Analysis of the impact of maximum platoon size of CAVs on mixed traffic flow: an analytical and simulation method

Abstract

This paper analyzes the characteristics of mixed traffic flow with maximum platoon size of connected automated vehicles (CAVs) based on analytical and simulation methods. First, we discuss the car-following and headway types of mixed traffic flows with human-driven vehicles (HDVs), connected human-driven vehicles (CHVs), automated vehicles (AVs), and CAVs. Second, the probability distribution of the CAVs platoon size is derived based on the Markov chain model. Subsequently, the capacity model and stability condition of mixed traffic flow are proposed based on the probability distribution of the CAVs platoon size, and some characteristics of the influence of the maximum platoon size on traffic capacity and stability are analyzed. A numerical experiment based on car-following models was conducted to investigate the effect of the maximum platoon size on capacity, stability, safety, fuel consumption, and emission of mixed traffic flow. The results show that (1) the capacity of mixed traffic flow increases with the maximum platoon size of CAVs; (2) the stability of mixed traffic flow deteriorates with the increase in the maximum platoon size of CAVs; (3) the safety risk of vehicles in mixed traffic flow increases with the maximum platoon size of CAVs; (4) the fuel consumption and emissions (e.g., CO, HC, and NOx) increase with the maximum platoon size of CAVs. This indicates that the larger maximum platoon size can only improve traffic capacity, which is not conducive to the stability, safety, fuel consumption, and emissions of mixed traffic flow.

Full text

Preprint submitted to Transportation Research Part C: Emerging Technologies
December 11, 2022
Analysis of the impact of maximum platoon size of CAVs on mixed traffic flow: an analytical and
simulation method
Zhihong Yao1,2,3*, Yunxia Wu1,2, Yi Wang1,2, Bin Zhao1,2, and 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 analyzes the characteristics of mixed traffic flow with maximum platoon size of
connected automated vehicles (CAVs) based on analytical and simulation methods. First, we discuss the car-
following and headway types of mixed traffic flows with human-driven vehicles (HDVs), connected human-
driven vehicles (CHVs), automated vehicles (AVs), and CAVs. Second, the probability distribution of the
CAVs platoon size is derived based on the Markov chain model. Subsequently, the capacity model and
stability condition of mixed traffic flow are proposed based on the probability distribution of the CAVs
platoon size, and some characteristics of the influence of the maximum platoon size on traffic capacity and
stability are analyzed. A numerical experiment based on car-following models was conducted to investigate
the effect of the maximum platoon size on capacity, stability, safety, fuel consumption, and emission of mixed
traffic flow. The results show that (1) the capacity of mixed traffic flow increases with the maximum platoon
size of CAVs; (2) the stability of mixed traffic flow deteriorates with the increase in the maximum platoon
size of CAVs; (3) the safety risk of vehicles in mixed traffic flow increases with the maximum platoon size of
CAVs; (4) the fuel consumption and emissions (e.g., CO, HC, and NOx) increase with the maximum platoon
size of CAVs. This indicates that the larger maximum platoon size can only improve traffic capacity, which
is not conducive to the stability, safety, fuel consumption, and emissions of mixed traffic flow.
Keywords: Mixed traffic flow; Platoon size; Connected automated vehicles (CAVs); Capacity; Stability; Safety; Fuel
consumption and emissions.
*Correspondence to: Zhihong Yao, School of Transportation and Logistics, Southwest Jiaotong University, Chengdu, Sichuan 610031,
China, E-mail: zhyao@swjtu.edu.cn

1
1. Introduction
As an emerging product of accelerating cross-border integration in transportation, communication,
automatic control, artificial intelligence, and other industries, connected automated vehicles (CAVs) have
developed rapidly worldwide in recent years (Shiwakoti et al., 2020; Yu et al., 2021). The development of
CAVs will result in significant changes in traffic patterns and usher intelligent transportation research into a
new stage. Existing research shows that the application of CAVs can improve transportation systems to a
certain extent by improving traffic capacity (Chen et al., 2017a; Ghiasi et al., 2017; Yao et al., 2022b), enhancing
the stability of traffic flow (Montanino and Punzo, 2021; Sun et al., 2020), promoting traffic safety
(Nascimento et al., 2020; Rahman and Abdel-Aty, 2018; Xiao et al., 2021), and reducing fuel consumption and
emissions (Yao et al., 2021a; Yu et al., 2018; Zhao and Zhang, 2021). However, upgrading the CAVs will take
some time. Therefore, the mixed traffic flows composed of CAVs, connected human-driven vehicles (CHVs),
automated vehicles (AVs), and human-driven vehicles (HDVs) will be present for a long time in the future
(Qin et al., 2021; F. Zheng et al., 2020c).
Extensive research has been conducted on new mixed traffic flow (i.e., traffic flow of CAVs, CHVs, AVs,
and HDVs) (Chen et al., 2017a; Ngoduy et al., 2021; F. Zheng et al., 2020c). These studies primarily involved
a mixed traffic flow model, capacity, stability, traffic safety, fuel consumption, and emissions (Kopelias et al.,
2020; Mahmassani, 2016; Shiwakoti et al., 2020; Sun et al., 2020, 2018). Al-Turki et al. (2021) indicated that
there would be a mixed traffic flow composed of CAVs, CHVs, AVs, and HDVs in the future. CHVs have a
communication function, but they fall under human driving; therefore, the communication function will only
affect their macro traffic behavior (e.g., route planning, navigation, etc.) (Mahmassani, 2016). The micro-
driving behavior of the CHVs is consistent with that of the HDVs. Therefore, in this paper, CHVs are
considered as HDVs. This implies that the new mixed traffic flow contains three types of vehicles, CAVs, AVs,
and HDVs (Li et al., 2020; Rahman et al., 2021). Subsequently, many studies have investigated the mixed
traffic flow model, and the details are available in the relevant reviews (Rios-Torres and Malikopoulos, 2017;
Shiwakoti et al., 2020; Tafidis et al., 2021; Yu et al., 2021). These studies have involved microscopic,
mesoscopic, and macroscopic traffic flow models (Yu et al., 2021). In terms of traffic capacity, microscopic
traffic parameters are generally used to describe the operating characteristics of different vehicles, and the
impact of CAVs on the capacity of mixed traffic flow has also been studied (Ghiasi et al., 2017; Sala and
Soriguera, 2021; Shang and Stern, 2021; Talebpour and Mahmassani, 2016). The results showed that the
application of CAVs can significantly improve traffic capacity. These studies on traffic capacity have
primarily involved road segments (Ghiasi et al., 2017; Liu et al., 2018a; Woo and Skabardonis, 2021) and
expressway ramps (Jing et al., 2019; Liu et al., 2018b; Pei et al., 2019; Xiao et al., 2018). In terms of traffic
stability, most previous studies were based on the micro-driving model. Based on car-following models, the
stability discrimination equation for mixed traffic flow was derived (Montanino and Punzo, 2021; Sun et al.,
2020, 2018). Moreover, the control models of CAVs and AVs were optimized to improve the stability of mixed
traffic flow (Wang et al., 2019). However, when a CAV cannot communicate with the front vehicle (i.e., the
front vehicle does not have a communication function), it degenerates, that is, the CAVs are degraded. Many
studies have shown that CAVs are conducive to improving the stability of mixed traffic flow, but degraded
CAVs and AVs worsen the stability of mixed traffic flow (Qin et al., 2019; Wang et al., 2019). Traffic safety
can be classified into two categories. The first type simulates mixed traffic flow based on the control models
of CAVs and AVs, and the safety evaluation index is adopted to evaluate the safety of mixed traffic flow
(Arvin et al., 2020; Rahman and Abdel-Aty, 2018). The second type optimizes and controls the trajectories of
CAVs and AVs to avoid vehicle collisions and improve traffic safety (Chen et al., 2021; Jing et al., 2020; Lin et
al., 2021; Qian et al., 2021). Therefore, most research results show that the application of CAVs can improve
traffic safety (Nascimento et al., 2020; Yang and Fisher, 2021). These studies can be divided into two categories,
in terms of fuel consumption and emissions: one is to simulate the mixed traffic flow using the micro control
models of CAVs and AVs, and then calculate the corresponding fuel consumption and emissions (Qin et al.,
2018, 2019; Yao et al., 2021a; Zhou et al., 2021). The second is to optimize the trajectory of CAVs to achieve
energy conservation and emission reduction (Han et al., 2020; X. T. Yang et al., 2021; Zhao and Zhang, 2021;
Zhao et al., 2018). Therefore, the results indicated that the application of CAVs can reduce fuel consumption
and emissions (Luo et al., 2021; Mersky and Samaras, 2016). However, the impact of CAVs platoon size was

2
not included in the above studies. Considering the safety and operational requirements, the actual platoon
size of CAVs is limited. Therefore, the impact of the maximum platoon size of CAVs on traffic flow
characteristics should be investigated.
This paper proposes a mixed traffic flow model that considers the maximum platoon size of CAVs. Based
on this, the impact of the maximum platoon size on the capacity, stability, traffic safety, fuel consumption,
and emissions of mixed traffic flow is investigated. The main contributions of this paper are as follows.
(1) The influence of the maximum platoon size on car-following characteristics is considered, and the
probability distribution of platoon size in mixed traffic flow is proposed based on the Markov chain. The
proposed model can calculate the probability of any platoon size of CAVs in mixed traffic flow.
(2) The capacity of mixed traffic flow considering the maximum platoon size of CAVs is derived based
on the mixed traffic flow model. Subsequently, the relevant characteristics of the mixed traffic flow capacity
are analyzed.
(3) The stability discrimination equation of mixed traffic flow considering the maximum platoon size of
CAVs is derived based on car-following models. Subsequently, the influence of the maximum platoon size of
CAVs on mixed traffic flow is theoretically proved. The theoretical results were verified based on numerical
simulation.
(4) Two safety evaluation metrics, the time-exposed time-to-collision (TET) and time-integrated time-to-
collision (TIT), are adopted to analyze the impact of the maximum platoon size of CAVs on the traffic safety
of mixed traffic flow.
(5) The fuel consumption, CO, HC, and NOx, are adopted to analyze the impact of the maximum platoon
size of CAVs on the fuel consumption and emissions of mixed traffic flow.
The remainder of this paper is organized as follows. Section 2 reviews related studies, including those
on the maximum platoon size, capacity, stability, safety, fuel consumption, and emissions of mixed traffic
flow. The problem statement, notations, and modeling assumptions are presented in Section 3. Section 4
proposes a mixed traffic flow model and platoon size distribution of CAVs based on a Markov chain. A
capacity model considering the maximum platoon size of CAVs is developed in Section 5. Section 6 analyzes
the stability of mixed traffic flow using theoretical derivation and numerical simulation. The impacts of the
maximum platoon size of CAVs on mixed traffic flow safety, fuel consumption, and emissions are discussed
in Sections 7 and 8, respectively. Section 9 concludes the paper and discusses future research.
2. Literature review
2.1. Maximum platoon sizes of CAVs
The maximum platoon size of CAVs is a critical parameter affecting the characteristics of mixed traffic
flow. When the platoon is large, the traffic capacity can be improved owing to the small headway between
CAVs (Yao et al., 2022a). However, a large platoon can affect the lane-changing of vehicles and is not
conducive to traffic flow stability (Shladover et al., 2015). Existing research on the maximum platoon size
primarily focuses on the traffic capacity. These studies can be divided into two categories. The first category
comprises simulation-based methods. To investigate the impact of CAVs on the multilane freeway merge
capacity, Liu et al. (2018a) suggested a maximum platoon size of 10 vehicles based on a preliminary analysis
of multiple candidate platoon sizes. Furthermore, Liu et al. (2018b) extended a simulation framework to
describe the interactions between CAVs and HDVs in mixed traffic flow. The results indicated that a
maximum platoon size of 10-20 is appropriate for freeway traffic operations. Xiao et al. (2018) investigated
the effects of CAVs on traffic flow at merging bottlenecks. To avoid string instability and potential problems
in the merging or weaving sections, they set the maximum platoon size of the simulation experiment to 10.
Luo et al. (2018) developed a coordinated platoon model with multiple speeds to minimize total fuel
consumption. In their numerical experiment, the maximum platoon size was set to 10. Zhou et al. (2021)
discussed the impact of CAV platoon management on fundamental diagrams, fuel consumption, and
pollutant emissions. The simulation results suggested that the optimal platoon size is 5, which has the most
apparent improvement in traffic capacity.
The abovementioned studies considered the maximum platoon size of CAVs from simulations, and none
provided a theoretical analysis of the impact of maximum platoon size on mixed traffic flow. To address this

3
gap, Chen et al. (2017a) proposed a capacity model for mixed traffic flow with CAVs and HDVs. The results
indicated that the capacity increases with the penetration rate and platoon size of CAVs. Sala and Soriguera
(2021) developed a generalized macroscopic model to calculate the average platoon size of CAVs when the
traffic demand and penetration rate of CAVs are known. Moreover, they discussed the impact of the
maximum platoon size on the distribution of platoon size of CAVs. However, the optimal platoon size was
not addressed in these studies. Zhu et al. (2021) analyzed the impact of the maximum platoon size of CAVs
on the capacity and stability of mixed traffic flow. The results showed that the maximum platoon size could
improve traffic capacity, but it is not conducive to the stability of mixed traffic flow. Therefore, the analysis
suggested that the traffic capacity and stability can be balanced when the optimal maximum platoon size is
4.
To investigate the impact of the maximum platoon size of CAVs on the characteristics of mixed traffic
flow, this paper reveals the impact of the maximum platoon size on traffic flow from five levels: capacity,
stability, safety, fuel consumption, and emissions, based on theoretical modeling and numerical simulation.
The influence of CAVs on the five characteristics of mixed traffic flow is discussed in the following sections.
2.2. Traffic capacity of mixed traffic flow
The application of CAVs can significantly improve traffic capacity. Many studies have conducted in-
depth research on this (Chen et al., 2017a; Qin et al., 2021). These studies primarily focused on theoretical
modeling and simulation analysis (Yu et al., 2021). Ghiasi et al. (2017) proposed an analytical capacity model
for mixed highway traffic flow. Their model used a Markov chain to describe the spatial distribution of the
heterogeneous and stochastic headways. The numerical experiment verified that the proposed model
accurately captured the mixed traffic capacity in various settings. Shi and Li (2021) investigated the impacts
of commercial AVs on the traffic flow capacity based on theoretical models and empirical experiments. The
results indicated that the shortest headway of the AVs can significantly improve traffic capacity, but some
headway settings may affect traffic flow stability, which may decrease traffic capacity. Shang and Stern (2021)
calibrated an intelligent driver model (IDM) with trajectory data collected on commercially available
adaptive cruise control (ACC) vehicles. A simulation experiment was designed to investigate the impact of
commercially available ACC vehicles on highway capacity. The analysis showed that the bottleneck capacity
was significantly affected by the string stability and inter-vehicle headways. Mirzaeian et al. (2021) analyzed
the effects of CAVs on highway congestion based on a queueing model. The results showed that CAVs can
significantly improve highway capacity because they can maintain smaller inter-vehicle gaps and travel
together on larger platoons than HDVs.
The abovementioned studies conducted in-depth investigations of the impact of CAVs on traffic capacity.
However, the effect of the platoon size of CAVs on the traffic capacity was not considered. Recent research
has attempted to study the impact of platoon size on the traffic flow capacity (Chang et al., 2020; Zhou and
Zhu, 2021). These studies assumed that mixed traffic flow included only CAVs and HDVs, without
considering the interaction between AVs and the platoon size of CAVs. Therefore, mixed traffic flow with
CAVs, AVs, and HDVs is investigated in this paper. Furthermore, the impact of the maximum platoon size of
CAVs on the capacity of the mixed traffic flow is discussed.
2.3. Stability of mixed traffic flow
Studies show that CAVs can improve traffic flow stability, but degraded CAVs are not conducive to the
stability of traffic flow (Yao et al., 2021b, 2019). Stability analysis can be classified into two aspects: simulation
and theoretical analysis. Regarding theoretical analysis, Talebpour and Mahmassani (2016) used car-
following models to describe vehicle types with various communication capabilities. Subsequently, the
stability of the mixed traffic flow was analyzed under different penetration rates of CAVs. The results
revealed that CAVs could improve the stability of mixed traffic flow. Sun et al. (2018) reviewed the primary
methods for analyzing the local and string stability of car-following models in conventional and connected
environments. Based on the review, they discussed the problems, challenges, and research requirements of
the stability analysis of car-following models in the era of CAVs. Sun et al. (2020) investigated the
complementary use of car-following instability and traffic oscillation. The case study suggested that the

