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Analysis of mixed traffic flow with different lane management strategy for
connected automated vehicles: a fundamental diagram method
Yi Wang
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;
E-mail: wangyi1227@my.swjtu.edu.cn
Zeqi Xu
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;
E-mail: zeqixu@my.swjtu.edu.cn
Zhihong Yao (Corresponding author)
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.
E-mail: zhyao@swjtu.edu.cn
Yangsheng Jiang
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.
E-mail: jiangyangsheng@swjtu.edu.cn
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Analysis of mixed traffic flow with different lane management strategy for
connected automated vehicles: a fundamental diagram method
Abstract
This paper proposes a fundamental diagram of the mixed traffic flow for multi-lane
road segments with connected automated vehicles (CAVs) dedicated lanes based on a
flexible and efficient lane management strategy for CAVs. First, two lane management
strategies of CAVs dedicated-HDVs lane strategy (i.e., C-H strategy) and CAVs
dedicated-Mixed lane strategy (i.e., C-M strategy) are introduced. The more efficient
C-M strategy is chosen based on the characteristics of the two strategies, and the five
stable states under this strategy are analyzed. Second, different car-following modes
in mixed traffic flows are analyzed, and general fundamental diagrams for
homogeneous and mixed traffic flows are derived based on car-following models.
Then, the stabilized traffic conditions are derived based on the traffic flow
characteristics in the five stable states. A fundamental diagram model for the multi-
lane road segment with CAVs dedicated lanes is developed based on the general
fundamental diagram, and some properties of the fundamental diagram are derived.
Finally, the impacts of different numbers of dedicated lanes and traffic conditions on
the traffic volume and capacity are analyzed based on the fundamental diagram, and
the optimal settings of dedicated lanes for CAVs under different lane number scenarios
are given. Numerical simulation results showed that: (1) there are five stable states for
the traffic flow on the road segment under the C-M strategy; (2) properly setting up
CAVs dedicated lanes can effectively improve the efficiency of road segments; (3)
under the C-M strategy, for any CAVs dedicated lane setting scenario, the traffic
capacity of the road segment shows a gradually increasing trend as the penetration
rate of CAVs increases. (4) compared to the C-H strategy, the capacity of the road
segment under the C-M strategy will be higher than the C-H strategy when the CAVs
penetration rate exceeds a certain threshold. Moreover, as CAVs penetration rate
increases further, the advantages of the C-M strategy in terms of traffic capacity
become more and more significant.
Keywords: mixed traffic flow; connected automated vehicles; dedicated lanes; fundamental
diagram; traffic volume; traffic capacity
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1. Introduction
Benefiting from the significant advantages of connected automated vehicles
(CAVs) technologies in improving traffic efficiency and safety, and reducing fuel
consumption and pollution (Razmi Rad et al., 2020), research and applications on
CAVs have received extensive attention in recent years. Unfortunately, although CAVs
are considered one of the effective solutions to achieve sustainable urban development,
there is still a long transition before CAVs can completely replace human-driven
vehicles (HDVs). Relevant studies have shown that the market penetration rate for
CAVs is expected to reach 50% only by 2040 (B. Chen et al., 2020). It is foreseeable that
the mixed traffic flow consisting of CAVs and HDVs will persist on the roads for
reasonably long periods. Therefore, how CAVs will influence the characteristics of the
mixed traffic flow at different stages during the transition period has become a hot
research topic.
To investigate the potential impact of CAVs on the mixed traffic flow at different
stages during the transition period, scholars have already investigated in terms of the
fundamental diagram (T. Chen et al., 2021; Ngoduy et al., 2021a; Tian et al., 2023; Wu
et al., 2022; Yao et al., 2019; J. Zhou & Zhu, 2020, 2021), stability (Gu et al., 2022; T.
Wang et al., 2022; Yao et al., 2021), safety (Garg & Bouroche, 2023; Nazir et al., 2023; X.
Wang et al., 2022), fuel consumption and emissions (Rahman & Thill, 2023; Shi et al.,
2022; Vellamattathil Baby et al., 2022; S. Zhou et al., 2023; Zong & Yue, 2023), etc.
However, it is difficult for CAVs to accurately identify and predict the driving behavior
of HDVs in the mixed traffic flow. In addition, the stochastic driving behavior of HDVs
limits the efficiency advantage of CAVs to some extent (S. Chen, Hu, et al., 2022; Razmi
Rad et al., 2021). To fully utilize the technical benefits of CAVs and further improve the
overall safety and operational efficiency, some scholars have suggested setting up
CAVs dedicated lanes and are committed to investigating the impacts of the settings
on road segments and even traffic networks. Only CAVs are allowed to drive in CAVs
dedicated lanes. Dedicated lanes for CAVs can significantly increase the possibility of
CAVs forming platoons compared to normal lanes, and can avoid potential conflicts
between CAVs and HDVs. Thus, CAVs dedicated lanes are more efficient and safe (Z.
Wang et al., 2022). Meanwhile, after setting up CAVs dedicated lanes, two different
lane management strategies can be formed depending on whether CAVs are allowed
to drive on normal lanes or not (D. Chen et al., 2017). In the first strategy, CAVs are
only allowed to drive in CAVs dedicated lanes, called the CAVs dedicated-HDV lane
management strategy (i.e., C-H strategy). In the second strategy, CAVs can drive in
both CAVs dedicated and normal lanes, which will be called the CAVs dedicated-
Mixed lane management strategy (i.e., C-M strategy). In some scenarios, the C-M
strategy is more flexible and efficient than the C-H strategy. When the CAVs
penetration rate is high, the C-H strategy restricts the normal lane to only a few HDVs,
which can waste roadway resources in the normal lane. In contrast, the C-M strategy
allows some CAVs to shift into the normal lanes to form a mixed traffic flow with
HDVs, improving the utilization efficiency of the normal lanes. The C-M strategy is
more suitable for future traffic scenarios from an efficiency point of view.
The impact of CAVs dedicated lane on the mixed traffic flow on road segments is
complex due to the variety of CAVs dedicated lane settings and management strategies.
Before a large-scale application of CAVs dedicated lanes, it is necessary to
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comprehensively investigate the impact of CAVs dedicated lane settings on traffic
volume on road segments at different periods, determining the optimal timing and
number of CAVs dedicated lanes deployed. Therefore, it is essential to model and
analyze the fundamental diagram of the mixed traffic flow with CAVs dedicated lanes.
Most relevant studies on CAVs dedicated lanes have focused on network design
optimization, traffic capacity analysis, traffic efficiency analysis, and safety analysis,
while the fundamental diagrams have received little attention. To our knowledge, only
Yao et al. (2022) have modeled the mixed traffic flow fundamental diagram with CAVs
dedicated lanes for the C-H strategy. However, compared with the C-H strategy, the
C-M strategy improves the utilization efficiency of the normal lanes. Therefore, the C-
M strategy is more suitable for future traffic scenarios.
To fill this gap, this paper aims to model the fundamental diagram of the mixed
traffic flow with CAVs dedicated lanes based on the C-M strategy and solves the
problem of CAVs dedicated lane setting on multi-lane road segments. First, two lane
selection principles for CAVs under the C-M strategy are given based on which five
stable states and their characteristics are analyzed. Second, the general fundamental
diagrams for homogeneous and mixed traffic flows are derived from the Intelligent
Driver Model (IDM). And the fundamental diagram model for the mixed traffic flow
with CAVs dedicated lanes is developed based on the general fundamental diagrams
and the stable state characteristics. Finally, a numerical analysis is adopted to discuss
the distribution characteristics of the stable state of the mixed traffic flow and the
effects of the CAVs dedicated lane settings and traffic conditions on the fundamental
diagram. The main contributions of this paper are as follows.
1) proposes a multi-lane fundamental diagram model of the mixed traffic flow
with CAVs dedicated lanes based on the C-M strategy.
2) analyzes the distribution characteristics of the stable states under different
traffic conditions and the applicability of the C-M strategy under different dedicated
lane setting scenarios.
3) analyzes the impact of different traffic conditions on the traffic volume and
capacity of the road segment based on the proposed model.
4) gives the optimal settings of CAVs dedicated lanes for maximizing the traffic
volume of the road segment under different traffic conditions.
The remainder of this paper is organized as follows. Section 2 briefly reviews the
research on the fundamental diagram modeling of the mixed traffic and the CAV
dedicated lanes. Section 3 gives the notation of the key variables and their descriptions.
Section 4 derives the fundamental diagram model for the mixed traffic flow with CAVs
dedicated lanes based on the C-M strategy. Section 5 analyzes the proposed
fundamental diagram model. Finally, conclusions and future works are discussed.
2. Literature review
2.1. Fundamental diagram of mixed traffic flow
With the emergence and development of connected automated vehicle technology,
many scholars have researched the fundamental diagram of the mixed traffic flow to
understand the impact of mixed traffic flow on traffic systems. In general, two main
methods exist to obtain the fundamental diagram of the mixed traffic flow. The first is
to obtain the fundamental diagram curves through traffic simulation, and the second
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is to derive the formula of the fundamental diagram through the car-following model.
