Lithium-Ion Battery Health State Prediction Based on Improved War Optimization Assisted-Long and Short-Term Memory Network

Abstract

It is essential that the state of health (SOH) for lithium-ion batteries is measured to ensure the safety and reliability of electric vehicles. However, an accurate prediction of SOH is still an art due to the complex degradation mechanisms. To address this challenge, a SOH prediction model based on Warfare Strategy Optimization-assisted hybrid mutual information in-former-Long Short-Term Memory neural network (IWSO-MILSTM) is proposed. First, both direct and virtual health indicators are derived from battery degradation curves. Building on this foundation, mutual information is applied to the correlation analysis of these health indicators, and the redundant health indicators can be filtered. Then, the selected health indicators are fed into the informer-LSTM to construct an interpretable predicted model for the health status of lithium-ion batteries. Notably, both redundancy of health indicators and the imprecision of model hyperparameters for LSTM affect the SOH prediction precision. IWSO is proposed to achieve co-optimization of filtering for health indicators and hyperparameters for the informer-LSTM based on developed initializing distribution methods and adaptive function so that the SOH prediction precision is ensured. Finally, the NASA dataset is used to validate the prediction precision of the IWSO-MILSTM, and the experimental results show that the IWSO-MILSTM can provide more competitive results, i.e., the R² value is improved by 25.68% and 3.63%, respectively, while the RMSE is reduced by 48.76% and 75.91% compared with XGBoost, LSTM, etc. Such results indicate the proposed method can predict SOH efficiently.

1. Introduction

Lithium-ion batteries, characterized by high energy density, extended lifespan, and cost-effectiveness, have demonstrated significant potential in electric vehicle (EV) applications [1]. However, during continuous usage, the depletion of lithium ions and the degradation of active electrode materials result in a reduction in battery capacity. This not only decreases the driving range of EVs, contributing to range anxiety, but also introduces safety concerns, exacerbating safety anxiety [2,3]. Therefore, achieving reliable prediction of the state of health (SOH) of lithium-ion batteries is essential for extending operational lifespan, mitigating risks, and ensuring effective management of EV power battery systems [4,5].

Currently, the primary approaches for assessing the SOH of lithium-ion batteries can be classified into direct measurement methods, numerical model-based methods, and data-driven methods [6,7]. The direct measurement method uses experimental methods to obtain information to characterize battery SOH [8,9]. However, this method requires specialized testing equipment and is difficult to apply in practice. The numerical model-based methods mainly involve constructing an accurate mathematical model to simulate the electrochemical reactions and physical changes that take place during the charging and discharging cycles of lithium-ion batteries. This type of method usually combines the Kalman filtering algorithm to estimate the internal resistance and capacity of the battery to represent battery SOH [10,11,12]. Hu et al. [13] proposed a fractional–order equivalent circuit model combined with a dual fractional–order extended Kalman filter to achieve concurrent estimation of SOC and SOH. Vennam et al. [14] designed a novel battery aging model that combines filtering techniques to? simultaneously predict the battery’sSOC, SOH, and temperature. Although the above method obtained accurate battery SOH under experimental conditions, it is heavily dependent on the precision of the battery model. Especially under complex working conditions, the precision of the battery model parameters cannot be guaranteed. These models allow for predicting the performance degradation trends of batteries under specific usage conditions, thereby estimating their SOH. Examples of such approaches include Kalman filtering [15] and equivalent circuit models [16]. Although model-based methods offer relatively accurate predictions under various operating conditions, their performance is often limited by the difficulty of accurately revealing the degradation mechanisms of lithium-ion batteries, making it challenging to construct precise mathematical and physical models. Data-driven methods have gained increasing attention owing to their simplicity, broad applicability, and rapid responsiveness. Li et al. [17] proposed an improved multilayer kernel extreme learning machine (KELM) model to evaluate the battery SOH. In their experiments, a multi-feature fusion approach was introduced to construct both a high-dimensional health feature (HF) dataset and a low-dimensional fusion feature (FF) dataset, enhancing the representational capacity of the original degradation features. Ma et al. [18] utilized a symbolic regression-based feature derivation method to discover potential new health degradation factors and employed a convolutional neural network (CNN) model to obtain deep degradation characteristics, enabling accurate SOH prediction. Xu et al. [19] introduced particle swarm optimization (PSO) into Gaussian process regression (GPR) for adaptive hyperparameter tuning, achieving adaptive SOH prediction for lithium-ion batteries.

