# A Carbon Benefits-Based Signal Control Method in a Connected Environment
## 1 Upper-level model
### Carbon Benefit Calculation Module  
This section provides a detailed explanation of the carbon emission benefit calculation for CBCVs, as illustrated in Figure 5.  
1. Data Preparation  
The data required for the carbon benefit calculation module are divided into two parts. The first part consists of the target speed provided to the CBCVs by the control system, which can be obtained from Equation (32). The second part includes the actual trajectory information of CBCVs as they traverse the controlled intersection area until they pass through. The complete set of these data is obtained after the vehicle has fully passed through the intersection. Execution Deviation Evaluation ModuleBased on the speed guidance from the signal control optimization module's output for CBCVs and the actual trajectory outputted by the CBCVs execution, the execution deviation evaluation module assesses whether the CBCVs have correctly executed the speed guidance instructions. It determines the IDs of the CBCVs to be rewarded based on the execution. The evaluation is as follows:  
\[t_{e}<TH \tag{34}\]  
\[t_{e}=\frac{t_{0}}{h_{\omega_{k}^{c}}}\cdot\sum_{t=l_{0}}^{h_{\omega_{k}^{c}}} \frac{\left|\mathbf{v_{gg}^{c}}(\mathbf{t})-\mathbf{v_{at}^{c}}(\mathbf{t})\right|}{v_{gg}^{c} (t)} \tag{35}\]  
where \(v_{at}^{\omega_{k}^{c}}(t)\) represents the actual speed of vehicle \(\omega_{k}^{c}\) at time \(t\). \(TH\) denotes the threshold for judging whether the vehicle has complied with the speed guidance instructions.  
CBCVs Carbon Benefit Calculation  
This model will calculate the corresponding carbon reward amounts based on the IDs of CBCVs. The calculation formula is as follows:  
\[R(\omega_{k}^{c})=CE(\omega_{k}^{c})\times CP \tag{36}\]  
\(R(\omega_{k}^{c})\) represents the carbon reward amount allocated to \(\omega_{k}^{c}\). \(CE(\omega_{k}^{c})\) denotes the reduction in carbon emissions by vehicle \(\omega_{k}^{c}\) at the intersection. \(CP\) represents the incentive amount per unit of carbon emission reduction, which is determined by the pricing of the Chinese carbon trading market.  
When quantifying the carbon benefits of CBCVs, there are two components involved. The first part pertains to the reduction in carbon emissions achieved through speed guidance applied to the CBCVs themselves. The second part concerns the emission reduction in non-CBCVs influenced by the guidance effect of the CBCVs. The proposed method determines the guidance effect of CBCVs by grouping vehicles, as shown in Equations (18) and (19). The IDs of non-CBCVs guided by each CBCV can be identified. Their corresponding trajectories can be obtained using Equations (22)-(26). Based on these vehicle trajectories, \(CE(\omega_{k}^{c})\) can be calculated, as shown below:  
\[CE(\omega_{k}^{c})=CE_{0}^{\omega_{k}^{c}}-CE_{opt}^{\omega_{k}^{c}} \tag{37}\]  
\[CE_{0}^{\omega_{k}^{c}}=\sum_{\mathbf{\Omega}^{k^{c}}}f_{ce}\big{(}\mathbf{x_{0}^{ \omega_{k}}}\big{)} \tag{38}\]  
\[CE_{opt}^{\omega_{k}^{c}}=\sum_{\mathbf{\Omega}^{k^{c}}}f_{ce}(\mathbf{x^{\omega_{k}}}) \tag{39}\]  
where \(CE_{0}^{\omega_{k}^{c}}\) represents the emissions of all vehicles within the group \(\Omega^{k^{c}}\) when vehicle \(\omega_{k}^{c}\) did not implement the speed guidance strategy. \(CE_{opt}^{\omega_{k}^{c}}\) denotes the emissions of all vehicles within the group \(\Omega^{k^{c}}\) when vehicle \(\omega_{k}^{c}\) had executed the speed guidance strategy. The function \(f_{ce}\big{(}x_{0}^{\omega_{k}^{c}}\big{)}\) is used to calculate the emissions produced by vehicle \(\omega_{k}\) as it passes through an intersection, typically employing a microsimulation model to accurately estimate emissions.