## 3 Semi-Empirical Model of Retired Battery Capacity Degradation Based on Interval Number Similarity
### Improvement of Empirical Model  
Reference [36] tested the capacity degradation performance of lithium iron phosphate batteries under various influencing factors through aging experiments and proposed a holistic aging model for lithium battery life estimation. In this model, the effect of average voltage on the decline in battery capacity is considered. The test results indicate that the decline factor has a linear relationship with DOD and a quadratic relationship with the average voltage. When the SOC is between 45% and 55%, the capacity degradation is minimal, and higher or lower \(\Delta\)SOCs will exacerbate the cyclic aging of the battery. The disadvantage lies in that this conclusion is obtained from the aging experiment results of new batteries and may not apply to the retired batteries.  
Most of the battery aging experiments conducted in research institutes are conducted on new batteries, but some experiments include part of the battery decommissioning phase. Combining the experimental cycle life test results of Li(NiMnCo)O\({}_{2}\) (NMC) batteries with different average SOC values from [PERSON] et al. [37] and the experimental test results of LiFePO4 (LFP) batteries from [PERSON] et al. [38], it can be concluded that the cycle life of a battery with an average SOC of close to 50% is the longest, and the cycle life decreases with the development of the average SOC towards both ends. Based on the experimental test data of NMC batteries [37], reference [39] adopted a piecewise Gaussian model to fit the stress factors caused by the mean value of SOC on capacity degradation. The downside is that none of the above references have improved and perfected their empirical models based on test data, and hence under different working conditions, it is impossible to use the general model to generate more capacity degradation models of retired batteries.  
Therefore, the empirical model of the retired battery shown in Equation (9) is improved and perfected in this paper. The \(\Delta\)SOC of the battery can be uniquely determined by DOD and average SOC, and the battery cycle aging factor \(\Delta\)SOC can be applied to the retired battery empirical model in the form of an aging factor \(c_{age}\); thus, the improved empirical model can be expressed as follows:  
\[SOH=0.8-c_{age}\cdot\alpha\cdot\exp\left[\frac{a\cdot C_{Rate}+b}{RT}\right] \cdot C_{Rate}\beta\cdot DOD^{\gamma}(N\cdot DOD\cdot Q_{b})^{z} \tag{18}\]  
Based on the above research results and conclusions, the quadratic function relationship is proposed in this paper to fit the relationship among the aging factor in the \(\Delta\)SOC and the average \(SOC\) and \(DOD\), as follows:  
\[c_{age}=\lambda_{1}+\lambda_{2}\cdot\left(SOC_{avg}-SOC_{0}\right)^{2}+ \lambda_{3}\cdot SOC_{avg}\cdot DOD+\lambda_{4}\cdot DOD+\lambda_{5}\cdot DOD ^{2} \tag{19}\]  
where \(\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\text{ and }\lambda_{5}\) denote the fitting parameter and \(SOC_{avg}\) is the average SOC. \(SOC_{0}\) is the median SOC value in the battery model, and the range is the interval [0, 100], which is obtained by repeated iterative solutions. At this point, the empirical model of retired batteries has been improved and the \(\Delta\)SOC is integrated into the retired battery model in the form of variables.