## 3. SOH Prediction Method Based on the PSO-GWO-LSSVM Model
### PSO-GWO Combined Optimization Algorithm
#### 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: \(\alpha\), \(\beta\), \(\delta\), and \(\omega\)[37]. The \(\alpha\) wolf is the leader of the pack, \(\beta\) and \(\delta\) are the assistants, and \(\omega\) represents the ordinary members. The \(\alpha\), \(\beta\), and \(\delta\) wolves update their positions relative to the target, while the \(\omega\) 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:  
\[\left\{\begin{array}{l}\stackrel{{\rightarrow}}{{W}}(t+1)= \stackrel{{\rightarrow}}{{W}}_{p}(t)-\stackrel{{ \rightarrow}}{{A}}\cdot\stackrel{{\rightarrow}}{{D}}\\ \stackrel{{\rightarrow}}{{D}}=|\stackrel{{ \rightarrow}}{{C}}\cdot\stackrel{{\rightarrow}}{{W}}_{p}(t)- \stackrel{{\rightarrow}}{{W}}(t)|\\ \stackrel{{\rightarrow}}{{A}}=2\stackrel{{ \rightarrow}}{{a}}\cdot\stackrel{{\rightarrow}}{{n}}_{1}- \stackrel{{\rightarrow}}{{a}}\\ \stackrel{{\rightarrow}}{{C}}=2\stackrel{{ \rightarrow}}{{n}}_{2}\end{array}\right. \tag{20}\]  
where \(\stackrel{{\rightarrow}}{{W}}(t+1)\) represents the position vector of the grey wolf at time \((t+1)\); \(\stackrel{{\rightarrow}}{{W}}_{p}(t)\) denotes the position vector of the prey; \(\stackrel{{\rightarrow}}{{A}}\) represents the attack target parameter; \(\stackrel{{\rightarrow}}{{D}}\) indicates the Euclidean distance; \(\stackrel{{\rightarrow}}{{n}}_{1}\) and \(\stackrel{{\rightarrow}}{{n}}_{2}\) are directionally bounded random change parameters, with values ranging from (0, 1); and \(\stackrel{{\rightarrow}}{{a}}\) 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 \(\alpha\) wolf, which helps accurately locate the prey's position, with the \(\beta\) and \(\delta\) wolves also assisting in the task. The \(\alpha\) wolf represents the optimal solution, while the \(\beta\) and \(\delta\) 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:  
\[\left\{\begin{array}{l}\stackrel{{\rightarrow}}{{D}}_{j}= \stackrel{{\rightarrow}}{{C}}_{n}\stackrel{{ \rightarrow}}{{W}}_{j}-\stackrel{{\rightarrow}}{{W}}_{p}\\ \stackrel{{\rightarrow}}{{W}}_{n}=\stackrel{{ \rightarrow}}{{W}}_{j}-\stackrel{{\rightarrow}}{{A}}_{n}\cdot \stackrel{{\rightarrow}}{{D}}_{j}\\ \stackrel{{\rightarrow}}{{W}}_{p}(t+1)=\frac{\sum\limits_{n} \stackrel{{\rightarrow}}{{W}}_{n}}{3}\end{array}\right. \tag{21}\]  
where j represents \(\alpha\), \(\beta\), and \(\delta\), and n takes the values 1, 2, and 3. \(\stackrel{{\rightarrow}}{{W}}_{p}(t+1)\) 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).