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IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS 1
Modeling Heterogeneous Traffic Mixing Regular,
Connected, and Connected-Autonomous Vehicles
Under Connected Environment
Shaohua Cui , Feng Cao, Bin Yu , and Baozhen Yao
Abstract— As inter-vehicle communication and automatic
driving technology continue to develop, but are not yet popular,
regular vehicles, connected vehicles and connected autonomous
vehicles (CAVs) will coexist on the road for a long time. This
mixed traffic environment highlights the need to theoretically
analyze the impacts of some connected and autonomous tech-
nologies (i.e., accurate detection technology, inter-vehicle com-
munication technology, data storage technology and inter-vehicle
cooperation technology) on the stable operation of heteroge-
neous traffic. According to the characteristics of the vehicles
equipped with different technologies, this paper extends the
corresponding car-following models based on the optimal velocity
model. Through these analytical models, these connected and
autonomous technologies are quantified and the linear stability
analyses are conducted. Numerical simulation shows that the
inter-vehicle communication between three vehicles, and two
previous time-step data storage or two future time-step inter-
vehicle cooperation are sufficient to stabilize the mixed traffic.
As CAV penetration rates increase, the stability of heterogeneous
traffic is improved. Furthermore, the stability of heterogeneous
traffic is weakened when the size of the largest single fleet
increases. These theoretical results can serve as a quantitative tool
for scholars and vehicle designers before drawing any qualitative
conclusions of related technologies on heterogeneous fleet stability
to avoid wasting resources such as data storage capacity and
inter-vehicle communication ranges.
Index Terms—Car-following models, stability analysis, con-
nected environment, heterogeneous traffic.
I. INTRODUCTION
WITH the development of wireless communication
and sensing technology, vehicle-to-vehicle (V2V) and
vehicle-to-infrastructure (V2I) communication technologies
have been rapidly developed [1], [2]. Through V2V and
Manuscript received December 3, 2019; revised June 6, 2020,
September 7, 2020, November 24, 2020, and February 25, 2021; accepted
April 17, 2021. This work was supported by the National Key Research
and Development Program of China under Grant 2018YFB1600500. The
Associate Editor for this article was B. Ayalew. (Corresponding authors:
Bin Yu; Baozhen Yao.)
Shaohua Cui is with the School of Transportation Science and Engineering,
Beihang University, Beijing 100191, China (e-mail: shaoh_cui@buaa.edu.cn).
Feng Cao is with the School of Transportation Science and Engineering,
Beihang University, Beijing 100191, China, and also with the School of
Automotive Engineering, Dalian University of Technology, Dalian 116024,
China (e-mail: caofeng@mail.dlut.edu.cn).
Bin Yu is with the Beijing Advanced Innovation Center for Big Data and
Brian Computing (BDBC), School of Transportation Science and Engineering,
Beihang University, Beijing 100191, China (e-mail: yubinyb@buaa.edu.cn).
Baozhen Yao is with the State Key Laboratory of Structural Analysis for
Industrial Equipment, School of Automotive Engineering, Dalian University
of Technology, Dalian 116024, China (e-mail: yaobaozhen@dlut.edu.cn).
Digital Object Identifier 10.1109/TITS.2021.3083658
V2I communication technologies, the transfer of multi-vehicle
status information can be achieved at low cost [3]. However,
in the case that V2V and V2I communication technologies
are not yet popular, the road will experience a long period
of the heterogeneous vehicles with different communication
capability [4]. According to [5]’s definition and classification
of vehicles, regular vehicles (RVs) (human-driven vehicles
with no connectivity), connected vehicles (CVs) (human-
driven vehicles with connectivity), and connected autonomous
vehicles (CAVs) (autonomous vehicles with connectivity) will
appear on the road together. To distinguish between CVs and
the other two types of vehicles, CVs are defined as human-
driven vehicles with driver assistance systems (DASs). DASs
can transmit vehicle status information (i.e., velocity, headway
and so on) to other CVs or CAVs.
Through the above definitions, CVs contribute to create an
internet of vehicles where individual vehicles communicate
with other CVs or CAVs. Hence, human driver’s decision mak-
ing may be influenced and generally enhanced [5]. CVs may
be conducive to the efficient operation of CAVs, especially
at low CAV penetration rates. Hence, for the heterogeneous
traffic mixing three different vehicle types, it is necessary and
interesting to discuss the following three problems: whether
the effect of different connected and autonomous driving
technologies on heterogeneous traffic could be quantified
and proven theoretically; whether CVs help CAVs play an
effective role to promote the stable operation of heterogeneous
traffic; and whether it is necessary to provide all vehicles’
status information for each vehicle in the fleet with data
connectivity.
To theoretically analyze the impact of various connected and
autonomous technologies on traffic stability under connected
environment, this paper uses modeling methods to character-
ize different vehicle types. In most studies, different initial
car-following models are used to simulate different vehicle
types [5]. The results of [5] showed that CAVs were more
able to prevent the shockwave formation and propagation than
CVs in heterogeneous traffic. However, it is difficult to derive
the advantage of a vehicle type from the sensitivity analysis
of model parameters when the initial car-following models of
the three vehicle types are different.
Therefore, there is some research using the same initial
model to characterize the car-following behavior of dif-
ferent vehicle types. The initial model is adjusted based
on the corresponding vehicle characteristics. According to
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2IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
whether the vehicles could realize inter-vehicle communica-
tion, Xie et al. [6] used the intelligent driver model (IDM)
proposed by [7] as the initial model to establish the car-
following models of RVs and CVs. The results showed that
inter-vehicle communication promotes traffic stability and
improves efficiency. Similarly, Davis [8] proposed two CACC
algorithms based on the IDM and proved that 50% CACC
vehicles quickly eliminate fluctuations.
In the study of [6], only the communication between two
consecutive vehicles and the different reaction time of CVs
and human drivers were considered, while the effect of the
cooperation between multiple CVs and the memory of human
drivers on heterogeneous traffic was not explored. Using the
IDM as the initial model to characterize various connected
and autonomous technologies is complicated, and the proof of
traffic flow stability is also hard.
The optimal velocity (OV) model presented by [9] is simple
in form where the vehicle’s acceleration is derived by the
difference between the vehicle’s speed and a complex optimal
velocity function with headway as an independent variable.
The OV model generates many typical traffic phenomena, such
as the evolution of traffic congestion, and stop-and-go traffic
waves. However, the OV car-following model may generate
excessive acceleration, unreasonable deceleration, and non-
physical short-period oscillations [10]. Therefore, the OV
model is adjusted by considering different vehicle characteris-
tics. The adjustment of the OV model can be divided into two
aspects. One aspect is the adjustment considering the driver’s
characteristics (timid or aggressive) and the surrounding envi-
ronment (the pedestrians and vehicles on adjacent lanes).
The other aspect is that when inter-vehicle communication is
considered, the OV model is adjusted based on multi-vehicle
information. Therefore, based on the classic OV car-following
model, this paper designs the corresponding car-following
models considering the characteristics of the three different
vehicle types.
For CAVs, four kinds of connected and autonomous tech-
nologies, named as accurate detection technology, inter-vehicle
communication technology, data storage technology and inter-
vehicle cooperation technology, are quantified. Based on
fixed or mobile detection equipment and inter-vehicle com-
munication equipment, CAVs not only accurately obtain the
vehicle’s current status information from other CVs or CAVs,
such as, velocity, headway and so on, but also transmit the
current status information to other CVs or CAVs to achieve
inter-vehicle cooperation. Xie et al. [11] modified the OV
model considering the current status information of multiple
vehicles. Through numerical simulation, they proved that the
OV model adjusted based on the multi-vehicle current status
information can suppress traffic congestion more effectively
than the initial OV model. In addition to the multi-vehicle cur-
rent status information, the inter-vehicle cooperation based on
the shared information of future multi-time steps is designed.
