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> T-ITS-20-02-0298.R3<
Abstract—This research aims to model system dynamics for
mixed traffic flow consisting of Connected and Automated
Vehicles (CAVs) and Human-driven Vehicles (HVs). It quantifies
the impact of CAVs’ speed change on the overall traffic state on a
real-time basis. The model describes the impedance of CAVs’
speed reduction on traffic flow and considers the impact of
potential additional lane change induced by the speed reduction.
To validate the effectiveness of the proposed model, a VISSIM
based microscopic simulation evaluation is performed. The
results confirm that the accuracy of the proposed model is
generally over 80% with the CAVs’ speed reduction constrained
within 20 km/h. Sensitivity analysis is conducted in terms of
various CAV penetration rates and congestion levels. The
proposed model demonstrates consistently good performance
across all CAV penetration rates and congestion levels. A
showcase is presented to show the effect of the system dynamics
in active traffic management. The proposed model could serve as
the foundation of CAV based traffic management applications,
such as variable speed limit and speed harmonization.
Index Terms—Active demand management; partially connected
and automated traffic; speed harmonization, system dynamics
I. INTRODUCTION
RAFFIC bottleneck congestion has been a bothering
problem for traffic operations. When upstream traffic
exceeds a bottleneck’s capacity, a queue appears and
problems follow. On one hand, traffic in the queue produces a
shock wave. This wave exacerbates vehicle speed oscillation,
further deteriorates the traffic flow, increases the queue and
delay. This brings a series of adverse impacts on the economy
and society, including higher fuel consumption, increased crash
risk, and worse driving experience [1-4]. On the other hand,
Manuscript received February 19, 2020; revised September 19, 2020;
accepted 31 December, 2021. This paper is partially supported by National
Key R&D Program of China (No. 2018YFB1600600), Shanghai Municipal
Science and Technology Major Project (No. 2021SHZDZX0100), Shanghai
Oriental Scholar (2018), Tongji Zhongte Chair Professor Foundation (No.
000000375-2018082), and the Fundamental Research Funds for the Central
Universities. (Corresponding author: Jia Hu)
L. An is with the Key Laboratory of Road and Traffic Engineering,
Ministry of Education, Tongji University, No.4800 Cao’an Road, Shanghai,
China, 201804 (e-mail: an_lianhua@163.com).
X. Yang is with Department of Civil & Environmental Engineering, The
University of Utah, 110 Central Campus Dr. Rm 2133, Salt Lake City, UT
84112 (e-mail: x.yang@utah.edu).
J. Hu is with the Key Laboratory of Road and Traffic Engineering,
Ministry of Education, Tongji University, No.4800 Cao’an Road, Shanghai,
China, 201804 (e-mail: hujia@tongji.edu.cn).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
traffic bottleneck congestion is usually accompanied by the
phenomenon named capacity drop. Capacity drop describes the
fact that the congestion discharge rate drops below its
maximum when a road segment becomes congested. The
magnitude of the drop ranges from 3% to 18% [5]. Although it
seems minimal at first sight, based on existing research, a 5%
decrease in capacity could lead to a 78% increase in maximum
queue length and a 135% increase in total delay time [6].
Therefore, it is critical to prevent a freeway from breaking
down by blocking upstream traffic to keep the bottleneck
operating at a proper congestion level.
Many studies have explored measures to mitigate traffic
bottleneck congestion. These measures improve traffic
performance by sending out speed guidance to drivers in order
to achieve a more homogenous and stable arrival flow.
Variable Speed Limit (VSL) is an example of them and has
long been recognized as a promising tool. The studies on VSL
are conducted in the forms of simulation, analytical analysis
and empirical studies [7-9]. However, the existing techniques
mainly rely on drivers’ obedience to realize the advisory speed,
while drivers usually have a low compliance rate. The random
behaviors of Human-driven Vehicles (HVs) may compromise
or even sometimes fail the VSL. For example, when only a
third of traffic complies with the dynamic speed limit, the
actual arrival rate at a bottleneck would be two third greater
than the projected value. In this case, the bottleneck would still
break down even with the introduction of the VSL system.
Fortunately, with recent advancements in wireless
communications and vehicle automation, VSL technology has
reached a new level of maturity and transformed into a new
technique named speed harmonization with Connected and
Automated Vehicles (CAVs), such as the technology
developed by Goulet [10]. In this context, CAVs are able to
receive and execute advisory speed provided wirelessly by the
control center. Since the compliance rate of CAVs is 100
percent, the obedience problem associated with conventional
VSL is overcome. A number of studies on speed harmonization
with CAVs have been conducted [11-16]. In most studies,
CAVs are merely information providers for improved traffic
state estimation and prediction [12, 17]. Only three studies take
one step further and actually take advantage of the proactive
control of CAVs. Malikopoulos proposed a speed
harmonization optimal control algorithm for pure CAV traffic
[13]. However, according to the American Association of State
Highway and Transportation Officials (AASHTO), the market
penetration of CAVs will not reach 100% until the 2060s [18].
Mixed traffic with CAVs and HVs will exist for a long time.
Modeling System Dynamics of Mixed Traffic
with Partial Connected and Automated Vehicles
Lianhua An, Xianfeng Yang, and Jia Hu, Member, IEEE
T
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Hence, it is critical to develop speed harmonization technology
that is compatible with mixed traffic. Recently, Ghiasi et al
developed a speed harmonization method under mixed traffic
with CAVs, Connected Vehicles (CVs) and HVs [14].
