers a discussion of the results, followed by concluding remarks.

2. Data Analysis and Features Extraction

As the number of charge–discharge cycles increases, the maximum usable capacity of LIBs gradually decreases, and the internal resistance increases [25]. SOH of LIBs is typically defined in terms of either capacity or internal resistance [26]. The SOH, serving as a key metric for quantifying LIBs degradation, is primarily characterized by capacity fade analysis, as defined below:

$$SOH={\displaystyle \frac{C_{aged}}{C_{rated}}}$$

(1)

where $C_{aged}$ and $C_{rated}$ are the current maximum available capacity and rated capacity of the battery, respectively.

2.1. Battery Experimental Data Analysis

This study uses battery aging cycle test data from the Center for Advanced Life Cycle Engineering (CALCE) Research Center at the University of Maryland for validation. The batteries selected for the experiment are labeled (CS2-35, CS2-36, CS2-37, and CS2-38). The battery dataset was collected using a (constant current constant voltage, CCCV) charging protocol, with a constant charging current of 0.5 A and a cutoff voltage of 4.2 V. The charging process continued under constant voltage until the charging current decreased below 0.05 A. The LIBs cells feature a nominal capacity of 1.1 Ah with lithium cobalt oxide (LiCoO₂) as the cathode material. The batteries provide complete charging data following the CCCV process. The charging process initiates with constant current charging at a 0.5 C rate until the terminal voltage reaches 4.2 V. Once this voltage is reached, the system switches to constant voltage charging, maintaining 4.2 V until the charging current drops below 0.05 A. The operational state of the batteries is divided into two stages: charging and discharging. Due to variability in usage patterns and operating conditions, the discharging stage presents issues in extracting stable input HFs. Therefore, in this study, input HFs are extracted from the more consistent charging phase.

2.2. Health Features Extraction

As shown in Figure 1, the SOH of the battery decreases nonlinearly with the increase in cycle count. This curve illustrates the fluctuations and partial rejuvenation characteristics, which are due to the natural recovery process of LIBs in a stationary state. During this process, unstable compounds near the electrodes re-decompose, releasing lithium-ion, which then recombines with the electrode’s active material, leading to a temporary capacity recovery in certain cycles [27]. The nonlinear relationship between battery capacity and cycle count has a considerable impact on the accuracy of SOH estimation. Therefore, it is crucial to extract reliable and accurate HFs from the battery’s external parameters to effectively predict its SOH. To accurately describe the battery SOH, this paper extracts four types of HFs from the charging process, namely time HFs, energy HFs, similarity HFs, and second-order HFs.

2.2.1. Time Health Features Extraction

Existing studies have shown a close correlation between the changes in the charging voltage curve and the battery capacity degradation. As shown in Figure 2, as the number of charge–discharge cycles increases, both the overall charging efficiency and duration decrease, resulting in a deterioration of the battery’s SOH. The constant current charging time directly reflects the charging speed during the constant current phase. As the battery undergoes aging, its capacity diminishes, resulting in a reduction in the duration of constant current charging. This feature is directly related to the degree of battery aging and is therefore chosen as the first health feature (HF1) to reflect SOH changes. During the constant voltage charging phase, the battery voltage is kept at a predetermined level. Under the same charge–discharge cycle, as the internal resistance of the battery rises, the charging current decreases at a slower rate, resulting in a longer constant voltage charging time compared to the constant current charging time. Therefore, the constant voltage charging time is selected as the second health feature (HF2) to reflect SOH changes. As shown in the zoomed-in section of Figure 2, the charging voltage curve exhibits the greatest variation within the range of 3.8~4.1 V [28]. Considering practical application scenarios such as range anxiety of electric vehicle drivers and charging habits, the battery charging time within the 3.8~4.1 V range is selected as the third health feature (HF3) to reflect SOH variations. This selection has important practical significance.

2.2.2. Energy Health Features Extraction

As the number of charge–discharge cycles increases, the available chargeable and dischargeable energy in the battery gradually decreases. The energy value can reflect the electrochemical reaction activity of the battery during the charging process. As shown in Figure 2, with battery aging, the electrochemical reaction activity declines, leading to a reduction in the constant-voltage charging energy. To reflect the SOH of the battery, the energy difference during constant-voltage charging (EDCV) is used as health feature 4 (HF4).

$$EDCV={\displaystyle {\int }_{t1 }^{t2}{\mathit{V}}_{\mathit{C}\mathit{C}}\cdot {\mathit{I}}_{\mathit{C}\mathit{C}}\left(t\right)dt}$$

(2)

where t₁ and t₂ represent the times when the charging voltage reaches 4.2 V and when the charging current drops to 0.05 A, respectively. V_CC and I_CC (t) represent the 4.2 V charging voltage and charging current within the time interval [t₁, t₂], respectively.