Based on data storage devices, the vehicle’s status data of
past and current time steps can be stored to promote the
efficient operation of CAVs. In the studies of [12] and
[13], the driver’s forecast effect and memory were consid-
ered respectively, which corresponded to the vehicle’s status
information in the past and future time steps, respectively.
Through numerical simulation, they proved that the proposed
models suppressed traffic congestion and were beneficial to
traffic stability. Based on the studies of [11]–[13], a multi-
vehicle and multi-time step cooperation mechanism (MMCM)
is designed to modify the OV model. Furthermore, in this
paper, information transmission failure and instability of CAVs
are discussed.
Compared with the CAVs using accurate status informa-
tion obtained through mobile or fixed detection equipment,
human drivers have a certain psychological difference in the
perception of headway [14]. The RVs driven by human drivers
cannot share RV status information and cooperate with other
RVs. Similar to CAV data storage equipment, human drivers
can memorize the past data. Zheng et al. [13] used the driver’s
memory of the velocity from the previous single time step to
extend the OV model, and found that the modified model can
suppress traffic congestion. For RVs, based on the modified
OV model presented by [13], we design a previous multi-time
step memory mechanism (PMMM) to modify the OV model.
The PMMM uses driver’s previous multi-time step memory
for the accurate and easily available velocity as a variable to
weaken the effects of the complex optimal velocity function
on the OV car-following model.
CVs are defined herein as the human-driven vehicles with
DASs. A CV driver combines the recommended motion pro-
vided by DASs with his own estimated vehicle motion through
the compliance rate of DASs. Therefore, the higher compliance
rate of DASs, the more similar operation of CVs is to DASs.
When the CV driver’s compliance rate of DASs is zero,
the CV car-following model is exactly the same as that of
RVs. However, in this case, the status information of CVs can
be transmitted to other CAVs or CVs. Therefore, the impact
of the CVs completely disobeying DASs on the stability of
heterogeneous traffic is different from that of RVs.
Although a lot of research has been done on the stability of
heterogeneous traffic, we try to answer some of the still open
questions through the above definitions of the three different
vehicle types: (1) what are the types and time ranges of
the effective information that should be transmitted between
CVs or CAVs to improve the stability of heterogeneous traffic?
(2) what is the impact of CVs on the stable operation of
heterogeneous traffic during the transition period from purely
RVs to purely CAVs? (3) is it necessary to achieve full
communication between vehicles in the internet of the fleet
including CAVs and CVs?
This study attempts to make three contributions to the
literature. Firstly, according to the characteristics of the three
different types of vehicles, some connected and autonomous
technologies are quantified in the corresponding car-following
models. Hence, from the two perspectives of theoretical analy-
sis and numerical simulation, the impacts of the types and
time ranges of the effective information transmitted between
CVs or CAVs on the stability of heterogeneous traffic are
studied. Secondly, the CVs with DASs have been further
developed to provide a visual field to promote the transition
from purely RVs to purely CAVs. Finally, the effective inter-
communication ranges in the internet of the fleet consisting
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CUI et al.: MODELING HETEROGENEOUS TRAFFIC MIXING REGULAR, CONNECTED, AND CAVs 3
of CAVs and CVs are simulated at different CAV penetration
rates.
The remainder of this paper is organized as follows:
In Section II, related research is summarized and orga-
nized. The car-following models of RVs, CVs and CAVs
are discussed in Section III. The linear stability analyses
of the three presented models are conducted in Section IV.
Section V investigates the impact of the size of a single
CAV or CV or RV fleet on the stability of heterogeneous traffic
flow under different CAV penetration rates. By comparing the
CVs completely disobeying DASs with RVs, the effect of inter-
vehicle communication on the stability of heterogeneous traffic
flow is analyzed. Section VI makes a simple summarization
and proposes research prospects.
II. LITERATURE REVIEW
We summarize related research from two aspects: the impact
of different connected and autonomous technologies on het-
erogeneous traffic, and the adjustment of the OV model based
on different characteristics.
A. The Impact of Different Connected and Autonomous
Technologies on Heterogeneous Traffic
As Varaiya [15] proposed the concept of smart vehicles,
adaptive cruise control (ACC) and cooperative adaptive cruise
control (CACC) strategies have been receiving attention in
the field of transportation. Through gradually mature V2V
and V2I communication technologies, vehicles or drivers can
obtain more vehicle status information (i.e., speed, headway,
and so on) to assist driving in a short time [16], [17]. Through
the assistance of multi-vehicle information and short reaction
time, fuel consumption and emission were reduced [18],
[19], the acceleration and deceleration of vehicles were
smoother [20], and traffic stability was improved [21]. Zhao
and Sun [22] used VISSIM simulation to study the effects of
ACC and CACC vehicles on traffic stability under different
fleet sizes and penetration rates. Using five CACC vehicles as
a platoon, Kato et al. [23] studied the possibilities and potential
of V2V communication technology
In addition to the above studies considering only a sin-
gle type of intelligent vehicles (ACC or CACC vehicles),
the traffic mixing single or multiple smart vehicles and RVs
has been studied. Ngoduy [24] studied the stability threshold
of the heterogeneous traffic mixing single intelligent vehi-
cles and RVs where intelligent vehicles were not defined
as CVs or CAVs. They assumed that compared with RVs,
intelligent vehicles had the smaller safe headway, delay time,
and desired deceleration. Through the same IDM and different
parameters, they pointed out that the stability of heterogeneous
traffic increased with the increase in the penetration rates
of intelligent vehicles. Based on the IDM, Schakel et al. [7]
proposed a generic car-following model studying the traf-
fic mixing CVs and RVs. Through linear stability analysis,
the results showed that CVs contributed to the improvement of
the efficiency and stability of heterogeneous traffic. They only
studied the impact of the inter-vehicle communication between
two CVs and communication delay on heterogeneous traffic.
Milanés and Shladover [25] tested the string stability using
the four vehicles equipped with ACC devices. Here the car-
following model was based on the IDM. They tested the
impact of the CACC devices adding inter-vehicle communi-
cation capacity on string stability. They found that the con-
secutive vehicles using ACC devices made the fleet unstable,
while the vehicles using CACC devices made the fleet stable.
However, there was no analysis model and theoretical proof
for the effective ranges of inter-vehicle communication and
the types of effectively transmitted information.
In the above-mentioned studies about the traffic mixing sin-
gle intelligent vehicles and RVs, most studies did not specify
the role of different connected or autonomous technologies.
Talebpour and Mahmassani [5] studied the stability of the het-
erogeneous traffic mixing CAVs, CVs and RVs where the three
different types of vehicles were modeled based on different
car-following models. For RVs, driver’s perception uncertainty
and differences in congested and non-congested environments
were considered using the model proposed by [26]. The inter-
vehicle communication of CVs was conducted through the
model presented by [27]. Based on [28], the accurate detec-
tion capability and quick reaction of CAVs were considered.
Although many characteristics were considered in modeling
the different types of vehicles, the impact of the types and
ranges of different connected and autonomous technologies
on fleet stability was not quantified and proven. Therefore,
the simple OV car-following model is applied to quantify some
connected and autonomous technologies. As the excessive
acceleration and unreasonable deceleration may be generated
by the OV car-following model [10], the OV model has been
adjusted from the two aspects of driver characteristics and
multi-vehicle communication. We summarize relevant studies
about the adjustment of the OV model in the next section.
