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Preprint submitted to Elsevier
May 3, 2024
Yangsheng Jiang
School of Transportation and Logistics, National Engineering Laboratory of Integrated Transportation Big
Data Application Technology, National United Engineering Laboratory of Integrated and Intelligent
Transportation, Southwest Jiaotong University, Chengdu, Sichuan 611756, China.
Email: jiangyangsheng@swjtu.edu.cn
Zipeng Man
School of Transportation and Logistics, National Engineering Laboratory of Integrated Transportation Big
Data Application Technology, Southwest Jiaotong University, Chengdu, Sichuan 611756, China.
Email: zpman@my.swjtu.edu.cn
Yi Wang
School of Transportation and Logistics, National Engineering Laboratory of Integrated Transportation Big
Data Application Technology, Southwest Jiaotong University, Chengdu, Sichuan 611756, China.
Email: wangyi1227@my.swjtu.edu.cn
Zhihong Yao (Corresponding author)
School of Transportation and Logistics, National Engineering Laboratory of Integrated Transportation Big
Data Application Technology, National United Engineering Laboratory of Integrated and Intelligent
Transportation, Southwest Jiaotong University, Chengdu, Sichuan 611756, China.
Email: zhyao@swjtu.edu.cn
1
Yangsheng Jiang1,2,3, Zipeng Man1,2, Yi Wang1,2, and Zhihong Yao1,2,3*
1. School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China;
2. National Engineering Laboratory of Integrated Transportation Big Data Application Technology, Southwest Jiaotong University,
Chengdu, Sichuan 611756, China;
3. National United Engineering Laboratory of Integrated and Intelligent Transportation, Southwest Jiaotong University, Chengdu,
Sichuan 611756, China.
Abstract
Connected and automated vehicles (CAVs) have enormous potential to enhance traffic safety,
efficiency, and emissions reduction. However, in the initial phases of CAV development, mixed
traffic comprising CAVs and human-driven vehicles (HDVs) will inevitably coexist in the traffic
system. To fully exploit the benefits of CAVs, dedicated lanes with independent rights of way will
be established. This paper proposes an optimal control strategy for coordinating the mandatory lane-
changing of CAVs from ordinary lanes to dedicated lanes. The strategy develops a centralized two-
stage cooperative optimal control model to optimize the lane-changing sequence and trajectories of
CAVs. In the first stage, a dynamic programming formulation is designed to determine the lane-
changing sequence decisions. The model predictive control (MPC) controller is adopted to
dynamically solve the optimal control problem with a fixed terminal state. In the second stage, we
dynamically and cooperatively designed the longitudinal trajectories of related CAVs. The lateral
trajectories of lane-changing CAVs are planned with a cubic polynomial. The objective function
considers driving comfort and state tracking to ensure traffic smoothness. Simulation results show
that: (1) the proposed strategy can improve the negative impact of lane-changing behavior under
different traffic demand levels. (2) Compared to the benchmark approach, the proposed strategy can
significantly enhance traffic efficiency and driving comfort, particularly in medium-traffic demand.
The strategy can improve the average speed of CAVs by approximately 12% and decrease the
average acceleration by over 45%. (3) The average fuel consumption is positively correlated with
traffic demands and the difference in arrival speeds between lane-changing and dedicated lane
CAVs. (4) The effectiveness of the strategy increases with the length of the lane-changing segment.
However, the marginal benefit becomes negligible when the segment exceeds 300 m. Therefore, the
findings of this paper can provide theoretical support for the cooperative control of CAVs in
dedicated lanes of highways in the future.
Keywords: mixed traffic; connected automated vehicles; dedicated lanes; cooperative lane-changing; model
predictive control; dynamic programming
Correspondence to: Zhihong Yao, School of Transportation and Logistics, Southwest Jiaotong University, Chengdu,
Sichuan 610031, China, E-mail: zhyao@swjtu.edu.cn
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1. Introduction
Connected autonomous vehicles (CAVs) technologies have developed rapidly in recent years
with the continuous advancement of sensing, communication, automation control, and other
technologies (Dong et al., 2022). CAVs are widely acknowledged as pivotal components of future
transportation systems, owing to their advantages for autonomous driving (Bichiou & Rakha, 2019),
reduced traffic emissions (Y. Chen et al., 2022), and enhanced traffic efficiency (He et al., 2022).
Despite the promising prospect of CAVs, their upgrading still takes a long time (Razmi Rad et al.,
2020). Therefore, there will be a long stage in which HDVs and CAVs coexist in traffic systems (Zhao
et al., 2020). Nevertheless, previous research has argued that the heterogeneity in behaviors between
HDVs and CAVs in mixed traffic flow may lead to dangerous accidents (Chityala et al., 2020).
To address this problem, current research and application mainly deploy CAV-dedicated lanes
to separate CAVs and HDVs (Hamilton et al., 2018; Y. Wang et al., 2024). However, in this case, the
coexistence of dedicated and ordinary lanes inevitably leads to interfaces where the two lanes switch
with each other. Specifically, to reduce the interference of HDVs on the dedicated lanes, the
dedicated lanes are physically segregated in most places, leaving only limited areas for the CAV to
perform switching operations (Jang et al., 2013). Obviously, these areas will undoubtedly turn into
bottlenecks affecting traffic flow. As a result, it is important to design cooperative lane-changing
strategies (i.e., determining when and how to change lanes) to minimize the effects of lane-changing
maneuvers. Numerous studies have addressed the cooperative lane-changing problem, primarily
categorized into decentralized and centralized approaches (Z. Wang et al., 2019). In the decentralized
approach, each CAV independently determines the lane-changing strategy and generates the lane-
changing trajectories (Nie et al., 2016). The optimal lane-changing cost for the target vehicle is
guaranteed, but the system cost may be poor. Instead, the centralized approach effectively addresses
the above issues (Yang C. et al., 2022). In the centralized approach, multiple vehicle trajectories have
to be coordinated. Therefore, many studies transform the problem into a vehicle trajectory planning
problem after fixing the lane-changing point. Taking the lane-changing point as a boundary, the
former part controls the longitudinal trajectory of the vehicle to reach the lane-changing point
optimally, while the latter part coordinates the vehicle to complete the lane-changing maneuver.
Despite these methodological advancements, most previous research scenarios concentrated on
CAVs merging from dedicated lanes into ordinary lanes (Xiong et al., 2022; Yang C. et al., 2022).
However, the lane-changing behavior of CAVs is influenced by the time-varying trajectories of HDVs
when merging into dedicated lanes. Failure to optimize the CAV trajectory will significantly impact
3
the steady traffic flow on dedicated lanes (Liu et al., 2023). Furthermore, previous studies have
focused on the safety of the lane-changing process, and mainly coordinated trajectory of the
following vehicle in the target lane (DFV) to decelerate to make space for the subject vehicle (SV) (D.
Yang et al., 2018; N. Li et al., 2021), as shown in Fig. 1 (a). This deceleration process will affect the
upstream traffic efficiency of the target lane, especially when the target lane is a dedicated lane, as
the sudden deceleration of vehicles leads to a continuous deceleration shockwave in the opposite
direction. To address this problem, we propose a new solution: the subject vehicle (SV) is directed
to position itself as close as possible to the leading vehicle of the target gap (DLV) before initiating
the lane-changing maneuver, and the leading vehicle of the target gap (DLV) is instructed to
accelerate during the lane-changing process, if possible, as shown in Fig. 1 (b). This strategy usually
impacts the subject vehicle and the leading vehicle, and reduces the negative effects of lane-changing
maneuvers on the target lane traffic.
