# A Carbon Benefits-Based Signal Control Method in a Connected Environment
## 1 Upper-level model
### Lower-level model  
The goal of the lower-level model is to reduce carbon emissions. The carbon emissions are quantified by the total number of stops and vehicles stop time. The reduction in carbon emissions is achieved by CBCVs' speed guidance. The lower-level model is a mixed-integer linear programming (MILP) problem. The objective of the lower-level model is as follows:  
\[\text{P2}:\underset{x^{\omega_{k}^{\xi}}}{min}=\sum_{t_{C}^{\xi}}p_{f}\left( x^{\omega_{k}^{\xi}}\right)+\alpha p_{t}\left(x^{\omega_{k}^{\xi}}\right) \tag{17}\]  
The decision variables are the trajectories of CBCVs, which are denoted as \(x^{\omega_{k}^{\xi}}\). \(p_{f}\left(x^{\omega_{k}^{\xi}}\right)\) represents the stop frequency, \(p_{t}\left(x^{\omega_{k}^{\xi}}\right)\) represents the duration of vehicle stops, and \(\alpha\) is the weight of the stop time, with \(\alpha\ll 1\). \(\mathbf{\Omega}^{k}\) is a regrouping of \(\mathbf{\Omega}^{k}\) based on CBCVs, as follows:  
\[\mathbf{\Omega}^{k}=\bigl{\{}\underbrace{1,2,3^{c},4,5,6,7^{c},8,9,10^{c},\ldots,\ldots,\underbrace{\left(\omega_{k}-m\right)^{c}\ldots,\omega_{k}-1,\omega_{k }}}_{\Omega_{\omega_{k}^{\xi}}^{k}}\bigr{\}} \tag{18}\]  
\[\mathbf{\Omega}^{k}=\Bigl{\{}\Omega_{\omega_{k}^{\xi}}^{k},i=0,1,2,\ldots\omega_{k }^{c}\Bigr{\}} \tag{19}\]where \(\Omega_{0}^{k}\) denotes the scenario where the lead vehicle of the group is a non-CBCV, occurring only when the first vehicle at the stop line is a non-CBCV. \(p_{f}(x^{\omega_{k}})\) and \(\alpha p_{t}(x^{\omega_{k}})\) can be calculated as follows:  
\[p_{f}\Big{(}x^{\omega_{k}^{c}}\Big{)}=\sum_{\omega_{k}^{c}}^{\Omega_{k}^{k}} \sum_{t=t_{0}}^{\omega_{k}}\delta^{\omega_{k}}(t) \tag{20}\]  
\[p_{f}\Big{(}x^{\omega_{k}^{c}}\Big{)}=\sum_{\Omega_{k_{k}^{c}}^{k}}\sum_{t=t_{0 }}\eta^{\omega_{k}}(t) \tag{21}\]  
where \(p_{f}\Big{(}x^{\omega_{k}^{c}}\Big{)}\) represents the total number of stops in the queue with vehicle \(\omega_{k}^{c}\) as the lead vehicle. \(p_{f}\Big{(}x^{\omega_{k}^{c}}\Big{)}\) denotes the total stop time in the queue with vehicle \(\omega_{k}^{c}\) as the lead vehicle. \(\delta^{\omega_{k}}(t)\) indicates the number of stops made by vehicle \(\omega_{k}\) while passing through the intersection. \(\eta^{\omega_{k}}(t)\) represents the stop time of vehicle \(\omega_{k}\). These two parameters can be obtained from \(x^{\omega_{k}}\). The calculation of \(p_{f}\Big{(}x^{\omega_{k}^{c}}\Big{)}\) and \(p_{t}(x^{\omega_{k}})\) can be derived from the optimized trajectories of CBCVs and the predicted trajectories of non-CBCVs. 1. Trajectory prediction for non-CBCVs  
The trajectories of non-CBCVs can be predicted based on car-following behavior and vehicle location information, as shown below:  
\[x^{\omega_{k}}(t)=f\Big{(}x^{\omega_{k}-1}(t),x^{\omega_{k}}(t_{0}),\tau_{cf},d_{cf}\Big{)} \tag{22}\]  
\(\tau_{cf}\) and \(d_{cf}\) are constants that satisfy the parameters of the [PERSON] car-following model. Equation (22) demonstrates that the trajectory of vehicle \(\omega_{k}\) is determined by vehicle \((\omega_{k}-1)\), the initial position, and the parameters of the car-following model. The prediction method is as follows:  
\[x^{\omega_{k}}(t)=\left\{ \begin{aligned} x^{\omega_{k}}(t-1)+ \mathbf{v}^{\omega_{k}}(t-1)\Delta t,u_{k}^{i}\leq t<o_{k}^{i+1}\\ 0,r_{\omega_{k}}(t)=1\end{aligned} \right. \tag{23}\]  
\[\mathbf{v}^{\omega_{k}}(t)=\left\{ \begin{aligned} v_{l},x^{\omega_{k}} (t)>x^{\omega_{k}-1}\Big{(}t-\frac{\tau_{cf}}{\Delta t}\Big{)}+d_{cf}\\ \mathbf{v}^{\omega_{k}-1},x^{\omega_{k}}(t)=x^{\omega_{k}-1}\Big{(}t- \frac{\tau_{cf}}{\Delta t}\Big{)}+d_{cf}\\ 0,r_{\omega_{k}}(t)=1\;and\;o_{k}^{i}\leq t<u_{k}^{i+1}\end{aligned} \right. \tag{24}\]  
\(x^{\omega_{k}}(t)\) can be divided into two scenarios: when the vehicle is the lead vehicle at the stop line facing a red light on the approach lane and when it is not the lead vehicle, as shown in Equation (23). The term \(r_{\omega_{k}}(t)=1\) indicates whether vehicle \(\omega_{k}\) is the leading vehicle at the stop line encountering a red light on the approach.