The aforementioned studies demonstrate that data-driven methods exhibit tremendous potential in predicting battery SOH. To further enhance the accuracy of SOH predictions, a novel approach for health state prediction is proposed based on an Improved Warfare Strategy Optimization-assisted hybrid mutual information informer-Long Short-Term Memory neural network (IWSO-MILSTM). First, this method extracts key metrics from the charging and discharging curves obtained from the current-voltage IC curve of lithium-ion batteries. These serve as both direct and indirect measurement indicators. On this basis, the mutual information method is employed to capture implicit relationships between various health indicators. Since not all health indicators contribute positively to the SOH prediction of batteries, the IWSO algorithm is proposed to ensure the selection of degradation indicators as inputs to the informer-LSTM. In the developed IWSO, an improved initialization distribution method and adaptive function are proposed to overcome the limitations of traditional warfare strategy optimization algorithms, which often suffer from slow convergence and a tendency to get stuck in local optima during the iterative process. Finally, the proposed method is compared with LSTM, CNN-LSTM, and SVR models to validate its effectiveness and superiority.

2. Feature Extraction

2.1. Experimental Data

This research employs the NASA lithium-ion battery dataset, which includes four 18650 lithium-ion batteries (B5, B6, B7, and B18) with a nominal capacity of 2 Ah [20,21]. The dataset was obtained through constant current-constant voltage (CC-CV) charge-discharge experiments conducted at 24 °C. Specifically, during the charging process, the batteries were subjected to a constant current (CC) of 1.5 A until reaching 4.2 V. Subsequently, charging proceeded in constant voltage (CV) mode until the charging current decreased to 20 mA. For discharging, batteries were discharged at a constant current (CC) of 2 A until the voltages of batteries B5, B6, B7, and B18 dropped to 2.7 V, 2.5 V, 2.2 V, and 2.5 V, respectively. Throughout the cyclic charge-discharge experiments, the failure threshold of the lithium-ion batteries was defined as 70% of the nominal capacity, and the experiments continued until this failure threshold was reached.

2.2. Health Indicator Extraction

The degradation of lithium-ion batteries is a multifaceted nonlinear process shaped by a variety of intrinsic and extrinsic influences. To accurately monitor and assess the health status of the battery, a series of key parameters were extracted through analysis of the battery’s charge-discharge cycles, ensuring that these parameters are both easily obtainable and effective in reflecting the degradation of lithium-ion batteries. These parameters are readily measurable and effectively reflect the performance degradation of the battery. Typically, as usage cycles increase, changes in current and voltage are used to evaluate the battery’s degradation state. In this work, statistical degradation features related to CC charging time, CV charging time, and CC discharging time were extracted as directly measurable health factors (HIs). Although these direct HIs provide useful insights into the battery’s characteristics, they have certain limitations in fully reflecting the battery’s degradation state. The IC (Incremental Capacity) curve, which records capacity changes at different discharge rates, can reveal subtle characteristics of the battery during the charge-discharge process. This curve analysis not only helps in identifying the overall health of the battery but also delves into changes in the battery’s internal structure and chemical properties. Hence, the IC curve is viewed as an indirect metric for assessing battery SOH, offering a deeper understanding of their degradation process. Original data information used in this work is shown in Figure 1. The capacity degradation curves for each battery are shown in Figure 1a. Figure 1b–d presents charge-discharge cycles rises, battery’s charging and discharging voltage, current, and time decrease correspondingly. The peak of the IC curve also diminishes with increasing usage cycles. As a result, a total of 28 HIs were cumulatively extracted, and the specifics are presented in Appendix A.