B. The Adjustment of the OV Model
There have been many studies only considering the influ-
ence of driver characteristics and other surrounding factors on
vehicle’s operation without using multi-vehicle information to
modify the OV model. Davis [8] introduced the reaction time
of drivers into the OV model and proved that the modified
OV model allowed longer reaction delay time and eliminated
non-physical short-period oscillations. Tang et al. [29] found
that when the speed of the preceding vehicle was too low, the
drivers would usually honk their horns to remind the driver of
the preceding vehicle. In this case, the driver of the preceding
vehicle would choose to accelerate or change lanes according
to the current traffic status. They defined the changed driving
behavior of the drivers hearing horns as the honk effect, and
modified the OV car-following model to study the effect of
honks on traffic stability. Then Wen et al. [30] studied the
influence of driver’s characteristics (timid or aggressive) on
traffic stability under the honk impact. Wang et al. [31] took
into account the differences in vehicle types and driver’s
driving skill, thereby introducing a random safety distance
with a certain probability into the OV model. They concluded
that traffic stability was reduced with the increase of the
safety distance of a single vehicle type. The effects of the
vehicles and pedestrians on adjacent lanes on the drivers on
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4IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
the main road were studied by [32] and [33], respectively. The
results showed that considering the impact of the vehicles and
pedestrians on adjacent lanes, the stability of the traffic on
main roads can be improved.
Through V2V and V2I communication technologies, each
driver can get multi-vehicle status information. The modified
OV model based on multi-vehicle information is called multi-
anticipative models. According to the differences in the use of
status information, multi-anticipative OV models are roughly
divided into two categories where the one category is based
on the speed and headway of multiple adjacent vehicles,
and the other category is based on the headway and speed
differences between the current vehicle and multiple preced-
ing vehicles. Nagatani [34] and Ge et al. [35] revised the
Newell car-following model by considering the headway of
the next-nearest-neighbor vehicle where the OV model can
be considered as the Taylor expansion of the Newell model.
Wilson et al. [36] extended the OV model by considering the
interaction of multiple preceding vehicles and proved that
multi-vehicle information reduced the unrealistic large accel-
eration and deceleration generated by the initial OV model.
Yu et al. [37] took the headway and speed differences of multi-
ple vehicles into consideration in the OV model, and linear and
nonlinear stability analyses were performed. Lenz et al. [38]
considered the speed and headway differences between the
current vehicle and multiple preceding vehicles in the OV
model, and proved that the traffic stability can be improved
by considering the multi-vehicle status information.
Therefore, the initial OV model is modified to quantify the
impact of different connected and autonomous technologies
on fleet stability according to the corresponding characteristics.
Based on the OV model, CAV’s accurate detection technology,
inter-vehicle communication technology, data storage technol-
ogy and inter-vehicle cooperation technology are quantified.
According to the corresponding CAV technology, the RV
driver’s psychological perception differences in headway and
memory of previous multi-time steps are quantified. The CVs
with DASs are designed to expect it to help transition from
purely RVs to purely CAVs.
III. CAR-FOLLOWING MODEL FORMULATION
In this section, the car-following models corresponding to
RVs, CAVs, and CVs are shown. The RV and CAV car-
following models are designed in Sections III-A and III-B,
respectively. For CVs, the CV driver combines the vehicle’s
motion recommended by DASs with the driver’s own esti-
mated motion by the compliance rate of DASs to derive the
operation of CVs. For the sake of simplicity, the vehicle
motion suggested by DASs is the same as that of CAVs.
Therefore, in Section III-C, the car-following model of CVs
is briefly introduced.
A. Modeling RVs
The initial OV model was presented by [9], i.e.,
dvn(t)
dt =1
τ[V(xn(t)) −vn(t)](1)
In (1), vn(t),xn(t)=xn+1(t)−xn(t)and V(xn(t))
are the nth vehicle’s velocity, headway and optimal velocity
depending on its headway at time t.xn+1and xnrepre-
sent the n+1th and nth vehicle’s positions respectively. τ
denotes the reaction time of RV drivers. As suggested by [32],
[33], the following optimal velocity function V(xn(t)) is
chosen:
V(xn(t)) =vmax
2[tanh(xn(t)−hc)+tanh(hc)](2)
where vmax refers to the vehicle’s maximum velocity and hc
presents the safe distance of consecutive vehicles.
The RV car-following model with the PMMM is designed
as:
dvRV
n(t)
dt =dvn(t)
dt +μ1
τRV
fRV
PMMM (vn(t−τRV ), ···,
vn(t−P1τRV))(3)
where P1is the memorized or stored previous time step
duration and μ1∈[0,1]is the weight of the PMMM.
We linearly weight the difference between the expected
velocity vexp
RV and the velocity vn(t−lτRV )at the previous
lth time step where l∈{1,2,···P1}, i.e.,
fRV
PMMM (vn(t−τRV ), ···,v
n(t−P1τRV))
=
P1
l=1
αlvexp
RV −vn(t−lτRV )(4)
vexp
RV may be related to the driver’s characteristics, travel
purpose, momentary risk perception, and so on [39]. vexp
RV is set
to a fixed value and equals the maximum speed of RVs. For the
time-varying vexp
RV , readers refer to references [39]–[40]. In (4),
αlrepresents the weight coefficient of difference between the
expected velocity and the velocity at the previous lth time
step. For αl, we assume that it satisfies the following two
properties:
(1) αlis monotonically decreasing. The farther the velocity
difference from the current time step, the smaller the influence
on the acceleration update of the current time step is.
(2)
P1
l=1
αl=1
Hence, if P1=1, α1=1. When P1>1, following [11],
the value of αlis defined as αl=6/7l,l= P1
1/7P1−1,l=P1
.
Therefore, Equation (3) is rewritten as follows:
dvRV
n(t)
dt =1
τRV
V(xn(t)) +μ1
P1
l=1
αlvexp
RV −vn(t−lτRV )−vn(t)
(5)
By Taylor expansion and ignoring high-order items, the vari-
able vn(t−lτRV )is simplified as:
vn(t−lτRV )=vn(t)−lτRV
dvRV
n(t)
dt (6)
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CUI et al.: MODELING HETEROGENEOUS TRAFFIC MIXING REGULAR, CONNECTED, AND CAVs 5
By combining (5) with (6), we get:
(1−μ1
P1
l=1
αll)dvRV
n(t)
dt
=1
τRV V(xn(t)) +μ1vexp
RV −(μ1+1)vn(t)(7)
The asymmetric forward difference is used to discretize (7):
xn(t+2τRV)=xn(t+τRV )+τRV
dxn(t+τRV )
dt
=xn(t+τRV)+τRV
dxn(t)
dt
+τ2
RV
d2xn(t)
dt2
=xn(t+τRV)+xn(t+τRV )−xn(t)
+τ2
RV dvRV
n+1(t)
dt −dvRV
n(t)
dt (8)
By combining (7) with (8), we get:
(1−μ1
P1
l=1
αll)xn(t+2τRV )−2xn(t+τRV)+xn(t)
τRV
=V(xn+1(t))−V(xn(t)) −(μ1+1)dxn(t)
dt (9)
B. Modeling CAVs
Compared with RVs, CAVs have more accurate data detec-
tion capability, greater data storage capacity, inter-vehicle
communication capability, and the possibility of data acquisi-
tion instability or even loss. Therefore, the CAV car-following
model with the MMCM is formulated as follows:
dvCAV
n(t)
dt =1
τCAV
fCAV
CM xn(t), ···,xn+P2−1(t)+μ2fCAV
PM
×(vn(t−τCAV ), ···,
×vn(t−P1τCAV))+μ3fCAV
FM (xn(t+στ
CAV), ··· ,
×xn+P3+1(t+στ
CAV )−vn(t)(10)
P2and P3respectively represent the number of the vehicles
providing the headway information at the current ttime step
and the number of the cooperation vehicles with the headway
information at the future σth time step where σ∈N∗.P2
and P3can take the same value equaling to the number of
the vehicles that the current vehicle communicates with. P1
defined in (3) is the stored previous time step duration. μ2is
the weight of the cooperation mechanism fCAV
PM based on the
velocity of previous multi-time steps, and μ3is the weight of
the cooperation mechanism fCAV
FM based on the future multi-
vehicle headway.
