## 3 Methodology
### Speed Control-Based Intersection Coordination Model (SICM)
#### 3.3.2 Model Development  
As mentioned above, we need to conduct the conflict resolution measure, which is used to adjust the speed of the vehicle in advance if there will be a collision (as shown in Figure 10). In this section, we will take vehicles A and B as an example to develop a conflict resolution model called the Speed Control-based Intersection Coordination Model (SICM).  
1. Assume that the serial numbers of vehicles A and B are 1 and 2, respectively. If there will be a collision, the intersection manager will adjust the default design speed of the vehicles to avoid occupying the same conflict point at the same time. Therefore, we take the modified design speed as a decision variable as follows: \[v^{\text{mod}}=\left[v_{1}^{\text{mod}},v_{2}^{\text{mod}}\right]\] (12)
2. The decision variables should not exceed the speed limit, and the acceleration affected by the design speed should not exceed the mobility performance limit. Therefore, \[0<v_{1}^{\text{mod}}\leq v^{\max}\] (13) \[0<v_{2}^{\text{mod}}\leq v^{\max}\] (14) \[-\left|a_{\max}\right|\leq(v_{k}^{1}-v_{k-1}^{1})\div dt\leq\left|a_{\max} \right|(k=1,2,\cdots,t_{1}^{m})\] (15) \[-\left|a_{\max}\right|\leq(v_{k}^{2}-v_{k-1}^{2})\div dt\leq\left|a_{\max} \right|(k=1,2,\cdots,t_{2}^{m})\] (16)  
Figure 10: The location of the time blocks of vehicle A and B.  
\[t_{1}^{m}=t_{1}^{cross}\div dt \tag{17}\] \[t_{2}^{m}=t_{2}^{cross}\div dt\] (18) \[v_{1}^{0}=v_{1}^{init}\] (19) \[v_{0}^{2}=v_{2}^{init} \tag{20}\]  
In addition, small errors in the model can lead to serious traffic accidents; thus, constraints on driving safety should also be set for the model, that is:  
\[[TB_{2\times j-1}^{1},TB_{2\times j}^{1}]\cap[TB_{2\times j-1}^{2},TB_{2\times j }^{2}]=\varnothing,\;\;j=1,2,\cdots,16 \tag{21}\]  
where \(j\) is the serial number of conflict points and \([TB_{2\times j-1}^{1},TB_{2\times j}^{1}]\) and \([TB_{2\times j-1}^{2},TB_{2\times j}^{2}]\) represent the duration of occupation at the \(j\)th conflict point respectively.
3. We want to improve traffic efficiency; therefore, we set the optimize goal as: \[\text{minmakespan}=P_{2\times p_{4}^{2}}^{2}-P_{2\times p_{1}^{1}-1}^{1}\] (22) where \(P_{2\times p_{1}^{1}-1}^{1}\) represents the start time of occupation at the first conflict point, \(P_{2\times p_{4}^{2}}^{2}\) represents the end time of occupation at the fourth conflict point, and makespan represents the period of two vehicles occupying the intersection area.
4. The number of vehicles entering the intersection at the same time will increase along with the traffic flow, which may increase the risk of collision. Therefore, we need to modify the constraint conditions to address this multi-vehicle conflict resolution problem. We take a three-vehicle conflict as an example to illustrate the modification of constraints. Assuming that the serial numbers of the three vehicles are 1, 2, and 3, we have: \[TB_{j}^{1}=[TB_{2\times j-1}^{1},TB_{2\times j}^{1}]\] (23) \[TB_{j}^{2}=[TB_{2\times j-1}^{2},TB_{2\times j}^{2}]\] (24) \[TB_{j}^{3}=[TB_{2\times j-1}^{3},TB_{2\times j}^{3}]\] (25) \[\left(TB_{j}^{1}\cap TB_{j}^{2}\right)\cup\left(TB_{j}^{1}\cap TB_{j}^{3} \right)\cup\left(TB_{j}^{2}\cap TB_{j}^{3}\right)=\varnothing,\;j=1,2,\cdots,16,\] (26) where \(j\) is the serial number of conflict points, and \(TB_{j}^{1}\), \(TB_{j}^{2}\) and \(TB_{j}^{3}\) represent the duration of occupation at the first, second and third conflict points, respectively.