## 5 Case Study
### Fitting of Improved Semi-Empirical Model Based on the Least Square Fitting Method  
In the battery charge/discharge cycle aging, not only the \(\Delta SOC\) will be changed, but also the charge/discharge rate, temperature, and other working conditions will be changed. Therefore, the semi-empirical model is expected to simulate performance degradation under various working conditions. The improved semi-empirical modeling method based on the least square method can meet the above application requirements. Based on the improved semi-empirical model fitting method proposed in this paper, Case 4 in Table 4 is taken as an example to solve the improved semi-empirical model. The known \(\Delta SOC\)s are [80, 100]%, [40, 60]%, [0, 20]%, [10, 90]%, and [0, 100]%, while the unknown \(\Delta SOC\) is [20, 100]%. The known data of Case 4 have the same \(DOD\) and different \(\Delta SOC\)s at the same time and two types of data of the same \(\Delta SOC\) and different DODs, which can more comprehensively fit the aging factor. First of all, the empirical model of Equation (9) is used for nonlinear least squares fitting, and then the empirical model fitting results are as follows:  
\[SOH=0.8-4.5750\times 10^{-4}\times\exp\left[\frac{0.0355\cdot C_{Rate}+0.8489}{8.3 14\cdot T}\right]\cdot C_{Rate}\prescript{0.9595}{}{}\cdot DOD\prescript{2.2140}{} \left(N\cdot DOD\cdot Q_{b}\right)^{0.8121} \tag{33}\]  
Based on the empirical model of Equation (33), the decline in battery capacity under various DODs, charge/discharge rates, and the temperature working conditions can be obtained correspondingly. The five conditions of the known measured data can be divided into two groups: the same DOD but different average SOCs and the same average SOC but different DODs. The aging factors of the two groups are calculated, and the relationship between the aging factor and the \(\Delta SOC\) is studied deeply. The results of aging factors obtained are shown in Figure 8 as follows. Figure 8: The aging factor varies with the average DOD under different average SOCs. It can be seen from the figure above that the relationship between the aging factor of the experimental data and the average SOC and DOD is approximately a quadratic function, which accords with the aging factor model in the improved semi-empirical model established in this paper. Based on five groups of measured data, the improved semi-empirical model of Equations (18) and (19) was fitted by the nonlinear least square method, and the fitting results of each unknown parameter were obtained, as shown in Table 6.  
Therefore, the improved semi-empirical model based on the least square fitting is built as follows:  
\[ \begin{split} c_{age}=26.01+0.0103\cdot(SOC_{avg}-37.26)^{2}-0.424 7\cdot SOC_{avg}.DOD-38.93\cdot DOD+33.49\cdot DOD^{2}\\ SOH=0.8-c_{age}\times 4.5750\times 10^{-4}\times\exp\left[\frac{0.0355 \cdot C_{Rate}+0.8489}{8.314\cdot T}\right]\cdot C_{Rate}\,^{0.9595}\cdot DOD ^{2.2140}(N\cdot DOD\cdot Q_{b})^{0.8121}\end{split}  \tag{34}\]  
Among them, \(SOC_{avg}\), \(DOD\), \(C_{Rate}\), and \(T\) can constitute the working conditions of the retired battery. The capacity degradation of the retired battery under various working conditions can be obtained from the above model. Figure 9 shows the capacity degradation of retired batteries in three different \(\Delta SO\)Cs obtained by the method in this paper when \(DOD\) = 20%, and it compares the original empirical model with the original data. It can be seen that compared with the original empirical model, which cannot distinguish the capacity decline of different \(\Delta SO\)Cs under the same DOD, the improved semi-empirical model in this paper improves the operation scenario of the empirical model and is more in line with the practical application.