# Virtual Belt Algorithm for the Management of Isolated Autonomous Intersection
## 7 Numerical Simulation  
In this section, two simulations are conducted to verify the effectiveness of the proposed VB algorithm. The layout of the center of the considered intersection is shown in Figure 20. In the first simulation, several scenarios with low traffic load are constructed to compare the performances of the VB algorithm, Multi-Agent (MA) algorithm [1] and four-phase traffic lights with different green light time. In the second simulation, several scenarios with high traffic load are constructed to further compare the performances of these algorithms.  
The VB and MA algorithms are performed using Matlab. Using the offline algorithm of the VB algorithm, the VBs designed at the intersection are illustrated in Figure 21. The simulations of traffic lights are conducted using VISSIM. The parameters of the intersection and CAVs in these simulations are listed in Table 1. A camera shot of the intersection controlled by the proposed VB algorithm is shown in Figure 22. The proposed VB algorithm consists of an offline algorithm and an online algorithm. The offline algorithm is used to model the considered intersection using several VBs and calculate the collision among the VGs in these VBs. As the collision of each VG with all VGs in other VBs will be checked according to Algorithms 1 and 2, the complexity of the offline algorithm is \(O(C_{g}\cdot C_{v}\cdot C_{g}\cdot(C_{v}-1))\approx O((C_{g}\cdot C_{v})^{2})\). Where \(C_{v}\) and \(C_{g}\) mean the count of VBs and the count of VGs in each VB, respectively. The online algorithm is used to calculate the trajectories of waiting vehicles in each circle one after another. The trajectory of a waiting vehicle can be obtained by successively checking the trajectories which are planned based on candidate VGs. As the candidate VGs of each vehicle is no more than \(C_{g}\), the complexity of the online algorithm is \(O(C_{g}\cdot C_{vch})\). Where \(C_{vch}\) is the count of the waiting vehicles in each control circle. In the simulation, the computation time of the offline algorithm is about one hour. As the offline algorithm can be used before the real-time application of the VB algorithm, the time cost in the offline algorithm has no effect to the real-time performance of the VB algorithm. For the online algorithm, the time used to plan the trajectory of a waiting vehicle is less than 1 ms, which indicates that the proposed VB algorithm can be used for real-time application.  
Figure 20: Center of the considered intersection.  
Figure 21: VBs of intersection, (**a**) VBs at center, (**b**) VBs on road W.  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Lane width & 3.75 m \\ Desired speed & 36 km/h \\ Length of virtual belt & 880 m \\ Length of virtual grid & 8 m \\ Length of CAV & 4.1 m\(-\)6.2 m \\ Width of CAV & 1.6 m\(-\)2.1 m \\ Maximum velocity of CAV & 30 m/s\(\sim\)35 m/s \\ Minimum velocity of CAV & 0 m/s \\ Maximum acceleration of CAV & 2.5 m/s\({}^{2}\)\(-\)4.5 m/s\({}^{2}\) \\ Minimum acceleration of CAV & \(-\)6 m/s\({}^{2}\)\(-\)4 m/s\({}^{2}\) \\ \hline \end{tabular}
\end{table}
Table 1: Parameters of intersection and CAVs.  
Figure 22: Camera shot.