However, this method can only be applied for one-lane freeway
which is not practical enough to be implementation-ready.
Hence, speed harmonization with CAVs needs to be further
enhanced to be compatible with partially connected and
automated traffic traveling on freeway with multiple lanes. To
achieve this goal, as with the development of any other control
algorithm, one needs to first understand the dynamic system
which in this case is the partially connected and automated
traffic and how the traffic responds to the speed reduction of
CAVs traveling within it. It is a quite complicated model since
it involves human behaviors which could be very stochastic
and random. When the speed reduction of CAVs is minor,
human drivers may keep following them and slow down with
them. As a result, the entire traffic slows down as well.
However, if the speed reduction of CAVs becomes significant,
human drivers may decide to overtake in order to keep their
speed. Then the traffic as a whole could either keep its original
speed or sometimes break down due to the additional lane
changes and overtakes. Therefore, constructing such a model to
understand the system dynamics of partially connected and
automated traffic on the freeway with multiple lanes is critical.
Existing system dynamics are mainly derived based on
fundamental diagram models in order to quantify the impact of
CAVs on traffic state [19-21]. They have limitations: i) the
impact of additional Lane Changes (LCs) potentially caused by
CAVs’ speed reduction is not considered; ii) the correlation
between the amount of speed drop and the additional lane
changes has not been found; iii) the impact of CAVs’
penetration rate has not been modeled; iv) existing system
dynamics are only applicable for steady-state traffic and
therefore not suitable for real-time control.
Hence, this paper aims to develop a system dynamics model
of partially connected and automated traffic which bears the
following features:
quantifying the state change of partially connected and
automated traffic caused by CAVs’ speed change;
applicable to fully connected and automated traffic as
well as partially connected and automated traffic;
considering the impact of lane changes induced by
CAVs’ speed reduction;
providing the foundation for future CAV-based traffic
management strategies such as VSL and speed
harmonization.
The remains of the paper are organized as follows: Section II
describes the problem. Section III proposes the system
dynamics model. Section IV presents the theoretical analysis
on the model. Section V performs an evaluation on the model.
Section VI present a showcase of the model application.
Section VII entails the conclusions and future works.
II. PROBLEM STATEMENTS
This study aims to provide a system dynamics model for
partially connected and automated traffic traveling on a road
segment, as shown in Fig. 1. CAVs are scattered amongst HVs
with no platoon formed. Desired speed commands are provided
from a control center to CAVs with an update frequency of
t
.
When CAVs travel on a road segment under control, all CAVs
receive an identical desired speed command. The task of this
paper is to model how outflow changes with CAVs’ speed
reduction. The impact of CAVs’ speed reduction on outflow
rate is two folds: i) flow rate would decrease as the entire
traffic would slow down with CAVs; ii) the number of lane
changes would increase as some HVs would conduct lane
change in order to keep their higher speed.
Fig. 1. The studied scenario of mixed traffic flow
III. MATHEMATICAL MODEL
In this section, system dynamics formulation is derived for
partially connected and automated traffic.
A. Parameters and Notations
TABLE I
NOTATIONS AND PARAMETERS
Notations
Explanation
j
t
k
The density of entire traffic flow on road segment
j
during
[ , ]t t t+
(passenger car unit per kilometer (pcu/km))
j
t
k−
The density of entire traffic flow on road segment
j
during
[ , ]t t t−
(pcu/km)
j
t
k
The phantom density (defined on page 4) of entire traffic flow on
road segment
j
during
[ , ]t t t+
(pcu/km)
jam
k
Jam density on the link (pcu/km)
L
Length of road segment (meters)
l
The average length of vehicles (meters)
n
The number of lanes on road segment
j
j
LCt
N
The number of LCs on road segment
j
during
[ , ]t t t+
j
t
N−
The number of vehicles on road segment
j
during
[ , ]t t t−
j
t
N
The number of vehicles on road segment
j
during
[ , ]t t t+
j
t
N
The number of phantom vehicles (defined on page 4) during
[ , ]t t t+
j
LC
N
The number of phantom vehicles caused by LCs on road
segment
j
during
[ , ]t t t+
j
CF
N
The number of phantom vehicles caused by the HVs following
CAVs on road segment
j
during
[ , ]t t t+
j
LCt
P
The lane change rate on road segment
j
during
[ , ]t t t+
j
t
q
The flow rate of entire traffic flow on road segment
j
during
[ , ]t t t+
(pcu/h)
j
t
q−
The flow rate of the entire traffic flow on road segment
j
during
[ , ]t t t−
(pcu/h)
-1j
t
q
The flow rate of traffic from upstream during
[ , ]t t t+
(pcu/h)
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q
The flow rate change of outflow (pcu/h)
t
The time when a new target speed command is sent (seconds)
t
T
Average travel time of all vehicles during
[ , ]t t t+
t
T−
Travel time of all vehicles during
[ , ]t t t−
t
Update frequency (seconds)
j
it
t−
The travel time of vehicle
i
on road segment
j
during
[ , ]t t t−
(seconds)
LC
t
Average lane-changing duration (seconds)
f
v
Free-flow speed (km/h)
j
t
v
The average speed of all vehicles on road segment
j
during
[ , ]t t t+
(km/h)
j
t
v−
The average speed of all vehicles on road segment
j
during
[ , ]t t t−
(km/h)
R
Congestion level (volume over capacity ratio)
j
t
The increased rate of travel time on road segment
j
during
[ , ]t t t+
j
t
The penetration rate of CAVs on road segment
j
j
t
The portion of vehicles under the influence of CAVs speed change
j
t
The intensity of density change during
[ , ]t t t+
1j
t
−
The flow rate of connected automated traffic entering from
upstream (pcu/h)
j
t
The target speed of CAVs on road segment
j
during
[ , ]t t t+
(km/h)
CAVs’ speed change (km/h)
B. Model Assumptions
Assumption 1: Without speed change command during
[ , ]t t t−
, CAVs’ speed equals to the average speed of entire
traffic:
=
jj
tt
v
−−
. When the speed change command is made at
time
t
, during time period
[ , ]t t t+
, the speed of CAVs will
be changed:
jj
tt
v
−
. This paper focuses on the scenarios
when the speeds of CAVs are lower than the previous mean
traffic speed
0jj
t t f
vv
−
.
