Joint optimization for autonomous intersection management and trajectory smoothing design with connected automated vehicles

Abstract

Trajectory smoothing design (TSD) may significantly reduce fuel consumption and improve driving comfort at intersections. In this paper, a mixed integer linear programming (MILP) model with discrete time is formulated to jointly optimize autonomous intersection management and TSD, aiming to improve traffic efficiency, fuel economy and driving comfort simultaneously. Driving safety of car-following and collision avoidance at conflict points, diverge points and converge points, as well as constraints of acceleration and jerk are considered. To reasonably describe vehicle movement within intersection areas, the vehicle trajectory within the intersection is treated as a channel considering the vehicle width. A rolling horizon framework is used to solve the model. We have compared the traffic efficiency, fuel economy, monetary cost and driving comfort of the joint optimization model with that of the state-of-the-art two-stage strategy. Finally, sensitivity analysis with respect to left-turn ratio, weighted coefficient of TSD and control zone length is conducted.

Full text

Full Terms & Conditions of access and use can be found at
https://www.tandfonline.com/action/journalInformation?journalCode=ttrb20
Transportmetrica B: Transport Dynamics
ISSN: (Print) (Online) Journal homepage: https://www.tandfonline.com/loi/ttrb20
Joint optimization for autonomous intersection
management and trajectory smoothing design
with connected automated vehicles
Guohong Wu & Rui Jiang
To cite this article: Guohong Wu & Rui Jiang (2023) Joint optimization for autonomous
intersection management and trajectory smoothing design with connected
automated vehicles, Transportmetrica B: Transport Dynamics, 11:1, 1234-1255, DOI:
10.1080/21680566.2023.2193314
To link to this article: https://doi.org/10.1080/21680566.2023.2193314
Published online: 30 Mar 2023.
Submit your article to this journal
Article views: 113
View related articles
View Crossmark data

TRANSPORTMETRICA B: TRANSPORT DYNAMICS
2023, VOL. 11, NO. 1, 1234–1255
https://doi.org/10.1080/21680566.2023.2193314
Joint optimization for autonomous intersection management and
trajectory smoothing design with connected automated vehicles
Guohong Wu and Rui Jiang
Key Laboratory of Transport Industry of Big Data Application Technologies for Comprehensive Transport, Ministry of
Transport, Beijing Jiaotong University, Beijing, People’s Republic of China
ABSTRACT
Trajectory smoothing design (TSD) may significantly reduce fuel consump-
tion and improve driving comfort at intersections. In this paper, a mixed
integer linear programming (MILP) model with discrete time is formulated
to jointly optimize autonomous intersection management and TSD, aiming
to improve traffic efficiency, fuel economy and driving comfort simulta-
neously. Driving safety of car-following and collision avoidance at conflict
points, diverge points and converge points, as well as constraints of accel-
eration and jerk are considered. To reasonably describe vehicle movement
within intersection areas, the vehicle trajectory within the intersection is
treated as a channel considering the vehicle width. A rolling horizon frame-
work is used to solve the model. We have compared the traffic efficiency,
fuel economy, monetary cost and driving comfort of the joint optimization
model with that of the state-of-the-art two-stage strategy. Finally, sensitiv-
ity analysis with respect to left-turn ratio, weighted coefficient of TSD and
control zone length is conducted.
ARTICLE HISTORY
Received 8 August 2022
Accepted 15 March 2023
KEYWORDS
Connected automated
vehicles (CAVs); autonomous
intersection management;
trajectory smoothing design;
mixed integer linear
programming; collision
avoidance
1. Introduction
Intersections are common and frequent bottlenecks for traffic flows in urban road networks. It is
reported that there are about 295 million vehicle-hours of delay on major roadways in the US, which
is mainly caused by intersections (Denney, Curtis, and Olson 2012). Meanwhile, according to statistics,
greenhouse gas emissions from transportation accounted for about 28.2 percent of total US green-
house gas emissions in 2018, making it the largest contributor (U.S.E.P.A. 2018). Traffic control and
trajectory planning have attracted continuous interests, which can improve the transportation sys-
tem efficiency and reduce fuel consumption and emissions. The emergence of Connected Automated
Vehicles (CAV) technologies provides promising solutions for traffic control and trajectory smoothing
design (TSD).
With the rapid development of V2V, V2I technologies, intersection controller can obtain real-time
vehicle trajectory information. Some studies optimize the signal phase and timing (SPaT) according
to real-time data and explore the benefits of using vehicle trajectory data for traffic signal control
(Sun, Zheng, and Liu 2018; Liang, Guler, and Gayah 2018). In addition, available SPaT information,
controllability of CAVs and connected vehicles can be utilized to make trajectory planning for vehi-
cles approaching an intersection. To achieve trajectory smoothing and speed harmonization, different
methods are applied in existing studies, e.g. model predictive control (Zhao et al. 2018), optimal
control theory (Jiang et al. 2017; Wan, Vahidi, and Luckow 2016; Lin et al. 2021), shooting heuristic
CONTACT Rui Jiang jiangrui@bjtu.edu.cn Key Laboratory of Transport Industry of Big Data Application Technologies
for Comprehensive Transport, Ministry of Transport, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
© 2023 Hong Kong Society for Transportation Studies Limited

