# Coordinated Control Method of Bus Signal Priority and Speed Adjustment Based on Stop-Skipping
## 3 Model Building
### Collaborative Control  
(1) Signal-priority method  
In order to reduce the interference of buses with other vehicles, the green-light time is generally advanced or extended by no more than 10% of the signal period [34]. According to Equation (16), the intersection signal is composed of two phases and the condition \(r_{1}=r_{2}\) is set in this study. It can be seen that the length of time allowed for signal priority in a cycle is \(\frac{1}{5}r_{1}\). The time when the bus arrives at the intersection is divided into four sections (where \(n^{\prime}\) refers to the number of signal cycles) to determine the passing time window of the priority application (see Figure 3). The signal-priority mode adopted in Section i is green extension (GE). The signal-priority mode adopted in Section iii is red truncation (RT). Signal priority is not carried out in Section ii. Section iv indicates that vehicles pass normally.  
\[t_{i,x}\in(n^{\prime}C,n^{\prime}C+\frac{1}{5}r_{1}),(n^{\prime}=0,1,2,\ldots) ;\,Y_{i,x}=0 \tag{23}\]  
\[t_{i,x}\in(n^{\prime}C+\frac{1}{5}r_{1},n^{\prime}C+\frac{4}{5}r_{1}),(n^{ \prime}=0,1,2,\ldots);\,Y_{i,x}=1 \tag{24}\]  
\[t_{i,x}\in(n^{\prime}C+\frac{4}{5}r_{1},n^{\prime}C+r_{1}),(n^{\prime}=0,1,2, \ldots);\,Y_{i,x}=0 \tag{25}\]  
\[t_{i,x}\in(n^{\prime}C+r_{1},n^{\prime}C+C),(n^{\prime}=0,1,2,\ldots);\,Y_{i,x }=0 \tag{26}\]  
According to the time period when the vehicle arrives at the intersection, there are the following priority methods:  
Combining Equation (23), it can be seen that the green-light extension method is adopted (section i). Considering the conversion of the yellow light time, the extension time is:  
\[r_{i,x}=(t_{i,x}-t_{0})-n^{\prime}C+3; \tag{27}\]  
Combining Equation (24), the vehicle cannot pass preferentially (section ii), and needs to wait for the remaining red-light time:  
\[\varphi_{i,x}=r_{1}-\big{[}(t_{i,x}-t_{0})-n^{\prime}C\big{]}; \tag{28}\]  
According to Equation (25), the early red-light mode is adopted (section iii), and the early red-light interruption time is:  
\[r_{i,x}=r_{1}-\big{[}(t_{i,x}-t_{0})-n^{\prime}C\big{]}; \tag{29}\]  
Vehicles are in the green-light time period and pass normally (section iv).  
(2) Speed adjustment method  
For the situation where the vehicle arrives at the intersection in Section ii, the method of adjusting the speed at the stop before the intersection is adopted, so that the vehicle can  
Figure 3: Division of time periods for vehicles arriving at intersections.  
pass directly when it arrives at the intersection. This method involves accelerating or decelerating at a station before the intersection and decelerating or accelerating correspondingly at the intersection, so that the running time between the two stations remains unchanged. The maximum running speed of an urban bus usually does not exceed 60 km/s [35]. (Note: \(v_{1}^{\prime}\), \(v_{2}^{\prime}\) is the change speed before and after the intersection)  
\[\frac{s_{j,x}}{v}-\frac{s_{j,x}}{v^{\prime}}\geq\frac{4}{5}r_{1}+3 \tag{30}\]  
According to Equation (30), the vehicle accelerates at the speed of \(v^{\prime}\) at the stop before the intersection:  
\[r_{1}-\varphi_{i,x}+3=\frac{s_{j,x}}{v}-\frac{s_{j,x}}{v_{1}^{\prime}} \tag{31}\]  
In order to ensure that the time to reach the next station remains unchanged, the speed will be reduced after crossing the intersection:  
\[r_{1}-\varphi_{i,x}+3=\frac{s_{x,j+1}}{v_{2}^{\prime}}-\frac{s_{x,j+1}}{v_{1}^ {\prime}} \tag{32}\]  
The vehicle decelerates at the speed of \(v^{\prime}\) at the stop before the intersection when the distance between the station and the intersection does not meet Equation (30):  
\[\varphi_{i,x}=\frac{s_{j,x}}{v_{1}^{\prime}}-\frac{s_{j,x}}{v} \tag{33}\]  
After passing the intersection, the vehicle speeds up to ensure that the time to reach the next stop remains unchanged:  
\[\varphi_{i,x}=\frac{s_{x,j+1}}{v_{1}^{\prime}}-\frac{s_{x,j+1}}{v_{2}^{\prime}} \tag{34}\]