# Coordinated Multi-Platooning Planning for Resolving Sudden Congestion on Multi-Lane Freeways
## 4 Proposed Platoon-Based Coordination Scheme to Minimize Traffic Congestion
### Grouping Vehicles into the Platoons  
When the vehicles in the segment \(X\) receive a warning message about the accident ahead (e.g., through a decentralized environmental notification message), the adjacent vehicles immediately activate the control mode for platoon forming. To join a platoon, vehicles can use (i) a straight-moving action if they are in the same lane or (ii) a lane-changing action if the vehicles are in different lanes. To maintain both efficiency and safety, the vehicles in a platoon are required to maintain a platoon gap (\(d_{pgap}\)) in the same lane and move at the same velocity as the platoon leader. Since the platoon can synchronously maintain the movement of the members and the leader, \(d_{pgap}\) is supposed to be much smaller than \(\varepsilon_{i}\). In addition, after the platoon-forming procedure is completed, platoon members will follow the instructions from the platoon leader, either moving forward on the same lane or changing to a neighboring lane.  
Suppose that \(j\) represents the index of platoon members in each platoon. Through vehicle status message exchange, each vehicle is able to obtain the position information of surrounding vehicles. When forming the platoon \(n\) at time \(t\) with \(L_{n}(t)\) members, the front-most vehicle will become the _platoon leader_\(P_{n,1}^{L}\). The position, velocity, and acceleration of the platoon leader \(P^{L}\) can be expressed as follows:  
\[\begin{cases}P^{L}=P_{n,j}^{L},j=\{1\}\\ v_{n,1}=v_{p}=\textit{target platoon velocity}\end{cases} \tag{3}\]  
The remaining vehicles are called _platoon members_\(P^{M}\). If the distance between a member and its preceding one is greater than \(d_{pgap}\), in order to reduce the distance between vehicles,the member \(P^{M}\) needs to accelerate. Generally, the position, velocity, and acceleration of vehicle platoon member \(j\) can be expressed as follows:  
\[\begin{cases}P^{M}=P^{M}_{n_{j},j}f=\{2,3,\dots,L_{n}(t)\},\\ \frac{1}{\pi_{n,j}}=x_{n,1}+(j-1)\times d_{gap},\\ \Delta d_{n,j}=\overline{x_{n,j}}-x_{n,j},\\ t^{1}_{n,j}=\frac{\varphi_{max}-\varphi_{n,j}}{d_{max}},\\ t^{3}_{n,j}=\frac{\varphi_{max}-\varphi_{n,j}}{d_{max}},\\ t^{2}_{n,j}=\frac{\Delta d_{n,j}-\frac{\varphi_{n,j}+\varphi_{max}}{2}\times t ^{1}_{n,j}-\frac{\max+\varphi_{0}}{2}\times t^{3}_{n,j}}{\varphi_{max}},\end{cases} \tag{4}\]  
where \(\Delta d_{n,j}\) is the distance needed for the \(j\)th platoon member to catch up with the preceding platoon member. We assumed that it takes \(t^{1}_{n,l}\) time to reach its maximum velocity limit with the maximum acceleration limit, \(a_{max}\), \(t^{2}_{n,l}\) time to be maintained at the maximum velocity, and \(t^{3}_{n,l}\) time to decelerate to the target platoon speed with the maximum deceleration limit, \(d_{max}\).  
In short, platoon members will need to accelerate when joining the platoon if the distance between a member and its preceding member is larger than the \(d_{pgap}\). When the platoon members can match their distance gap as the platoon gap \(d_{pgap}\), the velocity and acceleration of platoon members follow their leader as follows:  
\[\begin{cases}v_{n,j}=v_{n,1},\\ a_{n,j}=a_{n,1}\end{cases} \tag{5}\]  
Therefore, we can describe the safety constraints of the platoon at the time slot \(t\) as follows:  
\[\begin{cases}x_{n,j-1}(t)-x_{n,j}(t)-L_{j}=\gamma\geq d_{pgap}\\ v_{max}\geq v_{n,j}(t)\\ a_{max}\geq a_{n,j}(t)\end{cases} \tag{6}\]  
where \(j-1\) is the index of the preceding vehicle of the platoon, \(L_{j}\) is the vehicle length, and \(\gamma\) is the distance between the vehicle \(j\) and the preceding vehicle \(j-1\).