An Extended Car-Following Model Considering Generalized Preceding Vehicles in V2X Environment - MDPI
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future internet
Article
An Extended Car-Following Model Considering
Generalized Preceding Vehicles in V2X Environment
Junyan Han 1 , Jinglei Zhang 1 , Xiaoyuan Wang 2, * , Yaqi Liu 1,2 , Quanzheng Wang 2
and Fusheng Zhong 2
1 School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo 255000, China;
junyanhan1995@yeah.net (J.H.); zhang1jing2lei3@163.com (J.Z.); liuyaqi518@126.com (Y.L.)
2 College of Electromechanical Engineering, Qingdao University of Science & Technology, Qingdao 266000,
China; 0020030005@mails.qust.edu.cn (Q.W.); ttxhway@126.com (F.Z.)
* Correspondence: wangxiaoyuan@qust.edu.cn
Received: 30 October 2020; Accepted: 27 November 2020; Published: 28 November 2020
Abstract: Vehicle-to-everything (V2X) technology will significantly enhance the information
perception ability of drivers and assist them in optimizing car-following behavior. Utilizing V2X
technology, drivers could obtain motion state information of the front vehicle, non-neighboring
front vehicle, and front vehicles in the adjacent lanes (these vehicles are collectively referred to as
generalized preceding vehicles in this research). However, understanding of the impact exerted by the
above information on car-following behavior and traffic flow is limited. In this paper, a car-following
model considering the average velocity of generalized preceding vehicles (GPV) is proposed to
explore the impact and then calibrated with the next generation simulation (NGSIM) data utilizing the
genetic algorithm. The neutral stability condition of the model is derived via linear stability analysis.
Numerical simulation on the starting, braking and disturbance propagation process is implemented
to further study features of the established model and traffic flow stability. Research results suggest
that the fitting accuracy of the GPV model is 40.497% higher than the full velocity difference (FVD)
model. Good agreement between the theoretical analysis and the numerical simulation reveals that
motion state information of GPV can stabilize traffic flow of following vehicles and thus alleviate
traffic congestion.
Keywords: traffic flow theory; car-following model; generalized preceding vehicles; Vehicle-to-
everything (V2X) environment; genetic algorithm
1. Introduction
With the development of urbanization and motorization, the number of vehicles continues to
grow, and congestion has become one of the main problems existing in cities around the world.
Due to the limitation of urban space, previous methods of reducing traffic jams, such as building more
infrastructure, have been producing very little effect. Regarded as an effective technological approach to
improve transportation efficiency and alleviate traffic congestion, the intelligent transportation system
(ITS) has received increasing attention. As one of the most important parts of ITS, V2X technology,
which is the general term for communication and information technologies enabling vehicles to
connect to everything [1,2], can significantly broaden the driver’s information perception range,
enhance the driver’s information perception ability by enabling them to obtain the information about
movement state of vehicles on the road. Compared with the ordinary traffic environment, driver’s
car-following and other driving behavior, as well as traffic flow of following vehicles, will show
different characteristics in the V2X environment [3–5]. The impact of the information mentioned above
on car-following behavior and traffic flow was explored by scholars via constructing car-following
Future Internet 2020, 12, 216; doi:10.3390/fi12120216 www.mdpi.com/journal/futureinternet
Future Internet 2020, 12, 216 2 of 15
models. Nagatani [6] proposed an extended car-following model and explored the impact of the
non-neighboring front vehicle position information. Lenz et al. [7] and Ge et al. [8] respectively
established a car-following model considering the headway of an arbitrary number of vehicles ahead
in the current lane. Unlike Lenz believed that car-following behavior in the model was the result of
multiple optimal velocity functions related to each headway, Ge believed that car-following behavior
was the result of one optimal velocity function related to multiple headways. Chen et al. [9] further
incorporated the desired following distance and explored the impact of this information by developing
an improved car-following model. Li et al. [10] established an extended car-following model with
consideration of the relative velocity of an arbitrary number of vehicles ahead. Hu et al. [11] further
considered drivers’ reaction delay and extended the car-following model. Instead of drivers’ reaction
delay, Guo et al. [12] further investigated velocity fluctuation feedback information. Peng et al. [13]
presented an improved car-following model based on both headway and relative velocity information
of an arbitrary number of preceding vehicles. Li et al. [14] proposed an extended car-following
model to concurrently study headway, relative velocity and acceleration information of an arbitrary
number of vehicles ahead in the current lane. Compared with the motion state information of an
arbitrary number of vehicles ahead in the current lane, drivers incline to pay more attention to
the motion state of vehicles that are in their view. For vehicles outside the field of view, drivers
tend to focus on their overall motion state instead of individual motion state. Based on this, sun et
al. [15] established an extended car-following model considering headway of the front vehicle and
the average velocity of an arbitrary number of vehicles ahead in the current lane. Kuang et al. [16],
Guo et al. [17] and Zhu et al. [18] presented modified car-following models to explore the information
of average headway, average field velocity, the average desired velocity, respectively rather than
than the average velocity. Soon afterward, Kuang et al. [19] built an extended car-following model
with consideration of average velocity and average desired velocity in the meantime. Results of the
above research revealed that providing motion state information of vehicles ahead in the current
line to drivers could assist them to optimize car-following behavior and thus enhance the stability
of traffic flow. However, urban roads are not all one-lane roads, and traffic flow on each lane of
multi-lane roads is not independent of each other. Common driving experience also suggests that
drivers will pay attention to motion state of vehicles ahead in the current and adjacent lanes at the
same time. In recent years, a large number of research upon car-following behavior for different aims,
such as traffic flow prediction [20], feedback control [21] or the safety analysis [22,23], and based on
various idea, such as considering driver’ memory effect [24–26] or driver’s visual characteristics [27],
communication delay [28], reaction delay [29], have been worked out. Among them, several efforts
upon vehicle platoon control in V2X environment [30,31] or the influence of V2X technology on driving
behavior [32,33] have been conducted. However, understanding about the impact exerted by motion
state information, which is available for the drivers using V2X technology in ITS, of preceding vehicles
including ones in the adjacent lanes on car-following behavior and traffic flow is still limited.