Regarding the data-driven method for obtaining the fundamental diagram,
Talebpour et al. (2016) studied the effect of CAVs on the stability of the mixed traffic
flow. And through traffic simulation, the fundamental diagram of the mixed traffic
flow is obtained, which proves that CAVs can improve the mixed traffic volume under
certain CAVs penetration rates. Liu et al. (2017) proposed a mixed traffic flow cellular
automata model for multi-lane scenarios based on considering two types of lane-
changing behaviors. Based on many simulation experiments, the fundamental
diagram of the mixed traffic flow is gained, and the influence of the penetration rate
of CAVs and lane-changing behavior on the fundamental diagram is analyzed. Jiang
et al. (2021) developed a cellular automaton model for the mixed traffic flow under
single-lane scenarios based on considering CAVs platoon driving behavior. The
impacts of CAVs penetration rate on the fundamental diagram, capacity, and speed
fluctuation of the mixed traffic flow are analyzed by traffic simulation. Yang et al. (2022)
proposed a platoon cooperation strategy and analyzed the effect of CAVs penetration
rate on the fundamental diagram of the mixed traffic flow, oscillations, and safety.
Regarding the theoretical methods of obtaining fundamental diagrams by using
car-following model, these can be classified generally into two categories, respectively,
the deterministic fundamental diagram and the stochastic fundamental diagram. In
deterministic fundamental diagram models, it is generally assumed that the time
headways of vehicles are fixed, which is the most common assumption in fundamental
diagram modeling. Scholars have proposed many fundamental diagram models for
mixed traffic flow based on this assumption. Zhu et al. (2018) proposed a mathematical
model that can describe the mixed traffic flow and the fundamental diagram of the
mixed traffic flow was obtained by numerical simulation. Yao et al. (2019) modeled
different types of vehicle behaviors in a mixed traffic flow separately using different
car-following models, and then derived a fundamental diagram model of the mixed
traffic flow. The factors influencing the fundamental diagram of the mixed traffic flow
are analyzed by numerical experiments, and simulation experiments are adopted to
verify the correctness of the fundamental diagram model. The influence of different
factors on the fundamental diagram of the mixed traffic flow is analyzed through
numerical experiments, and the correctness of the fundamental diagram model is
verified based on traffic simulations. Chang et al. (2020) developed a mixed traffic flow
stability analysis method and a fundamental diagram model considering CAVs
platoons using IDM and a cooperative adaptive cruise control model (CACC model)
to capture car-following behavior. Some scholars have further considered the
microscopic characteristics of platoons, including platoon intensity, platoon size, etc.,
in modeling the mixed traffic flow. Zhou et al. (2021) proposed a fundamental diagram
model of the mixed traffic flow considering the degradation of CACC and platoons of
CAVs, then investigated the impact of the degradation of the fleet control model on
the fundamental diagram, fuel consumption, and emissions. Yao et al. (2022)
developed a fundamental diagram for the mixed traffic flow with CAVs dedicated
lanes based on the C-H strategy and analyzed the effects of different dedicated lane
number settings on the traffic volume and capacity of the road segment. On the other
hand, some studies have shown that in real traffic scenarios, the time headways of
CAVs and HDVs are heterogeneous and stochastic, especially for HDVs (Li & Chen,
2017; Qu et al., 2017). Scholars first studied stochastic fundamental diagrams in the
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traditional pure HDV environment and achieved many results (Cheng et al., 2024; Qu
et al., 2017; Siqueira et al., 2016). In recent years, some scholars have begun to pay
attention to the impact of the stochasticity of time headways on mixed traffic flow (S.
Chen, Wang, et al., 2022; T. Chen et al., 2023; Zheng et al., 2020). However, these studies
mainly focus on the traffic capacity rather than the fundamental diagrams. As for the
stochastic fundamental diagram for mixed traffic flow, a prospective study was first
carried out by Zhou et al (2020). Based on the concept of platoon intensity proposed
by Ghiasi et al. (2017), Zhou et al. (2020) used a hybrid Gaussian to model random
headway spacing and derived a stochastic fundamental diagram of the mixed traffic
flow considering CAV penetration and platoon intensity.
As can be observed from the above literature, most of the existing studies are
based on deterministic fundamental diagrams, and a few scholars have studied
stochastic fundamental diagrams for mixed traffic flow. Relevant studies have mainly
focused on the impact of CAVs penetration rate, platoon micro characteristics (platoon
distribution stochasticity, platoon intensity, platoon size, etc.), and degradation of
CACC on the fundamental diagram of the mixed traffic flow. However, scholars have
seldom investigated the influence of CAVs dedicated lane settings and lane
management strategies on the fundamental diagram of the mixed traffic flow. It is
noted that the focus of this paper is on the impact of dedicated lanes for CAVs on mixed
traffic flow under the C-M strategy. While the influence of factors such as platoon
intensity, platoon size, and the stochasticity of time headways on the mixed traffic flow
has been carefully and deeply investigated in previous studies (T. Chen et al., 2023;
Ghiasi et al., 2017; Yao et al., 2023; Yao, Gu, et al., 2022; J. Zhou & Zhu, 2020, 2021),
especially by Zhou and Zhu (2020, 2021) and T. Chen et al. (2023). Therefore, platoon
microscopic characteristics are not considered in this paper's fundamental diagram
modeling of mixed traffic flow considering CAVs dedicated lanes. Meanwhile, similar
to the previous studies of deterministic fundamental diagrams, in this paper, the
stochasticity of time headways is not considered, and the time headways of vehicles
will be set as a fixed value.
2.2. CAVs dedicated lane
In recent years, CAVs dedicated lanes have gradually become a hot research topic
as a new form of lane management. The current research on CAVs dedicated lanes
mainly consists of two research directions. The first research direction is to study
optimizing the layout of CAVs dedicated lanes on the traffic network to maximize the
road network's performance. The second research direction is investigating the
potential impacts of CAVs dedicated lanes on road segments.
For the problem of optimizing the layout of CAVs dedicated lanes on the traffic
network, researchers usually model it as a bi-level optimization model. Chen et al.
(2016) proposed an optimization model for designing CAVs dedicated lanes to
minimize the total social cost while considering the safety of dedicated lanes.
Subsequently, Chen et al. (2017) developed a mathematical framework for deploying
dedicated zone for CAVs in traffic networks. The model assumes that both CAVs and
HDVs follow the principle of user equilibrium outside the CAVs dedicated zone and
that CAVs follow the principle of system optimization within the CAVs dedicated zone.
Ye et al. (2018) proposed a bi- level network optimization model to minimize the total
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travel cost of the traffic network by optimizing CAVs dedicated links and congestion
pricing. Roy et al. (2022) offered a mathematical framework for deploying dedicated
areas for multi-lane CAVs and designed a new algorithm to evaluate the effectiveness
of the deployment model. Madadi et al. (2021) constructed a unified optimization
framework that allows for the simultaneous deployment of CAVs dedicated lanes,
dedicated links, and dedicated zones on the traffic network. Zhang et al. (2022)
proposed an optimization model for dedicated lanes of connected automated buses
(CABs), which was able to significantly increase the passengers of CABs by deploying
dedicated lanes of CABs. Zhang et al. (2023) proposed a model for the joint
deployment of dedicated lanes and roadside equipment for CAVs that considers the
roadway's stochastic traffic capacity. Unlike the general optimization model that
minimizes the travel cost of the road network as an objective, Seilabi et al. (2023)
developed a dedicated lane deployment model for CAVs with minimizing emissions
as an optimization objective. Meanwhile, the model considered the uncertainty of
CAVs market size and had good robustness.
Other scholars have dedicated their efforts to studying the impact of CAVs
dedicated lanes on road segments. Chen et al. (2017) provided a generalized formula
for calculating the traffic capacity of the mix traffic flow considering the penetration
rate of CAVs, the microscopic characteristics of different types of vehicles, and
different lane management strategies. Ghiasi et al. (2017) offered a formula for
highway capacity in a mixed traffic flow environment based on the Markov chain
approach and further proposed a framework for CAVs dedicated lane deployment to
maximize highway capacity. Mohajerpoor et al. (2019) derived a model for headway
in mixed traffic flow and considered the possible sequences of CAVs and HDVs in the
model. Meanwhile, the total delay of the roadway segment under several lane
management strategies, including setting up CAVs dedicated lanes is modeled
analytically based on the proposed model. Jiang et al. (2022) analyzed the impact of
CAVs dedicated lane setting strategy on traffic capacity based on considering platoon
intensity and platoon size. Wang et al. (2022) proposed three different lane
management strategies and derived the roadway capacity under different lane
management strategies based on the proposed lane management strategies. Chen et al.
