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IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS 1
Integrated Schedule and Trajectory Optimization for
Connected Automated Vehicles in a Conflict Zone
Zhihong Yao , Haoran Jiang, Yang Cheng ,Member, IEEE, Yangsheng Jiang, and Bin Ran
Abstract— The large-scale application of connected automated
vehicles (CAVs) provides new opportunities and challenges for
the optimization and management of traffic conflict zones.
To improve the traffic efficiency of conflict zones and reduce the
travel delay and fuel consumption of CAVs, this paper presents
a two-level optimization method of scheduling and trajectory
planning for CAVs. At the first level, a 0-1 mixed-integer linear
program (MILP) is proposed for vehicles entering scheduling.
At the second level, a multi-vehicle optimal trajectory control
model is developed based on the optimal vehicle schedule from
the first level. Then, to reduce the complexity of solving the
multi-vehicle optimal trajectory control model, we transform
this model into non-linear programming (NLP) based on the
infinitesimal method. Moreover, a rolling optimization strategy
is developed to facilitate field application. Numerical simulation
experiments of different traffic scenarios are conducted, and the
results show that the proposed method can effectively reduce
vehicle delays and fuel consumption, compared with the first-
in-first-out (FIFO) method. The numerical results show that the
vehicle delay can be reduced by up to 54% and fuel consumption
by up to 34% under different traffic demands. Sensitivity analysis
indicates that the performance of the proposed method is mainly
determined by the minimum safety time interval of vehicles
entering the conflict zone.
Index Terms—Connected automated vehicles, optimal schedul-
ing, MILP, conflict zone, trajectories planning.
I. INTRODUCTION
TRAFFIC conflict zones [1]–[4] (e.g. ramps, intersections,
work-zones) generally refer to areas of elevated collision
risks for vehicles. Studies show that conflict zones not only
have potential safety hazards but also are the primary nodes
causing vehicle delays and fuel consumption [3], [4]. There-
fore, most existing research [5]–[11] is to alleviate collision
Manuscript received February 19, 2020; revised June 28, 2020 and
August 16, 2020; accepted September 25, 2020. This work was supported in
part by the Chinese National Natural Science Fund under Grant 52002339,
Grant 71901183, and Grant 71771190, in part by the Innovation Center
Project of Chengdu Jiao Da Big Data Technology Co., Ltd. under Grant
JDSKCXZX202003, and in part by the Open Fund Project of Chongqing
Key Laboratory of Traffic and Transportation under Grant 2018TE01. The
Associate Editor for this article was I. Papamichail. (Corresponding author:
Zhihong Yao.)
Zhihong Yao is with the National Engineering Laboratory of Integrated
Transportation Big Data Application Technology, National United Engineering
Laboratory of Integrated and Intelligent Transportation, School of Transporta-
tion and Logistics, Institute of System Science and Engineering, Southwest
Jiaotong University, Sichuan 610031, China (e-mail: zhyao@swjtu.edu.cn).
Haoran Jiang and Yangsheng Jiang are with the National Engineering Labo-
ratory of Integrated Transportation Big Data Application Technology, School
of Transportation and Logistics, Southwest Jiaotong University, Sichuan
610031, China (e-mail: 1072135297@qq.com; jiangyangsheng@swjtu.cn).
Yang Cheng and Bin Ran are with the Department of Civil and Environmen-
tal Engineering, University of Wisconsin–Madison, Madison, WI 53706 USA
(e-mail: cheng8@wisc.edu; bran@wisc.edu).
Digital Object Identifier 10.1109/TITS.2020.3027731
through scheduling, which is controlling the time of vehicles
passing through the conflict zones. However, most of these
methods are only for human-driven vehicles (HDVs), and
cannot directly optimize and control the HDVs. In recent years,
the development of connected automated vehicles (CAVs)
technologies [12]–[14], have enabled the information exchange
among vehicles (V2V) and between vehicles and infrastructure
(V2I) through wireless communication by installing onboard
units (OBUs) and roadside units (RSUs). Moreover, the latest
research [15] shows that by 2045, the penetration rate of
CAVs on the road would reach a high level, which can
reach 87.2% if conditions permit. Therefore, the large-scale
application of CAVs presents new opportunities and challenges
for the operation, management, and optimization of the conflict
zones [16]–[21].
Many scholars have studied the application of CAVs in
transportation. These studies mainly focus on traffic flow char-
acteristics [22]–[24], vehicle trajectory optimization [25]–[28],
and traffic signal control optimization [9], [29], [30]. In this
study, we focus on the optimization of scheduling and tra-
jectory of CAVs. He et al. [25] proposed an optimal speed
model to provide speed guidance for CAVs considering the
influence of vehicle queues at intersections. The case study
showed that speed guidance could reduce vehicles’ travel
time and fuel consumption by 9% and 29%, respectively.
However, this study did not consider the interaction between
vehicles. Wan et al. [31] constructed a vehicle speed guidance
model with an analytical solution for the mixed traffic flow
for a fixed signal timing plan. The simulation result showed
that the fuel consumption of the mixed platoon could be
greatly reduced with the increase of the penetration rate
of CAVs. Zhao et al. [32] proposed a predictive control
model of speed guidance for the mixed platoon consisting
of CAVs and HDVs. The simulation results showed that the
proposed model could effectively smooth the vehicle trajectory
and reduce fuel consumption. These studies provide speed
guidance for a fixed signal timing plan only, not considering
optimizing traffic signal timing. Therefore, some scholars
studied coordinated optimization of traffic signals and vehicle
trajectories. Yu et al. [8] investigated a traffic signal and
vehicle trajectory optimization model based on mixed-integer
programming at an isolated intersection. Simulation results
validated the advantages of the proposed control method
over vehicle-actuated control in terms of intersection capacity,
vehicle delays, and CO2emissions. Feng et al. [33] pro-
posed a two-stage optimization model for traffic signal and
vehicle trajectory control, in which dynamic programming
algorithms and optimal control theories were used to obtain
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2IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
the signal timing plan and vehicle trajectory of the inter-
section, respectively. The simulation showed that compared
with fixed traffic signal timing and adaptive control, the pro-
posed model could reduce vehicle’s delays and fuel consump-
tion simultaneously. However, the basic assumption of these
studies is to use traffic signals to control vehicles, while
coordination and schedule optimization between vehicles are
not considered.
