## 4 Speed Guidance Strategy Considering the Uncertainty of Leading Vehicles
### Speed Guidance Model for Mixed Traffic Flow
The time \(t_{2j}(j=1,2,3,\cdots Q)\) through the rear line in the platoon shall meet the following constraints:  
\[t_{2j}=\tau_{c}+\frac{v_{\text{1j}}^{*}-v_{ij}}{b_{c,ij}^{*}}+\frac{L_{g}+x_ {ij}-v_{ij}\tau_{c}-\frac{\left(v_{\text{1j}}\right)^{2}-\left(v_{\text{1j}} \right)^{2}}{2v_{c,ij}^{*}}}{v_{\text{1j}}^{*}} \tag{13}\]  
\[t_{2j}\in \begin{cases}\left(0,\,t_{r,g}+T_{red}+T_{green}-h_{s}\right]&\text{ green phase}\\ \left[t_{r,g},t_{r,g}+T_{green}-h_{s}\right]&\text{red phase}\end{cases}  \tag{14}\]  
(3) Acceleration and deceleration guidance strategy  
When vehicle \(k\) in platoon \(j\) cannot keep up with the front portion of the platoon through the intersection, the platoon is divided into two segments, with vehicle \(k-1\)marking the end of the front segment and vehicle \(k\) leading the rear segment. Acceleration or deceleration guidance is applied to the front segment vehicles, whereas the rear segment vehicles undergo deceleration guidance. This approach constitutes a variable acceleration-deceleration guidance strategy. The variable acceleration-deceleration guidance strategy is applicable in the following two scenarios:  
When the target platoon enters the speed guidance zone and the signal light is green, with a preceding platoon \(j-1\) moving steadily in front of the signal, the acceleration process of some vehicles in the target platoon is affected during their attempt to proceed. Upon the target platoon's entry into the speed guidance zone under a red signal light, there is a relatively short remaining duration and a queued platoon \(j-1\) waiting ahead of the signal. In these scenarios, the guidance strategy involves the front portion of the platoon either accelerating or decelerating to align with the tail of platoon \(j-1\), following it through the green light phase to cross the stop line. Meanwhile, the remaining vehicles decelerate towards the stop line, preparing to proceed during the subsequent green light phase. The target speed for the lead vehicle of the preceding platoon \(j-1\) is set in accordance with the speed \(v_{j-1}(j=1,2,3,\cdots Q)\) of platoon \(j-1\), so:  
\[\left\{ \begin{array}{l}v_{j-1}\tau_{d}+\frac{\left(v_{1j}\right)^{2}-\left( v_{0j}\right)^{2}}{2\lambda_{ij}}+v_{j-1}t_{f}+s_{d}\geq L_{g}+x_{s},\text{ Front vehicle type is HDV}\\ v_{j-1}\tau_{c}+\frac{\left(v_{1j}\right)^{2}-\left(v_{0j}\right)^{2}}{2 \lambda_{ij}}+v_{j-1}t_{f}+s_{c}\geq L_{g}+x_{s},\text{ Front vehicle type is CAV}\end{array} \right. \tag{15}\]  
In the formula: \(x_{s}\) is the standard front spacing, take 15 m; \(d_{ij}\) represents the acceleration (or deceleration) of the \(i\)th vehicle in the \(j\)th platoon; \(t_{f}\) is the average dissipation time for vehicles, and all the other parameters can be found in Equations in the preceding text. From Formula (4), the best additional (decreased) speed of the rear car in the previous platoon is \(d_{ij}^{*}\)\((j=1,2,3,\cdots Q)\), so:  
\[\frac{\left(v_{ij}\right)^{2}}{2x_{c}}\leq d_{ij}^{*}\leq\left|d_{ij}^{*} \right|\leq a_{\max} \tag{16}\]  
1. Guide the front part of the platoon of vehicles to accelerate  
When guiding the leading vehicle of the front part of the target platoon to accelerate, the time \(t_{1j}(j=1,2,3,\cdots Q)\) through the stop line shall meet the following constraints:  
\[t_{1j}=\left\{ \begin{array}{l}\tau_{d}+\frac{v_{j-1}-v_{1j}}{d_{ij}^{*}}, \text{ Front vehicle type is HDV}\\ \tau_{c}+\frac{v_{j-1}-v_{1j}}{d_{1j}^{*}},\text{ Front vehicle type is CAV}\end{array} \right. \tag{17}\]  
\[t_{1j}\in\left\{ \begin{array}{l}\left(t_{f}+h_{s},\,t_{rg}\right]\text{ green phase}\\ \left[t_{f}+h_{s},t_{rg}+T_{green}\right]\text{ red phase}\end{array} \right. \tag{18}\]  
The time \(t_{2j}(j=1,2,3,\cdots Q)\) for the rear car in the platoon to pass through the stop line shall meet the following constraints:  
\[t_{2j}=\tau_{c}+\frac{v_{j-1}-v_{ij}}{d_{ij}^{*}}+\frac{L_{g}+x_{ij}-v_{ij} \tau_{c}-\frac{\left(v_{j-1}\right)^{2}-\left(v_{1j}\right)^{2}}{2d_{ij}^{*}}} {v_{j-1}} \tag{19}\]  
\[t_{2j}\in\left\{