## 4 Reliability Analysis of Pantograph System  
### System Reliability Assessment  
A pantograph system's reliability reflects the ability of the system components and subsystems to maintain regular operation within a specified period, and the DBN model can be used to positively infer the immediate reliability of the system to measure the system reliability. According to the equipment's failure rate and maintenance rate in Table 7, substitute the formula in Table 2 to obtain the state transfer table without and with the maintenance of each root node of DBN during the vehicle operation, respectively. The state transfer table without maintenance and with maintenance will be substituted into the root node, and the operation time will be set to obtain the system reliability curve with time, as shown in Figure 8. The system reliability of a pantograph system declined continuously during 720 days of operation. On day 720, the system reliability dropped to 0.881 for the system without maintenance and 0.952 for the system with maintenance.  
The dynamic reliability of the pantograph, arch frame, electrical parts, pneumatic parts, arch body, and frame are compared and analyzed, and the results are shown in Figure 9.  
By comparing the reliability of the overall system and intermediate components without and with maintenance in Figures 8 and 9, it is found that the reliability with maintenance is significantly higher than the reliability without maintenance. Therefore, different overhaul cycles can be formulated according to the reliability change curves of different sub-components so as to improve the maintenance rate of the system and reduce the probability of upgrading the fault state, thus improving the reliability of a pantograph system. The results of this reliability analysis can be used as a reference for the subsequent optimization of the maintenance of system components.  
Figure 10 shows that equipment failure rates change individually to affect system reliability in the following order: \(x_{4}\)Drop bow indicator \(>x_{5}\)Electrical control box \(>x_{3}\)Lifting bow motor \(>x_{6}\)Valve box \(>x_{8}\)Solenoid Valve \(>x_{7}\)Lifting bow cylinder \(>x_{1}\)Underframe insulator \(>x_{2}\)Lifting bow spring assembly \(>x_{11}\)Horn \(>x_{10}\)Spring Box \(>x_{12}\)Upper and lower frames \(>x_{9}\)Bracket \(>x_{13}\)Bearing \(>x_{14}\)Damper. The weak equipment is the drop bow indicator, the electrical control box, and the lifting bow motor.  
A comparative analysis of the serviced and non-serviced systems showed that the change in reliability of the serviced system was less than the change in reliability of the non-serviced system when the failure rate of an identical device was changed to 120% or 80%. Therefore, for a maintenance-free system, the change in the failure rate of a single device has a more significant impact on its reliability.  
Figure 8: Pantograph system reliability. (**a**) No maintenance. (**b**) Maintenance.  
Birnbaum's probabilistic importance is one of the important parameters in reliability analysis. It is used to describe the impact of the state change of each root node in the network on the overall reliability of the system. Birnbaum probabilistic importance is based on the static failure behavior, and based on the method of calculating the Birnbaum probability of importance of the T-S polymorphic fault tree [6], we obtain the Birnbaum probabilistic importance from \(x_{1}\)\(\sim\)\(x_{14}\), which are shown in Table 8.  
The events with greater probabilistic importance are \(x_{4}\)Drop bow indicator, \(x_{5}\)Electrical control box, and \(x_{3}\)Lifting bow motor. The comparison in Figure 9 reveals that the results of the weak components of the system obtained by sensitivity analysis and [PERSON]'s probabilistic importance analysis are consistent.  
Figure 9: Pantograph mid-component reliability. (**a**) No maintenance. (**b**) Maintenance.