Motivated by the above contents, a concept called GPV is proposed to stand for the vehicles group
consisted of the front vehicle, non-neighboring front vehicle, and neighboring front vehicles in the
adjacent lanes (also known as left/right front vehicle), and average velocity is employed to represent
motion state of GPV. Based on these, an extended car-following model is established in Section 2
and then calibrated with the NGSIM data set using the genetic algorithm in Section 3. The stability
condition of the model is derived through linear stability analysis in Section 4, and the performance of
our model is studied utilizing numerical simulation in Section 5. Based on these efforts, the impact of
GPV motion state information on car-following behavior and traffic flow in the V2X environment is
explored. Research results are discussed in Section 6, and the conclusion is given in Section 7.
2. Model
Car-following is the behavior, which is ubiquity in the traffic system, that driver manipulates
his/her vehicle to follow the vehicles ahead. To study the characteristics of the car-following behavior
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and traffic flow, multi car-following models [34–38] were proposed based on various modeling ideas.
Bando et al. [34] believe that drivers always attempt to keep a safe velocity depending on the headway
between two successive vehicles when following the front vehicle. According to this, Bando proposed
a car-following model called the optimal velocity (OV) model, and its motion equation is as follows:
dvn (t)
= a[V (∆xn ) − vn (t)] (1)
dt
where a represents the sensitivity of the driver. V (∆xn ) is the optimal velocity function, and
∆xn = xn+1 (t) − xn (t) denotes the headway of the two successive vehicles. xn (t) and vn (t) are,
respectively, the position and velocity of the n th vehicle where t represents time.
Helbing and Tilch [35] found that the OV model would work out excessive acceleration/deceleration
during calibration of the optimal velocity function. To improve the OV model, Helbing established the
generalized force (GF) model by introducing the negative velocity difference. Formulation of the GF
model is as follows:
dvn (t)
= a[V (∆xn ) − vn (t)] + λH [−∆vn (t)]∆vn (t) (2)
dt
where a and λ represent the sensitivity of the driver. H is the Heaviside function and ∆vn (t) = v j+1 − v j
is the velocity difference between the leading vehicle j + 1 and the following vehicle j.
The GF model improved the OV model by solving the problem of excessive acceleration/
deceleration. However, there are still some imperfections in the GF model. For instance, the following
vehicle will not slow down when the headway is less than the minimum safety headway, and the
preceding vehicle is going much faster. Motivated by these, Jiang et al. [36] constructed the full velocity
difference (FVD) model by further considering the positive velocity difference. Its motion equation is
as follows:
dvn (t)
= a[V (∆xn ) − vn (t)] + λ∆vn (t) (3)
dt
Compared with the OV and CF models, the FVD model shows higher performance in simulating
traffic flow and, especially, studying the stability of traffic flow.
However, the aforementioned models only reflect the interaction between the vehicle and its
front vehicle. In a realistic traffic system, drivers not only focus on the vehicle ahead but also pay
attention to multi preceding vehicles. Especially in a V2X environment, drivers can obtain massive
information (for example, the velocity of an arbitrary number of vehicles ahead). Compared with an
arbitrary number of vehicles ahead in the current lane, the driver would pay more attention to nearby
vehicles, particularly the GPV, which are in the driver’s field of view. Among vehicles that are of
GPV, the driver is primarily concerned with the vehicle in front of him/her to maintain a safe distance
and avoid a collision. On this basis, the driver would also focus on GPV to optimize car-following
behavior. Based on the above contents, an extended car-following model called the GPV model is
proposed by introducing the average velocity of GPV, which can reflect the whole traffic situation on
the segment [15]. The model’s dynamic equation is as follows:
dvn (t)
= p a[V (∆xn ) − vn (t)] + λvn (t) + (1 − p)(vn − vn (t))
(4)
dt
where a, λ and p respectively represent the sensitivity of driver about optimal velocity difference,
velocity difference and the difference between GPV’s average velocity and self-vehicle velocity, and the
drivers are assumed to be ideal and identical, which is expressed by a constant value of sensitivity
in the GPV model. vn is the average velocity of GPV and vn = (vn+1 (t) + vn+2 (t) + vl (t) + vr (t))/4,
where vn+1 (t), vn+2 (t), vl (t) and vr (t) are the velocity of the front vehicle, non-neighboring front
vehicle, left and right front vehicles in the adjacent lanes.
Future Internet 2020, 12, 216 4 of 16
v = ( v (t) + v (t) + v (t) + v (t)) 4
in the GPV model. v n is the average velocity of GPV and n n+1 n+2 l r
, where
vn +1 (t ) , vn+ 2 (t ) , vl (t ) and vr (t ) are the velocity of the front vehicle, non-neighboring front vehicle,
Future Internet 2020, 12, 216 4 of 15
left and right front vehicles in the adjacent lanes.
In this research, the optimal velocity function calibrated with empirical data by Helbing [35] is
employed:In this research, the optimal velocity function calibrated with empirical data by Helbing [35]
is employed:
( Δxnn) )== VV11 ++VV2 tanh
VV(∆x [CC11((Δ∆x
2 tanh xnn−−lclc))−−CC22] (5)
(5)
Parameters in
Parameters in Equation
Equation (5)
(5) are
are set,
set, as
as shown
shown in
in Table
Table 1.
1.
1. Parameters
Table 1.
Table Parameters value
value in
in Equation
Equation (5).
(5).
