## 2 Lifetime Prediction Model for Lithium-Ion Batteries
### The Improved Chicken Swarm Optimization Algorithm (ICSO)  
As the iteration goes deeper, the search ability of rooster particles becomes worse. In addition, the number of hens is the largest, and the search results of the hen particles have a great influence on the algorithm. Therefore, the chaotic search is introduced in the ICSO algorithm. The hens are sorted based on their fitness values and the first 10% of the hen particles with good fitness are searched. After the search, the first 10% of the hen particles with superior performance replace roosters with poor fitness which speeds up the search speed of the swarm. The neighborhood points of many local optimal solutions are generated in the iterative process due to the characteristics of ergodicity and randomness of chaotic search [48]. In the ICSO algorithm, the chaotic search is constructed by using the Logistic equation [49]:  
\[C_{i+1}=\mu C_{i}(1-C_{i})\ \ \ \ \ \ i=1,2,\cdots,n \tag{14}\]  
where \(C_{i}(C_{i}\in(0,1))\) is a chaotic factor, and \(\mu\) is the control parameter. When \(\mu=4\), the system is completely in the chaotic state. The foraging ability of the rooster is the strongest, the hen is second, and the chick is the worst in the CSO algorithm. The adaptive learning factor is introduced in the chick's location to enhance the foraging ability of the chick. The relative change rate of the fitness value of chicks is defined as:  
\[k=\frac{\left|f_{i}(t)-f_{best}\right|}{f_{best}} \tag{15}\]  
where \(f_{i}(t)\) is the fitness value of the \(i_{th}\) chick, and \(f_{best}\) is the fitness value of the best particle in the flock. In the \(t_{th}\) generation, the adaptive learning factor of the \(i_{th}\) chick is updated:  
\[s_{i}^{t}=\frac{1}{1+\exp(-k)} \tag{16}\]  
where \(k\) is the relative change rate. After the improvement is complete, the position of the \(i_{th}\) chick is shown in Equation (17):  
\[x_{ij}^{t+1}=s_{i}^{t}x_{ij}^{t}+x_{ij}^{t}(1+randn(0,\sigma^{2})) \tag{17}\]  
where \(randn(0,\sigma^{2})\) is a normal distribution, 0 is the mean value, \(\sigma^{2}\) is the standard deviation; and \(s_{i}^{t}\) is the adaptive learning factor of the \(i_{th}\) chick in the \(t_{th}\) generation. The relative change rate k of the chicks reflects the difference between the \(i_{th}\) chick and the optimum particle. The adaptive learning factor \(s_{i}^{t}\) reflects the deviation degree of the position of the \(i_{th}\) chick and the optimum particle position in the flock. When the relative change rate \(k\) increases, the adaptive learning factor \(s_{i}^{t}\) increases. When the relative change rate \(k\) decreases, the adaptive learning factor \(s_{i}^{t}\) decreases. In this way, the movement of chick particles to the best particles is accelerated. The mutation operator is introduced in position of the chick to further avoid the chicks falling into extreme values. The fitness variance of chicks is as shown in Equation (18):  
\[\sigma^{2}=\frac{1}{n_{k}}\sum_{i=1}^{n_{k}}\left(\frac{f_{i}-f_{avg}}{f} \right)^{2} \tag{18}\]  
where \(n_{k}\) is the number of chicks; \(f_{i}\) is the fitness value of the \(i_{th}\) chick; \(f_{avg}\) is the average fitness value of the chicks; and \(f=\max\{1,\left|f_{i}-f_{avg}\right|\}\) (\(i\in[1,n_{k}]\)) is the normalized factor. The fitness variance \(\sigma^{2}\) of the chicks reflects the aggregation degree of the fitness values of the chicks. The smaller the \(\sigma^{2}\) is, the higher the aggregation degree of the fitness value is. When the chick particles fall into the local extreme value, the fitness variance \(\sigma^{2}\) tends to 0. If the fitness variance \(\sigma^{2}\) reaches the threshold, the chicks are mutated. The narrowing search space of the chick can be broadened by the mutation operation. There are several steps in the optimization process of ICSO algorithm. (1) The parameters of the algorithm are defined, such as the number of roosters \(n_{r}\), hens \(n_{h}\), chicks \(n_{k}\), etc.