4
higher stability of some individual vehicles can alleviate the oscillation severity of the platoon. Moreover, the
desired headway is the most influential factor for smoothing traffic oscillations.
To consider driver’ and vehicle heterogeneity, Montanino and Punzo (2021) developed a unifying
framework to model the string stability of mixed traffic flow. In terms of simulation analysis, first, a
simulation experiment was constructed based on the car-following models. Subsequently, small fluctuations,
such as vehicle acceleration and deceleration, were added to the stable traffic flow. Finally, traffic flow
stability was discussed by analyzing the process of restoring the steady state of the traffic flow (Kesting and
Treiber, 2008; Treiber et al., 2007). These studies indicated that CAVs could improve the stability of mixed
traffic flows (Ruan et al., 2021; F. Zheng et al., 2020a).
The abovementioned studies conducted in-depth analyses of the stability of mixed traffic flow from two
aspects: theory and simulation. However, the influence of the platoon size of CAVs on the stability of mixed
traffic flows were not considered (Chang et al., 2020; Zhou and Zhu, 2021). Therefore, this paper analyzes the
impact of the maximum platoon size of CAVs on traffic flow stability using theoretical modeling and
numerical simulation.
2.4. Safety of mixed traffic flow
Traffic safety is a precondition for improving traffic efficiency (Evans, 2004). Considerable research has
been conducted on traffic safety (Rezaei and Caulfield, 2021; Wang et al., 2021; Xiao et al., 2021). The
application of CAVs provides new opportunities for traffic safety (Yang and Fisher, 2021). The existing
research on the impact of CAVs on traffic safety has been divided into two aspects. The first is to investigate
the improvement of CAVs on traffic safety based on the micro-driving model. Rahman and Abdel-Aty (2018)
used an IDM to describe the driving behavior of CAV platoons. The standard deviation of speed, time-
exposed TET, TIT, time-exposed rear-end crash risk index (TERCRI), and sideswipe crash risk (SSCR) were
utilized as indicators for safety evaluation. The results showed that CAVs significantly improved the
longitudinal safety of the expressways. Moreover, to better capture the car-following behavior of CAVs, Yao
et al. (2020a) simulated CAVs traffic flow using calibrated ACC and cooperative adaptive cruise control
(CACC). The simulation results suggested that degenerated CAVs significantly increase the safety risk of the
mixed traffic flow. The two research scenarios were road segments on the expressway. Zhu and Tasic (2021)
proposed a merging conflict model to estimate on-ramp merging safety in CAVs environments. The results
indicated that CAVs have apparent advantages in reducing the frequency and severity of crucial merger
events. Yang et al. (2021) developed an on-ramp cellular automata (CA) model to investigate the impact of
CAVs on traffic efficiency and safety of on-ramps. Time-to-collision (TTC) results showed that CAVs
promoted traffic safety under congested conditions. Arvin et al. (2020) developed a simulation framework to
evaluate the impact of CAVs on traffic safety at intersections. The simulation results showed that CACC
vehicles could communicate and collaborate better than ACC vehicles. As a result, CACC vehicles clearly
improve traffic safety. The second is to optimize the scheduling and trajectory of CAVs passing through the
conflict zone to avoid collisions and ensure vehicle safety. This method is considerably different from the
research in this paper and will not be discussed in detail here. For more details, please refer to (Chen et al.,
2021; Ding et al., 2020; Jing et al., 2019; Lin et al., 2021; Pei et al., 2021, 2019; Qian et al., 2021; Xu et al., 2019;
Yao et al., 2020b).
The abovementioned impact on CAV traffic safety can be summarized as CAVs can improve traffic safety,
but degraded CAVs increase the traffic safety risk. However, the effect of the maximum platoon size of CAVs
on traffic safety was not addressed in these studies. Therefore, this study analyzed the impact of the
maximum platoon size of CAVs on traffic safety based on numerical simulations.
2.5. Fuel consumption and emissions of mixed traffic flow
The development of CAVs provides new opportunities for energy conservation and emission reduction
in transportation systems. Two main methods are used to evaluate the fuel consumption and emissions of
CAVs. The first is to optimize the trajectory of CAVs with the objectives of fuel consumption and emissions.
This method can significantly reduce fuel consumption and emissions. Details are available in the literature
(Han et al., 2020; Song et al., 2021; Typaldos et al., 2020; Wan et al., 2016; Xu et al., 2021; X. T. Yang et al., 2021;
Yu et al., 2018; Zhao et al., 2018). The second is based on the micro control model of CAVs to simulate and

5
analyze the impact of CAVs on fuel consumption and emission in different traffic scenarios. Mersky and
Samaras (2016) modeled the effects of CAVs technology within the bounds of the current fuel economy test.
Subsequently, a drive cycle scenario was applied to estimate the changes in the fuel. The results showed that
CAVs control algorithms designed without considering efficiency could degrade the fuel economy by up to
3%, whereas efficiency-based control algorithms can reduce fuel consumption by up to 10%. Qin et al. (2019,
2018) analyzed the impact of CAVs on fuel consumption and emissions of transportation systems from the
perspective of stability. A numerical simulation showed that improving traffic flow stability reduces fuel
consumption and emissions. Yao et al. (2021a) evaluated the influence of CAVs on fuel consumption and
emissions of mixed traffic flow on the expressway. In this study, three models were adopted to describe the
car-following of HDVs, ACC, and CACC. The results indicated that the maximum reduction percentages of
HC, NOx, CO, and fuel consumption were 24.33%, 27.06%, 37.53%, and 40.58%, respectively, at a 100%
penetration rate of CAVs.
The abovementioned studies analyzed the fuel consumption and emissions of CAVs in the transportation
system. However, they did not consider whether the size of a CAV platoon affects fuel consumption and
emissions, when CAVs form a platoon. Therefore, this paper evaluates the impact of the maximum platoon
size of CAVs on fuel consumption and emissions based on numerical simulations.
Table 1 summarizes recent studies on the characteristics of mixed traffic flows, with three significant
highlights. First, many studies only considered two vehicle types (i.e., HDVs and CAVs) of mixed traffic flow,
and AVs were not involved. Second, most previous studies did not consider the impact of the platoon size of
CAVs on mixed traffic flow. Finally, the effect of CAVs and their platoon size on traffic flow characteristics
(i.e., capacity, stability, safety, fuel consumption, and emissions) have not been thoroughly studied.

6
Table 1. Summary of vehicle composition, platoon size, and characteristics of mixed traffic flow
Literature
HDVs
AVs
CAVs
CAVs degradation
Platoon size
Capacity
Stability
Safety
Fuel consumption
Emission
(Qin et al., 2018)






(Qin et al., 2019)





(Han et al., 2020)







(Yao et al., 2021a)






(Y. Zheng et al., 2020)



(Arvin et al., 2020)



(Xiao et al., 2021)



(Sun et al., 2020)



(Sun et al., 2018)



(Montanino and Punzo, 2021)



(Talebpour and Mahmassani, 2016)





(Lioris et al., 2017)





(Ghiasi et al., 2017)





(Zhou and Zhu, 2020)





(Shang and Stern, 2021)





(Shi and Li, 2021)



(Liu et al., 2018a, 2018b)





(Xiao et al., 2018)






(Chen et al., 2017a)





(Sala and Soriguera, 2021)






(Zhou and Zhu, 2021)






(Mirzaeian et al., 2021)



(Zhou et al., 2021)







This paper










Note: In this paper, ‘H’ stands for HDVs; ‘A’ stands for AVs or degraded CAVs; if the maximum CAVs platoon size is considered, the CAV can be divided into two cases: CAVs in the platoon and a
CAV following a CAV platoon as the preceding platoon size achieves the maximum size, namely, C-C (i.e., intra-platoon) and C-P (i.e., inter-platoon). ‘C’ stands for CAVs in intra-platoon, and ‘P’
stands for CAVs in inter-platoon.

7
3. Problem statement, notations, and modeling assumptions
3.1. Problem statement
Fig. 1. Sketch of the ten types of headways.
With the development of automatic driving and communication technology, the composition of mixed
traffic flow will change significantly in the future (Jiang et al., 2021; Yao et al., 2019). Traditional mixed traffic
flows are typically composed of cars, trucks, buses, or bicycles (Ambarwati et al., 2014; Jun, 2010). In the
future, the new mixed traffic flow will be composed of CAVs, AVs, CHVs, and HDVs (Mirzaeian et al., 2021;
Yao et al., 2021a; Zhou and Zhu, 2020). In the new mixed traffic flow, when multiple CAVs are driven together,
the front and rear CAVs can communicate directly, thus forming a stable platoon (SAE, 2020), as shown in
Fig. 1. Theoretically, the platoon size is unlimited. However, considering the stability and safety of traffic
flow, the platoon size of CAVs is typically limited. To investigate the impact of the maximum platoon size on
the characteristics of mixed traffic flow comprehensively and thoroughly, this study proposes a mixed traffic
flow model that considers the maximum platoon size of CAVs. Based on this, the impact of the maximum
platoon size of CAVs on the capacity, stability, traffic safety, fuel consumption, and emissions of mixed traffic
flow is investigated. Fig. 2 shows the relationships between the key contents of this study.
Fig. 2. Relationships between the key contents of this study.

8
3.2. Notations
Table 2 lists the general indices, sets, parameters, and variables used in this paper.
Table 2. Sets, indices, parameters and variables.
Indices
Definition
, , , 
Time headways of the four cases.
, , , 
Time gaps of the four cases.

The index of vehicles

A random variable/state variable at step

 in the Markov chain

A set of vehicles in the mixed traffic flow

A set of vehicle types (i.e., HDV, AV, and CAV)

A set of integers less than or equal to 
, , 
The penetration rates of CAVs, AVs, and HDVs.
, 
The initial state and transition matrix of the Markov chain model

The probability of a type-s vehicle being followed by a type-t vehicle

The maximum size of the platoon
󰇛󰇜
The probability of a vehicle being the -th () vehicle in a platoon

The average headway of mixed traffic flow

The approximate traffic capacity
, , 
The proportional coefficient related to headway
, 
Auxiliary variables

Discrete-time steps
󰇛󰇜, 󰇛󰇜, 󰇛󰇜
The position, speed, and acceleration of the -th vehicle at time 
󰇛󰇜, 󰇛󰇜
The distance and the speed gap between the -th vehicle and the preceding
vehicle
󰇛󰇜, 󰇗󰇛󰇜
The gap error and its differential of the -th vehicle
, , , ,







The discriminant number of traffic flow stability
, 
Time delay of vehicle  with respect to the distance headway, and speed with
the leader
, , 
The partial differentiation of the acceleration calculation equation for vehicle
speed, speed difference, and space headway
󰇛󰇜
The control equation of the car-following models
, , , 
The number of vehicles in four types of car-following modes

The steady-state speed of the mixed traffic flow
󰇛󰇜
The collision time at the time 

The threshold value of TTC

Time exposed time-to-collision

Time-integrated time-to-collision


The regression coefficient under the power

 of speed and the power  of
acceleration,

The fuel consumption (FC) and traffic emission (carbon monoxide (CO),
hydrocarbon (HC), and oxides of nitrogen (NOx)) rate of the -th vehicle
3.3. Modeling assumptions
Before introducing our proposed models, we present some key modeling assumptions to facilitate the
modeling process.
1) We consider only the longitudinal behavior of vehicles; that is, the impact of lane changing on the
maximum platoon size of CAVs is not considered in this paper (Chang et al., 2020; Zhou and Zhu,
2021).

9
2) CAVs are randomly distributed in a mixed traffic flow. Consequently, a platoon is randomly
generated. Therefore, CAVs are not considered to tend to form a platoon in this paper (Avedisov et
al., 2022; Li et al., 2022).
3) The time lag (e.g., sensor delay, communication delay, and reaction time) is not considered in this
paper. Previous studies have conducted in-depth research on the impact of time lag on traffic flow.
Details are available in (Ngoduy, 2013; Wang, 2018; Yao et al., 2021b).
4) We assume that AVs can be controlled longitudinally without human intervention. In this paper,
AVs are equipped with intelligent sensors (high-definition cameras, radar, etc.), which can recognize
changes in the driving behavior of the front vehicle. ACC control is adopted to capture the
longitudinal control of the AVs. Therefore, without considering lane changing (i.e., Assumption 1),
we assume that the ACC controller can manage the longitudinal behavior of AVs. Referring to SAE
(2020), the automation level of AVs is L1 in this paper.
5) In this paper, CAVs are considered to be AVs with communication functions. Compared with AVs,
CAVs can communicate with front and rear CAVs and share information (e.g., position, speed,
acceleration, and the next longitudinal driving behavior) (Shladover et al., 2012; Yao et al., 2021b).
As a result, CAVs can realize cooperation and connectivity, and can be described using CACC
control in terms of longitudinal control. Referring to SAE (2020), this cooperation belongs to class B,
and the automation level of CAVs is L1. Therefore, without considering lane changing (i.e.,
Assumption 1), cooperation and connectivity refer to the collaboration and communication between
front and rear CAVs in the car following.
4. Modeling of mixed traffic flow
4.1. Car-following types of mixed traffic flow
As shown in Fig. 1, many car-following types exist in the new mixed traffic flow. Moreover, CHVs are
driven by human drivers, and their driving behavior is the same as that of HDVs. Therefore, CHVs are not
considered separately in this paper.
In summary, the new mixed traffic flow contained three types of vehicles, CAVs, AVs, and HDVs.
According to the sequence of car-following vehicles, there will be nine different car-following types, namely,
(1) CAV-CAV (C-C), (2) CAV-AV (C-A), (3) CAV-HDV (C-H), (4) AV-CAV (A-C), (5) AV-AV (A-A), (6) AV-
HDV (A-H), (7) HDV-CAV (H-C), (8) HDV-AV (H-A), and (9) HDV-HDV (H-H). Note that if the maximum
CAVs platoon size is considered, the car-following type CAV-CAV can be divided into two cases: CAVs in
the platoon and a CAV following a CAV platoon as the preceding platoon size achieves the maximum size,
namely, C-C (i.e., intra-platoon) and C-P (i.e., inter-platoon), as shown in the innermost lane in Fig. 1. As a
result, this mixed traffic flow has a total of 10 car-following types. To capture different car-following types,
we adopted different headways between the front and rear vehicles in this study. The car-following types
and their corresponding headways are shown in Fig. 1 and Table 3.
Table 3. Car-following type and headways.
Index
The current vehicle
The following vehicle
Car-following type
Headway type
Headway
1
CAV
CAV
CAV-CAV (inter-platoon)


2
CAV
CAV-CAV (intra-platoon)


3
AV
CAV-AV


4
HDV
CAV-HDV


5
AV
CAV
AV-CAV


6
AV
AV-AV


7
HDV
AV-HDV


8
HDV
CAV
HDV-CAV


9
AV
HDV-AV


10
HDV
HDV-HDV



10
4.2. Headway types of mixed traffic flow
According to the analysis in Section 4.1, the new mixed traffic flow has 10 car-following types. As a result,
there are 10 headways in principle. In this paper, the headways refer to the time-headway between the
following vehicle and the front vehicle. Therefore, the headway is related only to the properties of the
following vehicle (Yao et al., 2022a; Zhou and Zhu, 2020, 2021). Note that if the following vehicle is a CAV, it
will degenerate into an AV if the front vehicle does not have a communication function (Qin et al., 2021; Yu
et al., 2021). In summary, headway types are divided into four cases.
(1) In the first case, the following vehicle is an AV or a degenerated CAV. Therefore, the car-following
types corresponding to this case are A-C, H-C, H-A, A-A, and C-A, which can be expressed as

(1)
where , , , , and  are the headways of car-following types A-C, H-C, H-A, A-A, and C-A,
respectively.
(2) In the second case, the following vehicle is an HDV. Therefore, the car-following types corresponding
to this case are A-H, C-H, and H-H, which can be expressed as
 
(2)
where , , and  are the headways of car-following types A-H, C-H, and H-H, respectively.
(3) In the third case, a CAV follows a CAV platoon as the preceding platoon size achieves the maximum
size, namely, inter-platoon car-following (Mirzaeian et al., 2021; Zhou and Zhu, 2021). Therefore, the
car-following type corresponding to this case is C-P, which can be expressed as

(3)
where  is the headway of the car-following type C-P.
(4) In the fourth case, a CAV is in a platoon and is not the leader, namely, intra-platoon car-following.
Unlike the first case, CAVs in the fourth case can communicate with the CAV in front; therefore, the
headway is generally smaller than that of the first case (Mirzaeian et al., 2021; Qin et al., 2021; Zhou
and Zhu, 2020). Therefore, the car-following type corresponding to this case is C-C, which can be
expressed as

(4)
where  is the headway of the car-following type C-C.
In summary, the new mixed traffic flow has four headway types, i.e., , , , and . Generally, to
ensure the safety of platoons, when a CAV follows a platoon of the maximum size, it maintains a considerable
distance (Chen et al., 2017b; Liu et al., 2018a; Zhou and Zhu, 2021), that is, . In addition, the time
headway of HDVs is the largest, while that of AVs is between CAVs and HDVs (Yao et al., 2022a; Zhou and
Zhu, 2020, 2021). Consequently, the size relationship of these four headways can be expressed as 
 Next, the probability distributions of the four headways are derived.
4.3. Markov chain model for mixed traffic flow
A Markov chain model was proposed by Ghiasi et al. (2017) to capture the characteristics of mixed traffic
flow with CAVs and HDVs. However, previous studies reported that there would be many types of vehicles
mixing in mixed traffic flow in the future, such as CAVs, AVs, CHVs, and HDVs (Fig. 1). Humans drive both
CHVs and HDVs; therefore, there is no difference in their micro-driving behavior. Therefore, CHVs are not
considered separately in this study, and they could be considered to be a special form of HDVs. In summary,
at least three vehicle types (i.e., CAVs, AVs, and HDVs) must be considered in mixed traffic flow.