The first part fCAV
CM xn(t), ···,xn+P2−1(t)is defined
as:
fCAV
CM xn(t), ···,xn+P2−1(t)
=V(
P2
j=1
ξjβjxn+j−1(t)) (11)
Fig. 1. Vehicle status information transfer between different vehicle types.
βjis the weight of the jth preceding vehicle headway with
the same properties and value as αlwhere j∈{1,2,··· ,P2}.
ξjis a binary variable. If the jth preceding vehicle infor-
mation is obtained, ξj=1; otherwise ξj=0. We assume
that the information of the first preceding vehicle must be
obtained, that is ξj=1. When the vehicle is a CAV or CV
and j>1, the value of ξjdepends on whether the information
transfer of this vehicle is stable at this time. If the jth
preceding vehicle is a RV and j>1, the vehicle’s information
cannot be obtained, i.e., ξj=0. We assume that the status
information of the CAVs and CVs in the fleet is transmitted
stably. The values of ξjin the heterogeneous fleet are shown
in Fig. 1.
Following (4), we define the cooperation machine based on
the velocity of previous multi-time steps as:
fCAV
PM (vn(t−τCAV), ··· ,v
n(t−P1τCAV))
=
P1
l=1
αl[vexp
CAV −vn(t−lτCAV)](12)
For the cooperation machine accounting for the multi-
vehicle headway of future time steps, we still use linear
weighting:
fCAV
FM xn(t+στ
CAV), ··· ,xn+P3−1(t+στ
CAV)
=V(
P3
j=1
ξjρjxn+j−1(t+στ
CAV )) (13)
where ρjis the weight coefficient of the jth preceding vehicle
headway at the future σth time step where σ∈N∗and
j∈{1,2,···,P3}. We assume that ρjand αlhave the same
characteristics and values.
Therefore, Equation (11) is written as:
dvCAV
n(t)
dt =1
τCAV
⎡
⎣V⎛
⎝
P2
j=1
ξjβjxn+j−1(t)⎞
⎠+μ2
P1
l=1
αl
vexp
CAV −vn(t−lτCAV)+μ3V
⎛
⎝
P3
j=1
ξjρjxn+j−1(t+στ
CAV)⎞
⎠−vn(t)⎤
⎦
(14)
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Through Taylor expansion and ignoring high-order terms,
we obtain:
V⎛
⎝
P3
j=1
ξjρjxn+j−1(t+στ
CAV)⎞
⎠
=V⎛
⎝
P3
j=1
ξjρjxn+j−1(t)⎞
⎠+στ
CAV
×V⎛
⎝
P3
j=1
ξjρjxn+j−1(t)⎞
⎠
P3
j=1
ξjρjvn+j−1(t)
(15)
By combining (14) with (15), we obtain:
(1−μ2
P1
l=1
αll)dvCAV
n(t)
dt =1
τCAV
⎡
⎣V⎛
⎝
P2
j=1
ξjβjxn+j−1(t)⎞
⎠+μ2vexp
CAV −
×(1+μ2)vn(t)+μ3⎡
⎣V⎛
⎝
P3
j=1
ξjρjxn+j−1(t)⎞
⎠
+στ
CAV V⎛
⎝
P3
j=1
ξjρjxn+j−1(t)⎞
⎠
×
P3
j=1
ξjρjvn+j−1(t)⎤
⎦⎤
⎦(16)
Using the asymmetric forward difference, Equation (16) is
discretized as (17), as shown at the bottom of the page.
C. Modeling CVs
We assume that the operation of CVs is determined by
two parts together where one part is the vehicle’s motion
based on the CV driver’s judgment, and the other part is the
operation provided by DASs. When the CV driver decides
the vehicle’s motion based entirely on his own judgment,
the driver can only obtain the information of a single preceding
vehicle and the previous data of his own vehicle. Therefore,
we assume that the car-following model decided by the CV
driver is the same as the RV car-following model dvRV
n(t)
dt
shown in Section III-A. The motion suggested by DASs
is based on a variety of information, such as multi-vehicle
previous and future status information, and previous data of
its own vehicle. We assume that the car-following model of
DASs is exactly the same as the CAV car-following model
dvCAV
n(t)
dt discussed in Section III-B. The CV driver linearly
weights the two car-following models through the compliance
rate λCof DASs to determine the operation of CVs, i.e.,
dvCV
n
dt =λCdvCAV
n
dt +(1−λC)dvRV
n
dt .
The higher λC, the closer the CV operation is to CAVs.
When λC=0, the CV motion is completely determined by its
driver, but the CVs with communication capacity can provide
vehicle status information for other CVs and CAVs. There is a
certain difference in the stability of the fleet mixing the same
proportion of the CVs completely disobeying DASs and RVs.
IV. LINEAR STABILITY ANALYSIS
In this section, homogenous fleet stability is analyzed.
As the car-following model of CVs combines the RV and
CAV car-following models by compliance rates, the stabil-
ity of homogenous RV traffic flow and homogenous CAV
traffic flow is studied in Section IV-A and Section IV-B,
respectively.
A. Linear Stability Analysis of Homogenous RV Traffic
The solution of steady-state RV traffic of (9) is:
xn,0(t)=hn +V(h)t,h=L/N(18)
where hdenotes the average headway, which is equal to the
road length Ldivided by the total number Nof RVs.
To generate a small disturbance, a small deviation yn(t)is
supposed from the steady-state solution xn,0(t), i.e., xn(t)=
xn,0(t)+yn(t). By this way, a linear equation is obtained
from (9):
(1−μ1
P1
l=1
αll)yn(t+2τRV)−2yn(t+τRV )+yn(t)
τRV
=V(h)(yn+1(t)−yn(t))−(μ1+1)dyn(t)
dt (19)
(1−μ2
P1
l=1
αll)(xn(t+2τCAV)−2xn(t+τCAV)+xn(t))τCAV
=V⎛
⎝
P2
j=1
ξjβjxn+j(t)⎞
⎠−V⎛
⎝
P2
j=1
ξjβjxn+j−1(t)⎞
⎠−(μ2+1)dxn(t)
dt
+μ3⎡
⎣V⎛
⎝
P3
j=1
ξjρjxn+c(t)⎞
⎠−V⎛
⎝
P3
j=1
ξjρjxn+j−1(t)⎞
⎠+στ
CAV ⎡
⎣V⎛
⎝
P3
j=1
ξjρjxn+j(t)⎞
⎠
×
P3
j=1
ξjρj
dxn+j(t)
dt −V⎛
⎝
P3
j=1
ξjρjxn+j−1(t)⎞
⎠
P3
j=1
ξjρj
dxn+j−1(t)
dt ⎤
⎦⎤
⎦(17)
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CUI et al.: MODELING HETEROGENEOUS TRAFFIC MIXING REGULAR, CONNECTED, AND CAVs 7
In (19), V(h)refers to the deviation of V(x)at x=
h. Assuming yn=Aexp(ikn +zt)and using the Fourier
Ansatz, we rewrite (19) as follows:
(1−μ1
P1
l=1
αll)e2zτRV −2ezτRV +1
τRV
=V(h)(eik −1)−(μ1+1)z(20)
yn=Aexp(ikn +zt)is the linear waves of strict
periodicity. i=√−1 represents the imaginary unit. Aand
zrespectively denote the oscillation amplitude and the growth
rate of linear waves. k∈[−π, π]is the dimensionless wave
number, which denotes the phase shift of traffic waves from
one vehicle to the next at a given time instant. By solving
(20) with respect to z, the leading term of zis the order of
ik.Whenik →0, z→0. We expand zas a long wave, i.e.,
z=z1(ik)+z2(ik)2+··· where z1and z2are the first- and
second-order terms of expanded zrespectively. By substituting
this long wave formulation into (20) and ignoring the high-
order terms, we obtain (21) and (22), as shown at the bottom
of the page.
