1. Introduction
Eco-driving is an emerging practice aimed at enhancing energy efficiency and curbing transport emissions. It involves optimizing vehicle operation by minimizing unnecessary stops, acceleration, and deceleration and maintaining a steady speed to reduce energy consumption, improve traffic mobility, and decrease emissions [
1,
2,
3]. The advent of connected autonomous vehicles (CAV) is also seen as a promising development that will further facilitate eco-driving by providing more space and opportunities for its implementation.
In the future, the adoption of CAVs is expected to trigger significant transformations in the safety, accessibility, and traffic flow dynamics of road transportation. By harnessing the power of traffic sensing, data collection, analysis, and processing, CAVs can employ eco-driving strategies to minimize vehicle energy consumption, conserve resources, and enhance economic efficiency [
4]. However, the development of effective traffic management and control mechanisms for CAVs on the road still presents a pressing challenge in the transportation sector.
Signalized intersections serve as critical nodes within the urban road network where the smooth transition of traffic between interrupted and uninterrupted flow is crucial. However, these intersections often cause traffic interruptions and delays. Implementing eco-driving control for CAVs at signalized intersections is, therefore, essential to mitigate the negative impact on vehicles and traffic flow.
Current research on eco-driving methods for connected autonomous vehicles (CAVs) at traffic signal intersections has primarily relied on rule-based control methods. These methods adjust the vehicle’s longitudinal speed based on signal state information to enable it to pass through the intersection without unnecessary stops. For example, Jin et al. [
5] developed a mathematical model that categorized intersection signal states into six situations and optimized the movement of a single vehicle upstream and downstream of the intersection. Similarly, Lu et al. [
6] computed appropriate constant velocities for continuous signalized intersections and employed smooth trigonometric curves to represent velocity changes during acceleration and deceleration, ensuring seamless passage through the intersection. However, as traffic environments become more complex, rule-based control methods have limitations, and optimizing control methods are computationally intensive. Hence, research on eco-driving for connected autonomous vehicles at signalized intersections now focuses on establishing a comprehensive optimization model that considers multiple objectives, including safety and efficiency.
Dynamic eco-driving modeling is an integrated framework that includes input space models, fuel consumption models, vehicle dynamics models, optimization objectives, and impact analyses of the eco-driving system on road areas and traffic signal control strategies [
7]. Optimal control methods are preferred due to their low computational complexity. Zhang et al. [
8] proposed a constrained optimization model that approximates optimal results similar to the pseudo-spectral method. Cheng et al. [
9] introduced a model-free control method based on the Monte Carlo search tree algorithm. Decentralized control strategies have been effective in improving road traffic efficiency while considering CAV eco-driving, such as the model proposed by Yao and Li [
10], which minimizes vehicle travel time, fuel consumption, and safety risks.
The behavior of multiple CAVs at signalized intersections has a significant impact on the application of eco-driving. Hence, studying connected vehicle queues is critical to reducing traffic congestion, improving road capacity, and decreasing energy consumption. Jiang et al. [
11] and Chen et al. [
12] developed a system for multi-vehicle driving at signalized intersections by combining a microscopic vehicle-following model with optimal control methods. Zhang et al. [
13] and Chen et al. [
14] proposed vehicle emission row schemes and the concept of “1 + n” mixed platoon to enhance overall traffic efficiency and fuel consumption at intersections. Wang et al. [
15] suggested that the front CAV’s speed trajectory provides a reference for the rear vehicle, leading to effective energy savings. Xu and Deng [
16] divided vehicles into fleets and calculated four speed guidance models to achieve optimal acceleration and deceleration and pass the signalized intersection at the same target speed as the lead vehicle. Various optimization control methods are employed in eco-driving research, including Pulse-and-Glide (PnG) cycle control [
17], model predictive control (MPC) [
18,
19,
20], deep learning algorithms [
21,
22,
23], and optimal control [
24,
25]. However, most of these approaches are computationally intensive and challenging to apply to the centralized control of traffic flow at signalized intersections. It is worth noting that, although Rad S R et al. [
26] proposed a conceptual framework for designing dedicated lanes for connected autonomous vehicles, it requires intelligent infrastructure, which may be challenging to implement. This paper proposes an optimal control method for CAVs at signalized intersections and a decentralized control strategy to mix traffic at existing signalized intersections, enabling CAVs to promote other environmentally friendly driving vehicles safely, smoothly, and indirectly, thus reducing total road energy consumption. The main contribution of our paper is to extend the optimal control method for intersections by considering mixed traffic flows and integrated energy consumption models. This method can be applied to mixed traffic flow and meet actual traffic conditions, and it can effectively reduce the energy consumption of human-driven vehicle (HDV) and CAV mixed traffic while balancing traffic efficiency. Moreover, we comprehensively analyze the energy-saving potential of connected autonomous vehicles at signalized intersections.