TRANSPORTMETRICA B: TRANSPORT DYNAMICS 1235
approach (Ma et al. 2017), dynamic programming (Wei et al. 2017), trigonometric speed profile (Altan
et al. 2017), and reinforcement learning (Li et al. 2021). Recent studies jointly optimize traffic signal
control and vehicle trajectory in a two-stage way under the 100% CAV scenario (Feng, Yu, and Liu
2018;Yuetal.2018;Guoetal.2019). The two-stage optimization model optimizes signal optimization
in the first stage and generates trajectories in the second stage.
Assuming fully CAV environment, autonomous intersection management (AIM) has been proposed
(Dresner and Stone 2004). In this approach, every vehicle and intersection controller are smart agents
and the internal intersection area is divided into identical cells. In the reservation-based method, each
cell cannot be occupied by two or more vehicles at the same time. The CAV sends a request to enter
intersection and reserves the cells it will pass through. The intersection controller will accept or reject
this request according to First-Come-First-Serve (FCFS) strategy in the intersection. Dresner and Stone
(2006) further investigated AIM with emergency vehicles. In addition, auction strategy (Carlino, Boyles,
and Stone 2013), platoon strategy (Tachet et al. 2016; Bashiri and Fleming 2017; Timmerman and
Boon 2021) and virtual platoon-based strategy (Medina, van de Wouw, and Nijmeijer 2018) have been
applied in AIM to gain more benefits. The above strategies are rule-based strategies, in which system
optimality is not guaranteed.
To address the above gaps, some optimization-based studies coordinated the trajectories of vehi-
cles approaching the intersection in AIM. Based on schedule trees, Li and Wang (2006)proposeda
safety driving pattern for unsignalized intersection. Kamal et al. (2015) used a model predictive control
method with a risk function to quantify the risk of collision. However, the trajectories within intersec-
tion are simplified into two straight lines. Moreover, the solution to the optimization problem is not
guaranteed to be the global optimum due to nonlinear modelling. Based on space–time slot, Chai
et al. (2018) applied an Acceleration Dynamically Adjusting based on Predicted Trajectory (ADAPT)
method to guide the vehicles. Zhao, Liu, and Ngoduy (2021) investigated the AIM problem with vehi-
cle trajectory planning at a simplified two-way intersection. They formulated the problem as a bi-level
optimization problem in which the upper level is designed to minimize the total travel time by a MILP
model and the lower level is a linear programming (LP) model with an objective function to maximize
the speeds entering the intersection. Fayazi and Vahidi (2018) proposed a MILP model to find the opti-
mal sequence then CAV guides its speed profile individually. Yu et al. (2019) proposed a MILP model to
cooperatively optimize the trajectories of CAVs along a corridor for system optimality in terms of total
vehicle delay. The car-following and lane-changing behaviour of each vehicle along the entire path
are optimized together. Nevertheless, vehicle dynamics are captured by the first-order model and the
optimal trajectories are over-simplified trajectories which allow sudden speed jumps and infinite accel-
erations. Wu et al. (2022) invented the ‘automated pedestrian shuttle’ (APS) to transport pedestrians
at internal autonomous intersection areas. Chen et al. (2021) proposed an innovative intersection con-
trol scheme, rhythmic control, in a fully CAV environment. Utilizing an appropriately designed layout
of conflicting points at an intersection, the RC assigns predetermined time spots in a rhythmic way
for vehicles entering an intersection from each lane. In addition, distributed method (Xu et al. 2018;
Liu et al. 2018) and reinforcement learning method (Wu, Chen, and Zhu 2019) have been applied into
vehicle trajectory planning and AIM problem with delay minimization.
In the above-mentioned studies, fuel consumption and driving comfort were not considered in
trajectory planning. Recently some studies have addressed this issue. Based on optimal control the-
ory, Malikopoulos, Cassandras, and Zhang (2018) developed the decentralized energy-optimal control
framework to improve the fuel economy performance after the service sequence of the vehicles is
determined. However, a constant safety distance is used in car-following constraints and turning vehi-
cles are not considered. Malikopoulos et al. further addressed the above limitations in Malikopoulos,
Beaver, and Chremos (2021), considering speed-dependent rear-end safety constraint. Mirheli et al.
(2019) proposed a distributed method to minimize each CAV’s travel time and speed variation. Xu
et al. (2020) applied Monte Carlo tree search (MCTS) to find a nearly global-optimal passing order. After
determining passing order, they treated trajectory planning problem as optimal control problem to
achieve speed harmonization. Moreover, considering lane-changing behaviour, Xu et al. (2022) further

1236 G. WU AND R. JIANG
employed a method combining optimal control and control barrier functions in a decentralized frame-
work. Jiang et al. (2022) proposed a classic two-stage method to coordinate mixed platoons passing
through signal-free intersection. In this study, a MILP model is used to determine service sequences
at the first stage then optimal control theory is applied to optimize CAVs’ trajectories at the second
stage. Recently, some studies applied joint optimization model to improve traffic signal control sys-
tem performance in terms of traffic efficiency and fuel economy (Niroumand et al. 2020; Liu et al. 2021;
Liu et al. 2022).
To our best knowledge, most studies decompose AIM-coupled vehicle trajectory optimization into
two problems: traffic efficiency and trajectory planning. In other words, the studies optimizing the
traffic efficiency and the vehicle trajectories of AIM system simultaneously are lacking. To address the
research gap, this paper proposed a discrete-time based MILP model to jointly optimize the AIM and
TSD on the approach lanes, aiming to improve traffic efficiency, fuel economy and driving comfort
simultaneously. An isolated 4-arm intersection is investigated. Collision avoidance at diverge points,
converge points and conflict points are considered. To more reasonably consider the safety constraint
within internal intersection areas, this study takes the vehicle width into account and assumes that the
vehicles move along a channel. The rolling horizon framework has been used to solve the proposed
MILP model. Performance of the joint optimization strategy has been compared with that of two-stage
strategy.
The remainder of this paper is organized as follows. Section 2 briefly introduces the integrated AIM
and TSD problem and basic assumptions in this study. Section 3 presents the joint optimization model
of integrated AIM and TSD problem in detail. Section 4 provides the rolling horizon framework to solve
the proposed model. Section 5 presents the details of simulation implementations, the performance
of the proposed model and the comparison with two-stage strategy. Section 6 concludes the paper
and lays out the direction of future work.
2. Problem statement
This section states the integrated AIM and TSD problem. We consider the typical intersection shown
in Figure 1, which has the same geometry as in (Kamal et al. 2015). The intersection has four arms. The
east and west arms have single lane, which allows vehicles to go through, turn left, and turn right. The
north and south arms have two lanes. The left lane is exclusive for left-turn vehicles and the right lane
is for go-through and right-turn vehicles.
As shown in Figure 2, taking arm 4 as an example, we regard the road segment with length L from
the stop bar to the upstream as ZONE 1 and the internal intersection area is ZONE 2. The control range
consists of ZONE 1 and ZONE 2. We define li(t)as the position of vehicle iat time t. The coordinate
represents the distance to the entrance of ZONE 1. Namely, coordinate of entrance to ZONE 1 is 0,
while the coordinate of stop bar is L. If vehicle iis upstream (downstream) of the entrance to ZONE 1 at
time t,li(t)<0(li(t)>0).Moreover,t0
idenotes the entry time of vehicle iinto ZONE 1, which is equal
to the arrival time of the front bumper of the vehicle iat the entrance of ZONE 1. The corresponding
speed is v0
i.tn
irepresents the arrival time of the front bumper of vehicle iat stop bar. The relationship
between vehicle position, time and the zone where the vehicle is located are as follows: If vehicle iis
in ZONE 1 at time t,0≤li(t)≤L,t0
i≤t≤tn
i; If vehicle iis in ZONE 2 at time t,li(t)>L,t>tn
i.
Given an CAV generation period [0, tg], the vehicles generated from the upstream of ZONE 1. The
CAVs in ZONE 1 and ZONE 2 are target vehicles. Considering driving safety (car-following constraint
in ZONE 1, collision avoidance at divergence points, converge points, and conflict points in ZONE 2)
and driving comfort (acceleration and jerk in ZONE 1), this paper jointly optimizes the AIM and the
TSD, aiming to improve benefits of traffic efficiency and fuel economy simultaneously. Some basic
assumptions are as follows:
(1) All vehicles on the road are CAVs with identical size;

