# Cooperative Roundabout Control Strategy for Connected and Autonomous Vehicles
## 3 Numerical Studies
Hence, the overall performance of the proposed model can be less efficient than the performance determined from the roundabout flow control component. In the following comparison, the idealized performance associated with the roundabout flow control is used as a benchmark. The proposed measure enables comparisons between intersection designs with different incoming approach capacities. The results for signal-free intersection control, fully actuated signal control, and fixed-time signal control are from [9] (simulated on intersections with four-lane roads). The proposed roundabout control strategy is simulated for an intersection with two-lane roads (one lane in each direction). The seven-minute (including a two-minute warm-up) simulation recording can be accessed online ([[https://youttu.be/XEZhP](https://youttu.be/XEZhP)]([https://youttu.be/XEZhP](https://youttu.be/XEZhP)))nrdeg) on 12 September 2022. As shown by Table 3, while the signal-free intersection control strategy from [9] has the highest throughput, its intersection design efficiency is only 0.188. This is because the \"cross\" flow pattern in conventional intersection design has significant conflicts associated with the flows from different directions, which limits the spatiotemporal occupancy within the intersection. By leveraging CAVs, the roundabouts' circular flow pattern can fully exploit road capacity inside the intersection. The overall performance of the proposed roundabout control yields a high intersection design efficiency, 0.343, which is almost twice that of the signal-free intersection control strategy. Additionally, note that the intersection design efficiency calculated from flow-level control is higher than that of the overall control, indicating that the proposed model suffers mild efficiency loss when converting optimal merge-in flows into vehicle-level controls. This is because the upper-level flow control neglects the vehicle level randomness. As the top-left part of Figure 8 shows, the queue length on approaches 1, 2, and 3 are 1, 2, 1, respectively, while the roundabout flow control components indicates that there should be no queues during the entire time period. Due to vehicle-level randomness, the queue lengths in the comprehensive simulation have small positive values all the time. The solution optimality is impaired as the flow control model addresses the problem in the continuous domain and an additional discretization step is introduced in the vehicle merge-in control. However, solving optimal control problems at the flow level mitigates the computational burden compared to signal-free intersection control in [9] which directly solves the high-dimensional vehicle-level optimal control problem. It is beneficial to sacrifice optimality slightly for real-time computational tractability. To summarize, the numerical results illustrate that: (i) the roundabout flow control can coordinate the merging flows from different directions to enhance efficiency; (ii) the vehicle platoon control can dampen the oscillations induced by the speed and position errors at the self-adjusting stage to ensure vehicle safety; (iii) the merge-in control is able to integrate the aforementioned two components with slight impairment to the solution optimality; and (iv) the proposed roundabout control strategy shows the advantage of the circular flow pattern over the \"cross\" flow pattern in terms of the intersection design efficiency. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Intersection Design**} & \multicolumn{2}{c}{**East and West Bound**} & \multicolumn{2}{c}{**North and South Bound**} \\ \cline{2-5}  & **Through Demand** & **Left-Turn Demand** & **Through Demand** & **Left-Turn Demand** \\  & **(veh/h/lane)** & **(veh/h/lane)** & **(veh/h/lane)** & **(veh/h/lane)** \\ \hline Two-lane roundabout & 1125 & 450 & 1125 & 450 \\ \hline Four-lane crossroad & 1500 & 600 & 1500 & 600 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Demand settings for performance comparison.