The paper is divided into four main sections. The first section describes the scenario and motivation for the study. The second section proposes an optimal control method using optimal control theory. In the third section, simulations and analyses of the proposed method are presented. Finally, the last section concludes the research and summarizes the main findings.
2. Scenario Description
We consider a mixed traffic scenario at a typical signal-controlled intersection, as shown in
Figure 1, where both HDVs and CAVs travel through the intersection in sequence. The intersection has fixed signal phase and timing (SPaT) information, and the signal is located at the center. To facilitate eco-driving, the intersection is divided into two areas based on previous research: the control zone and the merging zone. The control zone is where CAVs obtain SPaT and vehicle information, while the merging zone is the area within the green box line in the figure where the signals guide vehicles through the intersection.
Our study focused on optimizing the eco-driving of individual CAVs in a single-lane control zone at a signal-controlled intersection. We developed an optimal control method based on the vehicle’s dynamics, initial and final states, and constraints to minimize energy consumption by optimizing the acceleration of the vehicle phase. Each CAV obtains relevant information for optimal control upon entering the control zone and controls its acceleration through the intersection accordingly. We also used the Intelligent Driver Model (IDM) [
27] to model the trajectory of HDVs and follow them specifically. To facilitate information processing and calculations, we set the control zone distance as
l and used the stop line as the origin.
Our control design and analysis for the signal-controlled junction are based on the following assumptions.
All CAVs are connected, meaning they can transmit their own and surrounding vehicle information through wireless communication with real-time communication between vehicles and infrastructure. Communication delays or packet loss are not considered.
All CAVs can drive independently and follow the speed trajectory specified by the intelligent decision and algorithm system upon entering the control zone. HDVs are assumed to behave ideally.
To optimize vehicle efficiency in the control zone, overtaking or lane changing is not allowed. Furthermore, the impact of other roads on vehicle travel is not considered.
All vehicles passing through the signalized intersection are fueled vehicles, and other traffic disturbances, such as pedestrians or non-motorized vehicles, are not taken into account.
3. Methodology
Our study proposes an eco-driving method for CAVs that considers various practical driving situations in the control zone. We first define the vehicle state upon entering the control zone and then discuss the specific conditions that CAVs may encounter in this area. Finally, we develop an optimal control model for CAVs based on defined objectives and constraints.
3.1. Dynamical Modeling
Acquiring vehicle state information is vital for dynamic modeling; however, these states are not easily obtained directly. Numerous studies have been conducted to describe vehicle dynamics and design estimators by integrating the Global Navigation Satellite System (GNSS) [
28], Inertial Measurement Unit (IMU) [
29,
30], and cameras [
31] for state estimation. To provide an overview of the state descriptions commonly used in these studies, we assume that there are N vehicles entering and leaving the control zone at a specific time, numbered in chronological order. The position of CAV
i (
i ∈ N) at time t is represented as
; the velocity at time t is denoted by
; and the acceleration at time t is expressed as
. The system state vector of CAV
i is given by
, and the longitudinal dynamics can be represented in a second-order form, as follows:
Then, the system equation of state for CAV i is modeled as .
3.2. Cost Function
Eco-driving for CAVs at signalized intersections should prioritize driving safety and not hinder traffic maneuverability, as established in previous research and practical considerations. With these factors in mind, the primary control objective of vehicle eco-driving at signal intersections is to minimize total energy consumption. The optimal control cost function for CAV
i is defined as
where
is the time when CAV
i enters the control zone, i.e., reaches the boundary of control zone as shown in
Figure 1, and the time for CAV
i to reach the stop line is the final time,
, which will be discussed later in
Section 3.4.
As the terminal cost function in Equation (2),
represents the error between the vehicle and the desired state at the final time, which is expressed by Equation (3). The terminal cost ensures the CAV can enter the intersection on time at a preferred speed. Note that the stop line is set as the
position, and the first term of Equation (3) is expressed as the deviation of CAV
i from
at the final time,
; the second term of Equation (3) is expressed as the deviation of CAV
i from the desired velocity,
, at the final time,
. In Equation (3),
are the penalty weighting coefficients for the position deviation and speed deviation, respectively, which ensure a constraint on the vehicle’s final state. The discussion on the desired velocity at the final time is in
Section 3.4.
