## 4 Simulation Results
### Results
The above results indicate that the decrease in the PRA leads to a decline in the traffic efficiency of the intersection. To compare the performance of the proposed scheme with other methods used, we test the no-control (NC) operation and fixed-time signal control (FSC) strategy as benchmark baselines at the same intersection. The NC operation means allowing all vehicles to pass through the signal-free intersection by following the driving behaviors of MVs. The cycle time of the FSC system is set as 82 s. The green times of north- and southbound, and east- and westbound are chosen as 20 s and 15 s, respectively. The clearance time is 3 s. Figure 6 shows the trajectories of vehicles for case 3 in 5 min under the different strategies. It is observed that the vehicles with the proposed scheme in this paper adjust their trajectories after entering the CZ. Especially in the scenario of 100% PRA, the vehicles pass through the intersection smoothly. When the PRA is 70%, a few vehicles stop at the intersection. In contrast, the vehicles under NC or FSC frequently stop at the intersection, and lots of queues are formed. Figure 7 shows the speeds of vehicles for case 3 in 5 min under the different strategies. Corresponding to the results in Figure 6, the vehicles using the proposed scheme always pass the intersection without halting (speed greater than zero) under 100% PRA, and a small number of vehicles slow down to zero at the intersection under 70% PRA. When NC or FSC is adopted, more vehicles slow to a stop at the intersection. The above phenomena indicate that the proposed scheme in this paper notably outperforms NC and FSC in smoothing traffic flow and improving mobility. Figure 8 shows the acceleration  
Figure 4: Comparison of the mean speed relative rates for case 3 in 15 min under different penetration rates of AVs. Figure 5: Comparison of the number of stops for case 3 in 15 min under different penetration rates of AVs. curves of vehicles with respect to distance under the proposed scheme for case 3 in 5 min. We can easily observe that the vehicles adjust their acceleration significantly after entering the CZ. Notice that the maximum acceleration and deceleration of MVs are 3 m/s\({}^{2}\) and 4 m/s\({}^{2}\) in the scenario of 70% PRA, respectively, while the maximum acceleration and deceleration of AVs are both 3 m/s\({}^{2}\). Table 5 shows the results of the performance, i.e., travel time and fuel consumption, of the proposed scheme, NC operation, and the FSC strategy. The results show that the proposed scheme outperforms NC and FSC by decreasing the mean travel time by 45.5% and 49.2% under 100% PRA, and by 29.7% and 34.5% under 70% PRA, respectively. In addition, compared to NC and FSC, the proposed scheme reduces the mean fuel consumption by 35.3% and 29.9% under 100% PRA, and by 27.6% and 21.6% under 70% PRA, respectively. Overall, the proposed scheme of this paper significantly improves traffic efficiency and fuel economy compared to NC and FSC. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Performance**} & \multicolumn{3}{c}{**Proposed Scheme**} & \multirow{2}{*}{**NC**} & \multirow{2}{*}{**FSC**} \\ \cline{2-2} \cline{4-4}  & **100\% AV** & & & \\ \hline Travel time (s) & 30.8 & 39.7 & 56.5 & 60.6 \\ Fuel consumption (mL) & 34.7 & 38.8 & 53.6 & 49.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Summary of performance for case 3 under the proposed scheme, no-control operation, and the fixed-signal control strategy. Figure 6: The trajectories of vehicles for case 3 in 5 min under the different strategies. Figure 9 shows the mean speed of the road network with the different strategies for case 3 in 15 min, which reflects the traffic mobility to some extent. The vehicles with the proposed scheme in this study can always maintain a high mean speed in the scenario of 100% PRA. When the PRA is 70%, the mean speed declines, since more MVs are loaded onto the network and slow down at the intersection.