## 5 Case Study
### Fitting of Semi-Empirical Model Based on Interval Number Similarity  
Based on the semi-empirical modeling method of interval number similarity proposed in this paper, the empirical models of six conditions obtained from known data are analyzed below. For the six \(\Delta\)\(SO\)Cs of [80, 100]%, [40, 60]%, [0, 20]%, [20, 100]%, [10, 90]%, and [0, 100]%, calculate the similarity matrix between any two intervals, as shown in Figure 6.  
Figure 4: Use Equation (31) to determine the fitting parameter \(z\) for the empirical model in which \(\ln(\mathcal{Q}_{\mathrm{loss}})-\ln(\alpha)+\left(\frac{E_{z}}{R\cdot T}\right)\) is plotted as a function of \(\ln(\mathcal{A}h)\). Figure 5: Fitting results obtained from the empirical model under six conditions. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Condition**} & \multicolumn{6}{c}{**Fitting Parameters**} & \multicolumn{6}{c}{**Statistical Measure**} \\ \cline{2-9}  & \(\alpha\) & \(\beta\) & \(\gamma\) & \(a\) & \(b\) & \(z\) & \(R^{2}\) & \(RMSE\) \\ \hline Condition 1 & 2.1280 \(\times\)\(10^{-3}\) & 0.1622 & 0.8661 & 0.3167 & 0.5340 & 0.8121 & 0.9931 & 0.0008 \\ Condition 2 & 6.4090 \(\times\)\(10^{-4}\) & 0.4505 & 0.6615 & 0.2334 & 0.9178 & 0.8121 & 0.9185 & 0.0009 \\ Condition 3 & 2.2680 \(\times\)\(10^{-3}\) & 0.1067 & 1.1760 & 0.1053 & 0.8756 & 0.8121 & 0.9792 & 0.0009 \\ Condition 4 & 4.3810 \(\times\)\(10^{-4}\) & 0.4314 & 1.0530 & 0.1552 & 0.2371 & 0.8121 & 0.9906 & 0.0020 \\ Condition 5 & 3.3450 \(\times\)\(10^{-4}\) & 0.8530 & 0.8922 & 0.2951 & 0.4574 & 0.8121 & 0.9838 & 0.0022 \\ Condition 6 & 4.5930 \(\times\)\(10^{-4}\) & 0.4893 & 0.3377 & 0.9000 & 0.3692 & 0.8121 & 0.9812 & 0.0048 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fitting parameters and goodness of fit of the empirical model under six working conditions. To verify the effectiveness of the semi-empirical model of retired battery capacity degradation based on interval number similarity proposed in this paper, the measured data under six sets of test cases are used to verify one another, as shown in Table 4.  
To verify the advantages of the proposed method, the data-driven method based on interval number similarity proposed in reference [34] is used as the comparison method. We test the two interval number similarity-based semi-empirical modeling methods proposed in this paper, which are the model interval number method and the parameter interval number method. The results of six cases obtained by these two semi-empirical modeling methods and their comparison are shown in Figure 7. Among them, the original fitting curve is defined as the theoretical value of SOH for reference. It is not difficult to see from the results shown in Figure 7 that although the comparison method is close to the measured fitting data within 500 cycle times, the deviation of the results between the comparison method and the original fitting curve becomes larger and larger as the number of cycle times increases, and they gradually deviate far from the theoretical value. In contrast, the semi-empirical modeling method based on interval number similarity presented in this paper shows the same capacity degradation trend as the original fitting curve and can be more similar to the measured fitting curve in Case 4, Case 5, Case 6, and other examples. To quantify this degree of similarity, the values of \(R^{2}\) and RMSE between the two interval number similarity-based semi-empirical modeling methods and the measured fitting curves are shown in Table 5.