As z2>0, the flow is stable; if z2<0, the flow is unstable.
The neutral stability points are generated when z2=0, that is,
ηRV =(τRV )−1
=2(1−μ1
P1
l=1
αll)V(h)(μ1+1)2(23)
In (23), ηRV =(τRV )−1is the reactive coefficient of RV
drivers. According to the stability condition (z2>0),we
conclude that the unstable area ηUA
RV satisfies ηUA
RV <η
RV .
It is clear that with the increase of P1,ηRV decreases,
which means that the stable area increases with the increase
of previous velocity data. For the sensitivity analysis of μ1,
we set f(μ1)=1−μ1
P1
l=1
αll(μ1+1)2, and easily
obtain:
∂f(μ1)
∂μ1=
(μ1−1)
P1
l=1
αll−2
(μ1+1)3(24)
As 0 ≤μ1≤1, ∂f(μ1)
∂μ1<0 obviously. Therefore, the stable
area increases with the increase of μ1. To demonstrate the
effect of P1and μ1on the stable area, neutral curves are
showninFig.2wherevmax =2m/sand hc=4m.When
h=hc,V(h)reaches the maximum 0.5vmax.(hc,η
c
RV )is the
critical point of these curves with different parameters. ηc
RV
is the value of (23) when h=hc. It is clearly observed from
Fig. 2 that as P1and μ1increase, neutral curves move down-
ward and the stable area increases. Therefore, the PMMM
improves RV fleet stability.
Fig. 2. The neutral curves with different P1and μ1.
B. The Linear Stability Analysis of Homogenous CAV Traffic
Following the linear stability analysis of homogenous RV
traffic, a small deviation yn(t)at the steady state of CAV traffic
is supposed. A linear equation can be obtained from (17), i.e.,
(25), as shown at the bottom of the next page.
In (25), V(h)is the deviation of V(x)at x=h.
Assumingyn=Aexp(ikn +zt), we rewrite (25) as follows:
(1−μ2
P1
l=1
αll)e2zτCAV −2ezτCAV +1
τCAV =V(h)
⎛
⎝
P2
j=1
ξjβjejik−e(j−1)ik⎞
⎠−(μ2+1)z
+μ3⎡
⎣V(h)⎛
⎝
P3
j=1
ξjρjejik−e(j−1)ik⎞
⎠+zστ
CAV V(h)
⎛
⎝
P3
j=1
ξjρjejik−e(j−1)ik⎞
⎠⎤
⎦(26)
Equation (26) and Equation (20) are similar in the form,
so we also expand zinto a long wave form, i.e., z=z1(ik)+
z2(ik)2+···. By substituting this formulation into (26) and
neglecting the high-order terms, we obtain:
−(1−μ2
P1
l=1
αll)z2
1k2τCAV =V(h)
⎛
⎝
P2
j=1
ξjβjik +k2(1−2j)
2⎞
⎠−(μ2+1)
×z1(ik)+z2(ik)2+μ3
⎡
⎣V(h)⎛
⎝
P3
j=1
ξjρjik+k2(1−2j)
2⎞
⎠−z1στ
CAV V(h)k2
⎤
⎦
(27)
z1=V(h)(μ1+1)(21)
z2=V(h)(μ1+1)2−2(1−μ1
P1
l=1
αll)V2(h)τRV 2(μ1+1)3(22)
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Fig. 3. The neutral curves with different P2(μ2=0.1,μ
3=0.1,P1=
P2=1andσ=2).
By solving (27), we obtain (28) and (29), as shown at the
bottom of the next page.
If z2>0, the flow is stable, while if z2<0, the flow is
unstable. The neutral points are generated when z2=0, that
is, (30), as shown at the bottom of the next page.
ηCAV =(τCAV)−1denotes the reactive coefficient of CAVs.
Obviously, with the increase of μ2,P1,P2,P3or σ, neutral
curves drop. We set vmax =2m/sand hc=4m,andshow
the neutral curves with different P2in Fig. 3. It can be seen
from Fig. 3 that as P2increases, neutral curves decrease and
the stable area becomes larger. The stability of CAV traffic is
improved with the current multi-vehicle headway information.
The effects of μ2and P1on neutral curves shown in Fig. 4 are
similar to those in Fig. 2. As the previous time step duration
P1increases or μ2increases, the stability of CAV traffic is
improved. We analyze the effects of future cooperation time
steps σand the number P3of the vehicles providing future
cooperation headway information on neutral curves in Fig. 5.
From Fig. 5, the increases in future cooperation time steps and
in the number of the vehicles providing future cooperation
headway information lower neutral curves and expand the
stable area. Therefore, current multi-vehicle headway informa-
tion, future cooperation headway data and time steps, previous
velocity time duration, and the intensity μ2of the cooperation
mechanism based on previous velocity data contribute to CAV
traffic stability.
Fig. 4. The neutral curves with different μ2and P1(μ3=0.1,P2=P3=1
and σ=2).
Fig. 5. The neutral curves with different P3and σ(μ2=0.1,μ
3=0.2and
P1=P2=2).
To further compare CAVs with RVs, we assume that the
intensity of the cooperation mechanism based on future infor-
mation is 0, i.e., μ3=0. We can get (31), as shown at the
bottom of the next page.
We show the neutral curves without the impacts of the
cooperation mechanism based on future information in Fig. 6.
From the figure, it can be seen that as the number of the
vehicles providing information increases, the curves drop and
the stable area increases. This means that even if CAVs do
(1−μ2
P1
l=1
αll)yn(t+2τCAV)−2yn(t+τCAV)+yn(t)
τCAV =V(h)
P2
j=1
ξjβj
×yn+j(t)−yn+j−1(t)+μ3⎡
⎣V(h)⎛
⎝
P3
j=1
ξjρjyn+j(t)−yn+j−1(t)⎞
⎠
+στ
CAV ⎡
⎣V⎛
⎝
P3
j==1
ξjρjyn+j(t)⎞
⎠
P3
j=1
ξjρj
dyn+j(t)
dt −V⎛
⎝
P3
j=1
ξjρjyn+j−1(t)⎞
⎠
×
P3
j=1
ξjρj
dyn+j−1(t)
dt ⎤
⎦⎤
⎦−(μ2+1)dyn(t)
dt (25)
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CUI et al.: MODELING HETEROGENEOUS TRAFFIC MIXING REGULAR, CONNECTED, AND CAVs 9
Fig. 6. The neutral curves with different μ2and P2(μ3=0and P1=1).
not have future cooperation headway information, the CAV
traffic flow becomes more stable based on current multi-
vehicle headway.