With the large-scale application of CAVs, the vehicle would
be able to independently and cooperatively pass through the
conflict zones without a traffic signal [34]. The basic idea is
that the conflict zones do not need signal control, and the
time when the vehicle enters the conflict zone is controlled
and optimized to avoid the collision. Tachet et al. [35]
developed a slot-based intersection control method, which
is similar to slot-based systems used in aerial traffic. The
theoretical analysis results showed that compared with the
signal control system, the proposed method has the potential
to double the capacity and significantly reduce the delay.
Li and Wang [36] first conceptually proposed a future
intersection management model, in which the intersection has
no traffic signals and the vehicle could determine the timing
of entering through cooperation and interaction, thereby avoid-
ing collision between vehicles [37]. Fayazi and Vahidi [17],
Fayazi et al. [38] constructed a mixed-integer linear program
model to optimize the timing of vehicles entering intersections.
The model used the form of a multi-objective linear combi-
nation to achieve the effect of vehicle delay reduction. The
result showed that the MILP improved the average travel time
per vehicle and the average stopped delay per stopped vehicle
by 7.5% and 52.4%, respectively. Dresner and Stone [39]
developed an alternative mechanism for coordinating the
movement of autonomous vehicles through intersections based
on a multiagent system. The experimental results strongly
proved the effectiveness of the proposed method. However,
the above studies only studied how to control or optimize the
time when the vehicle entered the conflict zones and did not
consider the optimization of vehicle trajectory. He et al. [18]
designed a future intersection management control model in
CAVs environment. This model avoided collisions between
vehicles by controlling the time of the vehicle to enter the
intersection and optimizing vehicles’ trajectory. The simula-
tion results showed that the model significantly improved the
intersection capacity and reduced fuel consumption. However,
the basic assumption of this study was that the vehicle enters
the intersection based on the first-in-first-out (FIFO) rule.
Thus, the vehicle entry sequence was not optimized. To find
the optimal sequence and trajectory of vehicles, Rios-Torres
and Malikopoulos [40] presented an optimization framework
and an analytical closed-form solution to coordinate vehicles
at merging zones. The results showed that coordination of
vehicles can significantly reduce both fuel consumption and
travel time. Wang et al. [41] proposed a distributed consensus
protocol approach for CAVs to cooperate by V2X communica-
tions. The simulation study indicated that the approach benefits
traffic throughput and energy saving. However, the approach
manages CAV based on relevant rules, and does not use
Fig. 1. Common conflict zones.
the optimization method to optimize the vehicle timing and
trajectory as a whole.
As far as we know, most of the existing research does not
consider optimizing the timing and trajectory of vehicles enter-
ing the conflict zones simultaneously. Therefore, to address
this gap, this study proposes a two-level model to optimize
schedule and trajectories for CAVs in conflict zones.
The remainder of this paper is structured as follows.
Section 2 describes the problem of the optimization of the
conflict zones of CAVs. A two-level optimization model is pro-
posed in Section 3. Section 4 introduces a rolling optimization
strategy. Numerical simulation and discussion are conducted in
Section 5. Finally, Section 6 ends this paper with conclusions
and future work.
II. PROBLEM SETTINGS
Traffic conflict zones have a common function, which is
defined as separating vehicles with conflicting movements
into appropriate batches without inter-batch conflicts to let
them pass sequentially in a relatively fair manner. As shown
in Fig. 1, common conflict zones include freeway merges,
lane-closed work-zones, and one-way unsignalized intersec-
tions. In particular, we assume that vehicles do not change
lanes in the control zone, that is, the multi Lane problem can
be described as a one-lane problem, and we only consider
the one-lane problem in the Fig. 1c). The vehicle needs
to slow down and observe other vehicles from different
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YAO et al.: INTEGRATED SCHEDULE AND TRAJECTORY OPTIMIZATION FOR CAVs IN A CONFLICT ZONE 3
directions to ensure safety before it enters the conflict zone.
This behavior would directly lead to vehicle delays and extra
fuel consumption. In a CAV environment, CAVs can pass
through the conflict zone with a higher speed based on an
optimal arrival schedule. Therefore, vehicle delays would be
reduced significantly. Moreover, based on the optimal arrival
schedule, the vehicle trajectory is further optimized to reduce
fuel consumption. This study aims to optimize both sched-
ule and trajectories for CAVs in a conflict zone. Therefore,
the proposed method can not only avoid vehicle collisions but
also reduce vehicle delays and fuel consumption.
