# Cooperative Roundabout Control Strategy for Connected and Autonomous Vehicles
## 3 Numerical Studies  
To demonstrate the performance of the proposed CAV control strategy at roundabouts, several numerical studies are conducted for a four-approach single-lane roundabout with \(N=12\) vehicle slots. The desired speed \(v_{des}\) is set as 8 m/s (17.9 mph). The service capacity of the roundabout \(\mathcal{Q}_{S,max}\) is 60 vehicles/minute. The control time period \(\Delta\tau=5\) min. Note that in real-world applications, the control time period is related to the variation of incoming flows. The principle for the determination of \(\Delta\tau\) is that the incoming flows do not change significantly within the control time period because the optimal merge-in flows are solved based on the constant incoming flow assumption. First, the effectiveness of the roundabout flow control is investigated. The parameters and initial conditions for each approach are shown in Table 1.  
For \(\mathbf{\eta}=\left[\eta_{ij}\right]\), the experiment uses the following setting. \[\mathbf{\eta}= \begin{bmatrix}0.0&0.7&0.2&0.1\\ 0.2&0.0&0.6&0.2\\ 0.8&0.1&0.0&0.1\\ 0.1&0.5&0.4&0.0\end{bmatrix} . \tag{21}\]  
Figure 5 shows the optimal control trajectory of merge-in flows, queue lengths, and segment flows of the system. Five phases (separated by dashed lines) are observed in the 5 min optimal control solution. Within each phase, the control inputs (merge-in flows) remain constant, and the queue lengths change linearly. Figure 5c shows that the maximum capacity constraints for the four segments are all active in the first phase, indicating that the roundabout is operating at its maximum capacity. Each time a queue disappears, the system adjusts the merge-in flows, which demonstrates the adaptiveness of the control strategy. The number of active segment-maximum-capacity constraints decreases with the number of non-zero queues. Because when there is no queue on approach \(i\), the corresponding merge-in flow \(q_{i}=Q_{i}\). With fewer undetermined control inputs, fewer constraints are active. In the last phase, when there is no queue on four approaches, the merge-in flow \(q_{i}\) on each approach is equal to the incoming flow \(Q_{i}\). Compared to conventional roundabouts' first-come-first-serve principle, the cooperation among the four approaches is also reflected in the results. For example, the queue on approach 2 grows in the first phase to prioritize vehicles from other approaches with higher demand pressures (i.e., longer queues or higher incoming flows). However, in real-world applications, some approaches may have an upper bound on queue length to avoid spillbacks. Figure 6 illustrates how the queue lengths evolve with/without an upper-bound value of three, on the length of queue on approach 2, \(l_{2}\). Figure 6b illustrates that after \(l_{2}\) reaches 3, the system prioritizes vehicles from the other approaches. However, it adjusts the merge-in flows to meet the queue length constraint on approach 2. Next, the stability performance of the circular platoon control is validated. Note that the merge-in control and the self-adjusting stage are not simulated. It is assumed that the occurrence of merging-in follows a Poisson distribution (\(\lambda=1\)), because on average there will be one vehicle merging in when the roundabout operates at its maximum capacity \(\mathcal{Q}_{S,max}=60\) vehicles/minute. The initial speed and position disturbances are assumed to follow a uniform distribution \([-1,\ 1]\) m/s and a uniform distribution \([-1,\ 1]\) m, respectively, which are reasonable ranges for speed and position errors in the real world. Figure 7 shows the position errors, headways, and speed errors of 12 vehicles due to disturbances introduced by newly merging vehicles. The equilibrium state of the circular platoon changes each time a new vehicle merges in. For example, in Figure 7a, vehicles' equilibrium states in the platoon move in the same circular direction (counterclockwise in Figure 3) during \(t=30\ \sim 46\) s because of continuous positive position errors introduced by new merge-in vehicles.