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Impact of the heterogeneity and platoon size of connected vehicles on the capacity of mixed
traffic flow
Zhihong Yao1,2,3*, Yuqin Ma1,2, Tingting Ren1,2, Yangsheng Jiang1,2,3
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
This paper proposes a capacity analysis model for a mixed traffic flow environment that considers
the heterogeneity and maximum platoon size of connected vehicles. Firstly, we explore how the
organization mode of platoon affects car-following characteristics in mixed traffic flow with
connected and automated vehicles, connected human-driven vehicles, automated vehicles, and
human-driven vehicles. Different car-following models are used to describe the car-following
characteristics of different types of vehicles. Secondly, the probability distribution model is
developed for the size of platoons considering the penetration rate and the maximum platoon size
of connected and automated vehicles, connected human-driven vehicles. Thirdly, the fundamental
diagram of mixed traffic flow is further derived based on the probability distribution of platoon size
and car-following models. Then, the capacity model of mixed traffic flow is proposed. Finally, a
numerical experiment is designed to evaluate the impact of different penetration rates and
maximum platoon sizes of connected and automated vehicles, connected human-driven vehicles on
the capacity of the mixed traffic flow. The results show that (1) with a low penetration rate of
connected and automated vehicles, connected human-driven vehicles, the change in the maximum
platoon size has no noticeable effect on the distribution of platoon size of connected and automated
vehicles, connected human-driven vehicles. (2) Road capacity increases with the maximum platoon
size and the penetration rate of connected vehicles. (3) Compared with connected human-driven
vehicles, the penetration rate of connected and automated vehicles has a more significant impact on
road capacity.
Keywords: mixed traffic flow; maximum platoon size; fundamental diagram; road capacity;
penetration rate
* Correspondence to: Zhihong Yao, School of Transportation and Logistics, Southwest Jiaotong University, Chengdu,
Sichuan 610031, China, E-mail: zhyao@swjtu.edu.cn
2
1. Introduction
1.1. Background
In recent years, with the development of autonomous driving and wireless communication
technologies, connected automated vehicles (CAVs) have attracted increasing attention from
researchers. Equipped with onboard cameras, laser rangefinders, radar sensors, and other
equipment, CAVs can cooperate to share real-time information [1–3]. The real-time information
contains the status of the vehicle (i.e., position, velocity, acceleration), the traffic signals (i.e., traffic
signal timing, variable speed limit, traffic management strategy, etc.), the driving intention of the
vehicle (i.e., acceleration, deceleration, lane-changing), etc. Based on this real-time information,
CAVs can further complete collaborative driving tasks. These cooperative driving behaviors can
significantly improve road capacity, safety, stability and reduce traffic congestion and energy
consumption of transportation systems [4–6]. Many studies are optimistic about the application
prospect of CAVs, but the popularization of CAVs cannot be completed quickly. Moreover, the
reconstruction and upgrading of its supporting road infrastructure are time-consuming. At the same
time, the relevant laws and regulations of CAVs have not been improved. Therefore, in the future,
CAVs and human-driven vehicles (HVs) will coexist in a mixed traffic flow on the road for an
extended period [7]. Based on the classification of automation and connection level of CAVs in
relevant standards [8,9], it is confirmed that the mixed traffic flow will consist of CAVs, connected
human-driven vehicles (CVs), automated vehicles (AVs), and HVs on the road in the future [10].
To ease traffic congestion and other traffic problems, some scholars have conducted relevant
studies on road capacity under a mixed traffic flow environment. The main methods include setting
up automatic dedicated lanes and using CAVs' cooperation to form a platoon of vehicles. For
example, Chen et al. [11] developed a diffusion framework to predict the change in the penetration
rate (PR) of CVs by considering the benefits (i.e., reduced travel time) brought by the dedicated lanes.
Then, the dedicated lane configuration optimization model was proposed based on the traffic
equilibrium theory. However, when the traffic demand or the PR of CAVs is low, the setting of
dedicated lanes will increase the road construction cost and harm the overall road capacity [12].
Therefore, some scholars stated the concept of platoon management: the same type of connected
vehicles can form a certain size of the platoon to drive. Based on the homogeneity, the vehicles in
the platoon can realize cooperative driving with a smaller time headway. Therefore, this strategy
can improve road capacity and traffic flow stability to a certain extent, thus alleviating traffic
congestion. However, most existing studies focus on platoon size in mixed traffic flow with CAVs
and HVs [13,14], ignoring that CVs can also form flexible platoon driving through wireless
communication technologies. Thus, the proposed relevant models were insufficient in describing the
regularity of the mixed traffic flow, which couldn’t give theoretical support for intelligent traffic
planning and management. To solve this issue, this paper investigates the capacity of mixed traffic
flow from the perspective of the heterogeneity of connected vehicles and the platoon size. The
following will specifically analyze the shortcomings of existing literature to determine the research
objectives.
1.2. Literature review
1.2.1 Platoon size
Previous research mainly focused on how various platoon sizes affect road capacity, stability,
and safety. For example, Talebpour and Mahmassani [15] established a theoretical model
considering traffic stability. On this basis, they discussed the influence of platoon size on traffic
stability. Results showed that the stability decreased with the increase in platoon size. Shi et al. [4]
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proposed a lane-changing model which enables CVs to drive in a platoon through coordinated lane-
changing. The simulation results indicated that platoons could further improve road capacity and
stability. Besides, Jin et al. [16] believed that an appropriate platoon size could significantly improve
road capacity, but the capacity would decrease when the platoon size exceeded a threshold.
Considering the effect of the maximum platoon size (MPS) on mobility, safety, and environment,
Seraj et al. [17] proved that increasing platoon size can improve mobility and significantly improve
environmental gain. But the traffic safety gain would decrease. Based on the study, Zhou and Zhu
[14] further introduced the MPS and proposed an analytical model to explore the impact of CAVs
platoon size on mixed traffic flow. The results indicated that a larger platoon size was conducive to
improving the road capacity. However, with the increase in platoon size, the capacity gain decreased
gradually. Regarding traffic stability, both theoretical analysis and numerical experiment results
showed that the smaller the MPS, the better the stability. Additionally, the analysis results also
indicated that due to the random distribution of vehicles, only a small number of CAVs can form the
largest platoon, even if the PR of CAVs is high.
From the above analysis, it can be seen that many studies focused on CAVs platooning and
extensively researched road capacity, stability, and safety. However, CVs can also form a platoon to
realize cooperative driving, and few studies have been conducted on platooning of CVs. In addition,
the platoon size largely depends on traffic conditions (e.g., the arrival rate of vehicles and the PR of
CAVs) and management measures (e.g., lane management strategies) [18]. With the low PR of
connected vehicles, it is challenging for CAVs and CVs to form a platoon, and the platoon size is
random. Therefore, it is very important to investigate how the random distribution of platoon size
affects road capacity.