(2023) derived stochastic capacity formulas for single-lane highway scenarios and
merged-segment scenarios considering communication loss, and conducted
sensitivity analyses for factors such as stochastic communication loss and traffic arrival
pattern. Zhong et al. (2022) used a microsimulation platform to compare the impacts
of four different CAVs dedicated lane setting scenarios on traffic efficiency. Chen et al.
(2022) constructed a car-following model for mixed traffic flow in a multi-lane scenario
with CAVs dedicated lanes based on a cellular automata model. Through this model,
they analyzed the overall impacts of different numbers of dedicated lanes and
different traffic conditions on roadway segment efficiency and pollutant emissions.
Shao et al. (2022) proposed a lane management strategy called ''Dedicated Lane with
Intermittent Priority'' and developed a VISSIM-based simulation model to validate the
effectiveness of the strategy. Zhao et al. (2023) developed a CAVs dedicated lane setting
simulation platform to visualize the impact of dedicated lane settings on the traffic
volume and velocity of road segments. Zhang et al. (2020) evaluated the impact of
setting up dedicated lanes for CAVs on safety under different traffic demands and
different CAVs penetration rates based on a simulation-based method. Razmi Rad et
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al. (2021) collected the driving behaviors of CAV drivers in a scenario with CAVs
dedicated lanes through a driving simulator and analyzed the potential impact of
CAVs dedicated lanes setting on traffic safety and efficiency.
As can be observed from the above literature, most of the current research on
CAVs dedicated lanes focuses on road network layout optimization, road segment
traffic capacity analysis, efficiency analysis, and safety analysis. At the same time, there
is little relevant research on fundamental diagram modeling. For the CAVs lane setting
optimization problem, although some scholars (Ghiasi et al., 2017; Y. Jiang et al., 2023)
proposed a CAVs lane setting model that maximizes the traffic capacity, the maximum
capacity lane settings do not guarantee the maximum traffic volume of the road
segment under all traffic conditions. Therefore, it is still necessary to investigate the
fundamental diagram model of mixed traffic flow under different lane management
strategies to understand the overall impact of dedicated lane settings and traffic
conditions on the traffic volume of the road segment. For the fundamental diagram,
only Yao et al. (2022) modeled and analyzed the fundamental diagram of mixed traffic
flow with CAVs lanes based on the C-H strategy. However, the C-H strategy does not
fully utilize roadway capacity under conditions of high penetration rate of CAVs. The
C-M strategy is more efficient and flexible than the C-H strategy. Unfortunately, the
fundamental diagram of mixed traffic flow based on the C-M strategy has not been
studied.
3. Notation of Parameters and Variables
Table 1 lists the notation of the key parameters and variables used in this paper.
Table 1. Notation of key parameters and variables
Variables
Description
Unit
Common parameters
Time headway in HDV car-following mode
s
Time headway in AV car-following mode
s
Time headway in CAV car-following mode
s
Number of lanes on the road segment
—
Number of CAVs dedicated lanes on the road segment
—
The CAVs penetration rate of the road segment
—
Length of road segment
m
Free-flow velocity, i.e., the maximum speed allowed on the
road segment
m/s
Traffic density of the road segment
veh/km
Velocity of CAVs dedicated lane in stable states
m/s
Velocity of the normal lane in stable states
m/s
Traffic density of CAVs dedicated lanes in stable states
vehs/km
Traffic density of normal lanes in stable states
vehs/km
Virtual density of CAVs dedicated lanes when all CAVs are
assumed to be distributed in dedicated lanes
vehs/km
Virtual density of normal lanes when all HDVs are assumed
to be distributed in normal lanes
vehs/km
Traffic volume of the road segment
vehs/h
Traffic volume of a CAVs dedicated lane
vehs/h
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Traffic volume of a normal lane
vehs/h
CAVs penetration rate of normal lanes in stable states
—
Spillover ratio of CAVs in steady state 4
—
Spillover ratio of CAVs in steady state 5
—
Car-following model
Desired acceleration of vehicle
m/s2
Maximum acceleration
m/s2
General deceleration
m/s2
Velocity of vehicle
m/s
Velocity difference between vehicle and the preceding
vehicle
m/s
Minimum spacing when the vehicle is stopped
m
Headway between vehicle and the preceding vehicle
m
Safe time headway of vehicles
s
Length of vehicle
m
Acceleration coefficient
—
Fundamental diagram model
Stable headway of traffic flow
m
Stable velocity of traffic flow
m/s
Stable density of traffic flow
vehs/km
Traffic volume of a single lane under stable states
vehs/h
CAVs penetration rate of traffic flow
—
Average headway of vehicles
s
4. Modeling the Fundamental Diagram of Mixed Traffic Flow
4.1. Dedicated lane management strategies
Under the condition of setting up dedicated lanes for CAVs, the road segment
consists of dedicated lanes and non-dedicated lanes (referred to as normal lanes in the
paper). To ensure the operational efficiency of CAVs dedicated lanes, only CAVs are
allowed in the dedicated lanes. Depending on whether CAVs can drive on normal
lanes or not, two lane management strategies can be developed, as shown in Fig. 1.
The first strategy is called the C-H strategy. In this strategy, CAVs are only allowed to
drive in dedicated lanes, and HDVs are only allowed to drive in normal lanes. The
second strategy is called the C-M strategy. In this strategy, CAVs are allowed to drive
in both dedicated and normal lanes, and HDVs are only allowed to drive in normal
lanes. Compared to the C-H strategy, the C-M strategy is more flexible and efficient in
some scenarios. For example, in a two-lane scenario, there is one CAVs dedicated lane
and one normal lane. With a high penetration rate of CAVs, the number of CAVs on
the road segment is much larger than the number of HDVs. Adopting the C-H strategy
may result in a situation where there is congested-flow in the CAVs dedicated lane but
free-flow in the normal lane. The C-M strategy can allow a portion of the CAVs to shift
to the normal lane, thereby reducing the traffic pressure on the CAVs dedicated lane
and improving the utilization efficiency of the normal lane. Thus, the paper models
the fundamental diagram of mixed traffic flow with CAVs dedicated lanes for the C-
M strategy. This paper is aimed at helping the traffic administration to rationalize the
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timing and number of dedicated lanes to set up to improve the overall traffic efficiency
of road segments.
(a) C-H strategy
(b) C-M strategy
Fig. 1. Different lane management strategies
Under the C-M strategy, CAVs can drive in dedicated and normal lanes, while
HDVs can only drive in normal lanes. Therefore, CAVs can choose between the
different types of lanes. For the C-M strategy, the key is when CAVs will decide to
drive in the normal lanes. Two lane selection principles for CAVs are introduced here.
First, drivers generally focus more on the traffic efficiency of the lanes on the road.
Therefore, for CAVs, drivers will choose the lane with a higher speed. Second, CAVs
drivers prefer to drive in the CAVs dedicated lane when the speeds in the dedicated
and normal lanes are the same. This is due to the simpler and safer driving
environment in the dedicated lanes where only CAVs are allowed as opposed to the
normal lanes.
Table 2. The traffic flow characteristics of the road segment under different stable
states
States
Vehicles
in
dedicated
lanes
Traffic flow
in dedicated
lanes
Vehicles in
normal
lanes
Traffic flow
in normal
lanes
Velocity
relationships
State 1
CAVs
Homogenous
HDVs
Homogenous
State 2
HDVs
Homogenous
State 3
HDVs
Homogenous
State 4
CAVs and
HDVs
Mixed
State 5
CAVs and
HDVs
Mixed
Based on the lane selection principles, the roadway has five stable states under the
C-M strategy. Here, and are used to denote the traffic flow velocity in stable
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states of dedicated and normal lanes, respectively. is the free-flow velocity of the
roadway.
⚫ State 1: all CAVs are distributed in CAVs dedicated lanes, and all HDVs are
distributed in normal lanes. Traffic flow velocities of both CAVs dedicated and
normal lanes are equal to .
⚫ State 2: all CAVs are distributed in the dedicated lanes, and all HDVs are
distributed in the normal lanes. The traffic flow velocity of the CAVs dedicated
lanes is , and the traffic flow velocity of the normal lanes is lower than
.
⚫ State 3: All CAVs are distributed in the dedicated lanes, and all HDVs are
distributed in the normal lanes. Traffic flow velocities in both the CAVs dedicated
and normal lanes are lower than , but the traffic flow velocity of the dedicated
lanes is higher than or equal to the traffic flow velocity of the normal lanes
.
⚫ State 4: CAVs are distributed in both CAVs dedicated and normal lanes, and all
HDVs are distributed in normal lanes. The traffic flow velocities in both the CAVs
dedicated and normal lanes are equal to .
⚫ State 5: CAVs are distributed in both CAVs dedicated and normal lanes, and all
HDVs are distributed in normal lanes. The traffic flow velocities in the CAVs
dedicated and normal lanes are equal and lower than .