11
Referring to Ghiasi et al. (2017), a Markov chain is developed to model mixed traffic flow with CAVs,
AVs, and HDVs. We consider a mixed traffic flow with vehicles indexed by 󰇝󰇞. Let random
variable  represent whether vehicle  is an HDV, AV, or CAV, which is defined by Eq.(5).
 
 
 
(5)
Subsequently, we introduce an auxiliary random variable 󰇛󰇜, whose value can be divided into
three cases: case 1, if  , 󰇛󰇜 ; case 2, if  , 󰇛󰇜 ; and case 3,  ,
󰇛󰇜 . As a result, 󰇛󰇜
 is twice the number of CAVs in the mixed traffic flow.
Therefore, the penetration rate of CAVs can be defined using Eq. (6).
󰇧󰇛󰇜
 󰇨
(6)
Similarly, we introduce an auxiliary random variable . Subsequently,  when 
or  . When  ,  . Consequently, 󰇛󰇜
 is the number of CAVs in the mixed
traffic flow. Therefore, we can also obtain the penetration rates of AVs and HDVs, as shown in Eqs. (7) and
(8), respectively.
󰇧󰇛󰇜
󰇨
(7)
󰇧󰇛󰇜
 󰇨
(8)
In this Markov chain model,  is the state variable at step

, which also indicates the type of n-th
vehicle (i.e., CAV, AV, or HDV) The state space is 󰇝󰇞.

Combining Eqs. (6), (7), and (8), we can
define the initial state as
󰇟󰇠
(9)
Subsequently, the transition matrix of this Markov chain model is
  
  
  


(10)
where 󰇝󰇞, which is the probability of the car-following mode s-t.
Proposition 1. The Markov chain model defined by Eqs.(9) and (10) yields  as the invariant distribution probability.
Proof:
Basis: when , 󰇝󰇞, 󰇝󰇞, and 󰇝󰇞 based on Eq. (9).
Inductive step: We assume that 󰇝󰇞, 󰇝󰇞, and 󰇝󰇞, .
Subsequently, we only require to prove that when  , 󰇝󰇞 , 󰇝󰇞 , and
󰇝󰇞. The details of the proof are as follows:
Based on initial state Eq. (9) and transition matrix Eq. (10) of the Markov chain model, 󰇝󰇞,
󰇝󰇞, and 󰇝󰇞 are calculated using Eqs. (11)–(13).
󰇝󰇞󰇝󰇞󰇝󰇞󰇝󰇞
󰇛󰇜

(11)

12
󰇝󰇞󰇝󰇞󰇝󰇞󰇝󰇞
󰇛󰇜

(12)
󰇝󰇞󰇝󰇞󰇝󰇞󰇝󰇞
󰇛󰇜

(13)
Subsequently, Eq. (14) can be obtained using Eqs. (11)–(13):

(14)
Eq. (14) shows that  is the invariant distribution probability of the Markov chain model defined by
Eqs. (9) and (10). This completes the proof. □
4.4. Platoon size distribution
In this study, the platoon is defined as the real-time information (e.g., position, speed, acceleration)
shared by the leader and following vehicle through communication between the same platoon. Therefore,
HDVs and AVs cannot form a platoon as they have no communication function. To facilitate modeling and
derivation, we assume AVs and HDVs to be platoons with a size of 0, that is, not a platoon. Furthermore, if
the maximum size of the platoon is set to , the range of the actual platoon size is 0 to . Based on Zhou
and Zhu (2021), 󰇛󰇜 is defined as the probability of a vehicle being the -th () vehicle in a platoon
and 󰇛󰇜 is the probability of a vehicle being an HDV or AV. The vehicle index is then classified into three
scenarios:, ,  based on the car-following types.
(1) When , the vehicle is an HDV or AV, regardless of the type of vehicle in front. Therefore, 
can be expressed as
󰇛󰇜󰇝󰇞
(15)
(2) When 󰇝󰇞󰇝󰇞, which indicates the probability of the -th CAV in a platoon but
not the leader, which is defined in Eq. (16).
󰇛󰇜󰇝󰇞
(16)
Based on Proposition 1, we can obtain
󰇛󰇜󰇝󰇞
󰇝󰇞󰇝󰇞
󰇝󰇞󰇝󰇞
󰇛󰇜
(17)
Furthermore, Eq. (18) can be obtained by substituting Eq. (10) with Eq. (17).
󰇛󰇜󰇛󰇜
󰇛󰇜󰇛󰇜󰇝󰇞󰇝󰇞
(18)
(3) When , 󰇛󰇜 indicates the probability that the CAV is the platoon leader. In this case, the front
vehicle may be an HDV, AV, or maximum-size CAV platoon. First, when the front vehicle is an HDV
or AV, the probability is 󰇛󰇜. Subsequently, when its front vehicle is a maximum-sized
CAV platoon, the probability is

13
󰇝󰇞
󰇝󰇞
󰇝󰇞󰇝󰇞
󰇝󰇞󰇝󰇞
󰇝󰇞

󰇛󰇜
(19)
Subsequently, 󰇛󰇜 is the sum of the two probabilities, as shown in Eq. (20).
󰇛󰇜󰇛󰇜󰇛󰇜
󰇛󰇜󰇛󰇜󰇛󰇜
(20)
Therefore, 󰇛󰇜 can be obtained by solving Eq. (20), as shown in Eq. (21).
󰇛󰇜



󰇛󰇜
 

 
(21)
In summary, the distribution probability of the platoon size can be obtained by combining Eqs. (15), (18),
and (21).
󰇛󰇜





 

 󰇝󰇞
󰇛󰇜
 󰇝󰇞
(22)
Based on Eq. (22), the distribution probability of the platoon size under different maximum platoon sizes
and penetration rates of CAVs can be obtained (Fig. 3).
(a) Maximum platoon size ()

14
(b) Penetration rate of CAVs ()
Fig. 3. Distribution probability of CAV platoon size.
Fig. 3 shows the distribution probability of the CAVs platoon size. In particular, Fig. 3(a) displays the
distribution probability of platoon size with the CAV penetration rates when the maximum platoon size is 5.
From Fig. 3 (a), the same permeability is represented by the same color; therefore, the platoon size of the
same color from 0 to 5 equals 1. As shown in the red bar in Fig. 3(a), when the penetration rate of CAVs is 0,
all vehicles are HDVs or AVs, and the platoon cannot be formed at this time. As a result, 󰇛󰇜. In contrast,
when the penetration rate is 1, all vehicles are CAVs; therefore, all vehicles present the maximum platoon
size, as shown in the yellow bar in Fig. 3(a). Therefore, 󰇛󰇜. When the penetration rate of CAVs is ranges
0 to 1, the platoon size is between 0 and 5. With an increase in the penetration rate, the probability that the
platoon size is more significant than 2 will gradually increase.
Fig. 3(b) shows the probability distribution of platoon size under different maximum platoon sizes when
the penetration rate of CAVs is 0.5. In Fig. 3(b), the different colors represent different platoon sizes. When
the maximum platoon size is 1, the probabilities of a platoon size of 0 and 1 are equal, that is, 0.5. The
probability that the platoon size is 0 is independent of the maximum platoon size, with a probability of 0.5.
This result is consistent with theoretical results. The probability that the platoon size is 0 is only related to
the penetration rates of CAVs based on Eq. (22). In addition, with an increase in the maximum platoon size,
the platoon size will assume more forms. However, when the maximum platoon size of CAVs is given, the
area of the bar in Fig. 3(b) gradually decreases from the bottom to the top. This means that the larger the
platoon size, the smaller the probability of its occurrence.
5. Capacity
5.1. Capacity model
The capacity can be approximately estimated based on the average headway of mixed traffic flow, and
there are four types of headways based on the analysis in Section 4.2. Therefore, the approximate capacity is
󰇝󰇞󰇝󰇞󰇝󰇞󰇝󰇞
(23)
where 󰇝󰇞 is the probability of occurrence of headway .
Based on Eqs. (10), (19), (21), and (22), 󰇝󰇞 can be obtained, as shown in Eqs. (24)–(27).
(1) Case 1: the probability of occurrence of headway .
󰇝󰇞󰇛󰇜

 







󰇛󰇜


 
 
(24)

15
(2) Case 2: the probability of occurrence of headway .
󰇝󰇞







󰇛󰇜
 
(25)
(3) Case 3: the probability of occurrence of headway .
󰇝󰇞

󰇛󰇜󰇛󰇜
󰇛󰇜
(26)
(4) Case 4: the probability of occurrence of headway .
󰇝󰇞

󰇛󰇜

(27)
Here, 󰇝󰇞󰇝󰇞󰇝󰇞󰇝󰇞󰇝󰇞 . This equation
can be divided into two cases based on the value of .
(1) Case1: when ,
󰇝󰇞󰇝󰇞󰇝󰇞󰇝󰇞󰇝󰇞

󰇛󰇜
󰇛󰇜

󰇛󰇜
󰇛󰇜

(28)
(2) Case 2: when ,
󰇝󰇞󰇝󰇞󰇝󰇞󰇝󰇞󰇝󰇞



(29)
The results of Cases 1 and 2 show that 󰇝󰇞.
Furthermore, Eq.(30) can be obtained by substituting Eqs. (24)–(27) into Eq. (23).








 

󰇛󰇜
󰇛󰇜 
(30)
Let 


 and . Eq. (30) can be converted to Eq. (31).








 
󰇧
󰇛󰇜
󰇛󰇜󰇨 
(31)

16
Let 
 and 󰇛󰇜
; Eq.(31) can be rewritten as
󰇱

 
󰇛󰇜 
(32)
Remark 1.When the mixed traffic flow does not contain AVs, that is  . In this scenario, Eq. (32) is
consistent with the result when the platooning intensity was 0 in Zhou and Zhu (2021), in which the AVs
were not considered. This implies that the capacity model of Zhou and Zhu (2021) is a special case with 
. Furthermore, when both the maximum platoon size and AVs were not considered, that is, , 
, and , Eq.(32) is consistent with the results of Ghiasi et al. (2017) when the CAVs clustering strength
is random. Therefore, the capacity formulation in Ghiasi et al. (2017) can be considered a special case with
 and .
Two key parameters,  and  are the proportions of headways  and  in the mixed traffic
flow, respectively. To facilitate the subsequent analysis, we first deduce and prove the relevant properties of
 and .
Lemma 1. The sum of  and  is only related to  and is independent of . The proof is provided in Appendix
A.
Lemma 2. When  is given,  is an increasing function of . The proof is provided in Appendix B.
Corollary 1. 
 for any .
Lemma 3. When  is given,  is an increasing function of . The proof is provided in Appendix C.
Corollary 2. 
  for any 󰇟󰇠.
Lemma 4. When  is given,  is an increasing function of . The proof is provided in Appendix D.
Corollary 3. 
 for any .
Lemma 5. When  is given,  is a decreasing function of . The proof is provided in Appendix E.
Corollary 4. 
  for any 󰇟󰇠.
Based on Eq.(32), some propositions of capacity  can be obtained as follows:
Lemma 6. When  or  is given, capacity  is an increasing function of .
Proof:
First, the first derivative of the capacity  to  can be obtained by deriving Eq. (32) to , as shown in
Eq. (33).

󰇡󰇢
󰇡󰇛󰇜󰇢

(33)
Thus, there are two cases, where





is given. Therefore, we discuss the two cases separately.
(1) Case 1: When  is given, we obtain . Therefore, Eq. (33) can be rewritten as

󰇡󰇛󰇜󰇛󰇜󰇢

󰇡󰇛󰇛󰇜󰇜󰇢




(34)
Based on Corollary 1, we know that 
 . Furthermore, 

 based on  .
Therefore, Eq. (34) can be rewritten as

17







󰇧󰇛󰇜
󰇨
(35)
Subsequently, we obtain Eq. (36) by substituting Eq. (A.1) with Eq. (35).

󰇧󰇛󰇜
󰇨󰇛󰇜
(36)
where ,



, and . Thus,


󰇛󰇜.
(2) Case 2: When  is given, we obtain . Therefore, Eq. (33) can be rewritten as in Eq.
(37).

󰇡󰇛󰇜󰇢




(37)
Based on Corollary 1, we know that 
 . Furthermore, 

 based on  .
Therefore, Eq.(37) can be rewritten as







󰇧󰇛󰇜
󰇨
󰇛󰇛󰇜󰇜
(38)
where ,



, and . Thus,


󰇛󰇛󰇜󰇜.
In summary, 
. This means that capacity  is an increasing function of  when  or  is
given. This completes the proof. □
Lemma 7. Capacity  is an increasing function of  when  is given, but is a decreasing function of  when
 is given.
Proof:
First, the first derivative of the capacity  to  can be obtained by deriving Eq. (32) to , as shown in
Eq. (42).

󰇡󰇢
󰇡󰇛󰇜󰇢

(39)
Thus, there are two cases, where





is given. Therefore, we discuss the two cases separately.
(1) Case 1: When  is given, we obtain . Therefore, Eq. (39) can be rewritten as

󰇛󰇜
󰇛󰇜󰇛󰇜
(40)

18
where , , and . Thus, 
󰇛󰇜.
(2) Case 2: According to Lemma 6, we know that capacity  is an increasing function of  when 
is given. Furthermore, the relationship between , , and  is . This indicates
that  decreases as  increases. Therefore, capacity  is a decreasing function of  when 
is given.
In conclusion, capacity  is an increasing function of  when  is given, but it is a decreasing
function of  when  is given. This completes the proof. □
Lemma 8. When  or  is given, capacity  is a decreasing function of .
Proof:
The proof can be divided into two cases.
(1) Case 1: When  is given, the relationship between  ,  , and  is  . This
indicates that  decreases with an increase in . Moreover, according to Lemma 6, capacity  is
an increasing function of  when  is given. Therefore, capacity  is a decreasing function of 
when  is given.
(2) Case 2: When  is given,  decreases with an increase in . According to Lemma 7, capacity 
is an increasing function of  when  is given. Therefore, capacity  is a decreasing function of
 when  is given.
In summary, capacity  is a decreasing function of  when  or  is applied. This completes the
proof. □
Lemma 9. Capacity  is an increasing function of the maximum platoon size of CAVs ().
Proof:
The proof can be divided into two parts,  and .
(1) Case 1: When , based on Eq. (32), Eq. (44) can be obtained as

󰇡󰇢
 󰇧󰇡

󰇢󰇨
 
󰇛󰇜
󰇛󰇜
(41)
where  and , therefore, 

󰇛󰇜.
(2) Case 2: When ,

󰇡󰇢
 󰇡󰇛󰇜󰇢
 
(42)
Based on Corollary 2, we know that 
 . Furthermore, we have 
 
  based on .
Therefore, Eq. (45) can be rewritten as in Eq. (46).