(a) Existing strategies for the following vehicle to decelerate
(b) Cooperative lane-changing strategy in this paper
Fig. 1. Two cooperative lane-changing strategies
This paper suggests a dynamic control strategy that utilizes receding horizon optimal control
properties of MPC to resolve the challenge of CAVs merging into dedicated lanes. The objective is to
maximize driving comfort while enhancing lane-changing efficiency. The model has two main
components: Lane-changing Sequence Decision (LCSD) and Dynamic Cooperative Lane-changing
(DCLC). The first part formulates the lane-changing sequence decisions for the lane-changing CAVs
on the ordinary lane in a dynamic and centralized manner. The MPC controller dynamically solves
the optimal control problem with a fixed end state to plan the trajectories for the lane-changing CAVs.
In the second part, the MPC controller is utilized to control the SV and the DLV, with the DFV being
characterized by a car-following model. With safety as a priority, the acceleration of the DLV is
4
primarily focused on dynamically and collaboratively optimizing the longitudinal trajectories of the
relevant vehicles, while a cubic polynomial is employed for the lateral trajectory planning of the SV.
The main contributions of this paper are as follows.
(1) A cooperative lane-changing strategy is proposed for CAVs to merge into dedicated lanes. It can
exhibit strong robustness in optimal control using an iterative method. Meanwhile, allowing
the preceding vehicle to accelerate to mitigate the adverse impact of the lane-change maneuver
on the dedicated lane traffic.
(2) A new algorithm is developed for resolving MPC control issues that achieve precision and
effectiveness. The paper introduces the Hamiltonian function to convert the MPC optimal
control problem into a two-point boundary problem solving, aiming to enhance the accuracy,
efficiency, and adaptability to the dynamic control environment.
(3) In addition, the study examines the impact of different arrival speeds of lane-changing vehicles
on traffic performance in the lane-changing area, offering insights into potential speed limit
strategies.
The remainder of the paper is structured as follows. Section 2 provides a more detailed literature
analysis. Section 3 formulates the problem and the solution framework. The specific algorithm used
to solve the model is detailed in Section 4. Simulation experiments are analyzed in Section 5. Section
6 ends this paper with conclusions and future works.
2. Literature review
Since the proposal of “A model for the structure of lane-changing decisions” by Gipps (1986),
numerous microscopic lane-changing models for vehicles have been proposed. In recent years, there
has been significant development in vehicle-to-infrastructure (V2I) and vehicle-to-vehicle (V2V)
technology. CAVs equipped with on-board sensors can easily sense surroundings and make lane-
changing decisions while performing lane-changing maneuvers and collaborating with surrounding
vehicles to achieve lane-changing maneuvers (Hang et al., 2022).
Cooperative lane-changing decision-making approaches are mainly classified as decentralized
and centralized (Z. Wang et al., 2019). For the decentralized approach, Li et al. (2020) explored the
lane-changing patterns of CAVs in pure CAV traffic. This paper suggests dividing the CAVs in the
cooperative area into multiple sub-problems and devising individual lane-changing trajectories for
each group. Yao et al. (2023) has modeled the vehicle lane-changing process as an optimal control
problem, with the goal of minimizing the fuel consumption of each vehicle, and plans a complete
lane-change trajectory for autonomous vehicles. Other similar work, exemplified by Karimi et al.
5
(2020), explored the problem of highway merging areas. The study establishes a set of control
algorithms to optimize the trajectory of collaborative CAVs, using model predictive control (MPC)
to calculate the optimal control for the CAVs. The decentralized approaches mentioned above focus
primarily on the decision-making and trajectory optimization of lane-changing vehicles, but do not
consider the impact of lane-changing maneuvers on the whole traffic system.
On the other hand, the centralized approach solves this problem well. Hu and Sun (2019)
suggested a cooperative control model for solving the bottleneck problem in highway merging areas.
The model achieves safe and smooth lane-changing by adjusting the traffic flow distribution
upstream of the mainline to make space for the on-ramp CAVs. Z. Wang et al. (2021)proposed a
dynamic cooperative lane-changing model for CAVs. In this paper, the lane-changing trajectory of
SV is updated in real time through three steps: lane-changing decision, collaborative trajectory
planning and trajectory tracking. Improving the lane-changing success rate and reducing the
negative impact on dedicated lane traffic. Sun et al. (2021) proposed a two-stage optimized
cooperative lane-changing method in pure CAV traffic, which also considered the control inputs and
the terminal state tracking to achieve trajectory smoothing during the lane-changing process.
However, most studies have only been conducted in pure CAV traffic environments, and in fact,
cooperative lane-changing decision-making in mixed traffic is a much more applied study. Yang C.
et al. (2022) considered a scenario where a CAV transfers between dedicated lanes and ordinary
lanes, and proposed a dynamic optimal method to provide lane-changing gap decisions and
trajectories for CAVs with a coordinated approach. Xiong et al. (2022) constructed an optimal model
to optimize the lane-changing sequence and trajectories when CAVs merge into HDV lanes and also
considered the uncertain maneuver of HDVs during the lane-changing process using a stochastic
car-following model.
Most research focuses on the lane-changing maneuver of CAVs, leaving dedicated lanes in both
pure CAV traffic and mixed traffic. Only a few studies explored how CAVs merge into dedicated
lanes in mixed traffic environments. Davis (2020) presented a method for CAVs to merge into high
occupancy vehicle (HOV) lanes. The study determines the optimal merging position for the lane-
changing CAV by optimizing the vehicle speed differences and the deviation of the headway. Liu et
al. (2023) proposed a centralized control strategy that considers the speed difference between the
merging CAVs and the CAVs in dedicated lanes and the number of following vehicles in the
dedicated lane to provide a suitable lane-changing gap for the merging CAVs. However, these
studies did not consider the effect of HDVs and lacked efficient lane-changing strategies to adapt to
the time-varying traffic environment.
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The study above focuses on planning the longitudinal trajectory of vehicles during the lane-
changing process. Regarding lateral trajectory planning for vehicles, Yang et al. (2018) introduced a
dynamic lane-changing trajectory planning model. The research considered the dynamic changes of
the surrounding vehicles. In addition, to ensure safe lane-changing maneuvers for vehicles even in
emergency braking situations, the study proposes rollover and collision avoidance algorithms. Hou
et al. (2023a) developed a real-time cooperative lane-changing model for CAVs. The model utilizes a
sine function to describe the CAV's trajectory during the lane-changing process while employing
MPC to prevent potential collisions.
Table 1 compares the differences between previous research. This paper proposes a centralized
cooperative lane-changing model for a mixed traffic environment. In contrast to previous studies,
we focus on the scenario where CAVs merge from ordinary lanes into dedicated lanes, which has
not been studied extensively. At the same time, a dynamic optimization strategy is proposed to
improve the impact of lane-changing behavior on dedicated lanes. To achieve efficient and entire
lane-changing trajectory planning, we dynamically and cooperatively plan the trajectories of related
vehicles based on MPC and plan lateral trajectories for CAVs. Additionally, considering the
uncertain behavior of HDVs, we design a solution algorithm that achieves precision and
effectiveness to ensure the real-time performance of strategy updates.
Table 1. Summary of recent research
Literature
Strategy
Traffic
Scenario
Trajectory
(T. Li et al., 2020)
Decentralized
Pure
MTM
All
(Davis, 2020)
Decentralized
Mixed
MTD
Longitudinal
(Karimi et al., 2020)
Decentralized
Mixed
MTM
Longitudinal
(D. Yang et al., 2018)
Decentralized
Mixed
MTM
Lateral
(Nie et al., 2016)
Decentralized
Pure
MTM
Longitudinal
(Liu et al., 2023)
Centralized
Mixed
MTD
Longitudinal
(D. Yang et al., 2022)
Centralized
Mixed
MTM
All
(Yang C. et al., 2022)
Centralized
Mixed
DTM
Longitudinal
(Xiong et al., 2022)
Centralized
Mixed
DTM
Longitudinal
(K. Sun et al., 2021)
Centralized
Pure
MTM
Longitudinal
(Hu & Sun, 2019)
Centralized
Pure
MTM
Longitudinal
(Xu et al., 2019)
Centralized
Pure
MTM
All
(M. Wang et al., 2015)
Centralized
Pure
MTM
Longitudinal
DTM means dedicated lane changed to mixed lane. MTD means mixed lane changed to dedicated
lane. MTM means mixed lane changed to mixed lane. All means Lateral and Longitudinal.