To verify the effectiveness of the selected health indicators and capture the nonlinear relationships among these health factors, the mutual information method was applied to measure the correlation between these health indicators and the SOH [22,23]. The calculation formula is as follows:

$$MI([X,Y],x,y)={\displaystyle \int p(x,y){\log }_2\frac{p(x,y)}{p(x)p(y)}dxdy}$$

(1)

Based on the calculation of mutual information, $MIC([X,Y])$ can be written as:

$$MIC([X,Y])=\underset{ab<B}{{\displaystyle \max }}{\displaystyle \frac{\max MI\{x,y\}}{{\log }_2\min \{a,b\}}}$$

(2)

In this context, $a$ and $b$ represent the number of grid divisions along the x-axis and y-axis, respectively. Mutual information is assessed across different grid configurations, and the maximum mutual information result is used to indicate $MIC$. As shown in Figure 2, the majority of the extracted parameters demonstrate a correlation coefficient greater than 0.4 with the battery’s SOH. When the mutual information value is greater than 0, the health indicators can be considered to have some form of correlation with the SOH. However, since the prediction of battery SOH is closely related to the input health indicators, not all inputs provide useful information. Health indicators with weak correlations may interfere with the learning process of the model, necessitating a filtering of the original data. This filtering helps the SOH prediction model focus more on learning the latent relationships between lithium-ion health indicators and SOH, thereby improving robustness and generalization. Namely, the threshold for selecting the relevant features can be a hyperparameter selection. Given that battery degradation data can be considered time-series data, and to maintain a streaming data format, 70% of the historical data are defined as training data according to the bootstrapping principle, while 30% of the dataset is used as test data to assess the effectiveness of the proposed strategy.

3. SOH Prediction Method Based on Informer-LSTM Model

3.1. Informer-LSTM Network

Lithium-ion degradation is a long-term degradation process. Therefore, accurately obtaining and understanding the long-term degradation pattern of lithium-ion batteries can help improve the predictive precision of SOH prediction models.

As a variant of recurrent neural networks (RNNs), LSTM networks are designed to address the issues of vanishing or exploding gradients encountered by traditional RNNs when handling long-term dependencies [24,25]. It consists of the input gate, forget gate, and output gate. During the forward propagation process, the input gate selectively introduces new information, while the forget gate filters out historical information as needed (the mathematical representation of the information flow for LSTM see in Appendix B). This selective information flow enables LSTM to effectively process the degradation data of lithium-ion batteries. Notably, it is impossible for LSTM to thoroughly overcome the challenge of long-term dependence.

To overcome the limitations of LSTM in obtaining long-term dependence so that the degradation implicit characteristics of lithium-ion batteries can be obtained. Informer is used to process the degenerate information of the input LSTM [26]. Unlike previous work, the input to the LSTM in this work is not a direct degradation HIs, but the long-term dependencies obtained by Informer from the original degraded health indicators of lithium-ion batteries.

First, the variation patterns of input health indicators for lithium batteries are focused on after positional encoding using multi-head probabilistic sparse self-attention, in which health indicator of lithium-ion batteries is translated into the query, key, and value. The expression can be defined as follows:

$$\left\{\begin{array}{l}Q=X{W}_Q\\ K=X{W}_K\\ V=X{W}_V\end{array}\right.$$

(3)

where Q, K, V demonstrate the query matrix, key matrix, $X$ denotes the matrix consists of the HIs of lithium-ion batteries, and $W_Q,W_K,W_V$ denote the weights matrix. Then, sample the key-value matrix and obtain the sampled matrix $\overline{K}$. Building on this foundation, the m value of $\overline{K}$ from the i-th row in query matrix Q:

$$M(q_i,\overline{K})=\underset{j}{{\displaystyle \max }}\left\{{\displaystyle \frac{q_i{\overline{k}}_j^T}{\sqrt{d}}}\right\}-{\displaystyle \frac{1}{L_k}}{\displaystyle \sum _{j=1}^{L_k}\frac{q_i{\overline{k}}_j^T}{\sqrt{d}}}$$

(4)

where $M(q_i,\overline{K})$ is the evaluation metric for $q_i$; the larger it is, the more critical the $q_i$ is. ${\overline{k}}_j$ denotes the j-th row of $\overline{K}$. $\sqrt{d}$, $d$, and $L_k$ are the scaling factor, the columns of K, and the rows of K, respectively. The query matrix Q is reconstructed according to the $M(q_i,\overline{K})$ and the attention score of K is obtained as well. Thus, the probabilistic sparse self-attention gain can be defined as:

$$A(Q,K,V)=Soft\max ({\displaystyle \frac{\overline{Q}{K}^T}{\sqrt{d}}})V$$

(5)