2.2.3. Similarity Health Features Extraction

To accurately describe the voltage curve degradation trend and its extent, the voltage data during the battery charging phase are analyzed using both time-series analysis and probability distribution dispersion. Correlation analysis techniques are used to assess the similarity between the charging voltage curve of the initial cycle and that of the current cycle, facilitating the extraction of meaningful similarity features. In this study, multiple lithium-ion batteries of the same specifications were preliminarily measured in the laboratory to obtain the initial cycle charging voltage curves. The average of these curves was then calculated to serve as the standard initial cycle charging voltage curve for this specification of the battery. Dynamic time warping (DTW) is a method that adjusts the alignment of data points in two-time series to obtain an optimal matching strategy; a smaller DTW value indicates a higher similarity [29]. Thereby calculating the similarity between the two datasets. The time-series data S₁ = {a₁, a₂, a₃, …, a_m} and S₂ = {b₁, b₂, b₃, …, b_n} are given. The goal of DTW is to construct all possible warping paths based on the two time series, forming an m × n cost matrix R, where the element at the i-th row and j-th column, R_ij is defined as:

$$R_{ij}=\sqrt{{\left(a_i-b_j\right)}^2}1⩽i⩽m,1⩽j⩽n$$

(3)

The DTW distance between time-series data $a_i$ and $b_j$ is represented by the accumulated distance r (i, j) along the optimal warping path, which is calculated as follows:

$$r(i,j)=R_{ij}+\min \{r(i-1,j-1),r(i,j-1),r(i-1,j)\}$$

(4)

where r (i − 1, j − 1) represents the subsequence distance when a_i−1 and b_j−1 are matched; r (i − 1, j) represents the subsequence distance when a_i−1 and bj are matched; r (i, j − 1) represents the subsequence distance when ai and b_j−1 are matched.

The DTW algorithm selects the shortest distance from the three possible split options. By iteratively applying Equation (3), the optimal alignment between the variable sequences a and b can be obtained, resulting in the DTW distance V_DTW, $V_{\mathrm{DTW}}=r(m,n)$. The DTW distance can be employed to evaluate the similarity between the charging voltage curves of the initial and current cycles from a time-domain perspective. This similarity can be utilized as a health feature 5 (HF5) to represent the battery’s SOH.

To more intuitively reflect the reduction in the time required for the same voltage change as the battery’s SOH decreases, we introduce the coefficient of variation, a common statistical metric, as a similarity feature for characterizing the battery SOH, denoted as health feature 6 (HF6). The coefficient of variation is a normalized metric used to assess the dispersion of a probability distribution [30]. It quantifies the extent of variability relative to the mean across different sample data, with its mathematical expression provided in Equation (5).

$$C_{\mathrm{var}}={\displaystyle \frac{\sigma }{\mu }}={\displaystyle \frac{1}{\mu }}\sqrt{{\displaystyle \frac{{\displaystyle \sum _i^n{\left(x_i-\mu \right)}^2}}{n}}}$$

(5)

where $C_{\mathrm{var}}$ represents the coefficient of variation, $\sigma$ denotes the standard deviation of the sample, and $n$ indicates the sample size.

2.2.4. Second-Order Health Features Extraction

The second-order processing curve of the battery is derived by differentiating the monitoring data recorded by the BMS. The dV/dt curve during the constant current charging phase, as shown in Figure 3, provides a more intuitive visualization of the rate of voltage change per unit of time during the charging process [31]. From the perspective of the battery’s internal mechanisms, as SOH declines, both the positive and negative electrode active materials and lithium-ions are progressively lost, leading to reduced charge capacity. During charging, this manifests as a decrease in the voltage rise rate and a shortening of the stable charging time. Therefore, the maximum value of the voltage change rate during charging, as shown in Figure 3 (maximum value of dV/dt, CVT_MAX), is chosen as health feature 7 (HF7), and the duration of the stable voltage change rate during the constant voltage charging phase (Stable Time of dV/dt, CVT_ST) is chosen as health feature 8 (HF8).

2.3. Health Features Correlation Analysis and Processing

The correlation between HFs and capacity is rigorously quantified using Pearson ($\rho$) and Spearman ($\gamma$) correlation coefficients to assess their relationship with SOH. The calculation formulas are as follows:

$$\rho ={\displaystyle \frac{{\displaystyle \sum _{i=1}^n(X_{\mathrm{i}-SOH}}-X_{SOH}^{\ast })(Y_{\mathrm{i}-SOH}-Y_{SOH}^{\ast })}{\sqrt{{\displaystyle \sum _{i=1}^n(X_{\mathrm{i}-SOH}}-X_{SOH}^{\ast }{)}^2}\sqrt{{\displaystyle \sum _{i=1}^n(}{Y}_{\mathrm{i}-SOH}-Y_{SOH}^{\ast }{)}^2}}}$$

(6)

where n is the number of samples; $X_{SOH}^{\ast }$ and $Y_{SOH}^{\ast }$ represents the average value of the samples for the two different SOH groups, respectively, and $X_{\mathrm{i}-SOH}$ and $Y_{\mathrm{i}-SOH}$ represents the individual samples.

$$\gamma ={\displaystyle \frac{\frac{1}{n}{\displaystyle \sum _{i=1}^n[R(X_{\mathrm{i}-SOH})}-R(X_{SOH}^{\ast })][R(Y_{\mathrm{i}-SOH})-R(Y_{SOH}^{\ast })]}{\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n[R(X_{\mathrm{i}-SOH})}-R(X_{SOH}^{\ast }){]}^2}\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n[R(Y_{\mathrm{i}-SOH})-R(Y_{SOH}^{\ast })}{]}^2}}}$$

(7)

where n is the number of samples; $R(X_{\mathrm{i}-SOH})$ and $R(Y_{\mathrm{i}-SOH})$ represents the rank of the samples for the two different SOH groups, respectively; $R(X_{SOH}^{\ast })$ and $R(Y_{SOH}^{\ast })$ represents the rank of the average value of the samples for the two different SOH groups.

