# Contact Force Surrogate Model and Its Application in Pantograph-Catenary Parameter Optimization
## 3 Sensitivity Analysis of Pantograph-Catenary Parameters
### Sensitivity and Its Rating of Pantograph-Catenary Parameters  
Due to the strong non-linearity of the surrogate model of the SDCF, it is difficult to calculate the variance terms in Equations (2)-(4) through integration. Therefore, with the Monte Carlo method, two independent sample matrices with the same sample size were used for estimation. For two low-pass filtering frequencies of 20 Hz and 40 Hz and five speeds, including 250, 300, 350, 400, and 450 km/h, sample sizes and input parameter values are predefined (Table 1). Based on the two previously designed sample matrices, each column is replaced one by one with the corresponding column from the other matrix, resulting in an increase of 29 times in the sample size [28]. Using bootstrapping with 10,000 times resampling to calculate the sensitivities of pantograph-catenary parameters and their 95% confidence intervals [29; 30]. Statistical analysis of the first-order and total sensitivities of 14 pantograph-catenary parameters under five operating speeds and two low-pass filtering frequencies indicated that as the sample size increases, the sensitivities converge to specific values, and the 95% confidence intervals shrink, thus improving the accuracy and stability of sensitivities calculations (Figure 4). Considering the results for all operating conditions, a sample size of 21 was chosen when the values of sensitivities reached a stable convergence. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cut-Off Frequency** & \begin{tabular}{c} **Error Interval within** \\ **the Probability of** \\ **0.95 (N)** \\ \end{tabular} & \
\begin{tabular}{c} **Mean Squared Error** \\ **(N\({}^{2}\))** \\ \end{tabular} &
\begin{tabular}{c} **Coefficient of** \\ **Determination** \\ \end{tabular} \\ \hline
20 & \([-6.06,5.11]\) & 7.93 & 0.971 \\
40 & \([-5.67,6.12]\) & 7.66 & 0.982 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Prediction errors of surrogate models on test set. Figure 5 illustrates the first-order and total sensitivities of 14 pantograph-catenary parameters. The average sensitivity to each pantograph-catenary parameter under all operating conditions (Column 2 of Table 4) represents the level of the parameter's impact on the SDCF. Using the k-means clustering algorithm from prototype clustering, the 14 average sensitivities are divided into five categories, thus providing a sensitivity rating for the pantograph-catenary parameters (Column 3 of Table 4). Among these, \(T_{\rm c}\) and \(A_{\rm c}\) are rated as Class I, as their impact on the SDCF is significantly higher than that of the other parameters. \(A_{\rm m}\) and \(m_{3}\) are rated as Class II; \(c_{1}\), \(L\), and \(T_{\rm m}\) are rated as Class III; \(m_{1}\), \(m_{2}\), \(k_{3}\), and \(k_{2}\) are rated as Class IV; \(c_{3}\), \(c_{2}\), and \(k_{1}\) are rated as Class V, and their impact on the SDCF can be ignored. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Input Parameters** &
\begin{tabular}{c} **Average Value of** \\ **Sensitivities** \\ \end{tabular} & **Rating** \\ \hline \(T_{\rm c}\) & 0.713 & \\ \hline \(A_{\rm c}\) & 0.526 & I \\ \hline \(A_{\rm m}\) & 0.254 & II \\ \hline \(m_{3}\) & 0.230 & II \\ \hline \(c_{1}\) & 0.091 & \\ \hline \(L\) & 0.069 & III \\ \hline \(T_{\rm m}\) & 0.067 & \\ \hline \(m_{1}\) & 0.028 & \\ \hline \(m_{2}\) & 0.026 & IV \\ \hline \(k_{3}\) & 0.017 & IV \\ \hline \(k_{2}\) & 0.015 & \\ \hline \(c_{3}\) & 0.003 & \\ \hline \(c_{2}\) & 0.002 & V \\ \hline \(k_{1}\) & 0.001 & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Sensitivity rating of pantograph–catenary parameters. Figure 4: Convergence curve of the first-order and total sensitivities with sample size from \(2^{1}\) to \(2^{14}\).