## 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
In order to ensure the punctual arrival of the train and maximize the transportation efficiency, the train will be restricted by a minimum speed, \(v_{\min}\), during traction operation. When the train runs in coasting conditions, the train speed is close to the minimum speed, \(v_{\min}\). At this time, the train should be accelerated immediately to ensure that the running speed is greater than \(v_{\min}\), to guarantee that the train can reach the next station on time. * Train dwell time constraint  
When a train enters a station and stops, it needs to be given some time for passengers to fully disembark/embark before moving to the next station. Thus, the train dwell time (defined as \(d_{j}^{i}\)) should meet the following conditions. \[d_{\min}\leq d_{j}^{i}\leq d_{\max} \tag{30}\]Passenger waiting time constraint  
Considering the complex psychological characteristics of passengers, the train should meet their traveling and psychological needs. Therefore, the maximum waiting time that passengers can accept at the platform is defined as \(\left(\Delta t_{q}\right)_{\text{max}}\), i.e., passenger waiting time (defined as \(\Delta t_{q_{k}}^{i}\)) is calculated by Equation (31). \[0<\Delta t_{q_{k}}^{i}\leq\left(\Delta t_{q}\right)_{\text{max}} \tag{31}\]  
_Step 3_: Modeling  
Considering the cost benefit of the two main bodies, namely the operator and the passenger, a single train combination optimization model is established that aims to balance the traction energy consumption and the average passenger waiting time. The model control objectives are summarized as follows. \[ \begin{split}\min J&=\alpha\cdot E_{j}+\beta\cdot \overline{\Delta t_{q_{k}}^{i}}=\alpha\cdot H(\Delta t_{i},t_{i}^{1},t_{i}^{2},t_ {i}^{3},t_{i}^{4})+\beta\cdot K(\Delta t_{i})\\ s.t.&\left\{\begin{array}{l}\Delta t_{i}\in(0,60)\\ t_{i}^{1}+t_{i}^{2}+t_{i}^{3}+t_{i}^{4}+t_{B}\leq T_{j}^{i}+\Delta t_{i}\\ v_{\text{min}}\leq v_{s^{n}}\leq v_{\text{max}}\\ d_{\text{min}}\leq d_{i}^{j}\leq d_{\text{max}}\\ 0<\Delta t_{q_{k}}^{i}\leq\left(\Delta t_{q}\right)_{\text{max}}\end{array}\right.\end{split}  \tag{32}\]  
* _Step 4_: The model is solved by GA.  
_Step 4.1_: The process is initialized and basic data are input, including line name, train data model parameters, single side slope, curve radius, speed limit of running line, etc. _Step 4.2_: The control model and its objective function are determined based on Equation (32). _Step 4.3_: The encoding mode is determined. The real number encoding mode represents the conversion point \(s_{i}^{n}\) in train operating conditions. Since \(s_{i}^{n}\) is a model variable, the conversion point is represented according to genes. The chromosome represents the sequence of conversion points. A chromosome contains multiple genes: \(S_{I}=\left\{s_{i}^{1},s_{i}^{2},\cdots,s_{i}^{n}\right\}\). _Step 4.4_: The reciprocal of the objective function is selected as the fitness function to dynamically search the internal information, and then the individual evaluation is conducted. \[fit=\frac{1}{\alpha\cdot E_{j}+\beta\cdot\overline{\Delta t_{q_{k}}^{i}}} \tag{33}\]  
_Step 4.5_: GA is designed according to roulette selection and adaptive crossover/mutation operation. _Step 4.6_: Operation parameters, including population size, iteration times, selection operator, crossover operator, mutation operator, etc., are determined. _Step 4.7_: It is judged whether genetic algebra has been reached. If reached, the result is output and the algorithm ends. Otherwise, _Step 4.4_ should be repeated and the calculation should be continued. The flowchart of GA is shown in Figure 2.  
Multi-Train Cooperative Energy-Saving Control Model Based on Multi-Objective Combinatorial Optimization