By analyzing the correlation between the SOH of four batteries, the correlation matrix shown in Figure 4 is obtained. In Figure 4, higher correlation coefficients are represented by darker colors. The results indicate that the Pearson and Spearman correlation coefficients between the CS2_38 battery and the others are 0.96, 0.96, 0.99, and 0.96, 0.98, 0.99, respectively, as shown in Figure 4. Due to the higher correlation data for the CS2_38 battery and the availability of more cycle data, the CS2_38 battery is selected as the representative sample for further data analysis and model experimentation in this study [32]. Due to the varying trends of each HFs, it is challenging to directly determine their correlation with capacity. Therefore, the HFs were normalized, and the correlation coefficients between each HF and SOH are presented in

Table 1. The dual correlation coefficient of battery HFs and SOH.

Battery		HF1	HF2	HF3	HF4	HF5	HF6	HF7	HF8
CS2_35	$\rho$	−0.9331	−0.9567	1	0.9951	−0.8023	−0.7511	0.2074	0.2747
	γ	−0.9590	−0.9641	0.9910	0.9668	−0.7683	−0.7156	0.3648	0.5196
CS2_36	$\rho$	−0.9274	−0.9382	1	0.9913	−0.9067	−0.7907	0.1980	0.2633
	γ	−0.9332	−0.9478	0.9950	0.9518	−0.7485	−0.7279	0.4333	0.5533
CS2_37	$\rho$	−0.9349	−0.9468	1	0.9931	−0.7602	−0.7486	0.2358	0.2833
	γ	−0.9451	−0.9553	0.9970	0.9790	−0.7542	−0.7426	0.3349	0.4214
CS2_38	$\rho$	−0.9278	−0.9484	1	0.9977	−0.7760	−0.7162	0.2298	0.2923
	γ	−0.9304	−0.9502	0.9946	0.9755	−0.7598	−0.7165	0.3929	0.4906

, with the results shown in Figure 5. It can be observed that HF1~HF6 exhibits a clear correlation with the battery’s HFs, while it is difficult to ascertain whether HF7 and HF8 are correlated with the battery’s HFs.

To ensure that the extracted HFs are applicable to different batteries and operating conditions, the HFs with Pearson and Spearman correlation coefficients exceeding 0.7 were chosen for inclusion in this study. These features are HF1, HF2, HF3, HF4, HF5, and HF6, as shown in Figure 6, which presents the heatmap of the correlation between the six HFs and the SOH of the four batteries [32].

To remove noise and improve data smoothness, Gaussian filtering was applied to preprocess the battery health features. Additionally, the local outlier factor (LOF) algorithm was utilized to identify and correct potential outliers, thereby improving the reliability of the features [33]. For time-series data associated with the battery’s health features, Gaussian filtering can be applied using the following equation:

$$y(t)={\displaystyle \sum _{i=1}^{n}X(i)}\cdot {\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}\exp \left(-{\displaystyle \frac{{(t-i)}^2}{2{\sigma }^2}}\right)$$

(8)

where $X(i)$ represents the original health feature data, $y(t)$ denotes the filtered data, and $\sigma$ is the standard deviation of the Gaussian filter.

Despite the application of Gaussian filtering, extreme outliers may still remain in the battery state features, potentially due to measurement errors or other influencing factors. To address this issue, the LOF algorithm is employed to detect and manage anomalous data points. If LOF > 1, The LOF algorithm compares local data densities and calculates an outlier score for each point to identify those significantly deviating from the norm. Points with a higher outlier score are considered anomalies, which are effectively removed by this method. Then, point p is considered an outlier. For the outliers removed from the battery health features, Lagrange interpolation is used to fill in the missing values, mitigating their potential impact on battery health assessment.

$$LOF(p)={\displaystyle \frac{{\displaystyle \sum _{q\in N_k(p)}\frac{lrd(q)}{lrd(p)}}}{\left|N_k(p)\right|}}$$

(9)

where $LOF(p)$ is the local reachability density of point p, $N_k(p)$ is the k-nearest neighbor set of point p, and $lrd$ is the local reachability density of data point.

3. SOH Prediction Method Based on the PSO–GWO–LSSVM Model

3.1. Least Squares Support Vector Machine Model

With strong generalization capabilities and excellent robustness, LSSVM has been widely applied to problems such as nonlinear computations and function approximation. The regression principle of LSSVM is as follows: Let the sample be an n-dimensional vector, and let the training set be {(x_i, y_i)} (i = 1,2,…,n) where x_i represents the input vector and y_i is the corresponding output. Through a nonlinear mapping function $\phi (x)$, the input vector is mapped to a high-dimensional feature space, where linear regression is performed. The resulting function is given by:

$$y(x)={\omega }^T\phi (x)+b$$

(10)

where $\omega$ represents the weight vector, and $b$ denotes the bias term. LSSVM optimizes the model by minimizing the squared loss function, and determines $\omega$ and $b$ based on the principle of structural risk minimization.

$$R={\displaystyle \frac{1}{2}}{\omega }^T\omega +c{R}_{emp}$$

(11)

where ${\omega }^T\omega$ controls the complexity of the model, $c$ is the regularization parameter, and $R_{emp}$ is the loss function $R_{emp}={\displaystyle \sum _{i=1}^n{\xi }_i^2}$.