Assumption 2: CAVs conduct no discretionary lane change.
Assumption 3: When the portion of CAVs is more than
HVs (
1/ 2
j
t
), the movements of all HVs are enforced by
CAVs’ speed reduction. Therefore, the average speed of the
entire traffic is equal to the target speed of CAVs:
jj
tt
v
=
.
Otherwise, only the movements of HVs that are right behind
CAVs are influenced.
Assumption 4: The road segment under control has no
geometry change along its longitudinal direction. The arrival
flow at the road segment of interest is stable during each
modeling time period.
The reason why assumption 2 is proposed: the
assumption is introduced to simplify the proposed model.
Since the model is the foundation of a controller to be
developed, the complexity of the model significantly impacts
the efficiency and stability of the controller. Consider that
very few CAVs would change lanes with no clear incentive,
such a simplification is presumed to be reasonable.
The reason why assumption 3 is proposed: theoretically
speaking, considering the extreme condition, if the penetration
rate of CAVs is very low, CAVs can hardly cast their impact
on the speed of HVs. On the contrary, when the penetration
rate of CAVs is close to 1, the movements of all HVs would
be strictly constrained by CAVs. Therefore, there has to be a
transition threshold on the penetration rate of CAVs. In order
to identify this threshold, a simulation test is conducted.
Stochasticity is introduced to the distribution of CAVs in the
traffic stream. Through changing arrival sequence and
crowdedness of vehicles, how CAVs and HVs are mixed up
varies. With stochasticity introduced, the existence of a
threshold of 0.5 is confirmed (as shown in Fig. 1 in Appendix).
Moreover, this threshold has been adopted by existing studies
[22-24]. This is the reason why assumption 3 is proposed.
C. System dynamics formulation
The system dynamics reveals how the flow rate change with
CAVs’ speed reduction. The impact of CAVs’ speed reduction
on other vehicles is the key. In this section, the impact is
modeled from two perspectives: i) the increase of the travel
time of all vehicles; ii) the increased number of LC maneuvers.
Proposition 1. Speed reduction of CAVs leads to an increase
in the travel time of all vehicles. The increase is described by:
( )/
j
j j j j
t t t t t
v
−
=−
(1)
1, 1/ 2
2 , 1/ 2
j
jj
t
j
t
tt
=
(2)
where
j
t
is the target speed of CAVs during
[ , ]t t t+
.
j
t
is
the portion of vehicles under the influence of CAVs speed
change.
j
t
is the penetration rate of CAVs.
j
t
v−
is the average
speed of all vehicles during
[ , ]t t t−
.
Proof. The penetration rate of CAVs is:
11
/
j j j
t t t
q
−−
=
(3)
The increase rate of travel time during
[ , ]t t t+
is
( ) / ( ) ( ) /
j j j j
tt
t t t t
j j j
tt t t
TT L L L v v v
Tv v v
−
−
−−−
−
= = − = −
(4)
Based on Assumption 3, the increase rate of travel time
under various CAVs’ penetration rate is derived as follows:
i) when the number of CAVs is higher than HVs (
1/ 2
j
t
),
all the vehicles slow down to CAVs’ desired speed
j
t
, the
increased rate of travel time can be estimated by:
/jj
jtt
tj j j j
t t t t
v
L L L
vv
−
−−
−
= − =
(5)
ii) when the number of CAVs is less than HVs (
1/ 2
j
t
),
only the movements of HVs that are right behind CAVs are
influenced. Given the total number of vehicles
j
t
N−
, the
number of vehicles slow down to CAVs’ desired speed
j
t
is
2jj
tt
NL
−
. Others which are not influenced by CAVs would
keep their speed
j
t
v−
. The number of these vehicles is
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( )
12jj
tt
NL
−
−
. The travel time after CAVs speed change
is
( )
( )
1
2 1 2
jj
j j j j j
t t t t t t t
N L N L v N
−
− − − −
+−
. Combing with
(4), the increase rate of travel time is
( )
( )
( )
1
12
2+
12
2
= + /
2 ( )
j
j
j
j
j
j
jtt
jj
tt
tt
j j j j
t t t t
t
tj j j j
t t t t
jj
t t t
j
t
NL
NL LL
N
v v v
L
LLL
v v v
v
−−
−−
− − −
− − −
−
−
=−
−
−
−
=
(6)
For clear representation, define
j
t
as the portion of vehicles
under the influence of CAVs speed change. According to
assumption 3,
j
t
is changed with penetration rate: When
penetration rate is over 0.5,
1
j
t
=
. It means all the vehicles
are slowing down with CAVs. When penetration rate is less
than 0.5,
2j
j
tt
=
. It means only the vehicles behind CAVs
are influenced. The formulation of
j
t
is presented in (2).
