# A Carbon Benefits-Based Signal Control Method in a Connected Environment
## 1 Upper-level model  
The goal of the upper-level model is to reduce delay and indirectly maximize the average speed of vehicles. The objective function is defined as follows:  
\[\text{P1}:\;\underset{\text{Y}}{\text{min}}\text{T}\boldsymbol{D} \tag{1}\]  
Figure 4: The signal control optimization module. Figure 3: The process of identifying CBCVs and non-CBCVs. where _TD_ represents the total delay at the intersection. The calculation is as follows:  
\[\textbf{{TD}}=\sum_{k}\sum_{\omega_{k}\in\Gamma^{k}}D_{\omega_{k}} \tag{2}\]  
\[\textbf{{K}}=\left\{[\textbf{{k}_{1}},\textbf{{k}_{2}},\ldots,\textbf{{k}_{n}}]\right\} \tag{3}\]  
where \(k\) denotes the phase number and \(K\) is the set of all phases within the intersection, which is divided into \(n\) phases. Each phase change follows a predetermined sequence. The set \(\Omega^{k}\) represents the collection of vehicles within the control area corresponding to phase \(k\), and \(\omega_{k}\) represents the vehicle number, indicating the \(\omega_{k}\)-th vehicle in phase \(k\). \(D_{\omega_{k}}\) denotes the delay of vehicle \(\omega_{k}\) during passing through the intersection. The signal timing plan for the intersection is represented by \(Y\). \[\textbf{{Y}}=\left\{\left[\textbf{{u}^{i}_{k}},\textbf{{o}^{i}_{k}}\right] \right\},\forall k\in K;i=1,\ldots,N \tag{4}\]  
where \(u^{i}_{k}\) represents the start time of the \(i\)-th green time for phase \(k\) within the planning window, and \(o^{i}_{k}\) represents the end time of the \(i\)-th green time for phase \(k\). \(N\) denotes the number of optimization cycles within the model, meaning each optimization provides the control plan for the next \(N\) cycles. **a. ** **Calculation of Delay**  
The delay experienced by vehicles is jointly influenced by their trajectory and signal timing, captured by the following formula:  
\[\textbf{{D}}_{\omega_{k}}=f(\textbf{{x}}^{\omega_{k}},\textbf{{Y}})=\textbf{{ t}}_{\epsilon}^{\omega_{k}}+\textbf{{t}}_{a}^{\omega_{k}}-\textbf{{t}}_{ free}^{\omega_{k}},\forall\omega_{k}\in\Omega^{k} \tag{5}\]  
\[\textbf{{x}}^{\omega_{k}}=\left\{\textbf{{x}}^{\omega_{k}}(t)\big{|}t=t_{0},t _{0}+\Delta t,t_{0}+2\Delta t\ldots,h_{\omega_{k}}-\Delta t,h_{\omega_{k}}\right\} \tag{6}\]  
where \(\textbf{{x}}^{\omega_{k}}\) denotes the distance between vehicle \(\omega_{k}\) and the stop line, and \(h_{\omega_{k}}\) indicates the time when vehicle \(\omega_{k}\) crosses the stop line within this window. \(t_{0}\) indicates the start time of the control algorithm. \(\Delta t\) represents the planning time interval. \(t_{c}^{\omega_{k}}\) represents the time at which the vehicle passes the stop line, its calculation taking into account the impact of the signal plan changes on the vehicle's trajectory. \(t_{a}^{\omega_{k}}\) is the time taken by the vehicle to accelerate from crossing the stop line to its desired speed \(v_{l}\). \(t_{free}^{\omega_{k}}\) signifies the time the vehicle spends in free flow to reach the point where it achieves \(t_{a}^{\omega_{k}}\), which is calculated as follows:  
\[t_{c}^{\omega_{k}}=\left\{ \begin{matrix}max\left(u_{k}^{i}+\tau_{f},t_{c}^{ \omega_{k}-1}+\delta\left(d_{f}\right)\right),c_{k}^{i-1}<h_{\omega_{k}}<u_{k} ^{i}\\ max\left(h_{\omega_{k}},t_{c}^{\omega_{k}-1}+\delta\left(d_{f}\right)\right),u_{k}^{i}<h_{\omega_{k}}<o_{k}^{i}\end{matrix} \right. \tag{7}\]  
\[t_{a}^{\omega_{k}}=\frac{v_{l}-v_{c}}{a_{l}} \tag{8}\]  
\[v_{c}=\frac{\textbf{{x}}^{\omega_{k}}\left(t_{c}^{\omega_{k}}\right)-\textbf{{ x}}^{\omega_{k}}\left(t_{c}^{\omega_{k}}-\Delta t\right)}{2}+\frac{a_{l}\Delta t}{2} \tag{9}\]  
\[t_{free}^{\omega_{k}}=\frac{\left(v_{l}-v_{c}\right)t_{a}^{\omega_{k}}}{2v_{ l}}+\frac{L}{v_{l}} \tag{10}\]  
where \(v_{c}\) is the speed of the vehicle as it crosses the stop line, \(v_{l}\) denotes the speed limit of the road, and \(a_{l}\) represents the maximum acceleration capacity of the vehicle. \(\delta\left(d_{f}\right)\) indicates the time required for the vehicle \(\omega_{k}\) to travel a distance \(d_{f}\).