## 3 Pantograph System Reliability Analysis Model
### Dynamic Bayesian Network  
This paper's reliability data are obtained from a city metro 2018-2021 vehicle fault log and metro vehicle equalization repair manual for the whole network. The failure rate \(\lambda_{1}\), \(\lambda_{2}\), and the state repair rate \(u\) for equipment states from 0 to 0.5 and 0 to 1 are collated in Figure 2. In addition, the escalation of faults during equipment operation states from 0.5 to 1 cannot be observed practically, so \(\lambda_{3}=\lambda_{1}\)[14] is assumed, and the specific parameters are shown in Table 7.  
According to the principle of constructing a Bayesian network for T-S polymorphic fault tree construction in Figure 2, the T-S polymorphic fault tree of a pantograph system (Figure 6) is transformed into a DBN (shown in Figure 7). The conditional probability  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Rule \(l\)**} & \multirow{2}{*}{\(\mathbf{x_{6}}\)} & \multirow{2}{*}{\(\mathbf{x_{7}}\)} & \multirow{2}{*}{\(\mathbf{x_{8}}\)} & \multicolumn{3}{c}{\(\mathbf{y_{3}}\)} \\ \cline{5-6}  & & & & \(\mathbf{y^{b_{j}}=0}\) & \(\mathbf{y^{b_{j}}=0.5}\) & \(\mathbf{y^{b_{j}}=1}\) \\ \hline
1 & 0 & 0 & 0 & 1 & 0 & 0 \\
2 & 0 & 0 & 0.5 & 0.761 & 0.136 & 0.103 \\
3 & 0 & 0 & 1 & 0.131 & 0.508 & 0.361 \\
4 & 0 & 0.5 & 0 & 0.769 & 0.131 & 0.100 \\
5 & 0 & 0.5 & 0.5 & 0.200 & 0.644 & 0.156 \\
6 & 0 & 0.5 & 1 & 0.097 & 0.527 & 0.376 \\
7 & 0 & 1 & 0 & 0.197 & 0.496 & 0.307 \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\
27 & 1 & 1 & 1 & 0 & 0 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 6: T-S gate 4 rules.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Event**} & \multicolumn{2}{c}{**Failure Transfer Rate/h**} & \multicolumn{2}{c}{**Maintenance Rate/h**} \\ \cline{2-4}  & \(\mathbf{\lambda_{1}}\) & \(\mathbf{\lambda_{2}}\) & \(\mathbf{u}\) \\ \hline \(x_{1}\) & \(5.492\times 10^{-6}\) & \(4.145\times 10^{-7}\) & 0.004006 \\ \(x_{2}\) & \(3.938\times 10^{-6}\) & \(3.11\times 10^{-7}\) & 0.004004 \\ \(x_{3}\) & \(4.87\times 10^{-6}\) & \(4.145\times 10^{-7}\) & 0.004006 \\ \(x_{4}\) & \(5.907\times 10^{-6}\) & \(16.58\times 10^{-7}\) & 0.004007 \\ \(x_{5}\) & \(3.731\times 10^{-6}\) & \(10.363\times 10^{-7}\) & 0.004004 \\ \(x_{6}\) & \(5.389\times 10^{-6}\) & \(5.181\times 10^{-7}\) & 0.001339 \\ \(x_{7}\) & \(8.808\times 10^{-6}\) & \(7.254\times 10^{-7}\) & 0.001343 \\ \(x_{8}\) & \(12.23\times 10^{-6}\) & \(10.363\times 10^{-7}\) & 0.004013 \\ \(x_{9}\) & \(3.109\times 10^{-6}\) & \(1.036\times 10^{-7}\) & 0.004004 \\ \(x_{10}\) & \(3.523\times 10^{-6}\) & \(1.036\times 10^{-7}\) & 0.004006 \\ \(x_{11}\) & \(5.596\times 10^{-6}\) & \(3.109\times 10^{-7}\) & 0.004007 \\ \(x_{12}\) & \(4.56\times 10^{-6}\) & \(2.073\times 10^{-7}\) & 0.004005 \\ \(x_{13}\) & \(2.964\times 10^{-6}\) & \(1.036\times 10^{-7}\) & 0.001337 \\ \(x_{14}\) & \(2.383\times 10^{-6}\) & \(3.109\times 10^{-7}\) & 0.004004 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Pantograph basic component reliability parameters.  
table of each node in the DBN can be output according to the T-S gate rule constructed in Section 2.2. The interval \(\Delta t\) is set to 1 day in the DBN inference process. Each device's initial time interval \(t\) = 0 is entirely reliable, i.e., the prior probability of the root node is 1.