Therefore, the objective function and the constraint conditions of the optimization are as follows:

$$\min J(\omega ,\xi )={\displaystyle \frac{1}{2}}{\omega }^T\omega +{\displaystyle \frac{1}{2}}\gamma {\displaystyle \sum _{i=1}^n{\xi }_i^2}$$

(12)

$$y_i={\omega }^T\phi (x_i)+b+{\xi }_i(i=1,2,\dots ,n)$$

(13)

where ${\xi }_i$ represents the error, $\gamma$ is the regularization parameter, and the smaller $\gamma$ is, the stronger the generalization ability of the function.

In the LSSVM, the goal is to minimize the loss function while satisfying the constraints. To achieve this, Lagrange multipliers $\lambda$ are introduced, and the problem is solved using the Lagrange method:

$$L(\omega ,\xi ,\lambda ,b)=J(\omega ,\xi )-{\displaystyle \sum _{i=1}^n\lambda }\left[{\omega }^T\phi (x_i)+b+{\xi }_i-y_i\right]$$

(14)

According to the (Karush–Kuhn–Tucker, KKT) optimization conditions:

$${\displaystyle \frac{\partial L}{\partial \omega }}=0,{\displaystyle \frac{\partial L}{\partial {\xi }_i}}=0,{\displaystyle \frac{\partial L}{\partial {\lambda }_i}}=0,{\displaystyle \frac{\partial L}{\partial b}}=0$$

(15)

Obtain the system of linear equations:

$$\left[\begin{array}{l}b\\ \gamma \end{array}\right]={\left[\begin{array}{cc}0& E^T\\ E& K+I/Y\end{array}\right]}^{-1}\left[\begin{array}{c}0\\ D\end{array}\right]$$

(16)

where $\lambda ={\left[{\lambda }_1,{\lambda }_2\dots ,{\lambda }_n\right]}^T$ and $E={\left[1,1\dots ,1\right]}^T$ as n × 1-dimensional column vector; and $D={\left[d_1,d_2\dots ,d_n\right]}^T$ as the identity matrix.

Define the kernel function, $K(x_i,y_j)=\phi {(x_i)}^T\phi (y_j)$ replaces the dot product operation in the high-dimensional feature space. By optimizing the KKT conditions using the least squares method, the values of $\lambda$ and $b$ can be obtained. The resulting prediction model based on LSSVM is given by:

$$y(x)={\displaystyle \sum _{i,j=1}^n{\lambda }_i}K(x_i,x_j)+b$$

(17)

Due to the excellent interference resistance, strong nonlinear mapping capability, and good generalization ability of the (radial basis function, RBF) kernel, RBF is chosen as the kernel function for LSSVM in this study [34]. The expression for RBF is as follows:

$$K(x_i,y_j)=\exp \left[-{‖x_i-x_j‖}^2/\left(2{\sigma }^2\right)\right]$$

(18)

where $\sigma$ represents the radial basis kernel width. When constructing an RBF-based LSSVM model, the correct selection of $\sigma$ and $\gamma$ is crucial.

3.2. PSO–GWO Combined Optimization Algorithm

3.2.1. Particle Swarm Optimization Algorithm

PSO algorithm seeks the optimal solution by iteratively adjusting the positions and velocities of particles within the search space. In PSO, the initial population is randomly generated within the search domain, with each solution represented as a “particle” [35]. Each particle within the search space is characterized by a position and velocity. During the optimization process, particles update their position and velocity based on two sources of information: their own best-known position (local best) and the best-known position found by the entire swarm (global best) [36]. The strength of PSO lies in its strong local search ability, although it is prone to getting stuck in local optima.

All particles in the swarm update their positions in each iteration using the following equation:

$$\left\{\begin{array}{l}{u}_i^{t+1}=u_i^t+f_1{n}_1(pbes{t}_i-x_i^t)+f_2{n}_2(gbest-x_i^t)\\ x_i^{t+1}=x_i^t+u_i^{t+1}\end{array}\right.$$

(19)

where $u_i^{t+1}$ represents the velocity of particle i at iteration t, $x_i^t$ denotes the position of particle i at iteration t, $pbes{t}_i$ is the individual best position of particle, $gbest$ is the global best position, $f_1$ and $f_2$ are the learning factors, and $n_1$ and $n_2$ are random numbers.

3.2.2. Grey Wolf Optimizer Algorithm

GWO algorithm has strong global search capabilities, which helps to effectively avoid getting trapped in local optima. In the GWO algorithm, the wolf pack is divided into four hierarchical levels: α, β, δ, and ω [37]. The α wolf is the leader of the pack, β and δ are the assistants, and ω represents the ordinary members. The α, β, and δ wolves update their positions relative to the target, while the ω wolves and lower-ranked individuals follow the leaders. In each iteration, the new position of each wolf is determined by its current position, the prey’s position, and the positions of other wolves in the pack [38]. This updating process can be represented as:

$$\left\{\begin{array}{l}\overrightarrow{W}\left(t+1\right)={\overrightarrow{W}}_p\left(t\right)-\overrightarrow{A}\cdot \overrightarrow{D}\\ \overrightarrow{D}=|\overrightarrow{C}\cdot {\overrightarrow{W}}_p\left(t\right)-\overrightarrow{W}\left(t\right)|\\ \overrightarrow{A}=2\overrightarrow{a}\cdot {\overrightarrow{n}}_1-\overrightarrow{a}\\ \overrightarrow{C}=2{\overrightarrow{n}}_2\end{array}\right.$$

(20)

where $\overrightarrow{W}\left(t+1\right)$ represents the position vector of the grey wolf at time $\left(t+1\right)$; ${\overrightarrow{W}}_p\left(t\right)$ denotes the position vector of the prey; $\overrightarrow{A}$ represents the attack target parameter; $\overrightarrow{D}$ indicates the Euclidean distance; ${\overrightarrow{n}}_1$ and ${\overrightarrow{n}}_2$ are directionally bounded random change parameters, with values ranging from (0, 1); and $\overrightarrow{a}$ is a decreasing coefficient, which decreases from 2 to 0 based on the grey wolf’s local search behavior.

