## 3 Modeling Methodology
### MPC-Based CLC Control Strategy by Considering Preceding Vehicle Switching
#### 3.2.2 Constraints with Preceding Vehicle Switching of CLC Control Strategy  
(1) Speed limitation  
First, the longitudinal speed of vehicle M and its cooperative vehicle A should not exceed the maximum allowable speed in their current lanes, and they should always be positive. \[0<v_{M}(k)\leq\left\{ \begin{array}{l}v_{\text{Omax}}\quad if\ 0\leq y_{M}(k)\leq d/2\\ v_{\text{Tmax}}\quad\quad\quad\quad\text{otherwise}\end{array} \right.,\ k=1,2, \ldots,N \tag{5}\]  
\[0<v_{A}(k)\leq v_{\text{Tmax}},\ k=1,2,\ldots,N \tag{6}\]  
where \(v_{M}(k)\) and \(v_{A}(k)\) represent the longitudinal speeds of vehicles M and A, respectively, and \(v_{\text{Omax}}\) and \(v_{\text{Tmax}}\) represent the maximum allowable speeds in the original and target lanes, respectively. (2) Longitudinal safety distance requirements with preceding vehicle switching  
Second, the longitudinal distance between the controlled vehicles, i.e., vehicles M and A, should be more than the corresponding critical longitudinal safety distance during the two stages of the whole lane-changing process. In the LC-O stage, the main movement area of vehicle M is the original lane. So, the tracking-preceding vehicles of controlled vehicles M and A are vehicles C and B, respectively. In order to ensure a safe lane-changing process, the longitudinal safety distance constraints between vehicles C and M and vehicles B and A should be satisfied. These constraints are expressed using (7) and (8), respectively. Please note that the critical longitudinal safety distance is fixed due to the advantages of high-precision and high-response control of CAVs. \[S_{CM}(k)=x_{C}(k)-x_{M}(k)\geq S_{safe,\ CM}(k)=d_{0},\ k=1,2,\ldots,N \tag{7}\]  
\[S_{BA}(k)=x_{B}(k)-x_{A}(k)\geq S_{safe,\ BA}(k)=d_{0},\ k=1,2,\ldots,N \tag{8}\]  
where \(*(k)\) denotes the related predicted state at step k, \(S_{CM}(k)\), \(S_{BA}(k)\)\(S_{safe,\ CM}(k)\), and \(S_{safe,\ BA}(k)\) represent the actual longitudinal distance and the critical longitudinal safety distance between vehicles C and M, and vehicles B and A, respectively, \(x_{C}(k)\), \(x_{M}(k)\), \(x_{B}(k)\), and \(x_{A}(k)\) represent the longitudinal positions of vehicles C, M, B, and A, respectively, and \(d_{0}\) represents the fixed minimum safety distance. In the LC-T stage, the area where vehicle M is located in the target lane, the tracking-preceding vehicles of controlled vehicles M and A are switched from vehicles C and B to vehicles B and M, respectively. Therefore, the safe longitudinal distance constraints betweenvehics B and M, and vehicles M and A should also be guaranteed. These constraints are mathematically expressed in (9) and (10). \[S_{BM}(k)=x_{B}(k)-x_{M}(k)\geq S_{safe,\ BM}(k)=d_{0},\ k=1,2,\ldots,N \tag{9}\]  
\[S_{MA}(k)=x_{M}(k)-x_{A}(k)\geq S_{safe,\ MA}(k)=d_{0},\ k=1,2,\ldots,N \tag{10}\]  
where \(S_{BM}(k)\), \(S_{MA}(k)\), \(S_{safe,\ BM}(k)\), and \(S_{safe,\ MA}(k)\) represent the actual longitudinal distance and the critical longitudinal safety distance between vehicles B and M, and vehicles M and A, respectively. The aforementioned analysis shows that if the expressions presented in (7)-(10) are satisfied, then the driving safety of the four vehicles in two stages of the lane-changing process is ensured. However, it is noteworthy that these constraints are independent of each other and do not consider the switching problem for the preceding vehicles M and A. In addition, these constraints consider that the tracking of vehicles M and A can show sudden changes. Therefore, establishing an optimization problem based on the premise of a sudden change of the preceding vehicles for the controlled vehicles may lead to unsolvable optimization problems. In reality, vehicle M moving from the original lane to the target lane is a gradual process instead of being an instantaneous process.