󰇡󰇛󰇜󰇢

󰇡󰇛󰇜󰇛󰇜󰇢

󰇡󰇛󰇜󰇢
 
(43)
Therefore, both Eqs. (44) and (46) show that 
. This indicates that the capacity  is an increasing
function of . This completes the proof. □

19
5.2. Validation and analysis
According to existing research (Ghiasi et al., 2017; Mirzaeian et al., 2021; Zhou and Zhu, 2021), traffic
capacity is only related to the time headway of the following vehicle, and not to car-following models.
Therefore, we used the numerical simulation method to randomly generate different types of vehicles, and
obtain the traffic capacity in various scenarios by calculating the average time headway. In this study, we did
not consider the impact of lane-changing behavior on traffic capacity (Section 3.3, Assumption 1). Therefore,
the simulation here used a circular single-lane highway, which contains many different types of vehicles as
an example. The headways can be obtained based on the input of the relevant parameters, and the simulation
estimation value of the traffic capacity in this scenario were obtained.
In this study, the headways of all simulation scenarios were constant, which were 






 (Chen et al., 2017a; Milanes et al., 2014; Milanés and Shladover, 2014; Zhou
and Zhu, 2021). To avoid the influence of randomness, we randomly simulated the same scenario 100 times;
and took the average value as a result. The parameter settings for the different scenarios are listed in Table 4.
Fig. 4 to Fig. 7 present the comparison between the simulation and theoretical results under five scenarios.
Table 4. Parameter settings of different scenarios
Scenarios




1
0-0.8
0.2

2,4,6,8,10
2
0-0.8

0.2
2,4,6,8,10
3
0.5
0-0.5

2,4,6,8,10
4

0-0.5
0.5
2,4,6,8,10
5
0-1
󰇛󰇜
󰇛󰇜

Table 4 reports the parameter settings of five scenarios. Scenarios 1 and 2 were set to explore the impact
of the penetration rate of CAVs on the traffic capacity when  or  is fixed. The result is shown in Fig. 4.
Scenarios 3 and 4 were set to discuss the impact of the penetration rate of AVs on the traffic capacity when
 or  is fixed. The result is presented in Fig. 5. Scenarios 5 and 6 were set to analyze the impact of the
penetration rate of HDVs on the traffic capacity when  or  is fixed. The result is shown in Fig. 6. The
impact of maximum platoon size on traffic capacity was analyzed in scenario 5 and is shown in Fig. 7.
The results in Fig. 4 to Fig. 7 show that the theoretical results were highly consistent with the simulation
results, suggesting the effectiveness of the theoretical model in this paper.
(a) 
(b) 
Fig. 4. Impact of the penetration rate of CAVs on traffic capacity.
Fig. 4 shows that when the penetration rate of CAVs was low, such as less than 0.2, the maximum platoon
size had a minimal impact on the traffic capacity. With an increase in the penetration rate of CAVs, the effect
of the maximum platoon size on improving traffic capacity was more prominent. This is because when the

20
penetration rate of CAVs is high, more CAVs exist in the mixed traffic flow. Currently, with the increase in
the maximum platoon size, the number of the most significant platoons increases, resulting in an increase in
traffic capacity. Moreover, when the penetration rate of CAVs is given, the capacity increases with an increase
in the maximum platoon size, but the increase gradually slows. This indicates that when the platoon size
reaches a certain level, blindly increasing the platoon size will not result in a good gain, but will impact the
stability of traffic flow (Talebpour and Mahmassani, 2016). Fig. 4 (a) reveals that when  was given, the
traffic capacity increased exponentially with an increase in the penetration rate of CAVs. Similarly, when 
was given, the same result was observed between the traffic capacity and penetration rate of CAVs (Fig. 4
(b)). Therefore, the traffic capacity increases with an increase in the penetration rate of CAVs when  or 
is given. This result is consistent with Lemma 6.
(a) 
(b) 
Fig. 5. Impact of the penetration rate of AVs on traffic capacity.
(a) 
(b) 
Fig. 6. Impact of the penetration rate of HDVs on traffic capacity.
Fig. 5 (a) shows that the traffic capacity increased linearly with an increase in the penetration rate of AVs
when  was given. Moreover, the capacity was consistent with the increase in the maximum platoon size
under different AV penetration rates. This is because the platoon size is dependent only on the CAVs, and
the penetration rate of CAVs in this scenario was a fixed value of 0.5. Instead, when  was given, the traffic
capacity decreased with an increase in the penetration rate of AVs (Fig. 5 (b)). Therefore, the results in Fig. 5
were consistent with Lemma 7. In addition, the impact of the maximum platoon size on traffic capacity
decreased gradually with an increase in the penetration rate of AVs. The analysis indicates that in this
scenario,  was fixed, and  gradually decreased with an increase in . When  is small, it means
that there are fewer CAVs in the mixed traffic flow; thus, forming a large platoon is difficult. Therefore, setting
the maximum platoon size in this scenario did not affect the traffic capacity.

21
Fig. 6 (a) shows that when  was given, traffic capacity increased linearly with the penetration rate of
HDVs. In contrast, when  was given, the traffic capacity decreased with the penetration rate of HDVs (Fig.
6 (b)). This result is consistent with Lemma 8.
Fig. 7. Impact of maximum platoon size on traffic capacity.
Fig. 7 presents that when the penetration rate of CAVs was low, the maximum platoon size had a slight
impact on capacity. When the penetration rate of CAVs was less than 0.4, the traffic capacity remained
basically unchanged, with the maximum platoon size being more significant than 5. The analysis indicated
that when the penetration rate of CAVs was low, the mixed traffic flow contained only a few CAVs, which
were randomly distributed; therefore, it was difficult to form a large platoon. Therefore, increasing the
maximum platoon size cannot improve traffic capacity. Instead, when the penetration rate of CAVs is
significant, an increase in the maximum platoon size will increase the traffic capacity to a certain extent. The
penetration rate was 0.8, and the traffic capacity improved when the maximum platoon size was less than 15.
However, when the maximum platoon size was greater than 15, increasing the maximum platoon could not
further improve the traffic capacity. Moreover, when the penetration rate of CAVs was 1, the traffic capacity
was close to 6000 veh/h/lane, which was consistent with the results of other studies (Qin et al., 2021; Zhou
and Zhu, 2021, 2020), and the capacity increased with the increase in maximum platoon size. This indicated
that both the penetration rate of CAVs and the maximum platoon size determine traffic capacity. Compared
with the maximum platoon size, the effect of the penetration rate of CAVs was more pronounced. In summary,
the traffic capacity increases with an increase in the maximum platoon size, which is consistent with Lemma
9.
6. Stability
This section first introduces three car-following models to describe the car-following behaviors of CAVs,
AVs, and HDVs. Based on this, the stability conditions of the mixed traffic flow are derived. Finally, the effects
of relevant parameters on the stability of mixed traffic flow are analyzed, such as the penetration rate of CAVs
and AVs, and the maximum platoon size.
6.1. Car-following models
In this paper, car-following models are adopted to describe car-following behavior between vehicles. In
addition, we do not consider the randomness and uncertainty of different drivers; that is, we use unified

22
parameters to describe the same car-following behavior. To discuss the influence of the randomness of car-
following models, relevant research has been conducted (F. Zheng et al., 2020b; Zhou and Zhu, 2020).
6.1.1 Connected automated vehicles
The CACC model proposed by Milanes et al. (2014; 2014) is adopted to capture the car-following
characteristics of CAVs. The CACC model is
󰇛󰇜󰇛󰇜󰇛󰇜󰇗󰇛󰇜
(44)
󰇛󰇜󰇛󰇜󰇛󰇜
(45)
where  and  are the gains to adjust the time-gap error with respect to the preceding vehicle, and the
recommended values of the experiment were  and  (Milanes et al., 2014;
Milanés and Shladover, 2014). In this paper, if a CAV is in a platoon and is not the leader, the time gap of the
intra-platoon car following is  (Milanes et al., 2014; Milanés and Shladover, 2014). Unlike the intra-
platoon, if a CAV follows a maximum-size CAVs platoon, the inter-platoon car following is collision-free and
has a larger time gap, , , and  (Liu et al., 2018b; Xiao et al., 2018; Zhou
and Zhu, 2021).
The acceleration can be obtained using the first-order Taylor expansion of Eqs.(44) and (45),
󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜
 
(46)
where  is the iteration time step and is set to 0.01 s.
6.1.2 Automated vehicles
An ACC model based on experimental data (Milanes et al., 2014; Milanés and Shladover, 2014) is adopted
to model the car-following behavior of AVs and degraded CAVs (i.e., a CAV follows an HDV or AV). The
control model is expressed as follows:
󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜
(47)
where  and  are the gains on the positioning and speed errors, respectively, and are set to 
 and , the time gap

 is set to  (Milanes et al., 2014; Milanés and Shladover,
2014; Zhou and Zhu, 2021).
6.1.3 Human-driven vehicles
The IDM developed by Treiber et al. (2000) describes the car-following behavior of HDVs. The car-
following model is
󰇛󰇜





󰇧󰇛󰇜
󰇨󰇛󰇜󰇛󰇜󰇛󰇜

󰇛󰇜






(48)
where  is the free-flow speed, and .  is the minimum distance at a standstill, and is set to
.  and  are the maximum acceleration and the desired deceleration, respectively, and are set to
 and , respectively, and  is set to  (Ngoduy, 2013; Treiber et al., 2000).
6.2. Stability analysis
Traffic flow stability is an important index for measuring the smooth operation of traffic systems in
response to disturbances (Montanino and Punzo, 2021). For a car-following platoon in equilibrium on the
road, if one or more vehicles change their driving behavior, such as acceleration and deceleration, sudden

23
braking, lane change, etc., this change will affect the driving behavior of the vehicles behind and cause vehicle
speed fluctuation in the traffic flow. If the amplitude of this fluctuation gradually increases, it evolves into
the instability of the platoon system. In contrast, the system is stable if it decays, tends to zero over time, and
finally returns to equilibrium.
Generally, a system can be divided into nonlinear and linear stability analyses based on the magnitude
of the disturbance. Nonlinear stability analysis focuses on the stability characteristics under the influence of
considerable disturbances. Most researchers use the perturbation method to deduce the shock wave equation
of micro or macro traffic flow models, to describe the characteristics of traffic waves. The derivation
conditions of the nonlinear stability analysis are relatively strict and complex. Compared with linear stability,
nonlinear stability is often not analytical. Therefore, the stability analysis in this study refers to linear stability
(Sun et al., 2020; Yao et al., 2021b).
6.2.1 Theoretical analysis
6.2.1.1. Homogeneous traffic flow
The linear stability condition of homogeneous traffic flow (Ward, 2009) can be expressed as

󰇛󰇜

(49)
where,





󰇛󰇜

󰇛󰇜

󰇛󰇜
 
(50)
Note that existing research has proved that a time lag (e.g., sensor delay, communication delay, reaction
time, etc.) is not conducive to traffic flow stability (Ngoduy, 2013; Wang, 2018; Yao et al., 2021b). Considering
that this paper primarily focuses on the influence of the maximum platoon size on the stability of traffic flow,
the influence of the time lag on the stability of traffic flow is not investigated. As a result,


Therefore, the stability discriminant of the three car-following models can be obtained by substituting Eqs.
(46)–(48) into Eqs. (49) and (50):




(51)
where


󰇟󰇛󰇜󰇠



󰇟󰇛󰇜󰇠
󰇛󰇜and
󰇟󰇛󰇜󰇠󰇛󰇜





󰇛󰇜󰇛󰇜
(52)




󰇧
󰇨󰇧
󰇨


(53)
where  is the stability discriminant of the CAVs traffic flow, regardless of the maximum platoon size (i.e.,
intra-platoon), that is . If , we define  as the stability discriminant of the CAVs traffic
flow when the maximum platoon size is one (i.e., intra-platoon).
Based on Eqs. (51)–(53), we can obtain the stability of homogeneous traffic flow, as shown in Fig. 8.

24
Fig. 8. Stability analysis of homogeneous traffic flow.
Fig. 8 shows the variation in homogeneous traffic flow stability with speed. If the value is greater than 0,
homogeneous traffic flow is stable at this speed. From Fig. 8, we can observe that when the speed is between
1.2 and 21.5 m/s, the stability discrimination value  of HDVs traffic flow is less than 0. This means that
there is an unstable speed range for HDVs traffic flow. For AVs traffic flow, the stability discrimination value
, which is independent of speed and constant at less than 0. Therefore, the AVs traffic flow is
unstable. In addition, Fig. 8 shows that the stability discrimination values  and .
This means that the CAVs traffic flow is stable, and independent of the speed and maximum platoon size. In
summary, the application of CAVs can improve traffic flow stability; however, the application of AVs is not
conducive to traffic flow stability (Yao et al., 2020a, 2019). The impact of the two types of vehicles (i.e., CAVs
and AVs) on the stability of mixed traffic flow is analyzed in the next section.
6.2.1.2. Mixed traffic flow
For mixed traffic flow, Ngoduy (2013) proposed a discrimination formula for stability based on (Ward,
2009). However, Montanino and Punzo (2021) observed that the condition in (Ward, 2009) overestimated
string instability in heterogeneous traffic owing to its uniformity assumption. The stability condition in
(Ward, 2009) can be considered a weak condition. Considering its simple structure, it can capture the stability
characteristics of a mixed traffic flow, and the error is within 11% (Montanino and Punzo, 2021). Therefore,
a weak condition of this stability is adopted in this paper, as shown in Eq.(54).
󰇩󰇛󰇜


󰇪

 

(54)
where  is the vehicle index;  , 
 , and 
 are the partial differentiations of the acceleration
calculation equation for the vehicle speed, speed difference, and space headway, respectively.
According to the analysis in Section 4.1, the mixed traffic flow examined in this paper includes four car-
following modes. Eq. (55) can be obtained by substituting the probabilities of the four types of car-following
modes (i.e., Eqs. (24), (25), (26), and (27)) into Eq.(54).

25




󰇣



󰇤
󰇛󰇜



󰇣



󰇤
󰇛󰇜


󰇣



󰇤



󰇣



󰇤
(55)
where  is the number of vehicles in the mixed traffic flow, that is, .
Considering that  
, 
, 
, and 
, Eq. (55) can be reduced to
󰇛󰇜
(56)
where  is the discriminant value of mixed traffic flow stability, and














































(57)
Proposition 2.  is a decreasing function of maximum platoon size  with a given penetration rate of AVs and CAVs.
This indicates that when the penetration rate of AVs and CAVs are fixed, the larger the maximum platoon size of CAVs,
the worse the traffic flow stability.
Proof:
Based on Eq. (57), we know that  and  are independent of the maximum platoon size of CAVs.
This means that when penetration rates of AVs and CAVs are fixed, the values of 󰇛󰇜 and
 are also fixed. As a result, the change in  is related only to  and . Therefore, we only
require to discuss the influence of the maximum platoon size  on  and .
Firstly, we can calculate  and . 






 and 







, . Subsequently, we know that  is an increasing function of the maximum platoon
size  based on Lemma 3. Instead,  is a decreasing function of the maximum platoon size  based on
Lemma 5. Therefore, when the maximum platoon size  increases,  increases and  decreases.
Furthermore, because of ,  decreases. In summary,  is a decreasing function of the maximum
platoon size of CAVs. This completes the proof. □
Based on Eq. (56), we further discuss the impact of different parameters on the stability of mixed traffic
flow. The impact factors include the maximum platoon size and penetration rate of CAVs and AVs, as shown
in Fig. 9, Fig. 10, and Fig. 11.

26
(a) 
(b) 
(b) 
Fig. 9. Stability analysis of mixed traffic flow.
Fig. 9 shows that the stability of mixed traffic flow under different parameters, such as maximum platoon
size and the penetration rate of AVs and CAVs. Fig. 9 (a) depicts the stability of mixed traffic flow under
different maximum platoon sizes of CAVs, when  and . As shown in Fig. 9 (a), the stability
of mixed traffic flow gradually becomes worse with the increase in the maximum platoon size of CAVs. This
suggests that although the CAVs platoon can improve the capacity, it is not conducive to the stability of
mixed traffic flow. Fig. 9 (b) shows that the influence of the penetration rate of AVs on the stability of mixed
traffic flow. From Fig. 9 (b), we can observe that the stability of mixed traffic flow gradually deteriorates with
the increase in the penetration rate of AVs. This means AVs cannot improve the stability of mixed traffic flow
and have a negative impact on the stability. This is consistent with previous research findings (Wang et al.,
2019; Yao et al., 2020a). Fig. 9 (c) shows that when the maximum platoon size of CAVs is set to 4, the stability
of mixed traffic flow becomes gradually better with the increase in the penetration rate of CAVs, but the
unstable speed range becomes more considerable. When the penetration rate of CAVs is 1, the stability of
traffic flow will not be affected by speed and will always be in a stable state. In conclusion, CAVs are
conducive to the stability of mixed traffic flow, while the maximum platoon size and AVs will worsen the
stability of mixed traffic flow.
To further investigate the influence of the penetration rate of CAVs, maximum platoon size, and
equilibrium speed on the stability of the mixed traffic flow, Fig. 10 and Fig. 11 are obtained.
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
Fig. 10. Impact of the penetration rate of CAVs on stability.