Combining (5), (6) and (2), resulting in (1), the increase rate
of travel time caused by CAVs speed reduction is obtained.
This completes the proof of Proposition 1. □
Proposition 2. Speed change of CAVs leads to increased LC
maneuvers. The increased number of LCs caused by CAVs is:
(1 )( ) ( 1000)
jj j j j
jt t t t t
LCt f
v Exp lk k L
Nv
− − −
− − −
=
(7)
where
j
t
k−
is density during
[ , ]t t t−
.
f
v
is free-flow speed.
L
is length of road segment.
l
is average length of vehicles.
Proof. Based on the findings from past study [25, 26], the
probability of LC caused by CAVs’ speed reduction is:
( )(1 ) ( 1000)
j
j j j
jt t t t
LCt f
v Exp lk
Pv
−−
− − −
=
(8)
To be noted, Equation (8) is originally applicable to vehicles
whose front vehicles are HVs. Although it is CAVs that slow
down in this study, there is no solid evidence showing that
human drivers’ reaction to a slowing down front vehicle would
vary depending on whether it is a CAV or non-CAV. The
existing research shares the common understanding that HV
driving behavior does not change with the presence of CAVs
[24, 27, 28]. Hence, Equation (8) is adopted nevertheless.
The number of LC is
j j j
LCt LC t
N P k L
−
=
(9)
Combining (8) and (9), resulting in (7), the increased
number of LCs caused by CAVs’ speed reduction is obtained.
This completes the proof of Proposition 2. □
Definition 1. In this paper, the term phantom density refers
to the collective summation of both real vehicles and phantom
vehicles traveling on a unit distance.
jj
jtt
tNN
kL
−+
=
(10)
As illustrated in Proposition 1 and 2, CAVs’ speed
reduction leads to an increase of not only the travel time of all
vehicles but also the number of LCs. The essentials of these
impacts are explained by phantom density, as shown in Fig. 2.
(a)
(b)
Fig. 2. The sketch map of the phantom vehicles: (a) CAVs
without control; (b) CAVs under control
Phantom vehicles appear under two circumstances: when
HVs follow CAVs, and when HVs make a lane change. On one
hand, the speed-reducing CAVs impede the following HVs and
force them to decelerate together. As a result, the occupancy on
the road increases. It is as if the following HVs are stretched
and become a longer phantom vehicle which is equivalent to
more than one passenger car. On the other hand, during the
process a vehicle is making a lane change, the vehicle would
occupy both lanes. It is as if there are two phantom vehicles
occupying both lanes at the location where the lane change is
being made, as shown in Fig. 2 (b). These phantom vehicles
create void areas where real vehicles are not able to enter and
sometimes being impeded. On the macroscopic level, when
computing travel time of the entire traffic, the adverse effect of
these phantom vehicles should be considered.
The impact of phantom vehicles is overestimated when the
lane change and speed reduction are considered separately.
However, this may not be such a significant overestimation, as
vehicles in real world rarely choose to make a discretionary
lane change if there is a slow front vehicle in the target lane.
Moreover, it is hard to estimate the chance that both maneuvers
happen simultaneously. Therefore, in order to keep the
simplicity of the proposed model, the lane change and speed
reduction are considered separately. Hence, the number of
phantom vehicles after CAVs’ speed reduction is defined as
j j j
t LC CF
N N N=+
.
The phantom density during
[ , ]t t t+
is written as:
j j j j j j
jt t t t LC CF
tN N N N N N
kL L L
−−
+ + +
= = =
(11)
To be noted, Equation (11) is an additive formula proposed to
account for the combination of both effects from speed
decrease and lane changing increase. Why the additive
formula is proposed is as follows: i) By introducing the
concept of “phantom” vehicles, the effects of speed reduction
and lane change increase are unified into the measurement of
density. When there is lane change or speed reduction,
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> T-ITS-20-02-0298.R3<
“phantom” vehicles are generated. In this way, both effects
could be quantified by the same unit and are able to be
summed directly. ii) Both effects are considered independent
since they describe the maneuvers in different directions.
Speed decrease generates phantom vehicles in lateral direction.
Lane change generates phantom vehicles in longitude
direction. Hence, the interaction is minimal. iii) The approach
is agreed upon by previous studies [29, 30]. Therefore,
Equation (11) is adopted.
Theorem 1. The phantom density during
[ , ]t t t+
is
( ) ( )(1 ) ( 1000)
(1 )
j
j j j j j j
jj t t t t t t t LC
tt jf
t
v v Exp lk t
kk vt
− − −
−
− − − −
= + +
(12)
Proof. According to the definition given by Edie [31], the
average density in a space-time domain is vehicle-hours
traveled divided by the domain’s area:
1
j
t
Nj
it
ji
t
t
kLt
−
−
=
−=
(13)
Based on Proposition 1, 2 and (13), the increased number of
phantom vehicles during
[ , ]t t t+
is
1
j
t
N
j j j
t it LCt LC
ji
t
βt N t
Nt
−
−
=+
=
(14)
According to Definition 1, the phantom density during
[ , ]t t t+
is
11
jj
tt
NN
j j j j
it t it LCt LC
jii
t
t t N t
kLt
−−
−−
==
++
=
(15)
The relationship between phantom density during
[ , ]t t t+
and density during
[ , ]t t t−
is:
1j
j j j LCt LC
t t t t
Nt
kk k L t
−
−
= + +
(16)
Combining (16) with (1) and (7), resulting in (12), the
phantom density is acquired.