V1
ParametersParameters V1 V2 V 2 C
C11 C 2 C2lc lc
6.75 6.757.917.91 0.13
0.13 1.57 1.575 5
By substituting
By substituting Equation
Equation (5)
(5) into
into Equation
Equation (4),
(4), Equation
Equation (4)
(4) can
can be
be rewritten
rewritten as:
as:
d vn (t )
dt
{ }
=dvpn (at)V1 + V2 tanh C1 ( Δxn − lc ) − C 2 − vn ( t ) + λ vn ( t ) + (1 − p ) v n − vn ( t )
= p a[V1 + V2 tanh[C1 (∆xn − lc ) − C2 ] − vn (t)] + λvn (t) + (1 − p)[vn − vn (t)]
(6)
(6)
dt
3. Parameter Calibration
3. Parameter Calibration
NGSIM project initiated
NGSIM project initiated by
by the
the American
American Federal
Federal Highway
Highway Administration
Administration provides
provides large-scale,
large-scale,
high-precision vehicle trajectory
high-precision vehicle trajectory data
data for
for the
the study
study of
of traffic
traffic flow
flow theory,
theory, including
including the car-following
the car-following
model. The US101 data set of NGSIM is employed to celebrate the
model. The US101 data set of NGSIM is employed to celebrate the GPV model constructed inGPV model constructed in the
the
previous
previous section
sectionandandthenthenverify the the
verify celebration. To obtain
celebration. suitable
To obtain data fordata
suitable this for
specific
this work, selection
specific work,
of US101 of
selection data set need
US101 datatosetbeneed
takento according to the following
be taken according rules:
to the following rules:
Rule 1: Lane number. The highway section
Rule 1: Lane number. The highway section where the US101 where the US101 data
data setcollected
set is is collected is divided
is divided into
into
4 lanes, and 1 ramp numbered 1–5, as shown in Figure 1. Considering that lane 4 is next to rampto5
4 lanes, and 1 ramp numbered 1–5, as shown in Figure 1. Considering that lane 4 is next
ramp 5 and lane-changing
and lane-changing behavior behavior is frequent,
is frequent, vehiclesvehicles
in lane in laneramp
4 and 4 and5ramp 5 will
will not be not be regarded
regarded as
as object
object vehicle to eliminate interference on car-following behavior exerted
vehicle to eliminate interference on car-following behavior exerted by lane-changing behavior.by lane-changing behavior.
Further considering that
Further considering that the
the front
front vehicles
vehicles in in the
the adjacent
adjacent lanes
lanes are
are introduced
introduced into into the
the GPV
GPV model,
model,
vehicles only in lane 2 can be regarded as an object
vehicles only in lane 2 can be regarded as an object vehicle. vehicle.
1
2
3
4
5
Figure 1.
Figure 1. Lane
Lane setting
setting of
of US101
US101 collection
collection section.
section.
2: Integrity
Rule 2: Integrityof ofthe
thevehicle
vehiclegroup.
group.
GPVGPV is comprehensively
is comprehensively considered
considered in model,
in our our model, and
and thus
thusobject
the the object vehicle
vehicle forresearch
for this this research
shouldshould
have ahave a complete
complete GPV group.
GPV group. Based onBased
this,on
thethis, thevehicle
object object
vehicle
for for this research
this research should have should havevehicle,
the front the front vehicle, non-neighboring
non-neighboring front vehiclefront vehicle and
and left/right frontleft/right
vehicle
front
to meetvehicle to meet standard
the integrity the integrity standard
of the vehicleof the vehicle group.
group.
Rule 3: Time
Time headway between the object vehicle and its front vehicle. If time headway between
vehicle and
the vehicle and its
itsfront
frontvehicle
vehicleexceeds
exceeds5 5s s[39],
[39],movement
movementofof thethe vehicle
vehicle is no
is no longer
longer restricted
restricted by by
its
its front
front vehicle,
vehicle, andand
the the vehicle
vehicle will will notconsidered
not be be considered to beto
inbe in a car-following
a car-following state. state. Vehicles
Vehicles whichwhich
are in
are in
the the car-following
car-following state
state can becan be regarded
regarded as an vehicle
as an object object vehicle
for ourfor our study.
study.
Rule 4: Duration of car-following state. To ensure that the amount of data contained in each set of
trajectory data are consistent and sufficient, the duration of the car-following behavior is set as 30 s
considering characteristics of the US101 data set and its collection section.
Future Internet 2020, 12, 216 5 of 16
Future Internet
Rule2020, 12, 216
4: Duration of car-following state. To ensure that the amount of data contained in each set5 of 15
of trajectory data are consistent and sufficient, the duration of the car-following behavior is set as 30
s considering characteristics of the US101 data set and its collection section.
According to rules
According 1–4,1–4,
to rules data filtering
data progress
filtering progress can
canbebedetermined,
determined, asas shown
shown ininFigure
Figure2.2.Based
Based on
these,ona these,
selection of the US101 dataset is carried out, and 162 sets of trajectory data suitable
a selection of the US101 dataset is carried out, and 162 sets of trajectory data suitable for our for our
research are obtained.
research Half
are obtained. of of
Half the
thedatasets
datasetsare
are randomly selectedforfor
randomly selected calibrating
calibrating model
model parameters,
parameters,
and the
andothers are used
the others to verify
are used calibration
to verify calibrationresults.
results.
This candidate can This candidate can This candidate can
not be object vehicle not be object vehicle not be object vehicle
for our research. for our research. for our research.
No No No
If the If
If the the candidate This candidate can
Candidate object candidate has
Yes If the candidate Yes candidate Yes No
has non-neighboring has left front vehicle has right front vehicle not be object vehicle
vehicles in lane 2. front vehicle? front vehicle? in adjacent in adjacent for our research.
lane? lane?
This candidate can
not be object vehicle Yes
for our research.
No
The part of data
Divide trajectory Is time
before the limit is Is the
Date sets suitable marked as a set and
data of the candidate Yes duration of car Yes headway between No This candidate can
and its GPV into two the candidate and its not be object vehicle
for our research. the candidate is following state
parts with 30 seconds front vehicle for our research.
selected as an object ≥30s ?
as the limit. ≤5s ?
vehicle.
The part of data
after the limit is
marked as a
temporary set.
Figure Data
2. 2.
Figure Datafiltering
filtering progress ofthe
progress of theUS101
US101 data
data set.set.
Calibration of parameters
Calibration of parameters in in
the
thecar-following
car-following model
model isisaakind
kindofof optimum
optimum solution
solution for nonlinear
for nonlinear
programming
programming problems. In In
problems. this
thiswork,
work,thetheobjective functionisiscalculation
objective function calculation error
error between
between actual
actual data data
and model output, variables to be optimized are parameters in the model, and constraintsthe
and model output, variables to be optimized are parameters in the model, and constraints are are the
physical
physical boundaries
boundaries of these
of these parameters.The
parameters. The genetic
genetic algorithm
algorithmisiswidely
widely used andand
used has has
shown high high
shown
performance
performance in dealing
in dealing withwith
thisthis kind
kind of of problems[40,41];
problems [40,41];parameters
parameters inin the
thegenetic
geneticalgorithm
algorithm used
used in
in this research are set as follows:
this research are set as follows:
(a) population size: 60;
(a) population
(b) crossoversize: 60;
probability: 0.9;
(b) crossover
(c) mutationprobability:
probability:0.9;
0.2;
(c) (d) iteration
mutation number: 500;
probability: 0.2;
(d) iteration number: 500; a∈[0,2] λ ∈[ 0,1] p∈[ 0,1]
(e) value range of parameters to be celebrated: , ,
(e) value range of parameters to be celebrated: a ∈ [0, 2], λ ∈ [0, 1], p ∈ [0, 1].