The traffic flow characteristics of the road segment under different stable states
can be described in Table 2. CAVs and HDVs will change lanes according to the lane
selection principle, choosing the appropriate type of lane to travel in until some kind
of stable state is reached. These stable states are similar to Nash equilibrium. In a stable
state, CAVs with lane selection rights are not able to gain higher speeds by changing
to a different type of lane. Therefore, in a stable state, CAVs and HDVs do not change
lanes any more. This also implies that the distribution of CAVs and HDVs between
different types of lanes is constant in the stable state, which is an important premise
for the derivation of the fundamental diagram of a road segment with dedicated lanes
for CAVs. Based on this premise, we can treat the two types of lanes in stable state as
independent parts for fundamental diagram modeling separately, and then
superimpose the fundamental diagram models of the two types of lanes to obtain the
fundamental diagram model of the whole road segment. Also, we assume that the
distribution of vehicles in the same type of lanes is the same. Note that this paper
focuses on the distribution of vehicles on the road segment between two types of lanes
after a stable state has been reached by lane-changing behavior. This directly
determines the penetration of CAVs in normal lanes. While, the microscopic lane-
changing behavior of vehicles before reaching a stable state and the lane-changing
behavior of vehicles between the same types of lanes are beyond the scope of this paper.
4.2. Car-following modes in mixed traffic
According to the analysis in Section 4.1, there are five different traffic flow stable
states on the road segment under the C-M strategy. In State 4 and State 5, part of the
CAVs will choose the normal lane and form the mixed traffic flow with the HDVs. The
mixed traffic flow is more complex than the homogeneous traffic flow because the
sequence of vehicles can lead to the existence of multiple car-following behaviors at
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the same time in the mixed traffic flow. As shown in Fig. 1 (b), there are four car-
following behaviors in the mixed traffic flow: H-H, H-C, C-H, and C-C.
(1) H-H and H-C
For both the H-H and H-C car-following behaviors, the following vehicles are
HDV. HDV is not equipped with intelligent sensing devices, so the following vehicle
cannot detect the driving status of the leading vehicle through the onboard devices. In
this condition, whether the leading vehicle is HDV or CAV, the following vehicle
cannot exchange information with the leading vehicle. When the leading vehicle
accelerates or decelerates, the human driver in the following vehicle must spend some
time to perceive and judge the changes in the driving behavior of the leading vehicle,
and finally make the appropriate behavioral decisions. Here, the H-H and H-C
following behaviors are called HDV mode. The period between when the following
vehicle perceives the speed change of the leading vehicle and when it makes a
corresponding behavioral decision is referred to as the safety time interval. In HDV
mode, the HDVs safety interval is denoted by .
(2) C-H
For the C-H car-following behavior, the following vehicle is a CAV, and the
leading vehicle is an HDV. Since the leading vehicle is not equipped with intelligent
sensing devices, the leading vehicle cannot exchange information with the following
vehicle actively. In this condition, the information sensing function of the CAV will
degenerate, and the CAV can only sense the driving status of the leading vehicle
through onboard intelligent sensing devices. When the following vehicle perceives the
changes in the driving status of the leading vehicle, it can complete the perceiving,
judging, and decision process in a shorter time. Here the C-H following behavior is
referred to as AV mode. In AV mode, the safety interval for the following vehicle is
denoted by , which is less than .
(3) C-C
Both the leading and following vehicles are CAVs for the C- C car-following
behavior. In this condition, the following vehicle can utilize vehicle-to-vehicle (V2V)
technology to communicate with the leading vehicle to take full advantage of
cooperative driving. When the leading vehicle needs to accelerate or decelerate, the
leading vehicle will send the decision information of the next moment to the following
vehicle. Therefore, the driving behavior of the following vehicle can be almost
synchronized with the leading vehicle. Here the C-C following behavior is referred to
as CAV mode. In CAV mode, the safety interval for the following vehicle is denoted
by . can be a minimal value, much smaller than and , due to the sharing
of information between the leading and following vehicles.
4.3. Car-following model
To derive the fundamental diagram of mixed traffic flow with CAVs dedicated
lanes, the IDM is used here for modeling the car-following behavior of HDVs and
CAVs. The IDM has been widely used to model and simulate the car-following
behavior of HDVs since it was proposed. Since the IDM contains the parameter that
can reflect the physical meaning of the safe time headway of vehicles, in recent years,
scholars have also used the IDM to describe the car-following behavior of CAV for
exploring the characteristics of the mixed traffic flow(Talebpour & Mahmassani, 2016;
13
Yao, Wu, et al., 2022; J. Zhang et al., 2020). The IDM car-following model can be
expressed as Eq. (1).
(1)
Vehicles under different car-following modes have different safe time headway.
Therefore, just by changing the safe time headway in Eq. (1), the modeling of
vehicle car-following behavior in different car-following modes can be achieved based
on the IDM car-following model.
4.4. Fundamental diagram for mixed traffic with dedicated lanes
4.4.1 Fundamental diagram model for homogeneous traffic and mixed traffic without dedicated
lanes
Referring to (Yao, Wu, et al., 2022; L. Zhou et al., 2021), the headway of all vehicles
in the homogeneous traffic flow under the stable state is equal to the stable headway
. The velocity of traffic flow is the stable velocity , the velocity difference between
the leading and following vehicles is 0, and the acceleration of the vehicles is 0.
Therefore, when the traffic flow is stable, the vehicle driving state satisfies Eq. (2).
(2)
Substituting Eq. (2) into Eq. (1) obtains the fundamental diagram of homogeneous
traffic flow, which can be expressed as
(3)
(4)
(5)
For the convenience of calculation and discussion, is set to infinity here, when
the vehicle approaches with constant acceleration. In this situation, the
fundamental diagram of the homogeneous traffic flow is a linear fundamental
diagram, which can be expressed as
(6)
(7)
14
(8)
The traffic flow in stable states generally can be classified into two types. When
the traffic density is less than the critical density (i.e.,
), the traffic
flow is free-flow, and the vehicle drives at the free-flow velocity . When the traffic
density is greater than the critical density (i.e.,
), the traffic flow is
congested, and the velocity of the traffic flow is
. Thus, the fundamental
diagram of the homogeneous traffic flow can be expressed as Eq. (9).
(9)
The main difference for different homogeneous traffic flows containing different
types of vehicles is the safe time headway of the vehicles comprising the homogeneous
traffic flow. With only HDV car-following mode in a traffic flow composed entirely of
HDVs, the safe time headway is equal to . Thus, the fundamental diagram of
homogeneous traffic flow composed entirely of HDVs can be expressed as Eq. (10).
(10)
With only CAV car-following mode in a traffic flow composed entirely of CAVs,
the safe time headway is equal to . Thus, the fundamental diagram of homogeneous
traffic flow composed entirely of CAVs can be expressed as Eq. (11).
(11)
There are three car-following modes in the mixed traffic flow. Assuming that the
CAVs penetration rate in the mixed traffic flow is , the proportions of vehicles in
HDV mode, AV mode, and CAV mode are , ( ) , and , respectively
(Ngoduy et al., 2021b). The average time headway
for mixed traffic flow in stable
state can be expressed as Eq. (12).
(12)
Similarly, the fundamental diagram of mixed traffic flow can be expressed as Eq.
(13).
(13)
15
4.4.2 Fundamental diagram for mixed traffic with dedicated lanes
According to the analysis in Section 3.1, there are five possible different stable
states of traffic flow on the road segment under the C-M strategy. The traffic flow of
CAVs dedicated and normal lanes differs under different stable states. Therefore, for
each state, the traffic flows on different types of lanes must be modeled separately and
then superimposed to finally obtain the fundamental diagram model of the whole road
segment.
Assume that a road segment of length has lanes, where the number of
dedicated lanes is , . The number of normal lanes is . The
average traffic density of the whole road segment is denoted by ,
. The
penetration rate of CAVs on the road segment is denoted by ,
. Then, the
number of all vehicles on the road segment is , the number of CAVs and the
number of HDVs are and , respectively. Note that the
fundamental diagram model in this paper only considers mixed traffic flow scenarios
and does not consider the case where is 0 or 1. This is because when , there
is unnecessary to set up CAVs dedicated lanes, while when , all lanes can be
treated as CAVs dedicated lanes. Therefore, when is equal to 0 or 1, there is no
need to consider the setting of the number of CAVs dedicated lanes. Here, and
are used to denote the traffic flow velocities in the stable state for CAVs dedicated lanes
and normal lanes, respectively. and are the traffic flow densities in the stable
state for CAVs dedicated and normal lanes, respectively. In addition, two virtual
densities
and
are defined.
is the density of CAVs dedicated lanes when all
CAVs are assumed to be distributed in CAVs dedicated lanes.
is the density of
normal lanes when all HDVs are assumed to be distributed in normal lanes.
and
can be expressed as Eq. (14) and Eq. (15), respectively.