3. Methodology
3.1. Scenario statement
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We discuss the highway scenario of CAVs merging into a dedicated lane with mixed traffic of
CAVs and HDVs. As shown in Fig. 2, in the mixed traffic environment, to improve traffic safety and
efficiency on a two-lane highway, a road dedicated to CAVs is physically segregated, i.e., the
dedicated CAV lane (Razmi Rad et al., 2020). The ordinary lane permits a mixture of both CAVs and
HDVs. Typically, there are certain entrances or exits near the ramps for lane-changing maneuvers.
This paper concentrates on the practical scenario of CAVs transitioning from the ordinary lane into
the dedicated lane and examines the lane-changing process at the dedicated lane entrance. When
CAVs enter the from the on-ramp, they must frequently make a mandatory
lane-changing to the dedicated lane for a safer driving experience and higher traffic efficiency (Z.
Chen et al., 2017). Before the lane-changing segment, a of a certain length is
established to allow CAVs in the ordinary lane to adjust their trajectories before approaching the
lane-changing segment. Upon entry of the subject CAV (SV) from the ordinary lane into the lane-
changing segment, cooperative trajectory planning is conducted for the relevant vehicles to ensure
compliance with the lane-changing criteria for the SV to change lanes. Considering the randomness
of the driving process, there may be situations where failing to plan a feasible lane-changing
trajectory for the SV. Consequently, we have designed the termination of the lane-changing segment
as the . If the SV fails to initiate the lane-changing process even upon reaching the
mandatory zone, a special treatment will enforce the SV merger into the dedicated lane under such
circumstances.
It is assumed that all CAVs on the road can communicate with surrounding vehicles in real-
time using V2V technology. Detection devices placed strategically on the road can collect
information about vehicles within range and transmit it to the central controller for centralized
decision-making. Fig. 3 presents the framework proposed in this paper. The model consists of two
main parts: a lane-changing sequence decision ( ), and a dynamic cooperative
lane-changing model (). As shown in Fig. 2, in the planning segment, each
CAV has a certain probability of entering any gap in the dedicated lane. Nevertheless, not all
decisions are feasible, and the cost of selecting different gaps varies among CAVs. In the lane-
changing segment, we propose a dynamic cooperative lane-changing model. Upon making a lane-
changing decision, the longitudinal trajectories of the SV and the DLV are cooperatively optimized,
while the DFV follows the SV using a car-following model. The proposed lane-changing strategy
can reduce the speed oscillations of the relevant vehicles during the lane-changing process, thus
reducing the interference with the upstream traffic in the dedicated lane.
This study classifies vehicles into three categories: CAVs in the ordinary lane, HDVs in the
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ordinary lane, and CAVs in the dedicated lane. To distinguish the behaviors of the different types of
vehicles, the intelligent driver model (IDM) is used to describe the behaviors of the vehicles in the
ordinary lane, and cooperative adaptive cruise control (CACC) is used to describe the behaviors of
the CAVs in the dedicated lane. Specifically, the controlled CAVs are governed by the control
strategy outlined in this paper. Meanwhile, the following assumptions are required for modeling
purposes:
(1) CAVs in the dedicated lane are allowed to perform cooperative lane-changing maneuvers.
(2) CAVs in the ordinary lane must merge into the dedicated lane before exiting the lane-changing
segment.
(3) The lane-changing vehicles begin to change lanes as soon as the lane-changing criteria are met
in the lane-changing segment.
(4) The starting point of the planning segment is located at a certain distance from the entry of the
on-ramp to avoid frequent lane-changing by CAVs (Razmi Rad et al., 2020; Van Beinum et al.,
2018).
Fig. 2. Schematic and description of the lane-changing problem
Fig. 3. Dynamic optimization framework
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3.2. Notations
Table 2. The notation of parameters and variables used in this paper
Notation
Explanation
Unit
SV
Subject vehicle
—
DLV
Leading vehicle in the dedicated lane
—
DFV
Following vehicle in the dedicated lane
—
Time stamp
s
Vehicle longitudinal position
m
Vehicle longitudinal speed
m/s
Vehicle longitudinal acceleration
m/s2
Vehicle length
m
State variable of the MPC controller
—
Control variable of the MPC controller
—
Acceptable time gap for lane-changing
s
Speed limits
m/s
Maximum acceleration
m/s2
Minimum deceleration
m/s2
Objective function
—
Updated starting time instant of the dynamic programming
s
The CAVs index in the planning segment of ordinary lane and also the stage
index of dynamic programming
—
The gap chosen by the vehicle
—
The set of available dedicated lane gaps for the vehicle
—
Optimal cumulative cost of the stage
—
Cumulative cost of the system at a specific state in stage
—
the arc cost between two specific states at stage and stage
—
Number of CAVs in the planning segment of the dedicated lane at the time
—
Number of CAVs in the planning segment of ordinary lane at the time
—
Updated initiation time instant of the MPC controller
s
The terminal time instant of the MPC controller
s
The prediction time horizon of the MPC controller
s
Desired lane-changing speed
m/s
Lane-changing point position
m
The initiation time of the lane-changing process
s
The lane-changing period and the prediction time horizon of the MPC
controller
s
Updated initiation time instant of the MPC controller
s
Lateral position of CAVs
m
The terminal longitudinal position of the lane-changing trajectory
m
The terminal lateral position of the lane-changing trajectory
m
Lane length
m
Course angle of CAVs
—
The headway between DLV and SV vehicles
m
The headway between SV and DFV vehicles
m
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3.3. Lane-changing Sequence Decision(LCSD) Model
This section comprehensively overviews the lane-changing sequence decision (LCSD) model.
The LCSD model is a hierarchical optimal control model that employs dynamic programming in the
upper layer to determine the optimal sequence for the CAVs in the planning segment of the ordinary
lane. The time at which a new vehicle enters the planning segment of the ordinary lane is considered
as . At time , the dynamic programming problem is solved, and is updated whenever a new
lane-changing vehicle enters the planning segment from upstream. Subsequently, the lower layer
calculates the system cost for various decisions the upper layer makes using model predictive control
(MPC). It also dynamically plans the longitudinal trajectories of the CAVs in the ordinary lane, with
serving as the sampling period. As the process is iterative, we have developed a model better
suited to the uncertain behavior of surrounding vehicles. With the above description, we present a
comprehensive flowchart detailing the dynamic optimization process, as shown in Fig. 4. The timing
of the dynamic programming decision is denoted as , the initiation time of the MPC controller
as , and the departure time of the CAV from the planning segment as .
Fig. 4. Iterative optimization
3.3.1 Optimal lane-changing sequence model
In the planning segment, we obtain the optimal lane-changing sequence of the subject CAVs and
minimize the segment's cost. This problem involves assigning CAVs in ordinary lanes to
gaps in dedicated lanes. A dynamic programming model has been developed for this problem,
ensuring the system achieves an optimal solution in the current state (Z. Sun et al., 2020). As shown
: The central processor predicts the
feasibility of different gap decisions and
calculates the trajectory costs based on the
MPC controllers ()in each SV.
Based on the given lane-changing
decision, the MPC controller is used to
calculate the optimal trajectory and
output the control strategy within the
time ().
The SV adjusts its acceleration according
to the control strategy within the received
time ().
Updated： Until：
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in Fig. 5, this dynamic programming problem consists of n stages (representing the decision points)
and m+1 potential states (representing the available decisions at each stage). To prevent unnecessary
consideration of excessive vehicles and ensure high solution efficiency of the dynamic programming
problem, it is essential to establish the appropriate planning segment length. The dynamic
programming model is formulated by Eqs. (1)-(2).
(1)
(2)
where is the optimal system cost for stage , i.e., the optimal cost of the system from the initial
stage to the current stage. is the cost under a particular decision in stage . is the arc
cost between stage and a particular state , i.e., the cost of a particular gap decision at the time
, the specific calculations are described in Section 3.3.2.