Fig. 1 shows that the conflict zones studied in this paper
generally consist of two conflicting directions, each of which
is represented by i,wherei∈I={1,2}. The length of the
control zone is L, the free-flow speed of the road segment in
each direction is vf
i,i∈I. Given the optimization interval
T, if the current time is T0, the optimized time interval is
[T0,T0+T]. During this time interval, the set of vehicles
entering the control zone in the direction iis Ni. Therefore,
all vehicles entering the control zone during the time interval
are represented as j∈Ni={1,2,··· ,Ni},i∈I. Assume
that t−
ij represents the time when the jth vehicle enters the
control zone of the ith direction. t∗
ij and t+
ij are the ideal
time and the actual time when the jth vehicle enters the
conflict zone of the ith direction, respectively. The problem in
this study is expressed as a two-level optimization problem:
(1) the first level is to optimize the time t+
ij of all CAVs
arriving at the conflict zone for achieving the minimum delay;
(2) in the second level, based on the optimal arrival time t+
ij
in the first level, the safety distance between the vehicles in
the same direction, the start and end states (time, position
and speed, etc.) constraints are considered to optimize vehicle
trajectories for minimizing fuel consumption. The following
section would discuss in detail how to propose the two-level
optimization model.
III. TWO-LEVEL OPTIMIZATION MODEL
A. Assumptions and Limitations
To facilitate the development of our model, there are some
necessary assumptions and limitations.
(1) CAVs are in regular operation without considering its
failure, and all CAVs can be controlled and optimized
by infrastructure.
(2) Control and communication delay of CAVs are not
considered in this study.
(3) To reduce vehicle delay, refer to Zhao et al. [32],
all CAVs pass the downstream intersection by the
desired speed.
(4) In the control zone, vehicles are not allowed to
change lanes. Therefore, this study does not consider
lane-changing behavior in the control zone, which can
be referred to Yu et al. [8].
B. Scheduling Optimization Model
A one-way unsignalized intersections from Fig. 1c) is an
example. The east-west direction of the main road is set
TAB LE I
TIMING SCHEDULE ANALYSIS
to i=1, and the north-south direction of the secondary road
is set to i=2. A simple example is used to illustrate the
solution space for vehicles’ schedule optimization. We assume
that there are two vehicles on the main road and secondary
road to pass through the conflict zone in the optimized time
interval. The vehicles of the main road are labeled 1 and 2,
while the vehicles in the secondary road are labeled 3 and 4.
Meanwhile, vehicles in the same direction follow the first-
in-first-out (FIFO) rule, which means the vehicles entering
the conflict zone in the same direction are unchanged (i.e.,
Vehicle 1 arrives and passes the conflict zone before Vehicle
2). Therefore, the timing schedule optimization problem is to
optimize the timing of vehicles entering the conflict zone for
minimizing the average delay. The following is a brief analysis
of the timing schedule of the main road and the ramp with only
two vehicles, as shown in Table I.
Table I indicates that when there are two vehicles on the
main road and the secondary road, there are six kinds of timing
schedules for passing through the conflict zone. Furthermore,
if there are mvehicles on the main road and nvehicles on the
secondary road. According to the theory of arrangement and
combination, we can know that the number of possible timing
sequences is Am+n
m+nAm
m×An
n. Therefore, as the number of
vehicles increases, the number of timing sequences may be
huge. The enumeration method would not be applicable to
solve this problem. The following section would specifically
propose a mixed-integer linear program (MILP) model to
optimize the schedule of vehicles passing through the conflict
zones.
1) Time Parameter Analysis: The timing schedule opti-
mization model contains many time parameters, such as the
time vehicles enter the control zone, the ideal time, and the
actual time for vehicles to reach the conflict zone. In a CAV
environment, we assume that the time (t−
ij )when vehicles enter
the control zone can be estimated accurately [42]. Therefore,
this study assumes that the time when the vehicle enters the
control zone in the future time interval Tis known and
defined as t−
ij ,∀i∈I,j∈Ni. Then, the ideal time for
vehicles to reach the conflict zone can be estimated based
on the distance from the boundaries of the control zone to the
conflict zone and the free-flow speed of the road segment.
t∗
ij =t−
ij +L
Vi
,∀i∈I,j∈Ni.(1)
where t−
ij is the time for the jth vehicles of the ith direction
enters the control zone; t∗
ij is the ideal time for the jth
vehicles of the ith the direction enters the conflict zone; Viis
the free-flow speed in the ith direction; Lis the distance
from the boundaries of the control zone to the conflict zone,
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4IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
a common assumption is that the coverage of the communi-
cation is a spherical region, so the distance in each direction
is equal [43], [44].
2) Objective Function: After relevant time parameters
are determined, the objective function for vehicle delay is
obtained. The delay of vehicles is defined as the actual travel
time minus the ideal travel time (travel with free-flow speed).
Therefore, the total delay of vehicles in the ith direction is
formulated as
Di=
j∈N
it+
ij −t−
ij −L
Vi,∀i∈I.(2)
where Diis the total delay of vehicles in the ith direction;
t+
ij is the actual time when the jth vehicle in the ith direction
reaches the conflict zone, which is a variable that needs to be
optimized.
The average delay of all vehicles in the optimization time
interval is calculated based on Eq. (2), as shown in Eq. (3).
d=1
i∈I
Ni
i∈I
j∈N
it+
ij −t−
ij −L
Vi(3)
where dis the average delay of all vehicles within the
optimization time interval; Niis the number of vehicles in the
ith direction during the optimization time interval, Ni=|Ni|.
3) Constrains: The actual time of vehicles to reach the
conflict zone is not less than the ideal time, so this constraint
can be written in Eq. (4).
t+
ij ≥t∗
ij,∀i∈I,j∈Ni.(4)
In this study, we only consider a single lane, so there is no
lane change behavior. Therefore, the first-in-first-out (FIFO)
rule is adopted for vehicles in the same direction. To ensure
that two adjacent vehicles in the same direction pass through
the conflict zone safely, the time interval between them must
not be less than τ. This constraint is expressed in Eq. (5).
t+
ij ≥t+
i(j−1)+τ, ∀i∈I,j∈Ni.(5)
where τis the minimum time interval between two adjacent
vehicles in the same direction to pass through the conflict zone.