(14)
(15)
(1) State 1
In State 1, all CAVs are distributed in CAVs dedicated lanes, and all HDVs are
distributed in normal lanes, i.e., homogeneous traffic flow in both CAVs dedicated and
normal lanes. Therefore, the traffic densities of CAVs dedicated and normal lanes in a
stable state can be expressed as Eqs. (16) and (17). Meanwhile, both CAVs dedicated
lanes and normal lanes are in the free-flow state, . Therefore, the traffic
density on CAVs dedicated lanes and traffic density on normal lanes satisfy
Eq. (18) and Eq. (19), respectively.
(16)
(17)
16
(18)
(19)
Substituting Eq. (16) into Eq. (18) and Eq. (17) into Eq. (19) obtains the constraints
on the average traffic density for the whole road segment as Eqs. (20) and (21).
(20)
(21)
The traffic volume of a CAVs dedicated lane and a normal lane are shown in Eq.
(22) and Eq. (23), respectively.
(22)
(23)
Therefore, in State 1, the traffic volume of the road segment can be expressed
as Eq. (24).
(24)
(2) State 2
In State 2, all CAVs are distributed in CAVs dedicated lanes, and all HDVs are
distributed in normal lanes, i.e., homogeneous traffic flow in both CAVs dedicated and
normal lanes. The traffic densities of CAVs dedicated and normal lanes in a stable state
can be expressed as Eqs. (16) and (17) as well. Meanwhile, the CAVs dedicated lanes
are in a free-flow state, and the normal lanes are in a congested state, .
The traffic density on CAVs dedicated lanes and traffic density on normal
lanes satisfy Eq. (18) and Eq. (25), respectively.
(25)
Substituting Eq. (16) into Eq. (18) and Eq. (17) into Eq. (25) obtains the constraints
on the average traffic density for the whole road segment as Eqs. (20) and (26).
(26)
The traffic volume of a CAVs dedicated lane and a normal lane are shown in Eq.
(22) and Eq. (27), respectively.
(27)
Therefore, in State 2, the traffic volume of the road segment can be expressed
as Eq. (28).
17
(28)
(3) State 3
Similar to States 1 and 2, State 3 has homogeneous traffic flow in both CAVs
dedicated and normal lanes. Thus, the traffic densities of CAVs dedicated and normal
lanes in the stable state can be expressed as Eqs. (16) and (17). Also, both CAVs
dedicated and normal lanes are in congestion, and the velocity of the CAVs dedicated
lanes is higher than the velocity of the normal lanes, . The traffic density
on CAVs dedicated lanes and traffic density on normal lanes satisfy Eqs. (29) ,
(25) and (30).
(29)
(30)
Substituting Eqs. (16) and (17) into Eqs. (29), (25), and (30) yield the constraints on
the average traffic density for the whole road segment as Eqs. (31), (26) and (32).
(31)
(32)
The traffic volume of a CAVs dedicated lane and a normal lane are shown in Eq.
(33) and Eq. (34), respectively.
(33)
(34)
In State 3, the traffic volume of the road segment can be expressed as Eq. (35).
(35)
(4) State 4
In State 4, some CAVs shift to the normal lanes. There is a homogeneous traffic
flow in the CAVs dedicated lanes and a mixed traffic flow in the normal lanes. In this
state,
and
must satisfy Eq. (36) and Eq. (37).
(36)
(37)
The spillover ratio of CAVs in the stable state is denoted as here. The number
of CAVs in the normal lanes is and the number of vehicles in the normal lanes
18
is . The traffic density of the CAVs dedicated and normal lanes
can be expressed as
(38)
(39)
Both the dedicated and normal lanes are free-flow in State 4, and the stable density
of the CAVs dedicated lanes is exactly the critical density. Thus, and satisfy
Eq. (40) and Eq. (41).
(40)
(41)
In State 4, the CAVs spillover ratio is enough to keep the CAVs dedicated lanes in
a free-flow state. According to the CAVs' lane selection principles, CAVs in CAVs
dedicated lanes will not change lanes after reaching free-flow velocity. Therefore, the
spillover ratio corresponding to making the CAVs dedicated lanes shift from a
congested to a free-flow state is referred to here as the maximum possible spillover
ratio of CAVs, denoted as . Substituting Eq. (40) into Eq. (38), the maximum
spillover ratio is easily obtained. And the traffic density of the normal lanes
is obtained by Eq. (39) as Eq. (42) and Eq. (43).
(42)
(43)
The CAVs penetration rate and the average time headway
of the normal
lanes can be expressed as
(44)
(45)
According to Eqs. (36), (37), and (41), the constraints on the average traffic density
for the whole road segment are obtained as Eqs. (31), (46), and (47).
(46)
(47)
19
The traffic volume of a CAVs dedicated lane and a normal lane are shown in Eq.
(48) and Eq. (49), respectively.
(48)
(49)
In State 4, the traffic volume of the road segment can be expressed as Eq. (50).
(50)
(5) State 5
In State 5, some CAVs shift to the normal lanes. There is a homogeneous traffic
flow in the CAVs dedicated lanes and a mixed traffic flow in the normal lanes. Similar
to State 4,
and
satisfy Eq. (36) and Eq. (37).
In State 5, both the CAVs dedicated lane and the normal lane are in a congested
state, and the velocities of the CAVs dedicated lane and the normal lane are equal.
Assuming that the CAVs spillover ratio further increases to . At this time, the
CAVs dedicated lanes are just a free-flow state, the corresponding normal lanes are
inevitably congested, and the road segment is in an unstable state between the CAVs
dedicated lanes and normal lanes. In this unstable state, the traffic density of the road
segment satisfies Eq. (51).
(51)
Based on Eqs. (36), (37), and (51), the constraints on the average traffic density
for the whole road segment are obtained as Eqs. (31), (46), and (52).
(52)
The spillover ratio of CAVs in State 5 is denoted as here,
. The
traffic density of CAVs dedicated lanes and normal lanes in State 5 can be expressed
as Eq. (53) and Eq. (54). The CAVs penetration rate and the average time headway
of the normal lanes can be expressed as Eq. (55) and Eq. (56)
(53)
(54)
(55)
(56)
Since in State 5, the velocities of CAVs dedicated lanes and normal lanes are equal,
20
the density of traffic flow satisfies Eq. (57) when the CAVs spillover ratio .
(57)
To obtain the overflow ratio of CAVs in stable State 5, we construct a function
as Eq. (58).
(58)
When Eq. (57) holds, the corresponding spillover ratio makes .
However, the equation is complex, it is difficult to find the analytical solution of
directly from . Here, the bisection method is used to solve for , and
the solving precision is set to 0.01.
After obtaining the spillover ratio , the traffic volume of a CAVs dedicated lane
and a normal lane are shown in Eq. (59) and Eq. (60), respectively.
(59)
(60)
Meanwhile, the expression for
can be derived from Eq. (57) as in Eq. (61).
(61)
After substituting Eq. (61) into Eq. (60), the traffic volume on a normal lane
can be expressed as Eq. (62).
(62)
Substituting Eqs. (53) and (54) into Eqs. (59) and (62), and can be
expressed as Eqs. (63) and Eq. (64), respectively.
(63)
(64)
In State 5, the traffic volume of the road segment can be expressed as Eq. (65).
(65)
(6) Fundamental diagram
To facilitate the analysis, seven auxiliary variables are introduced here.
,
,
,
,
,
and
。Denote
the ratio of the number of CAVs dedicated lanes to the total number of lanes on the
road segment by ，
，. Then these seven auxiliary variables can
21
be expressed as Eq. (66).
(66)
Based on the above derivation and the introduced auxiliary variables, the
fundamental diagram of mixed traffic flow on a multi-lane road segment with CAVs
dedicated lanes under the C-M lane management strategy can be expressed as Eq. (67).
∞
∞
∞
∞
∞
∞
(67)
The following propositions provide some relevant properties of the proposed
fundamental diagram.
Proposition 1: In State 1, the traffic volumes of the CAVs dedicated lanes, the
normal lanes, and the whole road segment all gradually increase as the traffic density
increases.
Proof: In State 1,
,
, . The derivatives of ,
, and to can be obtained as Eqs. (68), (69), and (70).
(68)
(69)
(70)
In Eqs. (68), (69), and (70), the variables , , and are all greater than 0,
. Therefore,
,
and
are all greater than 0. It indicates that
increase with traffic density . This completes the proof.
Proposition 2: In State 2, with the traffic density increases, the traffic volume
of the CAVs dedicated lanes gradually increases, and the traffic volume of the normal
22
lanes gradually decreases. If
, with the traffic density increases, the
traffic volume of the road segment gradually decreases. If
, with the
traffic density increases, the traffic volume of the road segment gradually increases.
If
, with the traffic density increases, the traffic volume of the road
segment remains constant.
Proof: In State 2,
,
,
. The derivatives of , , and to can be obtained as Eqs.
(71), (72), and (73).