Fig. 5. Recursive diagram of lane-changing sequence decision
For the subject vehicle (SV), we define its dedicated lane following vehicle (DFV), and dedicated
lane leading vehicle (DLV) under each trajectory planning. Different gap selection scenarios have
different trajectory planning approaches. Fig. 6 illustrates two different gap selection scenarios.
(a) Two continuous cars choose different gaps.
m+1 Gaps on the Dedicated Lanes
n lane-changing CAVs on the Planning Segment
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(b) Two continuous cars choose the same gap.
Fig. 6. Two gap selection patterns
(1) Fig. 6(a) illustrates the general case where the subject CAVs select independent sink gaps.
(2) Fig. 6(b) illustrates the case where two continuous vehicles select the same gap. This situation
arises when there is a considerable distance between the two vehicles preceding and following
the target gap. A virtual vehicle is introduced into the target gap to prevent conflicts between
the optimized trajectories of continuous subject vehicles. As shown in Fig. 7, the virtual vehicle
is inserted between the DLV and DFV. Utilizing this virtual vehicle as the dedicated lane leader
for subject vehicle
enables planning its trajectory without impacting the optimized trajectory
of subject vehicle .
Fig. 7. Schematic diagram for adding the virtual vehicle
3.3.2 Optimal longitudinal trajectory control model
In this section, constraints and performance specifications are formulated for the MPC controller
in the LCSD model. This MPC controller is designed for two main purposes: predicting future states
and providing effective control. The specific details are outlined below.
(1). Using the prediction function of MPC. The controller is employed at a specific time to assess the
viability of lane-changing vehicles in the planning segment under each gap decision.
Subsequently, the cost for the current decision is computed, and the results are fed back to the
optimal lane-changing sequence model to determine the best lane-changing decision.
(2). Once the lane-changing decision has been made, the central controller iteratively optimizes the
longitudinal trajectory of the SV and sends the result to SV in real-time to control its longitudinal
trajectory.
Ideally, a minimal difference between the speed of SV at the lane-changing point and the speed
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of related vehicles in dedicated lanes would mitigate the impact of the lane-changing behavior on
the traffic efficiency in dedicated lanes. Since the lane-changing position has been fixed, we only
need to determine the time and speed of the lane-changing process. Considering the uncertainty
surrounding vehicles around the SV, predicting the behavior of the surrounding vehicles would lead
to an overly complex problem scale, while the model lacks certain robustness. Therefore, we propose
a more efficient predictive control model. This involves planning the longitudinal trajectory of SV at
the time (continuously updated with as the sampling step), under the assumption that the
related vehicles surrounding SV are operating at the current speed. Hence, the desired time and
speed of lane-changing behavior can be estimated (and continuously updated) when the SV obtains
information about the surrounding vehicles through V2V technology.
As shown in Fig. 8, the control process of the MPC controller, given a specific lane-changing
decision , SV acquires the relevant vehicle data at the time . Subsequently, the MPC controller is
employed to regulate SV’s acceleration within the time window , aiming to achieve the target state
at the optimal cost in time . The target state is as follows: When SV reaches the lane-changing point
with a desired lane-changing speed , it maintains a safe distance of with DLV.
are caculated by Eqs. (3)-(5), in which are the initial and terminal time of the MPC
controller , respectively, is the prediction time horizon of the MPC controller ,
are speeds of the leading and following vehicles in the dedicated lane at the time
(), is the lane-changing point (), is the DLV’s position at time (). Throughout
trajectory planning, is iteratively updated with , and the revised data is utilized to optimize the
longitudinal trajectory each time.
(3)
(4)
(5)
Fig. 8. The initial and final state of the MPC controller
To simplify the expression, we have omitted the time parameter. Therefore, we define
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as the state variable for each lane-changing vehicle (where ). is the
control variable, where denotes the system control input . Then, the MPC approach is
employed to construct the optimal control model, as shown in Eqs. (6)-(12). Eq. (6) represents the
objective function of the controller, which takes into account efficiency and control cost. The
efficiency cost implies that the lane-changing vehicle have a tendency to reach the desired lane-
changing speed. The control cost ensures driving comfort by penalizing excessive acceleration and
deceleration. Eq. (7) represents the system state-space equation, with as the state and control
variable coefficients. The longitudinal vehicle dynamics model is expressed with a third-order model
(S. H. Wang et al., 2022). where is the engine time constant (also known as inertial time lag). The
sampling period of the controller is , i.e., the optimization strategy is updated at a frequency of
in the prediction time horizon . Eq. (8) represents the constraint of the initial state, which is
obtained by the roadside monitoring unit and the vehicle detector. Eq. (9) represents the constraint
of the terminal state, which has been previously analyzed. Eq. (10) is the speed constraint. Eq. (11)
represents constraint of the control input constraint. The safety constraint is defined by Eq. (12). The
value of is calculated using the IDM introduced in Section 3.5, which serves as an upper
limit on acceleration, preventing the MPC controller from being too aggressive in achieving the
terminal goal and colliding with the preceding vehicle.
(6)
Subject to：
(7)
(8)
(9)
(10)
(11)
(12)
3.4. Dynamic Cooperative Lane-changing (DCLC) Model
This section briefly introduces the dynamic cooperative lane-changing (DCLC) model. Once
15
the SV enters the lane-changing segment, the model cooperatively and optimally controls the
longitudinal trajectories of related vehicles before performing the lane-changing maneuver,
provided SV satisfies the lane-changing decision criteria. Simultaneously, the lateral trajectory is
planned for SV to complete the lane-changing within the given time window.
3.4.1 Lane-changing decision model
Once the SV enters the lane-changing segment, the model is employed at each sampling time to
assess the feasibility of SV changing lanes. A lane-changing maneuver is permissible only when the
headway between SV and the leading and following vehicles in the target gap satisfies the minimum
safety gap requirement (Yang C. et al., 2022). The time when this requirement is met is considered
the initial time of the lane-changing process, denoted as . We have integrated a safety gap
constraint encompassing two lane-changing decision criteria, as delineated in Eqs. (13) and (14).
(13)
(14)
3.4.2 Longitudinal trajectory planning
For each lane-changing vehicle optimally controlled by the LCSD model in the planning
segment, it can arrive at the lane-changing point in either the target state or an intermediate state
as close to the target state as feasible. When the SV enters the lane-changing segment, the model
generates the desired acceleration of the relevant vehicle based on the information at time (and
continuously updated with ). When the SV meets the criteria for lane-changing and begins to do
so at time , the approach detailed in Section 3.4.3 is employed to plan the lateral trajectory for SV
within a designated time until the completion of the lane-changing maneuver.
In this case, vehicles related to the lane-changing process can be divided into two categories:
and , and .
(1) The MPC controller is set to control the optimal longitudinal trajectory for the DLV and SV.
(2) The DFV follows the SV with the IDM introduced in Section 3.5.
Similarly, for the MPC controllers of DLV and SV, we omit the time parameter and let “1”
represent , “2” represent Define as the state variable, denotes
the relative spacing of the vehicles , and denotes the relative speed of the vehicles
. is the control variable, denote the desired acceleration of DLV and SV,
respectively . The MPC controller is then employed to construct the optimization model using
Eqs. (15)-(22). Eq. (15) represents the objective function of the controller. In the objective function ,
16
the first term aims to maintain a minimum safe distance between vehicles, while achieving the lane-
changing position of SV as close to the DLV as possible, and reducing the impact of the lane-
changing maneuver on the DFV. The second term indicates that the SV will match the speed of the
DLV during the lane-changing process to ensure stability. The third and fourth terms control the
comfort of the lane-changing process by penalizing excessive acceleration and deceleration. Eq. (16)
presents the system state-space model. Eq. (17) represents the initial state of the SV when it arrives
at the lane-changing point. Eq. (18) represents the terminal state. Eqs. (19)-(21) represent the
acceleration constraints. Eq. (19) implements the acceleration of DLV during the lane-changing
process to maintain a safe lane-changing distance. Eq. (21) sets an upper limit for the acceleration to
ensure the safety of the vehicle in the current lane and the preceding vehicle during the control
process. The upper acceleration limit is calculated using CACC for CAVs in dedicated
lanes, and IDM for CAVs in ordinary lanes. Eq. (22) defines the speed limits.