In Equation (2),
is the operating cost, and the first term in Equation (4) indicates the immediate energy consumption of CAV
i at time t. The second term is a term created considering the driving comfort of CAV
i. The sharpness and duration of acceleration and deceleration of the vehicle have a significant effect on the fuel consumption of the vehicle. We chose to utilize the instantaneous fuel consumption model proposed by M.A.S. Kamal et al. [
32] based on vehicle dynamics, which is suitable for conventional vehicles. At the same time, we assume that all vehicles on the road are fuel-powered vehicles with the same construction and assembly. The energy consumption model is shown below:
and
The model uses standard vehicle engine characteristics and records velocity and acceleration to gather fuel consumption data, which are then used to create fuel consumption estimation equations through curve fitting. In Equation (5),
is the energy consumption caused by CAV
i when the speed is
at moment
t, and
is the additional energy consumption caused by CAV
i when the acceleration is
at moment
t. It is necessary to note that fuel consumption does not occur when a vehicle is decelerating. Additionally, fuel consumption is a constant value,
a, when the vehicle is idle. The coefficients in the equation are determined through curve fitting, and their values can be found in
Table 1.
3.3. Constraint Conditions
Previous studies have highlighted various constraints present in traffic conditions at signal-controlled junctions. These constraints include vehicle safety, kinematic, and control state constraints.
Vehicle safety constraints are a critical aspect of road traffic and typically involve maintaining a minimum safe spacing between vehicles to ensure stability and safety. In our study, the safety constraint is defined as the requirement for all CAVs to maintain a safe distance from the vehicle in front of them, and the distance between CAV
i and preceding vehicle
i − 1 is
. The safety constraint is described as
where
is the length of the vehicle, and
is the safe vehicle distance for CAV
i at moment
t. The safe vehicle distance can be obtained from
, where
is the safe headway.
Next, we take into account vehicle kinematics and constrain vehicle control based on speed, acceleration, and comfort requirements at the signalized intersection.
The speed limit sets a maximum allowable speed for vehicles on the road, while the minimum speed is constrained by road conditions and the movement of vehicles.
We specify the safe maximum acceleration and maximum deceleration of the vehicle.
We consider maintaining the comfort of driving the vehicle and describe it as the rate of change in acceleration,
, which represents the rate of change in the acceleration of CAV
i at moment
t.
Last, regarding the vehicle control state constraint, the terminal state constraint has been bounded by the terminal cost,
, of Equation (2), and the initial state constraint,
, is the speed of CAV
i at
, the initial moment; then, the initial state constraint is
3.4. Final Time and Desired Velocity
Upon entering the control zone for optimal control, the CAV must gather SPaT information and information on preceding vehicles to determine its desired velocity and the final time to reach the stop line. Previous studies [
33] often set the desired velocity as the restricted velocity to ensure maximum traffic volume at intersections. However, this approach may not balance vehicle fuel consumption and traffic efficiency in practical situations.
Therefore, we analyzed the optimal final time,
, and desired velocity,
, of CAV
i for various scenarios. The signal phase starts from the red phase at time
, and the signal period is
. The red phase time is
, and the green phase time is
. We started by discussing the final time, as it is directly linked to the desired velocity. Assuming that CAV
i arrives at the stop line at the maximum velocity, the earliest possible time to exit the control zone is
.
- (1)
Assuming that there are no vehicles in front of CAV i, the final time and desired speed of the CAV can be determined based on the signal state.
If the
is at the redlight phase with
, CAV i must wait until the next greenlight phase to proceed, and the final time is the start time of that green phase, provided by
If the is at the green phase with , CAV i can pass through in the current phase, and the final time is the earliest possible time, provided by .
- (2)
If there are other vehicles ahead of CAV i, its final time and desired speed will be adjusted based on the preceding vehicles. Assuming that there are n preceding vehicles, CAV i can only exit the intersection once all of the preceding vehicles have left.