TRANSPORTMETRICA B: TRANSPORT DYNAMICS 1237
Figure 1. Isolated intersection with 4 arms in this paper.
Figure 2. Lay out of study area.
(2) All CAVs are controlled by the intersection centralized controller, which can communicate with the
CAVs in real time. The information reception is good, and no communication delay or package loss
is considered in this paper;
(3) Lane allocation is shown in Figure 1. Lane-change behaviour is forbidden within the control range
(ZONE 1, ZONE 2);
(4) We assume that all vehicles’ entry boundary conditions (i.e. t0
irepresents entry time into ZONE 1
and l0
i,v0
iare the position and speed at entry time step into ZONE 1, respectively) can be exactly

1238 G. WU AND R. JIANG
predicted (e.g. with the advanced sensing and tracking technology in the future transportation
infrastructure);
(5) All CAVs pass through ZONE 2 at a constant speed. The constant speed vc
iis determined by
movement direction of the vehicle (directioni)within ZONE 2.
3. Model formulation
This section presents the proposed MILP model to optimize integrated AIM and TSD problem. Firstly,
the discrete-time trajectory planning problem is introduced. Then, the objective function and con-
straints of MILP model as well as trajectory design in ZONE 2 are explained in detail.
3.1. Discrete-time vehicle dynamics
Continuous-time vehicle dynamics result in nonlinear optimization problems with many decision vari-
ables, which greatly increases model complexity and implementation difficulty. To simplify the vehicle
dynamic, time discretization is widely used in trajectory planning. As pointed out in Li and Li (2019),
the modelling errors of discrete-time vehicle dynamics would not contribute too much for the loss of
objective function values, because the planning horizon is relatively long.
The discrete-time trajectory planning problem is considered as follows. Time horizon [mtr,mtr+tp]
is divided into Tpperiods with equal intervals. The interval is defined as the time step length t=tp
Tp.
Here tris rolling horizon, m is the number of rolling horizon, tpis planning horizon.
Based on the discrete-time model, the acceleration rate is updated every t. Note that the larger
the time step length is, the more parsimonious the trajectory is. If tis small enough, the discrete-time
vehicle trajectory can be approximated as the continuous vehicle trajectory. Assuming entry boundary
conditions (i.e. t0
i,l0
i,v0
i) are accurately predicted by the advanced technologies, we define j0
i=t0
i
tas
entry time step for vehicle i.Here is the ceiling function, and j0
iindicates the smallest integer value
which is larger than or equal to t0
i
t. For the sake of convenience, the set of time step indices for vehicle
iis defined as Ji={j0
i,j0
i+1, ...,Tp}.
3.2. AIM and TSD joint optimization model
3.2.1. Objective function
Urban transportation system is expected to achieve high performance in traffic efficiency and tra-
jectory smoothness. These two objectives may be in conflict. We can construct a multi-objective
optimization problem and solve the Pareto frontier. However, solving the multi-objective optimization
problem usually takes much longer time. Therefore, for real time application, we use the proposed lin-
ear weighted-sum model, which is widely used in many studies (Li and Li 2019; Yao et al. 2018;Maetal.
2017;Xuetal.2022). In order to make a balance between traffic efficiency and trajectory smoothness,
the objective function of our optimization model is to minimize total cost, which is defined as
min
tn
i,ai,j
Z=
i∈I
(tn
i−t0
i)+w
i∈I
j∈J1
i
|ai,j|(1)
Here |ai,j|is the absolute value of acceleration rate, J1
i={j,t0
i≤jt≤tn
i}indicates the set of time
step indices that vehicle iis in ZONE 1. In the objective function, i∈I(tn
i−t0
i)denotes AIM cost, i.e. the
total travel time within ZONE 1.i∈Ij∈J1
i|ai,j|denotes the total trajectory smoothing cost in ZONE 1
(Li and Li 2019;Berkovitz1974; Feng, Yu, and Liu 2018), aiming to eco-and comfort driving. Note that
explicit fuel consumption function is highly nonlinear, which would make joint optimization model
intractable if it was considered in the objective function. wis TSD cost weighted coefficient. By varying
w, we can make a trade-off between the AIM cost and TSD cost.