Utilizing MATLAB (Version 9.6) software, parameters in the GPV model is calibrated. The FVD
model constructed
Utilizing MATLAB in (Version
[36] is also calibrated
9.6) for parameters
software, comparison and further
in the GPVexploration in the following
model is calibrated. The FVD
model sections. Calibration
constructed results
in [36] arecalibrated
is also as shown inforTable 2.
comparison and further exploration in the following
sections. Calibration results are as shown in Table 2.
Table 2. Calibration results of parameters in the generalized preceding vehicles (GPV) model and the
Tablefull velocity difference
2. Calibration results(FVD) model.
of parameters in the generalized preceding vehicles (GPV) model and the
full velocity difference (FVD) model.
Parameters GPV Model FVD Model
a 0.767 0.852
Parameters GPV Model FVD Model
λ 0.301 0.389
a p 0.767
0.769 — 0.852
λ 0.301 0.389
p 0.769 —
To verify the calibration results of parameters in the GPV model and the FVD model, mean absolute
error (MAE) and mean absolute relative error (MARE) are employed as the performance index.
The equations of MAE and MARE are as follows:
n
1X
MAE = yi − ym
i (7)
n
i=1
absolute error (MAE) and mean absolute relative error (MARE) are employed as the performance
index. The equations of MAE and MARE are as follows:
1 n
MAE = yi − yim
n i =1
(7)
Future Internet 2020, 12, 216 6 of 15
m
1 n
yi − y
MARE = i
nn i =y1 i − yymi
(8)
1 X
i
MARE = (8)
where
y n
is the acceleration of the object vehicle.i = 1 y yi
i and y i
m
represent, respectively, the i th
and i thof the
y is thevalue yi and m represent, respectively, the i th measured
where
measured acceleration object vehicle.
calculated value with theymodel.
i The evaluation results of parameters
value and i th are
calibration calculated value
as shown with the
in Table 3. model. The evaluation results of parameters calibration are as
shown in Table 3.
Table 3. Evaluation Results of the parameter calibrations.
Table 3. Evaluation Results of the parameter calibrations.
Performance Index GPV Model FVD Model
Performance Index GPV Model FVD Model
MAE 1.4746 2.495
MAE 1.4746 2.495
MARE 0.1712 3.2896
MARE 0.1712 3.2896
From Table 3, one can obtain that the calibration results are solid, and all performance indexes
From
of the GPVTable 3, one
model can
are obtain that
superior the calibration
to those of the FVDresults
model.are solid, andtoallTable
According performance
3, we canindexes
obtain of
that
thethe
GPV model
fitting are superior
accuracy to those
of the GPV modelof the FVD
to the model.
data According
measured in theto Table
field 3, we can
is 40.497% obtainthan
higher thatthat
the of
fitting accuracy
the FVD of the
model. In GPV
ordermodel to theverify
to further data measured in thethe
and evaluate field is 40.497%
results higher than
of parameters that of the
calibration and
FVD model. In order to further verify and evaluate the results of parameters
explore the performance of the GPV model in fitting date measured in the field, we calculatecalibration and explore
theacceleration
performance of thethe
using GPVGPVmodel
modelin fitting
and the date
FVDmeasured in thecalibrated
model with field, we calculate
parametersacceleration
and compareusingthe
thecalculation
GPV model and the FVD model with calibrated parameters and compare the calculation
results with the 81 data sets which are randomly selected for verification in previous results
with the 81 Part
contents. data of
sets
thewhich are randomly
comparison results isselected
shown for verification
in Figure 3. in previous contents. Part of the
comparison results is shown in Figure 3.
Figure 3. Comparison of computational acceleration between the GPV model and the FVD model with
Figure 3. Comparison of computational acceleration between the GPV model and the FVD model
the verification data sets (part).
with the verification data sets (part).
The comparison results reveal that the GPV model has higher fitting accuracy to data measured in
the field than the FVD model. It is noteworthy that the fitting acceleration curve of the FVD model has
bigger curvature in several places, and there is a certain delay in acceleration calculation results of the
FVD model, especially in the deceleration phase. The causation of this phenomenon is that drivers
adjust car-following behavior only according to the motion state of their front vehicle in the FVD
model and cannot grasp the traffic situation further ahead, which will guide them to take measures in
advance and thus reduce reaction delay. The above results suggest that GPV’s motion state, such as
Future Internet 2020, 12, 216 7 of 15
average velocity, plays an important role in improving the performance of the car-following model in
fitting data measured in the field.
4. Stability Analysis
To explore the impact of average velocity information of GPV on traffic flow in the V2X environment,
linear stability analysis is conducted based on the perturbation method [34,42,43]. Assuming that all
three lanes in the system are in the same stable state, which means all vehicles maintain the same
headway h and velocity V (h), at the initial moment, the position of the n th vehicle can be expressed
as follows:
(0)
xn (t) = hn + V (h)t (9)
where h = L/N. L is the length of the road and N is the total number of vehicles on the road. V (h) is
the optimal velocity.
(0)
Suppose yn (t) to be a small deviation from the stable state solution xn (t)
(0)
xn (t) = xn (t) + yn (t) (10)
Substituting Equations (9) and (10) into Equation (6) and linearizing the equation, one can obtain
d2 yn (t) dyn (t) d∆y (t)
dt2
= p a V 0 (h)∆yn (t) − dt + λ dtn
dy +1 (t) dy +2 (t) dy (t) dy (t)
dyn (t)
(11)
+(1 − p) 41 ndt + ndt + dtl + dtr − dt
where ∆yn (t) = yn+1 (t) − yn (t) and V 0 (h) = dV (∆xn )/d∆xn ∆xn =h
. According to vehicles in all three
dyl (t) dyr (t) dyn+1 (t)
lanes of the road are at the same stable state, one can obtain dt = dt = dt .
