## 5 Case Study
### Fitting of Semi-Empirical Model Based on Interval Number Similarity
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{2}{c}{**Unknown \(\Delta SOC\)**} & \multicolumn{4}{c}{**Known \(\Delta SOC\)s (\%)**} \\  & \multicolumn{2}{c}{**(\%)**} & \multicolumn{2}{c}{**Known \(\Delta SOC\)s (\%)**} \\ \hline Case 1 & [80, 100] & [40, 60] & [0, 20] & [20, 100] & [10, 90] & [0, 100] \\ Case 2 & [40, 60] & [80, 100] & [0, 20] & [20, 100] & [10, 90] & [0, 100] \\ Case 3 & [0, 20] & [80, 100] & [40, 60] & [20, 100] & [10, 90] & [0, 100] \\ Case 4 & [20, 100] & [80, 100] & [40, 60] & [0, 20] & [10, 90] & [0, 100] \\ Case 5 & [10, 90] & [80, 100] & [40, 60] & [0, 20] & [20, 100] & [0, 100] \\ Case 6 & [0, 100] & [80, 100] & [40, 60] & [0, 20] & [20, 100] & [10, 90] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Known \(\Delta SOC\) and unknown \(\Delta SOC\)s for six sets of test cases. Figure 6: Interval similarity matrix. From the values of the goodness of fit in Table 5, we can see that in Case 2, Case 3, and Case 6, the parameter interval number similarity method performs better than the model interval number method; in Case 4 and Case 5, the model interval number similarity method performs better than the parameter interval number method. Combined with the interval number similarity matrix in Figure 6, it can be found that the model interval number similarity method has more advantages when the known \(\Delta SOC\) can fully cover the unknown \(\Delta SOC\); instead, when the known \(\Delta SOC\) can not adequately cover the unknown \(\Delta SOC\), the parameter interval number similarity method is more superior. In particular, in Case 6, when \(\Delta SOC=[0,100]\%\), the known \(\Delta SOC\) is not fully covered by the unknown \(\Delta SOC\), thus the parameter interval number method performs better than the modes. This is because the parameter interval number method approach requires fewer known data  
Figure 7: Comparison of the results of model interval number method and parameter interval number method for six groups of test cases. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Parameter Interval Number**} & \multicolumn{2}{c}{**Model Interval Number**} \\ \cline{2-5}
**Case** & \multicolumn{2}{c}{**Similarity Method**} & \multicolumn{2}{c}{**Similarity Method**} \\ \cline{2-5}  & \(R^{2}\) & \(RMSE\) & \(R^{2}\) & \(RMSE\) \\ \hline Case 1 & 0 & 0.0464 & 0 & 0.0806 \\ Case 2 & 0.0896 & 0.0111 & 0 & 0.1088 \\ Case 3 & 0.0235 & 0.0178 & 0 & 0.1153 \\ Case 4 & 0 & 0.0760 & 0.9998 & 0.0007 \\ Case 5 & 0 & 0.1047 & 0.5816 & 0.0287 \\ Case 6 & 0.8890 & 0.0297 & 0.1414 & 0.0827 \\ \hline \hline \end{tabular} Notes: \(R^{2}=0\) means that the fitting effect of the function is worse than the average value.  
\end{table} 
Table 5: The values of \(R^{2}\) and \(RMSE\) between the two interval number similarity-based semi-empirical modeling methods and the measured fitting curves. and is more adaptable, while the model interval number method approach relies more on the known data. Compared to the two interval number similarity-based semi-empirical modeling methods of parameter and model herein, the model-based method can only obtain the change in battery capacity degradation in the same \(\Delta SOC\) of the charge/discharge rate and temperature; it is not practical. The parameter-based method can obtain the change in battery capacity degradation in the desired \(\Delta SOC\) under different charge/discharge rates and temperatures of the working conditions. Taking Case 6 as an example, based on Equation (9), the semi-empirical model of \(\Delta SOC=[0,100]\%\) obtained by the parameter interval number similarity method is shown as follows:  
\[SOH=0.8-7.3885\times 10^{4}\times\exp\left[\frac{0.9702\cdot C_{Rate}+0.9568}{8.3 14\cdot T}\right]\cdot C_{Rate}\prescript{0.9}{}{\cdot}DOD\prescript{0.1756}{}{} \left(N\cdot DOD\cdot Q_{b}\right)^{0.8121} \tag{32}\]  
Equation (32) can be used to estimate the change of battery capacity degradation under other operating conditions in \(\Delta SOC=[0,100]\%\).