1.2.2 Time headway
The time headway of vehicles is significant in the traffic operation and management field, such
as traffic volume, stability, energy consumption, and safety studies [19]. The research on time
headway mainly discusses its distribution in HVs homogeneous traffic flow. In recent decades, the
rapid development of intelligent connected vehicles has led to increased attention and exploration
of time headway distribution in different car-following modes (e.g., CAV-CAV, CAV-HV, HV-CAV,
HC-HV). Relevant studies are shown in Table 1. In Table 1, we compare and analyze the values of
time headway in different car-following modes.
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Table 1
The time headway used in existing studies.
HV-HV
HV-AV
HV- CAV
AV-AV
AV-HV
AV- CAV
CV-CV
CAV-CAV
CAV-CAV platoon
CAV-HV
CAV- AV
References
1.5
1.5
0.6
[4]
1.5
1.5
0.6-2.2
1.1-2.2
[5]
1.6
1.6
1.2
1.4
[6]
1.0
2.0
2.0
1.5
1.5
1.5
0.6
1.0
1.5
1.5
[10]
2.0
2.0
1.0
0.7
[15]
1.0-1.8
1.0-1.8
0.5
0.8-1.0
[20]
0.8-2.2
0.8-2.2
0.6-1.1
0.7-1.5
[21]
1.8
1.8
0.5
[22]
0.5-1.1
[23]
1.8
1.8
0.9
1.2
[24]
1.6
1.6
0.5
1.6
[25]
2.0
0.6
[26]
5
Based on Table 1, the time headway in HV-HV car-following mode and HV-CAV car-following
mode are generally distributed in the range of 1-2 s; the time headway in CAV-HV car-following
mode is relatively smaller and distributed in the range of 1-1.5 s; and the time headway in CAV-CAV
car-following mode is relatively smallest and distributed in the range of 0.5-1.2 s. Besides, the same
type of time headway varies in different research assumptions. For example, the time headway in
CAV-CAV car-following mode is 1.2 s in conservative scenarios, while it can be 0.5s in radical
scenarios. Therefore, it is necessary to construct a more general road capacity model, which can
reflect the influence of various forms of stochastic time headway on road capacity. Furthermore,
Table 1 illustrates that most literature only studies the time headway in the mixed traffic flow
composed of HVs and CAVs. Nevertheless, there will be a considerable duration of a mixed traffic
flow with HVs, AVs, CVs, and CAVs in the future. In this scenario, the distribution of time headway
will be more diversified. Therefore, it is important to construct a capacity analysis model that
considers the heterogeneity of vehicles.
1.2.3 The capacity of mixed traffic flow
There are two research methods for road capacity, i.e., simulation analysis and theoretical
modeling [27,28]. The simulation analysis method can recreate traffic scenes by simulating the
microscopic behavior of vehicles. However, it lacks the analysis of the operation rules of traffic flow,
and it is challenging to adapt to the changing traffic environment. On the contrary, the theoretical
modeling method can comprehensively consider the influence of multiple factors (e.g., the PR of
connected vehicles, platoon size, and headway between vehicles) on road capacity. Therefore, most
scholars chose the theoretical modeling method to study the capacity of mixed traffic flow. For
example, van den Berg and Verhoef [29] believed that the road capacity was the weighted harmonic
average of the capacity of AVs and HVs. Chen et al. [27] further considered the influence of CAVs PR
and platoon size on capacity, and on this, a theoretical framework has been suggested that can be
used to examine the capacity response to changes in the PR of CAVs under an equilibrium state. It
was demonstrated by the experimental results that the capacity of mixed traffic flow increased with
the size of CAVs platoon, and the MPS should not exceed 10. Based on their study, Zhu et al. [30]
designed a cellular automata model for traffic flow mixed with CAVs and HVs. Besides, the impact
of PR and platoon size of CAVs on road capacity, congestion, and lane-changing frequency were
analyzed. Results showed an optimal CAVs PR and platoon size to maximize the road capacity.
Based on the queuing theory, Jin et al. [16] presented a fluid model for mixed traffic flow, and
established a platoon coordination strategy that could maximize road capacity and minimize delay.
In addition, by comparing the simulation results of capacity in coordinated and uncoordinated
conditions, the impact of the coordination strategy was further evaluated. To quantify the gain
degree of platoon driving on road capacity, Bujanovic and Lochrane [18] established an analytical
model considering the PR of cooperative adaptive cruise control (CACC) vehicles and MPS to
predict the road capacity on road segments. Three models of vehicle behavior were utilized in the
simulation of VISSIM. Their study indicated that when the traffic volume was less than 4000 veh/h,
and the PR of CACC vehicles was more than 70%, the gain of road capacity was not obvious as the
platoon size was more than 5. Similar work was also done by Xiao et al. [31], who used a realistic
CACC model to indicate how the CACC vehicles affect the capacity of bottleneck sections on the
expressway. Finally, 10 vehicles were selected as the MPS. Besides, Sharma et al. [32] considered the
influence of human drivers’ self-awareness on CVs and HVs, and further constructed the driving
strategy of CVs integrated with the intelligent driver model (IDM). At the same time, the influence
of CVs on mixed traffic flow was evaluated by three indexes, i.e., the disturbance, efficiency, and
safety of traffic flow. As for the management of mixed traffic flow at urban intersections, Wu et al.
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[13] modeled the capacity of signalized intersections as the objective function of linear programming,
and utilized non-collision constraints to maximize road capacity. Experimental results suggested
that a high PR of CAVs did not always indicate a high capacity of a signalized intersection. In
addition, the study also suggested that more attention should be paid to the PR of connected vehicles
and the queuing willingness of vehicles.
The above studies focused on how platoon size affects road capacity. However, most of them
ignored the influence of random changes in platoon size on road capacity. Moreover, mixed traffic
flow composition assumptions were too strict and did not incorporate the influence of CVs platoon.
To sum up, current research on road capacity involves factors such as the PR of vehicles and
platoon size. However, there are still two problems: (1) The composition of mixed traffic flow only
accounts for CAVs and HVs, disregarding the influence of CVs and AVs on mixed traffic flow; (2)
Current studies have not considered that CVs can use wireless communication technology to enable
the cooperative driving and form a platoon. Thus, the existing capacity models cannot portray how
the heterogeneity and the size of the platoon affect traffic flow. To fill this gap, this study focuses on
analyzing how the organization mode of the platoon affects car-following characteristics in mixed
traffic flow to propose a probability distribution model of the size of CAVs and CVs platoon. Based
on this, a capacity analysis model is further developed considering the heterogeneity and MPS of
connected vehicles. Then the numerical experiments verify the validity of the proposed model,
which can provide theoretical support for traffic planning and management in the connected
automated environment.
1.3. Contributions
To address the gaps of existing research, this study constructs a capacity analysis model
focusing on the heterogeneity and limitation of the MPS of connected vehicles in the mixed traffic
flow environment. Firstly, we discuss the impact of platoon organization mode on the car-following
characteristics in mixed traffic flow with CAVs, CVs, AVs, and HVs. Different car-following models
are introduced to describe the following characteristics of different vehicles. Secondly, the
probability distribution model of CAVs and CVs platoon size is deduced considering the PR of
connected vehicles and the MPS. Then, the fundamental diagram of the mixed traffic flow is further
derived based on the probability distribution model of platoon size and the car-following model.
