## 4 Speed Guidance Strategy Considering the Uncertainty of Leading Vehicles
### Speed Guidance Model for Mixed Traffic Flow  
To guide the CAV platoon in mixed traffic flow, the platoon is divided, and the headway is used to determine the platoon's boundary. If the headway between adjacent vehicles is less than a standard value, the vehicles are classified as the same platoon; otherwise, they form different platoons. When the vehicle platoon size is 1, the traffic strategy and expected acceleration are consistent with the algorithm of the CAV traffic model constructed in the previous chapter. When the vehicle platoon size is greater than 1, the entire platoon is regarded as a whole entity, and the strategy of leading one vehicle is used to lead multiple vehicles, reducing the complexity of the algorithm and improving the traffic efficiency. Using the car-following model constructed for CAV above, the target acceleration for the leading vehicle passing through the stop line is calculated. Subsequently, the speeds of the preceding vehicle and the improved CACC model are combined to keep a tight car-following distance, ensuring that as many vehicles in the platoon as possible pass the stop line. If a vehicle in the platoon cannot pass the stop line within the current green light phase, the platoon needs to be disbanded. Figure 5: Schematic of speed guidance. The time \(t_{2j}(j=1,2,3,\cdots Q)\) for the following vehicles in the platoon to cross the stop line should satisfy the following constraints:  
\[t_{2j}\in \begin{cases}\left(0,\,t_{r,g}-h_{s}\right]&\text{Green light phase}\\ \left[t_{r,g},t_{r,g}+T_{green}-h_{s}\right]&\text{Red light phase}\end{cases}  \tag{8}\]  
\[t_{2j}=\tau_{c}+\frac{v_{\text{max}}-v_{ij}}{a_{cjj}^{*}}+\frac{L_{g}+x_{ij}-v _{ij}\tau_{c}-\frac{v_{\text{max}}^{2}-\left(v_{ij}\right)^{2}}{2a_{cjj}^{*}}} {v_{\text{max}}} \tag{9}\]  
All parameters can be found in Equations in the preceding text. (2) The deceleration guidance strategy  
When the target platoon enters the speed guidance area, when the remaining time of the green light phase is shorter or the remaining time of the red light phase is longer, guide the platoon leader to slow down to the target speed \(v_{\text{1j}}^{*}\left(v_{\text{min}}<v_{\text{1j}}^{*}<v_{\text{max}},j=1,2,3,\cdots Q\right)\) to achieve the purpose of the green light phase just starting when the leading vehicle drives to the stop line, then the time \(t_{\text{1j}}(j=1,2,3,\cdots Q)\) through the stop line shall meet the following constraints:  
\[t_{\text{1j}}= \begin{cases}\tau_{d}+\frac{v_{\text{1j}}^{*}-v_{\text{1j}}}{b} +\frac{t_{\text{2s}}-v_{\text{1j}}\tau_{d}-\frac{\left(v_{\text{1j}}^{*} \right)^{2}-\left(v_{\text{1j}}\right)^{2}}{2b}}{v_{\text{1j}}^{*}},\text{ Front vehicle type is HDV}\\ \tau_{c}+\frac{v_{\text{1j}}^{*}-v_{\text{1j}}}{b}+\frac{t_{\text{2s}}-v_{ \text{1j}}\tau_{c}-\frac{\left(v_{\text{1j}}^{*}\right)^{2}-\left(v_{\text{1j }}\right)^{2}}{2b}}{v_{\text{1j}}^{*}},\text{ Front vehicle type is CAV}\end{cases}  \tag{10}\]  
\[t_{\text{1j}}\in \begin{cases}\left(t_{r,g}+T_{red},t_{r,g}+T_{red}+T_{green} \right)&\text{Green light, no traffic at signal}\\ \left[t_{j-1}+h_{s},t_{r,g}+T_{red}+T_{green}\right]&\text{Green light, vehicles present at signal}\\ \left[t_{r,g},t_{r,g}+T_{green}\right]&\text{Red light, no traffic at signal}\\ \left[t_{j-1}+h_{s},t_{r,g}+T_{green}\right]&\text{Red light, vehicles present at signal}\end{cases}  \tag{11}\]  
In these formulas, \(T_{red}\) is the total duration of the red light phase at the intersection signal (including the yellow light's duration), and all the other parameters, \(v_{\text{max}},\,a_{\text{max}},\,\tau_{d},\,\tau_{c},\,\text{b}\), can be found in Equations in the preceding text. From Formula (4), the following deceleration of the car is \(b_{c,ij}\) (\(j=1,2,3,\cdots Q\)), and the best deceleration \(b_{c,ij}^{*}\) of the car is:  
\[b_{c,ij}^{*}=\min\left[\left|b_{c,ij}\right|,\left|b\right|\right] \tag{12}\]  
The min represents that the best deceleration of the rear car is the minimum value between the two.