Furthermore, a specific time interval should be ensured for
two adjacent vehicles in different directions to pass through
the conflict zone. We assume that the minimum time interval
between two adjacent vehicles in different directions passing
through the conflict zone is ω. To facilitate our model devel-
opment, we use the time interval of any two vehicles passing
through the conflict zone in different directions to formulate
this constraint, as shown in Eq. (6).
t+
1,m−t+
2,n≥ω, m∈N1,n∈N2.(6)
where ωis the minimum time interval between two adjacent
vehicles in different directions to pass through the conflict
zone.
The absolute value in Eq. (6) makes the optimization
problems nonlinear, which is not easy to solve. To transform
nonlinear constraints into linear constraints, a 0-1 variable
Bm,nis introduced. Therefore, Eq. (6) is rewritten as follows:
t+
1,m−t+
2,n≥ω, if Bm,n=1
t+
2,n−t+
1,m≥ω, if Bm,n=0,m∈N1,n∈N2.(7)
where Bm,nis a 0-1 variable, Bm,n=1 indicates that the
vehicle numbered nin direction 1 passes the conflict zone
first compared with the vehicle numbered min direction 2;
otherwise Bm,n=0.
Further, Eq. (7) is converted into the following two equiv-
alent constraint formulas based on a larger value M.
Bm,n×M+t+
2,n−t+
1,m≥ω, m∈N1,n∈N2.(8)
1−Bm,n×M+t+
1,m−t+
2,n≥ω, m∈N1,n∈N2.(9)
4) Solution Algorithm: From the above analysis, the objec-
tive function Eq. (3) and the constraints Eqs. (4-9) of the
proposed model are linear. Thus, the proposed optimization
model is linear programming (LP), which is directly solved by
commercial software, such as MATLAB, CPLEX, and Lingo.
However, the value of Mhas a significant influence on the
efficiency of the solving algorithm. To improve the calculation
efficiency of the algorithm, we assume that the actual travel
time of the vehicle is not more than θtimes the free-flow
travel time, where θis greater than 1. To sum up, Eq. (4) can
be specified as
t−
ij +θL
Vi
≥t+
ij ≥t−
ij +L
Vi
,∀i∈I,j∈Ni.(10)
Besides, the value of Monly needs to be higher than the
maximum possible time, which equals to the vehicle to reach
the conflict zone within the optimized time interval minus the
minimum possible time. Therefore, Mis defined as follows
M=max t−
1,N1+θL
V1
,t−
2,N2+θL
V2
−min t−
1,1+L
V1
,t−
2,1+L
V2.(11)
where N1and N2are the last vehicle entering the control
zone in direction 1 and direction 2 within the optimized time
interval, respectively.
In summary, the scheduling of the vehicle entering the
conflict zone is a linear programming model, as shown in
Eq. (12).
min D=1
i∈INi
i∈I
j∈N
it+
ij −t−
ij −L
Vi
s.t.
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
t−
ij +θL
Vi
≥t+
ij ≥t−
ij +L
Vi
,∀i∈I,j∈Ni.
t+
ij ≥t+
i(j−1)+τ,∀i∈I,j∈Ni.
Bm,n×M+t+
2,n−t+
1,m≥ω, m∈N1,n∈N2.
(1−Bm,n)×M+t+
1,m−t+
2,n≥ω, m∈N1,n∈N2.
Bm,nis a 0,1variable
(12)
When the flow rate is low, the vehicle can pass through
the conflict zone without delay, so the value of θis very
close to 1 in this case; otherwise, the value of θshould be
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YAO et al.: INTEGRATED SCHEDULE AND TRAJECTORY OPTIMIZATION FOR CAVs IN A CONFLICT ZONE 5
Fig. 2. Schematic diagram of trajectory optimization.
more significant. To ensure the efficiency of the algorithm,
we set reasonable θunder different flow conditions.
C. Trajectory Optimization Model
1) Problem Analysis: Based on the optimal schedule,
the space-time diagram of the optimal trajectory is shown in
Fig. 2. In the trajectory optimization model, the optimized
trajectory minimizes the fuel consumption of all vehicles
during the optimized time interval based on the initial and
end state of vehicles. Besides, the minimum safety interval
between vehicles in the same direction must be considered.
Therefore, the trajectory optimization is an optimal control
problem [25], [31], [32], [45], [46].
2) Objective Function: Based on existing research
[46]–[50], the fuel consumption of vehicles is related to
its speed and acceleration. The instantaneous gasoline
consumption function proposed by Akcelik [47] is adopted
in this study. The gasoline consumption rate is formulated
according to the instantaneous velocity and acceleration of a
vehicle.
F(v,a;t)
=⎧
⎪
⎨
⎪
⎩
α+β1P(t)+β2ma(t)2v(t)
1000 a>0
,P(t)P>0
α, P(t)≤0.
(13)
where Fis the instantaneous fuel consumption of vehicles
at the time t;αis the fuel consumption rate at the time of
vehicle idling; mis the mass of the vehicle; β1is an efficiency
parameter which relates fuel consumed to the energy provided
by the engine; β2is an efficiency parameter, which relates fuel
consumed during positive acceleration to the product of inertia
energy and acceleration; v(t)and a(t)represent speed and
acceleration of vehicle at the time t, respectively; P(t)is the
total power of vehicles, which is expressed by Eq. (14).
P(t)=d1v(t)+d2v(t)2+d3v(t)3+ma (t)v(t)
1000 .(14)
where d1,d2and d3are represent rolling, engine and aerody-
namic drag, respectively.
Based on the literature [47], the values of the relevant
parameters in Eqs. (13) and (14) are: α=0.666mL/s, β1=
0.072 ml/kJ, β2=0.0344mL/(kJ(m/s)2),d1=0.269kN,
d2=0.0171kN/(m/s), d3=0.000672 kN/(m/s)2,m=1680 kg.