Substituting yn (t) = eikn+zt into Equation (11), one can obtain
1
n h i o
z2 = p a V 0 (h) eik − 1 − z + λz eik − 1 + (1 − p) 3zeik + ze2ik − z (12)
4
By expanding yn (t), where z = z1 (ik) + z2 (ik)2 + · · · , and inserting it into Equation (12), the first-
and second-order terms of ik can be obtained as follows:
z1 = V 0 (h) (13)
2paV 0 (h) − 4z1 2 + 5(1 − p)z1 + 4pλz1
z2 = (14)
4pa
For long wavelength modes, the uniformly stable state traffic flow becomes unstable if z2 < 0,
while the uniformly stable state traffic flow remains stable if z2 > 0. Therefore, the neutral stability
condition is given as:
4V 0 (h) − 5(1 − p) − 4pλ
a= (15)
2p
For small disturbances with long wavelengths, the uniform traffic flow is stable if
4V 0 (h) − 5(1 − p) − 4pλ
a> (16)
2p
Based on Equation (15) and parameters calibrated in Section 3, the neutral stability curves of the
GPV and FVD models in the headway-sensitivity space are as shown in Figure 4. From Figure 4, one can
see that the headway-sensitivity phase space is divided into two regions by the neutral stability curves.
The first is the stable region, which is above the corresponding neutral stability curve, and the second
is the unstable region, which is below the corresponding neutral stability curve. In a stable region,
4V ' ( h ) − 5(1 − p) − 4 pλ
a> (16)
2p
Based on Equation (15) and parameters calibrated in Section 3, the neutral stability curves of the
GPV and FVD models in the headway-sensitivity space are as shown in Figure 4. From Figure 4, one
Future Internet that12,
can see2020, the216
headway-sensitivity phase space is divided into two regions by the neutral stability 8 of 15
curves. The first is the stable region, which is above the corresponding neutral stability curve, and
the second is the unstable region, which is below the corresponding neutral stability curve. In a stable
traffic flow is stable, which means that small disturbances will be suppressed and, thus, traffic jams
region, traffic flow is stable, which means that small disturbances will be suppressed and, thus, traffic
will not occur.
jams will In
notthe unstable
occur. In the region,
unstabletraffic
region,flow is unstable,
traffic and density
flow is unstable, waveswaves
and density emerge. In this
emerge. region,
In this
small disturbances cannot be suppressed
region, small disturbances effectively,
cannot be suppressed and, on and,
effectively, the contrary, it will itgradually
on the contrary, enlarge
will gradually
with propagation, which couldwhich
enlarge with propagation, lead to congestion
could eventually.
lead to congestion eventually.
FigureFigure
4. The 4. The neutral
neutral stabilitystability
curves curves of the
of the GPV GPVand
model model and the
the FVD FVD
model model
with with calibrated
calibrated parameters.
parameters.
To explore the impact of the sensitivity parameter p on the stability of traffic flow, the neutral
stability curves of thethe
To explore GPV model
impact with
of the sensitivity p, as p = p
differentparameter 0.9,
on0.8,
the0.7, 0.6, 0.5
stability of respectively, areneutral
traffic flow, the obtained
when λ = 0.2 ascurves
stability shown inthe
Figure
GPV 5. Fromwith
Figure p
5, one can
, asobtain
p = 0.9that
，0.8with
，0.7， the
0.6， decrease
0.5 p, the9 neutral
Future Internet 2020, of
12, 216 model different respectively, are
of 16
stability curve gradually
obtained when λ=0.2 moves down, and the stable region keeps enlarging.
as shown in Figure 5. From Figure 5, one can obtain that with the decrease
p
, the neutral stability curve gradually moves down, and the stable region keeps enlarging.
5. The 5.neutral
Figure Figure stability
The neutral curves
stability of theofGPV
curves model
the GPV withwith
model different
different
pcompared
p compared with that
with of of
that thethe
FVD
model FVD
whenmodel when λ=0.2 .
λ= 0.2.
From Figures 4 and 5, one can obtain that the stable region of the GPV model is larger than that
of the FVD model. the
5. Numerical Simulation
To further verify analysis results in previous sections and study characterize features of the GPV
Future Internet 2020, 12, 216 9 of 15
From Figures 4 and 5, one can obtain that stable region of the GPV model is larger than that of the
FVD model. This is because motion state of GPV considered in our model can assist driver with better
grasping traffic condition ahead and taking measures in advance to maintain stable state as much as
possible, and thus enhance the stability of traffic flow, which suggests that motion state such as average
velocity of GPV plays an important role in enhancing the stability of traffic flow.
5. Numerical Simulation
To further verify analysis results in previous sections and study characterize features of the GPV
model, numerical simulation on three typical traffic scenarios with comparison to the FVD model is
carried out utilizing MATLAB (Version 9.6) software in this section. The three typical traffic scenarios,
including the starting process, braking process as well as disturbance process, are constructed as shown
in the following contents, and the motion state of vehicles in the scenarios are determined by the GPV
model or the FVD model via numerical computation.
5.1. Simulation of Starting Process
To simulate the car-following behavior of vehicles in the starting process at the intersection when
the traffic light turns from red to green in a realistic traffic system, the simulation scenario about vehicle
starting process is set as the following: At an intersection with a traffic light, 10 identical vehicles stop
and wait in every single of three lanes with the same headway of 10 m between any two consecutive
vehicles, and all vehicles are about to start when the traffic light turns from red to green and move in
the same direction. The vehicles in the middle of all three lanes are selected as object vehicles and
marked as 1 to 10 according to the distance to the intersection from near to far. Considering that GPV
is introduced in our model, the first object vehicle of the fleet is following its GPV in the scenario.
At the beginning of the simulation, the traffic light turns green, and the vehicles start in sequence.
The velocity limit of all object vehicles is set as 5 m/s, and the termination condition of this simulation
is set as all object vehicles reach the velocity limit. The velocity and acceleration of all object vehicles
are studied, as shown in Figures 6 and 7.
