## 2 Framework of DRP Strategy and Problem Formulation
### CAVs Trajectory Optimization Problem Formulation  
Upon establishing the sequence of all CAVs entering the MZ, additional computation of the arrival times of CAVs at the MZ, symbolized as \(t^{m}\in\mathbb{R}^{+}\), becomes imperative to furnish essential temporal constraints for tackling the TOP. By taking into account the arrival time \(t^{m}_{j}\) of the preceding CAV \(j\) in the sequence at the MZ, along with the collision avoidance costs \(C_{j,i}\) between CAVs \(j\) and \(i\), recursively determining the arrival time \(t^{m}_{i}\) of CAV \(i\) at the MZ emerges as a straightforward and efficient approach [8]. Following this, to derive the precise trajectories of the CAVs, it is essential to establish the corresponding models. To accomplish this, we postulate that the movement of each CAV is subject to second-order dynamics:  
\[\dot{p}_{i}=v_{i}(t),\ \dot{v}_{i}=u_{i}(t) \tag{10}\]  
Here, \(p_{i}(t)\), \(v_{i}(t)\), and \(u_{i}(t)\) correspond to the position (the distance traveled upon entering the CZ), velocity, and acceleration (control input) of CAV \(i\). To ensure that the velocity and control input of CAV \(i\) remain within acceptable ranges, constraint (11) is imposed:  
\[0\leq v_{\text{min}}\leq v_{i}(t)\leq v_{\text{max}}=v_{0},\ u_{\text{min}} \leq u_{i}(t)\leq u_{\text{max}},\ \forall t\in\left[t^{0}_{i},\ t^{x}_{i}\right] \tag{11}\]  
These constraints are effective within the interval \(\left[t^{0}_{i},t^{x}_{i}\right]\), where \(t^{0}_{i}\) and \(t^{x}_{i}\) denote the instants when the CAV \(i\) enters the CZ and exits the MZ, respectively. In accordance with the assumptions outlined in Section 2.1, all CAVs traverse the OZ at a consistent velocity \(v_{0}\) and maintain a uniform velocity \(v^{m}_{i}\) upon entering the MZ. Given the arrival time \(t^{e}_{i}\) of any  
Figure 4: Network representation of centralized CSP. CAV \(i\) in the OZ, the time \(t_{i}^{0}\) when CAV \(i\) enters the CZ can be ascertained by incorporating the respective travel time to \(t_{i}^{e}\). Similarly, the departure time \(t_{i}^{x}\) when CAV \(i\) exits the MZ can be directly computed by adding the travel time to \(t_{i}^{m}\), as illustrated in the ensuing equation:  
\[t_{i}^{0}=t_{i}^{e}+\frac{v_{0}}{H},\ t_{i}^{x}=t_{i}^{m}+\frac{v_{i}^{m}}{S} \tag{12}\]  
A primary focus of research in TOP is the reduction in energy consumption [9]. For HEVs, scholarly investigations have proposed leveraging [PERSON]'s minimum principle for analysis and resolution, effectively converting electrical energy into prospective fuel consumption equivalents [23]. As indicated by the study cited in [24], the energy consumption rate of CAV \(i\) is contingent upon its control input \(u_{i}(t)\), exhibiting a monotonic increase with the augmentation of \(u_{i}(t)\). By amalgamating the aforementioned studies, TOP can be reformulated into the subsequent energy-optimal control problem for each CAV \(i\)[9]:  
\[\min_{u_{\min}\leq u_{i}(t)\leq u_{\max}}\int_{t_{i}^{0}}^{t_{i}^{m}}C_{i}(u_{ i}(t))dt \tag{13}\]  
\[\text{subject to }(\ref{eq:10}),\ (\ref{eq:11}),\]  
\[p_{i}(t_{i}^{0})=0,\ p_{i}(t_{i}^{m})=L,\ v_{i}(t_{i}^{0})=v_{0}. \tag{14}\]  
Here, \(C_{i}(u_{i}(t))\) represents the energy consumption rate functional, and, when \(C_{i}(u_{i}(t))=\frac{1}{2}\ u_{i}^{2}(t)\), the goal of problem (13) is to minimize transient engine operation [11,12]. Since internal combustion engines achieve their maximum efficiency at steady-state operating points [24], pursuing this objective directly leads to advantages in terms of fuel consumption and emissions [19]. Problem (13) is subject to multiple constraints, encompassing vehicle dynamics (10) and velocity and control input (11). In our investigation, \(p_{i}(t)\), \(v_{i}(t)\), and \(u_{i}(t)\) are defined to denote the position (the distance traveled upon entering the CZ), velocity, and acceleration (control input) of CAV \(i\), respectively. Additionally, \(t_{i}^{0}\) and \(t_{i}^{m}\) signify the times at which CAV \(i\) enters CZ and exits MZ, respectively.