## 5 Case Study
### Comparison of Three Semi-Empirical Modeling Methods  
The feasibility of the three semi-empirical modeling methods proposed in this paper is verified above, and the accuracy of the same set of measured data under various methods is calculated at the same time. The following gives a comprehensive comparison of the results of six cases. From Figure 11, it can be seen that in the six cases above, the fitting curve obtained by the improved semi-empirical modeling method is closest to the original fitting curve, and the fitting accuracy is higher compared to the two interval number similarity-based semi-empirical modeling methods of parameter and model. Especially for Case 1, Case 3, and Case 4, the improved semi-empirical model performs the same as the original fitting curve. This is because in these three cases, it is known that the \(\Delta SOC\) includes two sets of cyclic data with the same DOD and different \(\Delta SOC\)s and the same \(\Delta SOC\) and different DODs. Under this data premise, the improved semi-empirical modeling method can obtain more accurate fitting models for retired battery capacity degradation under different operating conditions. The above results can indicate that when the \(\Delta SOC\) data are relatively comprehensive, the improved semi-empirical model has better accuracy than the interval number method in all cases. To verify the applicability of different methods in data scenarios, the following is a fitting prediction of battery capacity degradation for the above three methods in extreme cases with limited measured data. Thus, the following scenarios in Table 8 are covered. In the above three scenarios, the measured data are fitted and analyzed based on the three methods proposed in this paper, and the results are as follows: In the above two scenarios, the measured data are fit and analyzed based on the three methods proposed in this article. Among them, due to the limited known data, it is not sufficient to meet the fitting conditions for the aging factor proposed in Equation (16). Therefore, in the improved semi-empirical model below, the optimal polynomial is used to fit the aging factor. The final numerical results of goodness of fit for the two scenarios are shown in the table below. From the goodness of fit values in Table 9, it can be seen that in both scenarios, the \(R^{2}\) values of the interval number similarity-based semi-empirical modeling method are not all equal to 0. However, the \(R^{2}\) values of the improved semi-empirical modeling method are all equal to 0, and their fitting accuracy is completely inferior to the interval number similarity-based semi-empirical modeling method. Therefore, under the premise that the \(\Delta SOC\) of the known data is very limited, the method based on interval number similarity has better fitting performance than the method based on improved semi-empirical models. Comparing the two types of methods, it can be found that the method based on interval number similarity has low dependence on data and can perform with high accuracy when known data are limited. For example, the unknown intervals of Scenario 1 and Scenario 2 can be accurately fitted through parameters or modeling methods. The method based on an \begin{table}
\begin{tabular}{c c c} \hline \hline \(\Delta SOC\) **(\%)** & \(\mathbf{R^{2}}\) & \(\mathbf{RMSE}\) \\ \hline
[80, 100] & 1.0000 & \(1.1353\times 10^{-5}\) \\
[40, 60] & 1.0000 & \(1.8958\times 10^{-7}\) \\
[0, 20] & 0.9999 & \(1.6002\times 10^{-4}\) \\
[20, 100] & 1.0000 & \(3.7004\times 10^{-5}\) \\
[10, 90] & 1.0000 & \(2.0161\times 10^{-4}\) \\
[0, 100] & 1.0000 & \(5.5489\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The goodness of fit of the improved semi-empirical model.