(15)
Subject to:
(16)
(17)
(18)
(19)
(20)
(21)
(22)
3.4.3 Lateral trajectory planning
When the SV satisfies the lane-changing decision criteria in the lane-changing segment, a cubic
polynomial is employed to describe its lateral trajectory during the lane-changing process. This
selection is grounded in that cubic polynomials entail fewer model parameters than higher-order
polynomials, making them better suited for real-time trajectory planning. Furthermore, the cubic
polynomial also exhibits second-order smoothness, guaranteeing the continuity of lateral position
and velocity during lane changes (D. Yang et al., 2018).
17
(23)
(24)
(25)
(26)
(27)
Fig. 9(a) shows the lane-changing trajectory curve starting from position . Eq. (23) shows
the cubic polynomial curve, where and represent the longitudinal and lateral positions of the
SV during the lane-changing process , respectively. The polynomial coefficients are denoted by
and . To enhance the efficiency of trajectory planning, we simplify the calculation by
assuming that trajectory planning starts at with the vehicle course angle of , and ends at
with the vehicle course angle of 0. The curve at the initial and terminal positions must
satisfy the conditions represented by Eqs. (24) and (25). By solving the system of Eqs. (23)-(25), the
coefficients of each cubic polynomial can be obtained to derive the final cubic polynomial in Eq.(26).
Based on Eq. (27), we can calculate curvature of the cubic polynomial lane-changing
trajectory for the longitudinal position of SV.
and
are the first and second order
derivatives of the cubic polynomial. Fig. 9 (b) shows the curvature variations of cubic polynomials.
The curvature at the end of the lane-changing process is not exactly zero, but it is still a value close
to zero. Therefore, the CAVs can adjust the course angle to zero to keep traveling parallel to the lane
line shortly after completing the lane-changing process. Previous studies have supported this idea
by using cubic polynomials to plan the lane-changing trajectory (Zhang et al., 2013).
(a) cubic polynomial trajectory
(b) curvature variation.
Fig. 9. Schematic diagram of a cubic polynomial trajectory
18
To plan the lane-changing trajectory of the SV dynamically, we move the coordinate system
mentioned above dynamically. The lane-changing period is divided into four-time steps, with the
initial coordinates considered as at the beginning of each step. SV can acquire its initial state at
the beginning of each time step. As mentioned previously, in Eq. (27), the initial course angle and
the terminal lateral position of the trajectory
are known variables for each time step. In addition,
from the MPC controller in Section 3.4.2, we obtain the longitudinal position for the remaining
lane-changing duration, and the terminal longitudinal position of the trajectory,
. Obviously, the
cubic polynomial lane-changing trajectory of the SV can be easily fitted. The final lane-changing
trajectory comprises only the curves within the current step, while discarding the remainder. To
ensure a comfortable lane-changing process (Hou, Zheng, Liu, & Guo, 2023), the duration of lane-
changing maneuver was determined to be . The iterative planning process is illustrated in
Fig. 10, with the black dashed line representing the actual lane-changing trajectory of the SV.
Fig. 10. Iterative trajectory planning
3.5. Car-following models
The selection of an appropriate car-following model is crucial in mixed traffic, given the
presence of both CAVs and HDVs in ordinary lanes. In this paper, the IDM is chosen as the car-
following model for vehicles in ordinary lanes. The core concept of IDM is that, in the absence of
external interference, each vehicle wishes to travel at a desired speed while maintaining a safe
distance from the preceding vehicle. The model is as follows.
,
(28)
,
(29)
where Eq. (28) is the acceleration formula for vehicles n, is the acceleration exponent, which is
step1
step2
step3
step4
Final trajectory
19
taken as 4 here (Kesting et al., 2010), is the desired speed of the vehicle, is the
vehicle n-1 and n with minimum safe distance .
Cooperative adaptive cruise control (CACC) between vehicles increases traffic capacity in
dedicated lanes. This keeps CAVs operating with a constant gap. CACC vehicles can anticipate
downstream traffic and respond quickly to possible disruptions with small following gaps, due to
the widespread use of V2V technology (Xiao et al., 2018).
(30)
(31)
(32)
Eq. (30) is the cruising controller. Eq. (31) shows the acceleration for the gap-regulating
controller and gap-closing controller. is the gap error between the self-vehicle and the
preceding vehicle , which is calculated by Eq. (32).
Table 3 gives the values of relevant parameters (Xiao et al., 2018).
Table 3. The IDM and CACC parameters
Notation
Explanation
Values
Maximum acceleration
Minimum deceleration
Safe time headway of IDM
Speed limits
Minimum safe distance
Safe time headway of CACC
Control gain coefficient used to regulate the acceleration error
0.4
Control gain coefficients
0.23/0.04
Control gain coefficients
0.07/0.8
4. Solution algorithms
In this section, we focus on the algorithms for solving the model. In Section 4.1, we present
dynamic programming methods to solve the lane-changing sequence problem, ensuring that the
system reaches the optimal current state. In Section 4.2, an algorithm is developed to resolve MPC
control issues and achieve precision and effectiveness. The original problem is solved in real-time
by combining the two algorithms. Fig. 11 shows the solution algorithm framework.
Computational complexity is a crucial factor in engineering application problems. In this study,
the dynamic programming algorithm was used, and it is observed from Fig. 11 that there are at most
states in each stage. Therefore, there are at most state combinations in two
consecutive stages, and the computational complexity of this problem is . To simplify
20
the problem-solving process, we limit the length of the planning segment. On the other hand, it is
important to note that the actual computational complexity is reduced when considering the
feasibility of the gap decision. Similar issues have been proved in the relevant literature (Z. Sun et
al., 2020). Additionally, efficient resolution of the optimal control problem is crucial for this study.
To this end, a numerical solution algorithm based on Pontryagin's Minimum Principle has been
introduced, which simplifies computation by solving a set of ordinary differential equations. The
study shows that using both the “scipy.integrate” computational package in Python and the “BVP4c”
solver in MATLAB, the optimal control problem can be solved within 0.01-0.03s. The real-time
performance of the method is assured by its low-order polynomial complexity and efficient solution
of the optimal control problem.
Fig. 11 The proposed solution algorithm framework.
4.1. Algorithm for optimal lane-changing sequence
As the number of vehicles in the planning segment is limited, the optimal lane-changing
sequence of the lane-changing CAVs can be calculated efficiently using dynamic programming. In
addition, the conceptual diagram of the dynamic programming solution is illustrated in Fig. 5.
The steps of dynamic programming are as follows.
Update the speed and position of all CAVs in the planning segment. Let is the set of
feasible decisions in the current stage, is the set of feasible decision costs in the current stage, and
is the set of optimal decisions up to the current stage. Let .
For each gap decision , the longitudinal trajectory optimization control
algorithm is applied to calculate the trajectory. If there exists a lane-changing trajectory, the feasible
gap decision is deposited into the set , and the corresponding cost is deposited
Start: collect information of
CAVs in the studied area
Problem #1: allocate the n CAVs into
the m+1 gaps.
Determine: formulates the lane-
changing sequence decisions and plan
the trajectories for the lane-changing
CAVs
Problem #2: plan the lane-changing
trajectories for CAVs
Decompose: decomposition of the
lane-changeing traj ectory into lateral
and longitudinal trajectories.
