# Lithium-Ion Battery State of Health Estimation Based on CNN-LSTM-Attention-FVIM Health Features
## 3 Algorithm Principle Explanation
### Four-Vector Optimization Algorithm
\(\xi_{1}\), \(\xi_{2}\), \(\xi_{3}\) represent random numbers uniformly distributed in the interval [0,1][38]. Step 4: Identify the positions of the top four agents with the best performance, \(P_{1}\), \(P_{2}\), \(P_{3}\), and \(P_{4}\), and evaluate the fitness function _f(x)_ for each agent. If this step fails to find the best point for the four agent positions, return to step 3 and re-evaluate the fitness function _f(x)_ for each agent. Otherwise, proceed to the next step to output the best \(P\) and end. In the FVIM algorithm, the adaptive parameter \(\alpha\) serves a homologous purpose to the inertia weight (W) in Particle Swarm Optimization (PSO), specifically engineered to bridge the exploration-exploitation trade-off. Exploration denotes the algorithm's capacity to probe diverse regions for potential optima, whereas exploitation emphasizes refining search precision within promising domains to enhance solution quality. This parameter undergoes linear decay as agents approach the optimal solution. During FVIM initialization, \(\alpha\) is strategically initialized at a substantial value of 1.5 to prioritize expansive search space exploration. Subsequently, it systematically diminishes to zero through progressive iterations as the algorithm transitions toward exploitation-dominant phases near convergence-critical regions. \[\alpha=1.5-1\times\left(\frac{2}{\frac{Max\text{ }iter}{2}}\right) \tag{19}\]  
The parameter \(\alpha\) is used to balance global exploration and local optimization. When \(\alpha\) is too large, the adjustment range increases, which may cause the solution to deviate from the optimal area; when \(\alpha\) is too small, it focuses on local optimization but limits the exploration ability. The specific adjustment needs to be determined according to the characteristics of the problem. The Four-Vector Optimization Algorithm (FVIM) significantly improves the optimization performance by introducing a four-guide point strategy and a dynamic adaptive mechanism. Its core advantages include: (1) using four optimal individuals to guide the search direction, enhancing population diversity and effectively avoiding local optimal traps; (2) dynamically balancing global exploration and local development through adaptive coefficients \(\alpha\) and random perturbation parameters \(\xi\), scanning the solution space extensively in the early stage, and focusing on fine optimization in the later stage; (3) employing a mean vector mechanism to integrate multi-guide point information and improve global convergence efficiency; (4) achieving fast convergence speed and strong robustness, especially in multimodal and high-dimensional scenarios, significantly outperforming traditional algorithms (such as PSO, GWO) and providing an efficient and reliable optimization solution for practical applications. The time series prediction model of the CNN-LSTM-Attention hybrid neural network model with FVIM added is mainly divided into the following parts: CNN-LSTM-Attention training and prediction, FVM optimization, and various error calculations. In the initialization stage of FVIM optimization, the basic FVIM parameters are randomly initialized, and upper and lower limits are set for each problem. By setting the parameter range, randomly generating the initial population,determining the number of iterations and particle size, and assigning the baseline fitness value and extreme value variables, the algorithm is provided with a flexible search starting point to ensure that it can adapt to the optimization needs of minimization or maximization problems. In the iteration stage, the algorithm guides the movement of particles based on the dynamic motion equation and gradually converges to the global optimal solution by continuously evaluating the quality of the solution and comparing it with the objective function, finally outputing the best solution consistent with the optimization goal when the stop condition is met.