Figure 6 illustrates the simulated velocity of the two models. As indicated in Figure 6a, it takes
19 s for all object vehicles to reach the preset velocity (5 m/s) in the simulation with the GPV model.
By comparison, it takes 21 s to reach the same state with the FVD model, as shown in Figure 6b.
(The lines with different color in Figure 6 as well as Figures 7–9 respectively represents the object
vehicles in the
Future Internet scenario.)
2020, 12, 216 10 of 16
(a) GPV model (b) FVD model
Figure6.6.Comparison
Figure Comparisonof
ofvelocity
velocityduring
duringthe
thestarting
startingprocess
processbetween
betweenthe
thetwo
twomodels.
models.(a) GPV model (b) FVD model
Future Internet 2020, 12, 216 10 of 15
Figure 6. Comparison of velocity during the starting process between the two models.
Future Internet
Future Internet 2020,
2020, 12,
12, 216
216 11 of
11 of 16
16
(a) GPV model (b) FVD model
simulation
simulation is set
is
Figure
Figure
set as
as all object
all objectofof
7.7.Comparison
Comparison vehicles
vehicles have
have
acceleration
acceleration
stopped.
stopped.
during
during the
The velocity
The
thestartingvelocity
starting and
and thethe
the
processbetween
process between acceleration
acceleration
the twomodels.
two
during the
during
models. the
braking process
braking process of of object
object vehicles
vehicles are
are studied,
studied, as
as shown
shown in in Figures
Figures 88 and
and 9.9.
Figure 6 illustrates the simulated velocity of the two models. As indicated in Figure 6a, it takes
19 s for all object vehicles to reach the preset velocity (5 m / s ) in the simulation with the GPV model.
By comparison, it takes 21 s to reach the same state with the FVD model, as shown in Figure 6b. (The
lines with different color in Figure 6 as well as Figures 7–9 respectively represents the object vehicles
in the scenario.)
Figure 7 shows simulated acceleration with the two models. One can see that acceleration and
accelerating time of the GPV model are less than those of the FVD model during the starting process.
2
As shown in Figure 7a, the maximum acceleration is 1.6 m / s , and the acceleration process lasts
19 s with the GPV model. In contrast, the maximum acceleration is 1.8 m / s 2 , and it cost an extra 3
s (total 21 s) to complete the acceleration process with the FVD model, as shown in Figure 7b.
5.2. Simulation of Braking
(a) GPVProcess
(a) GPV model
model (b) FVD
(b) FVD model
model
Figure8.
Figure
To simulate
Figure 8.Comparison
8.
the Comparison
Comparison ofvelocity
of
car-following
of velocity
velocity
behaviorduring
during thebraking
the braking
braking
of vehicles
during the process
process
in the
process between
between
starting theat
the
process
between the two
two
twothemodels.
models.
intersection when
models.
the traffic light turns from red to green in a realistic traffic system, the simulation scenario about
vehicle starting process is set as following: 10 identical vehicles in every single of three lanes are
moving in the same direction with the same initial velocity 5 m / s and headway of 10 m between
any two consecutive vehicles. The vehicles in the middle of all three lanes are selected as object
vehicles and marked as 1 to 10 according to the distance to the intersection from near to far.
Considering that GPV is introduced in our model, the first object vehicle of the fleet is following its
GPV in the scenario. Moreover, regarding the comparability of simulation results, the scenario for
simulation with the FVD model is set as the same. At the beginning of the simulation, the traffic light
turns from green to red, and all vehicles brake in sequence. The termination condition of this
(a) GPV
(a) GPV model
model (b) FVD
(b) FVD model
model
Figure9.9.
Figure
Figure 9.Comparison
Comparisonof
Comparison ofdeceleration
of decelerationduring
deceleration duringthe
during thebraking
the brakingprocess
braking processbetween
process betweenthe
between thetwo
the twomodels.
two models.
models.
Figure788shows
Figure
Figure depicts
depicts the simulated
the
simulated simulated velocity
velocity
acceleration of the
of
with object
object vehicles
twovehicles
models.with with
Onethethe
canGPVGPV
see that model
model and the
and
acceleration the FVD
FVD
and
model,
model, respectively,
respectively, during
during the
the braking
braking process.
process. As
As shown
shown in
in Figure
Figure
accelerating time of the GPV model are less than those of the FVD model during the starting process. 8a,
8a, it
it costs
costs 16
16 ss that
that the
the first
first
As shown
vehicle
vehicle ofin
of theFigure
the fleet 7a, the maximum
fleet decelerates
decelerates to 00 m
to macceleration
// ss ,, and allis
and all 1.6 m/s
object
object
2 , and stop
vehicles
vehicles the acceleration
stop at 75
at fromprocess
75 ss from the initial
the lasts
initial 19 in
time
time sin
with the GPV model. In contrast, the maximum acceleration is 1.8 m/s 2 , and it cost an extra 3 s
the simulation
the simulation with with the
the GPV
GPV model.
model. By By comparison,
comparison, itit takestakes 35 35 ss that
that the
the first
first vehicle
vehicle of of the
the fleet
fleet
(total 21 s) to complete the acceleration
m // ss ,, and process with the FVD model, as shown in Figure 7b.
decelerates to
decelerates to 00 m and all
all object
object vehicles
vehicles stopstop atat 87
87 ss from
from thethe beginning
beginning time time in
in the
the simulation
simulation
with the
with the FVD
FVD model
model as as shown
shown in in Figure
Figure 8b.8b.