27
Fig. 10 compares the stability of the mixed traffic flow under different penetration rates of CAVs and the
equilibrium speed. The vertical axis in Fig. 10 shows the penetration rate of CAVs, while the horizontal axis
represents the equilibrium speed. The colors in Fig. 10 are the G values of the stability of the mixed traffic
flow. Fig. 10 shows that when the maximum platoon size of CAVs is given, the unstable speed range of mixed
traffic flow is gradually expressed with an increase in the penetration rate of CAVs. The analysis showed that
this is because the functional degradation of CAVs is considered in this paper; that is, CAVs degenerate into
AVs in some scenarios. With an increase in the CAVs penetration rate, the AVs gradually increase. Therefore,
the unstable speed range of mixed traffic flows gradually becomes more extensive. However, Fig. 10 (a)
shows that when the penetration rate of CAVs is more significant than 0.8, no unstable speed region occurs
in the mixed traffic flow. This means that when the penetration rate of CAVs reaches a high level, the
degraded CAVs gradually decrease, and the mixed traffic flow returns to a stable state. Fig. 10 (a)–(f) show
that the area covered by the black curve gradually becomes more prominent with the increase of the
maximum platoon size of CAVs. The black curve represents the baseline of the stability of the mixed traffic
flow, i.e.,. Therefore, the phenomenon indicates that the stability of the mixed traffic flow gradually
weakens with an increase in the maximum platoon size. This result is consistent with the findings shown in
Fig. 9 (a).
(a) 
(b) 
(c) 
(d) 
(e) 
(f) 
(g) 
(h) 
(i) 
Fig. 11. Impact of maximum platoon size on stability.
Fig. 11 shows the impact of the maximum platoon size on the mixed traffic flow stability under different
penetration rates of CAVs. As shown in Fig. 11, when the penetration rate of CAVs is low, there are two stable
speed intervals (i.e., Fig. 11 (a) and (b)) for mixed traffic flow, and when the penetration rate of CAVs is
significant, there is only one stable speed interval (Fig. 11 (c) to (i)). Moreover, Fig. 11 (a)–(d) show that when

28
the penetration rate of CAVs is low, the vertical line in the figures is straight. This indicates that the maximum
platoon size has a minimal impact on the stability of the mixed traffic flow when the penetration rate of CAVs
is low. However, Fig. 11 (e)–(i) show that the vertical line in the figure begins to bend, and bending occurs
only at a small maximum platoon size. This means that the maximum platoon size has an impact on the
stability of the mixed traffic flow when the CAV penetration rate is high, but the impact is more noticeable
when the maximum platoon size is small. This is because forming a large platoon when vehicles are
randomly distributed in the mixed traffic flow is difficult. Therefore, setting a large maximum platoon has
no effect on the actual platoon size. Moreover, Fig. 11 (h)–(i) show that the stability value increases
significantly when the penetration rate of CAVs is close to 1. This means that large-scale application of CAVs
can significantly improve the stability of mixed traffic flows.
Fig. 10 and Fig. 11 also show that when the speed is high, such as greater than 20 m/s, the mixed traffic
flow will be in a stable state, which is independent of speed. This indicates that the mixed traffic flow is more
stable at higher speeds.
6.2.2 Simulation analysis
In this section, we describe the design of a numerical simulation to analyze the stability of mixed traffic
flow. The basic concept of the numerical simulation is to investigate whether the mixed traffic flow could
recover to the steady state and recovery process by adding small fluctuations to the steady-state traffic flow.
First, to study the influence of the speed change of the leader car on the car following vehicles, we run
the simulation under the open boundary condition. One hundred vehicles were randomly generated and
distributed on a one-lane freeway in the simulations (Wang et al., 2019). The car-following model of each
vehicle was determined according to the penetration rate of CAVs and AVs, maximum platoon size, and other
parameters. Subsequently, we obtained the distance between all vehicles at a steady-state speed based on the
car-following models, which are presented in Section 6.1. Finally, we changed the speed of the first vehicle
to break this steady state, and studied the process of restoring the steady state of the mixed traffic flow (Sun
et al., 2018; Zhou and Zhu, 2021).
The duration of the simulation was 300 s. The leader vehicle decelerated from 60 to 63 s and then returned
to steady-state speed after acceleration (Sun et al., 2018; Zhou and Zhu, 2021). This process can be described
by Eq.(58). To investigate the stability of mixed traffic flow under different traffic conditions, we set three
steady-state speeds, , , and , representing the congestion, normal, and
free-flow scenarios, respectively.
󰇛󰇜 


 
(58)
where 󰇛󰇜 is the speed of the leader vehicle at time .
Based on the simulation parameters, two experiments were designed to investigate the effects of
different parameters on the stability of mixed traffic flow.
(1) The penetration rate of AVs was constant, and the effects of the penetration rate of CAVs and
maximum platoon size on mixed traffic flow were studied. The relevant parameters of the scene were
set as follows:   , and  . The relevant results for this
scenario are shown in Fig. 12, Table 5, Table 6, and Table 7.
(2) The penetration rate of CAVs was constant, and the effects of the penetration rate of AVs and
maximum platoon size on mixed traffic flow were studied. The relevant parameters of the scene were
set as follows:  , and . The relevant results for this scenario
are shown in Fig. 13, Table 8, Table 9, and Table 10.

29
Fig. 12. Speeds and accelerations of vehicles in the simulation. ()
Fig. 12 shows the speed and acceleration of all vehicles over time when the equilibrium speed was 20
km/h. As shown in Fig. 12, the time required for the vehicle speed to return to the equilibrium speed is
differed for different maximum platoon sizes. When the maximum platoon sizes of CAVs were 2, 4, 6, and 8,
the recovery times were 210, 230, 240, and 245 s, respectively. Therefore, the speed recovery time gradually
increased with an increase in the maximum platoon size. This suggested that the stability of the mixed traffic
flow gradually deteriorated with an increase in the maximum platoon size of CAVs. Moreover, Fig. 12 shows
the acceleration fluctuation of the vehicles with time. When the maximum platoon sizes were 2 and 4, the
acceleration was stable (i.e., zero) at approximately 200 s, but the time increased to 240 s when the maximum
platoon sizes were 6 and 8. This phenomenon also indicated that a larger platoon size affects the stability of
mixed traffic flow. These conclusions are consistent with the theoretical research results presented in Section
6.2.1.
Table 5. Mean and deviation of speeds ().


Mean (m/s)
Standard deviation (m/s)






0.2
2
9.9606
19.9695
29.9925
0.8038
0.1458
0.0418
4
9.9606
19.9695
29.9925
0.8038
0.1458
0.0418
8
9.9606
19.9695
29.9925
0.8038
0.1458
0.0418
16
9.9606
19.9695
29.9925
0.8038
0.1458
0.0418
32
9.9606
19.9695
29.9925
0.8038
0.1458
0.0418
0.4
2
9.9692
19.9705
29.9908
0.8508
0.1521
0.0415
4
9.9700
19.9705
29.9908
0.9824
0.1586
0.0415
8
9.9700
19.9705
29.9908
0.9824
0.1586
0.0415
16
9.9700
19.9705
29.9908
0.9824
0.1586
0.0415
32
9.9700
19.9705
29.9908
0.9824
0.1586
0.0415
0.6
2
9.9698
19.9699
29.9847
0.7770
0.2474
0.0663
4
9.9701
19.9701
29.9846
1.1078
0.2936
0.0674
8
9.9701
19.9701
29.9845
1.2080
0.3007
0.0674
16
9.9701
19.9701
29.9845
1.2080
0.3007
0.0674
32
9.9701
19.9701
29.9845
1.2080
0.3007
0.0674
0.8
2
9.9701
19.9703
29.9738
0.1472
0.1231
0.0676
4
9.9701
19.9703
29.9734
0.2768
0.1590
0.0711
8
9.9699
19.9702
29.9733
0.4172
0.1896
0.0720
16
9.9699
19.9702
29.9732
0.4879
0.2053
0.0725
32
9.9699
19.9702
29.9732
0.4879
0.2053
0.0725

30
Furthermore, based on Fig. 12, we investigated the effects of different CAVs penetration rates and
maximum platoon size on the stability of mixed traffic flow. The average and standard deviations of the
speed, speed variation, and acceleration are shown in Table 5, Table 6, and Table 7, respectively.
Table 6. Mean and deviation of accelerations ().


Mean ( m/s)
Standard deviation (m/s)






0.2
2
0.8609
0.0838
-0.0285
0.1446
0.0232
0.0171
4
0.8609
0.0838
-0.0285
0.1446
0.0232
0.0171
8
0.8609
0.0838
-0.0285
0.1446
0.0232
0.0171
16
0.8609
0.0838
-0.0285
0.1446
0.0232
0.0171
32
0.8609
0.0838
-0.0285
0.1446
0.0232
0.0171
0.4
2
-0.2244
-0.0210
-0.0358
0.1667
0.0258
0.0188
4
-0.3020
-0.0251
-0.0348
0.1903
0.0264
0.0188
8
-0.3020
-0.0251
-0.0348
0.1903
0.0264
0.0188
16
-0.3020
-0.0251
-0.0348
0.1903
0.0264
0.0188
32
-0.3020
-0.0251
-0.0348
0.1903
0.0264
0.0188
0.6
2
-0.0178
-0.0015
-0.0565
0.1788
0.0495
0.0235
4
-0.0185
-0.0006
-0.0566
0.2386
0.0574
0.0238
8
-0.0179
-0.0006
-0.0565
0.2541
0.0583
0.0238
16
-0.0179
-0.0006
-0.0565
0.2541
0.0583
0.0238
32
-0.0179
-0.0006
-0.0565
0.2541
0.0583
0.0238
0.8
2
-0.0035
-0.0018
-0.0613
0.0346
0.0295
0.0238
4
0.0007
0.0003
-0.0633
0.0679
0.0360
0.0245
8
0.0008
0.0014
-0.0585
0.1082
0.0423
0.0247
16
0.0028
0.0017
-0.0605
0.1288
0.0459
0.0247
32
0.0028
0.0017
-0.0605
0.1288
0.0459
0.0247
Table 7. Mean and deviation of speed variation ().


Mean (m/s)
Standard deviation (m/s)






0.2
2
0.1042
0.0130
0.0023
0.2612
0.0437
0.0337
4
0.1042
0.0130
0.0023
0.2612
0.0437
0.0337
8
0.1042
0.0130
0.0023
0.2612
0.0437
0.0337
16
0.1042
0.0130
0.0023
0.2612
0.0437
0.0337
32
0.1042
0.0130
0.0023
0.2612
0.0437
0.0337
0.4
2
0.1045
0.0122
0.0018
0.2862
0.0369
0.0214
4
0.1184
0.0129
0.0018
0.3282
0.0384
0.0214
8
0.1184
0.0129
0.0018
0.3282
0.0384
0.0214
16
0.1184
0.0129
0.0018
0.3282
0.0384
0.0214
32
0.1184
0.0129
0.0018
0.3282
0.0384
0.0214
0.6
2
0.1166
0.0268
0.0043
0.2871
0.0878
0.0413
4
0.1549
0.0327
0.0045
0.3864
0.1028
0.0419
8
0.1628
0.0334
0.0045
0.4091
0.1040
0.0419
16
0.1628
0.0334
0.0045
0.4091
0.1040
0.0419
32
0.1628
0.0334
0.0045
0.4091
0.1040
0.0419
0.8
2
0.0098
0.0061
0.0027
0.0371
0.0293
0.0206
4
0.0249
0.0102
0.0029
0.0822
0.0392
0.0225
8
0.0390
0.0139
0.0030
0.1330
0.0479
0.0228
16
0.0464
0.0159
0.0030
0.1598
0.0530
0.0231
32
0.0464
0.0159
0.0030
0.1598
0.0530
0.0231
Note: Speed variation between two adjacent vehicles is defined as the absolute value of the speed difference between
the latter and the former vehicles. This indicator is similar to the existing study (Zhou et al., 2022) and can reflect the
stability of traffic flow.

31
Table 5 and Table 6 report that the standard deviation of speeds and accelerations changed slightly with
maximum platoon sizes when the penetration rate of CAVs was low. For example, Table 5 shows that when
 was 0.2, the speed standard deviations under three different equilibrium velocities were 0.08038, 0.1458,
and 0.0418 m/s, respectively. However, when  was 0.8, the variation ranges of the speed standard
deviation were [0.1472, 0.4879] m/s, [0.1231, 0.2053] m/s, and [0.0676, 0.0725] m/s, respectively. This meant
that the maximum platoon size of CAVs had a slight impact on the stability of the mixed traffic flow when
the penetration rate of CAVs was low. The analysis showed that when the penetration rate of CAVs was low,
forming a large-scale CAV platoon was difficult. Therefore, the maximum platoon size did not restrict the
CAVs platoon. In contrast, when the penetration rate of CAVs was high, the maximum platoon size became
one of the factors affecting the stability of the mixed traffic flow. Moreover, the standard deviations of the
speed and acceleration decreased with an increase in the equilibrium speed, as shown in Table 5 and Table 6.
This indicated that when the speed of mixed traffic flow was high, the stronger the anti-interference, the
easier it was to stabilize, which was consistent with our theoretical analysis in Fig. 10 and Fig. 11. In addition,
Table 5 presents that when  was given, the average speed decreased with an increase in the maximum
platoon size. This phenomenon was more apparent at a high penetration rate of CAVs and high speed (i.e.,
 , and  ). This suggested that under the high speed and penetration rate of CAVs, the
influence of the maximum platoon size on the stability of mixed traffic flow is more apparent.
Table 7 shows the speed variation between the preceding and following vehicles. When the penetration
rate of CAVs was given, the average and standard deviation of vehicle speed variation increased with an
increase in the maximum platoon size. This suggested that the larger the maximum platoon size, the worse
the traffic flow stability. This is also consistent with the conclusions presented in Table 5 and Table 6. It is
worth noting that when the penetration rate of CAVs was less than 0.6, the vehicle speed variation increased
with an increase in the penetration rate of CAVs. However, when the penetration rate of CAVs reached 0.8,
the vehicle speed variation decreased. The analysis showed that when the penetration rate of CAVs reached
0.8, they had a dominant role in the mixed traffic flow. Compared with HDVs, CAVs can maintain lower
speed variations. This was consistent with the original intention of the CACC design (Milanes et al., 2014;
Milanés and Shladover, 2014).
Fig. 13. Speeds and accelerations of vehicles in the simulation. ()
Fig. 13 shows that when the maximum platoon size was given, the fluctuation of speed and acceleration
is gradually increased with an increase in the penetration rate of AVs. This means that the application of AVs
deteriorates the stability of mixed traffic flow, which is consistent with the theoretical analysis in Section
6.2.1.2. Table 8 and Table 9 report that the deviation in speed and acceleration increased with an increase in
the penetration rate of AVs. This indicated that the stability of the traffic flow worsened with an increase in
AVs permeability. Moreover, Table 8 and Table 9 show that the deviation of speed and acceleration decreased
with the increase of the equilibrium speed of the traffic flow. This means that, compared with low-speed
traffic flow, high-speed traffic flow has better stability, which is consistent with the theoretical analysis
shown in Fig. 10 and Fig. 11.
Moreover, Table 10 shows that when the penetration rate of AVs was given, the average and standard
deviation of the vehicle speed variation increased with an increase in the maximum platoon size. This meant

32
that the larger the maximum platoon size, the worse the traffic flow stability. Unlike Table 7, Table 10 reports
that the vehicle speed variation increased with an increase in the penetration rate of AVs. This is because AVs
are not conducive to the stability of traffic flow. This conclusion is consistent with previous studies (Wang et
al., 2019; Yao et al., 2020a, 2019).
Table 8. Mean and deviation of speeds ().


Mean (m/s)
Standard deviation (m/s)






0
2
9.9700
19.9706
29.9844
0.1790
0.1182
0.0565
4
9.9698
19.9704
29.9842
0.2651
0.1344
0.0579
8
9.9699
19.9704
29.9842
0.2980
0.1394
0.0580
16
9.9698
19.9704
29.9841
0.3155
0.1427
0.0581
32
9.9698
19.9704
29.9841
0.3155
0.1427
0.0581
0.1
2
9.9700
19.9703
29.9789
0.3683
0.1527
0.0581
4
9.9699
19.9702
29.9786
0.7614
0.2089
0.0595
8
9.9698
19.9702
29.9786
0.9085
0.2295
0.0597
16
9.9699
19.9702
29.9785
0.9795
0.2420
0.0597
32
9.9699
19.9702
29.9785
0.9795
0.2420
0.0597
0.2
2
9.9710
19.9700
29.9742
0.8312
0.4120
0.0685
4
9.9699
19.9699
29.9739
3.1865
0.9811
0.0724
8
9.9698
19.9704
29.9739
4.2552
1.1822
0.0734
16
9.9698
19.9706
29.9738
4.2983
1.2294
0.0741
32
9.9699
19.9706
29.9738
4.3056
1.2294
0.0741
Table 9. Mean and deviation of accelerations ().


Mean (m/s)
Standard deviation (m/s)






0
2
0.0033
0.0017
0.0618
0.0360
0.0267
0.0229
4
0.0015
0.0020
0.0567
0.0523
0.0287
0.0231
8
0.0011
0.0005
0.0578
0.0591
0.0293
0.0231
16
0.0008
0.0005
0.0550
0.0628
0.0297
0.0231
32
0.0008
0.0005
0.0550
0.0628
0.0297
0.0231
0.1
2
0.0015
0.0023
0.0700
0.0903
0.0324
0.0229
4
0.0324
0.0006
0.0671
0.1909
0.0438
0.0231
8
0.0610
0.0008
0.0661
0.2209
0.0483
0.0231
16
0.0711
0.0012
0.0680
0.2332
0.0512
0.0231
32
0.0711
0.0012
0.0680
0.2332
0.0512
0.0231
0.2
2
0.0801
0.0850
0.0727
0.3891
0.1176
0.0239
4
1.8645
0.5735
0.0656
0.8054
0.2971
0.0247
8
2.8548
0.8585
0.0651
1.2686
0.3593
0.0249
16
3.2252
0.9285
0.0673
1.5879
0.3724
0.0250
32
3.2252
0.9285
0.0673
1.5879
0.3724
0.0250
Table 10. Mean and deviation of speed variation ().


Mean (m/s)
Standard deviation (m/s)






0
2
0.0130
0.0062
0.0022
0.0416
0.0258
0.0191
4
0.0221
0.0077
0.0023
0.0675
0.0303
0.0205
8
0.0253
0.0081
0.0023
0.0772
0.0314
0.0205
16
0.0273
0.0084
0.0023
0.0828
0.0323
0.0206
32
0.0273
0.0084
0.0023
0.0828
0.0323
0.0206
0.1
2
0.0451
0.0114
0.0022
0.1315
0.0383
0.0191
4
0.0971
0.0196
0.0023
0.3022
0.0593
0.0205
8
0.1139
0.0227
0.0023
0.3561
0.0672
0.0206

33
16
0.1224
0.0246
0.0023
0.3804
0.0723
0.0206
32
0.1224
0.0246
0.0023
0.3804
0.0723
0.0206
0.2
2
0.1888
0.0577
0.0031
0.6252
0.1676
0.0211
4
0.3984
0.1330
0.0037
1.1596
0.4234
0.0235
8
0.7847
0.1575
0.0039
2.6489
0.5073
0.0240
16
0.8269
0.1680
0.0041
2.8545
0.5334
0.0244
32
0.8269
0.1680
0.0041
2.8545
0.5334
0.0244
Note: Speed variation between two adjacent vehicles is defined as the absolute value of the speed difference between
the latter and the former vehicles.
7. Traffic safety
To evaluate the impact of maximum platoon size on the traffic safety of mixed traffic flow, we adopted
two safety evaluation indexes in this study: TET and TIT. TTC is an important indicator of traffic safety and
is widely used in traffic safety evaluation (Arvin et al., 2020; Das and Maurya, 2020; Rahman and Abdel-Aty,
2018; Xiao et al., 2021). The two quantitative indicators TET and TIT of TTC, were used to evaluate traffic
safety.