TRANSPORTMETRICA B: TRANSPORT DYNAMICS 1239
3.2.2. Vehicle dynamics
Constraints (2) and (3) are used to update the vehicle speeds and positions. Here li,jindicates the
position of vehicle iat time step j(at time jt)vi,jand ai,jare the corresponding vehicle speed and
acceleration rate. (4) and (5) are speed and acceleration bound constraints, where vmax,amax and amin
are the maximum speed, maximum acceleration rate and minimum acceleration rate, respectively.
Moreover, Jidenotes the time step set ranging from t0
ito the planning horizon tp.
vi,j+1=vi,j+ai,jt,∀i∈I,j∈{j0
i,...,Tp−1}(2)
li,j+1=li,j+(vi,j+vi,j+1)
2t,∀i∈I,j∈{j0
i,...,Tp−1}(3)
0≤vi,j≤vmax,∀i∈I,j∈Ji(4)
amin ≤ai,j≤amax,∀i∈I,j∈Ji(5)
Constraint (6) ensures that jerk of the vehicle in ZONE 1 is bounded by its maximum jerk and min-
imum jerk. Although jerk is not considered in objective function, jerk bound constraint also improves
driving comfort.
jerkmin −M(wi,j+wi,j+1)≤(ai,j+1−ai,j)
t≤jerkmax +M(wi,j+wi,j+1),∀i∈I,j∈Ji(6)
3.2.3. Entry boundary conditions
Vehicle ihas entered ZONE 1 at time step j0
iand has not entered ZONE 1 at time step j0
i−1. The speed
and position of the vehicle iat time step j0
iare formulated as
vi,j0
i=v0
i,∀i∈I(7)
li,j0
i=l0
i,∀i∈I(8)
3.2.4. Zone identification
The following constraints determine whether vehicle ihas entered ZONE 2 or not.
(j·t−tn
i)−ε≥−M(1−ωi,j),∀i∈I,j∈Ji(9)
(j·t−tn
i)≤Mωi,j,∀i∈I,j∈Ji(10)
Here ωi,jis an auxiliary binary variable. Mis a sufficiently large positive number, and εis a sufficiently
small positive number. If vehicle ihas entered ZONE 2 at time step j, one has j·t−tn
i>0. Thus, to
ensure the two constraints are satisfied, ωi,jshould equal to 1. Similarly, if vehicle ihas not entered
ZONE 2 at time step j, one has j·t−tn
i≤0. Thus, to ensure the two constraints are satisfied, ωi,j
should equal to 0. Therefore, auxiliary binary variable ωi,jare used for vehicle’s zone identification.
−M(1−ωi,j+1+ωi,j)≤vi,j−vc
i≤M(1−ωi,j+1+ωi,j),
∀i∈I,j∈{j0
i,...,Tp−1}(11)
−M(1−ωi,j+1+ωi,j)≤L−li,j−vc
i(tn
i−j·t)≤M(1−ωi,j+1+ωi,j),
∀i∈I,j∈{j0
i,...,Tp−1}(12)
Constraints (11) and (12) are used to determine the speed in ZONE 2 and the location when the vehi-
cle enters ZONE 2. For ωi,jand ωi,j+1, there are three cases: (i) ωi,j=0, ωi,j+1=0; (ii) ωi,j=1, ωi,j+1=1;
(iii) ωi,j=0, ωi,j+1=1. In cases (i) and (ii), one can easily see that constraints (11) and (12) are always

1240 G. WU AND R. JIANG
met. In case (iii), constraint (11) and (12) become
0≤vi,j−vc
i≤0, ∀i∈I,j∈{j0
i,...,Tp−1}(11a)
0≤L−li,j−vc
i(tn
i−j·t)≤0, ∀i∈I,j∈{j0
i,...,Tp−1}(12a)
which yield vi,j=vc
i,andli,j=L−vc
i(tn
i−j·t). This means that when the vehicle enters ZONE 2, the
speed equals vc
iand the location is L−vc
i(tn
i−j·t). Note that the location is not equal to Lbecause
the time is discrete and it tends to Lwhen ttends to 0.
Constraints (13) and (14) guarantee that the vehicle acceleration rate in ZONE 2 is 0 and the vehicle
speed is vc
i, which is determined by the movement direction of the vehicle within ZONE 2.
−M(1−ωi,j)≤ai,j≤M(1−ωi,j),∀i∈I,j∈Ji(13)
−M(1−ωi,j)≤vi,j−vc
i≤M(1−ωi,j),∀i∈I,j∈Ji(14)
One can easily see that when ωi,j=0, the two constraints are always met. When ωi,j=1, constraints
(13) and (14) become
0≤ai,j≤0, ∀i∈I,j∈Ji(13a)
0≤vi,j−vc
i≤0, ∀i∈I,j∈Ji(14a)
which yield ai,j=0andvi,j=vc
i.
3.2.5. Arrival time constraints
Given entry boundary conditions, tmin
i, the minimum travel time of vehicle iin ZONE 1, considering
restriction of maximum acceleration and maximum speed, is formulated as
tmin
i=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
L−(vmax)2−(v0
i)2
2amax 
vmax
+vmax −v0
i
amax
,L≥(vmax)2−(v0
i)2
2amax
2amaxL−(v0
i)2−v0
i
amax
, otherwise
(15)
Obviously, the difference between tn
iand t0
ishould not be smaller than minimum travel time tmin
i.
Thus we have constraint (16) to narrow the feasible region.
tn
i≥t0
i+tmin
i,∀i∈I(16)
On the other hand, the movement of vehicle is also constrained by that of its preceding vehicle i∗.
Thus, one has the constraint (17) as follow.
tn
i−tn
i∗≥τ+d
vc
i∗
,∀i∈I\Ilead (17)
We also discuss how to determine the planning horizon tp. Firstly, we set an upper limit of tp.Asin
Zhao, Liu, and Ngoduy (2021), we select 30 s as the maximum delay for each CAV. For simplicity, we let
CAVs in the same rolling horizon share identical planning horizon. Thus, the planning horizon can be
determined by tp=max t0
i+L
vmax +30
ΔtΔt,i∈I.Here is ceiling function. Given planning horizon
tp, constraint (18) guarantees that all target vehicles have left ZONE 1 and have entered ZONE 2 at tp
(Yu et al. 2019).
tp>tn
i,∀i∈I(18)

TRANSPORTMETRICA B: TRANSPORT DYNAMICS 1241
Figure 3. Potential collisions within ZONE2. (a)conflict points (b) converge points (c) diverge points
3.2.6. Safety constraints
For vehicles in ZONE 1 (i.e. ωi,j=0), it is necessary to maintain a safe space headway from its preceding
vehicle in the same approach lane. The rear-end constraint is as follows.
li∗,j−d−li,j−τvi,j≥−Mωi,j,∀i∈I\Ilead,j∈Ji(19)
Here τindicates CAVs’ reaction time, dindicates minimum space headway between two vehicles,
i∗denotes preceding vehicle and Ilead is the set of leading vehicles of each lane.
Besides rear-end constraint, collision avoidances within intersection are also considered. Most stud-
ies only consider the conflict points in ZONE 2 (see Figure 3(a)) but ignore the diverge points and
converge points as shown in Figure 3(b,c). While it is easy to understand that there is a collision risk
of vehicles from different approach lanes at the converge points, also note that for a leading vehicle
and a following vehicle with different directions in the same approach lane, the two vehicles may also
collide at the diverge point when the leading vehicle’ s speed is smaller than that of the following
vehicle.
Based on reservation-based method, ZONE 2 is divided into cells with size c×c(as shown in
Figure 4). Each cell can be occupied by no more than one car at any time. Constraints (20) and (21)
can guarantee that vehicle i1and vehicle i2pass through cell λsafely and sequentially.
tn
i1+xλ+
i1
vc
i1
+γ−tn
i2−xλ−
i2
vc
i2
≤Mμi1,i2
λ,
∀i1,i2∈{i1∩i2=λ}∩{lanei1= lanei2∪directioni1= directioni2}(20)
tn
i2+xλ+
i2
vc
i2
+γ−tn
i1−xλ−
i1
vc
i1
≤M(1−μi1,i2
λ),
∀i1,i2∈{i1∩i2=λ}∩{lanei1= lanei2∪directioni1= directioni2}(21)
Here xλ+
iindicates the travel distance of vehicle ifrom stop bar when it leaves cell λ,xλ−
iindi-
cates the travel distance of vehicle ifrom stop bar when it arrives cell λ,andγindicates the safety time
gap. Given a constant speed continuous trajectory design, xλ+
iand xλ−
icanbeeasilycalculated
and our model is still linear. i1indicates the set of cells passed by vehicle i1.μi1,i2
λis an auxiliary vari-
able used to indicate the sequence of vehicles passing through the cell λ. If vehicle i1pass through cell
λbefore vehicle i2,μi1,i2
λ=0, otherwise μi1,i2
λ=1. Note that if vehicle i1and i2areinthesameapproach
lane and have the same movement direction within ZONE 2. Constraint (19) ensures the safety of the
two vehicles because they move with the same constant speed within ZONE 2.
For the vehicle trajectory design within ZONE 2, vehicle width is considered so that the safety
constraint is more realistic. The vehicles are supposed to move along a channel, see Figure 4.The
vehicles’movement direction is always perpendicular to the cross-section of the channel.