Finally, the capacity model is proposed, and numerical simulation experiments were designed to
explore the effect of the different PR of CAVs, CVs, and MPS on road capacity. Therefore, the main
contributions of this paper can be stated as follows.
(1) This paper fully considers that with the development of connected technology and market
demand, different levels of connected vehicles will be put into use in the future, so CAV is further
subdivided into CAV and CV, which is more suitable for future application scenarios.
(2) A probability distribution model of platoon size is proposed by fully considering the impact
of CAVs and CVs platoon on the overall platoon size distribution. At the same time, the probability
of any following mode can be obtained based on the model.
(3) A fundamental diagram of the mixed traffic flow is developed, and the road capacity under
each equilibrium state is obtained through numerical experiments.
(4) The capacity analysis model considering the heterogeneity and the limitation of platoon size
of connected vehicles in the mixed traffic flow environment is proposed. Contrasted with the current
models, this model has universality and can represent the road capacity in the future when the PR
of connected vehicles is random.
(5) A numerical experiment is designed to explore the effect of different PR of CAVs, CVs, and
MPS on the road capacity.
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1.4. Framework
The framework of this paper is organized as follows. Section 1 discusses the research
background, reviews the platoon size of connected vehicles, time headway, and capacity of the
mixed traffic flow, and summarizes the research contribution of this paper. Section 2 gives the
modeling background, such as the definitions and assumptions for models, analyzes the
characteristics of vehicle behavior, and provides three corresponding models. In Section 3, the
probability distribution model of the size of CAVs and CVs platoons is deduced considering the PR
and MPS of connected vehicles. Based on the probability distribution model of platoon size and car-
following model, the fundamental diagram of the mixed traffic flow is further derived in Section 4.
Section 5 proposes the capacity model, and numerical experiments are designed to investigate the
effect of the different PR of CAVs, CVs, and MPS on capacity in mixed traffic flow. Finally, Section 6
ends this work with a summary of the conclusions and a brief introduce of future works.
2. Modeling background
2.1. Definitions and modeling assumption
2.1.1 Definitions
Table 2 presents the parameters, variables, and definitions used to create the proposed model in
this paper.
2.1.2 Modeling assumption
To simplify the modeling process, we present some assumptions for modeling before
introducing the proposed models.
(1) The horizontal vehicle behavior, such as changing the lane, is not considered in this paper,
and we only consider the longitudinal vehicle behavior.
(2) The effect of time delay on the communication of CVs is not considered.
(3) According to the definition of automation and connection degree of CAVs in relevant
standards, the degree of vehicle automation is divided into L0-L5 levels, and the degree of vehicle
connection is divided into C1-C3 levels [9]. This paper supposes that the CAV automation level is
above L3, and the connection level is above C2. They can interconnect and realize large-scale
cooperative driving. The difference between CVs and CAVs is that human drivers can take over the
driving of CVs. Thus, the CV connection level is the same as CAV, but the automation level is below
L3. At the same time, information interaction between CVs can be realized, but the operation ability
of human drivers is limited. As a result, only small-scale cooperative driving can be realized between
CVs. AVs are vehicles with a high level of automation, but a low level of connection, so their
automation level is the same as CAV, but the connection level is below C2. HVs are vehicles with the
lowest level of automation and connection.
(4) The platoon size of CAVs and CVs depends only on their PR, and is distributed randomly in
mixed traffic flow.
(5) The vehicles of the same type (e.g., CAV and CAV, CV and CV) will spontaneously form a
platoon when they meet. Different types of vehicles (such as CAV and CV) cannot form a platoon.
When the platoon reaches maximum size, extra vehicles will drive in another platoon.
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Table 2
Parameters, variables, and definitions.
Parameters
Definition
The index of vehicles.
The length of vehicles.
The average time headway of new mixed traffic flow.
The time headway of a type-j vehicle to be followed by a type-i vehicle,
nd 、 represent the time headway of CAV and CV following
the CAVs and CVs platoon that have reached the MPS limitation.
Time headways of the eight cases, s.
Respectively represent the expected time gap of vehicles using IDM, ACC and
CACC following models.
Respectively represent the expected time gap of vehicles with and
following modes under IDM, ACC and CACC following models in the
equilibrium state,.
The PR of type-i vehicle and connected vehicles, .
The MPS of CAVs and CVs platoon.
The probability of a vehicle being the -th and the -th vehicle in the CAVs
and CVs platoon, where 、 represent the probability of the leader in
the CAVs and CVs platoon, 、 represent the probability of in the
CAVs and CVs platoon but not the leader.
Respectively represent the probability of IDM, ACC and CACC following
models for vehicles with and following modes.
The probability of the -th time headway.
The spacing and the average spacing.
Respectively represent the spacing of vehicles with , and following
modes under IDM, ACC, and CACC following models in the equilibrium
state.
The current and previous time speed of vehicle n.
The desired acceleration of the vehicle using IDM and ACC model, where
represents the derivative form of the original variable.
The control parameters in the ACC model.
Respectively represent the difference between the actual gap distance and the
expected gap distance, its derivative form, and are their control
parameters in the CACC model.
Respectively represent the maximum acceleration, desired deceleration, free-
flow speed, and minimum distance at a standstill of the vehicle using the IDM
model.
Capacity of mixed traffic flow.
Traffic flow density in an equilibrium state.
Traffic flow in an equilibrium state.
The equilibrium speed.
The free-flow speed of the road segment.
Maximum capacity of new mixed traffic flow.
2.2. The car-following modes
2.2.1 The analysis of car-following characteristics
Within the mixed traffic flow are four distinct vehicle classifications, i.e., CAVs, CVs, AVs, and
HVs, which are expressed as 1, 2, 3, and 4, respectively. Considering the same type of vehicles with
9
the same degree of automation and connection, it is assumed that when a CAV follows a CAV, they
can form a CAVs platoon with a lower time headway. Similarly, when a CV follows a CV, the time
headway of a CV platoon is slightly higher than that of the CAVs platoon. To ensure traffic safety,
the platoon cannot be formed between CAVs and CVs due to their difference in the degree of
automation. In addition, AVs and HVs cannot connect due to the low degree of connection, so they
cannot drive in a platoon. Therefore, only CAVs and CVs can operate in a platoon within a mixed
traffic flow. Considering the MPS, there are a totally 18 modes of car-following on the road according
to the arrangement and combination, as illustrated in Fig. 1.
Fig. 1. The car-following modes in mixed traffic flow.
In Fig. 1, represent the time headway between vehicles when type j vehicle follows type i
vehicle, . In addition, and represent the time headway when CAV and CV
follow a CAV platoon and a CV platoon that have reached the MPS, respectively.
