## 2 Multiple Objective-Driven Energy-Saving Control Model of a Single Train
### Energy-Saving Control Model of a Single Train
#### 2.2.4 Single-Train Control Model  
The process of urban rail transit operation can be divided into two main bodies: passengers and operators. From the perspective of passengers, there is a \"benefit paradox\" relationship between the high demand of passengers for a certain service level and train operation energy consumption. Passengers always hope to board the train as soon as it arrives as well as for the waiting time at the platform to be shortened. However, such hopes lead to an increase in train running speed and traction energy consumption, which, in turn, increases operation cost. From the perspective of the operator, the operator wants to appropriately reduce the train's running speed and traction energy consumption in order to lower operation costs while transporting as many passengers as possible and ensuring the safety of train operation. The contradiction between supply and demand between the two main bodies is becoming increasingly prominent and influential. Therefore, this paper comprehensively considers strategies to meet the high service level requirements for passengers and co-optimize the traction energy consumption of trains. _Step 1_: Control optimization objectives  
* Average passenger waiting time  
In an actual situation, when passengers arrive at the platform to wait for the train, they automatically disperse according to the current queuing situation. Finally, the number of people queuing in front of each door tends to be balanced. Assuming that the travel time of passengers between the gate and the platform is ignored, the passenger waiting time is calculated as follows. The departure time of the train at the \(i\)th platform is defined as \(r_{j}^{i}\), which is calculated by Equation (23). \[r_{j}^{i}=T_{j}^{i}+\Delta t_{i}+d_{j}^{i} \tag{23}\]  
Similarly, the departure time of the previous train at the \(i\)th platform is defined as \(r_{j-1}^{i}\), which is shown as Equation (24). \[r_{j-1}^{i}=T_{j-1}^{i}+\Delta t_{i}+d_{j}^{i} \tag{24}\]The maximum passenger waiting time at the \(i\)th platform is calculated by Equation (25). \[\Delta T_{j}^{i}=T_{j}^{i}+\Delta t_{i}-r_{j-1}^{i}=T_{j}^{i}+t_{i}-(T_{j-1}^{i}+ \Delta t_{i}+d_{j}^{i})=T_{j}^{i}-T_{j-1}^{i}-d_{j}^{i} \tag{25}\]  
The number of passengers waiting to board the train at the \(i\)th platform is the number of passengers arriving during the time difference between the departure of the previous train from the platform and the arrival of this train at the platform. \[Q_{up}^{i}=f_{i}(\Delta t_{i})=\int_{1}^{T_{j}^{i}+\Delta t_{i}}h_{i}(t)dt \tag{26}\]  
The time when the \(k\)th passenger arrives at the \(i\)th platform is defined as \(\Delta t_{q_{k}}^{i}\); the passenger waiting time at the \(i\)th platform is calculated by Equation (27). \[\Delta t_{q_{k}}^{i}=T_{j}^{i}+\Delta t_{i}-t_{q_{k}}^{i} \tag{27}\]  
Therefore, the average passenger waiting time is calculated by Equation (28). \[\frac{\sum\limits_{k=1}^{Q_{up}^{i}}\Delta t_{q_{k}}^{i}}{Q_{up}^{i}}=\frac{ \sum\limits_{k=1}^{T_{j}^{i}+\Delta t_{i}}h_{i}(t)dt}{\int_{1}^{T_{j}^{i}+ \Delta t_{i}}h_{i}(t)dt} \tag{28}\]  
* Traction energy consumption  
The total traction energy consumption of the train on the whole line is the comprehensive energy consumption between every two adjacent stations. It consists of traction energy consumption and cruise energy consumption. Assuming that the train line has \(I\)-1 station spacing, it can be seen from Equation (22) that the total traction energy consumption of the \(j\)th train on this line is calculated by Equation (29). \[E_{j}=\sum\limits_{1}^{I-1}E^{3}=\sum\limits_{1}^{I-1}H(\Delta t_{i},t_{i}^{1 },t_{i}^{2},t_{i}^{3},t_{i}^{4}) \tag{29}\]  
_Step 2_: Constraint condition  
To ensure safe train operation, the train operating status needs to be subject to the following constraints:  
* Speed constraint  
According to multiple factors, such as the physical conditions of the train line, train operating safety and passenger comfort, the train operation will be limited by the maximum speed, \(v_{\max}\).