## 2 Multiple Objective-Driven Energy-Saving Control Model of a Single Train
### Methodology  
#### 3.3.1 Modeling  
_Step 1_: The \"queuing theory\" is introduced to study the relationships among time-varying passenger flow prediction, arrival interval and dwell time. Service time is defined as train dwell time. Train arrival interval (\(\Delta T_{j}^{i}\)) is defined as vacation time. Therefore, trains that enter and leave the stations and passengers who embark/disembark belong to a service queuing model, i.e., a single vacation system in which every service enters a vacation. The average queue length of waiting passengers is defined as \(L\). Arrival interval is defined as \(\Delta T_{j}^{i}\). Dwell time is defined as \(d_{j}^{i}\). According to [PERSON], [PERSON] and [PERSON] [53]:  
\[L=\widetilde{\rho}+\frac{\lambda^{2}[b^{2}+\frac{2}{d_{j}^{i}}E(\Delta T_{j}^{ i})+E({(\Delta T_{j}^{i})}^{2})]}{2[1-\rho-\widetilde{\lambda}E(\Delta T_{j}^{i})]}+ \frac{\widetilde{\lambda}E({(\Delta T_{j}^{i})}^{2})}{2E(\Delta T_{j}^{i})} \tag{35}\]  
where \(\overline{\lambda}\) represents the input conversion parameter of passenger arrival rate \(\lambda\). \(\rho\) and \(\widetilde{\rho}\) both represent the service intensity, i.e., \(\rho=\frac{\widetilde{\lambda}}{\widetilde{\mu}}\) and \(\widetilde{\rho}=\rho+\widetilde{\lambda}E(\Delta T_{j}^{i})\). \(b^{2}\) represents the second moment of service time distribution. Considering Equation (26):  
\[Q_{up}^{i}=f_{i}(\Delta t_{i})=\int_{T_{j-1}^{i}+\Delta t_{i}+d_{j}^{i}}^{T_{j }^{i}+\Delta t_{i}}h_{i}(t)dt \tag{36}\]  
The average passenger arrival rate during the arrival interval is calculated by Equation (37). \[\overline{\lambda_{i}}=\frac{Q_{up}^{i}}{\Delta T_{j}^{i}}=\frac{\int_{T_{j- 1}^{i}+\Delta t_{i}+d_{j}^{i}}^{T_{j}^{i}+\Delta t_{i}}h_{i}(t)dt}{T_{j}^{i}- T_{j-1}^{i}-d_{j}^{i}} \tag{37}\]  
The average passenger waiting time is calculated by Equation (38). \[\Delta T=\frac{L_{i}}{\overline{\lambda_{i}}} \tag{38}\]  
The service time for one passenger is defined as \(Y_{k}\); the total service time for all arriving passengers is calculated by Equation (39). \[X(t)=\sum_{1}^{Q_{up}^{i}}Y_{k} \tag{39}\]  
As the dwell time of the train is certain, the service time of each passenger is the same, and is calculated by Equation (40). \[E\{X(t)\}=\lambda tE(Y_{k}) \tag{40}\]  
The average service time of passengers per unit time is calculated by Equation (41). \[H=\frac{E\{X(t)\}}{t} \tag{41}\]  
The minimum service intensity of the train is Ds/person within the dwell time of a train, which is calculated by Equation (42). \[\widetilde{\lambda}=\frac{H}{D}=\frac{\lambda E(Y_{k})}{D} \tag{42}\]If the arrival time of the train obeys the negative exponential distribution, the dwell time obeys the deterministic distribution, and the second moment is \(b^{2}=D^{2}\). Service time is \(D=u_{j}^{i}\). The arrival interval is expressed based on the actual arrival and departure times:  
\[E(\Delta T_{j}^{i})=u_{j}^{i}-u_{j-1}^{i}=T_{j}^{i}+\Delta t_{j}^{i}-T_{j-1}^{i} -\Delta t_{j-1}^{i} \tag{43}\]  
\[E[(\Delta T_{j}^{i})^{2}]=D(\Delta T_{j}^{i})+[E(\Delta T_{j}^{i})]^{2}=2(T_{j }^{i}+\Delta t_{j}^{i}-T_{j-1}^{i}-\Delta t_{j-1}^{i})^{2} \tag{44}\]  
Equations (36)-(44) are substituted into Equation (45), giving Equation (45). \[\Delta T =\frac{1}{\lambda}\left\{ \begin{array}{l}\frac{\widetilde{ \lambda}}{d_{j}^{i}}+\widetilde{\lambda}(T_{j}^{i}+\Delta t_{j}^{i}-T_{j-1}^{i }-\Delta t_{j-1}^{i})+\frac{2\widetilde{\lambda}(T_{j}^{i}+\Delta t_{j}^{i}-T _{j-1}^{i}-\Delta t_{j-1}^{i})^{2}}{2(T_{j}^{i}+\Delta t_{j}^{i}-T_{j-1}^{i}- \Delta t_{j-1}^{i})}\\ +\frac{\left(\widetilde{\lambda}\right)^{2}([d_{j}^{i})^{2}+2(T_{j}^{i}+ \Delta t_{j}^{i}-T_{j-1}^{i}-\Delta t_{j-1}^{i})^{2}+2(T_{j}^{i}+\Delta t_{j} ^{i}-T_{j-1}^{i}-\Delta t_{j-1}^{i})^{2}}{2[1-\frac{\widetilde{\lambda}}{d_{j }^{i}}-\widetilde{\lambda}(T_{j}^{i}+\Delta t_{j}^{i}-T_{j-1}^{i}-\Delta t_{j -1}^{i})]}\end{array} \right\} \tag{45}\] \[=P(\widetilde{\lambda},\Delta t_{j}^{i})\]  
where the service time is the minimum dwell time and the number of passengers that the train can serve is \(Q\).