Once the hunting target is identified, the wolf pack is guided by the α wolf, which helps accurately locate the prey’s position, with the β and δ wolves also assisting in the task. The α wolf represents the optimal solution, while the β and δ wolves correspond to the next-best solutions. As a result, the remaining wolves in the pack adjust their positions based on the positions of these three leading wolves. This concept is mathematically expressed in the following equation:

$$\left\{\begin{array}{l}{\overrightarrow{D}}_j={\overrightarrow{C}}_n{\overrightarrow{W}}_j-{\overrightarrow{W}}_p\hfill \\ {\overrightarrow{W}}_n={\overrightarrow{W}}_j-{\overrightarrow{A}}_n\cdot {\overrightarrow{D}}_j\hfill \\ {\overrightarrow{W}}_p(t+1)={\displaystyle \frac{{\displaystyle \sum _n{\overrightarrow{W}}_n}}{3}}\hfill \end{array}\right.$$

(21)

where j represents α, β, and δ, and n takes the values 1, 2, and 3. ${\overrightarrow{W}}_p(t+1)$ is the new position of the prey, which is represented as the average of the three optimal positions in the population, with A taking values in the range of (−2a, 2a).

3.3. The PSO–GWO–LSSVM Prediction Model

Appropriate parameter settings can significantly improve the model’s predictive accuracy and generalization capabilities. This paper proposes a PSO–GWO hybrid algorithm to optimize the width coefficient $\sigma$ of the LSSVM kernel function and the regularization parameter $\gamma$, in order to select the optimal parameters and achieve the best prediction results [39]. The integration of these two methods effectively mitigates the issue of local optima, thereby enhancing the accuracy of parameter optimization. The PSO–GWO algorithm integrates the particle position update formula from PSO to enhance the memory ability of the wolf pack during the optimization process. Additionally, an inertia constant is introduced to enhance the balance between global search ability and local exploitation in the PSO–GWO hybrid algorithm. The updated formulas for the velocity and position of the wolf pack in the hybrid algorithm are as follows:

$$\left\{\begin{array}{l}{u}_i^{k+1}=r^{\ast }\left[u_i^k+f_1{n}_1\left(x_1-x_i^k\right)\right.\\ +f_2{n}_2\left(x_2-x_i^k\right)+f_3{n}_3\left(x_3-x_i^k\right)]\\ x_i^{k+1}=x_i^k+u_i^{k+1}\end{array}\right.$$

(22)

$${\overrightarrow{D}}_j=\left|{\overrightarrow{C}}_m\cdot {\overrightarrow{X}}_j-w^{\ast }\overrightarrow{X}\right|$$

(23)

where j represents α, β, and δ; m takes values 1, 2, and 3; k denotes the current iteration number; f₁, f₂, and f₃ represent the learning factors; n₁, n₂, and n₃ are random numbers in the range of (0, 1); and $r$ denotes the inertia weight coefficient.

The core concept of the PSO–GWO algorithm is to rank particles in each iteration based on their fitness values. The three particles with the highest fitness values are referred to as α, β, and δ, and these particles help estimate the approximate range and potential location of the optimal solution. The remaining particles explore the optimal solution within the predicted range. This strategy allows the particles to find the optimal solution more quickly and effectively while also enhancing the convergence rate and path-planning capability of the PSO algorithm. As a result, this method enables a more efficient search for the width coefficient $\sigma$ and regularization parameter $\gamma$ of the LSSVM kernel function. The basic process of PSO–GWO–LSSVM is shown in Figure 7.

3.4. SOH Estimation Based on PSO–GWO–LSSVM and HFs Extraction

As shown in Figure 8, this section outlines an SOH prediction framework using the PSO–GWO–LSSVM model. Step 1: Extract eight features from the charging voltage profiles. Step 2: Normalize processing for eight HFs. Step 3: Analyze the HFs correlations and handle outliers with Gaussian filtering and LOF. Step 4: Train the model using the training set and optimize LSSVM hyperparameters with PSO–GWO. Step 5: Predict SOH and evaluate performance for four battery estimation results.

4. Experimental Results and Analysis

4.1. Evaluation Indicators

To evaluate the SOH prediction accuracy and performance of the proposed method, the model’s performance is measured using the (mean absolute error, MAE), (mean absolute percentage error, MAPE), (root mean squared error, RMSE), and (coefficient of determination, R²) [40]. The mathematical expressions for these metrics are as follows:

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|SO{H}_i-SO{H}_i^{\prime }\right|}\times 100\%$$

(24)

$$\mathrm{MAPE} ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|\frac{SO{H}_i-SO{H}_i^{\prime }}{SO{H}_i^{\prime }}\right|}\times 100\%$$

(25)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(SO{H}_i-SO{H}_i^{\prime }\right)}^2}}\times 100\%$$

(26)

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(SO{H}_i-SO{H}_i^{\prime }\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(SO{H}_i-{\overline{SOH}}_i\right)}^2}}}$$

(27)

where n represents the number of samples, while $SO{H}_i$, $SO{H}_i^{\prime }$ and ${\overline{SOH}}_i$ correspond to the estimated value, actual value, and average value of the SOH for the i-th group of batteries, respectively.