If the initial time is at the redlight phase with
, the final time depends on whether all preceding vehicles can leave at the same greenlight phase. If all preceding vehicles can leave at the same greenlight phase, the final time is provided by
. If
, the final time is provided by
However, if the final time is not affected by the preceding vehicles with
, the final time is
. If preceding vehicles fail to cross the intersection all at once with
, CAV
i must wait until the next green phase to cross, and the final time is provided by
If the initial time is at the green light phase with
, the final time must also be determined based on the preceding vehicles. All preceding vehicles can leave at once with
. If
, the final time is provided by Equation (12). If
, the final time can be expressed as Equation (13). However, if the preceding vehicles cannot cross the intersection all at once with
, the final time is provided by
Based on the final time, the desired velocity through the junction is determined. If the final time is the earliest time,
, then the desired velocity is set to the limit velocity, and we have
. However, if the final time is later than the earliest time, the desired velocity should be lower than the limit velocity to better reflect the actual situation of CAV
i passing through the junction:
3.5. Optimal Control Model
The optimal control model for CAV
i in the control zone of a signal-controlled junction can be expressed as a nonlinear optimization problem:
subject to constraints (6)–(10).
To solve this problem, the CAV must first collect SPaT and traffic information and determine the initial and final states, including the final time and desired velocity. The optimal control problem can then be solved using the Gaussian pseudospectra method in GPOPS-II [
34]. GPOPS-II transforms the continuous-time optimal control problem into a nonlinear programming problem (NLP) and uses an NLP solver to obtain the optimal control solution.
3.6. Control Method for Connected Automated Vehicle
We made the assumption that the CAV performs optimal control planning as soon as it enters the control zone and this planning process only occurs once. In fact, due to the low traffic flow and the low influence between vehicles, the eco-driving of the CAV is almost undisturbed. However, the traffic flow is constantly changing, and higher traffic flow can affect the normal driving of adjacent vehicles, thereby disturbing the eco-driving of the CAV.
When the CAV enters the control zone, we considered two scenarios. In the first scenario, if there is no vehicle in front of the CAV or if the CAV and the preceding vehicle meet the conditions stated in Equation (6), the CAV can perform optimal control and drive through the control zone according to the optimized trajectory. In the second scenario, if the CAV and the preceding vehicle do not meet the conditions in Equation (6), the CAV cannot perform eco-driving and must follow the preceding vehicle through the signal intersection using the following model of HDV.
While driving through the control zone, the CAV continuously detects the safe headway from the preceding vehicle in real-time. If the safe headway is satisfied, the CAV follows the optimized trajectory until it leaves the intersection. However, if the CAV detects that it is not at a safe headway from the preceding vehicle, it interrupts the eco-driving and follows the preceding vehicle through the signal intersection using the following model of HDV.
Figure 2 illustrates the driving methods of CAVs in different states.
5. Conclusions
This paper proposed an eco-driving method for CAVs in signalized intersections. When the CAV enters the control zone, it collects SPaT information and makes driving decisions. The eco-driving control in the control zone is determined based on a cost function derived from the vehicle dynamics model, and an optimal control model is constructed according to constraints. The optimal control problem is solved using GPOPS-II, and the CAV passed through the signalized intersection according to the optimal trajectory.
Simulation results were compared and analyzed under different conditions, including varying saturation levels, MPRs, and green ratios. The results demonstrate that the eco-driving method can smooth CAV trajectories, reduce the number of vehicle stops, and guide following vehicles more effectively. Compared with uncontrolled traffic flow, the intervention of CAVs at signalized intersections can improve energy efficiency without affecting traffic efficiency. Under unsaturated conditions (V/C = 0.6), a CAV has a weak effect on the ATTD and average energy consumption, as the intersection can fully release the vehicles on the road. However, under saturated conditions (V/C = 1.0), the eco-driving method can reduce the ATTD to a certain extent and significantly reduce energy consumption. The study also found that increasing the green ratio and MPR can improve energy consumption efficiency by more than 40%. In addition, a larger green ratio and a CAV permeability of 40–60% can guide other vehicles to achieve stable eco-driving, further improving the overall fuel consumption efficiency of the road.
While the research in this paper focused on a single lane at a single signal intersection with a mix of HDVs and CAVs, future studies will consider multi-lane driving and more complex traffic environments, such as lane changes and turning. The impact of multiple consecutive signal intersections on CAV eco-driving will also be studied to establish a more realistic eco-driving method.
Overall, this paper provides valuable insights into the potential benefits of eco-driving for CAVs at signalized intersections and highlights the need for further research in this area to fully consider the complexities of real-world traffic environments.