# A Carbon Benefits-Based Signal Control Method in a Connected Environment
## 1 Upper-level model
### Lower-level model
Conversely, \(r_{\omega_{k}}(t)=0\) suggests the opposite scenario. \(\mathbf{v}^{\omega_{k}}(t)\) denotes the predicted speed of vehicle \(\omega_{k}\) at time \(t\). This prediction can be categorized into three types: free-flow state, car-following state, and the state where the vehicle is the lead vehicle at the stop line facing a red light on the approach lane, as shown in Equation (24). The linearization forms of Equations (23) and (24) are as follows:  
\[x^{\omega_{k}}(t)=\big{[}x^{\omega_{k}}(t-1)+\mathbf{v}^{\omega_{k}}(t-1)\Delta t \big{]}\big{(}1-r_{\omega_{k}}(t)\big{)} \tag{25}\]  
\[\mathbf{v}^{\omega_{k}}(t)=\big{[}v_{l}\big{(}1-s_{\omega_{k}}(t)\big{)}+\mathbf{v}^{ \omega_{k}-1}s_{\omega_{k}}(t)\big{]}\;\big{(}1-r_{\omega_{k}}(t)\big{)} \tag{26}\]  
Equation (26) incorporates the logical variable \(s_{\omega_{k}}(t)\). The condition \(s_{\omega_{k}}(t)=0\) holds when \(\mathbf{x}^{\omega_{k}}(t)=x^{\omega_{k}-1}\Big{(}t-\frac{\tau_{cf}}{\Delta t} \Big{)}+d_{cf}\). Otherwise, \(s_{\omega_{k}}(t)=1\). 2. Target speed optimization for CBCVsThe results of trajectory optimization in the lower-level model will be sent to the vehicle end of CBCVs for execution. However, trajectory-level commands are challenging for CBCVs to execute with precision. Consequently, it is essential to transform these trajectory commands into speed guidance instructions, as shown below:  
1. Trajectory planning of CBCVs: The kinematic equation for CBCVs is \[\mathbf{v}^{\omega_{k}}(t)=\dot{\mathbf{x}}^{\omega_{k}}(t)\] (27) \[\mathbf{a}^{\omega_{k}}(t)=\dot{\mathbf{v}}^{\omega_{k}}(t))\] (28) where \(\mathbf{v}^{\omega_{k}}(t)\) represents the speed of vehicle \(\mathbf{\omega_{k}}\) at time \(t\), and \(\mathbf{a}^{\omega_{k}}(t)\) represents the acceleration of vehicle \(\mathbf{\omega_{k}}\) at time \(t\). The following constraints apply: Speed limits constrain: \[0\leq\mathbf{v}^{\omega}\leq\mathbf{v}_{l}\] (29) Acceleration limits constrain: \[-a_{L}^{\omega_{k}}\leq\mathbf{a}^{\omega_{k}}(t)\leq a_{U}^{\omega_{k}}\] (30) Vehicle following constraint: \[x^{\omega_{k}}(t)\geq x^{\omega_{k}-1}\bigg{(}t-\frac{\mathbf{\tau}_{cf}}{\Delta t }\bigg{)}+d_{cf},\forall\omega_{k}\in\Omega_{k}\] (31)  
The trajectory planning for CBCVs is conducted sequentially based on the distance from the vehicle's initial position to the stop line from the nearest to the farthest vehicle. Once the trajectory of the first vehicle at the stop line is determined, the subsequent vehicle's identity is assessed: if the following vehicle is a CBCV, its trajectory is planned to ensure compliance with constraints (29) to (31); if the following vehicle is a non-CBCV, its trajectory is predicted according to Equations (25) and (26). Speed guidance for CBCVs: \[\mathbf{v}^{\omega_{k}^{c}}_{\mathbf{sg}}(\mathbf{t})=\mathbf{v}^{\omega_{k}^{c}}\big{(}t_{0} +i\cdot N_{sg}\Delta t\big{)},\ for\ i=0,1,2,\ldots,\bigg{[}\frac{h_{\omega_{ k}^{c}}-t_{0}}{N_{sg}\Delta t}\bigg{]}\] (32) \[\mathbf{v}^{\omega_{k}^{c}}(t)=\frac{x^{\omega_{k}^{c}}\big{(}t+N_{sg} \Delta t\big{)}-x^{\omega_{k}^{c}}(t)}{N_{sg}\Delta t}\] (33) where \(\mathbf{v}^{\omega_{k}^{c}}_{sg}(t)\) represents the target speed for vehicle \(\omega_{k}^{c}\) at time \(t\). The term \(N_{sg}\Delta t\) denotes the transmission interval for speed instructions, during which the vehicle receives a single speed guidance command. \(\mathbf{v}^{\omega_{k}^{c}}(t)\) provides a calculation method for speed instructions with a period of \(N_{sg}\Delta t\).