4.2. SOH Estimation of Four Batteries Under Different Proportions of Training Data

Four battery samples (CS2_35, CS2_36, CS2_37, and CS2_38) from the Maryland dataset are used to evaluate the proposed method. For each battery, the dataset is split into training set and testing set subsets with varying proportions: 50%, 60%, and 70% of the data is used for training, while the remainder is reserved for testing. Figure 9 presents the SOH estimation results along with the corresponding errors for each battery across different training set proportions. In the box plot of errors, the horizontal line represents the mean error. It can be observed that the errors are concentrated around 0%, following a normal distribution pattern. As depicted, the proposed method closely follows the actual SOH values and demonstrates high predictive accuracy for various training data splits. Additionally, the method successfully captures the capacity regeneration effect during battery aging. The R² values for all cases exceed 0.985.

Table 2. SOH estimation errors for different batteries in different proportions of the training dataset.

Battery	Training Data	MAE (%)	MAPE (%)	RMSE (%)
	50%	0.3321	0.4371	0.4417
CS2_35	60%	0.3431	0.4597	0.4603
	70%	0.3375	0.4643	0.4412
	50%	0.4753	0.6183	0.5929
CS2_36	60%	0.4674	0.6153	0.5725
	70%	0.5205	0.7031	0.6697
	50%	0.3844	0.4955	0.4878
CS2_37	60%	0.4066	0.5331	0.5151
	70%	0.4191	0.5545	0.4953
	50%	0.3446	0.4286	0.4618
CS2_38	60%	0.3584	0.4512	0.4741
	70%	0.4128	0.5264	0.5307

summarizes the error values for SOH predictions across the four batteries using different training set proportions.

4.3. The Comparison of SOH Prediction Results Using Different Prediction Models

In the field of battery SOH estimation, deep learning (DL) and transfer learning (TL), as data-driven approaches, have demonstrated significant application potential. Deep learning models, particularly those based on recurrent neural networks (RNN) such as LSTM and GRU, are capable of effectively capturing the nonlinear characteristics of battery performance over time. However, their high computational complexity, substantial data requirements, and poor interpretability limit their widespread use in real-time applications [41]. The transfer learning-based multistage LIBs’ SOH estimation model proposed in reference [42] demonstrates strong transferability under small sample data, but its modeling and training processes are relatively complex. The model reveals the hidden dynamic characteristics of batteries through phase space reconstruction, relying on precise parameter selection and high-dimensional data decomposition. Cycling discrepancy learning (CDL) extracts domain-invariant features using Kullback–Leibler divergence (KLD), resulting in high computational complexity. The multistage estimation framework combines LSTM and temporal capsule network (TemCap), with the latter having a complex structure and consuming significant training resources. Although the online transfer strategy enhances adaptability, it relies on error compensation from shallow neural networks, further increasing complexity. Overall, while the model performs well with small samples, its complex modeling and training processes limit its practical application. Reference [43] proposes a transfer learning-based method for estimating the SOH of LIBs, which can achieve high-precision prediction even with small data samples. The method automatically extracts features using a CNN and reduces the distribution difference between training and testing battery data through an improved maximum mean discrepancy (MMD). Although the method demonstrates strong transferability with small samples, its modeling and training processes are complex. The computational costs of CNN and MMD are high, especially when processing high-dimensional data, significantly increasing training time. Additionally, model hyperparameters need to be adjusted through trial and error, further increasing modeling complexity and limiting its widespread use in real-time applications. Reference [44] proposes a battery SOH estimation method based on degradation pattern recognition and transfer learning, suitable for small sample data. The method identifies battery degradation patterns using K-means clustering and improves estimation accuracy by combining LSTM networks with transfer learning. Although the method exhibits good transferability with small samples, its modeling and training processes are complex. LSTM training is time-consuming, and the fine-tuning process in transfer learning increases computational complexity. Moreover, model hyperparameters need to be adjusted through trial and error, further increasing modeling complexity and limiting its widespread use in real-time applications.

In contrast, LSSVM, which optimizes the objective function through the least squares method, significantly improves computational efficiency, particularly excelling in small-sample data scenarios [45]. LSSVM exhibits not only high generalization capability and robustness, effectively handling noise and uncertainties in battery data but also offers strong interpretability due to its simple model structure, facilitating intuitive analysis of feature weights [46]. In this study, we employed bidirectional LSTM (Bi-LSTM) and bidirectional GRU (Bi-GRU) models for battery SOH prediction, using 60% of each battery’s data for training and 40% for testing. The prediction results are shown in Figure 10, and the prediction error results are presented in

Table 3. SOH prediction errors resulting from different RNN methods under 60% training dataset.