CAVs can receive information from CVs and other CAVs through communication equipment,
and transmit information simultaneously, which can be controlled automatically. However, the
automation level of CVs is limited. Although they can send and receive information, they cannot
quickly react to alterations in the driving patterns of the leading vehicles. Therefore, the time
headway of the CVs platoon should be slightly higher than that of the CAVs platoon. Meanwhile,
compared with HVs and AVs, CAVs and CVs can drive with a lower time headway due to the
communication equipment. In addition, as autonomous driving relies on satellite navigation,
positioning system, and other technologies, compared with HVs, AVs can reduce the uncertainty of
driving behavior caused by sensory errors. However, compared with CAVs and CVs, the connection
level of AVs is lower. Therefore, the time headway between AVs is slightly higher than CAVs and
CVs, and smaller than HVs. It is also proved by Shi et al. [4], who believed that due to the distrust of
human drivers of autonomous driving technology, drivers would deliberately maintain a higher
time headway with connected vehicles to improve self-safety. Nevertheless, this situation will be
improved as autonomous driving and wireless communication technologies advance. In addition,
vehicles in a platoon can drive at a smaller time headway, while vehicles between platoons drive at
a slightly higher time headway. After the above analysis, a total time headway size relationship can
be summarized as follows: , the time headway inside the CAV platoon should
be the smallest, followed by the time headway inside the CV platoon. When AV is the following
vehicle, the time headway is smaller, while when HV is the following vehicle, the corresponding
time headway is the largest. And as this paper is the first to study the mixed traffic capacity of four
types of vehicles, including CAV, CV, AV, and HV, it is not easy to find a reference for the value of
some types of time headway. So, the corresponding value can be assigned according to the size
relationship. Therefore, based on the above analysis and the method for evaluating the time
headway of connected vehicles in Table 1, the time headway for all modes of car-following can be
categorized into eight distinct types.
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(1) Type 1: the following vehicle is an HV, while the vehicle ahead is a CAV, CV, or AV, which
can be expressed as .
(2) Type 2: the following vehicle is an HV, while the vehicle ahead is an HV, which can be
expressed as .
(3) Type 3: the following vehicle is a CAV, CV, or AV, while the vehicle ahead is an HV, which
can be expressed as .
(4) Type 4: the following vehicle is a CV, while the vehicle ahead is an AV, which can be
expressed as .
(5) Type 5: the following vehicle is an AV, while the vehicle ahead is a CAV, CV, or AV; or the
following vehicle is a CV, while the vehicle ahead is a CV in the leading platoon or a CAV; or the
following vehicle is a CAV, while the vehicle ahead is an AV; which can be expressed as
.
(6) Type 6: the following vehicle is a CV, while the vehicle ahead is a CV in the same platoon,
which can be expressed as .
(7) Type 7: the following vehicle is a CAV, while the vehicle ahead is a CAV in the leading
platoon or CV, which can be expressed as .
(8) Type 8: the following vehicle is a CAV, while the vehicle ahead is a CAV in the same platoon,
which can be expressed as .
2.2.2 The car-following model
Based on the above analysis, the car-following model calibrated by the PATH laboratory in a
field test, known as the CACC [33,34] model, has been selected as the adopted model for certain
CAVs and CVs. Additionally, the adaptive cruise control (ACC) [34,35] model is adopted as the car-
following model for AVs and some CAVs and CVs. At the same time, the IDM [36] has been widely
used and can well describe the status of traffic flow and drivers’ habits, which is selected as the
following model for HVs. The equation describing each model is given below:
(1) IDM
(1)
where is the desired acceleration of IDM; is the maximum desired acceleration; is the
desired deceleration; is the current speed; is the free-flow speed; is the spacing; is the length
of vehicle; is the minimum distance at a standstill; is the expected time gap.
(2) ACC model
(2)
where is the desired acceleration of the ACC model; and are the control parameters; is
the expected time gap.
(3) CACC model
(3)
where is the previous speed of vehicle ; is the difference between the actual gap distance and
the expected gap distance, and the derivative form of is represented by ; are their control
parameters; is the expected time gap.
According to the references [10,26], the parameters of each car-following model can be found in
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Table 3.
Table 3
The parameters of each model.
Models
Parameters
Value
Unit
CACC
0.45
0.25
ACC
0.23
0.07
IDM
1.0
2.0
33.0
5.0
2.0
In Section 2.2.1, 18 car-following modes are found on the road segment in the mixed traffic flow
environment. When the following vehicle is an HV/AV, it can only rely on the driver's
senses/onboard detection equipment to further control the driving behavior. Therefore, no matter
what type of the leading vehicle, the car-following model remains the IDM/ACC model. If the
following vehicle is a CAV, the realization of the vehicle control system needs to rely on wireless
communication technology. While the vehicle ahead is a CAV, the car-following model is the CACC
model. However, when the leading vehicle is a CV, HV, or AV, it cannot transmit its own driving
information, such as acceleration, to the following vehicle. In this case, the communication
technology of CAVs is ineffective, and the car-following model will degenerate into an ACC model.
It can only rely on onboard detection equipment to sense the driving state of the leading vehicle. At
the same time, if the following vehicle is a CV, the vehicle-vehicle communication technology can be
used to receive the acceleration and other state information of the leading vehicle only when the
leading vehicle is a CAV. In other cases, the vehicle driving state can only be controlled by its own
onboard detection equipment. In this scenario, the car-following model degenerates into an ACC
model. To sum up, the car-following models under various scenarios in mixed traffic flow are shown
in Fig. 2.
Fig. 2. The division of the car-following model in mixed traffic flow.
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Table 4
Summary of all car-following modes and models in mixed traffic flow.
Car-following modes
The following
car
The leading car
Car-following models
Time headway/
Time gap
HV-HV
HV
HV
IDM
1.6
HV-AV
AV
1.8
HV-CV
CV
1.8
HV-CAV
CAV
1.8
AV-HV
AV
HV
ACC
1.4
AV-AV
AV
1.2
AV-CV
CV
1.2
AV-CAV
CAV
1.2
CV-HV
CV
HV
1.4
CV-AV
AV
1.3
CV-CV (intra-platoon)
CV
1.0
CV-CV (inter-platoon)
CV
1.2
CV-CAV
CAV
CACC
1.2
CAV-HV
CAV
HV
ACC
1.4
CAV-AV
AV
1.2
CAV-CV
CV
0.8
CAV-CAV (intra-
platoon)
CAV
CACC
0.6
CAV-CAV (inter-
platoon)
CAV
0.8
From the analysis of time headway described above, all car-following modes and their
corresponding car-following models and time headway can be obtained, as shown in Table 4.
3. The probability distribution model of platoon size
It is known that the mixed traffic flow is composed of CAVs, CVs, AVs, and HVs. Let ,
and represent the PR of CAVs, CVs, AVs, and HVs, respectively. Then, , and
represent the PR of connected vehicles. From the previous analysis in Section 2.2.1, CAVs
and CVs will drive on the road in a platoon at random size (not exceeding the maximum size )
in order to enhance the road capacity and give full play to the advantages of connected vehicles in
mixed traffic flow. and respectively represent the -th and -th CAV and CV in the platoon
are composed of CAVs and CVs. Due to the limited communication range, the platoon size of CAVs
cannot exceed the maximum , that is, . Similarly, the platoon size of CVs cannot
exceed the maximum , that is, (Note: if or , it means that the vehicle is
non-CAV or non-CV). In addition, due to the different levels of connection, CAVs have a slightly
greater communication range compared to CVs. Thus, under the same conditions, a larger platoon
is allowed to be formed, i.e., . However, due to the lack of vehicle-vehicle communication or
vehicle-infrastructure communication technology, platoons could not be formed by HVs or AVs. As
a result, they are distributed randomly on the road. The probability distribution model of platoon
size under arbitrary PR of CAVs and CVs can be derived below.