(71)
(72)
(73)
In Eqs. (71), (72), and (73), the variables , , , , , and are all
greater than 0, ,
. Therefore,
is greater than 0,
is less than
0. It indicates that increases with traffic density and decreases with traffic
density . Meanwhile, for
, three cases can be discussed. Case 1, if
,
is less than 0. In this case, decreases with traffic density . Case 2, if
,
is greater than 0. In this case, increases with traffic density . Case 3,
if
,
is equal to 0. In this case, does not change with increasing
traffic density . This completes the proof.
Proposition 3: In State 3, the traffic volumes of the CAVs dedicated lanes, the
normal lanes, and the whole road segment gradually decrease as the traffic density
increases.
Proof: In State 3,
,
,
. The derivatives of , , and to can be
obtained as Eqs. (74), (75), and (76).
(74)
(75)
(76)
In Eqs. (74), (75), and (76), the variables , , , , , and are all
23
greater than 0, ,
. Therefore,
,
and
are all less than 0. It
indicates that decrease with traffic density . This completes the
proof.
Proposition 4: In State 4, the traffic volumes of the normal lanes, and the whole
road segment all gradually increase as the traffic density increases. The traffic
volume in the CAVs dedicated lanes is constant.
Proof: In State 4,
,
, . The
derivatives of , , and to can be obtained as Eqs. (77), (78), and (79).
(77)
(78)
(79)
In Eqs. (77), (78), and (79), the variables , , , , , and are all
greater than 0, ,
. Therefore,
and
are greater than 0. It
indicates that increase with traffic density . And
, which
indicates that is a constant, independent of traffic density . This completes the
proof.
5. Results and Discussions
5.1. Analysis of auxiliary variables
According to Eq. (67), the auxiliary variables determine the stable state
distribution of the road segment. To analyze the stable state distribution of the road
segment, the auxiliary variable curves were analyzed numerically. Before proceeding
with the numerical analysis, we determine some traffic flow parameters first. Referring
to the study by Ngoduy et al. (Ngoduy et al., 2021b), The safety time headway ,
and are taken as 0.5s, 1.0s and 2.0s respectively. The free-flow velocity is set to
33.3 m/s. The minimum vehicle spacing is set to 2.0 m. The vehicle length is set
to 5.0 m.
The curves of the seven auxiliary variables for different CAVs dedicated lane
setting scenarios under different penetration rates are shown in Fig. 2. According to
Eq. (66), The curve of the auxiliary variable is independent of the number of the CAVs
dedicated lanes and the number of the road segment lanes. It is determined only by
the ratio of the number of the CAVs dedicated lanes to the number of the road
segment lanes. Thus Fig. 2 (a) and Fig. 2 (e) are completely identical.
In Fig. 2, the stable state distributions of road segments under different CAVs
penetration rates and traffic densities are represented by color shadows, with different
colored shaded regions representing different stable states. In addition, some non-
shaded regions are shown in white in Fig. 2. These areas represent that the traffic
densities of the normal or CAVs dedicated lanes have exceeded their jammed density
under that CAVs penetration rate and traffic density condition. Note that the arrival
24
and distribution of different types of vehicles in the mixed traffic flow are uniform.
When any lane in the road segment reaches a totally jammed state, the traffic density
of all road segments cannot continue to increase. The traffic density at this point is the
critical jammed density of the whole road segment. In this paper, we only discuss the
traffic flow characteristics when the lanes are in the free-flow and congested states,
without discussing the case when the road segment is jammed.
The density of the shaded regions of different colors at a given CAVs penetration
rate will be the effective traffic density for that stable state. The traffic density
corresponding to the boundary between the non-shaded region and the colored
shaded region is the critical jam density of the road segment. As shown in Fig. 2, the
critical jammed density is determined by auxiliary variables and . In general,
the critical jammed density tends to increase and then remain constant as the CAVs
penetration rate increases. When the CAVs penetration rate increases to , the critical
jammed density reaches a maximum value of
. The critical jammed density
remains constant as the CAVs penetration rate increases further. It can be seen from
Fig. 2 that the total area of the shaded region is determined by . As increases, the
total area of the shaded region decreases. It also indicates that the fewer traffic
conditions for which the C-M strategy is feasible, the lower the applicability of the C-
M strategy.
It can be clearly observed in Fig. 2 that the stable state of the road segment under
the identical penetration rate condition shifts with the variation of traffic density. If the
composition of stable states of traffic flow under the same CAVs penetration rate is
used as a differentiation criterion, the penetration rate change process can be divided
into four stages, each consisting of a different stable state. The boundaries of the stages
are determined by the intersection of the curves of the auxiliary variables, e.g., the final
penetration rate of Stage 1 is the penetration rate corresponding to the intersection of
the curves of the auxiliary variables and . Therefore, the initial and final
penetration rates for each stage and the range of each stage can be easily obtained
through the formulae for the auxiliary variables, as shown in Table 3. In Table 3, State
4 and State 5 only exist in Stage 3 and Stage 4. It indicates that under the C-M strategy,
only when the penetration of CAVs is higher than , some of the CAVs may shift to
the normal lanes to form mixed traffic flow. When the CAVs penetration rate is less
than , all CAVs are distributed in the CAVs dedicated lanes, at wthis point, there is
homogeneous traffic flow in both the CAVs dedicated lanes and the normal lanes.
The stages appear in the same order for the different CAVs dedicated lane setting
scenarios as the CAVs penetration rate increases. Stage 1 is first, followed by Stage 2,
Stage 3, and Stage 4 in that order.
(1) During Stage 1, only two stable states exist on the road segment. Under the
same CAVs penetration rate, the stable state of the road segment shifts from
State 1 to State 2 as the traffic density increases. The range of effective densities
for State 1 and 2 gradually increases with the increasing CAVs penetration rate.
(2) During Stage 2, there are three stable states on the road segment. Under the
same CAVs penetration rate, the stable state of the road segment shifts from
State 1 to State 2 and finally to State 3 as the traffic density increases. Moreover,
as the CAVs penetration rate increases, the range of effective densities of States
1 and State 3 gradually increases, and the range of effective densities of State
25
2 gradually decreases.
(3) During Stage 3, there are four stable states on the road segment. Under the
same CAVs penetration rate, the stable state of the road segment shifts in order
from State 1, to State 2, to State 3, to State 5, as the traffic density increases. As
the CAVs penetration rate increases, the range of effective densities of States 1
and 5 gradually increases, and the range of effective densities of States 2 and
3 gradually decreases.
(4) During Stage 4, there are three stable states on the road segment. Under the
same CAVs penetration rate, the stable state of the road segment shifts from
State 1 to State 4 and finally to State 5 as the traffic density increases. As the
CAVs penetration rate increases, the range of effective densities of State 4
gradually increases, and the range of effective densities of State 1 and State 5
gradually decreases.
(a)
(b)
(c)
(d)
26
(e)
(f)
Fig. 2. The auxiliary variables with different penetration rates of CAVs and number of
lanes
27
Table 3. Range of CAVs penetration rate in different stages
Stages
composition of states
Initial CAVs penetration rate
Final CAVs penetration rate
Range of CAVs penetration rate
Stage 1
State 1, State 2
0
Stage 2
State 1, State 2, State 3
Stage 3
State 1, State 2, State 3, State 5
Stage 4
State 1, State 4, State 5
1
28
5.2. The fundamental diagram with dedicated lanes
It is worth noting that the fundamental diagram model proposed in this paper
applies to arbitrary multi-lane scenarios. To analyze the impacts of different CAVs
penetration rates, traffic densities, and CAVs dedicated lane settings on the
fundamental diagram of mixed traffic flow, the three-lane scenario is used as a case
for analysis and demonstration in this section. Based on the parameter settings in
Section 4.1, the traffic volumes under different conditions are numerically computed
according to Eqs. (67), and finally, the fundamental diagram of the mixed traffic flow
for the three-lane scenario with CAVs dedicated lanes is obtained, as shown in Fig. 3.
Combining the slope of the traffic volume curve in Fig. 3 and the stable state
distribution in Fig. 2, it is clear to observe the traffic volume changes and the shift of
the stable state of the road segment. For example, in Fig. 3 (a), when the road section
is equipped with one CAVs dedicated lane and the CAVs penetration rate is 0.2, the
road segment goes through three stable states, State 1, State 2, and State 3, as the traffic
density increases, until it reaches the critical jammed density. In Fig. 3 (a), the traffic
volume of the road segment shows a trend of increasing and then decreasing as the
density increases. The traffic volume corresponding to the highest point of the traffic
volume curve is the traffic capacity of the roadway for that CAVs penetration rate
condition. It can be observed that as the CAVs penetration rate increases, the traffic
capacity gradually increases. In addition, the variation of the critical jammed density
can be observed in Fig. 3. For example, in Fig. 3 (b), when the road segment is equipped
with two dedicated lanes, the critical jammed densities corresponding to penetration
rates of 0.2, 0.4, and 0.6 are 60 veh/km, 80 veh/km, and 120 veh/km, respectively. When
the traffic density exceeds the critical jammed density, the corresponding traffic
volume is null, indicating that this situation is impossible. In addition, when the
penetration rate of CAVs is 0.2, the traffic volume curve has been showing an
increasing trend without a turning point. This indicates that the traffic density reaches
the critical jamming density before the turning point in the traffic volume curve occurs.
