## 3 WD-IWOA-DELM Prediction Framework
### Implementation of WD-IWOA-DELM Estimation  
The overall framework of the proposed WD-IWO-DELM estimation method is illustrated in Figure 4. The implementation of this framework involves the following specific steps:  
1. Collect information on the capacity degradation sequence of lithium batteries and calculate the SOH using Equation (1).
2. Apply Wavelet Decomposition to the original SOH sequence of lithium batteries. In this study, the db10 wavelet basis function is selected to perform a four-layer decomposition of the capacity, resulting in an approximate component (a4) representing the overall degradation trend of the lithium battery's SOH, and detail components (d1 to d4) containing localized regeneration signals. The decomposition results for Battery B0005 and CS2_35 are shown in Figure 5. After proportionally dividing the sequence components into training and testing sets, normalization is performed. Figure 6 compares the decomposition errors of Wavelet Decomposition, EEMD, and CEEM-DAN, demonstrating that Wavelet Packet Decomposition outperforms the EEMD decomposition method.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Battery Number** & **Charging Protocol** & **Rated Capacity** & **Temperature** & **(C/D) Current** & **(C/D) Voltage/V** \\ \hline CS2\_35 & CC-CV & 1.1 Ah & 24 \({}^{\circ}\)C & 0.55/0.55 A & (4.2, 2.7) V \\ CS2\_36 & CC-CV & 1.1 Ah & 24 \({}^{\circ}\)C & 0.55/0.55 A & (4.2, 2.7) V \\ CS2\_37 & CC-CV & 1.1 Ah & 24 \({}^{\circ}\)C & 0.55/1.1 A & (4.2, 2.7) V \\ CS2\_38 & CC-CV & 1.1 Ah & 24 \({}^{\circ}\)C & 0.55/1.1 A & (4.2, 2.7) V \\ \hline \hline \end{tabular}
\end{table}
Table 2: CALCE lithium-ion battery experimental specifications.  
Figure 3: CALCE’s lithium-ion battery capacity variation curve with the number of discharge cycles.  
3. Initialize the parameters for the IWOA-DELM model algorithm. The detailed parameter information is available in Table 3. To enhance the global search capability and local exploration ability of the whale optimization algorithm, a nonlinear convergence factor is introduced. Different weighted strategies (13) and (14) are adopted for position updating in the two stages of the WOA, balancing global search and local exploration to simultaneously improve convergence accuracy and speed. The IWOA algorithm is employed to optimize the parameters of the DELM model, using the mean squared error as the fitness function to measure the discrepancy between the model's predicted values and the actual observed values. Before dividing the dataset, approximately 50% serves as the training set, and the remaining 50% as the test set. By defining a sliding window size and utilizing the improved whale optimization algorithm to optimize the network parameters of the deep extreme learning machine, an IWOA-DELM network model is established for forecasting sequence components. Finally, an effective ensemble of individual forecasted components yields the ultimate prediction outcome. The fitness function is expressed as follows: \[\text{Fitness}= \frac{1}{T}\sum\limits_{1}^{T}{{{(y_{i}-\widetilde{y}_{i})}^{2}}}\] (18) where \(T\) represents the number of validation data points, \(y_{i}\) denotes the true values, and \(\tilde{y}_{i}\) represents the predicted values. (4) Performance evaluation: Several performance evaluation metrics have been constructed to effectively assess the proposed method.