13
3.1. Models
3.1.1 Connected and automated vehicles
is set to represent the probability of a vehicle occupying the -th position in a CAVs
platoon. Obviously, the value of can be divided into three scenarios based on vehicle types and
car-following modes: (1), in this scenario, the size of the CAVs platoon is 0, and the vehicles
on the road include CVs, AVs, and HVs. (2), it indicates that the CAV is
a part of the CAVs platoon but not the leader. (3) , it indicates that the CAV is at the front of
the platoon. The probability of with different values are discussed as follows.
(1), the vehicle is a CV, HV, or AV. Therefore, we can obtain
(4)
(2), the vehicle is the -th CAV in a CAVs platoon but not the leader,
. It can be deduced that the probability of the -th CAV in a CAVs platoon is
(5)
Furthermore, we can obtain Eq. (6) based on Eq. (5).
(6)
(3) , in this scenario, the vehicle is CAV which leads a platoon. As previously analyzed,
the vehicle has four possible car-following modes, and the leading vehicle may be an HV, AV, CV,
or maximum-size CAVs platoon. Therefore, the probability of the CAVs platoon leader can be
calculated as
(7)
where is the probability of a CAVs platoon leader following an HV, AV, or CVs platoon,
and is the probability that its leading vehicle is a maximum-sized CAVs platoon.
Therefore, the total probability of a CAVs platoon leader is
(8)
Therefore, can be derived by solving Eq. (8), as shown in Eq. (9).
(9)
By substituting Eq. (9) with Eq. (6), we can obtain
14
(10)
In brief, the probability distribution of the size of the CAVs platoon can be derived by combining
Eqs. (4), (9), and (10).
(11)
3.1.2 Connected human-driven vehicles
is set to represent the probability of a vehicle occupying the -th position in a CVs
platoon. Similar to the derivation process of the CAV platoon size probability distribution model,
the value of also can be divided into three scenarios based on vehicle types and car-following
modes: (1) , (2) , (3) , respectively, indicates the non-CV
vehicles, CV being inside the platoon, and CV being at the front of the platoon. The probability of
with different values can be calculated as follows.
(12)
(13)
(14)
In brief, the probability distribution of the size of the CVs platoon can be derived by combining
Eqs. (12), (13), and (14).
(15)
Due to space limitations, the derivation process of the CV platoon size probability distribution
model is not repeated. The reader can refer to the derivation process of the CAV model above.
3.2. Numerical experiments
To further study platoon size distribution across different MPS and PR of CAV and CV, based
on Eqs. (11) and (15), this paper calculates probability distribution values for CAV and CV platoon
sizes, as demonstrated in Table 5 and Table 6. In Table 5, the probability distribution values of
15
platoon size under various PR combinations ( ) in mixed traffic flow and fully connected
environment are calculated respectively, while the MPS unchanged. In addition, the probability
distribution values under different MPS combinations ( ) while the PR of CAVs and CVs
unchanged are also calculated, as shown in Table 6. The results can further reveal the influence of
the MPS on the probability distribution of the platoon.
Table 5
The probability distribution values of platoon size under various PR combinations () in mixed
traffic flow and a fully connected environment.
(a) mixed traffic flow environment ()
Probability
Parameters
The probability distribution of CAVs platoon size
The probability distribution of CVs platoon size
0.0
0.1
0.2
0.3
0.4
0.5
0.5
0.4
0.3
0.2
0.1
0.0
0
1
0.9000
0.8000
0.7000
0.6000
0.5000
0
0.5000
0.6000
0.7000
0.8000
0.9000
1
1
0
0.0900
0.1600
0.2102
0.2410
0.2540
1
0.2667
0.2463
0.2117
0.1603
0.0900
0
2
0
0.0090
0.0320
0.0630
0.0964
0.1270
2
0.1333
0.0985
0.0635
0.0321
0.0090
0
3
0
0.0009
0.0064
0.0189
0.0386
0.0635
3
0.0667
0.0394
0.0191
0.0064
0.0009
0
4
0
0.0001
0.0013
0.0057
0.0154
0.0317
4
0.0333
0.0158
0.0057
0.0013
0.0001
0
5
0
0.0000
0.0003
0.0017
0.0062
0.0159
6
0
0.0000
0.0001
0.0005
0.0025
0.0079
(b) Fully connected environment ()
Probability
Parameters
The probability distribution of CAVs platoon size
The probability distribution of CVs platoon size
0.5
0.6
0.7
0.8
0.9
1.0
0.5
0.4
0.3
0.2
0.1
0.0
0
0.5000
0.4000
0.3000
0.2000
0.1000
0
0
0.5000
0.6000
0.7000
0.8000
0.9000
1
1
0.2502
0.2415
0.2161
0.1792
0.1382
0.1
1
0.2510
0.2402
0.2100
0.1600
0.0900
0
2
0.1251
0.1449
0.1513
0.1434
0.1244
0.1
2
0.1255
0.0961
0.0630
0.0320
0.0090
0
3
0.0626
0.0869
0.1059
0.1147
0.1119
0.1
3
0.0627
0.0384
0.0189
0.0064
0.0009
0
4
0.0313
0.0522
0.0741
0.0918
0.1007
0.1
4
0.0314
0.0154
0.0057
0.0013
0.0001
0
5
0.0156
0.0313
0.0519
0.0734
0.0907
0.1
5
0.0157
0.0061
0.0017
0.0003
0.0000
0
6
0.0078
0.0188
0.0363
0.0587
0.0816
0.1
6
0.0078
0.0025
0.0005
0.0001
0.0000
0
7
0.0039
0.0113
0.0254
0.0470
0.0734
0.1
7
0.0039
0.0010
0.0002
0.0000
0.0000
0
8
0.0020
0.0068
0.0178
0.0376
0.0661
0.1
8
0.0020
0.0004
0.0000
0.0000
0.0000
0
9
0.0010
0.0041
0.0125
0.0301
0.0595
0.1
10
0.0005
0.0024
0.0087
0.0241
0.0535
0.1
Table 6
The probability distribution values of platoon size under various MPS combinations ( ) in
mixed traffic flow and a fully connected environment.