This completes the proof of Theorem 1. □
Definition 2. The intensity of density change during
[ , ]t t t+
is
( )
( ) 1 ( 1000)
() j
j j j
j j j j j t t t t LC
jt t t t t
tjj f
tt
vψ ξ Exp lk t
kk γvψ
εvt
kψ
−−
−−
−
− − −
−−
= = +
(17)
Theorem 2. System dynamics formulation for partially
connected and automated traffic is
2
(1 ) ( 1000)
() (1 ) ( 1000)
(1 ) ( 1000)
2
j
j
j
jj
jt t t LC
tf jf
t
jj
j
ft t t LC
j
jam f
t
jj
t t t LC
jf
t
Exp lk t
q k v vt
vk Exp lk t
k v t
Exp lk t
vt
−
−
−−
−
− −
= − +
− −
++
− −
− +
(18)
Proof. According to the fundamental diagram of Green
Shields, the flow-density relation is
(1 / ) (1 / )
j j j j j
t t f t jam t f t jam
q k v k k k v k k= − = −
(19)
Given Theorem 1, the flow rate exiting road segment
j
during
[ , ]t t t+
is:
2
( ) ( )(1 ) ( 1000)
1
( ) ( )(1 ) ( 1000)
1
j
j
j j j j j j
jj t t t t t t t LC
t t f
jf
t
j j j j j j
fjt t t t t t t LC
tj
jam f
t
v v Exp lk t
q k v
vt
vv v Exp lk t
k
k v t
− − −
−
− − −
−
− − − −
= + +
− − − −
− + +
(20)
Define the CAVs’ speed change
==
j j j j
t t t t
v
−−
− −
, the
change on the flow rate
jj
tt
q q q −
= −
, combining (19) and
(20), resulting in (18), system dynamics formulation is derived.
This completes the proof of Theorem 2. □
Corollary 1. The number of lane changes could also be
calculated using flow rate and update frequency:
=
j j j
LCt LC t
N P q t
−
(21)
If substituting (9) into (18), the relation between update
frequency and segment length could be found. Hence, the
system dynamics formulation is rewritten as a function of
segment length instead of update frequency:
2
(1 ) ( 1000)
() (1 ) ( 1000)
(1 ) ( 1000)
2
j
j
j
j j j
jt t t t LC
tf jf
t
jj j j
ft t t t t LC
j
jam f
t
j j j
t t t t LC
jf
t
Exp lk v t
q k v vL
vk Exp lk v t
k v L
Exp lk v t
vL
−−
−
−−−
−−
− −
= − +
− −
++
− −
− +
(22)
IV. MODEL THEORETICAL ANALYSIS
This section evaluates the proposed model from an
analytical perspective.
From the derivation of system dynamics formula, the
following observations are founded:
The outflow rate after CAVs’ speed change is affected by
speed control command, including CAVs’ speed change
,
CAVs target speed
j
t
, and update frequency
t
, as shown in
Theorem 2. In addition, the outflow rate is affected by traffic
state before CAVs’ speed change, including density
j
t
k−
, speed
j
t
v−
and penetration rate of CAVs
j
t
ξ
, as well as road
characteristics, including lane-changing duration
LC
t
, jam
density
jam
k
and free-flow speed
f
v
.
The change in the outflow rate due to CAVs’ speed
reduction depends on the density during
[ , ]t t t−
. Equation
(18) shows that: the outflow rate increases (
0q
) when the
initial density
( )
2j
t jam t
kk
−+
. This means CAVs’ speed
reduction could increase the outflow rate under light traffic
conditions. Otherwise, CAVs’ speed reduction leads to an
outflow rate decrease. To better illustrate the theory, a three-
dimensional diagram of the outflow change rate is plotted
against CAVs’ speed change and penetration rate (PR), as
6
> T-ITS-20-02-0298.R3<
shown in Fig. 3. The following settings are adopted for the plot:
90
jam
k=
pcu/km/lane,
3n=
,
=120
f
v
km/h,
100t=
seconds,
2.5
LC
t=
seconds [32] ,
26,55,110,130
t
k−
pcu/km,
where the density
t
k−
values are corresponding to congestion
levels of 0.2, 0.4, 0.6, 0.8 respectively.
Fig. 3. Outflow change rate under various congestion levels
Fig. 3 confirms that the change on flow rate is under the
influence of not only CAVs’ speed reduction but also
congestion levels. When traffic is light, CAVs’ speed reduction
may bring up outflow when the reduction is within a certain
threshold. When traffic becomes heavier, CAVs’ speed
reduction reduces outflow rate. This finding is critical as it
demonstrates the possibility of not only reducing traffic but
also increase demand by only asking CAVs to slow down.
V. EVALUATIONS
The experiment design, simulation settings, Measurement of
Effectiveness (MOE) and results are discussed in this section.
A. Experiment design
The study area is presented in Fig. 4. It is a simulated road
segment of a three-lane freeway. The length of the freeway is
7.2 km. The first 3.6 km of the freeway is used as a warm-up
section. The following freeway segment is used as the segment
under control. The desired speed of CAVs can only be altered
when they are traveling on the segment under control. Two
groups of detectors are placed at the entrance and exit of the
segment under control (illustrated as D1 and D2 in Fig. 4). The
detectors are to collect flow rate, occupancy and speed data.