Section 4.1: find optimal merging
sequence and trajectories
Section 4.2: calculate feasible gap
decision and merging trajectory by
solving the MPC problem
Section 4.2: optimize the
longitudinal trajectories of the
relevant CAVs by solving the MPC
problem
Section 3.4.3: generate the lateral
trajectory with cubic polynomial
End
Complexity
m+1 states in each
stage
final execution:
solve a numerical
solution algorithm
within 0.01-0.03s
21
into the set . Let to get the optimal gap decision in the current state, and the
corresponding optimal decision is deposited into the set , let .
For each new gap decision in the current stage, using the longitudinal
trajectory optimization control algorithm to calculate the trajectory. If there exists a lane-changing
trajectory, look for each decision in the set that satisfies , let
as the cumulative cost under the current decision. Take the optimal cumulative cost as the
cost under the current gap decision and deposit it into the set . Simultaneously, the
corresponding preceding gap decision
is saved as the optimal preceding gap decision under
the current decision.
Let
, we get the optimal decision for stage , and store the optimal
decision for the current stage into the position in set , and update the position in set
to be the optimal preceding decision
in the current state.
If , return to step 3. Otherwise, end the loop and output System optimal cumulative
cost and optimal lane-changing sequence set .
Table 4. Lane-changing Sequence Decision (LCSD) Algorithm
Number of ordinary lane CAVs in the planning segment: , Number of dedicated lane
CAVs in the planning segment: , Speed and Position of the above CAVs.
System optimal cumulative cost: , The set of optimal lane-changing sequence: .
The set of feasible gaps: ; The set of feasible gaps cost: ; The set of optimal decisions:
.
;
add it to the set ;
The optimal cost of the current state:
;
;
;
:
The cumulative cost:
add it to the set ;
22
The optimal cost of the current state:
;
Update:
4.2. Algorithm for MPC optimal control problem
This section explains how to use Pontryagin's Minimum Principle to solve the MPC optimal
control problem and generate the optimal control inputs formulaically (M. Wang et al., 2014).
The first step is to establish the Hamiltonian function that corresponds to the optimization
problem.
(33)
where is the system cost function, is the co-state of the state variable, the
co-state must be the dynamic equation of Eq. (34), and is the system state-space model. Using
the Hamiltonian function, we can get the necessary conditions to satisfy the optimal control in Eq.
(35), i.e., the optimal control minimizes the Hamiltonian function within the permissible range ,
and the optimal control variabl can be obtained.
(34)
(35)
In this study, the above system of differential equations satisfies the initial state constraints
and the terminal state constraints, and the possibility of transforming such a problem
into a two-point boundary value problem is described in detail in (M. Wang et al., 2014). Table 5
gives an algorithm of the MPC optimal control problem involved in this paper.
Table 5. Optimal Control Problem Algorithm
Initial state
; Terminal state
; Prediction horizon; The system state-space model
; The cost function .
Optimal control output .
⚫ Import the co-state variable , establish the Hamiltonian Function:
⚫ Calculate the dynamic differential equation of the state variable and the co-state variable :
⚫ Give the requirement of optimal control:
⚫ Transform the optimal control into the two-point boundary value problem, and solve the
23
problem with Python.
⚫ Obtain the output of Optimal control:
5. Simulation analysis
5.1. Simulation setup
This section presents simulation experiments using SUMO to validate the performance of the
proposed optimal control strategy. The vehicle behavior and cooperative control involved in the
study are programmed in Python and integrated with SUMO through the TraCI interface. As shown
in Fig. 12, the vehicle passes through the on-ramp and adjusts its driving behavior freely in the
warm-up segment before entering the studied merge area. The studied merge area consists of a
planning segment of m and a lane-changing segment of 300 m. With the planning strategy being
implemented at and the lane-changing strategy at , the final 100 m of the lane-
changing segment are called the mandatory zone. As the CAV enters the mandatory area, it will
gradually decelerate to zero to stop and wait for a suitable gap. The jerk is set as
during the
process of acceleration adjustment. Considering the proximity of the studied scenario to the highway
on-ramp area, we define the limiting speed . In addition, the remaining parameter
values, based on related studies (M. Wang et al., 2018; Xiong et al., 2022; Yang C. et al., 2022), are
shown in Table 6. Furthermore, we analyze the average value after 50 simulations for each scenario
to account for randomness.
In addition, a benchmark scenario for comparison is included, featuring a lane-changing
scenario without applied control. In this benchmark scenario, we employ a simplified rule-based
decentralized approach (i.e., the benchmark approach), where each vehicle independently makes
lane-changing decisions. After reaching the lane-changing point , the CAV in the ordinary lane
compares its actual gaps with CAVs of the target gap to the minimum safe distance given by Eqs.
(13) and (14). If the criteria are met, it will change lanes. Otherwise, it decelerates until finding a
suitable gap that meets the requirements before performing the lane-changing maneuver.
Furthermore, considering that the main focus of this paper is to optimize the longitudinal trajectory
of the CAVs to reduce the impact of the lane-changing behavior on dedicated lanes. The lateral
trajectory planning is only used as a complement to make the lane-changing trajectory more
complete. Therefore, we still adopt the method in Section 3.4.3 to describe lateral trajectories in the
benchmark scenario.
24
Fig. 12. Illustration of a simple lane-changing network in simulation
Table 6. The values of variables and parameters
Notation
Explanation
Value
Vehicle length
Acceptable time gap for lane-changing
Engine time constant
maximum speed limit
Maximum acceleration
Minimum deceleration
Lane length
The lane-changing period
5.2. Results evaluation
This section analyses the results of the optimal control model. Three simulation scenarios with
different traffic demand of 800 veh/h, 1200 veh/h, and 1600 veh/h are selected to simulate low,
medium, and high traffic demands, respectively. The benchmark scenario compares the results from
two aspects: (1) speed spatiotemporal trajectories, and (2) average travel time. In this scenario, the
penetration rate of CAVs in ordinary lanes is 40%, meaning that 40% of the vehicles in ordinary lanes
require change lanes. The arrival speed of CAVs in the dedicated lane is 26 m/s, while the arrival
speed of vehicles in the ordinary lane does not exceed 22 m/s. A detailed analysis process is provided
in the subsequent subsections of this section.
5.2.1 Speed spatiotemporal trajectories
To analyze the impact of lane-changing behavior on dedicated lane traffic, we plot the
longitudinal spatiotemporal trajectories of CAVs in the dedicated lane, and the lane-changing CAVs
in the ordinary lane after the implementation of the lane-changing maneuver. Fig. 13 illustrates the
spatiotemporal trajectories of the relevant CAVs. Where (a) is the benchmark scenario, (b) is the
scenario after applying the control strategy, with color shades indicating vehicle speeds.
As shown in Fig. 13(a), for the low traffic demand scenario (800 ), only three small low-
speed zones are observed, and the lane-changing maneuver of CAVs in the ordinary lane has no
Ordinary lane
Dedicated lane
On-ramp
Mandatory
zone
Planning Segment Lane-changing Segment
Lane-changing
point()
Warm-up
Segment
Start point()
The Studied Merge Area
Driving direction
25
discernible impact on the dedicated lane traffic. For the medium traffic demand scenario (1200
), it is clear that the lane-changing maneuver of the CAVs notably affects the dedicated lane
traffic. The lane-changing segment exhibits numerous extended low-speed zones. However, the
deceleration shockwave does not spread upstream of the lane-changing segment. In the high traffic
demands scenario (1600 ), the lane-changing maneuver of the CAVs significantly affected the
traffic in the dedicated lane. As a result, the lane-changing segment became entirely composed of
low-speed zones, and the deceleration shockwave began to spread toward the planning segment.
As shown in Fig. 13(b), compared to the benchmark scenario, the strategy proposed in this paper
brings different improvements to the impact of lane-changing behavior at different traffic demands.
