# Coordinated Control Method of Bus Signal Priority and Speed Adjustment Based on Stop-Skipping
## 3 Model Building
### Objective Function  
This research establishes a model with the lowest passenger time cost and the lowest company stopping cost as the optimization goal. The model is divided into five parts:  
(1) The waiting time  
The running time between stations is not considered here. The waiting time is mainly the stopping time and the acceleration and deceleration time of the vehicle at the station. Due to the random arrival of passengers at each station, the waiting station coefficient \(\eta\) is set here. \[T_{1}=\sum_{i=1}^{m}\sum_{j=1}^{n}\left[D_{i,j}^{p}\cdot\eta\left(\tau_{i,j}+ \mu\right)\cdot Z_{i,j}\right] \tag{1}\]  
Among them, the stop time at the station:  
\[\tau_{i,j}=\left\{ \begin{array}{c}0;Z_{i,j}=0\\ \max\left\{D_{i,j}^{p}\cdot\overline{t},A_{i,j}^{p}\cdot\overline{t}\right\};Z _{i,j}=1\end{array} \right. \tag{2}\]  
It can be seen that the stop time \(\tau_{i,j}\) depends on \(Z_{i,j}\) and the number of passengers entering and exiting. (2) The extra waiting time for the sites to be crossed  
The extra waiting time is the sum of the headway of the skipping vehicle and the adjacent vehicle at the station:  
\[T_{2}=\sum_{i=1}^{m}\sum_{j=1}^{n}\left[D_{i,j}^{p}\cdot h_{j}\cdot\left(1-Z_ {i,j}\right)\right] \tag{3}\]  
The headway satisfies:  
\[h_{j}=t_{i+1,j}-t_{i,j} \tag{4}\]  
The time \(t_{i,j}\) for bus \(i\) to arrive at station \(j\) is:  
\[t_{i,j}=t_{0}+(i-1)\cdot h+\frac{d_{i,j}}{v}+\sum_{j=1}^{n}\left(\tau_{i,j-1}+ \mu\right)\cdot Z_{i,j-1}+\sum_{x=1}^{q}\left(\varphi_{i,x}+\mu/2\right)\cdot Y _{i,x} \tag{5}\]  
(3) The stopping time of the stations  
The stopping time is the stay time of the passengers on the bus, including entering and exiting the bus, as well as the acceleration and deceleration time:  
\[T_{3}=\sum_{i=1}^{m}\sum_{j=1}^{n}\left[\left(L_{i,j}^{p}-A_{i,j}^{p}\right) \cdot\left(\tau_{i,j}+\mu\right)\cdot Z_{i,j}\right] \tag{6}\]  
The number of passengers on the bus is determined by the number of people entering and exiting at the previous stop:  
\[L_{i,j}^{p}=L_{i,j-1}^{p}+D_{i,j-1}^{p}-A_{i,j-1}^{p} \tag{7}\]  
The vehicle is not fully loaded; that is, waiting passengers can enter the bus when the vehicle is parked:  
\[L_{i,j}^{p}\leq Q,\text{ and }D_{i,j}^{p}=W_{i,j}^{p} \tag{8}\]  
(4) The delay time at intersections  
\[T_{4}=\sum_{i=1}^{m}\sum_{j=1}^{n}\sum_{x=1}^{q}\left[\left(L_{i,j}^{p}+D_{i, j}^{p}-A_{i,j}^{p}\right)\cdot\left(\varphi_{i,x}+\mu\right)\cdot Y_{i,x}\right] \tag{9}\]  
When bus \(i\) passes at intersection \(x,\ Y_{i,x}=0;\text{ otherwise, }Y_{i,x}=1.\) (10)When \(Y_{i,x}=1\), wait for the remaining red-light time (where \(n^{\prime}\) refers to the number of signal cycles):  
\[\varphi_{i,x}=r_{1}-\left[\left(t_{i,x}-t_{0}\right)-n^{\prime}C\right] \tag{11}\]  
The time when bus \(i\) arrives at intersection \(x\):  
\[t_{i,x}=t_{0}+(i-1)\cdot h+\frac{d_{i,j}+s_{j,x}}{v}+\sum_{j=1}^{n}\left(\tau _{i,j}+\mu\right)\cdot Z_{i,j}+\sum_{x=1}^{q}\left(\varphi_{i,x-1}+\mu\right)Y _{i,x-1} \tag{12}\]  
(5) The stopping costs of company  
The stopping costs of company mainly refers to the loss and fuel consumption of vehicles when they are parked. \[C_{5}=\sum_{i=1}^{m}\sum_{j=1}^{n}\Delta C\cdot Z_{i,j}+\sum_{i=1}^{m}\sum_{x=1 }^{q}\Delta C\cdot Y_{i,x} \tag{13}\]  
The lowest total cost:  
\[h_{j}=t_{i+1,j}-t_{i,j} \tag{4}\]0  
s.t.