Then, the redundant combinations of the final output feature maps are eliminated by max-pooling and 1D convolution networks. To eliminate the final redundant long term implicit degenerate relations while ensuring computational efficiency, distillation is introduced, which is given by:

$$y_r=MaxPool(ELU(Conv1d(F_X)))$$

(6)

where $ELU(·)$ denotes the activation function. $MaxPool(·)$ and $Conv1d(·)$ are the max pooling and one-dimensional convolution operation, respectively, in which the stride of max pooling is set to 2 and the convolution kernel is defined as 3. $y_r$ denotes the implicit degeneracy relations obtained by distillation. $y_r$ is used as input to the long probability sparse self-attention to focus on long term degenerate relations $y_r^1$, which is fed into the LSTM and obtains the final predicted SOH results.

Figure 3 illustrates the detailed neural network architecture of SOH estimation. From this figure, it can be seen that the input degradation data, like IC curve, voltage, etc., consist of the selected HIs. The long-term degenerate implicit relation of lithium-ion batteries is obtained by probabilistic sparse self-attention in Encoder. On this basis, distillation is introduced to eliminate the final redundant and process long sequences information. Decoder is used to obtain potential relationships from Encoder, which is fed into the LSTM to obtain the final SOH prediction results.

The weights and biases of the neural network are updated using the Adam optimizer, a first-order optimization algorithm known for its high computational efficiency and low memory requirements [27]. Adam is widely used to solve large-scale parameter optimization problems, making it a popular choice in deep learning tasks due to its ability to adapt learning rates and improve convergence stability.

3.2. Improved Warfare Strategy Optimization Algorithm

While the proposed informer-LSTM demonstrates efficacy in predicting battery SOH, the selection of neural network hyperparameters is an optimization dilemma. Identifying the optimal solution generally necessitates conducting numerous experiments, making the process heavily reliant on experience. In recent years, swarm intelligence optimization algorithms have been introduced to optimize LSTM hyperparameters. As a meta-heuristic optimization algorithm, the warfare strategy optimization (WSO) algorithm draws inspiration from military tactics and defense strategies employed in ancient warfare, including the attack strategy and the defensive strategy [28,29] (for details, see in Appendix A).

In the WSO, the position of the optimal solution is determined by the commander, soldiers, and king. Additionally, dynamic weights are introduced in the calculations to adjust the influence of soldiers of different ranks on the optimal solution. Compared to other meta-heuristic optimization algorithms, this enhances WSO’s global search capability. However, like other algorithms, WSO has some shortcomings, such as a tendency to get trapped in local optima early in the iterative process and slow development during iterations, which can lead to premature convergence. Therefore, this study proposes an improved mathematical expression to enhance the iterative process of the Warfare Strategy Optimization algorithm, thereby increasing its global search capability.

First, the traditional WSO initialization is a random process, which can result in uneven distribution of randomly initialized soldiers in the search space, limiting the algorithm’s ability to identify the optimal solution. To ensure uniform coverage of the solution space and enhance the algorithm’s capacity to locate the optimal solution, Logistic mapping [13] is employed for the initialization of soldiers. This mapping has advantages such as ergodicity and non-repetition, which help generate an orderly and dispersed initial population. The mathematical expression can be defined as follows:

$$\begin{array}{l}{X}_{i,j}=(X_{\max ,j}-X_{\min ,j})·x(r+1).\\ x(r+1)=\mu ·x(r)·(1-x(r))\end{array}$$

(7)

In the equation, the initial value $x(0)$ and the arbitrary value $x(r)$ both fall within the range [0, 1], while $\mu$ represents a parameter in the chaotic mapping. When $\mu \in (3.57,4]$ takes a specific value, the system enters a chaotic state.

