# Dynamic Preventive Maintenance Optimization of Subway Vehicle Traction System Considering Stages
## 2 Maintenance Strategy Ideas
### The Idea of a Group Preventive Maintenance Strategy  
Suppose the system is composed of \(W\) key components. According to the single-component preventive maintenance strategy method, the preventive maintenance interval of key components is obtained. When the system runs to a certain time \(t\), the component \(l\) is preventively maintained. It is judged whether the component \(k\) needs preventive maintenance. The reliability of the component \(k\) reaches the preventive maintenance reliability threshold before time \(t+\Delta t\). In order to reduce the downtime loss caused by preventive maintenance, the component \(k\) should be preventively maintained at time \(t\), that is, opportunistic maintenance. If the number of preventive maintenance for the component \(k\) reaches \(N\), it is preventively updated for maintenance. After completing a preventive maintenance, the system continues to operate. The components \(l\) and \(k\) go to the next maintenance interval. Until a certain time \(t\), a component is preventively maintained again. At this time, it is also judged whether it is necessary to perform opportunistic maintenance on other components. Taking a simple system consisting of 2 components as an example, the specific process of the group preventive maintenance strategy is described, as shown in Figure 3. It is to clarify the idea of group preventive maintenance strategy. Suppose the system consists of components A and B. Its preventive maintenance intervals are \(T_{i}^{A}\) and \(T_{i}^{B}\), respectively. When a preventive maintenance program is carried out on either component A or B, it results in downtime of the system. Therefore, in the \((0,t_{6})\) period, according to the preventive maintenance plan of the components A and B, A is preventively maintained 5 times and B is preventively maintained 3 times. Then, the system will have 8 shutdowns. At time \(t_{4}\), the second preventive maintenance is performed on the component B. At this point, the reliability of the component A is below the opportunistic maintenance reliability threshold. According to the group preventive maintenance strategy, A is preventively maintained. The preventive maintenance schedule for the component A is advanced. After implementing a group preventive maintenance strategy on the system, the total number of system downtimes was reduced from 8 to 6, so as to achieve the purpose of reducing maintenance costs. Figure 2: Schematic diagram of a single-component preventive maintenance strategy. According to the group preventive maintenance strategy, the key components of the subway traction system are preventively maintained. By seeking the best \(\Delta t\) value, the total maintenance cost of the system is minimized. The established group preventive maintenance model is considered in a certain operating time of the system. Preventive maintenance plans that combine multi-component maintenance activities into groups are optimized, and components are required to meet specified reliability requirements. A group preventive maintenance strategy results in fewer system downtimes, so as to successfully complete the maintenance work and reduce maintenance costs. To build the model, the following assumptions are required:  
1. When the operational reliability of components does not reach the preventive maintenance threshold \(R_{m}\), preventive maintenance work can be carried out in advance, and preventive maintenance work can be carried out for multiple components at the same time. 2. Maintenance methods may be preventive maintenance, opportunistic maintenance, and preventive update maintenance. Components are restored to a \"non-new and non-old\" condition after preventive and opportunistic maintenance. The recovery result is jointly determined by the service age decreasing factor and the failure rate increasing factor. 3. Non-preventive failure maintenance does not lead to opportunistic maintenance. 4. There are a total of \(W\) key components.