Next, we analyze the impact of μ3on the stable area. Some
symbols are defined to simplify (30):
A1=2(1+μ2)V(h)σ (32)
A2=2(1−μ2
P1
l=1
αll)V(h)(33)
A3=(1+μ2)2
P2
j=1
ξjβj(2j−1)(34)
A4=(1+μ2)2
P3
j=1
ξjρj(2j−1)(35)
Because P2and P3depend on the number of the vehicles
providing status information, they usually take the same value,
i.e., P2=P3. Hence, A3=A4>0, so (30) is simplified as:
f(μ3)=−μ3(1+μ3)A1+(1+μ3)2A2(A3+μ3A3)
(36)
It is easy to prove that:
df(μ3)
dμ3
=μ2
3A3(A2−A1)+2μ3A3(A2−A1)+A3(A2−A1)
(μ3A3+A3)2
(37)
As the denominator is definitely greater than 0, we define:
F(μ3)=μ2
3A3(A2−A1)+2μ3A3(A2−A1)
+A3(A2−A1)(38)
Because σ∈N∗, we obtain:
A2−A1=2V(1−μ2
P1
l=1
αll)−(1+μ2)σ
=2V(1−σ)−μ2(
P1
l=1
αll+σ)
<0 (39)
Hence, F(μ3)is a quadratic function. The symmetry axis of
this quadratic function is XSA =−2A3(A2−A1)
2A3(A2−A1)=−1. F(μ3)
decreases with the increase of μ3∈[0,1].Weget F(μ3=
0)=(A2−A1)A3<0andF(μ3=1)=4A3(A2−A1)<0.
Therefore, F(μ3)<0anddf(μ3)
dμ3<0 obviously when
μ3∈[0,1]and σ∈N∗, which means that when there are
future cooperation time steps (i.e., σ∈N∗),μ3∈[0,1]has a
positive effect on CAV traffic stability. In other words, as μ3∈
[0,1]increases, f(μ3)decreases and the stable area increases.
The neutral curves with different μ3are shown in Fig. 7. From
z1=(1+μ3)(1+μ2)V(h)(28)
z2=−2(1−μ2
P1
l=1
αll)(1+μ3)2V(h)2τCAV −V(h)
P2
j=1
ξjβj(1−2j)(1+μ2)2
2(1+μ2)3
+−μ3V(h)
P3
j=1
ξjρj(1−2j)(1+μ2)2+2μ3(1+μ3)(1+μ2)V(h)2στ
CAV
2(1+μ2)3(29)
ηCAV =1
τCAV =−2μ3(1+μ3)(1+μ2)V(h)σ +2V(h)(1−μ2
P1
j=1
αll)(1+μ3)2
(1+μ2)2
P2
j=1
ξjβj(2j−1)+(1+μ2)2μ3
P3
j=1
ξjρj(2j−1)
(30)
ηCAV =(τCAV)−1=2V(h)(1−μ2
P1
l=1
αll)⎛
⎝(1+μ2)2
P2
j=1
ξjβj(2j−1)⎞
⎠(31)
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10 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
Fig. 7. The neutral curves with different μ3(μ2=0.1,P1=P2=P3=1
and σ=2).
the figure, we conclude that when there are future cooperation
time steps, the CAV traffic stability increases with μ3increase.
V. N UMERICAL SIMULATION
In this section, the numerical simulation is performed based
on periodic boundary conditions. In Section V-A, the stability
of homogenous and heterogeneous traffic is studied. In Section
V-B, taking a fleet of RVs and CAVs in different mixing ratios
as an example, the influence of the size of a single CAV /
RV fleet on the stability of heterogeneous traffic is explored.
In Section V-C, how inter-vehicle communication affects the
stability of heterogeneous traffic is explored.
A. Stability Analysis of Homogenous/ Heterogeneous Traffic
1) Homogenous RV Traffic Flow: For the RV car-following
model, the PMMM intensity μ1and the time duration P1of
previous velocity data are the two main parameters. Therefore,
the influence of these two parameters on the stability of
homogeneous RV traffic is explored. Under periodical bound-
ary conditions where there is no ramp on the circle road,
we assume that initial conditions are:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
xn(0)=xn(1)=x0,n= 0.5N,
n= 0.5N+1
xn(0)=xn(1)=x0+0.1,n=0.5N
xn(0)=xn(1)=x0−0.1,n=0.5N+1
(40)
In (40), N=200 is the number of RVs. The tests
are conducted under two typical traffic jam scenarios with
different maximum speed. In the first scenario, the maximum
speed is vmax =vexp
RV =3.5m/s(e.g., 12.6km/h) where the
low maximum speed falls into the speed range of vehicles
during traffic jam in cities, such as Zurich, Switzerland [41].
In the second scenario, the maximum speed is vmax =
vexp
RV =33.6m/s(e.g., 120.96km/h) [42]. In the first scenario,
according to the references [33] and [43], we assume that the
average headway x0=4.0m, the road length L=800m,
τRV =0.5s,hc=4.0m,andP1=1. In the second scenario,
based on the reference [42], x0=25.0m,L=5000m,
τRV =0.1s,hc=25.0m,and P1=1. In order to avoid the
TAB L E I
THE HEADWAY STATI S T I CS WITH LOW MAXIMUM SP EED
(UNIT:METERS)
TAB L E I I
THE HEADWAY STATI S T I CS WITH HIGH MAXIMUM SPEED
(UNIT:METERS)
Fig. 8. Velocity profiles at the 10,000th time step with different P1and μ1.
interference of the initial disturbance designed in (40) on test
results, the headway statistics (i.e., the maximum, minimum,
median, and upper and lower quartiles of headways) at the
10,000th time step under the two scenarios are respectively
showninTableIandTableII.
From Table I and Table II, we observe that headway is no
longer equal due to the small initial disturbance designed in
(40). The gradual decline of congestion with the increase of
μ1indicates that the PMMM contributes to RV traffic stability.
When μ1is small, traffic congestion cannot be completely
eliminated. We should consider the influence of a strong
PMMM on RV traffic stability. The headway evolution law has
nothing to do with the maximum speed. Hence, the following
tests are based on the parameters with low maximum speed.
To further investigate the impact of memory time duration
P1and PMMM intensity μ1on RV traffic stability, we test four
cases using the parameters with low maximum speed. In the
first case, there is no the PMMM (i.e., μ1=0), which means
that the RV car-following model is equivalent to the OV model
(see Section III-A). The other three cases are: μ1=0.18 is
fixed and P1is equal to 1, 2, and 3, respectively. We show the
results at the 10,000th time step for the four cases in Fig. 8.
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CUI et al.: MODELING HETEROGENEOUS TRAFFIC MIXING REGULAR, CONNECTED, AND CAVs 11
TABLE III
THE HEADWAY STATISTICS FOR THE FOUR
HCAV-CASES (UNIT:METERS)
From Fig. 8, we see that even if there is only one memory
time step (i.e., P1=1), the RV traffic stability is improved
greatly. The maximum velocity difference under two and three
memory time steps is less than 0.5×10−3m/s, which means
that both cases have almost the same impact on RV traffic
stability. Hence, the PMMM contributes to RV traffic stability,
and two past memory time steps achieve stable RV traffic.
2) Homogenous CAV Traffic Flow: To compare with the RV
car-following model, we test four cases for the homogenous
CAV traffic where we assume that CAVs only communicate
with two CAVs, i.e., P2=2. For the first and second cases
(i.e., HCAV-Case I and HCAV-Case II), we assume that CAVs
only update acceleration based on previous data, and cannot
use future cooperative headway information, i.e., μ3=0.
Similar to μ1,μ2is the weight of the cooperation mechanism
based on the velocity of previous multi-time steps. Let μ2
equal 0.05 and 0.1 for the first and second cases respec-
tively. For the third and fourth cases (i.e., HCAV-Case III
and HCAV-Case IV), we consider the impact of the inter-
vehicle cooperation based on future headway information on
traffic stability, and fix μ2as a constant 0.05. The third case
is a strong cooperation mechanism (μ3=0.15)based on
future headway information with short future cooperation time
duration (σ=3), and the fourth case is a weak cooperation
mechanism (μ3=0.10)with long cooperation time duration
(σ=5). Other parameters are the same as those with
low maximum speed set in Section V-A-1. Under periodical
boundary conditions, the headway statistics at the 10,000th
time step for the four cases are shown in Table III.