1242 G. WU AND R. JIANG
Figure 4. Illustration of channel trajectory.
The trajectory centre line and the boundary of the channel of the go-through vehicle are straight
lines. The trajectory centre line and the boundaries of the channel of the left/right-turn vehicle are
assumed as elliptical/circle arcs. In constraints (20) and (21), xλ+
iand xλ−
iof go-through vehicles
can be calculated easily. Take a left-turn vehicle on arm 1 as an example, we explain how to calculate
xλ+
iand xλ−
iof turning vehicles in the Appendix.
3.2.7. Linearization
The sum of absolute value of the acceleration rates of the vehicle at each time step is used to evalu-
ate the trajectory smoothness in our model. Note that because of |ai,j|, our model is a Mixed Integer
Nonlinear Programming (MINLP), which is difficult to solve. We linearize the problem via constraints
(22)–(23).
si,j≥−ai,j,∀i∈I,j∈Ji(22)
si,j≥ai,j,∀i∈I,j∈Ji(23)
Constraints (22) and (23) show that |ai,j|is lower bound of si,j. Together with the objective function
(1), one can derive that si,jequals to |ai,j|
Using auxiliary variable si,j, we reformulated objective (1) as (24) and our model is treated as MILP.
min
tn
i,ai,j
Z=
i∈I
(tn
i−t0
i)+w
i∈I
j∈J1
i
si,j(24)
s.t. constraints (2)–(23).

TRANSPORTMETRICA B: TRANSPORT DYNAMICS 1243
4. Solution procedure
We investigate a continuous traffic flow process whose duration is defined as CAV generation period
tg. During the process, CAVs enter ZONE1 consecutively. The "curse of dimensionality "is inevitable if
we treat all CAVs entering ZONE1 as target vehicles. Therefore, we use a rolling horizon framework
to balance optimality and computational burdens (Levin and Rey 2017). The CAV generation period
tgis divided into nidentical time windows whose duration is equal to rolling horizon tr. We firstly
select the upcoming vehicles during the 1st time window as the target vehicles and optimize their
arrival times at stop bar and trajectories. Then we repeat above procedure for following time windows
sequentially. We set 1 s as the solving time limit for each time window, considering the potential of
real-time application. The solution procedure for proposed MILP model is as follows:
Step1: Initialization, time step length t, CAV generation period tg, rolling horizon tr, planning
horizon tp. Time that first CAV entering ZONE 1 is selected as the starting optimization time, which
is indicated by t0
opt. Then we set number of rolling horizon p=1.
Step2: If p>1, the solution information of previous time windows is stored by intersection con-
troller for the pth rolling horizon.
Step3: CAVs entering ZONE 1 during [t0
opt +(p−1)tr,t0
opt +ptr] are treated as target vehicles. The
intersection controller is expected to accurately predict the target vehicles’ entrance boundary con-
ditions (i.e. the entry times, positions and speeds of CAVs into ZONE 1: {t0
i,i∈I},{l0
i,i∈I},{v0
i,i∈I})
based on the control algorithm of CAVs (i.e. the Intelligent driver model (Zhao, Liu, and Ngoduy 2021),
) . Once the trajectory of a CAV is optimized, intersection controller does not re-optimize its trajectory
in the following time windows. The optimized vehicle trajectories in the previous time windows are
considered in the constraints to avoid collision within ZONE 1 and ZONE 2. The MILP model is solved
in the pth time window.
Step4: If all CAVs obtain their optimal trajectory. end the procedure. Otherwise, update p=p+1,
then go to Step 2.
5. Numerical results
5.1. Simulation settings
We investigate a 160 s continuous traffic flow process in our study. The simulation setup is as follows.
The lane width is 3.5 m. The width of vehicle trajectory channel is set to 2 m. The curb radiuses are
Dx=Dy=10.5 m. The median strip width is Dc=3.5 m. The size length of each cell is c=3.5 m. The
internal intersection area is divided into 88 identical cells. The length of ZONE 1 is L=200 m. Consid-
ering the turning radius, the desired constant speeds of go-through, left-turn and right-turn vehicles
within ZONE 2 are 10, 8 and 7.5 m/s, respectively.
Traffic demand is one of the most important factors affecting performance of the control strategy.
We vary traffic demand from 1920 veh/h to 3840 veh/h with an increment of 640 veh/h. Traffic flow
composition is shown in Table 1. The through, left-turn, right turn ratio of arm 1 and arm3 are 60%,
30%, 10%, respectively, while those are 80%, 10%, 10% for arm 2 and arm 4.
In the AIM problem, two-stage strategy is widely used, in which AIM cost is optimized firstly
by setting w=0 in the objective function (24). Then acceleration profile of each CAV is optimized
Tab le 1 . Traffic composition at 3200 veh/h.
Demand To
From Arm 1 Arm 2 Arm 3 Arm 4
Arm 1 – 96 576 288
Arm 2 64 – 64 512
Arm 3 576 288 – 96
Arm 4 64 512 64 –