To sum up, the total fuel consumption Giof the vehicle in
the ith conflict direction during an optimized time interval is
shown as
Gi=
j∈N
itij+
t−
ij
F˙xij (t),¨xij (t);tdt,∀i∈I.(15)
where ˙xij (t)and ¨xij (t)are the speed and acceleration of the
jth vehicle in the ith direction at the time t, respectively.
The average fuel consumption of all vehicles is obtained,
as shown in Eq. (16).
g=1
i∈INi
i∈I
j∈N
it+
ij
t−
ij
F˙xij (t),¨xij (t);tdt.(16)
where gis the average fuel consumption of all vehicles during
the optimized time interval.
3) Constraints: The vehicle should satisfy the following
dynamic equations at any time.
˙xij (t)=dxij (t)
dt ,∀t∈t−
ij ,t+
ij ,i∈I,j∈Ni.(17)
¨xij (t)=d˙xij (t)
dt ,∀t∈t−
ij ,t+
ij ,i∈I,j∈Ni.(18)
The vehicle’s initial and the termination state satisfy the
following constraints
xij t−
ij =0,˙xij t−
ij =v0
ij,¨xij t−
ij =0,∀i∈I,j∈Ni.
(19)
xij t+
ij =L,˙xij t+
ij =vf
ij,¨xij t+
ij =0,∀i∈I,j∈Ni.
(20)
Meanwhile, to ensure the safety of the vehicle, the minimum
safety spacing requirements must be met between the adjacent
vehicles in the same direction (segment), which is written as
follows:
xi(j−1)(t)−xij (t)≥s0+l,
∀i∈I,j∈Ni\{1},tij ∈t−
i(j−1),t+
ij .(21)
where s0is the minimum safe spacing between adjacent
vehicles, lis the length of vehicle, which is generally set at
5m.
Besides, the speed and acceleration of the vehicle must meet
the relevant constraints during driving, which is formulated as
follows.
0≤˙xij (t)≤vmax,∀t∈t−
ij ,t+
ij ,i∈I,j∈Ni.(22)
amin ≤¨xij (t)≤amax,∀t∈t−
ij ,t+
ij ,i∈I,j∈Ni.(23)
where amin and amax are the minimum and maximum accel-
eration of the vehicle, respectively.
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6IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
4) Solution Algorithm: Based on the above analysis,
the vehicle trajectory optimization model is an optimal control
model. We transform it into nonlinear programming (NLP)
to solve. Therefore, the infinitesimal method can be used to
process the optimal control model discretely.
We assume δis a sufficiently small number, so t−
ij ≤
kδ≤t+
ij ,k−
ijδ=t−
ij ,k+
ijδ=t+
ij . The objective function Eq. (16)
is defined as follows.
g=1
i∈I
Ni
i∈I
j∈N
i
k∈k−
ij ,k+
ij
F˙xij [k],¨xij [k];tδ(24)
Similarly, the constraints of dynamic equations are formu-
lated as follows.
˙xij [k]=xij [k]−xij [k−1]
δ,
∀k∈k−
ij +1,k+
ij,i∈I,j∈Ni.(25)
¨xij [k]=˙xij [k+1] −˙xij [k]
δ
=xij [k+1] −2xij [k]+xij [k−1]
δ2,
∀k∈k−
ij +1,k+
ij −1,i∈I,j∈Ni.(26)
The initial and termination state constraints are
xij k−
ij=0,˙xij k−
ij=v0
ij,¨xij k−
ij=0,∀i∈I,j∈Ni.
(27)
xij k+
ij=L,˙xij k+
ij=vf
ij,¨xij k+
ij=0,∀i∈I,j∈Ni.
(28)
The positional constraints of adjacent vehicles can be
expressed as follows.
xi(j−1)[k]−xij [k]≥s0,
∀i∈I,n∈Ni\{1},tij ∈k−
i(j−1),k+
ij.(29)
The speed and acceleration constraints of the vehicle can
be written as follows.
0≤˙xij [k]≤vmax,∀k∈k−
ij,k+
ij,i∈I,j∈Ni.(30)
amin ≤¨xij [k]≤amax,∀k∈k−
ij,k+
ij,i∈I,j∈Ni.(31)
Above all, the objective function of the model is non-
linear, as shown in Eq. (24); the constraint conditions are
Eqs. (25)-(31), all of which are linear formulas. Therefore,
this model is nonlinear programming (NLP), which can be
solved by IPOPT or MATLAB.
IV. ROLLING OPTIMIZATION STRATEGY
In the actual optimization, a rolling optimization strategy
can be adopted [51]. There are two main reasons for con-
sideration. Firstly, we assume that the time that vehicles reach
the control boundary can be calculated accurately in the CAVs
environment [42]; but the longer the prediction time interval,
the more difficult the prediction is. Therefore, short-term (such
as 10 seconds) predictions are easier to implement. Secondly,
when the optimization time interval (T)is higher than a spe-
cific value, the vehicles in different optimization time intervals
would not affect the optimal sequence. Therefore, the sequence
of vehicles in different optimization time intervals can be
optimized independently. To sum up, we assume that the
schedule and trajectories of CAVs passing through the conflict
zone are optimized every Ttime interval, and the total
optimization duration is T. The specific rolling optimization
strategy is as follows.