Fig. 4. Simulated road segment with detector locations
Sensitivity analysis was conducted for CAVs’ speed
reduction, congestion levels, and the PR of CAVs. Speed
reduction varies from 10 km/h to 60 km/h by a 10 km/h interval.
Congestion levels varies from 0.2 to 1 by a 0.2 interval. The
PR of CAVs varies from 0.1 to 0.9 by a 0.1 interval.
MOEs adopted are the relative error of the outflow rate and
fundamental diagram. They are averaged over all output runs.
The relative error of the outflow rate is calculated as follows:
1
1Nsi ci
risi
qq
ENq
=
−
=
(23)
where
r
E
is the relative error of outflow rate.
si
q
is the outflow
rate collected from simulation
i
.
ci
q
is the outflow rate
estimated using the proposed model (20).
B. Simulation set-up
The validation of the proposed model is conducted on a
simulation platform developed by this research team. The
platform is based on VISSIM but has been enhanced explicitly
for partially connected and automated traffic environment. The
driving model of HVs adopted is VISSIM internal model and
calibrated using naturalistic driving data collected from
Shanghai. The movement of CAVs are controlled by a
commercialized controller which is developed for China’s
Original Equipment Manufacturer (OEM) Shanghai
Automotive Industry Corporation (SAIC) by this research team.
Trajectories of simulated CAVs are validated with results of
California Partners for Advanced Transportation Technology
(PATH) from their field experiment in terms of speed,
acceleration and time gap. Hence, the simulation platform has
been validated with the capability of replicating the traffic with
the mixture of Advanced Driver-Assistance Systems (ADAS)
equipped vehicles. The work has been peer-reviewed and
published in Transportation Research Part C [24, 33].
In terms of the consideration of vehicle dynamics, the
simulated CAVs do not change speed instantaneously. The
speed commands (from the proposed model) are fed to CAVs
as desired speeds. CAVs may or may not fully comply with the
speed commands due to the traffic around. The VISSIM
external driver model is adopted to develop modules that
precisely replicate automated functions that are commercially
available. Modules included in the external driver model
include decision-maker, controller, and vehicle dynamics
model. It is designed as such, since the goal is to develop a
technology that could be applied in the near future. As of now,
the most mature longitudinal automated function is Adaptive
Cruise Control (ACC) which is assumed to be adopted by the
CAVs under investigation.
The stochasticity of CAV-HV mixed state is introduced by
setting a random seed. The mixed state of traffic in simulation
is dependent on arrival sequence and crowdedness of vehicles:
i) The arrival sequence of vehicles is influenced by random
seeds. When random seed varies, the arrival sequence of
vehicles changes. Hence the relative locations of CAVs in the
traffic changes, so does the mixed state of traffic. ii) The
crowdedness of vehicles is described by vehicle headway. The
distribution of headway varies with random seeds. This is how
the stochasticity of CAV-HV mixed state is introduced by
setting a random seed. The evaluations are simulated with
different random seeds. A minimum sample size requirement
was checked to make sure that a sufficient number of
simulation runs was achieved to ensure statistical significance.
To ensure the simulation platform realistic, road capacity
is calibrated following the definitions from Highway
Capacity Manual 2010 [34]. The capacity was calibrated to
7
> T-ITS-20-02-0298.R3<
be 2380 pcu/h/lane. The parameters are set as following:
3n=
,
2.5
LC
t=
seconds,
3.6L=
km,
120
f
v=
km/h,
100t=
seconds,
90
jam
k=
pcu/km/lane.
C. Results
The results are averaged from 5 simulation runs. The sample
size has passed the statistical test. The accuracy of the proposed
model is presented in Fig. 5. It confirms the performance of the
proposed model. The accuracy level is generally over 80% with
the speed reduction constrained within 20 km/h. The range of
speed change could be extended with the increase in
congestion level.
Generally, the performance of the proposed model improves
with the increase of congestion level. This is because, when
the traffic is light, the distances between vehicles are large.
Hence, the CAVs can hardly influence their surrounding HVs.
On the other hand, as the traffic becomes more congested, the
distances between vehicles close up. Therefore, the influence
of CAVs on their surrounding HVs becomes increasingly
significant. However, when congestion level approaches its
boundary saturated condition (equals 1), the model accuracy
deteriorates and is only about 65%. This means the proposed
model is not applicable to this condition. This makes sense as
there is not much room left for CAVs to hold the traffic under
the extreme congested condition. This phenomenon indicates
that the density level does influence the performance of the
proposed model. To best consider this influence, a compound
piecewise model in terms of both density level and penetration
rate could be proposed. However, since the impact of density
level and penetration rate is interactive, the compound
piecewise model may be complex. This ultimate piecewise
model may not be a good idea. Since the proposed model
serves as the foundation of a controller to be developed, the
compound piecewise structure would deteriorate the stability
of the controller. As the penetration rate has more significant
influence on the model (as shown in Fig. 1 in Appendix), the
density level is simplified from the piecewise structure in
order to balance the performance of the proposed model and
the controller to be developed. Fortunately, the consequence
of the aforementioned simplification is measured to be not
significant. According to the model evaluation, the accuracy is
generally over 80%. Hence, this simplification is adopted.
The accuracy of the proposed model deteriorates as the
speed reduction of CAVs increases. Therefore, it is suggested
to limit CAVs’ speed reduction to be within 20 km/h. On one
hand, the reason why the accuracy of the proposed model
limits the boundary of the controller to be developed is that
estimation capability of a system dynamics is critical to control
performance. In the worst case, control with an inaccurate
system dynamics could lead to the breakdown of a controller.
