## 3 Signal Control Method
### Coordination of Main and Pre-Signal Control
#### 3.2.2 Model of Coordination Control  
(1) Objective function  
To satisfy the constraints of the saturation of approaches at the intersection, it is notable that the shorter the phase time for the through movement and the left-turn is, the shorter the cycle time will be. As well, the operation efficiency of the intersection can be improved highly when the number of the right-turn vehicles are relatively small. Other than the ending time difference \(t_{f}\) between the main-signal and pre-signal green phase in the integrated waiting area discussed in Section 3.1, the starting time of the pre-signal must be earlier for a phase equal to the releasing time of the cross-direction traffic than that of the main signal. To meet the requirement of traffic in the cross-direction, this phase time is employed as the starting time difference \(t^{\prime}\) between the main-signal and pre-signal phase in the waiting area [29]. Usually, the smaller \(t^{\prime}\) is, the smaller the queuing time for the vehicles waiting to enter the waiting area will be. This will also ensure a higher operation efficiency of the waiting area as well as shortening the cycle time. Based on the above analysis, the objective function of the coordinated control model of the main and pre-signal in the waiting area is expressed as:  
\[\left\{ \begin{array}{c}\min(G_{T}+G_{L})\\ \min{t_{T}}^{\prime}\\ \min{t_{L}}^{\prime}\end{array} \right. \tag{2}\]  
where: \(G_{T}\) is the duration of green time for through movement at the main signal (s); \(G_{L}\) is the green time of the main signal for left-turn vehicles(s); \(t_{T}\)' is the difference of the start time for the through movement between main and pre-signal phase (s); \(t_{L}\)' is the difference of the start time for the left-turn movement between main and pre-signal phase (s). (2) Constraints  
For the convenience of expression, Table 1 explains the related variables. The convention is as follows: The variable subscript T represents the through movement, while the variable subscript L denotes the left-turn movement. To coordinate the control of the main and pre-signal in the integrated waiting area, except for the starting and ending time of the green light between the main and pre-signal, both signal cycles should be the same. They are all constrained by the length of the total cycle of the intersection, and expressed as follows:  
\[G_{T}+G_{L}+R=T \tag{3}\]  
\[g_{T}^{s}=0 \tag{4}\]  
\[g_{T}^{s}+g_{T}+t_{f}=g_{L}^{s} \tag{5}\]  
\[g_{L}^{s}+g_{L}+t_{f}=T \tag{6}\]  
\[g_{T}^{s}+{t_{T}}^{\prime}=G_{T}^{s} \tag{7}\]  
\[g_{L}^{s}+{t_{T}}^{\prime}=G_{L}^{s}. \tag{8}\]  
To ensure that all the waiting vehicles pass through the waiting area during the phase of the release, a clearance time should be met as expressed in Formula (9):  
\[t_{f}\geq\frac{l}{v_{s}} \tag{9}\]  
The saturation requirements of the traffic flow from all directions should be satisfied under the control plan of the main and pre-signals. The saturation degree before the pre-signal in the integrated waiting area should not be greater than the maximum saturation the road can carry. \[\frac{Q_{T}T}{S_{T}G_{T}N}\leq X_{max},\ \frac{Q_{L}T}{S_{L}G_{L}N}\leq X_{max} \tag{10}\]  
\[\frac{Q_{T}T}{s_{T}g_{T}n_{T}}\leq x_{max},\ \frac{Q_{L}T}{s_{L}g_{L}n_{L}}\leq x _{max}. \tag{11}\]  
For the through and left-turn traffic, the capacity of the main signal should be the same or greater than the capacity of the pre-signal, constraints are as follows:  
\[n_{T}\frac{g_{T}}{T}s_{T}\leq N\frac{G_{T}}{T}S_{T} \tag{12}\]  
\[g_{T}^{s}=0 \tag{4}\]0According to the signal timing rule of the conventional intersection, duration of green for through and left-turn movement should be constrained as the maximum value and minimum value. \[g_{T}^{s}=0 \tag{4}\]1  
\[g_{T}^{s}=0 \tag{4}\]2  
\[g_{T}^{s}=0 \tag{4}\]3  
\[g_{T}^{s}=0 \tag{4}\]4  
If the length of the integrated waiting area is larger than the maximum queuing length of vehicles in the waiting area, it will cause a waste of resources.