(a) mixed traffic flow environment ()
Probabilit
y
Parameters
The probability distribution of CAVs platoon size
The probability distribution of CVs platoon size
6
7
8
9
10
4
5
6
7
8
0
0.7000
0.7000
0.7000
0.7000
0.7000
0
0.4000
0.4000
0.4000
0.4000
0.4000
1
0.2102
0.2100
0.2100
0.2100
0.2100
1
0.2757
0.2602
0.2517
0.2469
0.2441
2
0.0630
0.0630
0.0630
0.0630
0.0630
2
0.1654
0.1561
0.1510
0.1481
0.1465
3
0.0189
0.0189
0.0189
0.0189
0.0189
3
0.0993
0.0937
0.0906
0.0889
0.0879
4
0.0057
0.0057
0.0057
0.0057
0.0057
4
0.0596
0.0562
0.0544
0.0533
0.0527
5
0.0017
0.0017
0.0017
0.0017
0.0017
5
--
0.0337
0.0326
0.0320
0.0316
6
0.0005
0.0005
0.0005
0.0005
0.0005
6
--
--
0.0196
0.0192
0.0190
7
--
0.0002
0.0002
0.0002
0.0002
7
--
--
--
0.0115
0.0114
8
--
--
0.0000
0.0000
0.0000
8
--
--
--
--
0.0068
9
--
--
--
0.0000
0.0000
10
--
--
--
--
0.0000
(b) Fully connected environment ()
Probability
Parameters
The probability distribution of CAVs platoon size
The probability distribution of CVs platoon size
6
7
8
9
10
4
5
6
7
8
0
0.5000
0.5000
0.5000
0.5000
0.5000
0
0.5000
0.5000
0.5000
0.5000
0.5000
1
0.2540
0.2520
0.2510
0.2505
0.2502
1
0.2667
0.2581
0.2540
0.2520
0.2510
16
2
0.1270
0.1260
0.1255
0.1252
0.1251
2
0.1333
0.1290
0.1270
0.1260
0.1255
3
0.0635
0.0630
0.0627
0.0626
0.0626
3
0.0667
0.0645
0.0635
0.0630
0.0627
4
0.0317
0.0315
0.0314
0.0313
0.0313
4
0.0333
0.0323
0.0317
0.0315
0.0314
5
0.0159
0.0157
0.0157
0.0157
0.0156
5
--
0.0161
0.0159
0.0157
0.0157
6
0.0079
0.0079
0.0078
0.0078
0.0078
6
--
--
0.0079
0.0079
0.0078
7
--
0.0039
0.0039
0.0039
0.0039
7
--
--
--
0.0039
0.0039
8
--
--
0.0020
0.0020
0.0020
8
--
--
--
--
0.0020
9
--
--
--
0.0010
0.0010
10
--
--
--
--
--
Based on the data in Table 5 and Table 6, the trend of the probability of platoon size with PR
and MPS of CAVs, and CVs under different conditions can be obtained, as shown in Fig. 3-Fig. 5.
Fig. 3. Probability distribution diagram of CAVs and CVs platoon size under fixed MPS and
various PR combinations () in mixed traffic flow.
Fig. 3 and Fig. 4 show the trend of the probability of platoon size with CAVs and CVs PR in
mixed traffic flow () and fully connected environment () when the MPS is
fixed, respectively.
Fig. 4. Probability distribution diagram of CAVs and CVs platoon size under fixed MPS and
various PR combinations () in fully connected environment.
The changes in platoon size distribution with various PR combinations () when the MPS
is (6,4) are illustrated in Fig. 3. Fig. 4 shows the trend of platoon size distribution with various PR
combinations () when the MPS is (10,8). From Fig. 3 and Fig. 4, we can know that when the PR
17
of CAVs and CVs is 0, vehicles on the road include AVs and HVs. In this case, the platoon cannot be
formed, so , as shown in Fig. 1. On the contrary, when the PR of CAVs or CVs
is 1, all vehicles on the road are fully composed of CAVs or CVs. According to model mentioned
above, at this moment, the distribution of the platoon is in an equilibrium state, and the probability
of various platoon sizes is equal, as shown in Fig. 4. In addition, while the PR of CAVs and CVs
increases from 0 to 1, the distribution of platoon size begins to show a diversified trend, and the
probability of platoon size (greater than or equal to 1) begins to increase. Combined with the road
environment, when the total number of vehicles is certain, a larger platoon size means that more
connected vehicles of the same type meet and spontaneously form a platoon. However, it is difficult
to achieve when the PR of connected vehicles is low, so the probability value is small. As the PR of
connected vehicles continues to rise, there is an increasing potential for larger platoons to form.
Hence, the probability of the platoon size (greater than or equal to 1) begins to increase.
Fig. 5 shows the trend of the probability distribution of platoon size with the MPS while the PR
of CAVs and CVs are unchanged. In particular, Fig. 5 illustrates that with the increase of CAVs and
CVs PR, the distribution of platoon size presents a diversified trend, which is supported by the
findings in Fig. 3 and Fig. 4. In addition, when the PR of CAVs and CVs is lower, the change of MPS
has no significant influence on the distribution of CAVs and CVs platoon size, as shown in Fig. 5(a)-
(e). However, when the PR of CAVs and CVs is higher, the impact of MPS change on the distribution
of CAVs and CVs platoon size becomes significant. It can be found that with the increase of the MPS,
the probability of platoon size (greater than or equal to 1) begins to decrease, but the distribution is
more diverse, as shown in Fig. 5(a)-(d) of CVs’ platoon size. After analyzing the data, it can be
inferred that with the increase in the MPS, the platoon size has a variety of states to choose from.
Therefore, the probability transition occurs. Fig. 5 also indicates that the probability of having a
platoon size of 0 is only influenced by the PR of CAVs and CVs, while the MPS appears to have no
significant impact on this outcome, which is consistent with the theoretical results.
(a)
(b)
(c)
(d)
18
(e)
Fig. 5. Probability distribution diagram of CAVs and CVs platoon size under fixed PR and various
MPS combinations ().
4. Fundamental diagram of mixed traffic flow
4.1. Fundamental diagram
As the basis of traffic flow theory, the fundamental diagram model describes the characteristics
of the continuous traffic flow by using traffic volume , density and speed . The relationship
between the three parameters is significant for studying mixed traffic flow states, which is adopted
in many studies [26,37]. Based on the traffic density curve in the fundamental diagram, the
researchers can know the corresponding traffic density when the traffic volume is maximum. This
is also the purpose of using the model of the fundamental diagram in this paper. Considering the
definition of traffic density, when all vehicles are driving with the equilibrium speed (generally
taking ), the traffic density and traffic volume can be obtained.
(16)
where is the average spacing in the equilibrium state under mixed traffic flow.
It is known that in the equilibrium state of traffic flow, all vehicles will travel at equilibrium
speed. From the analysis in Section 2.2.2, in mixed traffic flow, IDM, ACC, and CACC are used in 4,
11, and 3 cases, respectively. The corresponding spacing of each model under the equilibrium state
of traffic flow are
(17)
where and are the equilibrium spacing and the expected time gap corresponding to
the IDM in the - th car-following mode, respectively. Other symbols have similar meanings.
Combined with the probability distribution model of platoon size in Section 3, the probability of
using a specific following model under each following mode can be obtained. By summarizing, we
can obtain Table 7.
