This section provides a detailed representation of the proposed method, including parameters and notations, control structure, model formulation, and solution method.
4.4. Signal Control Optimization Module
The goal of this module is to reduce carbon emissions at intersections. In this scenario, the additional emissions originate from speed alterations and stops necessitated by red lights while navigating intersections. The lower the speed of vehicles passing through an intersection, the greater the emissions produced as vehicles accelerate to regain their desired speed.
Building on this analysis, a bi-level optimization model is proposed. At the upper level, signal timing optimization aims to maximize the average speed of vehicles. For computational efficiency, minimizing the vehicles’ delay is employed as a surrogate goal for maximizing average speed, which equivalently contributes to reducing carbon emissions. The lower-level model focuses on optimizing speed recommendations for CBCVs, leveraging the guiding efficacy of CBCVs to minimize the number and duration of vehicle stops. This integrated approach, optimizing both signal timing and vehicle trajectory, significantly reduce emissions, aligning intersection control with environmental sustainability goals. Details are illustrated in
Figure 4.
- 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:
where
represents the total delay at the intersection. The calculation is as follows:
where
denotes the phase number and
is the set of all phases within the intersection, which is divided into
phases. Each phase change follows a predetermined sequence. The set
represents the collection of vehicles within the control area corresponding to phase
, and
represents the vehicle number, indicating the
-th vehicle in phase
.
denotes the delay of vehicle
during passing through the intersection.
The signal timing plan for the intersection is represented by
.
where
represents the start time of the
-th green time for phase
within the planning window, and
represents the end time of the
-th green time for phase
.
denotes the number of optimization cycles within the model, meaning each optimization provides the control plan for the next
cycles.
- a.
Calculation of Delay
The delay experienced by vehicles is jointly influenced by their trajectory and signal timing, captured by the following formula:
where
denotes the distance between vehicle
and the stop line, and
indicates the time when vehicle
crosses the stop line within this window.
indicates the start time of the control algorithm.
represents the planning time interval.
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.
is the time taken by the vehicle to accelerate from crossing the stop line to its desired speed
.
signifies the time the vehicle spends in free flow to reach the point where it achieves
, which is calculated as follows:
where
is the speed of the vehicle as it crosses the stop line,
denotes the speed limit of the road, and
represents the maximum acceleration capacity of the vehicle.
indicates the time required for the vehicle
to travel a distance
.
represents the length of the control area.
- b.
Constraints on Signal timing.
In the process of signal optimization, the constraints that must be adhered to include the minimum green time constraint, the maximum green time constraint, the phase sequence constraint, and the clearance time constraint. They are respectively represented as follows:
where
represents the clearance time required during phase switching.
denotes the minimum green time duration,
specifies the maximum green time duration. Equations (11) and (12) are the constraints for the minimum and maximum green time duration of a phase. Equation (13) constrains the start and end times of the signal phase. Equation (14) addresses the constraints on the phase clearance time.
- c.
Constraints on the smoothness of signal timing.
This research adopts a rolling horizon scheme [
39]. It is able to continuously collect and update real-time traffic conditions during the optimization process, reducing errors caused by vehicle uncertainties.
In the process of rolling-horizon optimization, the signal timing optimization is repeated in every
. To mitigate the adverse effects of abrupt changes in signal timing updates on traffic, a portion of the timing plan from the previous optimization is retained during each optimization. Specifically, the first optimization occurs at time
, which is followed by a second optimization at
. During the second optimization, a safety interval
is introduced to preserve the green phases planned before
. To accommodate this, the model incorporates constraints:
where
represents the signal timing solutions from the previous optimization.
- 2.
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:
The decision variables are the trajectories of CBCVs, which are denoted as
.
represents the stop frequency,
represents the duration of vehicle stops, and
is the weight of the stop time, with
.
is a regrouping of
based on CBCVs, as follows:
where
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.
and
can be calculated as follows:
where
represents the total number of stops in the queue with vehicle
as the lead vehicle.
denotes the total stop time in the queue with vehicle
as the lead vehicle.
indicates the number of stops made by vehicle
while passing through the intersection.
represents the stop time of vehicle
. These two parameters can be obtained from
.
The calculation of and can be derived from the optimized trajectories of CBCVs and the predicted trajectories of non-CBCVs.
- a.
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:
and are constants that satisfy the parameters of the Newell car-following model. Equation (22) demonstrates that the trajectory of vehicle is determined by vehicle , the initial position, and the parameters of the car-following model.
The prediction method is as follows:
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 indicates whether vehicle is the leading vehicle at the stop line encountering a red light on the approach. Conversely, suggests the opposite scenario. denotes the predicted speed of vehicle at time . 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:
Equation (26) incorporates the logical variable . The condition holds when . Otherwise, .
- b.
Target speed optimization for CBCVs
The 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:
- (a)
Trajectory planning of CBCVs:
The kinematic equation for CBCVs is
where
represents the speed of vehicle
at time
t, and
represents the acceleration of vehicle
at time
. The following constraints apply:
Acceleration limits constrain:
Vehicle following constraint:
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).
- (b)
Speed guidance for CBCVs:
where
represents the target speed for vehicle
at time
. The term
denotes the transmission interval for speed instructions, during which the vehicle receives a single speed guidance command.
provides a calculation method for speed instructions with a period of
.
4.5. Carbon Benefit Calculation Module
This section provides a detailed explanation of the carbon emission benefit calculation for CBCVs, as illustrated in
Figure 5.
- a.
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.
- b.
Execution Deviation Evaluation Module
Based 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:
where
represents the actual speed of vehicle
at time
.
denotes the threshold for judging whether the vehicle has complied with the speed guidance instructions.
- c.
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:
represents the carbon reward amount allocated to . denotes the reduction in carbon emissions by vehicle at the intersection. 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,
can be calculated, as shown below:
where
represents the emissions of all vehicles within the group
when vehicle
did not implement the speed guidance strategy.
denotes the emissions of all vehicles within the group
when vehicle
had executed the speed guidance strategy. The function
is used to calculate the emissions produced by vehicle
as it passes through an intersection, typically employing a microsimulation model to accurately estimate emissions.
4.6. Solution
In this section, a GA-PSO algorithm is designed to solve the proposed bi-level optimization model, optimizing the signal timing and the optimal trajectories of CBCVs. The methodology proceeds as follows:
Step 1: Initialization. Initialize parameters for the GA-PSO, encompassing population size , crossover rate , mutation rate , inertia weight , cognitive constant , social constant , and the maximum number of iterations .
Step 2: Generation. Generate initial populations of signal plans, ensuring compliance with signal control constraints Equations (11)–(16).
Step 3: Solution. For each population, solve the standard MILP model P2 to optimize the trajectories of CBCVs using common solvers and subsequently predict the trajectories of non-CBCVs .
Step 4: Fitness Calculation. Evaluate the fitness of each upper-level model configuration using the predicted trajectories and corresponding signal plans.
Step 5: Iteration Check. Assess whether the optimization iteration limit R has been reached. If so, output the optimal result. If not, advance to step 6.
Step 6: Selection and Update. Select individuals with high fitness scores and update the GA-PSO population. This includes selection, crossover, and mutation processes (GA components) as well as updates to velocity and position (PSO components). Subsequently, cycle back to Step 3.
The algorithm flow chart is depicted in
Figure 6.