# Cooperative Roundabout Control Strategy for Connected and Autonomous Vehicles
## 3 Numerical Studies
However, it is related more to coordinate shifts and does not influence vehicle safety. Figure 7b shows that vehicles' headways remain in a safe range (the desired time headway is set as 1 s) during the entire simulation. Figure 7c illustrates that all newly introduced speed errors can be mitigated in a short time. The spikes in Figure 7b,c mostly occur on on-ramps because position and speed errors are introduced by virtual vehicles. It allows the system to converge closer to the equilibrium state before real vehicles merge into the circular platoon, which enhances smoothness and safety. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Approach ID** & \(\mathcal{Q}_{i}\) **(Vehicles/min)** & \(l_{i}(0)\) \\ \hline
1 & 30 & 2 \\
2 & 25 & 1 \\
3 & 35 & 3 \\
4 & 20 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Experiment settings. Figure 5: (a) Optimal merge-in flows; (b) queue lengths; (c) segment flows. Figure 6: Queue length evolution: (**a**) without queue length upper bound; (**b**) with queue length upper bound. Figure 7: Stability and safety analysis of virtual platoon control: (**a**) position errors; (**b**) headways; (c) speed errors. In summary, the performances of both upper-level flow control and lower-level vehicle control are illustrated separately. Next, these two components are integrated with the proposed vehicle merge-in control and a comprehensive simulation (microscopic simulation) is performed. Figure 8 shows a screenshot of the simulation interface, which illustrates the integration of the proposed control components. The blue dots denote incoming vehicles and platooning vehicles, while the orange ones are vehicles that have passed through the roundabout. In the top-right zoomed-in view, desired positions of platooning vehicles (slot positions) are marked with the slot number labeled. The bottom-right of Figure 8 shows that the speed errors and headway errors of the platooning vehicles in the last 250 steps (the step size is 0.05 s) are updated dynamically. Damping patterns can be observed similar to those in the validation of the platoon stability in Figure 7.  
To compare the overall performance of the proposed roundabout control strategy with other intersection control strategies, the demand pattern associated with the high demand case (case 6) in [9] is used. Table 2 shows that though the demands are different, the proportions of corresponding demands in the two intersection designs are identical (0.75). As the roundabout model uses only a two-lane roadway setting, it is not meaningful to compare its performance using the demand on various approaches of typical four-lane crossroads. To address this issue, and account for the effects of different time headway settings and different numbers of lanes on incoming approaches, a new intersection design efficiency measure labeled as \(\backslash\) is proposed, which is the throughput of the intersection divided by the sum of the capacity of all input lanes. The intersection design efficiency quantifies the capability of the intersection design (both flow pattern design and control strategy design) to serve the capacity of all incoming approaches. The proposed unit-free measure is justified as follows. The most efficient intersection design is to use separate facilities (for example, grade-separated roads) to connect all approaches. Thereby, vehicles traveling in different directions will not affect each other, which implies that the intersection can serve the capacity of all incoming approaches (regardless of the number of lanes and how large the capacities are); thus \(\backslash=1\). This \"perfect\" intersection design serves as an idealized benchmark to evaluate the efficiency of various intersection designs using \(\backslash\). Figure 8: Simulation of the proposed roundabout control strategy. Note that in flow-level control, the vehicle-level randomness in the incoming flow is neglected. Additionally, the vehicle merge-in control assumes instantaneous changes of within-roundabout flow due to merge-in flows.