The simulated
The simulated deceleration
deceleration of of object
object vehicles
vehicles with
with thethe two
two models
models are are asas shown
shown in in Figure
Figure 9. 9. In
In
contrast with
contrast with thethe FVD
FVD model,
model, thethe deceleration
deceleration of of the
the GPV
GPV model
model is is more
more rapid,
rapid, and
and the the response
response
delay time
delay time isis much
much shorter.
shorter. As As shown
shown in in Figure
Figure 9a,9a, the
the first
first vehicle
vehicle ofof the
the fleet
fleet reaches
reaches thethe maximum
maximum
deceleration of
deceleration of 0.83 m // ss22 at
0.83 m at2.8
2.8 s,
s, and
and the
the last
last vehicle
vehicle of
of the
the fleet
fleet reaches
reaches the
the maximum
maximum deceleration
deceleration
m // ss22
mFuture Internet 2020, 12, 216 11 of 15
5.2. Simulation of Braking Process
To simulate the car-following behavior of vehicles in the starting process at the intersection when
the traffic light turns from red to green in a realistic traffic system, the simulation scenario about vehicle
starting process is set as following: 10 identical vehicles in every single of three lanes are moving in the
same direction with the same initial velocity 5 m/s and headway of 10 m between any two consecutive
vehicles. The vehicles in the middle of all three lanes are selected as object vehicles and marked as 1 to
10 according to the distance to the intersection from near to far. Considering that GPV is introduced in
our model, the first object vehicle of the fleet is following its GPV in the scenario. Moreover, regarding
the comparability of simulation results, the scenario for simulation with the FVD model is set as the
same. At the beginning of the simulation, the traffic light turns from green to red, and all vehicles brake
in sequence. The termination condition of this simulation is set as all object vehicles have stopped.
The velocity and the acceleration during the braking process of object vehicles are studied, as shown in
Figures 8 and 9.
Figure 8 depicts the simulated velocity of object vehicles with the GPV model and the FVD model,
respectively, during the braking process. As shown in Figure 8a, it costs 16 s that the first vehicle of
the fleet decelerates to 0 m/s, and all object vehicles stop at 75 s from the initial time in the simulation
with the GPV model. By comparison, it takes 35 s that the first vehicle of the fleet decelerates to 0 m/s,
and all object vehicles stop at 87 s from the beginning time in the simulation with the FVD model as
shown in Figure 8b.
The simulated deceleration of object vehicles with the two models are as shown in Figure 9.
In contrast with the FVD model, the deceleration of the GPV model is more rapid, and the response
delay time is much shorter. As shown in Figure 9a, the first vehicle of the fleet reaches the maximum
deceleration of 0.83 m/s2 at 2.8 s, and the last vehicle of the fleet reaches the maximum deceleration of
0.5 m/s2 at 61.1 s. By comparison, the first vehicle of the fleet reaches the maximum deceleration of
0.63 m/s2 at 23.8 s, and the last vehicle of the fleet reaches the maximum deceleration of 0.63 m/s2 at
76.1 s in this simulation with the FVD model as shown in Figure 9b.
From Figures 8 and 9, one can see that there is a certain brake delay in the simulation with FVD
and this result consistent with the results of data fitting in Section 3. Furthermore, it is worth noting
that there are two deceleration fluctuations of each object vehicle during the braking process with the
GPV model, while there is only one deceleration fluctuation with the FVD model. This phenomenon
will be discussed in the following section.
5.3. Simulation of Disturbance Propagation Process
The neutral stability curves of the GPV model and the FVD model are obtained in Section 4.
According to the conclusion of the section, the headway–sensitivity phase diagram is divided into two
regions. The region above neutral stability curves is the stable region, in which a small disturbance
can be suppressed or absorbed. Simulation of the disturbance propagation process can represent
the operation characteristics of traffic flow when an incident or accident occurs in a realistic traffic
system and thus is employed to verify the above theoretical analysis results. The simulation scenario
on propagation process of disturbance with the GPV model and the FVD model is set as following:
100 identical vehicles with a length of 5 m in each lane of three are moving towards the same direction
on a 1500 m circular road with a constant velocity of 2 m/s and the same headway of 10 m. Then,
a small disturbance of 1 m/s (half of the initial velocity) and 2 m (one-fifth of initial headway) is exerted
on the vehicles, and the propagation process of this disturbance in the vehicle fleet is simulated as
shown in Figure 10.
From Figure 10a, one can see that the same disturbance is rapidly suppressed to a small amplitude
and finally absorbed with the GPV model. As shown in Figure 10b, this disturbance can be absorbed
eventually with the FVD model. However, both the amplitude of the disturbance during the propagation
process and the time for the disturbance to be absorbed are significantly greater than those of the GPV
model, which has good agreement with the theoretical analysis results in Section 4.following: 100 identical vehicles with a length of 5 m in each lane of three are moving towards the
same direction on a 1500 m circular road with a constant velocity of 2 m / s and the same headway
of 10 m. Then, a small disturbance of 1 m / s (half of the initial velocity) and 2 m (one-fifth of initial
headway) is exerted on the vehicles, and the propagation process of this disturbance in the vehicle
Future Internet 2020, 12, 216 12 of 15
fleet is simulated as shown in Figure 10.
(a) GPV model (b) FVD model
Figure
Figure 10.
10. Propagation
Propagation process
process of
of disturbance
disturbance in
in the
the vehicle
vehicle fleet
fleet with
with the
the two models.
two models.
6. Discussion
From Figure 10a, one can see that the same disturbance is rapidly suppressed to a small
amplitude and finallyperception
The information absorbed ability
with the
of GPV model.
the driver As shown inenhanced
is significantly Figure 10b, thisV2X
in the disturbance can
environment.
be absorbed eventually with the FVD model. However, both the amplitude of the disturbance during
Utilizing V2X technology, drivers can obtain massive traffic information, and based on the information,
the propagation
adjust process
and optimize and
their the time for behavior.
car-following the disturbance to be absorbed
The impact are significantly
of information greater
about motion than
state of
those of the GPV model, which has good agreement with the theoretical analysis results in
vehicle individuals such as headway, velocity and acceleration of preceding vehicles in the current Section 4.
lane on car-following behavior and traffic flow was studied in [6–14], and the impact of information
6. Discussion
about motion state of vehicles group including the average velocity of preceding vehicles in the current
lane The information
was explored perception
in [15–19]. ability
Although of the driver
information aboutismultisignificantly
precedingenhanced in the V2X
vehicles, including left
environment.
and the right Utilizing V2X in
front vehicle technology,
the adjacentdrivers can
lanes, areobtain massive
available traffic information,
for drivers and based on
in the V2X environment,
the information,about
understanding adjusttheand optimize
influence oftheir car-followingonbehavior.
