## 2 Multiple Objective-Driven Energy-Saving Control Model of a Single Train
### Energy-Saving Control Model of a Single Train
#### 2.2.2 Operating Condition Analysis  
In the force analysis of the train, it can be clearly seen that under the action of different forces, the train travels in different operating conditions. The operating condition of a train during operation can be divided into four types: traction, cruise, coasting and braking. In the traditional sense, there are only three train operating conditions: traction, coasting and braking. Traction conditions include primary traction, secondary traction and multiple traction. Cruising conditions mean that the running resistance and traction force of the train are balanced without resultant force, and the train runs at a fixed speed. Thus, the resultant force is analyzed under each operating condition.  
\[F=\begin{cases}F_{traction}-W&traction\;operating\;condition\\ 0&cruise\;operating\;condition\\ -W&coasting\;operating\;condition\\ -(B+W)&braking\;operating\;condition\\ \end{cases} \tag{12}\]  
where \(F_{traction}\) is the traction; \(W\) is resistance; \(B\) is braking force.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(i\) & The station and station spacing number \\ \(j\) & The train number \\ \(Q_{up}^{i}\) & The number of passengers getting on at the \(i\)th station \\ \(Q_{off}^{i}\) & The number of passengers getting off at the \(i\)th station \\ \(Q_{i}^{i}\) & The number of passengers on the train after departure from the \(i\)th station \\ \(T_{j}^{i}\) & The scheduled operation time of the \(j\)th train to the next station \\ \(\Delta t_{i}\) & The optimization time at the \(i\)th station \\ \(m\) & The total weight of the train \\ \(m_{p}\) & The weight of the train \\ \(m_{Q_{i}}\) & The weight of the passengers on the train \\ \(S_{i}\) & The distance between the \(i\)th and \(i+1\)th stations \\ \(s_{i}^{h}\) & The position of the \(n\)th operating condition conversion point at the \(i\)th station, \(n\in N\) \\ \(t_{i}^{n}\) & The conversion time of the \(n\)th operating condition at the \(i\)th station spacing, \(n\in N\) \\ \(v_{s_{i}^{n}}\) & The speed at which the position of the \(n\)th operating condition conversion point at the \(i\)th station \\ \(a_{s_{i}^{n}}\) & The acceleration at which the position of the \(n\)th operating condition conversion point at the \(i\)th station \\ \(t_{B}\) & The time between the train braking with maximum braking force and stopping \\ \(F_{s_{i}^{n}}\) & The force of the \(n\)th operating condition at the \(i\)th station spacing \\ \(E^{i_{i}^{n}}\) & The energy consumption of the \(n\)th operating condition at the \(i\)th station spacing \\ \(d_{j}^{i}\) & The stopping time of the \(j\)th train at the \(i\)th station \\ \(u_{j}^{i}\) & The arrival time of the \(j\)th train at the \(i\)th station \\ \(r_{j}^{i}\) & The departure time of the \(j\)th train at the \(i\)th station \\ \(\Delta T_{j}^{i}\) & The maximum passenger waiting time at the \(i\)th platform \\ \(t_{i_{B}}^{i}\) & The time of arrival of the \(k\)th passenger at the \(i\)th platform, \(k=1,2,\cdots,Q_{up}^{i}\) \\ \(\Delta t_{q_{k}}^{i}\) & The waiting time of the \(k\)th passenger at the \(i\)th platform \\ \(F_{traction}\) & The traction \\ \(W\) & The resistance \\ \(B\) & The braking force \\ \hline \hline \end{tabular}
\end{table}
Table 1: Notation explanation.