This is because of the low penetration rate of CAVs and the fact that there is only 1
normal lane, as the traffic density increases, the jamming condition is reached very
quickly in the normal lane.
The CAVs spillover ratios for different CAVs penetration rates and traffic densities
are illustrated in Fig. 4. As shown in Fig. 4, and the CAVs spillover ratio is greater than
0 only when the traffic conditions are in State 4 and State 5. For the rest of the stable
states, the CAVs spillover ratio is equal to 0, and there are no CAVs spillovers from the
CAVs dedicated lanes to the normal lanes. Under the same traffic density, the CAVs
spillover ratio increases with CAVs penetration rate. Similarly, under the same CAVs
penetration rate, the CAVs spillover ratio increases with traffic density. In addition,
when CAVs penetration rate is at a high level, changes of density in the range of State
5 (i.e., the region above the curve of ) do not have a significant effect on the CAVs
spillover ratio compared to State 4. For example, in Fig. 4 (a), when the CAVs
penetration rate exceeds 0.8, the CAVs spillover ratio rises rapidly in State 4 as the
density increases. When the density reaches the range of effective densities in State 5,
further increases in density have little impact on the CAVs spillover ratio. In Fig. 4 (b),
when the penetration rate reaches 0.95 or more, the change in the spillover ratio also
shows the same feature.
29
(a)
(b)
Fig. 3. Fundamental diagram under different CAVs penetration rates for the three-
lane scenario with CAVs dedicated lanes
(a)
(b)
Fig. 4. Spillover ratio of CAVs under different traffic conditions
To address the problem of when dedicated lanes should be set up and how many
lanes should be set up, the fundamental diagrams of the different dedicated lane
setting scenarios are compared. Fig. 5 shows the traffic volumes under different
dedicated lane setting scenarios with CAVs penetration rates of 0.2, 0.4, 0.6, and 0.8,
respectively. The optimal timing and number for setting up CAVs dedicated lanes can
be quickly determined by Fig. 5. A detailed explanation of how to set up CAVs
dedicated lanes via the fundamental diagram is illustrated using Fig. 5 (a) and (b) as
an example.
For facilitating the discussion, for the three-lane road segment, we denote no
CAVs dedicated lane, setting up one CAVs dedicated lane and setting up two CAVs
dedicated lanes as Setting 0, Setting 1, and Setting 2, respectively. And the CAVs
dedicated lane setting that maximizes the traffic volume of the road segment under a
particular traffic condition is referred to as the optimal setting. As shown in Fig. 5 (a),
when the penetration rate of CAVs is 0.2, the traffic volume of Setting 1 and Setting 0
shows an increasing and then decreasing trend as the density increases, while the
traffic volume of Setting 2 always keeps an increasing trend. When the density is less
than 30veh/km, the traffic volume of Setting 0 is always greater than or equal to the
traffic volume of Setting 1 and Setting 2. To maximize the traffic efficiency of the
30
roadway, CAVs dedicated lanes should not be set up at this point. As the density
increases further, the traffic volume of Setup 1 begins to be greater than that of Setup
0 and Setup 2. When the density reaches 60 veh/km, Setup 2 reaches the critical
jammed density. Setup 2 will become infeasible for higher densities. When the density
ranges from 30 veh/km to 120 veh/km, Setting 1 is the optimal setting. During this
period, setting up one dedicated lane for CAVs can maximize the traffic volume of the
entire road segment. When the density is more significant than 120 veh/km, both
Setting 1 and Setting 2 have reached the critical jammed density, so both Setting 1 and
Setting 2 are infeasible, and Setting 0 is the optimal setting. Similar to Fig. 5 (a), in Fig.
5 (b), Setting 0 is optimal when the density is less than 20 veh/km. When the density is
between 20 veh/km and 44 veh/km, Setting 1 is the optimal setting. When the density
is between 44 veh/km and 80 veh/km, Setting 2 is the optimal setting. When the density
is greater than 80 veh/km, Setting 1 is the optimal choice.
In addition, in Fig. 5 (b), when the CAVs penetration rate is 0.4, and the density is
70 veh/km, Setting 2 is the optimal setting with a traffic volume of 10,282 veh/h. The
traffic volume of Setting 0 and Setting 1 are 3624 veh/h and 4979 veh/h, respectively.
Compared with Setting 0 and Setting 1, Setting 2 improves the traffic volume by 184%
and 107%, respectively. Therefore, the appropriate setting of dedicated lanes under
certain traffic conditions is beneficial and necessary for improving traffic volume on
roadway segments. Meanwhile, as the CAVs penetration rate increases, the difference
in traffic volume between different settings becomes smaller. At a CAVs penetration
rate of 0.8, the traffic volume curves of Setting 0 and Setting 1 have largely overlapped,
and Setting 2 has seen a limited improvement in traffic volume compared to Setting 0
and Setting 1.
Utilizing the fundamental diagram, the optimal setting of the CAVs dedicated
lanes that maximize the traffic volume of the road segment under different CAVs
penetration rates and different densities can be obtained, as shown in Fig. 6. In the
two-lane scenario, the setting up of CAVs dedicated lanes is possible to have a gain on
the overall traffic volume of the road segment only when the traffic density is greater
than 22veh/km, or the CAVs penetration rate is greater than 0.1. As the number of lanes
on the road segment rises, the minimum density and the minimum CAVs penetration
rate for setting up dedicated lanes gradually decrease. Meanwhile, as the CAVs
penetration rate increases, the optimal number of CAVs dedicated lanes gradually
increases. Fig. 7 shows the traffic volume gain rates of the optimal setting for the multi-
lane scenario compared to that without dedicated lanes. As shown in Fig. 7, compared
with no CAVs dedicated lane setting, the rational CAVs dedicated lane settings can
significantly improve the traffic volume of the road segment. In the two-lane scenario,
three-lane scenario, and four-lane scenario, the optimal CAVs dedicated lane settings
can improve the traffic volume of the road segment by 243.4%, 269.2%, and 276.4% at
maximum.
31
(a)
(b)
(c)
(d)
Fig. 5. Fundamental diagram of different CAVs dedicated lane settings for a three-
lane scenario
(a)
(b)
(c)
Fig. 6. Optimal CAVs dedicated lane settings in different multi-lane scenarios
(a)
(b)
(c)
Fig. 7. Traffic volume gain rates of the optimal setting for the multi-lane scenario
compared to that without dedicated lanes
32
5.3. Traffic capacity
Traffic capacity refers to the maximum traffic volume of a road segment under
certain traffic conditions, and the traffic density corresponding to the maximum traffic
volume is called the optimal density. Fig. 8 shows the traffic capacity and optimal
density under different CAVs dedicated lane setting scenarios. As shown in Fig. 8 (a),
under the C-M strategy, the traffic capacity of all three CAVs dedicated lane setting
scenarios gradually increases as the CAVs penetration rate increases. When the CAVs
penetration rate is less than 0.13, the traffic capacity without CAVs dedicated lanes is
greater than with one CAVs dedicated lane and two CAVs dedicated lanes. It is because
when the penetration rate is low, the number of CAVs on the road segment is minimal,
and the setup of dedicated lanes will waste road resources and reduce the overall
operational efficiency of the road segment. When the CAVs penetration rate is between
0.13 and 0.3, at this point, Setting 1 has a greater traffic capacity than the remaining
two settings. Setting 2 has the greatest traffic capacity when the CAVs penetration rate
is greater than 0.3. As the CAVs penetration rate increases further, the traffic capacity
advantage of Setting 2 shows a trend of increasing and then decreasing. When the
CAVs penetration rate reaches 0.95 or more, the traffic capacity of the three CAVs
dedicated lane settings is basically equal. As shown in Fig. 8 (b), the optimal density
of Setting 0 gradually increases as the CAVs penetration rate increases. And the
optimal densities of Setting 1 and Setting 2 show four stages of change with increasing
CAVs penetration rates, and are illustrated with the example of Setting 1. The optimal
density remains constant when the CAVs penetration rate is less than 0.1. Secondly,
the optimal density gradually increases with CAVs penetration rates between 0.1 and
0.13. Then, the optimal density gradually decreases with CAVs penetration rates
between 0.13 and 0.61. Finally, the optimal density gradually increases with CAVs
penetration rates greater than 0.61.
Fig. 9 shows the impact of different lane management strategies on traffic capacity.