Comparing Table III with Table I, we see that the CAVs
with multi-vehicle communication more easily achieve traffic
stability than RVs. The stability of the CAV traffic with
μ2=0.1 under HCAV-Case II is the same as that of the RV
traffic with μ1=0.2. Therefore, inter-vehicle communication
improves traffic stability. Comparing the first, third and fourth
cases, we find that the unstable traffic under the first case
becomes stable when the cooperation mechanism based on
future headway information is considered. The cooperation
mechanism based on future headway information contributes
to traffic stability.
3) Homogenous CV Traffic Flow: As the car-following
model of CVs combines the CAV car-following model with
the RV car-following model by the compliance rate λC, based
on the above tests, we select the parameters of unstable CAV
and RV traffic to study the impacts of λCon CV traffic
stability. We set μ1=μ2=0.05, P2=2, and μ3=0. The
parameters with low maximum speed set in Section V-A-1 are
adopted. The headway statistics at the 10,000th time step are
shown in Table IV. Comparing Tables I, III and IV, we see
TAB L E I V
THE HEADWAY STATI S T I CS WITH DIFFERENT
COMPLIANCE RATES (UNIT:METERS)
Fig. 9. The headway SD for heterogeneous traffic.
that the headway status generated by the CV car-following
model is between that of CAV and RV car-following models.
Furthermore, with the increase of λC, the headway status of
CVs is closer to that of CAVs.
4) Heterogeneous Traffic Flow: Theimpactofthesizeofa
single fleet on heterogeneous traffic stability at different CAV
penetration rates will be studied in Sections V-B and V-C. Here
we take the fleet of evenly mixed 50% CAVs and 50% RVs,
and the fleet of evenly mixed 50% CAVs and 50% CVs as
an example. To show the role of inter-vehicle communication,
we assume that CAVs do not get future headway information,
i.e., μ3=0 and the compliance rate λCof CVs is zero. For the
fleet of evenly mixed 50% CAVs and 50% RVs, CAVs only
get the information of two vehicles, namely its preceding RV
and a CAV in front of the RV. Although the fleet of evenly
mixed 50% CAVs and 50% CVs achieve full connectivity,
we assume that CAVs acquire the information of three pre-
ceding vehicles for updating acceleration. We set the previous
data duration as 1 and 2 time steps, i.e., P1=1and P1=2
respectively.
We use the standard deviation (SD) of headway as the basis
for judging heterogeneous traffic stability where the SD unit
is in meters. Test results are shown in Fig. 9 where ‘PV =2’
denotes the test results of the fleet of mixed CAVs and RVs,
and ‘PV =3’ denotes the test results of the fleet of mixed
CAVs and CVs. It can be seen from Fig. 9 that increasing
the intensity of the mechanism based on previous data, the
contribution of the CAVs with inter-communication capacity
to the stability of heterogeneous traffic is higher than that of
RVs. Moreover, the increase in previous data duration helps to
improve traffic stability. Comparing Fig. 8 with Fig. 9, we find
that when inter-vehicle communication is achieved for some
vehicles in the heterogeneous fleet, the memorized or stored
past time duration can be further reduced to 1 time step
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12 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
Fig. 10. The headway SD of the heterogeneous traffic of mixed RVs, CVs
and CAVs.
to make heterogeneous traffic stable. Increasing the same
intensity of the mechanism based on previous data, the impact
of CAVs on heterogeneous traffic stability is higher than that
of CVs. The stability of the heterogeneous traffic of CAVs
and CVs is higher than that of the fleet of CAVs and RVs
at the same intensity of the mechanism based on previous
data. Hence, inter-vehicle communication helps to stabilize
heterogeneous traffic.
Based on the above-mentioned test results, the stability of
a heterogeneous fleet with a mix of RVs, CVs and CAVs
is tested. The first half of the heterogeneous fleet evenly
mixes 50% CAVs and 50% CVs, and the second half of
the heterogeneous fleet evenly mixes 50% CAVs and 50%
RVs. The headway SD is used to judge the heterogeneous
fleet stability. μ1and μ2are the weights of the PMMM
and the cooperation mechanism fCAV
PM in the car-following
model of RVs and CAVs, respectively. Both the PMMM and
the cooperation mechanism fCAV
PM are related to the mem-
orized or stored previous time step duration P1. Therefore,
μ1and μ2are adjusted simultaneously. The test results of the
heterogeneous fleet are shown in Fig. 10 where ‘FH’ represents
the headway SD of the first half of the heterogeneous fleet,
‘SH’ represents the headway SD of the second half of the
heterogeneous fleet, and ‘EF’ represents the headway SD of
the entire heterogeneous fleet.
Comparing Fig. 10 with Fig. 9, we get the similar results.
With the increase of μ1,μ2,orP1, the stability of the
heterogeneous fleet is improved. Under the same parameters
μ1,μ2,andP1, ‘FH’ is lower than ‘SH’, which means that
the stability of the heterogeneous fleet of evenly mixed CAVs
and CVs is higher than that of the heterogeneous fleet of
evenly mixed CAVs and RVs. Comparing Fig. 9 and Fig. 10,
we find that under the same parameters μ1,μ2,andP1,
the stability of the heterogeneous fleet with a mix of RVs, CVs
and CAVs is better than that of a fleet of only mixed CAVs and
RVs, but worse than that of a fleet of only mixed CAVs and
CVs. In addition, ‘FH’ is greater than the headway SD of a
fleet of only mixed CAVs and CVs, and ‘SH’ is less than the
headway SD of a fleet of only mixed CAVs and RVs. Hence,
inter-vehicle communication makes the heterogeneous fleet
stable.
Fig. 11. The reduced percentages of headway/velocity SD at low CAV
penetration rates.
Fig. 12. The increased percentages of headway/velocity SD at high CAV
penetration rates.
B. Impact of the Size of a Single Fleet on Heterogeneous
Fleet Stability
Taking a fleet of RVs and CAVs with different mixing ratios
as an example, the impact of the size of a single fleet on the
stability of heterogeneous traffic is studied. To facilitate the
reader’s observation, in this section, we adjust the number
Nof vehicles in Section V-A-1 to 100. We first study the
impact of the size of a single CAV fleet on heterogeneous
fleet stability at low CAV penetration rates. According to the
suggestions in Fig. 9, we choose the parameters of the unstable
fleet of mixed RVs and CAVs. Let μ1=0.1 be unchanged
and μ2equals 0.03, 0.04 and 0.05 respectively. The remaining
parameters are the same as those with low maximum speed
set in Section V-A-1. It is worth noting that under periodical
boundary conditions the first vehicle is connected to the 100th
vehicle. Heterogeneous fleet stability is not related to the
spatial distribution of CAV fleets. Hence, we test the following
two cases at a 10% CAV penetration rate: Case 1-5: the first
to fifth vehicles and the 50th to 55th vehicles are CAVs, and
Case 1-10: the first to tenth vehicles are CAVs. At a 20%
CAV penetration rate, Case 2-10: the first to tenth vehicles
and the 50th to 60th vehicles are CAVs, and Case 2-20: the
first to twentieth vehicles are CAVs, are tested. Compared
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CUI et al.: MODELING HETEROGENEOUS TRAFFIC MIXING REGULAR, CONNECTED, AND CAVs 13
Fig. 13. The headway SD for 5 cases at 10% and 20% CAV penetration rates.
Fig. 14. The headway SD for 5 cases at 80% and 90% CAV penetration rates.
with the headway and velocity SDs of a fleet of RVs only,
the reduced percentages of headway and speed SDs at 10%
and 20% CAV penetration rates are shown in Fig. 11 where
‘H’ and ‘V’ respectively denote the changed headway and
velocity SD percentage.