Specifically, low traffic demand significantly diminishes the occurrence of small low-speed zones in
the dedicated lane. For scenarios with medium traffic demand, the strategy optimizes numerous
extensive low-speed zones in the dedicated lane, enhancing its traffic efficiency. However, as the
flow rate approaches saturation, the reduced headway of vehicles makes it more challenging for
vehicles to change lanes, resulting in many vehicles being unable to adhere to the optimized
trajectory. Consequently, these vehicles may enter the mandatory zone, unexpectedly obstructing
subsequent lane-changing vehicles and diminishing the strategy's effectiveness in improving the
dedicated lane's traffic efficiency.
Traffic demand: 800
26
Traffic demand: 1200
Traffic demand: 1600
(a) Without control
(b) With control
Fig. 13. Spatiotemporal trajectory diagram in the dedicated lane
5.2.2 Average travel time
Table 7 shows the average travel times for vehicles passing through the studied zone for
different traffic demands. For each scenario, two different travel times are measured: (1) the average
travel time for dedicated lane vehicles, and (2) the average travel time for ordinary lane vehicles.
Table 7. Average travel time
Traffic demand (veh/h)
Average travel time (s)
Without control
With control
Improve ratio
Dedicated lane
800
24.57
23.51
4.31%
1200
27.75
24.58
11.42%
1600
47.78
43.74
8.46%
Ordinary lane
800
25.01
24.03
3.92%
1200
28.43
24.93
12.31%
1600
47.54
43.42
8.67%
For both the benchmark scenario and the scenario with the control strategy, the average travel
time for vehicles on both lanes increases as traffic demands rise. In particular, there is a surge when
the traffic capacity is near saturation. It is well-known that increased traffic demand leads to longer
travel times. Compared to the benchmark scenario, the optimal control strategy proposed in the
study reduces average travel times across all scenarios. In low traffic demand, the lane-changing
maneuver of the CAVs is easier to perform, and therefore, the optimization strategy is not significant.
In contrast, substantial improvement is evident in the medium traffic demand, leading to an
reduction in average travel time for CAVs in the dedicated lane and a reduction for vehicles
in the ordinary lane. However, as traffic demand approaches saturation, the complexity of lane-
changing maneuvers prevents the optimization of each vehicle's trajectory, diminishing the
27
effectiveness of the optimization strategy.
5.3. Model performance evaluation
As analyzed in Section 5.2, the strategy proposed in this study does not completely eliminate
the traffic impact caused by lane-changing maneuvers in the lane-changing segment. This limitation
arises because the model only provides optimal or near-optimal strategies for the current sampling
time of relevant CAVs. Furthermore, not all CAVs conform to the proposed optimal control strategy,
and the unpredictable behavior of HDVs in ordinary lanes also limits the optimization effect.
Considering the optimization objective of the model, this evaluation criterion consists of two metrics:
(1) traffic efficiency and (2) comfort. This section examines various scenarios with different traffic
demands. Specifically, the traffic demand of the two lanes is increased from 600 to 1600
with an interval of
. The simulation scenarios include low, medium, and high traffic
demands for detailed analyses. The CAV penetration rate in the ordinary lane is 40%. The arrival
speed of CAVs in the dedicated lane is 26 m/s, while the arrival speed of vehicles in the ordinary lane
does not exceed 22 m/s. The subsequent subsections of this section provide a detailed analysis
process.
5.3.1 Traffic efficiency
The average speed of vehicles traversing the lane-changing segment indicates traffic efficiency
in this study. In addition, we introduced the indicator, the average speed increment rate (), to
measure the specific improvement in traffic efficiency. is calculated by Eq. (36), where is the
average speed with control, and is the average speed without control.
(36)
28
(a) under different traffic demands.
(b) The average speed in the dedicated lane
(c) The average speed of lane-changing CAVs
Fig. 14. Distribution of average speed for different traffic demand levels
Fig. 14 illustrates the traffic efficiency indicator across various levels of traffic demand. As
shown in Fig. 14(a), the proposed strategy improves traffic efficiency in the lane-changing segment.
Specifically, the effect of the strategy exhibits an increasing and then decreasing trend within the
studied traffic demand range. By combining Fig. 14(c) and Fig. 14(d), it is evident that the average
speeds of dedicated lane CAVs and lane-changing CAVs are and
, respectively,
at medium demand without control. Following the adoption of the control strategy, the average
speeds increased to and . In this case, the strategy can increase the average
speeds of the vehicles in the lane-changing segment by , which is the most significant advantage.
In addition, we also found that when the demand is low, this strategy can improve the traffic
efficiency of lane-changing CAVs better than dedicated lane CAVs. However, when the traffic
demand exceeds 900 , the dedicated lane vehicles exhibit better improvement in traffic
600 800 1000 1200 1400 1600
0
2
4
6
8
10
12
14
Iv (%)
Traffic demand (veh/h)
CAVs of dedicated lane
CAVs of lane-changing
800
1200
1600
0
4
8
12
16
20
24
28
Traffic demand(veh/h)
Average speed(m/s)
without control
with control
800
1200
1600
0
4
8
12
16
20
24
28
Traffic demand(veh/h)
Average speed(m/s)
without control
with control
29
efficiency. This is because the spacing between adjacent vehicles in the dedicated lane is large at low
traffic demand, and the lane-changing maneuver of CAVs has less impact on the dedicated lane
traffic. In this case, the Proposed strategy mainly optimizes the traffic efficiency of lane-changing
CAVs. As traffic demand increases, the impact of lane-changing maneuvers on dedicated lane traffic
becomes increasingly apparent. The proposed strategy optimizes traffic efficiency in dedicated lanes.
5.3.2 Comfort
In this study, the comfort experienced during the lane-changing process is defined as the
average longitudinal acceleration of CAVs. The duration involved the period from when the CAVs
enter the lane-changing segment until exiting it. The average acceleration increment rate () serves
as an indicator utilized to quantify the enhancement in comfort. The calculation of is calculated
by Eq. (37), where is the absolute value of the average acceleration with control, and is the
absolute value of the average acceleration without control.
(37)
As shown in Fig. 15(a), the comfort indicator is similar to the indicator of traffic efficiency. The
impact of the proposed strategy on comfort exhibits a trend of initially increasing and then
subsequently decreasing as traffic demand rises. Furthermore, the proposed strategy significantly
improves the lane-changing CAVs and enhances the comfort level by approximately , even in
high traffic demand. In the scenario studied in this paper, CAVs in ordinary lanes must make
frequent acceleration adjustments to complete the lane-changing maneuver. The primary objective
of the proposed strategy is to optimize the acceleration of the CAVs during the lane-changing process.
Furthermore, the data from the simulation experiments also support the performance of the model.
As shown in Fig. 15(c) and (d), it is evident that the proposed strategies effectively limit the
average acceleration of both dedicated lane CAVs and lane-changing CAVs to below across
all three levels of traffic demand. The proposed strategies plan smoother trajectories for these
vehicles. At the same time, it is evident that increasing the demand to 1600 results in a more
dispersed distribution of acceleration values. In such cases, as the lane-changing behavior of the
preceding CAVs becomes more challenging, more CAVs enter the mandatory zone and become
unexpected obstacles for the following CAVs. Consequently, this requires the following CAVs to
adjust their traveling status through larger acceleration and deceleration.
30
(a) under different traffic demands.
(b) The average acceleration in the dedicated
lane
(c) The average acceleration of lane-changing
CAVs.
Fig. 15. Distribution of average acceleration for different traffic demand levels
5.4. Speed difference and infrastructure impact
In addition to proving the performance of the proposed strategy by comparing it with a
benchmark method. We also analyze the impact of the strategy under different arrival speeds of
vehicles and road layouts. A detailed analysis process is provided in the subsequent subsections
within this section.