Therefore, in order to ensure the performance and stability of a
controller, the boundary of a system dynamics determines the
limit of the associated controller. On the other hand, the
boundary of 20 km/h is acceptable since the speed reduction of
CAVs shall not be too significant. Otherwise, the CAVs with
sudden speed changes might become a safety hazard on the
road. The proposed model is of value even with the speed
reduction constrained within 20 km/h. The specific reason is as
follows: In the situation where the proposed model is applied
on two consecutive segments, the speed reduction between
two segments should not be too great. Otherwise, when a
vehicle enters into a downstream road segment, the significant
speed reduction would cause concerns in comfort and safety.
A speed reduction of 20 km/h is very likely able to cover the
majority of the cases, as the speed change of CAVs is
completed over a very short time. Imagining one CAVs is
crossing into a downstream road segment with a desired
maximum deceleration of 3
2
ms
[29, 35]. In order to reduce
speed by 20 km/h, it would take the CAV at least 2 seconds. It
would be quite a significant deceleration process. In addition,
consider the dynamic speed limit system that is in today’s
practice, a typical speed limit increment between two adjacent
segments would be about 5 km/h to 10 km/h. Hence, the
proposed model is of value as it is able to achieve (generally)
at least 80% accuracy when the speed reduction is constrained
within 20 km/h, no matter the PR and congestion level.
Fig. 5 Model accuracy sensitivity analysis
Under the condition that speed reduction is constrained
within 20 km/h, the accuracy of the proposed model generally
increases with PR of CAVs. This makes sense because the
partially connected and automated traffic becomes less
stochastic and increasingly similar to CAVs traffic. Therefore,
the phantom density concept introduced in this paper becomes
less suitable to describe the traffic. To be noted, under the
8
> T-ITS-20-02-0298.R3<
condition that R = 0.8, PR = 0.1 and a speed change of 10 km/h,
the accuracy level is about 78%. This is where the only
exception happens for the statement that the accuracy level is
over 80% with the speed reduction constrained within 20 km/h.
This makes sense as it is hard for thin scattered CAVs to
influence the entire traffic. With this being stated, with the
speed reduction constrained within 20 km/h, the accuracy of the
proposed model is acceptable for implementation.
The fundamental diagram has been adopted as an additional
MOE. A comparison has been made between the micro
(simulation) and the non-micro (analytical model) approaches,
as demonstrated in Fig. 6. Sample data collected from when
CAVs speed reduction is 20 km/h are presented for
demonstration purposes. As shown in Fig. 6, the accuracy rate
is generally over 80% across all PRs and densities. In addition,
the performance of the proposed model improves with PR.
Fig. 6 Performance on fundamental diagrams
VI. SHOWCASE
To demonstrate the potential of the proposed model, a
simple case study is conducted and presented in this section.
The experiment design, simulation settings, Measurement of
Effectiveness (MOE) and the results are discussed in the
following.
A. Experiment design
The testbed is demonstrated in Fig. 7. It is a freeway
segment consists of three lanes. The first 500 meters of the
freeway is a warm-up segment. The next 1200 meters is the
segment under control. The warm-up time is 600 seconds.
CAVs change speed after the warm period ends. The desired
speed of CAVs is only altered when traveling on the segment
under control. The experiment is conducted on a traffic flow
with a speed of 120 km/h, flow rate of 6500 pcu/h, and jam
density of 90 pcu/km/lane. The penetration rate of CAVs is 0.5.
Control update frequency
t
is 5 seconds. Average lane-
changing duration is 2.5 seconds. A simple feedforward
controller is adopted. It computes a desired speed for CAVs
based on the proposed model. The goal is to regulate the
approaching traffic flow to match the capability of bottleneck.
Fig. 7 The testbed
It is one good idea to compare the proposed model against
an existing traffic state estimation model. However, there is no
common understanding on which traffic state estimation model
is the most prevalent. Hence, if the proposed model were
compared with a random traffic state estimating model, it
would be challenging for future readers to compare the
proposed model against another newly developed model which
is evaluated against a different baseline. Therefore, the scenario
without control is adopted as a universal benchmark.
The MOEs adopted are flow rate, cumulative delay of all
vehicles and cumulative number of stops. The MOEs are
collected over the entire segments under control.
B. Results
The results demonstrate that, with the help of the proposed
model, traffic breakdown could be mitigated. The average
increase in flow rate is 6%. The cumulative number of stops is
reduced by almost 45%. The cumulative delay of all vehicles
is reduced by almost 5%.
Fig. 8 presents the flow rate over time. It quantifies the
throughput in the entire duration when the control is activated.
As shown in Fig. 8, the flow rate is increases significantly
with the help of speed harmonization. The average increase in
flow rate is 6%. This is because the arrival rate of traffic is
reduced due to CAVs’ speed reduction. It prevents the
bottleneck from a drop in capacity. Hence, the throughput is
maintained at a high level.
Fig. 8 Performance on flowrate
Fig. 9 presents the trajectory of the cumulative number of
stops over time. It portrays the entire duration when the
control is activated. As shown in Fig. 8, the cumulative
number of stops decreases significantly with the help of speed
harmonization. The decrease in stops is as high as 45%. The
reason behind this is as follows: on one hand, due to the CAVs
speed reduction, the arrival rate of traffic is reduced. This
prevents the bottleneck from a drop in capacity. The
throughput is maintained at a high level. On the other hand,
the speed oscillation of traffic flow is reduced through speed
harmonization. Hence, the traffic flow becomes smoother and
steadier. Therefore, the cumulative number of stops drops.
