# Dynamic Preventive Maintenance Optimization of Subway Vehicle Traction System Considering Stages
## 3 Single-Component Maintenance Model
### Preventive Maintenance Strategy Modeling
The total cost of component maintenance from time zero to preventive update maintenance is composed of preventive maintenance costs, non-preventive failure maintenance costs, and preventive update maintenance costs, namely:  
\[C=CP+CF+CU \tag{6}\]  
\[CF=\left(C_{f}+C_{lf}\right)\sum_{i=1}^{N}\int_{0}^{T_{i}}\lambda_{i}(t)dt= \left(C_{f}+C_{lf}\right)N(-\ln R_{m}) \tag{7}\]  
where \(C_{f}\) is the maintenance cost required to perform a non-preventive maintenance; \(C_{lf}\) is the loss caused by a non-preventive maintenance; \(N\) is the preventive maintenance times on the component before the preventive update in the life cycle, \(i=1,2,\cdots,N\). From Formulas (5)-(7), the total maintenance cost in the life cycle of the component can be obtained as:  
\[C=N\left(C_{p}+C_{lp}\right)+N\left(C_{f}+C_{lf}\right)(-\ln R_{m})+C_{u} \tag{8}\]  
where \(C_{p}\) is the maintenance cost required to perform a preventive maintenance; \(C_{lp}\) is the loss caused by a preventive maintenance; \(C_{u}\) is the cost of a preventive update maintenance. The total time \(T\) from the start time to the preventive update maintenance time is composed of the component normal operation time \(TI\), preventive maintenance time \(TP\), non-preventive failure maintenance time \(TF\), and preventive update maintenance time \(TU\), namely:  
\[T=TI+TP+TF+TU \tag{9}\]\[T=\sum_{i=1}^{N}T_{i}+NT_{p}+NT_{f}(-\ln R_{m})+T_{u} \tag{10}\]  
where \(T_{i}\) is each preventive maintenance interval, \(i=1,2,\cdots,N\); \(T_{p}\) is the time required to perform a preventive maintenance; \(T_{f}\) is the time required to perform a non-preventive maintenance; \(T_{u}\) is the time required to perform a preventive update maintenance. At present, the minimum unit in the maintenance schedule formulated by subway operation companies is \"day\" as the unit. Therefore, all the time \(T\) in this article is in \"days\". To sum up, the preventive maintenance cost per unit time for a single component is:  
\[\overline{C}=\frac{C}{T}=\frac{N\Big{(}C_{p}+C_{lp}\Big{)}+N\Big{(}C_{f}+C_{lf }\Big{)}(-\ln R_{m})+C_{u}}{\sum_{i=1}^{N}T_{i}+NT_{p}+NT_{f}(-\ln R_{m})+T_{u}} \tag{11}\]  
Each preventive maintenance interval \(T_{i}\) of the key components can be calculated from Formulas (1)-(5). After \(T_{i}\) is determined, given the preventive maintenance threshold \(R_{m}\), there is only one variable in the model. It is the preventive maintenance times \(N\). Each \(N\) value corresponds to a \(\overline{C}\) value. By minimizing the objective function \(\overline{C}\), the optimal maintenance plan in the life cycle of key components can be obtained, and the optimal preventive maintenance times \(N\) can be calculated. The optimization model of preventive maintenance strategy for the key components in subway traction system is as follows:  
\[\begin{split}&\min\bigg{\{}\overline{C}=\frac{N\big{(}C_{p}+C_{lp} \big{)}+N\big{(}C_{f}+C_{lf}\big{)}(-\ln R_{m})+C_{u}}{\sum_{i=1}^{N}T_{i}+NT_{ f}+NT_{f}(-\ln R_{m})+T_{u}}\bigg{\}}\\ & s.t.\bigg{\{} \begin{split}&\int_{0}^{T_{i}} \lambda_{i}(t)dt=-\ln R_{m}\\ & R_{m}>0,N>0,T_{i}>0,i=1,2,3,\cdots,N\end{split}  \tag{12}\]