󰇛󰇜




󰇟󰇛󰇜󰇠󰇛󰇜



 
(59)
where  varies from 1 to 3 s (Das and Maurya, 2020; Rahman and Abdel-Aty, 2018);

󰇛󰇜 is the
collision time at time of ,
󰇛󰇜󰇛󰇜󰇛󰇜
󰇛󰇜󰇛󰇜󰇛󰇜󰇛󰇜
(60)
where  is the length of the vehicle, .
Consistent with the simulation experiment described in Section 6.2.2, when Eq. (58) was adopted, we
observed that the evaluation indexes TET and TIT were 0 at a high steady-state speed. The analysis showed
that when the steady-state speed was high, the impact on traffic safety was small, owing to slight acceleration
and deceleration. Therefore, to reflect the difference in the safety indicators more clearly at different steady-
state speeds, Eq. (58) can be rewritten as
󰇛󰇜





 




 
(61)
Eq. (61) indicates that the acceleration is directly proportional to the steady-state speed. When the steady-
state speed is considerable, the acceleration and deceleration are also significant. The other parameters of the
simulation remained unchanged. Traffic safety is only related to the car-following models of vehicles. In the
mixed traffic flow, the maximum platoon size only affects the proportion of CAV-CAV (inter-platoon) and
CAV-CAV (intra-platoon) in CAV-CAV based on the analysis in Section 4.1. Moreover, it does not affect the
proportion of other vehicles. This means the characteristic that the penetration rate is 1 is consistent with
other values. Therefore, to avoid the impact of different factors on the maximum platoon size, we set the
penetration rate of CAVs to 1. Based on the simulation data, we calculated the two safety evaluation metrics
for different scenarios as shown in Table 11.

34
Table 11. Safety evaluation metrics under different maximum platoon sizes.
()




TET ()
TIT ()
TET ()
TIT ()
TET ()
TIT ()
1
2
5
6.12
16
15.00
19
14.92
4
8
8.18
23
25.86
23
20.66
8
8
8.18
29
42.18
29
34.18
16
8
8.18
35
50.56
34
160.57
32
8
8.18
35
50.56
34
160.57
2
2
7
12.47
23
35.73
28
36.44
4
10
17.45
31
52.20
36
48.14
8
14
19.67
41
76.69
43
70.42
16
14
19.67
45
91.05
44
199.84
32
14
19.67
46
91.44
49
200.90
3
2
10
21.10
35
64.13
33
67.22
4
17
31.77
45
90.01
47
89.19
8
22
38.22
53
124.19
54
118.08
16
23
38.23
56
141.40
56
250.09
32
23
38.23
64
145.75
61
256.82
(a) 
(b) 

35
(c) 
Fig. 14. Safety evaluation indexes under different maximum platoon sizes. ()
Table 11 reports that when the equilibrium speed of mixed traffic flow was low, TET and TIT were small
under the same maximum platoon size; when the equilibrium speed was high, TET and TIT were largely
under the same maximum platoon size. For example, when the threshold of TTC was 2 s and the maximum
platoon size was 8, the TET corresponding to equilibrium speeds of 10 and 20 m/s were 14 and 41 s, and the
TIT was 19.67 and 76.69 s2, respectively. This indicates that the safety performance of mixed traffic flow is
better when the speed is low.
Furthermore, Table 11 shows that TET and TIT gradually increased with an increase in the maximum
platoon size. This meant that the safety risk of vehicles in a mixed traffic flow increased gradually with the
maximum platoon size of CAVs. As shown in Table 11, TET and TIT did not increase infinitely with an
increase in the maximum platoon size, and a critical platoon size was observed. When the maximum platoon
size was more significant than the critical platoon size, the TET and TIT did not increase. Moreover, the
critical platoon size differed under different conditions. For example, when the equilibrium speed was 10
m/s, Table 11 shows that the thresholds of TTC were 1, 2, and 3 s, and the corresponding critical platoon sizes
were 4, 8, and 16, respectively. When the equilibrium speed was 20 m/s, and the threshold of TTC was 1 s,
the critical platoon size was 16. Therefore, when the equilibrium speed is given, the larger the TTC threshold,
the larger the corresponding critical platoon size. This indicates that the TTC threshold directly affects the
safety assessment of vehicles. When the TTC threshold is given, the larger the equilibrium speed, the larger
is the corresponding critical platoon size.
To further discuss the specific critical platoon size, we set the step size of the maximum platoon size to
1 (Fig. 16). Fig. 14 shows the safety evaluation indexes under different maximum platoon sizes when  was
1. As shown in Fig. 14, when the equilibrium speed was 10 m/s, the critical platoon sizes were 4, 6, and 9
under the three TTC thresholds based on TET or TIT. However, when the equilibrium speeds were 20 m/s
and 30 m/s, the critical platoon sizes were 16, 18, and 20 under the three thresholds of TTC based on TET or
TIT. Moreover, as shown in Fig. 14, when the maximum platoon size was larger than the critical platoon size,
the TET and TIT did not change and were fixed values. When the maximum platoon size was smaller than
the critical platoon size, the TET and TIT increased with an increase in the maximum platoon size. This
indicated that the larger the maximum platoon size, the lower the traffic safety. Fig. 14 shows that the two
safety evaluation indexes (i.e., TET and TIT) increase with the maximum platoon size overall, but some
fluctuations occurred. Compared with TET, the fluctuation in TIT was more prominent. Moreover, compared
with the low equilibrium speed, the two indexes fluctuated significantly at high equilibrium speeds. There
are possible reasons for this finding. First, we set higher acceleration and deceleration at a high equilibrium
speed. Second, when the equilibrium speed was high, the transmission of small fluctuations between the
vehicles was faster.
8. Fuel consumption and emission
Many studies have proposed vehicle fuel consumption and traffic emission models (Ahn, 1998; Ahn et
al., 2002; Akcelik, 1989; Hooker, 1988; Rakha and Ahn, 2003). The VT-Micro model was first proposed by Ahn

36
et al. (Ahn, 1998; Ahn et al., 2002; Rakha and Ahn, 2003) and was fitted with the observed data. It is widely
used for fuel consumption and emissions (Qin et al., 2019, 2018; Tang et al., 2015; Yao et al., 2021a). Therefore,
the VT-Micro model was adopted to calculate the fuel consumption and emissions of the mixed traffic flow.
The model can be described as follows:

󰇛󰇜󰇛󰇜



 
(62)
where  and  are the exponential coefficients of speed and acceleration, respectively.  and  represent
the speed and acceleration of the vehicle, respectively. 
 was calibrated by Ahn et al. (1998) using field
data collected at the Oak Ridge National Laboratory. The calibration regression coefficients have been widely
used in various studies (Qin et al., 2019, 2018; Tang et al., 2015; Yao et al., 2021a), as shown in Appendix F.
Similar to Section 7, we set the penetration rate of CAVs to 1. The speed and acceleration of the vehicle
at each time step were obtained based on the results of the simulation. Subsequently, we calculated fuel
consumption and traffic emissions for different scenarios (Table 12).

37
Table 12. Fuel consumption and emission under different maximum platoon sizes.




FC /
CO /
HC /
NOx /
FC /
CO /
HC /
NOx /
FC /
CO /
HC /
NOx /
2
0
0
0
0
0
0
0
0
0
0
0
0
4
0.723
3.400
2.669
3.323
2.960
4.610
2.874
1.957
5.517
1.401
3.737
1.140
6
1.935
6.043
3.610
7.473
3.976
7.366
4.147
2.770
7.914
2.323
5.413
1.610
8
2.471
7.428
4.253
9.415
4.928
8.694
5.061
3.395
9.522
2.805
6.464
1.897
10
2.987
8.514
4.600
11.162
5.490
9.619
5.627
3.784
10.530
3.127
7.122
2.074
12
3.246
9.166
4.871
12.096
5.673
10.441
5.940
3.976
11.261
3.335
7.584
2.197
14
3.407
9.609
5.089
12.690
5.702
11.170
6.118
4.079
11.793
3.482
7.915
2.284
16
3.263
9.672
5.331
12.392
6.303
11.101
6.498
4.375
12.315
3.529
8.193
2.354
18
3.432
10.009
5.434
12.951
6.333
11.513
6.601
4.440
12.548
3.649
8.362
2.399
20
3.597
10.317
5.493
13.493
6.376
11.807
6.694
4.499
12.714
3.753
8.490
2.434
22
3.641
10.469
5.568
13.673
6.495
11.946
6.804
4.577
12.866
3.823
8.595
2.463
24
3.590
10.508
5.670
13.575
6.678
12.004
6.932
4.677
13.258
3.756
8.756
2.499
26
3.514
10.504
5.758
13.396
6.940
11.915
7.084
4.800
13.338
3.817
8.821
2.517
28
3.677
10.737
5.756
13.894
6.777
12.236
7.050
4.759
13.365
3.889
8.867
2.531
30
3.387
10.482
5.913
13.092
7.349
11.792
7.312
4.992
13.687
3.796
8.980
2.555
32
3.726
10.893
5.827
14.087
6.838
12.435
7.129
4.815
13.559
3.929
8.978
2.559
34
3.710
10.916
5.870
14.062
7.017
12.350
7.226
4.896
13.650
3.938
9.026
2.571
36
3.751
10.996
5.871
14.201
6.944
12.489
7.216
4.878
13.632
3.986
9.039
2.576
38
3.833
11.111
5.868
14.450
6.845
12.667
7.190
4.851
13.663
4.013
9.066
2.583
40
3.906
11.216
5.866
14.673
6.770
12.814
7.173
4.832
13.695
4.033
9.090
2.590
42
3.812
11.142
5.918
14.419
7.081
12.548
7.312
4.956
13.755
4.031
9.118
2.597
44
3.843
11.194
5.933
14.515
7.004
12.684
7.291
4.934
13.879
3.994
9.161
2.606
46
3.966
11.353
5.893
14.884
6.880
12.856
7.252
4.894
13.705
4.111
9.130
2.602
48
3.926
11.331
5.921
14.782
6.976
12.792
7.301
4.935
13.823
4.071
9.168
2.610
50
3.905
11.324
5.945
14.728
6.967
12.840
7.306
4.937
13.861
4.072
9.187
2.615
Note: The value in this table is the growth rate, and the specific calculation is (fuel consumption or emission at maximum platoon size N - fuel consumption or emission with
maximum platoon size 2) / (fuel consumption or emission with maximum platoon size 2).

38
(a) Fuel consumption
(b) HC
(c) CO
(d) NOx
Fig. 15. Fuel consumption and emission under different maximum platoon sizes. ()
Table 12 reports that fuel consumption and emissions (e.g., CO, HC, and NOx) increased with an increase
in the maximum platoon size. This differed from other studies (Tsugawa et al., 2016; L. Zhang et al., 2020).
Existing research shows that the platoon can reduce fuel consumption and emissions (Liang et al., 2016).
Analysis indicated that there are two possible reasons for this. First, previous studies optimized the platoon
trajectory, and the optimization objective was fuel consumption or emissions. Second, this study considered
that when vehicles form a platoon, the air resistance of vehicles other than the first vehicle in the platoon
should be smaller to conserve energy conservation and reduce emissions (Bian et al., 2021). In this study, the
trajectory of the platoon was not optimized, but the car-following model was used to describe the vehicle
operation in the platoon, and the influence of wind speed, air, and other factors on vehicle operation was not
considered. Therefore, the conclusions of this study differ from that of previous studies. Moreover, as shown

39
in Table 12, when the equilibrium speed was 10 or 30 m/s, the increase rate of NOx with the maximum platoon
size was the highest. However, the increase rate of HC with the maximum platoon size was the highest when
the equilibrium speed was 20 m/s. This means that when the equilibrium speed is low or high, the maximum
platoon size has a greater impact on NOx. Table 12 presents that compared with the three emission indicators
(e.g., CO, HC, and NOx), the maximum platoon size has the least impact on fuel consumption. Moreover,
when the maximum platoon size is given, fuel consumption and emissions increase with the increase in
equilibrium speed, which is consistent with Eq. (62).
Fig. 15 shows that fuel consumption and emissions (i.e., CO, HC, and NOx) under different maximum
platoon sizes when  was 1. As shown in Fig. 15, the fuel consumption and emissions increase with an
increase in the maximum platoon size under different equilibrium speeds. However, the increased rate of
fuel consumption and emissions gradually slowed with the maximum platoon size of CAVs. Therefore, in
theory, a critical platoon size also exists. When the maximum platoon size is larger than the critical platoon
size, fuel consumption and emissions are not affected by the platoon size. As shown in Fig. 15 that the critical
platoon size should exceed 50. However, when the platoon size is excessively large, it is detrimental to both
the stability of traffic flow and traffic safety. Therefore, an excessively large platoon will be meaningless; we
do not discuss this further here.
9. Conclusions and future work
In this paper, we analyze the characteristics of a mixed traffic flow with the maximum platoon size of
CAVs. First, we discuss the car-following and headway types of mixed traffic flow with HDVs, CHVs, AVs,
and CAVs. The analysis results show that there are ten car-following types but only four headway types.
Second, we derive the probability distribution of the CAVs platoon size based on the Markov chain model
(Ghiasi et al., 2017). Through numerical analysis, some interesting conclusions are obtained: (1) the
probability of the platoon size increases with the penetration rate of CAVs; and (2) the probability of platoon
size occurrence decreases with an increase in platoon size when the maximum platoon size is given. Third,
the capacity model and stability condition of the mixed traffic flow are proposed based on the probability
distribution of the CAVs platoon size. Theoretical analysis shows that the maximum platoon size is conducive
to improving traffic capacity but weakens the stability of mixed traffic flow. A numerical experiment based
on car-following models was conducted to investigate the effect of the maximum platoon size on capacity,
stability, safety, fuel consumption, and emission of mixed traffic flow. The following conclusions were drawn
based on the numerical experiments and sensitivity analysis.
(1) When the penetration rate of CAVs is low, the maximum platoon size has a slight impact on traffic
capacity. The traffic capacity of the mixed traffic flow increases with the maximum platoon size of CAVs.
Moreover, when the maximum platoon size reaches a certain level, blindly increasing the maximum platoon
size will not result in good gain.
(2) CAVs have a positive effect on traffic flow stability; however, the stability of mixed traffic flow
deteriorates with the maximum platoon size of CAVs or the penetration rate of AVs. This means that the
maximum platoon size of CAVs and AVs have a negative impact on traffic flow stability. Moreover, this
indicates that the impact of maximum platoon size on traffic capacity and stability is the opposite. There may
be an optimal maximum platoon size to balance the two metrics.
(3) The two safety evaluation indices, TET and TIT, increased with the maximum platoon size of CAVs.
This suggests that the safety risk of vehicles in mixed traffic flow increases with the maximum platoon size
of CAVs. Moreover, there is a critical platoon size; TET and TIT will not continue to increase when the
maximum platoon size is more significant than the critical platoon size of CAVs.
(4) The fuel consumption and emissions (e.g., CO, HC, and NOx) increase with the maximum platoon
size of CAVs. However, the increased rate of fuel consumption and emissions gradually slows down with an
increase in the maximum platoon size of CAVs. As a result, when the maximum platoon size is larger than
the critical platoon size, the fuel consumption and emissions are not affected by the platoon size of CAVs.
These conclusions suggest that a larger maximum platoon size can only improve traffic capacity, which
is not conducive to the stability, safety, fuel consumption, and emissions of mixed traffic flow.

40
In this paper, the characteristics of mixed traffic flow considering the maximum platoon size of CAVs
are analyzed and discussed in detail. The future research of this study can be extended to several interesting
directions. First, the impact of the maximum platoon size on traffic systems in more complex traffic scenarios
(e.g., ramps and intersections) (Arvin et al., 2020; Liu et al., 2018a, 2018b; Mersky and Samaras, 2016) can be
investigated. Second, for fuel consumption and emissions, when vehicles form a platoon, a smaller air
resistance of vehicles other than the first vehicle in the platoon was not considered in this study (Liang et al.,
2016; Tsugawa et al., 2011). This problem can be addressed in future studies. Third, this paper only discusses
the results from the data level on the impact of maximum platoon size on traffic safety, fuel consumption,
and emissions. In the future, the relationship equations between the maximum platoon size of CAVs and
traffic safety, fuel consumption, and emission can be fitted based on simulation or observed data (He et al.,
2020). Fourth, this study adopted fixed car-following models without optimizing the parameters to study
stability, safety, and emissions. In the future, we can consider optimizing the parameters of the CAV
controller (i.e., ACC (Zhou et al., 2022) and CACC (Naus et al., 2010; Y. Zhang et al., 2020)), and investigate
the impact of the maximum platoon size on stability, safety, and emissions of mixed traffic flow. Fifth, a weak
stability condition of the mixed traffic flow was adopted in this study (Montanino and Punzo, 2021), and
time lag, such as reaction time and communication delay, were not reflected. These problems should be
addressed in future studies. Finally, this study was primarily based on theoretical derivation and simulation
data. The proposed model results can be further verified using actual observational data (Shi and Li, 2021)
in the future.
Acknowledgments
This study received research funding support from the National Natural Science Foundation of China
(52002339), Sichuan Science and Technology Program (2021YJ0535, 2022YFG0152), Fundamental Research
Funds for the Central Universities (2682021CX058), and Guangxi Science and Technology Program
(21AA01007AA).
Appendix
Appendix A
Proof:
To facilitate the analysis of the capacity characteristics, we analyze the properties of  and . First,
both  and  represent the current vehicle and the following vehicle is a CAV. Therefore, the sum of 
and  is independent of , as shown in Eq.(A.1).