Method	CS2_35	CS2_36	CS2_37	CS2_38	Mean Value
	MAE (%)
Bi-LSTM	0.9443	1.0926	0.7391	0.6115	0.8474
Bi-GRU	1.8981	2.136	1.5513	1.2082	1.6984
	MAPE (%)
Bi-LSTM	1.3633	1.4955	1.0022	0.7572	1.1543
Bi-GRU	2.1572	2.6017	1.8723	1.4424	2.0184
	RMSE (%)
Bi-LSTM	1.3914	1.3855	0.9876	0.7241	1.1224
Bi-GRU	2.8875	2.6792	1.9671	1.4536	2.2463

. The prediction curve of LSSVM is shown in Figure 11, and the prediction error results are presented in

Table 4. SOH prediction errors resulting from different methods under 60% training dataset.

Method	CS2_35	CS2_36	CS2_37	CS2_38	Mean Value
	MAE (%)
M1	1.8135	1.8533	2.4597	0.3447	1.6178
M2	1.6059	1.8032	1.2836	0.3347	1.2569
M3	0.6733	0.9779	0.9154	0.3208	0.7219
MD	0.3430	0.4675	0.4066	0.3584	0.3939
	MAPE (%)
M1	2.7036	2.5551	3.3765	0.4336	2.2672
M2	2.3773	2.4907	1.7553	0.4164	1.7599
M3	0.9467	1.3251	1.2402	0.3993	0.9778
MD	0.4597	0.6153	0.5331	0.4512	0.5148
	RMSE (%)
M1	2.7351	2.3641	3.3294	0.4567	2.2213
M2	2.6235	2.3379	1.7205	0.4263	1.7771
M3	0.9218	1.1777	1.1673	0.4117	0.9196
MD	0.4603	0.5725	0.5151	0.4747	0.5057

From the prediction results, the Bi-LSTM model outperformed the LSSVM and Bi-GRU models. However, LSSVM has a simpler structure with fewer parameters and lower computational complexity, especially when dealing with short sequence data, where it demonstrates higher computational efficiency and significantly reduces the time cost for training and prediction. Additionally, LSSVM can effectively handle nonlinear relationships through kernel functions and maintains good generalization ability even with small sample datasets. It also exhibits strong robustness against noise, and its prediction accuracy can be further enhanced by combining appropriate preprocessing methods and optimization algorithms. In contrast, although Bi-LSTM and Bi-GRU models perform well in processing long sequence data, their complex network structures lead to high computational resource consumption and increased sensitivity to noise. Considering factors such as computational complexity, hyperparameter design, and training cost, the LSSVM model, with its simple structure, fewer parameters, and low computational complexity, especially its superior computational efficiency for short sequence data and the significant reduction in training and prediction time cost, was selected as the basic model after a comprehensive evaluation.

To validate the effectiveness of the proposed model, we compare its SOH predictions with those from LSSVM, PSO–LSSVM, and GWO–LSSVM, using 60% of each battery’s data for training and 40% for testing. The kernel parameter and regularization parameter are key hyperparameters in LSSVM that play a decisive role in the overall performance of the model. The PSO–GWO algorithm optimizes these critical hyperparameters to obtain their optimal values, thereby enhancing the model’s prediction accuracy for battery SOH. For example, the CS2_38 battery’s particle dimension for PSO, GWO, and PSO–GWO was set to 2, with a population size of 20 and 50 iterations. The regularization parameter for the single LSSVM model was set to 25, and the radial basis kernel parameter was set to 50. The specific parameter settings for the PSO–GWO–LSSVM model are shown in

Table 5. Algorithm/Mode parameter settings.

Algorithm/Model	Parameter	Symbol	Range	Result
PSO–GWO	population size	N	/	20
	maximum number of iterations	imax	/	50
LSSVM	regularization parameter	γ	(0.001, 1000)	186.37
	kernel parameter	σ	(0.001, 1000)	1000

.

As shown in Figure 11, the SOH estimation results for various batteries using four different models. Compared to the other three models, the proposed method more accurately captures the actual SOH, particularly during the stages of capacity regeneration and local fluctuations. Table 5 presents the error values for SOH predictions across the four batteries using the different models, where M1, M2, M3, and MD represent LSSVM, PSO–LSSVM, GWO–LSSVM, and PSO–GWO–LSSVM, respectively.

The fitness curve typically reflects the evolutionary trajectory of fitness values with respect to iteration numbers during optimization processes. In PSO and GWO algorithms that optimize model performance through adjustment of LSSVM kernel parameters and regularization parameters, fitness curves exhibit rapid decline during initial iterations followed by gradual stabilization. However, inherent limitations in PSO and GWO optimization mechanisms predispose these algorithms to local optima entrapment, thereby impeding their ability to yield globally optimal hyperparameters. In contrast, the PSO–GWO–LSSVM framework synergistically integrates the strengths of PSO and GWO algorithms, effectively circumventing local optima while generating superior LSSVM hyperparameters. Using the CS2_38 battery as a representative case, Figure 12 illustrates the optimization trajectories and hyperparameter optimization outcomes of the three algorithms. As demonstrated in the figure, PSO–GWO–LSSVM achieves a minimum fitness value below 0.001 during the stabilization phase, indicating superior optimization efficacy. Furthermore, evaluation metrics (MAE, MAPE, and RMSE) for four batteries optimized through different methodologies consistently maintain low magnitudes, with maximum values remaining below 5.2%. This empirical evidence confirms that the selected HFs effectively characterize degradation patterns across diverse battery specimens and demonstrate applicability in SOH estimation frameworks. Collectively, these findings substantiate that PSO–GWO–LSSVM exhibits significant advantages in mitigating local optima convergence while enhancing optimization precision and stability, thereby establishing its suitability for complex optimization tasks.