1244 G. WU AND R. JIANG
Tab le 2 . Parameters used in simulations.
Parameter Description Value Unit
tTime step length in optimization 1 s
tgCAV generation period 160 s
trRolling horizon 10 s
tpPlanning horizon 54 s
wWeighted coefficient of TSD cost 0.8
γSafety time gap in ZONE 2 0.5 s
τCAVs’ reaction time 0.5 s
dMinimum space headway 6 m
vmax Maximum speed 15 m/s
sjam Jam gap 2 m
amax Maximum acceleration 3 m/s2
amin Minimum acceleration −3m/s2
jerkmax Maximum jerk 2 m/s3
jerkmin Minimum jerk −2m/s3
tsim Trajectory generation step length 0.1 s
individually to minimize its TSD cost. This strategy is selected as benchmark case for comparison
in numerical simulations. For a fair comparison, rolling horizon framework is also applied in this
strategy.
CAVs update acceleration rates with interval Δt=1s. The CAV generation period and rolling hori-
zon are tg=160 s, tr=10 s, respectively. We define tdas maximum allowed delay within ZONE 1, and
it is set to 30 s. Therefore, tp=tr+L
vmax +td
Δt×Δt=54 s, which is long enough to satisfy constraint
(18). The weighted coefficient of trajectory smoothing cost is set to w=0.8. The parameters used in
simulations are summarized in Table 2.
The proposed MILP optimization algorithm is written in Yalmip. The algorithm is solved by Cplex
12.8 through MATLAB COM (Component Object Model, COM) interface. All the experiments are per-
formed in a desktop computer with an Intel i7-3.20 GHz CPU and 16 GB memory. An upper limit of
1 s is set for real-time application in this study. A sub-optimal solution produced by the solver will be
accepted if the solving time exceeds the time limit. Considering the stochastic vehicle arrivals, the
simulation will be run three times in each demand scenario.
5.2. Model performance
This section compares performance of the two control strategies in terms of traffic efficiency, fuel
economy, monetary cost and driving comfort. Note that all measures are summarized as average
values.
5.2.1. Traffic efficiency
Travel time (TT) has been defined in section 3.2 as (tn
i−t0
i), which is widely used to measure traffic effi-
ciency and mobility. As shown in Table 3, the TT in the two-stage strategy is smaller than that in the joint
optimization strategy, because two-stage strategy prioritizes traffic efficiency over fuel economy. With
the growth of traffic demand, the TT in both strategies increases, but the increment ratio decreases.
The increment ratio is 23.78% at the low demand 1920 veh/h. In contrast, at the high demand 3840
veh/h, the increment ratio decreases to 14.66%.

TRANSPORTMETRICA B: TRANSPORT DYNAMICS 1245
Tab le 3 . Performance of the two strategies under different traffic demand.
Demand Two-stage Joint optimization
(veh/h) TT (s) FC (mL)
MC in China
(10−2)
MC in USA
(10−2$) TT (s) FC (mL)
MC in China
(10−2)
MC in USA
(10−2$)
1920 15.02 29.88 34.75 10.96 18.59 20.29 30.16 12.34
(23.78%) (−32.10%) (−13.20%) (12.66%)
2560 15.42 28.73 34.18 11.11 18.69 20.67 30.53 12.43
(21.18%) (−28.04%) (−10.66%) (11.89%)
3200 16.00 27.36 33.58 11.35 18.87 21.15 31.04 12.57
(17.91%) (−22.71%) (−7.56%) (10.76%)
3840 17.88 25.18 33.37 12.29 20.50 21.56 32.63 13.55
(14.66%) (−14.39%) (−2.21%) (10.32%)
Note: Bracket indicates the change ratio of proposed strategy compared with benchmark strategy.
5.2.2. Fuel economy
The VT-Micro model is used to calculate fuel consumption (FC), in which instantaneous FC is a
polynomial function of the speed and acceleration:
e(vi(t),ai(t)) =exp 3

m=0
3

n=0
Km,n(vi(t))m(ai(t))n(25)
where coefficient Km,ndepends on the sign of ai(t), the vehicle type and the measure-of-effectiveness
(MOE) type. The values of coefficient Km,nare referred to Refs. (Ahn et al. 2002) and (Guo et al. 2019).
As shown in Table 3, FC in the joint optimization strategy is always smaller than that in the two-
stage strategy. The difference is as high as 32.10% at the low demand 1920 veh/h. To explain the
difference in fuel economy between the two control strategies, Figure 5presents typical speed profiles
and acceleration profiles within ZONE 1 under the two strategies. Figure 5(a) shows that vehicle under
the two-stage strategy first accelerates to the maximum speed within ZONE 1 and then decelerates to
the target constant speed vcto pass through ZONE 2. In contrast, vehicle does not accelerate to the
maximum speed under the joint optimization strategy. Figure 5(b) shows that vehicle under the two-
stage strategy have aggressive and extreme accelerations and deceleration rates, compared with that
of joint optimization strategy. With the increment of demand, FC decreases in the two-stage strategy,
but increases in the joint optimization strategy. This is because FC is related to both travel time and
trajectory smoothness. In the two-stage strategy, the positive contribution of trajectory smoothness
to fuel economy exceeds the negative contribution of travel time growth. Therefore, FC decreases. In
contrast, in the joint optimization strategy, positive contribution of trajectory smoothness to fuel econ-
omy cannot counterbalance the negative contribution of travel time growth. Therefore, FC increases.
Moreover, at the high demand 3840 veh/h, the reduction ratio decreases to 14.39%.
5.2.3. Monetary cost
As shown in Table 3and Figure 5, since the joint optimization strategy improves fuel economy with
the sacrifice of traffic efficiency, we make an integrated evaluation of the system cost by convert-
ing the TT cost and FC cost into monetary cost (MC). In China, the mean salary is 4829.88 /month
(Insurance_Information 2021), legal working time is 40 h per week. Therefore, the unit cost of travel
time is k1=25.61 /h (1 ≈0.1582 US $). The price of gasoline is k2=7.69 /L (Eastmoney 2022).
Therefore, MC is weighted sum of TT and FC, i.e. MC =k1TT +k2FC.
Due to the contribution of weight k2, like FC, MC in China is always smaller in the joint optimization
strategy than in the two-stage strategy. The reduction ratio is 13.20% at the low demand 1920 veh/h.
With the increment of demand, MC in China decreases in the two-stage strategy, but increases in the
joint optimization strategy. At the high demand 3840 veh/h, the reduction ratio decreases to 2.21%.
However, the results about MC might change in different countries. For example, in the USA, the
mean salary is 3607.03 $/month (Numbeo 2022), thus, the unit cost of travel time is k1=21.04 $/h.