Fig. 10 presents the trajectory of cumulative delay over time.
It portrays the entire duration when the control is activated. It
demonstrates that the cumulative delay is reduced with the help
of speed harmonization. The reduction rate is 5%. However,
compared to the decrease in the average number of stops, the
decrease in cumulative delay is less. The less reduction in delay
is due to the cancel-out effect between throughput increase and
speed reduction. On one hand, speed harmonization prevents
the bottleneck from breaking down by reducing the speed of
traffic. This is beneficial for delay reduction. On the other hand,
the speed reduction brings up the delay of the traffic. Hence,
9
> T-ITS-20-02-0298.R3<
the benefit in delay reduction is not as significant as that in
cumulative number of stops.
Fig. 9 Performance on cumulative number of stops
Fig. 10 Performance on cumulative delay
VII. CONCLUSION
This research proposed a system dynamics model of mixed
flow consisting of CAVs and HVs. It quantifies the impact of
CAVs’ speed change on traffic state. The model describes the
impedance of CAVs’ speed reduction on traffic flow and
considers the impact of potential additional lane changes
induced by the speed reduction. A VISSIM based microscopic
simulation evaluation was performed. The evaluation showed:
• The accuracy of the proposed model is generally over
80% with the CAVs’ speed reduction constrained within
20 km/h. The range of speed reduction is extended with
the increasing congestion level.
• Generally, the performance of the proposed model
improves with the increase of congestion level.
• The accuracy of the proposed model deteriorates as the
speed reduction of CAVs increases.
• Under the condition that CAVs’ speed reduction
constrained within 20 km/h, the accuracy of the proposed
model generally increases with PR of CAVs.
It is suggested to limit the speed reduction of CAVs to be
within 20 km/h.
The proposed model is not applicable for the saturated
condition when congestion level approaches 1.
The proposed model could serve as the foundation of CAV
based traffic management applications, such as variable speed
limit and speed harmonization. Future research should consider
investigating the system dynamics for oversaturated traffic
flow. In addition, the model proposed in this research assumes
that the road segment under control has no geometry change
along its longitudinal direction. It would be interesting to
enhance the generality of the model by enabling its capability
of handling road segments with a bottleneck.
APPENDIX
This section presents the impact of penetration rate on traffic
speed. Fig. 1 presents the speed of the entire traffic with CAVs
reducing their speed. The speed difference between CAVs and
the entire traffic is very minimal (only 0.7 km/h) when the
penetration rate of CAVs is greater than 0.5. The marginal
reduction of speed of the entire traffic is insignificant after PR
reaches 0.5. This means there exist a threshold of 0.5
penetration rate.
Fig. 1 The impact of penetration rate on traffic speed
ACKNOWLEDGMENT
This paper is partially supported by National Key R&D
Program of China (No.2018YFB1600600), Shanghai
Municipal Science and Technology Major Project
(No.2021SHZDZX0100), Shanghai Oriental Scholar (2018),
Tongji Zhongte Chair Professor Foundation (No. 000000375-
2018082), and the Fundamental Research Funds for the Central
Universities.
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Lianhua An received the B.S. and M.S. degree in
transportation engineering from Shandong University
of Science and Technology, Qingdao, China. Since
2018, she has worked as a Researcher Assistant at the
Key Laboratory of Road and Traffic Engineering of
the Ministry of Education, Tongji University,
Shanghai, China. Her research interests are in the
areas of traffic flow theory, active traffic control and management
with connected and automated vehicles.
Xianfeng Yang received B.S. degree in Civil
Engineering from Tsinghua University, Beijing,
China in 2009, M.S. and Ph.D. degrees in Civil
Engineering from University of Maryland, College
Park, MD, USA in 2012 and 2015, respectively. He
is an Assistant Professor in the Department of Civil
and Environmental Engineering at the University of Utah (UU). His
research is sponsored by multiple funding agencies such as Utah
Department of Transportation (UDOT), National Science Foundation
(NSF), US Department of Transportation (USDOT), and Federal
Highway Administration (FHWA). He has published over 100
journal and conference papers in developing dynamic evacuation
systems, designing traffic incident management systems, and using
connected vehicle technology in traffic operations. He has been
serving as the member of the Traffic Signal System committee and
Emergency Evacuation committee of the Transportation Research
Board (TRB). He is also the elected vice chair of INFORMS SIG-ITS
group and the associate editor and editorial board member of several
journals.
Jia Hu works as a ZhongTe Distinguished Chair in
Cooperative Automation in the College of
Transportation Engineering at Tongji University.
Before joining Tongji, he was a research associate at
the Federal Highway Administration, USA (FHWA).
He is an Associate Editor of the American Society of
Civil Engineers Journal of Transportation Engineering, IEEE Open
Journal in Intelligent Transportation Systems and an editorial board
member of the International Journal of Transportation. Furthermore,
he is a member of TRB (a division of the National Academies)
Vehicle Highway Automation Committee, Freeway Operation
Committee and Simulation subcommittee of Traffic Signal Systems
Committee, and a member of Advanced Technologies Committee of
ASCE Transportation and Development Institute. He is also Chair of
Vehicle Automation and Connectivity Committee of the World
Transport Convention.
