## 4 Simulation Results
### Results  
The proposed algorithm is tested under the different demand cases and penetration rates of AVs (PRAs) to investigate the performance of the scheme in this study. Table 3 shows the comparison of the mean travel time for the different demand and PRAs. The results show that the mean travel time increases slightly when the PRA decreases in cases 1 and 2. In contrast, the mean travel time for case 3 rises substantially with the reduction in PRA. On the other hand, by comparing the results vertically, it slightly differs in the mean travel time for the different cases under 80% to 100% PRA. Conversely, the mean travel time increases significantly for case 3 under 70% PRA. The results indicate that when the PRA is as low as 70% and the total traffic demand reaches 1800 vehicles per hour, there is a significant adverse effect on the traffic efficiency of the intersection. Compared to the results for under 100% PRA, the mean travel time increases by 28.9% under 70% PRA. Table 4 shows the results for fuel consumption under the different traffic demand and PRAs, respectively. Similarly, the differences in fuel consumption under different PRAs are less distinct for cases 1 and 2. There are even slight drops in fuel consumption when the PRA decreases, since the additional control cost for the desired performance indexes is reduced. The fuel consumption rises significantly with the decline in PRA in case 3. The results indicate that the proposed scheme has less apparent impact on fuel consumption with different PRAs under low demand, while the differences are distinct under high demand. Figure 4 shows the mean speed relative rates for case 3 in 15 min under different PRAs. The mean speed relative rate is defined as the ratio of the actual average speed to the desired speed. Figure 4 indicates that the mean speed under higher PRAs is generally closer to the desired speed than that under lower PRAs. In particular, during the 700 to 900 s period, the mean speed relative rate maintains a relatively high value under 90% and 100% PRA, while these drop dramatically under 70% and 80% PRA. This phenomenon occurs since more uncontrolled MVs are loaded onto the network during this period. Multiple \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{5}{c}{**Penetration Rate of AVs**} \\ \cline{2-5}  & **100\%** & **90\%** & **80\%** & **70\%** \\ \hline
1 & 31.8 & 31.6 & 31.4 & 31.5 \\
2 & 33.4 & 33.3 & 33.5 & 33.7 \\
3 & 34.7 & 35.3 & 36.4 & 38.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Summary of mean fuel consumptions (mL) for the three cases under different penetration rates of AVs. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{3}{c}{**North- and Southbound**} & \multicolumn{2}{c}{**East- and Westbound**} \\ \cline{2-5}  & **Through** & **Left-Turn** & **Through** & **Left-Turn** \\
**Demand** & **(veh)/h/lane)** & **(veh)/h/lane)** & **(veh)/h/lane)** & **(veh)/h/lane)** & **(veh)/h/lane)** \\ \hline
1 & 250 & 125 & 150 & 75 \\
2 & 300 & 150 & 200 & 100 \\
3 & 350 & 175 & 250 & 125 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Demand for different cases. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{3}{c}{**Penetration Rate of AVs**} \\ \cline{2-5}  & **100\%** & **90\%** & **80\%** & **70\%** \\ \hline
1 & 30.4 & 30.8 & 31.1 & 31.7 \\
2 & 30.6 & 31.5 & 32.1 & 32.7 \\
3 & 30.8 & 32.3 & 34.6 & 39.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of mean travel time (s) for the three cases under different penetration rates of AVs. MVs slow down or even stop at the intersection for safe passing, resulting in reducing the mean speed of the road network. Figure 5 shows the number of stops for case 3 in 15 min under different PRAs. No stops and a few stops occur in the scenarios of 100% and 90% PRA, respectively. When the PRA drops to 70% and 80%, during the 700 to 900 s period, more vehicles stop at the intersection for safe passing, which leads to queuing.