From Fig. 11, we find that when CAVs are mixed into the
RV fleet, the headway and speed SDs of the RV fleet are
reduced at the same time. The reduced percentage of speed
SD is lower than that of headway SD. The mixed CAVs make
the RV fleet stable. With the increase of CAV penetration rates
and the intensity μ2of the mechanism based on previous data,
the stability of heterogeneous traffic is improved. Comparing
H-Case 1-5 and H-Case 1-10, as well as H-Case 2-10 and
H-Case 2-20, we find that under the same CAV penetration
rate, the reduced percentages of headway and velocity SDs are
lowered as the size of the largest single CAV fleet increases.
For high CAV penetration rates, the mixing strategy is
changed and the impact of the size of a single RV fleet on the
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14 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
stability of heterogeneous traffic is studied. The parameter μ2
is fixed as 0.05, and μ1equals 0.03, 0.04 and 0.05 respectively.
We test the following two cases at an 80% CAV penetration
rate: Case 3-10: the first to tenth vehicles and the 50th to
60th vehicles are RVs, and Case 3-20: the first to twentieth
vehicles are RVs. At a 90% CAV penetration rate, Case 4-5:
the first to fifth vehicles and the 50th to 55th vehicles are
RVs, and Case 4-10: the first to tenth vehicles are RVs, are
tested. Compared with the headway and velocity SDs of a
fleet of CAVs only, the increased percentages of headway
and velocity SDs at 80% and 90% CAV penetration rates are
shown in Fig. 12 where ‘H’ and ‘V’ respectively denote the
increased headway and velocity SD percentage. We observe
that the stability of heterogeneous traffic is related to the size
of the largest single RV fleet and the impact of RV fleets on
headway/velocity SD is just opposite to that of CAV fleets.
C. Impact of Inter-Vehicle Communication on
Heterogeneous Fleet Stability
In this section, we further study the impact of inter-
vehicle communication on the stability of heterogeneous traf-
fic. We assume that the CV driver’s compliance rate of DASs
is zero, that is, the CV car-following model is exactly the
same as that of RVs. The only advantage of CVs over RVs is
that CVs achieve data transfer between vehicles, which helps
CAVs optimize their acceleration. The parameters with low
maximum speed set in Section V-A-1 are adopted. We study
the stability of heterogeneous traffic under five cases where
the size of a single CAV fleet is 1, 2, 5, 10, and 20 CAVs
at 10% CAV and 20% CAV penetration rates. We change the
value of μ1and show the headway SD in Fig. 13. In Fig. 13,
‘CV data’ and ‘RV data’ are the test results when CAVs are
mixed into the CV and RV fleet respectively, and ‘CV line’
and ‘RV line’ are the fit curves of corresponding tested results.
Observing Fig. 13, we find that with the increase of the
size of a single CAV fleet, ‘RV line’ is closer to or even
coincides with ‘CV line’, because the difference between CAV
fleets mixed into the CV and RV fleet is whether the first
vehicle of CAV fleets communicates with other vehicles. When
the size of a single CAV fleet is small, more CAVs take
advantage of inter-vehicle communication. Hence, as the size
of a single CAV fleet increases, the communication advantage
of CVs is gradually weakened compared with RVs, and even
is ignored. Furthermore, we find the similar results to those in
Section V-B. The stability of heterogeneous traffic is related to
the size of the largest single fleet, and the larger the size of the
largest fleet, the more unstable heterogeneous traffic is. The
headway SD curves at a 20% CAV penetration rate are lower
than the curves at a 10% CAV penetration rate. Hence, as CAV
penetration rates increase, the stability of heterogeneous traffic
is improved.
For 80% and 90% CAV penetration rates, we study the
effect of the size of a single RV or CV fleet on the stability
of heterogeneous traffic where the size of a single RV or CV
fleet is 1, 2, 5, 10, and 20 corresponding vehicles respectively.
All results are shown in Fig. 14. We find that under the same
size of a single fleet, the curves of high CAV penetration
rates are low than the curves of low CAV penetration rates,
TAB L E V
NOMENCLATURE
indicating that heterogeneous fleet stability increases with the
increase of CAV penetration rates. Also, the smaller the size
of the largest single RV or CV fleet, the more stable the
heterogeneous fleet is, which is the same as the results at low
CAV penetration rates. In Fig. 14 (b), we even find that the
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CUI et al.: MODELING HETEROGENEOUS TRAFFIC MIXING REGULAR, CONNECTED, AND CAVs 15
stability of the heterogeneous traffic of 80% CAVs and 20%
CVs is higher than that of the heterogeneous fleet of 90%
CAVs and 10% RVs, which means that inter-communication
contributes to the stability of heterogeneous traffic.
VI. CONCLUSION
V2V and V2I communication technologies are rapidly
evolving. Due to the existence of a large number of RVs,
the road will be in the stage of mixing RVs, CVs and
CAVs for a long time. To theoretically analyze the impact of
different connected and autonomous technologies on the sta-
bility of mixed traffic, corresponding car-following models are
established based on the characteristics of vehicles. Through
stability analysis, we find that for the RV car-following model,
the PMMM is beneficial to the stability of homogeneous RV
traffic. Previous two-time step velocity data achieve stable
RV traffic. For CAVs, we also find that multi-vehicle com-
munication, previous multi-time step velocity data, and the
cooperation based on the future multi-time step headway are
all beneficial to the stability of homogeneous CAV traffic.
It can be concluded that the communication of three vehicles
can achieve stable CAV traffic. Previous two-time step velocity
data contribute to the stability of homogeneous CAV traffic.
For CVs, as the compliance rate of DASs increases, CV fleet
stability increases. For the impacts of the size of a single
fleet and penetration rates of CAVs on the stability of mixed
traffic, we conclude that as CAV penetration rates increase,
the stability of heterogeneous traffic is improved. The size of
the largest single fleet has an impact on heterogeneous traffic
stability. Furthermore, as the size of the largest single fleet
increases, the stability of heterogeneous traffic is weakened.
Finally, we find that even if the CV driver’s compliance rate of
DASs is zero where the car-following model of CVs is exactly
the same as that of RVs, due to the inter-communication
capacity of CVs, the stability of the heterogeneous traffic
mixed with the same proportion CVs is higher than that of the
heterogeneous traffic mixed with the same proportion RVs.
In the future, this paper will be further deepened in the fol-
lowing aspects. First, the parameters in the RV, CV and CAV
car-following models will be calibrated by driving simulation.
Secondly, RV drivers will be further distinguished, including
the driver’s personality, and the driver’s perceived differences
in congested and non-congested situations.
APPENDIX
See Table V.
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Shaohua Cui received the M.S. degree from the
System Science Institute, School of Traffic and
Transportation, Beijing Jiaotong University. He is
currently pursuing the Ph.D. degree with the School
of Transportation Science and Engineering, Beihang
University. He has focused on traffic flow analysis
and transportation planning.
Feng Cao received the M.S. degree from the
School of Automotive Engineering, Dalian Univer-
sity of Technology. He is currently pursuing the
Ph.D. degree with the School of Transportation
Science and Engineering, Beihang University. His
current research interests include intelligent trans-
portation systems, traffic assignment, and big data
for transportation.
Bin Yu received the Ph.D. degree from the
Dalian University of Technology in 2006. From
October 2006 to June 2016, he served at Dalian
Maritime University. He is currently a Professor
with the School of Transportation Science and Engi-
neering, Beihang University. His research activity
concerns the application of innovative models.
Baozhen Yao received the Ph.D. degree from
Beijing Jiaotong University in 2011. She is currently
a Professor with the School of Automotive Engineer-
ing, Dalian University of Technology, Dalian, China.
Her current research interests include public trans-
portation, vehicle automation, swarm intelligence,
and vehicle routing problems.
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