5.4.1 Speed difference
In the preceding analyses, the arrival speed of CAVs in ordinary lanes was limited to 22. It
is crucial to note that the arrival speed of lane-changing CAVs significantly influences the traffic
performance in the lane-changing segment. This section introduces the average fuel consumption
600 800 1000 1200 1400 1600
10
15
20
25
30
35
40
45
50
55
Ia (%)
Traffic demand (veh/h)
CAVs of dedicated lane
CAVs of lane-changing
800
1200
1600
0.0
0.5
1.0
1.5
2.0 without control
with control
Traffic demand(veh/h)
Average acceleration(m/s^2)
800
1200
1600
0.0
0.5
1.0
1.5
2.0 without control
with control
Traffic demand(veh/h)
Average acceleration(m/s^2)
31
(AFC) metric to evaluate the impact of varying arrival speeds of lane-changing CAVs on the total
traveling cost under the proposed strategy. The study considers 30 simulation scenarios in which
two lanes have average traffic demands ranging from 600
to 1400
at intervals of 200
. The arrival speed of dedicated lane CAVs is 26
, while the arrival speed of lane-changing
CAVs is 16 to 26
with an interval of 2.
In this section, the VT-micro model mentioned in (X. Li et al., 2014) is used to calculate the
average fuel consumption (AFC).
(38)
where is all the vehicles entering the studied area, is the simulation step ,
is the time when
the vehicle enters the studied area ,
is the time when the vehicle leaves the studied area.
and are the speed and acceleration of the vehicle at time , is the fuel consumption
coefficient, and Table 8 is the matrix of the coefficients.
Fig. 16 shows the average fuel consumption (AFC) of lane-changing CAVs and dedicated lane
CAVs at different speed differences for various traffic demands. As shown in Fig. 16, the average
fuel consumption of vehicles is positively correlated with traffic demands and the difference in
arrival speeds between lane-changing and dedicated lane CAVs. Furthermore, it has been observed
that the speed difference of less than 4
does not significantly affect the AFC value. However,
when the speed difference exceeds 4
, any further increase leads to a rapid rise in the AFC value.
Analyzed in terms of traffic demand variability, although the correlation between AFC and the
change of traffic demand is weak when the traffic demand does not exceed 1200
, the AFC still
rises with increases in traffic demand. Fig. 16 illustrates the distribution of AFC indicators at varying
traffic demands and speed differences. It is evident that as traffic flow increases, the impact of arrival
speed difference on fuel consumption indicators also increases significantly. Traffic managers can
establish distinct speed limit modes for ordinary lanes based on traffic demand, thereby regulating
the arrival speed of lane-changing CAVs and enhancing traffic flow performance in the lane-
changing segment.
Table 8. The matrix of coefficients
is positive
is negative
-7.735
0.2295
-5.61E-3
9.773E-5
-7.735
-0.01799
-4.27E-3
1.8829E-4
0.02799
0.0068
-7.722E-4
8.38E-6
0.02804
7.72E-3
8.375E-4
3.387E-5
-2.228E-4
-4.402E-5
7.90E-7
8.17E-7
-2.199E-4
-5.219E-5
-7.44E-6
2.77E-7
1.09E-6
4.80E-8
3.27E-8
-7.79E-9
1.08E-6
2.47E-7
4.87E-8
3.79E-10
32
Fig. 16. Fuel consumption in different traffic demands
5.4.2 Infrastructure
Furthermore, we investigated the impact of varying lane-changing segment lengths on the
strategy. In this subsection, the average speeds of dedicated lane CAVs and lane-changing CAVs are
simulated for different lane-changing segment lengths in a medium traffic demand (1200). As
shown in Table 9, it is evident that the average speeds of both CAVs rise with the increasing length
of the lane-changing segment. However, once the length exceeds 300
, further increases in the
length of the lane-changing segment do not significantly improve the model performance.
Table 9. The average speed of related CAVs under different lane-changing segment length
Length ()
200
250
300
350
400
450
DL ()
24.25
25.06
25.16
25.17
25.23
25.24
LC ()
23.43
24.45
24.53
24.55
24.55
24.57
: DL denotes the average speed of dedicated lane CAVs, LC denotes the average speed of lane-
changing CAVs
5.5. Trajectory evolution in the DCLC Model
As mentioned above, the proposed lane-changing strategy consists of two models. Among
them, the DCLC model involves longitudinal trajectory control and lateral trajectory planning of the
lane-changing CAVs during the process. Therefore, it is necessary to track the trajectory evolution
process for lane-changing CAVs.
To demonstrate the dynamic behavior of CAVs during the lane-changing process, a subject
vehicle (SV) is randomly selected for analysis in the simulation. Fig. 17(a) illustrates the longitudinal
position-time diagrams of the SV, the DLV, and the DFV at the lane-changing point. The DCLC
model ensures stability during the lane-changing process. Fig. 17 (b) and (c) show the speed and
0 2 4 6 8 10
0.06
0.09
0.12
0.15
0.18
0.21
0.24
0.27
Fuel consumption (Liters/km)
Speed diff (m/s)
600veh/h
800veh/h
1000veh/h
1200veh/h
1400veh/h
600 800 1000 1200 1400
0
2
4
6
8
10
Speed diff(m/s)
Traffic demand(veh/h)
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
33
acceleration, respectively. The MPC controller in the DCLC model can regulate the acceleration of
the DLV during the lane-changing process, making space for the SV to change lanes. Additionally,
the SV should smoothly adjust its speed to match the DLV. The DFV can utilize the IDM to follow
the SV smoothly, ensuring a safe and seamless lane-changing maneuver.
(a) Longitudinal position evolutions
(b) Longitudinal speed evolutions
(c) Longitudinal acceleration evolutions
Fig. 17. Trajectory evolution of SV and related vehicles in lane-changing segment
6. Conclusions and Future Work
This paper proposes a dynamic optimal control strategy for planning CAV lane-changing
trajectories in mixed traffic, aiming to plan the lane-changing trajectory for each CAV in an ordinary
lane until it completes its lane-changing maneuver. To address this issue, we utilized a dynamic
programming approach to determine the lane-changing sequences of the CAVs in ordinary lanes,
and a controller based on MPC is designed to solve the optimal control problem of CAV trajectories.
We adopted the Hamiltonian function to convert the MPC optimal control problem into a two-point
boundary problem solving. This allows the established model to better adapt to a dynamic control
environment. The simulation experiments led to the following conclusions:
34
(1) Based on the proposed strategy, it is possible to eliminate small low-speed zones on dedicated
lanes under low traffic demand. At medium-traffic demand, multiple large-scale low-speed
zones on dedicated lanes can be optimized to improve traffic efficiency. However, when the
traffic demand is close to saturation, the optimization strategy reduces the improvement in
traffic efficiency of the dedicated lanes.
(2) The proposed strategy has enhanced both traffic efficiency and comfort. Specifically, under
medium traffic demand, it can improve the average speeds of the CAVs in the studied area by
about 12%. Meanwhile, the proposed strategy can significantly reduce the average and relevant
CAVs' acceleration by about , even in high traffic demand.
(3) The arrival speed difference between lane-changing CAVs and dedicated lane CAVs affects the
average travel cost through the studied area. In addition, the impact of the arrival speed on the
AFC indicator also increases significantly with the traffic demand.
(4) The effectiveness of the proposed strategy increases with the length of the lane-changing
segment. However, if the segment exceeds 300, the marginal benefit becomes negligible.
We identified several issues that could be explored in depth. First, the simulation results
indicate that the proposed strategy is advantageous in medium-traffic demand. However, its
performance decreases as traffic demand increases due to oversaturated traffic demand, which
prevents the strategy from optimizing the trajectory of each lane-changing CAV. In the future, we
may explore optimizing the management and layout of highways near the studied merge area to
adjust the density of vehicles approaching the study area and fully exploit the advantages of the
strategy. Second, given that CAVs in dedicated lanes prefer to travel in platoon mode, our future
research will explore the strategy for CAVs in ordinary lanes merging into CAV platoons in
dedicated lanes.
Acknowledgments
The paper received research funding support from the National Natural Science Foundation
of China (52002339), the Sichuan Science and Technology Project (2023ZHCG0018), the Fundamental
Research Funds for the Central Universities (2682023ZTPY034), and Chengdu Soft Science Research
Project (2023-RK00-00029-ZF).
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