In addition, an improved nonlinear decay function is proposed to enhance the convergence performance during the search phase of WSO. It is defined as follows:

$$s_1=d_r\times r_2\times e^{({\displaystyle \frac{t}{\max iter}})}$$

(8)

In the equation, $r_2$ and $d_r$ represent a random number between 0 and 1 and a decay factor, respectively, being defined as 0.5. Therefore, the position of soldiers under attack strategy and the defensive strategy are reconstructed as follows:

$$\begin{array}{c}{X}_i(t+1)=X_i(t)+2\times \rho \times s_1\times (K-X_r(t))\\ +r_1\times W_i\times (c-X_i(t))\end{array}$$

(9)

$$\begin{array}{ll}{X}_w(t+1)=& -(1-randn)\times s_1\times \\ & (X_w(t)-median(X))+K\end{array}$$

(10)

3.3. Construction of the IWSO-MILSTM Lithium-Ion Battery SOH Prediction Model

The retained degradation information and the parameter configurations of the proposed informer-LSTM substantially influence the predictive accuracy of the constructed model, particularly the volume of retained degradation information, the learning rate, and the number of hidden units in the LSTM. Therefore, the Improved Warfare Strategy Optimization (IWSO) algorithm is used to improve the model performance. This approach forms the IWSO-assisted informer-LSTM modeling method for lithium-ion battery health state estimation. Compared to the standard WSO, IWSO not only avoids the problem of getting trapped in local optima but also enhances the rate of search and convergence of the algorithm. To better understand the flow of the algorithm, pseudo-code is provided, as shown in Algorithm 1.

Algorithm 1. The execution process of SOH prediction for LIBs by using the proposed IWSO-MILSTM

Data: The obtained HIs and the MIC values for these HIs. The population size S.

The upper and lower limits for Thr, lr, and $num\_hidden$. The maximum

iteration T.

Output: The normalized results of obtained HIs, initialized hyperparameters

results and initialized fitness values.

while stopping criteria is not satisfied do

for each soldier in population do

The hyperparameters are updated using Equations (7)–(10).

Calculate fitness values by using Equation (13).

end for

Sorting hyperparameters according to calculate fitness values and the worst

hyperparameters are replace randomly.

end while

The optimal hyperparameter space for MILSTM are obtained and training the SOH

prediction model based on them

Output: The SOH prediction results.

The specific steps for constructing the IWSO-MILSTM-based lithium-ion battery SOH prediction model are as follows:

Step 1: Split the Training Set and Test Set and Perform Normalization. The training and test sets are first divided, followed by normalization of the data:

$$x_{norm}={\displaystyle \frac{x_i-x_{\min }}{x_{\max }-x_{\min }}}$$

(11)

Step 2: Initialization of IWSO Parameters. This step involves initializing the parameters of the IWSO algorithm, including the size S of the soldier population U and the maximum iteration count T. The soldier population is initialized using Equation (7). It is important to note that the hyperparameters in U are defined as follows:

$$U=[Thr,lr,num\_hidden]$$

(12)

In the equation, $Thr,lr,num\_hidden$ represents the predefined values for filtering redundant information, the learning rate, and the number of hidden units.

Step 3: Calculation of the Fitness Function. In this work, −R² (the negative coefficient of determination) is used as the optimization objective.

$$-R^2={\displaystyle \frac{{{\displaystyle {\sum }_{k=1}^K(y_k^{\prime }-y_k)}}^2}{{{\displaystyle {\sum }_{k=1}^K(y_k^{\prime }-\overline{y})}}^2}}-1$$

(13)

This metric ensures that the optimization process minimizes the error in the model’s predictions.

Step 4: Update Soldiers’ Positions. The positions of the soldiers (i.e., the hyperparameters) are updated using Equations (7)–(10). These equations represent the core update mechanisms in the IWSO algorithm, adjusting the positions of the soldiers based on the current solution space to explore more optimal hyperparameter configurations.

Step 5: Calculate Fitness Values. After updating the soldiers’ positions, the fitness values are recalculated. If the current fitness value is lower (better) than the previous one, it is retained. Otherwise, the prior fitness value is maintained.

Step 6: Termination Check. The optimization loop terminates either upon reaching the maximum iteration count or upon the fulfillment of a predefined early stopping criterion. At this point, the process exits the loop and outputs the optimal hyperparameter space.

Step 7: Model Construction Using Optimized Parameters. The optimized parameters are used as inputs to the proposed method. The training set is used to develop the prediction model, while the test set is employed to provide the final prediction results.

Figure 4 depicts the flowchart of the proposed IWSO-MILSTM strategy. In this study, the soldier population is configured to comprise 20, and the total of iterations is established at 100.