19
Table 7
The probability of using a specific car-following model under each car-following mode
IDM
1
2
3
4
ACC
1
2
3
4
5
6
7
8
9
10
11
CACC
1
2
3
where is the probability that the CAV leads a platoon, and is the probability that the
CAV is in a platoon but not the leader. Similarly, and respectively represent the probability
of leader and non-leader in a CVs platoon. Therefore, Eqs. (18)-(21) can be obtained.
(18)
(19)
(20)
(21)
In summary, in the equilibrium state of mixed traffic flow, the average spacing can be expressed
20
as
(22)
where is the probability of using the IDM in the - th car-following mode, and other
symbols have similar meanings. By substituting Eq. (22) into Eq. (16), the fundamental diagram of
the mixed traffic flow can finally be derived.
(23)
From Eq. (23), it can be seen that the fundamental diagram depends on the PR () and the
MPS (, ) of CAVs and CVs.
4.2. Numerical experiments
Based on Eq. (16) and Eq. (23), we can obtain the curves of the fundamental diagram in the
traffic flow of a fully connected environment (composed of CAVs and CVs) and all kinds of
homogeneous traffic flow, as shown in Fig. 6. At the same time, a numerical experiment is designed
to analyze the influence of PR and MPS on the fundamental diagram. The figures illustrating the
results can be viewed in Fig. 7 and Fig. 8.
Fig. 6. The curves of the fundamental diagram in all kinds of homogeneous traffic flow.
Fig. 6 presents that when the equilibrium density of each homogeneous traffic flow reaches its
critical density, road capacity reaches its maximum. For example, in a fully connected environment
( ), when the equilibrium density is , the maximum capacity is
. In addition, when the equilibrium density is in a fully CAVs environment,
the maximum capacity is , which is consistent with the conclusion of most studies. In
fully HVs environment, when the equilibrium state density is , the maximum capacity is
21
, which is only about 2/5 of the maximum capacity in a fully CAVs environment.
Fig. 7 shows that when the MPS is (6, 4), the higher the PR of connected vehicles, the larger the
capacity under the equilibrium state. When , with increasing PR, the increase in road
capacity is not significant. However, when , with the increase of PR, the road capacity
will be substantially improved. For example, when increases from (0.1, 0.1) to (0.2, 0.2), and
the maximum capacity in the equilibrium state only increases by 1.56%. But when increased
from (0.4, 0.4) to (0.5, 0.5), and the maximum capacity increased by 24.23%.
Fig. 7. Influence of the PR on the fundamental diagram ().
(a)
(b)
22
(c)
Fig. 8. Influence of MPS on fundamental diagram under different PR.
Fig. 8 shows that under the same PR of connected vehicles, the larger the MPS, the greater the
capacity under the equilibrium state. When the PR of connected vehicles is low, the MPS has little
impact on improving road capacity. With the continual rise in the PR of connected vehicles, the
influence of the change in platoon size on road capacity is increasingly significant.
5. Capacity analysis model
5.1. Capacity model
According to Zhou and Zhu [26], road capacity can be calculated based on the average time
headway of mixed traffic flow, as shown in Eq. (24).
(24)
where is the free-flow speed of the road segment; is the minimum distance at a standstill,
generally taking ; is the average time headway. According to Eq. (24), when and are
fixed, road capacity depends only on the average time headway, rather than the car-following model.
From the analysis in Section 2.2.1, when both CAVs and CVs drive with a platoon, there are 18
car-following modes in mixed traffic flow, which can be further divided into 8 car-following types.
Then, the average time headway can be expressed as Eq. (25).
(25)
where , is the time headway of -th car-following type, is the probability
of the time headway of -th car-following type. Combined with Section 3, can be obtained,
as shown in Eqs. (26)-(33).
(1) Type 1: the following vehicle is an HV, while the vehicle ahead is a CAV, CV, or AV. Therefore,
the probability of can be expressed as Eq. (26).
(26)
23
(2) Type 2: the following vehicle is an HV, while the vehicle ahead is an HV. Therefore, the
probability of can be expressed as Eq. (27).
(27)
(3) Type 3: the following vehicle is a CAV, CV, or AV, while the vehicle ahead is an HV. Therefore,
the probability of can be expressed as Eq. (28).
(28)
(4) Type 4: the following vehicle is a CV, while the vehicle ahead is an AV. Therefore, the
probability of can be expressed as Eq. (29).
(29)
(5) Type 5: the following vehicle is an AV, while the vehicle ahead is a CAV, CV, or AV; or the
following vehicle is a CV, while the vehicle ahead is a CV in the leading platoon or a CAV; or the
following vehicle is a CAV, while the vehicle ahead is an AV. Therefore, the probability of can be
expressed as Eq. (30).
(30)
(6) Type 6: the following vehicle is a CV, while the vehicle ahead is a CV in the same platoon.
Therefore, the probability of can be expressed as Eq. (31).
(31)
(7) Type 7: the following vehicle is a CAV, while the vehicle ahead is a CAV in the leading
platoon or a CV. Therefore, the probability of can be expressed as Eq. (32).
(32)
(8) Type 8: the following vehicle is a CAV, while the vehicle ahead is a CAV in the same platoon.
Therefore, the probability of can be expressed as Eq. (33).
(33)
Based on Eqs. (25)-(33), the average time headway can be calculated. Then by substituting Eq.
(25) into Eq. (24), the road capacity can be obtained when both CAVs and CVs drive with a platoon.
According to the above analysis, when and are fixed, the road capacity depends only on
the average time headway. Furthermore, is related to the PR () and the MPS (, ) of CAVs
and CVs. Therefore, numerical simulation experiments have been devised to examine the impact of
parameter variations on road capacity.
24
5.2. Numerical experiments
This section aims to verify the capacity analytical model established in Section 5.1 through
numerical experiments. The experiments only consider the basic segment in a single lane, neglecting
the influence of lane-changing behavior. All simulation experiments are completed in MATLAB.
Eq. (24) indicates that the parameters involved in the capacity analytical model (i.e., , and
) are all constants, as shown in Table 8. Additionally, as mentioned in the introduction, achieving
a fully connected automated environment will take a long time due to technology limitations and
imperfect infrastructure. During this period, as CAV, CV, and AV continue to enter the market, the
PR of connected vehicles will continue to increase until they fully occupy the market. And the model
in the previous section also shows that the capacity depends only on the PR () and the MPS
(, ) of CAVs and CVs. Therefore, to further reveal the impact of the PR and the MPS of CAVs
and CVs on road capacity, five scenarios with different parameters are designed to achieve different
research purposes, as shown in
Table 9. Based on Eqs. (24)-(33), the calculation results are displayed in Fig. 9(a)-(e).
Table 8
The values of basic parameters.
Parameters
Values
Table 9
The settings of experimental scenarios.
Scenarios
Parameters
Research purposes
1
To discuss how changes in the PR of CAVs and CVs affect
road capacity in a fully connected environment.
2
To discuss how changes in the PR of CAVs and CVs affect
road capacity from mixed traffic flow to a fully connected
environment.
3
To discuss how changes in the PR of CAVs and CVs (at a low
level) affect road capacity in a mixed traffic flow
environment with a fixed PR of AVs and HVs.
