# Priority of Emergency Vehicle Dynamic Right-Of-Way Control Method in Networked Environment
## 2 PDR-EVs Control Method
### Avoidance of EVs Guidance Control Method
#### 2.3.2 Guided Control Methods  
As illustrated in Figure 4, the \(j\)-th vehicle is an obstacle vehicle for the EVs and should be guided into the inner lane. The following three scenarios may exist for its guidance control process:  
Case 1: In the interval of \(DMS(t_{s,j})\forall t_{s,j}\in\Psi\), \(\forall k\in\mathrm{K}\), \(\exists H_{i+k}(t_{s,j})\) satisfies the inequality of Equation (9), and the spatial position of the \(j\)-th vehicle satisfies:  
\[S_{i+k}(t_{s,j})+S_{0}(D_{i+k}(t_{s,j}),D_{j}(t_{s,j}))\geq S_{j}(t_{s,j}), \forall j\in J \tag{14}\]The \(j\)-th vehicle must accelerate to a position that can converge into the gap, thus assuming that all CV shifting phases are uniformly variable. Then,  
\[x_{j}(\theta_{j}+\tau)= \begin{cases}\begin{array}{c}v_{j}(t_{s,j})*(t_{c,j}-t_{ s,j})+\frac{1}{2}a_{j}*(t_{c,j}-t_{s,j})^{2}\\ v_{max}*(t_{c,j}-\frac{v_{max}-v_{j}(t_{s,j})}{a_{j}}-t_{s,j})+\frac{v_{max}^{2} -v_{j}^{2}(t_{s,j})}{2a_{j}},\forall t_{s,j}\in\Psi,t_{c,j}\in\Omega,\theta_{j }\in\Theta\end{cases} \end{cases} \tag{15}\]  
If the following inequality is satisfied, the \(j\)-th vehicle can safely and smoothly merge into the gap \(H_{i+k}(t_{s,j})\), and subtracting \(X_{S}(t_{s,j})\) from both sides of the inequality can be obtained:  
\[ \begin{cases}\begin{array}{c}S_{j}(t_{s,j})+x_{j}(\theta_{j}+\tau)\geq S_{i+k }(t_{s,j})+S_{0}(D_{i+k}(t_{s,j}),D_{j}(t_{s,j}))+\int_{t_{s,j}}^{t_{c,j}}v_{i+ k}(t)dt\\ S_{j}(t_{s,j})+x_{j}(\theta_{j}+\tau)\leq S_{i+k+1}(t_{s,j})-S_{0}(D_{i+k+1}(t_{ s,j}),D_{j}(t_{s,j}))+\int_{t_{s,j}}^{t_{c,j}}v_{i+k+1}(t)dt\\ S_{j}(t_{s,j})+x_{j}(\theta_{j}+\tau)\leq S_{j+1}(t_{s,j})-S_{0}(D_{j+1}(t_{s,j }),D_{j}(t_{s,j}))+\int_{t_{s,j}}^{t_{c,j}}v_{j+1}(t)dt\end{array}\end{cases}  \tag{16}\]  
In the given formula, \(t_{s,j}\) and \(\tau\) represent the known system parameters, and the range of \(t_{c,j}\) could be easily available from \(\theta_{j}\). \[t_{s,j}+\theta_{j,max}+\tau\geq t_{c,j}\geq t_{s,j}+\theta_{j,min}+\tau,\forall j\in J \tag{17}\]  
By combining the above equations and inequalities to form a linear programming model, it can be seen that there exist only two unknown model parameters, i.e., \(t_{c,j}\) and \(a_{j}\). The traversal search approach can be exploited to solve the model [19]. Since the range of \(t_{c,j}\) has been determined, the value of \(t_{c,j}\) can be substituted into the model from small to large to solve the corresponding \(a_{j}\) until a solution set of \(a_{j}\) is obtained for the model. Notably, the optimal solution method is to give priority to minimize \(t_{c,j}\) and then minimize \(a_{j}\), while considering \(a_{j}^{*}=min(a_{j}),\forall j,a_{j}\in A_{\star}\). In the actual system operation, the CVs of the obstacle in front of the EVs merge into the inner lane as soon as possible, thus ensuring the comfort of the shifting process. Case 2: In an interval of \(DMS(t_{s,j})\forall t_{s,j}\in\Psi,\forall k\in K,\)\(\exists H_{i+k}(t_{s,j})\) satisfies the inequality given in Equation (9), and the spatial position of the \(j\)-th vehicle satisfies:  
\[S_{i+k}(t_{s,j})+S_{0}(D_{i+k}(t_{s,j}),D_{j}(t_{s,j}))<S_{j}(t_{s,j}),\forall j\in J \tag{18}\]  
The \(j\)-th vehicle must decelerate to reach the position where it can merge into the gap. Before reaching the merging position, its displacement can be evaluated by:  
\[x_{j}(\theta_{j}+\tau)= \begin{cases}\begin{array}{c}v_{j}(t_{s,j})*(t_{c,j}- t_{s,j})-\frac{1}{2}a_{j}*(t_{c,j}-t_{s,j})^{2}\\ v_{min}*(t_{c,j}-\frac{v_{j}(t_{s,j})-v_{min}}{a_{j}}-t_{s,j})+\frac{v_{j}^{2} (t_{s,j})-v_{min}^{2}}{2a_{j}},\forall t_{s,j}\in\Psi,\forall t_{c,j}\in \Omega,\forall\theta_{j}\in\Theta\end{cases} \end{cases}\end{cases} \tag{19}\]  
Similarly, \(t_{c,j}\) and \(a_{j}\) are calculated by taking \(a_{j}*=\max(a_{j}),\forall j,a_{j}\in A\).