The results show that the developed model accurately predicts the SOH of different batteries, with mean errors of 0.3939% (MAE), 0.5148% (MAPE), and 0.5057% (RMSE). In contrast, the LSSVM model demonstrates lower accuracy due to its inability to dynamically adjust the regularization and kernel parameters. The mean errors for this model are 1.6178% (MAE), 2.2672% (MAPE), and 2.2213% (RMSE).

Although PSO is capable of optimizing the parameters of LSSVM, its local search strategy faces limitations when dealing with highly nonlinear data, as it fails to fully capture the complexity of the data. This results in the model often falling into local optima, leading to higher prediction errors. The mean errors for the PSO–LSSVM model are 1.2569% (MAE), 1.7599% (MAPE), and 1.7771% (RMSE). While GWO enhances the global search ability and helps prevent overfitting and underfitting by optimizing LSSVM parameters, its exploration capability in complex data environments remains limited, which can lead to the overlooking of significant patterns. The mean errors for the GWO–LSSVM model are 0.7219% (MAE), 0.9778% (MAPE), and 0.9196% (RMSE). The proposed PSO–GWO–LSSVM model combines the advantages of both global and local search strategies, effectively addressing issues such as overfitting, underfitting, and local optima. The mean errors for this model are 0.3939% (MAE), 0.5148% (MAPE), and 0.5057% (RMSE), demonstrating superior prediction accuracy and enhanced generalization capabilities.

4.4. SOH Prediction Results Using Different Battery Types

To validate the model generalization capability, cross-type battery experiments were conducted using cycle data from two 18650-type lithium-ion cells (cell B0005 and B0006) obtained from the Prognostics Center of Excellence at the NASA Ames Research Center. The cells with a nominal capacity of 2 Ah underwent accelerated aging tests under controlled ambient temperature (24 °C) to simulate typical operational scenarios. A standardized charge–discharge protocol was implemented: (1) CC-CV charging at 1.5 A until reaching 4.2 V followed by voltage holding until current decayed to 20 mA; (2) CC discharging at 2 A to cutoff voltages of 2.7 V (B0005) and 2.5 V (B0006), deliberately set below manufacturer specifications to induce accelerated degradation through deep discharge cycles.

The prediction curves are shown in Figure 13, and the prediction results are presented in

Table 6. SOH estimation errors for different battery types in different proportions of the training dataset.

Battery	Training Data	MAE (%)	MAPE (%)	RMSE (%)
	50%	0.2651	0.3817	0.4499
B0005	60%	0.1950	0.2881	0.2479
	70%	0.1672	0.2512	0.2098
	50%	0.5628	0.8710	0.7389
B0006	60%	0.3142	0.4919	0.4216
	70%	0.2574	0.4127	0.3381

. The prediction errors are all within 1%, which demonstrates that the model proposed in this paper has a certain degree of generalization ability.

5. Conclusions

To improve the accuracy of SOH prediction and achieve precise estimation for electric vehicle applications, this study presents an innovative SOH estimation approach that combines the PSO–GWO–LSSVM prediction model with multi-dimensional HFs extraction. Initially, eight HFs were extracted from battery charge segments to accurately capture the aging process. These features were categorized into four groups: time-based, energy-based, similarity-based, and second-order HFs. A subsequent correlation analysis identified six highly correlated HFs, optimizing the system’s efficiency. The proposed methodology employs a PSO–GWO hybrid algorithm for precision tuning of LSSVM hyperparameters, where the grey wolf optimizer enhances the local search capability of particle swarm optimization during parameter identification. This approach leverages the strengths of both algorithms, improving the balance between global and local search functions and addressing challenges such as overfitting, underfitting, and local optima. The resulting PSO–GWO–LSSVM model, which is characterized by high prediction accuracy and low computational cost, was developed for SOH estimation, making it well-suited for practical applications in electric vehicles.

The proposed method was validated using a publicly available dataset containing four battery samples from the CALCE Research Center at the University of Maryland dataset (CS2_35, CS2_36, CS2_37, and CS2_38). Experimental results demonstrate that the method consistently achieves a prediction error margin of approximately 0.5%, with R² values for all four batteries exceeding 0.985, highlighting the model’s high accuracy. The average SOH prediction errors—MAE, MAPE, and RMSE—are 0.3939%, 0.5148%, and 0.5057%, respectively, outperforming other models. Compared to the LSSVM model and three alternative approaches, the proposed method shows superior accuracy, with maximum SOH prediction errors around 0.5%. To verify the model’s generalization capability, experiments using cycle data from two 18650-type lithium-ion cells (B0005 and B0006) from the NASA Ames Research Center were conducted. The prediction errors are all within 1%, which demonstrates the generalization capability of the proposed model.

Future research will extend the application of this method to other battery types, such as lithium-ion and sodium-ion batteries, to further evaluate its effectiveness. Another focus will be on SOH estimation for battery packs, with a particular emphasis on real-world applications. The primary goal is to provide accurate SOH assessments for battery packs, addressing the growing demands of energy storage systems and the electric vehicle industry. This work aims to tackle the issue associated with achieving highly accurate SOH estimations with limited data, thereby contributing to ongoing advancements in performance and reliability in the fields of energy storage and electric vehicles.