## 4 Model Formulation
### Lower-Layer Model
#### 4.3.2 First Vehicle Trajectory Planning in the CAV Queue  
The acceleration profile of the first CAV \(\omega\) at the front of the CAV queue in the passing zone and variable virtual waiting zone is optimized to minimize the trip delay. Similar to the research ideas of [PERSON] and [PERSON] [11, 46], this study continues to simplify three-segment trajectory planning, as shown in Figure 11. To ensure that the CAV moves through the passing zone and changeable virtual waiting zone without halting, three portions of the trajectory are identified, with the CAV running at maximum deceleration \(a_{1}\), constant velocity (acceleration of \(a_{2}\)), and maximum acceleration \(a_{3}\). \(T_{3}\) is the optimized model. \[\mathit{mint}_{pass}^{\omega} \tag{27}\]  
\[\mathit{maxv}^{\omega}\left(t_{pass}^{\omega}\right) \tag{28}\]  
\[\mathit{s.t.} \tag{29}\]  
\[\mathit{x}^{\omega}\left(t_{pass}^{\omega}\right)=L_{p}+L_{C} \tag{30}\]  
\[0<v^{\omega}(t)\leq v_{max} \tag{31}\]  
\[a_{1}^{\omega}\in\left\{a^{U},0,a^{L}\right\},i=1,2,3 \tag{32}\]  
\[v^{\omega}(t_{1}^{\omega})-v^{\omega}(t_{0}^{\omega})=(t_{1}^{\omega}-t_{0}^{ \omega})a_{1}^{\omega} \tag{33}\]  
\[v^{\omega}(t_{2}^{\omega})-v^{\omega}(t_{1}^{\omega})=(t_{2}^{\omega}-t_{1}^{ \omega})a_{2}^{\omega} \tag{34}\]  
\[v^{\omega}\left(t_{pass}^{\omega}\right)-v^{\omega}(t_{2}^{\omega})=\left(t_{ pass}^{\omega}-t_{2}^{\omega}\right)a_{3}^{\omega} \tag{35}\]  
\[\mathit{mint}_{pass}^{\omega} \tag{27}\]0  
\[\mathit{mint}_{pass}^{\omega} \tag{27}\]1 CAV trajectory queue. (**a**) Case 1 utilizing two platoons. (**b**) Case 2 utilizing one platoon. \[\mathit{mint}_{pass}^{\omega} \tag{27}\]2  
\[\mathit{mint}_{pass}^{\omega} \tag{27}\]3  
\[\mathit{mint}_{pass}^{\omega} \tag{27}\]4  
\[\mathit{mint}_{pass}^{\omega} \tag{27}\]5  
\[\mathit{mint}_{pass}^{\omega} \tag{27}\]6  
where \(t_{0}^{\omega}\) indicates when CAV \(\omega\) has just reached the passing zone; \(t_{1}^{\omega}\) and \(t_{2}^{\omega}\) indicate the end points of the trajectory of lead CAV \(\omega\) during the deceleration phase and the uniform speed phase, respectively; \(t_{pass}^{\omega}\) is the moment when lead CAV \(\omega\) crosses the stop line ahead of the variable virtual waiting zone; \(\
u^{\omega}(t)\) and \(x^{\omega}(t)\) are lead CAV \(\omega\) at time point \(t\); \(a_{i}^{\omega}\) is the acceleration of lead CAV \(\omega\) during the deceleration phase, the uniform velocity phase, and the acceleration phase; and \(a^{L}\) is the maximum deceleration. The term \(a^{U}\) is the maximum acceleration of lead CAV \(\omega\) during the acceleration phase; \(v_{0}^{\omega}\) is the initial velocity of lead CAV \(\omega\); when the phase index is \(P\) in group \(j\), the starting time point of CAV \(\omega\) crossing the intersection is \(t_{p,j}^{g}\); decision variables include \(t_{i}^{\omega}(i=1,2)\), \(t_{pass}^{\omega}\), \(a_{i}^{\omega}(i=1,2,3)\), \(\
u^{\omega}(t)\), and \(x^{\omega}(t)\). The passing zone length is \(L_{P}\), and the variable virtual waiting zone length is \(L_{C}\). The other parameters are known. In Equation (27), the objective function is minimizing \(t_{pass}^{\omega}\) since the vehicle arrival time has been determined. It is important to note in this study that, as shown in Figure 12, there may be multiple optimal solutions \(t_{pass}^{\omega}\) for the same amount of time to cross the passing zone and the variable virtual waiting zone. For the selection of the optimal solution, preference is given to the trajectory plan that passes the stop line in front of the virtual waiting zone at a higher speed because this will enable a CAV to pass the intersection at a higher safe speed; thus, maximizing the passing speed also becomes the optimization objective, which is Equation (28). The moment \(t_{0}^{\omega}\) at which CAV \(\omega\) enters the passing zone is related to the rolling repetition barrier scheme described previously.