1246 G. WU AND R. JIANG
Figure 5. Typicalspeed and acceleration profiles within ZONE 1 under the two strategies. (a) speed profiles, (b) acceleration profiles
The price of gasoline is k2=0.73 $/L (Numbeo 2022). The MC of two-stage strategy is always smaller
than that of joint optimization strategy, which is opposite to results in China. However, note that one
can improve performance of the joint optimization strategy in terms of MC in USA by tuning TSD cost
weight w, as presented in sensitivity analysis below.
5.2.4. Driving comfort
Since frequent change of speed and acceleration leads to passenger discomfort (Ntousakis, Nikolos,
and Papageorgiou 2016;LiandLi2019; Liu, Hoogendoorn, and Wang 2020), we introduce two indices
p|a|and p|Δa|to measure the mean driving comfort over the vehicles
p|a|=
i∈I
j∈{j,0≤xi,j≤L}
|ai,j|/|I|(26)
p|Δa|=
i∈I
j∈{j,0≤xi,j≤L}
|ai,j−ai,j−1|/|I|(27)
where |I|is the total number of vehicles. The larger p|a|and p|Δa|are, the worse the driving comfort is.
As shown in Table 4, the two indices are significantly larger in the two-stage strategy than in
the joint optimization strategy, indicating the joint optimization strategy far outperforms in terms of
driving comfort. With the increment of traffic demand, the two indices decrease remarkably in the
two-stage strategy. In contrast, in the joint optimization strategy, the two indices only mildly change
under different traffic demand. Both p|a|and p|Δa|slightly increase when demand increases from 1920
veh/h to 3200 veh/h and slightly decrease when demand further increases to 3840 veh/h.
For conventional vehicles, stop-and-go traffic occurs frequently at signalized intersection because
of red signal phase. However, for CAVs, the optimized trajectories are different. They usually firstly slow
down and then accelerate to pass the intersection, always during the green signal. Thus, stop-and-go
traffic is eliminated. Unlike signalized intersection, there is no red phase in autonomous intersection
so that CAVs could flexibly coordinate their motions to alleviate traffic congestion and improve fuel
economy. Figure 6shows the optimized trajectories under different demands. There is no stop-and-go
wave at the stop bar (orange line in Figure 6.) even if in high demand scenario.
5.3. Sensitivity analysis
This section performs sensitivity analysis of the joint optimization model with respect to left-turn ratio,
weighted coefficient w, length of ZONE1 L.

TRANSPORTMETRICA B: TRANSPORT DYNAMICS 1247
Tab le 4 . Driving comfort of different strategies under different traffic demand.
Traffic demand Two-stage Joint optimization
(veh/h) p|a|(m/s2)p|Δa|(m/s2)p|a|(m/s2)p|Δa|(m/s2)
1920 91.37 10.53 27.30 4.51
(−70.12%) (−57.21%)
2560 84.47 10.03 29.28 4.85
(−65.34%) (−51.68%)
3200 75.57 9.33 30.54 4.96
(−59.58%) (−46.90%)
3840 59.14 7.77 29.56 4.85
(−50.01%) (−37.60%)
Note: Bracket indicates the change ratio of proposed strategy compared with benchmark
strategy.
Figure 6. (a) The optimized trajectories on the approaching lane on ARM 2 under low traffic demand; (b) The optimizedtrajec tories
under high traffic demand. The total demand of all approaching lanes is 1920 and 3840 veh/h in panel (a) and (b).
5.3.1. Left-turn ratio
Traffic composition is a factor that affects model performance. Due to conflict point is the major col-
lision point within intersection area, we set right-turn ratio as zero. We test 4 left-turn ratios: 0.15, 0.3,
0.45, 0.6. Two traffic demand levels are considered: the low demand 1920 veh/h and the high demand
3840 veh/h.
When traffic demand is in low level, performance of the two control strategies only slightly depends
on the left-turn ratio, see Table 5. The TT and thus MC in USA are always smaller in the two-stage strat-
egy. In contrast, the FC and thus MC in China are always smaller, and driving is always more comfortable
in the joint optimization strategy.
When traffic demand is in high level, the TT in the two-stage strategy and the joint optimization
strategy increases by 3.08 and 3.19 s, respectively, when left-turn ratio increases from 0.15 to 0.6. This
is because the constant speed of left-turn vehicle within intersection area is smaller than that of go-
straight vehicle. As a result, more left-turn vehicles have to decelerate to lower speed on the approach
lanes. The FC in the two-stage strategy decreases by 2.3 mL. However, the FC in the joint optimization
strategy only slightly changes. It decreases by 0.47 mL when left-turn ratio increases from 0.15 to 0.45
and increases by 0.48 mL when left-turn ratio further increases to 0.6, see Table 6. As a result, MC in
China would become larger in the joint optimization strategy than in the two-stage strategy when the
left-turn ratio increases to 0.45 and 0.6. Generally speaking, the advantage of joint optimization strat-
egy in terms of MC in China is more remarkable at small left-turn ratio and low traffic demand. However,

