## 2 Multiple Objective-Driven Energy-Saving Control Model of a Single Train
### Passenger Flow Prediction of URT
#### 2.1.1 The Change-Point Algorithm  
The passenger flow data are random in nature; these data present continuous variation in a sawtooth shape. Some change points are ignored when predicting directly with the original data, which affects the prediction accuracy. Therefore, the set of change points is defined as \(b(w)\). The change-point algorithm is used to obtain the \(b(w)\) in the time series of passenger flow. The change-point algorithm can improve the fitting accuracy of the model by dividing the time series of passenger flow into time periods with different characteristics based on qualitative changes in the observed function values at a certain time point. The specific steps of the algorithm are as follows:  
_Step 1_: The cycle number is set as \(u\), the number of change point as \(w\) and the auxiliary \(a\) variable as \(e\). It is supposed that \(u=1\), \(w=1\), \(e=1\), \(\beta_{j}=l_{j}\) (\(j=1\), \(2\), \(\ldots\), k), where \(l_{j}\) is the valley point on the peak curve (or the peak point of the valley curve). _Step 2_: On the curve, \(\beta_{u}\) is the starting point. Two convex waves are taken along the direction of the time axis, which are denoted as \((\beta_{u}-\beta_{u+1}-\beta_{u+2})\). The wave variance of \(b(w)\) is calculated within this range. Each point in the wave is taken as the middle point, and then evaluated. The next step is to calculate the maximum and minimum values of \(t^{*}\). \[Y(\beta_{u},\beta_{u+1},\beta_{u+2})=\sum_{t=\beta_{u}}^{\beta_{u+1}}\left(x_{ t}-\overline{x}\Big{|}_{\beta_{u}}^{\beta_{u+1}}\right)^{2}+\sum_{t=\beta_{u+1}+1}^{ \beta_{u+2}}\left(x_{t}-\overline{x}\Big{|}_{\beta_{u+1}+1}^{\beta_{u+2}} \right)^{2} \tag{1}\]  
\[\max(\beta_{u},t_{1}{}^{*},\beta_{u+2})=\max\left[\sum_{t=\beta_{u}}^{t}\left( x_{t}-\overline{x}\Big{|}_{\beta_{u}}^{t}\right)^{2}+\sum_{t=t+1}^{\beta_{u+2}} \left(x_{t}-\overline{x}\Big{|}_{t+1}^{\beta_{u+2}}\right)^{2}\right] \tag{2}\]  
\[\min(\beta_{u},t_{2}{}^{*},\beta_{u+2})=\min\left[\sum_{t=\beta_{u}}^{t}\left( x_{t}-\overline{x}\Big{|}_{\beta_{u}}^{t}\right)^{2}+\sum_{t=t+1}^{\beta_{u+2}} \left(x_{t}-\overline{x}\Big{|}_{t+1}^{\beta_{u+2}}\right)^{2}\right] \tag{3}\]where \(x_{t}\) is the value of the passenger flow time series at time \(t\). \({t_{1}}^{*}\) represents the maximum values. \({t_{2}}^{*}\) represents the minimum values. _Step 3_: Change point estimation:  
\[R_{u}=\frac{2Y(\beta_{u},\beta_{u+1},\beta_{u+2})}{\max(\beta_{u},{t_{1}}^{*}, \beta_{u+2})}+\min(\beta_{u},{t_{2}}^{*},\beta_{u+2}) \tag{4}\]  
\[c_{ru}=\frac{\max(\beta_{u},{t_{1}}^{*},\beta_{u+2})}{\min(\beta_{u},{t_{2}}^{ *},\beta_{u+2})} \tag{5}\]  
_Step 3.1_: If \(\max(\beta_{u},{t_{1}}^{*},\beta_{u+2})\approx\min(\beta_{u},{t_{2}}^{*}, \beta_{u+2})\), when \(1\leq c_{ru}<1.3\), the time series of passenger flow is slightly changed. There is no need to delineate the change point in the range of \((l_{u}l_{u+2})\). _Step 3.2_: If \(R_{u}\leq\frac{1}{4}+\frac{3}{2}(2c_{ru}+1)\), whether it satisfies \({t_{1}}^{*}=\beta_{u+1}\) or not, then \(b^{(w)}=\beta_{u+1}\), where \(b^{(w)}\) represents a set of change points on the curve. _Step 3.3_: If \(\frac{1}{4}+\frac{3}{2}(2c_{ru}+1)\leq R_{u}\leq 1\), search for \(\beta_{u+1}\). _Step 3.4_: If \(\exists t_{c}\in\left(\beta_{u+1}^{L},\beta_{u+1}^{R}\right)\), the following condition satisfies \(Dev(\beta_{u},t_{c},\beta_{u+2})=\min\left[Dev(\beta_{u},\beta_{u+1}^{L}, \beta_{u+2}),Dev(\beta_{u},\beta_{u+1}^{R},\beta_{u+2})\right]\), where \(Dev(\cdot)\) indicates variance. \(b^{(w)}=t_{c}\) can be obtained. _Step 3.5_: If \(R_{u}>1\), so get \(b^{(w)}=t_{2}^{*}\). The validity of \(b^{(w)}\) is determined; if \(T(w-1)>\varepsilon+1\), the change point is retained. Otherwise, it is deleted. _Step 3.6_: If \(b^{(w)}\) exists, so \(u=u+1\), \(b^{(w)}=\beta_{u}\), \(\beta_{u+1}=\beta_{u+1}\), \(\beta_{u+2}=\beta_{u+2}\), \(w=w+1\), \(e=0\). Otherwise, \(e=1\), \(\beta_{u}=\beta_{u}\), \(\beta_{u+1+e}=\beta_{u+1+e}\), \(\beta_{u+2+e}=\beta_{u+2+e}\).