Step 1. Initializing parameter: θ,andt=0;
Step 2. Calculate the current optimization time interval
[t,t+T];
Step 3. Predict the time of CAVs arriving in the control zone
within the current optimization time interval, t−
ij ,∀i∈
I,j∈Ni;
Step 4. The first-level of linear programming model in Eq. (12)
is adopted to optimize the optimal schedule of CAVs
passing through the conflict zone: t+
ij ,∀i∈I,j∈Ni
and vehicle delay D;
Step 5. Based on the optimal scheduling t+
ij ,∀i∈I,j∈Ni
calculated by Step 4, the second-level nonlinear opti-
mization model is adopted to solve the optimal vehicle
trajectories and fuel consumption: xij,∀i∈I,j∈Ni
and G;
Step 6. Let t=t+T.Ift>T, the optimization ends,
output the optimal schedule and trajectories, the vehicle
delay and fuel consumption; otherwise, go to Step 2.
To ensure the safety distance of any adjacent vehicle, the last
vehicle in the previous optimization time interval and the
first vehicle in the current optimization time interval should
satisfy the minimum safety head time constraint. If it is not
satisfied, the time sequence needs to be time-shifted, as shown
in Eqs. (32) and (33).
t+
ij =⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
t+
ij ,
if t+
i1≥Tp+ξ
t+
ij +(Tp+ξ−t+
i1),
if t+
i1<Tp+ξ,
∀i,p∈I,j∈Ni.(32)
ξ=τ, if i=p
ω, if i= p(33)
where t+
ij is the optimal timing without considering the rolling
optimization strategy; Tprepresents the time of the last vehicle
from direction ppassing through the conflict zone during the
previous optimized time interval; ξindicates the minimum safe
time headway. If the first vehicle within the next optimized
time interval belongs to the same direction as the last vehicle
within the previous optimized time interval, then ξ=τ;
otherwise, ξ=ω.
Based on the rolling optimization strategy, the algorithm
can find the optimal solution in 1 second on a general laptop
computer. Besides, with the rise of computational power and
edge computing, the efficiency of the algorithm would be
further improved. Therefore, the algorithm can be used in
practical applications.
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YAO et al.: INTEGRATED SCHEDULE AND TRAJECTORY OPTIMIZATION FOR CAVs IN A CONFLICT ZONE 7
TAB LE I I
SIMULATION PARAMETER SETTINGS
TABLE III
DIFFERENT TRAFFIC DEMANDS
V. N UMERICAL SIMULATION AND ANALYSIS
A. Simulation Parameter Settings
Referring to the literature [8], [33], we assume that vehicle
arrivals follow a Poisson process, the total simulation time
is 900 s, and the optimization interval is 10 s. Based on
the Dedicated Short-Range Communication (DSRC) tech-
nique [43], [44], the control area length is set to 300 m.
Take one-way unsignalized intersections in Fig. 1c) as an
example, the parameters are shown in Table II. In particular,
the simulation experiment is a numerical simulation. There-
fore, pre-warm time is not included in the duration of T,which
only contains optimization and data collection time. More-
over, for other conflict zones in Fig.1, different parameters
(e.g., desired speed, desired gap, location constraints) need to
be set according to the actual situation. Besides, we assume
that all CAVs can be controlled in the simulation experiment,
regardless of the actual vehicle uncontrolled or communication
failure and other abnormal factors.
To verify the performance of the proposed model and
algorithm, the proposed method is tested in different traf-
fic demands (Table III) and compared with the FIFO-based
method [41]. Specifically, different simulation traffic scenarios
are shown in Table IV.
All simulation scenarios are implemented by the MATLAB
platform in the computer with CPU i7-6500U and 8GB
the memory. Meanwhile, each set of simulation scenarios is
simulated 10 times with different random seeds to capture the
influence of randomness. Therefore, combined with the rolling
optimization strategy in Section IV, the detailed pseudocode
of the simulation experiment is presented in the Appendix.
Based on the simulation, the results are shown in Table IV
and Table V.
As seen from Table IV, the optimized vehicle delays in the
six scenarios are less than the delays obtained from imple-
menting the FIFO-based method. Moreover, the percentage of
delay reduction for all six scenarios is above 10%, and the
maximum vehicle delay can be reduced by 54.23%. Besides,
Fig. 3 shows the space-time trajectory of CAVs. There is no
intersection trajectory in Fig. 3, which verifies the correctness
of the optimization results of the proposed method. Moreover,
as shown in the A areas in Fig. 3, when the schedule of
vehicles is not optimized, vehicles from different conflict
directions cross the conflict zone (A areas in Fig. 3). This
means ωwill appear more frequently. However, the proposed
method can optimize the schedule of vehicles and avoid the
occurrence of ωand increase the occurrence of τ(B areas in
Fig. 3). Moreover, because ωis larger than τ, vehicle delay of
the proposed method is smaller than the FIFO-based method.
Table V presents the proposed method effectively reduces
fuel consumption compared with the FIFO-based method.
With the increase of flow rate, the reduction of fuel con-
sumption is more significant, and the fuel consumption can
reduce by up to 34.36%. The result illustrates that although
vehicle trajectories optimized by the same method, the fuel
consumption of the proposed method is smaller. In other
words, the vehicle trajectories in the space-time figure are
shorter, and the time interval in which the vehicle accelerates
and decelerates during driving is shorter. From the existing
research [8], [33], the fuel consumption of vehicles is related
to the magnitude and duration of the acceleration and decel-
eration of vehicles. Therefore, the proposed method not only
can significantly reduce the average delay of vehicles but also
effectively save fuel consumption of vehicles.
B. Sensitivity Analysis of Parameters
In the above simulation scenarios, the optimization time
interval is 10 s. The result shows that when the optimized
time interval is larger, the space for scheduling optimization
is larger; that is, the percentage of delay reduction is more
evident under the same simulation duration. Meanwhile, in this
study, the minimum safe time interval of adjacent vehicles in
the conflict and in the same direction, passing through the
conflict zone also play a vital role in the proposed model.