In (Yao, Wu, et al., 2022), Yao et al. provided the fundamental diagram formulation
under the C-H strategy, through which the traffic capacity under the C-H strategy can
be obtained. The calculation method of traffic capacity under C-H strategy is not
repeated here, and interested readers can refer to (Yao, Wu, et al., 2022). Different from
the C-M strategy, the traffic capacity of the C-H strategy shows a trend of increasing
and then decreasing with the CAVs penetration rate. As shown in Fig. 9, for the
scenario with one CAVs dedicated lane, when the CAVs penetration rate is less than
0.61, both the C-H and the C-M strategies have the same traffic capacity. However,
when the CAVs penetration rate is greater than 0.61, the traffic capacity of the C-H
strategy gradually decreases, while the traffic capacity of the C-M strategy gradually
increases. Similarly, for the scenario with two CAVs dedicated lanes, when the CAVs
penetration rate is less than 0.87, both the C-H and C-M strategies have the same traffic
capacity. When the CAVs penetration rate is greater than 0.87, the traffic capacity of
the C-M strategy starts to be higher than the traffic capacity of the C-H strategy. And
the traffic capacity gap gradually increases with the increase of CAVs penetration rate.
When the difference in traffic capacity between the different lane management
strategies starts to appear, the CAVs penetration rate at this point is exactly the initial
penetration rate of Stage 4 in Fig. 2 (b) and (c). Therefore, the difference in lane
management strategies only impacts the traffic capacity within Stage 4. As
33
increases, the range of Stage 4 will gradually decrease. At the same time, the gap
between two lane management strategies in terms of traffic capacity becomes smaller
as increases.
(a) Traffic capacity
(b) Optimal density
Fig. 8. Traffic capacity and optimal density of different CAVs dedicated lane settings
under a three-lane scenarios
(a)
(b)
Fig. 9. Traffic capacity of different lane management strategies in a three-lane
scenario
5.4. The effect of headways on traffic capacity
In general, the time headways and of CAVs depend mainly on the vehicle
communication system, the automatic driving system, and the consideration of safety.
With the advancement of communication and control technologies, and can be
changed. To investigate the impact of different time headways on the traffic capacity
of the road segment, two numerical experiments were conducted for a three-lane road
segment scenario. In the first numerical experiment, we fixed at 1.0s and varied
from 0.2 to 1.0. In the second numerical experiment, we fixed at 1.0s and varied
from 0.6 to 1.4. The remaining parameters, such as , and , are the same as in
Section 5.1.
Fig. 10 shows the effect of on the traffic capacity of the three-lane road
segment. From Fig. 10, it can be observed that has almost no effect on traffic
34
capacity when CAVs penetration rate is low. Only when CAVs penetration rate reaches
a certain level will the change in have a significant impact on traffic capacity. And
as the penetration rate of CAVs increases, the impact of on traffic capacity becomes
more and more significant. For example, in Fig. 10 (a), when the penetration rate of
CAVs is lower than 0.2, there is almost no change in traffic capacity as is raised.
The effect of on traffic capacity becomes observable when the penetration rate of
CAVs is higher than 0.2. The higher the penetration rate of CAVs, the greater the effect
of on traffic capacity. Compared to the setting of 1 CAVs dedicated lane, the
change in has a more significant effect on traffic capacity when 2 CAVs dedicated
lanes are set up. In addition, as a whole, under the same penetration rate of CAVs,
traffic capacity gradually increases as decreases. Similar to Fig. 10, Fig. 11 shows
the impact of the change in on traffic capacity. From Fig. 11, it can be found that at
the same penetration rate of CAVs, there is almost no change in traffic capacity with
the increase of , even though the penetration rate of CAVs is high. This indicates
that the impact of on traffic capacity is not significant.
(a)
(b)
Fig. 10. Traffic capacity with different
(a)
(b)
Fig. 11. Traffic capacity with different
6. Conclusions and Future Work
To rationally deploy CAVs dedicated lanes, this paper proposes the fundamental
diagram model of mixed traffic flow with CAVs dedicated lanes based on the C-M lane
management strategy. Numerical simulations and analyses based on the proposed
35
fundamental diagram model were conducted to investigate the impacts of different
CAVs dedicated lane settings on the traffic volume and capacity of the road segment.
The following significant conclusions were obtained.
(1) Under the C-M lane management strategy, the road segment has five stable
states of traffic flow. The distribution of each stable state under different traffic
conditions (CAVs penetration rate, traffic density) is only related to the ratio of the
number of CAVs dedicated lanes to the total number of lanes.
(2) The applicability of the C-M lane management strategy is influenced by .
As increases, the fewer traffic conditions for which the C-M strategy is feasible, the
less applicable the C-M strategy is to different traffic conditions.
(3) Under the C-M strategy, CAVs may shift into the normal lanes only if the
CAVs penetration rate is greater than . Otherwise, CAVs will all be distributed in
CAVs dedicated lanes.
(4) Using the proposed fundamental diagram model, the optimal number and
timing of setting up CAVs dedicated lanes in multi-lane scenarios can be rationally
determined to maximize the efficiency of the road segment. As the total number of
lanes on the road segment rises, the minimum density and minimum CAVs
penetration rate allowed for setting up CAVs dedicated lanes decrease gradually. At
the same time, the optimal number of CAVs dedicated lanes is gradually increased as
the CAVs penetration rate increases.
(5) In the two-lane scenario, three-lane scenario, and four-lane scenario, the
optimal CAVs dedicated lane settings can improve the traffic volume of the road
segment by 243.4%, 269.2%, and 276.4% at maximum.
(6) Under the C-M strategy, for the different CAVs dedicated lane setting
scenarios, the traffic capacity of the road segment shows a trend of gradual rise as the
CAVs penetration rate increases.
(7) Unlike the C-M strategy, the traffic capacity of the C-H strategy shows a trend
of increasing and then decreasing with the increase in the CAVs penetration rate. When
the CAVs penetration rate is lower than the initial penetration rate of State 4, the traffic
capacity of the C-M and C-H strategies are identical. When the CAVs penetration rate
is higher than the initial penetration rate of State 4, the traffic capacity of the C-M
strategy is gradually higher than that of the C-H strategy, and the traffic capacity gap
between the two strategies gradually increases as the CAVs penetration rate increases.
With the increase of , the advantage of the C-M strategy in terms of traffic capacity
diminishes compared to the C-H strategy.
(8) has almost no effect on traffic capacity when CAVs penetration rate is low.
Only when CAVs penetration rate reaches a certain level will the change in T_C have
a significant impact on traffic capacity. the impact of on traffic capacity is not
significant.
This paper analyzes five stable states of the road segment and the traffic flow
characteristics under each stable state based on the flexible and efficient C-M lane
management strategy. Then, the fundamental diagram model of mixed traffic flow
with CAVs dedicated lanes under the C-M strategy is derived based on the car-
following model. Finally, numerical simulations of the fundamental diagram model
were conducted to analyze the impacts of different CAVs dedicated lane settings on
the traffic volume and capacity of the road segment under the C-M strategy, and some
important and valuable conclusions were discovered. In summary, the fundamental
36
diagram model of mixed traffic flow with CAVs dedicated lanes proposed in this paper
can provide new insights and guidance for CAVs dedicated lane management under
different CAVs development stages.
However, this work still has some limitations. Based on these limitations, this
work could be extended to the following interesting directions in the future.
(1) This paper analyzes the impact of CAVs dedicated lanes on traffic efficiency
only based on the proposed fundamental diagram model. The influence of different
lane management strategies and different numbers of CAVs dedicated lanes on traffic
safety, fuel consumption, and emissions will be further analyzed via traffic simulation
in the future.
(2) In this paper, platoon micro characteristics (e.g., platoon intensity and platoon
size) were not considered. In future studies, the impact of platoon intensity and
platoon size on the fundamental diagram and traffic capacity of mixed traffic flow with
CAVs dedicated lanes will be further investigated.
(3) The fundamental diagram model proposed in this paper belongs to the
deterministic fundamental diagram, i.e., it is assumed in the model that the time
headways of vehicles are fixed. In actual traffic, the time headways are always
stochastic. In future research, the time headways’ stochasticity will be considered in
the process of fundamental diagram modeling, and the effect of the time headways’
stochasticity on the fundamental diagram with CAVs dedicated lanes for mixed traffic
flow will be investigated.
(4) In this paper, the lane-changing and merging behaviors of vehicles are not
considered when the mixed traffic flow on a road segment is in a stable state. It is
assumed that the traffic flow on different types of lanes is stable when in a steady state,
which is the premise for deriving the fundamental diagram using the car-following
model. In real traffic, vehicles may make lane changes or merge. Therefore, in future
studies, traffic simulation methods will be considered to investigate the effect of lane-
changing behavior on mixed traffic flow.
Acknowledgments
The paper received research funding support from the National Natural Science
Foundation of China (52002339), the Sichuan Science and Technology Program
(2024NSFSC0179, 2023ZHCG0018), the Fundamental Research Funds for the Central
Universities (2682023ZTPY034), and Chengdu Soft Science Research Project (2023-
RK00-00029-ZF).
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