# Lithium-Ion Battery State of Health Estimation Based on CNN-LSTM-Attention-FVIM Health Features
## 3 Algorithm Principle Explanation
### Four-Vector Optimization Algorithm  
The Four-Vector Intelligent Metaheuristic (FVIM) is a new type of metaheuristic algorithm (intelligent optimization algorithm) proposed by [PERSON] et al. [38] in 2024, which is inspired by the mathematical modeling of four vectors. The FVIM optimization process mainly consists of three stages, namely the initialization stage, the iteration stage, and the stage of finding the optimal solution. It uses the four best points in the group to determine the direction of movement of the entire group. The FVIM algorithm first identifies the four best individuals in the current group, then it calculates the average position of these four individuals to obtain a new vector position. This new position may provide a better solution for the previously determined solution, thereby guiding the group to move in a better direction. The specific stages of FVIM are as follows:  
Phase 1. Initialization phase  
Initialize four vector populations X and four agent positions, P1, P2, P3, P4. Define and randomly initialize basic FVIM parameters. To maintain efficiency, FVIM sets upper and lower limits for each problem to limit the search space. At the same time, the number of algorithm iterations and the number of particles used by the algorithm are determined according to the complexity, size, dimension, and other characteristics of the optimization problem. Additionally, the reference fitness value is assigned 0 to assess the performance of different solutions. Certain variables are assigned extreme values to guarantee algorithm adaptability, enabling it to address both minimization and maximization objectives. Phase 2. Iteration phase  
This phase is the core of FVIM, and the main steps are as follows. Step 1: Evaluate the fitness function f(xi) for each agent,  
Step 2: Identify the positions of the top four agents with the best performance, P1, P2, P3, P4,  
Step 3: Update i in X(1,2,3,4) using Equations (1)-(4), and update the average position P using Equation (5), which can be described by the mathematical model as follows:  
\[ \begin{array}{l}\{X_{1,i}^{\mathrm{X}_{1,i}}\mathrm{=}\mathrm{P}_{\mathrm{i },\mathrm{+}}(\mathrm{a}\times\mathrm{2}\times\mathrm{5}_{1,\mathrm{-}})\times \mathrm{|\zeta_{2}\times\mathrm{P}_{\mathrm{i},\mathrm{-}}\overline{\mathrm{P} }_{\mathrm{i}}|}\text{ if }\mathrm{\zeta_{3}\times 0.5}\\ \{X_{1,i}^{\mathrm{X}_{2,i}}\mathrm{=}\mathrm{P}_{\mathrm{i},\mathrm{-}}( \mathrm{a}\times\mathrm{2}\times\mathrm{5}_{1,\mathrm{-}})\times\mathrm{| \zeta_{2}\times\mathrm{P}_{\mathrm{i},\mathrm{-}}\overline{\mathrm{P}}_{ \mathrm{i}}|}\text{, otherwise}\end{array}  \tag{14}\]  
\[ \begin{array}{l}\{X_{2,i}^{\mathrm{X}_{2,i}}\mathrm{=}\mathrm{P}_{\mathrm{i },\mathrm{+}}(\mathrm{a}\times\mathrm{2}\times\mathrm{5}_{1,\mathrm{-}})\times \mathrm{|\zeta_{2}\times\mathrm{P}_{\mathrm{i},\mathrm{-}}\overline{\mathrm{P} }_{\mathrm{i}}|}\text{ if }\mathrm{\zeta_{3}\times 0.5}\\ \{X_{2,i}^{\mathrm{X}_{2,i}}\mathrm{=}\mathrm{P}_{\mathrm{i},\mathrm{-}}( \mathrm{a}\times\mathrm{2}\times\mathrm{5}_{1,\mathrm{-}})\times\mathrm{| \zeta_{2}\times\mathrm{P}_{\mathrm{i},\mathrm{-}}\overline{\mathrm{P}}_{ \mathrm{i}}|}\text{, otherwise}\end{array}  \tag{15}\]  
Figure 4: CNN-LSTM-Attention model flow chart. \[\{\chi_{\text{x}_{i}=P_{i,j}+(\alpha\times 2\times\zeta_{1}-\alpha)\times|\zeta_{2 }\times P_{3,i}-\overline{P}_{i}|}\text{ }if\text{ }\xi_{3}<0.5 \tag{16}\] \[\{\chi_{\text{x}_{i}=P_{3,j}}^{\chi_{\text{x}_{i}=P_{3,j}}}-(\alpha \times 2\times\zeta_{1}-\alpha)\times|\zeta_{2}\times P_{3,j}-\overline{P}_{i}| \text{ }if\text{ }\xi_{3}<0.5\] (17) \[\{\overline{P}_{i}=\frac{X_{1,i}+X_{2,i}+X_{3,i}+X_{4,i}}{4} \tag{18}\]  
where \(X_{n,i}\) represents the updated position of the nth best individual in the \(i\)-th dimension. \(P_{n,i}\) represents the current position of the nth best individual in the \(i\)-th dimension. \(P_{i}\) represents the current average position of all individuals in the \(i\)-th dimension. \(\alpha\) is an adaptive coefficient, which is equivalent to the search step size.