󰇛󰇜

(A.1)
From Eq.(A.1), the sum of  and  is , which is only related to  and is independent of . This
completes the proof. □
Appendix B
Proof:
First, we define  as a function of ,
󰇛󰇜

(A.2)
Subsequently, assuming 󰇛󰇜, and , Eq.(A.3) is obtained.

41
























(A.3)
where  is a positive integer, i.e., 





󰇛󰇜

This means that 




 . Subsequently, we have  . Therefore,  is an increasing
function of . This completes the proof. □
Appendix C
Proof:
First, we define  as a function of ,
󰇛󰇜

(A.4)
Subsequently, we only require to compare the sizes of 󰇛󰇜 and 󰇛󰇜:
󰇛󰇜󰇛󰇜
 

󰇛󰇜
󰇛󰇜󰇛󰇜
󰇛󰇜
󰇛󰇜󰇛󰇜󰇛󰇜
󰇛󰇜󰇛󰇜
(A.5)
where 󰇛󰇜 and  . We know that 󰇛󰇜 and 󰇛󰇜 . Thus,
󰇛󰇜󰇛󰇜. This indicates that  is an increasing function of . This completes the proof. □
Appendix D
Proof:
First, we define  as a function of ,
󰇛󰇜󰇛󰇜

(A.6)
Subsequently, assuming 󰇛󰇜, and , Eq.(A.7) is obtained.




















(A.7)
where  is a positive integer, i.e., 





󰇛󰇜

This means that




 . Thus, we have  . Therefore,  is an increasing
function of . This completes the proof. □

42
Appendix E
Proof:
First, we define  as a function of ,
󰇛󰇜󰇛󰇜

(A.8)
Subsequently, we only require to compare the sizes of 󰇛󰇜 and 󰇛󰇜:
󰇛󰇜󰇛󰇜󰇛󰇜
 󰇛󰇜

󰇛󰇜󰇛󰇜󰇛󰇜
󰇛󰇜󰇛󰇜
󰇛󰇜
󰇛󰇜󰇛󰇜 󰇛󰇜
󰇛󰇜󰇛󰇜
(A.9)
where 󰇛󰇜 and  . We know that 󰇛󰇜 and 󰇛󰇜 . Thus,
󰇛󰇜󰇛󰇜. This indicates that  is a decreasing function of. This completes this proof. □
Appendix F
Table 13. Regression coefficients 
in Eq. (62).
Parameters
Fuel
CO
HC
NOx


-0.679439
0.887447
-0.728042
-1.067682


0.135273
0.148841
0.012211
0.254363


0.015946
0.03055
0.023371
0.008866


-0.001189
-0.001348
-0.000093243
-0.000951


0.029665
0.070994
0.02495
0.046423


0.004808
0.00387
0.010145
0.015482


-0.000020535
0.000093228
-0.000103
-0.000131


5.5409285E-8
-0.000000706
0.000000618
0.000000328


-0.000276
-0.000786
-0.000205
-0.000173


0.000083329
-0.000926
-0.000549
0.002876


0.000000937
0.000049181
0.000037592
-0.00005866


-2.479644E-8
-0.000000314
-0.000000213
0.00000024


0.000001487
0.000004616
0.000001949
0.000000569


-0.000061321
0.000046144
-0.000113
-0.000321


0.000000304
-0.00000141
0.00000331
0.000001943


-4.467234E-9
8.1724008E-9
-1.739372E-8
-1.257413E-8
References
Ahn, K., 1998. Microscopic Fuel Consumption and Emission Modeling (Thesis). Virginia Polytechnic
Institute and State University, Blacksburg, Virginia.
Ahn, K., Rakha, H., Trani, A., Van Aerde, M., 2002. Estimating Vehicle Fuel Consumption and Emissions
based on Instantaneous Speed and Acceleration Levels. J. Transp. Eng. 128, 182–190.
https://doi.org/10.1061/(ASCE)0733-947X(2002)128:2(182)
Akcelik, R., 1989. Efficiency and drag in the power-based model of fuel consumption. Transp. Res. Part B
Methodol. 23, 376–385. https://doi.org/10.1016/0191-2615(89)90014-3

43
Al-Turki, M., Ratrout, N.T., Rahman, S.M., Reza, I., 2021. Impacts of Autonomous Vehicles on Traffic Flow
Characteristics under Mixed Traffic Environment: Future Perspectives. Sustainability 13, 11052.
https://doi.org/10.3390/su131911052
Ambarwati, L., Pel, A.J., Verhaeghe, R., van Arem, B., 2014. Empirical analysis of heterogeneous traffic flow
and calibration of porous flow model. Transp. Res. Part C Emerg. Technol. 48, 418–436.
https://doi.org/10.1016/j.trc.2014.09.017
Arvin, R., Khattak, A.J., Kamrani, M., Rio-Torres, J., 2020. Safety evaluation of connected and automated
vehicles in mixed traffic with conventional vehicles at intersections. J. Intell. Transp. Syst. 1–18.
https://doi.org/10.1080/15472450.2020.1834392
Avedisov, S.S., Bansal, G., Orosz, G., 2022. Impacts of Connected Automated Vehicles on Freeway Traffic
Patterns at Different Penetration Levels. IEEE Trans. Intell. Transp. Syst. 23, 4305–4318.
https://doi.org/10.1109/TITS.2020.3043323
Bian, Y., Du, C., Hu, M., Li, S.E., Liu, H., Li, C., 2021. Fuel Economy Optimization for Platooning Vehicle
Swarms via Distributed Economic Model Predictive Control. IEEE Trans. Autom. Sci. Eng. 1–13.
https://doi.org/10.1109/TASE.2021.3128920
Chang, X., Li, H., Rong, J., Zhao, X., Li, A., 2020. Analysis on traffic stability and capacity for mixed traffic
flow with platoons of intelligent connected vehicles. Phys. Stat. Mech. Its Appl. 557, 124829.
https://doi.org/10.1016/j.physa.2020.124829
Chen, D., Ahn, S., Chitturi, M., Noyce, D.A., 2017a. Towards vehicle automation: Roadway capacity
formulation for traffic mixed with regular and automated vehicles. Transp. Res. Part B Methodol. 100, 196–
221. https://doi.org/10.1016/j.trb.2017.01.017
Chen, D., Ahn, S., Chitturi, M., Noyce, D.A., 2017b. Towards vehicle automation: Roadway capacity
formulation for traffic mixed with regular and automated vehicles. Transp. Res. Part B Methodol. 100, 196–
221. https://doi.org/10.1016/j.trb.2017.01.017
Chen, X., Li, M., Lin, X., Yin, Y., He, F., 2021. Rhythmic Control of Automated Traffic—Part I: Concept and
Properties at Isolated Intersections. Transp. Sci. trsc.2021.1060. https://doi.org/10.1287/trsc.2021.1060
Das, S., Maurya, A.K., 2020. Defining Time-to-Collision Thresholds by the Type of Lead Vehicle in Non-
Lane-Based Traffic Environments. IEEE Trans. Intell. Transp. Syst. 21, 4972–4982. https://doi.org/10/ghqdrk
Ding, J., Li, L., Peng, H., Zhang, Y., 2020. A Rule-Based Cooperative Merging Strategy for Connected and
Automated Vehicles. IEEE Trans. Intell. Transp. Syst. 21, 3436–3446.
https://doi.org/10.1109/TITS.2019.2928969
Evans, L., 2004. Traffic safety, Science Serving Society. ed. Bloomfield Hills, MI.
Ghiasi, A., Hussain, O., Qian, Z. (Sean), Li, X., 2017. A mixed traffic capacity analysis and lane management
model for connected automated vehicles: A Markov chain method. Transp. Res. Part B Methodol. 106, 266–
292. https://doi.org/10/gcqbj5
Han, X., Ma, R., Zhang, H.M., 2020. Energy-aware trajectory optimization of CAV platoons through a
signalized intersection. Transp. Res. Part C Emerg. Technol. 118, 102652.
https://doi.org/10.1016/j.trc.2020.102652
He, Z., Zhang, W., Jia, N., 2020. Estimating Carbon Dioxide Emissions of Freeway Traffic: A Spatiotemporal
Cell-Based Model. IEEE Trans. Intell. Transp. Syst. 21, 1976–1986. https://doi.org/10.1109/TITS.2019.2909316
Hooker, J.N., 1988. Optimal driving for single-vehicle fuel economy. Transp. Res. Part Gen. 22, 183–201.
https://doi.org/10.1016/0191-2607(88)90036-2
Jiang, Y., Wang, S., Yao, Z., Zhao, B., Wang, Y., 2021. A cellular automata model for mixed traffic flow
considering the driving behavior of connected automated vehicle platoons. Phys. Stat. Mech. Its Appl.
126262. https://doi.org/10.1016/j.physa.2021.126262
Jing, S., Hui, F., Zhao, X., Rios-Torres, J., Khattak, A.J., 2019. Cooperative Game Approach to Optimal
Merging Sequence and on-Ramp Merging Control of Connected and Automated Vehicles. IEEE Trans.
Intell. Transp. Syst. 20, 4234–4244. https://doi.org/10.1109/TITS.2019.2925871
Jing, S., Zhao, X., Hui, F., Khattak, A.J., Yang, L., 2020. Cooperative CAVs optimal trajectory planning for
collision avoidance and merging in the weaving section. Transp. B Transp. Dyn. 0, 1–18.
https://doi.org/10.1080/21680566.2020.1845852

44
Jun, J., 2010. Understanding the variability of speed distributions under mixed traffic conditions caused by
holiday traffic. Transp. Res. Part C Emerg. Technol. 18, 599–610. https://doi.org/10.1016/j.trc.2009.12.005
Kesting, A., Treiber, M., 2008. How Reaction Time, Update Time, and Adaptation Time Influence the
Stability of Traffic Flow: Influence of reaction time, update time, and adaptation time. Comput.-Aided Civ.
Infrastruct. Eng. 23, 125–137. https://doi.org/10.1111/j.1467-8667.2007.00529.x
Kopelias, P., Demiridi, E., Vogiatzis, K., Skabardonis, A., Zafiropoulou, V., 2020. Connected & autonomous
vehicles – Environmental impacts – A review. Sci. Total Environ. 712, 135237.
https://doi.org/10.1016/j.scitotenv.2019.135237
Li, K., Wang, J., Zheng, Y., 2022. Cooperative Formation of Autonomous Vehicles in Mixed Traffic Flow:
Beyond Platooning. IEEE Trans. Intell. Transp. Syst. 1–16. https://doi.org/10.1109/TITS.2022.3146612
Li, T., Guo, F., Krishnan, R., Sivakumar, A., Polak, J., 2020. Right-of-way reallocation for mixed flow of
autonomous vehicles and human driven vehicles. Transp. Res. Part C Emerg. Technol. 115, 102630.
https://doi.org/10.1016/j.trc.2020.102630
Liang, K.-Y., Martensson, J., Johansson, K.H., 2016. Heavy-Duty Vehicle Platoon Formation for Fuel
Efficiency. IEEE Trans. Intell. Transp. Syst. 17, 1051–1061. https://doi.org/10.1109/TITS.2015.2492243
Lin, X., Li, M., Shen, Z.-J.M., Yin, Y., He, F., 2021. Rhythmic Control of Automated Traffic—Part II: Grid
Network Rhythm and Online Routing. Transp. Sci. trsc.2021.1061. https://doi.org/10.1287/trsc.2021.1061
Lioris, J., Pedarsani, R., Tascikaraoglu, F.Y., Varaiya, P., 2017. Platoons of connected vehicles can double
throughput in urban roads. Transp. Res. Part C Emerg. Technol. 77, 292–305.
https://doi.org/10.1016/j.trc.2017.01.023
Liu, H., Kan, X. (David), Shladover, S.E., Lu, X.-Y., Ferlis, R.E., 2018a. Modeling impacts of Cooperative
Adaptive Cruise Control on mixed traffic flow in multi-lane freeway facilities. Transp. Res. Part C Emerg.
Technol. 95, 261–279. https://doi.org/10.1016/j.trc.2018.07.027
Liu, H., Kan, X. (David), Shladover, S.E., Lu, X.-Y., Ferlis, R.E., 2018b. Impact of cooperative adaptive cruise
control on multilane freeway merge capacity. J. Intell. Transp. Syst. 22, 263–275.
https://doi.org/10.1080/15472450.2018.1438275
Luo, F., Larson, J., Munson, T., 2018. Coordinated platooning with multiple speeds. Transp. Res. Part C
Emerg. Technol. 90, 213–225. https://doi.org/10.1016/j.trc.2018.02.011
Luo, Y., Xiang, D., Zhang, S., Liang, W., Sun, J., Zhu, L., 2021. Evaluation on the Fuel Economy of
Automated Vehicles with Data-Driven Simulation Method. Energy AI 3, 100051.
https://doi.org/10.1016/j.egyai.2021.100051
Mahmassani, H.S., 2016. 50th Anniversary Invited Article—Autonomous Vehicles and Connected Vehicle
Systems: Flow and Operations Considerations. Transp. Sci. 50, 1140–1162.
https://doi.org/10.1287/trsc.2016.0712
Mersky, A.C., Samaras, C., 2016. Fuel economy testing of autonomous vehicles. Transp. Res. Part C Emerg.
Technol. 65, 31–48. https://doi.org/10.1016/j.trc.2016.01.001
Milanés, V., Shladover, S.E., 2014. Modeling cooperative and autonomous adaptive cruise control dynamic
responses using experimental data. Transp. Res. Part C Emerg. Technol. 48, 285–300.
https://doi.org/10.1016/j.trc.2014.09.001
Milanes, V., Shladover, S.E., Spring, J., Nowakowski, C., Kawazoe, H., Nakamura, M., 2014. Cooperative
Adaptive Cruise Control in Real Traffic Situations. IEEE Trans. Intell. Transp. Syst. 15, 296–305.
https://doi.org/10/f5sf8v
Mirzaeian, N., Cho, S.-H., Scheller-Wolf, A., 2021. A Queueing Model and Analysis for Autonomous
Vehicles on Highways. Manag. Sci. 67, 2904–2923. https://doi.org/10.1287/mnsc.2020.3692
Montanino, M., Punzo, V., 2021. On string stability of a mixed and heterogeneous traffic flow: A unifying
modelling framework. Transp. Res. Part B Methodol. 144, 133–154. https://doi.org/10/gh5q4q
Nascimento, A.M., Vismari, L.F., Molina, C.B.S.T., Cugnasca, P.S., Camargo, J.B., Almeida, J.R. d, Inam, R.,
Fersman, E., Marquezini, M.V., Hata, A.Y., 2020. A Systematic Literature Review About the Impact of
Artificial Intelligence on Autonomous Vehicle Safety. IEEE Trans. Intell. Transp. Syst. 21, 4928–4946.
https://doi.org/10.1109/TITS.2019.2949915

