## 4 Experimental Results and Analysis
### The Comparison of SOH Prediction Results Using Different Prediction Models
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.  
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. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Algorithm/Model** & **Parameter** & **Symbol** & **Range** & **Result** \\ \hline PSO-GWO & population size & N & / & 20 \\  & maximum number of iterations & i\({}_{\text{max}}\) & / & 50 \\  & regularization parameter & \(\gamma\) & (0.001, 1000) & 186.37 \\  & kernel parameter & \(\sigma\) & (0.001, 1000) & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Algorithm/Mode parameter settings. 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).