4. Experimental Validation

4.1. Evaluation Metrics

To assess the efficacy and progress of proposed methodology for predicting battery SOH, models including LSTM, BiLSTM, Random Forest [30], and Extreme Gradient Boosting (XGBoost) [31] are also used for SOH prediction. Additionally, R² and Root Mean Square Error (RMSE) are introduced as evaluation metrics to assess the prediction outcome.

$$R^2=1-{\displaystyle \frac{{{\displaystyle {\sum }_{k=1}^K(y_k^{\prime }-y_k)}}^2}{{{\displaystyle {\sum }_{k=1}^K(y_k^{\prime }-\overline{y})}}^2}}$$

(14)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{(y_i^{’}-y_i)}^2}}$$

(15)

In the equations, $y_k^{\prime }$ and $y_k$ represent the k-th predicted result and the actual result, respectively, and n denotes the count of cycles from the start of prediction to the end of the loop. R² ranges from 0 to 1, with values approaching 1 signifying a stronger fitting accuracy. When the predicted values perfectly match the actual values, the metric values will be 0. R² represents the goodness of fit and reflects the influence of the independent variables in the constructed battery SOH model on the SOH prediction results. RMSE quantifies the discrepancy between the predicted and observed values of the constructed SOH prediction model, indicating the model’s robustness. In this study, all experiments were conducted in the PyCharm environment using Python 3.7. The execution environment is shown in

Table 1. Hardware Environment for Program Execution.

Hardware Component	Model/Specification
CPU	I7-10700
GPU	NVIDIA GeForce GTX 1650 (Hsinchu City, Taiwan)
RAM	16 GB

and the hyperparameter space and optimization results are shown in

Table 2. The searching space of hyperparameters and optimization results.

Hyperparameter	The Searching Space	Optimization Results
		40% testing set	30% testing set
$Thr$	(0.4, 2.0)	1.51 (19 HIs)	1.84 (9 HIs)
$lr$	(0.001, 0.01)	0.0024	0.0018
$num\_hidden$	(32, 128)	64	96

.

4.2. Comparative SOH Prediction Results of Different Optimization Algorithms for Lithium-Ion Battery Under Different Training Data

To verify the advancement of IWSO, devise optimization algorithms were utilized to optimize the hyperparameters in this study, including WSO, Sparrow Search Algorithm (SSA) [32], and Whale Optimization Algorithm (WOA) [33]. The initial population is set to 20 and maximum number of 100 iterations for these algorithms. Figure 5 and Figure 6 illustrate that the SOH prediction results using the proposed method under these optimization algorithms are illustrated. From these figures, the prediction results of SOH using the IWSO-MSLTM method are highly consistent with the actual results compared to other optimization algorithms. This demonstrates the efficacy and progress of the IWSO-MSLTM strategy in SOH prediction task. The detailed evaluation metrics are shown in

Table 3. Comparison of Results for Different Optimization Algorithms with 40% Testing Set.

Evaluation Metrics	IWSO-MILSTM	WSO-MILSTM	WOA-MILSTM	IWSO-LSTM	SSA- LSTM
R2	0.9781	0.9667	0.9603	0.9516	0.9435
RMSE	0.0278	0.0376	0.0411	0.0454	0.0491

and

Table 4. Comparison of Results for Different Optimization Algorithms with 30% Testing Set.

Evaluation Metrics	IWSO-MILSTM	WSO-MILSTM	WOA-MILSTM	IWSO-LSTM	SSA- LSTM
R2	0.9875	0.9732	0.9695	0.9608	0.9595
RMSE	0.0206	0.0302	0.0322	0.0366	0.0372

. Take Table 3 as an example. Under equivalent iteration thresholds, IWSO exhibits stronger search capabilities and more efficient convergence speeds during the optimization process. Specifically, the R² and RMSE of WSO are both optimized in the traditional methods used in this work, i.e., R² is 0.9667 and RMSE is 0.0376. In contrast, the R² and RMSE of IWSO-MILSTM are still improved by 1.17% and 26.06%. The experimental results depict that the IWSO can enhance the global searching ability so that more accurate SOH prediction results can be provided by MILSTM due to the proposed improvements. At the same time, while other optimization algorithms also achieve good prediction results in certain cases, IWSO clearly performs better in terms of avoiding premature convergence and escaping local optima. In addition, IWSO is used for the optimization of MILSTM and LSMT under the same hyperparameters. It can be observed that the R² of MILSTM is increased by 2.78% and RMSE of MILSTM is reduced by 38.77% compared to LSTM. This indicates that the long-term degradation information of lithium-ion batteries can be more fully utilized in MSLTM.