45
Naus, G.J.L., Vugts, R.P.A., Ploeg, J., van de Molengraft, M.J.G., Steinbuch, M., 2010. String-Stable CACC
Design and Experimental Validation: A Frequency-Domain Approach. IEEE Trans. Veh. Technol. 59, 4268–
4279. https://doi.org/10.1109/TVT.2010.2076320
Ngoduy, D., 2013. Analytical studies on the instabilities of heterogeneous intelligent traffic flow. Commun.
Nonlinear Sci. Numer. Simul. 18, 2699–2706. https://doi.org/10.1016/j.cnsns.2013.02.018
Ngoduy, D., Hoang, N.H., Vu, H.L., Watling, D., 2021. Multiclass dynamic system optimum solution for
mixed traffic of human-driven and automated vehicles considering physical queues. Transp. Res. Part B
Methodol. 145, 56–79. https://doi.org/10/gjkqw4
Pei, H., Feng, S., Zhang, Y., Yao, D., 2019. A Cooperative Driving Strategy for Merging at On-Ramps Based
on Dynamic Programming. IEEE Trans. Veh. Technol. 68, 11646–11656.
https://doi.org/10.1109/TVT.2019.2947192
Pei, H., Zhang, Yuxiao, Zhang, Yi, Feng, S., 2021. Optimal Cooperative Driving at Signal-Free Intersections
With Polynomial-Time Complexity. IEEE Trans. Intell. Transp. Syst. 1–13.
https://doi.org/10.1109/TITS.2021.3118592
Qian, G., Guo, M., Zhang, L., Wang, Y., Hu, S., Wang, D., 2021. Traffic scheduling and control in fully
connected and automated networks. Transp. Res. Part C Emerg. Technol. 126, 103011.
https://doi.org/10/gjf4qf
Qin, Y., Wang, H., Ni, D., 2021. Lighthill-Whitham-Richards Model for Traffic Flow Mixed with
Cooperative Adaptive Cruise Control Vehicles. Transp. Sci. 55, 883–907.
https://doi.org/10.1287/trsc.2021.1057
Qin, Y., Wang, H., Ran, B., 2019. Impacts of cooperative adaptive cruise control platoons on emissions
under traffic oscillation. J. Intell. Transp. Syst. 1–8. https://doi.org/10/ghsbp5
Qin, Y., Wang, H., Ran, B., 2018. Stability Analysis of Connected and Automated Vehicles to Reduce Fuel
Consumption and Emissions. J. Transp. Eng. Part Syst. 144, 04018068. https://doi.org/10/ggcch5
Rahman, M.H., Abdel-Aty, M., Wu, Y., 2021. A multi-vehicle communication system to assess the safety
and mobility of connected and automated vehicles. Transp. Res. Part C Emerg. Technol. 124, 102887.
https://doi.org/10.1016/j.trc.2020.102887
Rahman, M.S., Abdel-Aty, M., 2018. Longitudinal safety evaluation of connected vehicles’ platooning on
expressways. Accid. Anal. Prev. 117, 381–391. https://doi.org/10/gdvkvz
Rakha, H., Ahn, K., 2003. Closure to “Estimating Vehicle Fuel Consumption and Emissions based on
Instantaneous Speed and Acceleration Levels” by Kyoung Ahn, Hesham Rakha, Antonio Trani, and Michel
Van Aerde. J. Transp. Eng. 129, 579–581. https://doi.org/10.1061/(ASCE)0733-947X(2003)129:5(579)
Rezaei, A., Caulfield, B., 2021. Safety of autonomous vehicles: what are the insights from experienced
industry professionals? Transp. Res. Part F Traffic Psychol. Behav. 81, 472–489.
https://doi.org/10.1016/j.trf.2021.07.005
Rios-Torres, J., Malikopoulos, A.A., 2017. A Survey on the Coordination of Connected and Automated
Vehicles at Intersections and Merging at Highway On-Ramps. IEEE Trans. Intell. Transp. Syst. 18, 1066–
1077. https://doi.org/10.1109/TITS.2016.2600504
Ruan, T., Zhou, L., Wang, H., 2021. Stability of heterogeneous traffic considering impacts of platoon
management with multiple time delays. Phys. Stat. Mech. Its Appl. 583, 126294.
https://doi.org/10.1016/j.physa.2021.126294
SAE, 2020. Taxonomy and Definitions for Terms Related to Cooperative Driving Automation for On-Road
Motor Vehicles (No. SAE J3216). SAE International.
Sala, M., Soriguera, F., 2021. Capacity of a freeway lane with platoons of autonomous vehicles mixed with
regular traffic. Transp. Res. Part B Methodol. 147, 116–131. https://doi.org/10.1016/j.trb.2021.03.010
Shang, M., Stern, R.E., 2021. Impacts of commercially available adaptive cruise control vehicles on highway
stability and throughput. Transp. Res. Part C Emerg. Technol. 122, 102897.
https://doi.org/10.1016/j.trc.2020.102897
Shi, X., Li, X., 2021. Constructing a fundamental diagram for traffic flow with automated vehicles:
Methodology and demonstration. Transp. Res. Part B Methodol. 150, 279–292.
https://doi.org/10.1016/j.trb.2021.06.011

46
Shiwakoti, N., Stasinopoulos, P., Fedele, F., 2020. Investigating the state of connected and autonomous
vehicles: a literature Review. Transp. Res. Procedia 48, 870–882. https://doi.org/10.1016/j.trpro.2020.08.101
Shladover, S.E., Nowakowski, C., Lu, X.-Y., Ferlis, R., 2015. Cooperative Adaptive Cruise Control:
Definitions and Operating Concepts. Transp. Res. Rec. J. Transp. Res. Board 2489, 145–152.
https://doi.org/10.3141/2489-17
Shladover, S.E., Su, D., Lu, X.-Y., 2012. Impacts of Cooperative Adaptive Cruise Control on Freeway Traffic
Flow. Transp. Res. Rec. J. Transp. Res. Board 2324, 63–70. https://doi.org/10.3141/2324-08
Song, M., Chen, F., Ma, X., 2021. Organization of autonomous truck platoon considering energy saving and
pavement fatigue. Transp. Res. Part Transp. Environ. 90, 102667. https://doi.org/10.1016/j.trd.2020.102667
Sun, Jie, Zheng, Z., Sun, Jian, 2020. The relationship between car following string instability and traffic
oscillations in finite-sized platoons and its use in easing congestion via connected and automated vehicles
with IDM based controller. Transp. Res. Part B Methodol. 142, 58–83. https://doi.org/10/ghqdr6
Sun, Jie, Zheng, Z., Sun, Jian, 2018. Stability analysis methods and their applicability to car-following
models in conventional and connected environments. Transp. Res. Part B Methodol. 109, 212–237.
https://doi.org/10.1016/j.trb.2018.01.013
Tafidis, P., Farah, H., Brijs, T., Pirdavani, A., 2021. Safety implications of higher levels of automated
vehicles: a scoping review. Transp. Rev. 1–23. https://doi.org/10.1080/01441647.2021.1971794
Talebpour, A., Mahmassani, H.S., 2016. Influence of connected and autonomous vehicles on traffic flow
stability and throughput. Transp. Res. Part C Emerg. Technol. 71, 143–163.
https://doi.org/10.1016/j.trc.2016.07.007
Tang, T., Huang, H., Shang, H., 2015. Infuences of the driver’s bounded rationality on micro driving
behavior, fuel consumption and emissions 41, 423–432. https://doi.org/10/gjkqw3
Treiber, M., Hennecke, A., Helbing, D., 2000. Congested traffic states in empirical observations and
microscopic simulations. Phys. Rev. E 62, 1805–1824. https://doi.org/10.1103/PhysRevE.62.1805
Treiber, M., Kesting, A., Helbing, D., 2007. Influence of Reaction Times and Anticipation on Stability of
Vehicular Traffic Flow. Transp. Res. Rec. J. Transp. Res. Board 1999, 23–29. https://doi.org/10.3141/1999-03
Tsugawa, S., Jeschke, S., Shladover, S.E., 2016. A Review of Truck Platooning Projects for Energy Savings.
IEEE Trans. Intell. Veh. 1, 68–77. https://doi.org/10.1109/TIV.2016.2577499
Tsugawa, S., Kato, S., Aoki, K., 2011. An automated truck platoon for energy saving, in: 2011 IEEE/RSJ
International Conference on Intelligent Robots and Systems. IEEE, San Francisco, CA, pp. 4109–4114.
https://doi.org/10.1109/IROS.2011.6094549
Typaldos, P., Papamichail, I., Papageorgiou, M., 2020. Minimization of Fuel Consumption for Vehicle
Trajectories. IEEE Trans. Intell. Transp. Syst. 21, 1716–1727. https://doi.org/10.1109/TITS.2020.2972770
Wan, N., Vahidi, A., Luckow, A., 2016. Optimal speed advisory for connected vehicles in arterial roads and
the impact on mixed traffic. Transp. Res. Part C Emerg. Technol. 69, 548–563.
https://doi.org/10.1016/j.trc.2016.01.011
Wang, C., Xie, Y., Huang, H., Liu, P., 2021. A review of surrogate safety measures and their applications in
connected and automated vehicles safety modeling. Accid. Anal. Prev. 157, 106157.
https://doi.org/10.1016/j.aap.2021.106157
Wang, H., Qin, Y., Wang, W., Chen, J., 2019. Stability of CACC-manual heterogeneous vehicular flow with
partial CACC performance degrading. Transp. B Transp. Dyn. 7, 788–813.
https://doi.org/10.1080/21680566.2018.1517058
Wang, M., 2018. Infrastructure assisted adaptive driving to stabilise heterogeneous vehicle strings. Transp.
Res. Part C Emerg. Technol. 91, 276–295. https://doi.org/10.1016/j.trc.2018.04.010
Ward, J.A., 2009. Heterogeneity, Lane-Changing and Instability in Traffic: A Mathematical Approach.
University of Bristol, Bristol, GB, Bristol United Kingdom.
Woo, S., Skabardonis, A., 2021. Flow-aware platoon formation of Connected Automated Vehicles in a mixed
traffic with human-driven vehicles. Transp. Res. Part C Emerg. Technol. 133, 103442.
https://doi.org/10.1016/j.trc.2021.103442

47
Xiao, G., Lee, J., Jiang, Q., Huang, H., Abdel-Aty, M., Wang, L., 2021. Safety improvements by intelligent
connected vehicle technologies: A meta-analysis considering market penetration rates. Accid. Anal. Prev.
159, 106234. https://doi.org/10.1016/j.aap.2021.106234
Xiao, L., Wang, M., Schakel, W., van Arem, B., 2018. Unravelling effects of cooperative adaptive cruise
control deactivation on traffic flow characteristics at merging bottlenecks. Transp. Res. Part C Emerg.
Technol. 96, 380–397. https://doi.org/10.1016/j.trc.2018.10.008
Xu, H., Feng, S., Zhang, Y., Li, L., 2019. A Grouping-Based Cooperative Driving Strategy for CAVs Merging
Problems. IEEE Trans. Veh. Technol. 68, 6125–6136. https://doi.org/10.1109/TVT.2019.2910987
Xu, Z., Wang, Y., Wang, G., Li, X., Bertini, R.L., Qu, X., Zhao, X., 2021. Trajectory Optimization for a
Connected Automated Traffic Stream: Comparison Between an Exact Model and Fast Heuristics. IEEE
Trans. Intell. Transp. Syst. 22, 2969–2978. https://doi.org/10.1109/TITS.2020.2978382
Yang, C.Y.D., Fisher, D.L., 2021. Safety impacts and benefits of connected and automated vehicles: How real
are they? J. Intell. Transp. Syst. 25, 135–138. https://doi.org/10/gjkqww
Yang, S., Du, M., Chen, Q., 2021. Impact of connected and autonomous vehicles on traffic efficiency and
safety of an on-ramp. Simul. Model. Pract. Theory 113, 102374. https://doi.org/10.1016/j.simpat.2021.102374
Yang, X.T., Huang, K., Zhang, Z., Zhang, Z.A., Lin, F., 2021. Eco-Driving System for Connected Automated
Vehicles: Multi-Objective Trajectory Optimization. IEEE Trans. Intell. Transp. Syst. 1–13.
https://doi.org/10.1109/tits.2020.3010726
Yao, Z., Gu, Q., Jiang, Y., Ran, B., 2022a. Fundamental Diagram and Stability of Mixed Traffic Flow
Considering Platoon Size and Intensity of Connected Automated Vehicles. Phys. Stat. Mech. Its Appl. 23.
Yao, Z., Hu, R., Jiang, Y., Xu, T., 2020a. Stability and safety evaluation of mixed traffic flow with connected
automated vehicles on expressways. J. Safety Res. 75, 262–274. https://doi.org/10.1016/j.jsr.2020.09.012
Yao, Z., Hu, R., Wang, Y., Jiang, Y., Ran, B., Chen, Y., 2019. Stability analysis and the fundamental diagram
for mixed connected automated and human-driven vehicles. Phys. Stat. Mech. Its Appl. 533, 121931.
https://doi.org/10.1016/j.physa.2019.121931
Yao, Z., Jiang, H., Cheng, Y., Jiang, Y., Ran, B., 2020b. Integrated Schedule and Trajectory Optimization for
Connected Automated Vehicles in a Conflict Zone. IEEE Trans. Intell. Transp. Syst. 1–11.
https://doi.org/10.1109/TITS.2020.3027731
Yao, Z., Wang, Y., Liu, B., Zhao, B., Jiang, Y., 2021a. Fuel Consumption and Transportation Emissions
Evaluation of Mixed Traffic Flow with Connected Automated Vehicles and Human-driven Vehicles on
Expressway. Energy 230, 120766. https://doi.org/10.1016/j.energy.2021.120766
Yao, Z., Wu, Y., Jiang, Y., Ran, B., 2022b. Modeling the Fundamental Diagram of Mixed Traffic Flow With
Dedicated Lanes for Connected Automated Vehicles. IEEE Trans. Intell. Transp. Syst. 1–13.
https://doi.org/10.1109/TITS.2022.3219836
Yao, Z., Xu, T., Jiang, Y., Hu, R., 2021b. Linear stability analysis of heterogeneous traffic flow considering
degradations of connected automated vehicles and reaction time. Phys. Stat. Mech. Its Appl. 561, 125218.
https://doi.org/10.1016/j.physa.2020.125218
Yu, C., Feng, Y., Liu, H.X., Ma, W., Yang, X., 2018. Integrated optimization of traffic signals and vehicle
trajectories at isolated urban intersections. Transp. Res. Part B Methodol. 112, 89–112.
https://doi.org/10.1016/j.trb.2018.04.007
Yu, H., Jiang, R., He, Z., Zheng, Z., Li, L., Liu, R., Chen, X., 2021. Automated vehicle-involved traffic flow
studies: A survey of assumptions, models, speculations, and perspectives. Transp. Res. Part C Emerg.
Technol. 127, 103101. https://doi.org/10.1016/j.trc.2021.103101
Zhang, L., Chen, F., Ma, X., Pan, X., 2020. Fuel Economy in Truck Platooning: A Literature Overview and
Directions for Future Research. J. Adv. Transp. 2020, 1–10. https://doi.org/10.1155/2020/2604012
Zhang, Y., Bai, Y., Hu, J., Wang, M., 2020. Control Design, Stability Analysis, and Traffic Flow Implications
for Cooperative Adaptive Cruise Control Systems with Compensation of Communication Delay. Transp.
Res. Rec. J. Transp. Res. Board 2674, 638–652. https://doi.org/10.1177/0361198120918873
Zhao, S., Zhang, K., 2021. Online predictive connected and automated eco-driving on signalized arterials
considering traffic control devices and road geometry constraints under uncertain traffic conditions.
Transp. Res. Part B Methodol. 145, 80–117. https://doi.org/10/gjkqxf

48
Zhao, W., Ngoduy, D., Shepherd, S., Liu, R., Papageorgiou, M., 2018. A platoon based cooperative eco-
driving model for mixed automated and human-driven vehicles at a signalised intersection. Transp. Res.
Part C Emerg. Technol. 95, 802–821. https://doi.org/10.1016/j.trc.2018.05.025
Zheng, F., Liu, C., Liu, X., Jabari, S.E., Lu, L., 2020a. Analyzing the impact of automated vehicles on
uncertainty and stability of the mixed traffic flow. Transp. Res. Part C Emerg. Technol. 112, 203–219.
https://doi.org/10.1016/j.trc.2020.01.017
Zheng, F., Liu, C., Liu, X., Jabari, S.E., Lu, L., 2020b. Analyzing the impact of automated vehicles on
uncertainty and stability of the mixed traffic flow. Transp. Res. Part C Emerg. Technol. 112, 203–219.
https://doi.org/10.1016/j.trc.2020.01.017
Zheng, F., Menendez, M., Li, X., Guler, I., van Zuylen, H., 2020c. Modeling and managing mixed traffic
with human-driven and automated vehicles. Transp. Res. Part C Emerg. Technol. 121, 102825.
https://doi.org/10.1016/j.trc.2020.102825
Zheng, Y., Ran, B., Qu, X., Zhang, J., Lin, Y., 2020. Cooperative Lane Changing Strategies to Improve Traffic
Operation and Safety Nearby Freeway Off-Ramps in a Connected and Automated Vehicles Environment.
IEEE Trans. Intell. Transp. Syst. 21, 4605–4614. https://doi.org/10.1109/TITS.2019.2942050
Zhou, H., Zhou, A., Li, T., Chen, D., Peeta, S., Laval, J., 2022. Significance of low-level control to string
stability under adaptive cruise control: Algorithms, theory and experiments. Transp. Res. Part C Emerg.
Technol. 140, 103697. https://doi.org/10.1016/j.trc.2022.103697
Zhou, J., Zhu, F., 2021. Analytical analysis of the effect of maximum platoon size of connected and
automated vehicles. Transp. Res. Part C Emerg. Technol. 122, 102882. https://doi.org/10/ghqdr4
Zhou, J., Zhu, F., 2020. Modeling the fundamental diagram of mixed human-driven and connected
automated vehicles. Transp. Res. Part C Emerg. Technol. 115, 102614. https://doi.org/10/gjkqwc
Zhou, L., Ruan, T., Ma, K., Dong, C., Wang, H., 2021. Impact of CAV platoon management on traffic flow
considering degradation of control mode. Phys. Stat. Mech. Its Appl. 581, 126193.
https://doi.org/10.1016/j.physa.2021.126193
Zhu, J., Tasic, I., 2021. Safety analysis of freeway on-ramp merging with the presence of autonomous
vehicles. Accid. Anal. Prev. 152, 105966. https://doi.org/10.1016/j.aap.2020.105966