Therefore, this section would perform a sensitivity analysis of
these three parameters. Traffic scenario number 4 in Table III
is selected as the fixed scenario to avoid the impact of traffic
volume. Firstly, the optimized time interval is set as 8 to 20 s
with 2 s step. The vehicle delay is selected as the evaluation
index to analyze the influence of different optimized time
intervals on the performance of the proposed method. The
sensitivity analysis results are shown in Fig. 4.
Fig. 4 illustrates that with the increase of the rolling opti-
mization time interval, the average delay under the proposed
method generally shows a decreasing trend. The average
delay of vehicles under the FIFO method fluctuates around
90 s/veh. Theoretically, the delay of the FIFO method is not
affected by the optimization interval. This is because different
random numbers are used in the simulation experiment with
different optimization intervals, which leads to different delay
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8IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
TAB LE I V
COMPARISON OF VEHICLE DELAYS
TAB LE V
COMPARISON OF VEHICLE FUEL CONSUMPTION UNDER DIFFERENT METHODS AND TRAFFI C DEMANDS
Fig. 3. Space-time trajectory (Experiment No. 3).
fluctuations in different optimization intervals. At the same
time, compared with the FIFO-based method, the percentage
reduction of the average delay under the proposed method
increases with the increase of the rolling optimization time
interval. Therefore, this is consistent with theoretical analysis,
the larger the rolling optimization time interval, the larger the
optimization space is. However, the actual rolling optimization
time interval should be selected according to the length of
the predicted time interval. With the large-scale application
of CAVs, the time interval of vehicle arrival prediction would
continue to increase [42].
Secondly, the minimum safe time interval on the per-
formance of the proposed method is analyzed. Moreover,
the minimum safe time interval of the conflict direction is
1 to 2 s, and the minimum safe time interval in the same
Fig. 4. Sensitivity analysis of rolling optimization time interval.
direction is 0.5 to 1.5 s. The analysis results are shown in
Fig. 5.
Fig. 5a) shows that as the minimum safety interval of
vehicles decreases, the vehicle delay would also decrease.
According to the analysis, the minimum safety interval of
vehicles limits the time difference between the front and
rear vehicles passing through the conflicts zone, which leads to
vehicle delay. Therefore, the smaller the minimum safety inter-
val is, the smaller vehicle delays would be. If the minimum
safety time interval is zero, the vehicle will pass through the
conflict zone without delay. Moreover, the dark blue region on
both sides of the black straight line in Fig. 5b) shows that when
the values of τand ωare close, the average delay reduction
percentage is smaller than the FIFO-based method. Especially,
when τis equal to ω, the percentage of delay reduction is zero.
This result illustrates that when τis equal to ω, the average
delay of the vehicle is equal in both the proposed method and
FIFO-based method. From the above analysis, we could know
that when τ<ω, the basic principle of the proposed method
is to avoid the occurrence of ωand increase the occurrence
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YAO et al.: INTEGRATED SCHEDULE AND TRAJECTORY OPTIMIZATION FOR CAVs IN A CONFLICT ZONE 9
Fig. 5. Sensitivity analysis of vehicle minimum safety interval.
of τ. Thus, the vehicle delay will be reduced by the proposed
method. Nevertheless, if τ=ω, the proposed method cannot
find the gap of τand ω. In this case, the results of the proposed
method are consistent with the FIFO-based method; that is,
the average delay reduction percentage is zero.
The yellow area in Fig. 5b) shows that when the values
of τand ωdiffer greatly, the percentage of vehicle average
delay reduction under the proposed method is more significant
than that of the FIFO-based method. Therefore, this is also
consistent with the above analysis. The proposed method can
optimize the scheduling of vehicles and avoid the occurrence
of ωand increase the occurrence of τ, thereby achieving the
purpose of reducing vehicle delay. Besides, the value of ω
depends on the type and size of the conflict zone, while τis
the gap for car-following. Therefore, in practical application,
the reasonable value of ωshould be designed according to the
type and size of the conflict zone. In addition, the design of
smaller safe following gap τis conducive to reduce vehicle
delay.
VI. CONCLUSION AND FUTURE WORK
This study proposes a two-level model to optimize schedul-
ing and trajectories for CAVs in a conflict zone, in which the
first level is a MILP to optimize the timing of vehicles entering
the conflict zone, and the second level is a multi-vehicle
Pseudocode of the Simulation Experiment
Initialize:
1: Total simulation time T, time plan-
ning horizon T, current time t=
0,L,Vi,τ,ω,θ,v0
ij ,vf
ij,v
min,v
max,amin ,amax,s0,δ
in arm i,∀i∈I.
2: Generate the arrival times of CAVs at arm i,∀i∈I.
Iterate:
3: While t+T≤Tdo
4: Get the arrival times (t−
ij ,∀i∈I,j∈Ni)of CAVs
in time planning horizon [t,t+T].
5: Calculate t+
ij ,∀i∈I,j∈Nibased on Eq.(1).
6: Optimize the scheduling of CAVs entering the
conflict zone based on Eq.(12).
7: Obtain d
8: Obtain the scheduling of CAVs.
9: For i=1→2do
10: Obtain Ni,t−
ij ,t+
ij ,∀i∈I,j∈Ni.
11: Optimize the CAV trajectory based on
Eqs.(24-33).
12: Obtain Gi.
13: End
14: Calculate g.
15: Save vehicle trajectories, the scheduling of
CAVs,average vehicle’s delay, and gasoline
consumption at the current time planning
horizon [t,t+T].