In addition, ablation experiments with different HIs are explored as well, and the details are shown in Figure 7. From this figure, both R² and RMSE are optimal when the HIs are selected as 19 and 9 for 2 different testing sets. Notably, when the number of HIs is more than 19 and 9, the metrics all tend to decrease and level off. Indeed, the selected numbers of HIs by using the proposed IWSO-MILSTM for two testing sets are also 19 and 9, respectively. Thus, it can be determined that the best subset selected contains 19 HIs and 9 HIs for 2 different testing sets. It also indicates which HIs contribute to the construction process of the SOH prediction model.

4.3. Comparative Analysis of SOH Prediction Results Across Various Algorithms for Lithium-Ion Batteries

To further evaluate efficacy of the proposed methodology, alternative methods were introduced for comparison, including Gate Recurrent Unit (GRU) [34], CNN-LSTM [35], AM-seq2seq [36], and CMGRU [37]. The prediction results for various batteries are shown in Figure 8 and Figure 9. While standard LSTM and GRU perform well in predicting battery SOH, they exhibit larger prediction errors in the slope of capacity degradation and greater fluctuations in SOH prediction. The detailed evaluation metrics are presented in

Table 5. Comparison of Results for Different Algorithms with 40% Testing Set.

Evaluation Metrics	IWSO-MILSTM	GRU	LSTM	CNN-LSTM	AM-seq2seq	CMGRU	XGBoost	SVR
R2	0.9781	0.9401	0.9349	0.9618	0.9693	0.9574	-	-
RMSE	0.0278	0.0505	0.0527	0.0403	0.0361	0.0381	0.2416	0.3435

and

Table 6. Comparison of Results for Different Algorithms with 30% Testing Set.

Evaluation Metrics	IWSO-MILSTM	GRU	LSTM	CNN-LSTM	AM-seq2seq	CMGRU	XGBoost	SVR
R2	0.9875	0.9534	0.9526	0.9681	0.9803	0.9742	0.7857	-
RMSE	0.0206	0.0399	0.0402	0.0329	0.02592	0.02966	0.0855	0.1630

. For Table 5, 60% of the overall dataset was allocated as training settings for comparison under different training data scenarios. Compared to Table 6, it is evident that the model fails to fully grasp the degradation patterns of lithium-ion batteries; this is likely Comparison has been revised attributable to the reduced amount of training data, resulting in a decline in prediction accuracy. Nevertheless, it is noteworthy that the proposed strategy still achieves competitive results. Specifically, the R² decreases by 0.951% and the RMSE increases slightly by 34.9514%, with the absolute error of SOH not exceeding 3%. This meets the prediction requirements for battery health state estimation. As the training dataset increases, the prediction performance can be improved as listed in Table 6. The IWSO-MILSTM secures an R² of 0.9875 and RMSE of 0.0206. Compared to single LSTM, the R² values are improved by 3.663%. The RMSE values are reduced by and 48.7521%.Notably, AM-seq2seq achieved competitive results among the contrasting approaches. The R² of the IWSO-MILSTM is improved by 0.9078% while RMSE is reduced by 27.0341%, demonstrating the superior prediction performance of the IWSO-MILSTM algorithm.

5. Conclusions

To achieve accurate SOH prediction for batteries, this study proposed the IWSO-MILSTM approach. First, statistical degradation features such as constant current charging time, constant voltage charging time, and constant current discharging time were identified as directly measured health indicators, while the peak value of the IC curve was used as an indirect health indicator. Based on this, an improved WSO method was introduced to achieve the co-optimization of redundant parameters and informer-LSTM model parameters. IWSO was employed to mitigate issues like premature convergence and getting trapped in local optima based on developed initializing distribution methods and adaptive function, which further improved the model’s predictive accuracy and interpretability. Finally, the efficacy of proposed strategy was validated using a battery dataset from NASA. Experimental results demonstrated that the proposed model’s root mean square error and absolute error were both below 3%, verifying the feasibility and high accuracy.