4
To discuss how the fluctuation of PR of CAVs, CVs, AVs, and
HVs affect road capacity in a mixed traffic flow environment,
while the PR of connected vehicles is low.
5
To discuss how changes in the PR of CAVs and CVs (at a low
level) affect road capacity in a mixed traffic flow
environment with a fixed PR of AVs and HVs.
25
(a)
(b)
(c)
(d)
(e)
Fig. 9. Impact of different parameters on road capacity.
Combining the results in Scenario 1 and 2, the road capacity is significantly higher in a fully
connected environment than in mixed traffic flow, as indicated. And the results in Scenarios 3 and 4
26
show that when the PR of connected vehicles () is low, the road capacity increases with increases
and decreases. At the same time, the results in Scenarios 3 and 5 indicate that when the PR of AVs
and HVs remains constant, the road capacity increases with the PR of connected vehicles. In addition,
under different scenarios, the increase rate of maximum road capacity with MPS can be calculated
based on Fig. 9, as shown in Table 10. Through further analysis of Table 10, the following conclusions
can be obtained.
(1) When the PR of connected vehicles is fixed, the road capacity increases with the MPS, while
the increase rate of capacity decreases.
(2) Under the same MPS, the road capacity increases gradually with the PR of connected
vehicles, and the capacity in a fully CAVs environment is much greater than that in a fully CVs
environment.
(3) In a mixed traffic flow environment, when the PR of CAVs and CVs is low (),
the impact of MPS on road capacity is not obvious; when the PR of CAVs and CVs is high (
), the road capacity increases with the MPS.
Table 10
The increase rate of maximum capacity under different scenarios.
(a)
()
()
2647
2935
3000
3029
3045
3056
3063
()
2827
3057
3105
3124
3133
3139
3142
()
3058
3242
3272
3280
3283
3284
3284
()
3349
3507
3533
3540
3542
3543
3543
()
3716
3886
3933
3954
3965
3972
3976
()
4186
4444
4538
4586
4615
4635
4649
(b)
()
()
2241
2246
2246
2246
2246
2246
2246
()
2366
2389
2390
2390
2390
2390
2390
()
2553
2607
2611
2611
2611
2611
2611
()
2818
2920
2931
2933
2933
2933
2934
()
3195
3363
3389
3394
3396
3396
3396
(c)
()
()
2326
2418
2431
2434
2434
2434
2434
()
2367
2431
2437
2438
2438
2438
2438
()
2420
2463
2466
2466
2466
2466
2466
()
2485
2518
2520
2520
2520
2520
2520
()
2560
2598
2603
2603
2603
2604
2604
()
2647
2707
2718
2721
2722
2722
2722
(d)
()
()
2169
2256
2268
2270
2271
2271
2271
()
2256
2317
2323
2324
2324
2324
2324
()
2377
2419
2422
2422
2422
2422
2422
()
2536
2570
2572
2572
2572
2572
2572
()
2744
2784
2789
2790
2790
2790
2790
()
3017
3086
3099
3102
3103
3103
3103
(e)
27
()
()
2493
2696
2738
2755
2764
2769
2771
()
2632
2772
2795
2802
2804
2805
2805
()
2818
2920
2931
2933
2933
2933
2934
()
3055
3158
3179
3186
3188
3189
3189
()
3354
3525
3582
3608
3623
3631
3636
In addition, for a fully connected environment and mixed traffic flow environment, the
influence of MPS and PR of connected vehicles on road capacity is also analyzed, respectively, as
shown in Fig. 10(a) and Fig. 10(b). Fig. 10 shows that with the increase in the PR of connected vehicles
or MPS, the road capacity gradually increases, and the PR of connected vehicles significantly impacts
road capacity. In addition, the road capacity in a fully CAVs environment is much larger than that
in a fully CVs environment. Therefore, in fully connected environment, the road capacity can be
further improved by increasing the PR of CAVs.
(a) fully connected environment
(b) mixed traffic flow environment
Fig. 10. The heat map of road capacity in the fully connected and mixed traffic flow environment
6. Conclusion
In this paper, we examine the traits of mixed traffic flow with CAVs, CVs, AVs, and HVs. Firstly,
we discuss the different car-following modes observed in mixed traffic flow, which involves the
interaction of various types of vehicles mentioned above. According to the different characteristics
of car-following behaviors, it can be divided into 18 car-following modes. On this basis, 8 types of
time headway can be further obtained. Secondly, the probability distribution model of platoon size
is derived, and numerical simulation experiments are conducted to explore the effect of various PR
of CAVs, CVs, and MPS on the probability distribution of platoon size. Then, based on the probability
distribution model, the fundamental diagram of mixed traffic flow is further derived. Finally, a
capacity analysis model considering heterogeneity and MPS is constructed. Furthermore, the
numerical experiments are designed to discuss how the PR and MPS affect road capacity under
several scenarios. From the results of the numerical experiments, the following conclusions can be
drawn.
(1) When the PR of CAVs and CVs is low, the changes in MPS have no obvious effect on the
distribution of platoon size. When the PR is high, with the increase of MPS, the probability of platoon
size greater than or equal to 1 begins to decrease. Meanwhile, the distribution of platoon size shows
a diversified trend.
(2) When the equilibrium density of each traffic flow reaches its critical density, the road
28
capacity reaches its maximum. For example, in a fully CAVs environment, when the equilibrium
state density is , the maximum road capacity is , more than two times the
maximum road capacity in a fully HVs environment.
(3) In summary, under the same conditions, the road capacity increases with the MPS limitation.
When the PR of connected vehicles is low, the increase in platoon size has no obvious effect on
enhancing road capacity. With the increase in the PR of connected vehicles, the change in platoon
size has a more and more obvious influence on the traffic flow capacity. In addition, when the PR of
connected vehicles is low, the increase in PR has minimal influence on the capacity in the equilibrium
state. However, when the PR is high, the road capacity in the equilibrium state changes significantly
under the influence of the PR. For example, when the PR () increases from (0.1,0.1) to (0.2,0.2),
and the maximum capacity in the equilibrium state only increases by 1.56%. But when it increases
from (0.4,0.4) to (0.5,0.5), the maximum capacity of the equilibrium state increases by 24.23%.
(4) When the PR of connected vehicles remains unchanged, the road capacity increases with
MPS, while the increased rate of road capacity decreases.
(5) Under the same MPS, the road capacity increases with the PR of connected vehicles, and the
road capacity in a fully CAVs environment is much larger than that in a fully CVs environment.
(6) With the increase of the PR of connected vehicles or MPS, the road capacity of mixed traffic
flow gradually increases. And the PR of connected vehicles has a more significant influence on the
capacity.
Acknowledgments
The paper received funding from the National Natural Science Foundation of China (52002339),
the Fundamental Research Funds for the Central Universities (2682023ZTPY034), and Chengdu Soft
Science Research Project (2023-RK00-00029-ZF). The authors would like to thank Mrs. Yunxia Wu
from Southwest Jiaotong University for her valuable comments on the revised manuscript.
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