16: t←− t+T
17: End
Output:
18: Output vehicle trajectories, the scheduling of CAVs,
average vehicle delay, and gasoline consumption at total
simulation time.
nonlinear optimal control model to optimize vehicle trajec-
tories. Considering the actual application requirements and
complexity of the proposed method, we introduce a rolling
optimization strategy. Results from a simulation experiment
indicate that:
(1) Compared with the FIFO-based method, the proposed
method reduces the vehicle delay and fuel consumption by
optimizing the schedule of vehicles in the collision direc-
tion through the conflict zone. Through simulation results
of different traffic scenarios, the vehicle delay can be
reduced by up to 54.23% and the fuel consumption by up
to 34.36%.
(2) The rolling optimization time interval has a significant
influence on the performance of the proposed method. More-
over, the percentage reduction of the average delay increases
with the increase of the rolling optimization time interval
compared with the FIFO-based method.
(3) As the minimum safety interval of the vehicle decreases,
vehicle delays would also decrease. When the minimum time
interval τ=ω, the proposed method cannot find the gap of τ
and ωto reduce vehicle delay.
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10 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
(4) τis a very important parameter of the conflict zones.
The reasonable value of τshould be designed according to
the type and size of the conflict zone.
This paper reveals the mechanism of improving the traffic
efficiency of the conflict zones by CAVs, and puts forward a
rolling optimization framework suitable for practical applica-
tion. However, this work still has some limitations:
(1) Only one conflict zone, a one-way unsignalized inter-
section, is discussed in case study. The setting and selection of
parameters for the other two conflict zones need to be further
discussed in combination with the actual cases.
(2) In this study, we did not consider the possibility that
vehicles cannot enter the approach because of queues. If the
traffic demand is very big, the queue length over the border
of the conflict zone. In this case, the demand is greater than
capacity. To avoid this situation, the smaller time gap between
two vehicles will be designed to improve the capacity of
conflict zone.
(3) The conflict zone is too simple and contains only
two conflict directions. Therefore, future work would include
further investigation of scheduling and trajectory collaborative
optimization methods for more complex conflict zones (such
as multi-turn intersections), and traffic signals and trajectory
optimization methods for mixed traffic flow of CAVs and
HDVs.
APPENDIX
See Pseudocode of the Simulation Experiment.
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Zhihong Yao received the B.S. and Ph.D. degrees
in transportation engineering from Southwest Jiao-
tong University, Chengdu, China, in 2014 and 2019,
respectively.
He also spent one year as a joint Doctoral Student
at the University of Wisconsin–Madison, Madison,
WI, USA. He is an Assistant Professor with the
School of Transportation and Logistics, Southwest
Jiaotong University. He has coauthored more than
40 articles that have been published in the I EEE
TRANSACTIONS ON INTELLIGENT TRANSPORTA-
TION SYSTEMS,theJournal of Intelligent Transportation Systems,IET Intelli-
gent Transport Systems,Physica A: Statistical Mechanics and Its Applications,
the Journal of Advanced Transportation,andTransportation Research Record.
His research interests include traffic signal control and connected automated
vehicles.
Haoran Jiang received the B.S. degree in trans-
portation engineering from Southwest Jiaotong Uni-
versity, Chengdu, China, where he is currently
pursuing the Ph.D. degree with the School of
Transportation and Logistics. His research interests
include intelligent transportation systems and con-
nected automated vehicles.
Yan g Ch en g (Member, IEEE) received the B.S.
degree in automation and the M.S. degree in intelli-
gent transportation systems from Tsinghua Univer-
sity, Beijing, China, in 2004 and 2006, respectively,
and the Ph.D. degree in transportation engineering
from the University of Wisconsin–Madison, Madi-
son, WI, USA, in 2010.
From 2010 to 2011, he was a Teaching Assistant
with the University of Wisconsin–Madison. From
2011 to 2013, he was a Research Associate with the
Traffic Operations and Safety Laboratory, University
of Wisconsin–Madison, where he has been an Assistant Researcher with the
Traffic Operations and Safety Laboratory since 2013. His research interests
include traffic modeling, traffic information, and intelligent transportation
systems.
Yangsheng Jiang received the B.S. degree in
mechanical engineering from Yanshan University,
Qinhuangdao, China, in 1998, and the Ph.D. degree
in transportation engineering from Southwest Jiao-
tong University, Chengdu, China, in 2004.
He has been working as an Assistant Professor,
an Associate Professor, and a Professor with the
School of Transportation and Logistics, Southwest
Jiaotong University, where he is currently a Pro-
fessor and the Deputy Director of the National
Engineering Laboratory of Integrated Transportation
Big Data Application Technology. His research interests include transportation
systems optimization and traffic big data.
Bin Ran received the B.S. degree in civil engi-
neering from Tsinghua University, Beijing, China,
in 1986, the M.S. degree in civil engineering from
the University of Tokyo, Tokyo, Japan, in 1989,
and the Ph.D. degree in civil engineering from the
University of Illinois, Chicago, IL, USA, in 1993.
From 1993 to 1994, he worked as a Post-Doctoral
Researcher at the University of California at Berkley.
From 1994 to 1995, he served as a Lecturer at the
Massachusetts Institute of Technology. Since 1995,
he has been working as an Assistant Professor,
an Associate Professor, and a Professor with the Department of Civil and
Environmental Engineering, University of Wisconsin–Madison, Madison, WI,
USA. He became a Guest Professor at Southeast University in 2009 and
has been a Professor since 2010. His research has focused on five major
areas such as intelligent transportation system (ITS) technology development
and system evaluation, dynamic transportation network and traffic modeling,
the development of mobile probe technologies for traffic state estimation and
passenger flow estimation, connected automated vehicle highway (CAVH),
vehicle-highway coordination, intelligent vehicles, automated highway sys-
tems, and big data applications for multimodal transportation databases.
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