this information The impact
car-following behaviorof information about
and traffic flow is
motion
limited.state
Motivedof vehicle individuals
by this, a concept such
namedas headway,
GPV wasvelocity
proposed andtoacceleration
represent theof preceding
precedingvehicles
vehicle
in the current
group consistinglaneofonfront
car-following behavior and traffic
vehicle, non-neighboring frontflow was and
vehicle studied in [6–14],
left/right frontand the impact
vehicle in the
of information
adjacent lanes, andabout motionvelocity
average state ofwasvehicles
employedgroup to including
represent the motion
averagestate
velocity
of GPV.of preceding
Based on
vehicles
these, anin the current
extended lane was model
car-following exploredwasinestablished
[15–19]. Although
and theninformation
used to exploreaboutthemulti
impact preceding
exerted
vehicles,
by motion including left and the
state information of right
GPV front vehicle in the
on car-following adjacentand
behavior lanes, areflow.
traffic available for drivers in the
V2X Research
environment, understanding
results aboutstate
reveal that motion the influence
information of this information
of GPV can optimizeon car-following behavior
driver’s car-following
and trafficand
behavior flow is limited.
enhance Motivedofby
the stability this,flow.
traffic a concept named
In Section GPV
3, the was proposed
acceleration to represent
of object the
vehicles was
preceding
calculated vehicle
with thegroup
GPV consisting
model andofthe front
FVD vehicle,
model,non-neighboring
respectively andfront vehiclewith
compared andtheleft/right front
verification
vehicle in The
data sets. the adjacent
comparison lanes,
(asand average
shown velocity
in Figure was employed
3) shows that the model to represent the motion
established state of
in this research
GPV.consideration
with Based on these, an extended
of GPV car-following
motion state can bettermodel
fit the was
data established
sets and reduce and acceleration/deceleration
then used to explore the
impact exertedinbythe
delay existing motion state information
FVD model. of GPV
This illustrates thaton car-following
information aboutbehavior and traffic
GPV motion state flow.
can enable
Research
drivers to grasp results reveal
traffic that motion
situation ahead on state
theinformation
road instead of GPV
of in can optimizelane
the current driver’s car-following
and guide drivers
behavior and enhance
to take measures the stability
in advance of traffic
to decrease theflow. In Section
response delay. 3,The
thecomparison
accelerationresults
of object
alsovehicles
reveal was
that
taking GPV motion state account into the car-following model can make the model more in line with
driving behavior characteristics, which is that drivers not only focus on the front vehicle but also pay
attention to multi-preceding vehicles, including left/right front vehicles, and thus fit the measured
data more accurately. In Section 4, the neutral stability condition of the GPV model was derived via
linear stability analysis and then compared with that of the FVD model. The stability analysis results
infer that considering the motion state of GPV can enhance the stability of traffic flow on a certain
scale, and traffic flow will be more stable as more attention of drivers attached to the motion state of
GPV. Explanation of these results are as follows: On one hand, information about the motion state of
GPV can assist drivers with better understanding traffic situation ahead and guide them take measure
earlier, which can effectively reduce reaction delay. On the other hand, with a better understanding
of the traffic situation ahead, drivers can complete the maneuvering process with a relatively small
acceleration/deceleration value to achieve a new stable driving state. Results in Sections 3 and 4
suggest that GPV motion state information can effectively optimize driver’s car-following behavior andFuture Internet 2020, 12, 216 13 of 15
enhance traffic flow stability. Those results also confirm that it is necessary to take GPV into account
into studying car-following behavior. To verify the above theoretical analysis results and to present
the characteristics of the GPV model in an intuitive way, the numerical simulation based on three
typical traffic scenarios with GPV and FDV model was conducted for contrastive analysis in Section 5.
The results of numerical simulation agree well with the above analysis. Among the simulation results,
one is noteworthy that there are two deceleration fluctuations during the braking process with the GPV
model, and there is one fluctuation in the same scenario with the FVD model. This may be caused
by drivers adopt larger deceleration values to maintain a safe distance and avoid a collision as the
headway between the object vehicle and its front vehicle decreases, which in line with that, safety is
the first primary interest for all drivers.
With these results, we believe that in the V2X environment, information of GPV motion state
can assist drivers in optimizing car-following behavior and, thus, enhance the stability of traffic flow,
which infer that traffic efficiency will be improved and energy consumption will be reduced with
V2X technology in ITS. The above results also suggest that GPV should be taken into account in
car-following research. Finally, it must be pointed out that we assumed all vehicles and their drivers
are ideal and identical to eliminate the influence caused by the heterogeneity of the vehicles and their
drivers, and we look forward to exploring this influence in our future research.
7. Conclusions
The impact of information about GPV motion state in V2X environment upon car-following
behavior and traffic flow was studied by establishing an extended car-following model (called the
GPV model) in this work. The fitting accuracy of the GPV model to the data measured in the field
is 40.497% higher than that of the FVD model. Research results reveal that the motion state of GPV,
which should be considered in research about car-following behavior, can assist drivers to optimize their
car-following behavior and enhance the stability of traffic flow efficiently. These results confirm that
the application of V2X technology in ITS will alleviate traffic jams, improve transportation efficiency,
and thus reduce the energy consumption of the transportation system to a certain extent.
Author Contributions: Conceptualization, J.H.; methodology, J.H.; software, J.Z. and Q.W.; validation, X.W.
and Y.L.; formal analysis, J.H. and Y.L.; investigation, J.H.; resources, X.W.; data curation, J.Z. and Y.L.;
writing—original draft preparation, J.H.; writing—review and editing, X.W. and F.Z.; visualization, J.Z. and Q.W.;
supervision, X.W. and J.Z.; project administration, X.W.; funding acquisition, X.W. All authors have read and
agreed to the published version of the manuscript.
Funding: This study was supported by the National Key Research and Development Project (Grant No. 2018YFB1601500),
the Qingdao Top Talent Program of Entrepreneurship and Innovation (Grant No. 19–3–2–11-zhc), the Joint Laboratory
for the Internet of Vehicles, Ministry of Education-China Mobile Communications Corporation (ICV-KF2018-03), and the
National Natural Science Foundation of China (Grant No. 61074140). Corresponding authors: Xiao-yuan Wang,
wangxiaoyuan@qust.edu.cn.
Conflicts of Interest: The authors declare no conflict of interest.
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