## 2 Multiple Objective-Driven Energy-Saving Control Model of a Single Train
### Energy-Saving Control Model of a Single Train
#### 2.2.3 Energy Consumption Calculation  
Most studies only calculate the traction energy consumption of the train under the four operating conditions while ignoring the secondary traction, which reduces the accuracy of the results. Thus, due to comprehensive consideration of secondary traction, it is assumed that the train operation process consists of six stages in Figure 1: traction, coasting, secondary traction, cruise, coasting and braking. The energy consumption is calculated, respectively, as follows.  
_Step 1_: Traction stage 0-\(s_{1}\): the train starts with a certain acceleration traction to overcome the resistance, resulting in traction energy consumption.  
\[F_{s_{i}^{1}}-m\cdot(1+\gamma)\cdot g\cdot W=m\cdot(1+\gamma)\cdot a_{s_{i}^{1}} \tag{13}\]  
\[v=a_{s_{i}^{1}}\cdot t_{i}^{1} \tag{14}\]  
\[\begin{array}{ll}E^{s_{1}^{1}}=\int_{0}^{t_{1}^{1}}F_{s_{i}^{1}}\cdot vdt&= \int_{0}^{t_{1}^{1}}[m(1+\gamma)(g\cdot W+a_{s_{i}^{1}})]a_{s_{i}^{1}}\cdot tdt \\ &=\int_{0}^{t_{1}^{1}}M(\Delta t)\cdot(1+\gamma)(g\cdot W+a_{s_{i}^{1}})\cdot a _{s_{i}^{1}}\cdot dt\end{array} \tag{15}\]  
where \(\gamma\) is the rotary mass coefficient. \(\gamma=0.08\).  
_Step 2_: Coasting stage \(s_{1}\)-\(s_{2}\): the train is only affected by resistance during this process. The speed gradually decreases, and there is no traction energy consumption.  
_Step 3_: Secondary traction stage \(s_{2}\)-\(s_{3}\): the train tracts for the second time with a certain acceleration, causing traction energy consumption.  
\[v_{s_{i}^{2}}=v_{s_{i}^{1}}-a_{s_{i}^{1}}t_{i}^{2} \tag{16}\]  
\[v_{s_{i}^{3}}=v_{s_{i}^{2}}+a_{s_{i}^{1}}\cdot t_{i}^{3}=v_{s_{i}^{1}}-a_{s_{i }^{1}}t_{i}^{2}+a_{s_{i}^{3}}\cdot t_{i}^{3} \tag{17}\]  
\[F_{s_{i}^{3}}-m\cdot(1+\gamma)\cdot g\cdot W=m\cdot(1+\gamma)\cdot a_{s_{i}^{ 3}} \tag{18}\]  
\[\begin{array}{ll}E^{s_{i}^{3}}&=\int_{t_{1}^{2}}^{t_{1}^{3}}F_{s_{i}}\cdot vdt =\int_{t_{1}^{2}}^{t_{2}^{3}}m\cdot(1+\gamma)\cdot(g\cdot W+a_{s_{i}^{3}}) \cdot(v_{s_{i}^{1}}-a_{s_{i}^{1}}\cdot t_{i}^{2}+a_{s_{i}^{3}}\cdot t_{i}^{3} )dt\\ &=\int_{t_{1}^{2}}^{t_{2}^{3}}M(\Delta t)\cdot(1+\gamma)\cdot(g\cdot W+a_{s_{i }^{3}})\cdot(v_{s_{i}^{1}}-a_{s_{i}^{1}}\cdot t_{i}^{2}+a_{s_{i}^{3}}\cdot t_{ i}^{3})dt\end{array} \tag{19}\]  
Figure 1: Schematic diagram of train operating conditions.  
_Step 4_: Cruise stage \(s_{3}\)-\(s_{4}\): the resultant force on the train is zero. At the same time, the train runs at a constant speed of \(v_{s_{i}^{3}}\). The traction energy consumption is calculated according to the resistance of the train.  
\[F_{s_{i}^{4}}=m\cdot(1+\gamma)\cdot g\cdot W=M(\Delta t)\cdot(1+\gamma)\cdot g\cdot W \tag{20}\]  
\[E^{s_{i}^{4}}=\int_{t_{i}^{2}}^{t_{i}^{4}}F_{s_{i}^{4}}\cdot vdt=\int_{t_{i}^{3 }}^{t_{i}^{4}}M\cdot(1+\gamma)\cdot g\cdot W\cdot v_{s_{i}^{3}}dt \tag{21}\]  
_Step 5_: Coasting stage \(s_{4}\)-\(s_{5}\): the train is only affected by resistance in this process and performs a deceleration motion; the traction energy consumption is zero.  
_Step 6_: Breaking stage \(s_{5}\)-\(s_{6}\): the train is only affected by two forces, resistance and braking force, and the traction force does not do any work, so the traction energy consumption is zero.  
In summary, the energy consumption of train operation in this zone, defined as \(E^{3}\), is the sum of the energy consumption of operation, consisting of primary traction, secondary traction and the cruising stage, which is calculated by Equation (22).  
\[\begin{split} E^{3}=E^{s_{i}^{1}}+E^{s_{i}^{3}}+E^{s_{i}^{4}}& =\int_{0}^{t_{1}^{1}}M(\Delta t)\cdot(1+\gamma)(g\cdot W+a_{s_{i}^{1} })\cdot a_{s_{i}^{1}}\cdot tdt\\ &+\int_{t_{i}^{2}}^{t_{i}^{3}}M(\Delta t)\cdot(1+\gamma)\cdot(g \cdot W+a_{s_{i}^{3}})\cdot(v_{s_{i}^{1}}-a_{s_{i}^{2}}\cdot t_{i}^{2}+a_{s_{i }^{3}}\cdot t_{i}^{3})dt\\ &+\int_{t_{i}^{3}}^{t_{i}^{4}}M(\Delta t)\cdot(1+\gamma)\cdot g \cdot W\cdot(a_{s_{i}^{1}}\cdot t_{i}^{1}-a_{s_{i}^{2}}\cdot t_{i}^{2}+a_{s_{ i}^{3}}\cdot t_{i}^{3})dt\\ &=H(\Delta t_{i},t_{i}^{1},t_{i}^{2},t_{i}^{3},t_{i}^{4})\end{split} \tag{22}\]