1248 G. WU AND R. JIANG
Tab le 5 . Impact of left-turn ratio on performance of the two strategies under low traffic demand.
Two-stage Joint optimization
Left-turn ratio Left-turn ratio
Measure 0.15 0.3 0.45 0.6 0.15 0.3 0.45 0.6
TT (s) 15.02 15.21 15.35 15.41 18.73 18.75 18.67 18.85
(24.69%) (23.33%) (21.66%) (22.37%)
FC (mL) 29.87 29.35 28.89 28.92 20.33 20.39 20.40 20.31
(−31.94%) (−30.55%) (−29.40%) (−29.77%)
MC in China (10−2)34.75 34.49 34.25 34.32 30.31 30.37 30.31 30.39
(−12.78%) (−11.96%) (−11.49%) (−11.45%)
MC in USA (10−2$) 10.96 11.03 11.08 11.12 12.43 12.45 12.40 12.50
(13.42%) (12.86%) (11.94%) (12.47%)
p|a|(m/s2)84.84 88.09 88.03 89.07 21.98 25.29 29.69 31.32
(−74.09%) (−71.30%) (−66.28%) (−64.84%)
p|Δa|(m/s2) 10.16 10.31 10.27 10.32 3.97 4.29 4.75 4.97
(−60.94%) (−58.35%) (−53.72%) (−51.80%)
Note: Bracket indicates the change ratio of proposed strategy compared with benchmark strategy.
Tab le 6 . Impact of left-turn ratio on performance of the two strategies under high traffic demand.
Two-stage Joint optimization
Left-turn ratio Left-turn ratio
Measure 0.15 0.3 0.45 0.6 0.15 0.3 0.45 0.6
TT (s) 16.99 18.87 18.64 20.07 19.27 20.78 21.21 22.46
(13.44%) (10.12%) (13.79%) (11.91%)
FC (mL) 26.01 24.72 23.83 23.71 21.85 21.67 21.38 21.86
(−16.00%) (−12.36%) (−10.28%) (−7.81%)
MC in China (10−2)33.31 33.79 32.93 33.96 31.90 32.93 33.06 34.40
(−4.25%) (−2.54%) (0.38%) (1.31%)
MC in USA (10−2$) 11.83 12.83 12.63 13.46 12.86 13.72 13.96 14.72
(8.72%) (6.96%) (10.48%) (9.37%)
p|a|(m/s2)60.18 52.06 49.41 48.36 28.39 27.09 27.04 30.34
(−52.83%) (−47.97%) (−45.26%) (−37.26%)
p|Δa|(m/s2) 7.99 7.24 6.86 6.83 4.55 4.60 4.87 5.23
(−54.09%) (−52.07%) (−48.49%) (−45.66%)
Note: Bracket indicates the change ratio of proposed strategy compared with benchmark strategy.
MC in USA is still always larger, and driving is still always more comfortable in the joint optimization
strategy.
5.3.2. Weighted coefficient w
We test 4 scenarios: w=0.01, 0.25, 0.8 and 10,000, and also consider low demand 1920 veh/h and
high demand 3840 veh/h. The basic settings are the same as in section 5.1. The simulation results are
shown in Table 7and Table 8.
In the low demand level, TT increases and FC decreases with the increment of w. As a result, MC in
China decreases from 0.3463 to 0.3016 when wincreases from 0.01 to 0.8, and slightly increases
to 0.311 when wfurther increases to 10,000. MC in USA is almost the same when w increases from
0.01 to 0.25, and increases from 0.1094 $ to 0.135 $ when wfurther increases to 10,000. Note that the
joint optimization strategy slightly outperforms the two-stage strategy in terms of MC in USA when
w=0.01 and 0.25. The two comfort driving indices decrease with the increment of w.Inparticular,
when wincreases from 0.25 to 0.8, p|a|decreases significantly from 90.17 to 27.30 m/s2and p|Δa|
decreases dramatically from 9.96 to 4.51 m/s2. Note that when w=0.01, p|a|becomes larger in the
joint optimization strategy than in the two-stage strategy.

TRANSPORTMETRICA B: TRANSPORT DYNAMICS 1249
Tab le 7 . Impact of coefficient won performance of the two strategies under low traffic demand.
w
Measure 0.01 0.25 0.8 10,000
TT (s) 15.02 15.10 18.59 20.68
(0.03%) (0.57%) (23.81%) (37.73%)
FC (mL) 29.73 28.97 20.29 19.37
(−0.51%) (−3.06%) (−32.10%) (−35.18%)
MC in China (10−2)34.63 34.11 30.16 31.10
(−0.34%) (−1.84%) (−13.21%) (−10.50%)
MC in USA (10−2$) 10.95 10.94 12.34 13.50
(−0.07%) (−0.16%) (12.62%) (23.20%)
p|a|(m/s2)94.32 90.17 27.30 10.51
(3.23%) (−1.31%) (−70.12%) (−88.50%)
p|Δa|(m/s2) 10.21 9.96 4.51 1.99
(−3.08%) (−5.45%) (−57.19%) (−81.11%)
Note: Bracket indicates the change ratio of proposed strategy compared with benchmark strategy.
Tab le 8 . Impact of coefficient won performance of the two strategies under high traffic demand.
w
Measure 0.01 0.25 0.8 10,000
TT (s) 17.73 18.44 20.50 23.49
(−0.82%) (3.16%) (14.68%) (31.41%)
FC (mL) 24.92 24.37 21.56 22.32
(−1.03%) (−3.22%) (−14.38%) (−11.36%)
MC in China (10−2)33.05 33.18 32.63 35.57
(−0.96%) (−0.57%) (−2.21%) (6.60%)
MC in USA (10−2$) 12.18 12.56 13.55 15.36
(−0.86%) (2.23%) (10.29%) (25.02%)
p|a|(m/s2)58.62 54.10 29.56 28.34
(−0.88%) (−8.52%) (−50.02%) (−52.08%)
p|Δa|(m/s2) 7.59 7.36 4.85 4.89
(−2.33%) (−5.29%) (−37.59%) (−37.07%)
Note: Bracket indicates the change ratio of proposed strategy compared with benchmark strategy.
In the high demand level, TT increases with the increment of w.NotethatTTcanbesmallerinthe
joint optimization strategy than in the two-stage strategy when w=0.01. This is because global opti-
mal cannot be guaranteed in the rolling horizon framework. FC decreases by 3.36 mL when wincreases
from 0.01 to 0.8. However, FC increases by 0.76 mL when wfurther increases to 10,000 because the
negative contribution of travel time growth to fuel economy exceeds the positive contribution of tra-
jectory smoothness. As a result, MC in China is smaller in the joint optimization strategy whenw=0.01,
0.25, and 0.8, but becomes larger when w=10, 000. MC in USA is smaller in the joint optimization
strategy when w=0.01, but becomes larger when w=0.25, 0.8, and 10,000. From the perspective of
MC (not only in China but also in USA), joint optimization strategy outperforms two-stage strategy by
selecting suitable weighted coefficient w. Moreover, the two comfort driving indices are always smaller
in the joint optimization strategy. The driving comfort improves with the increment of w, except that
p|Δa|slightly increases from 4.85 to 4.89 m/s2when w increase from 0.8 to 10,000.
5.3.3. Length of ZONE 1
This subsection investigates the impact of the length of ZONE 1 on the performance of proposed con-
trol strategy. Additional numerical experiments are conducted under L=150, 250 and 300 m at both
low and high traffic demand levels. The simulation results are presented in Table 9and 10. One can see
that almost all measures increase with L, except that p|a|slightly decreases from 29.6 to 29.56 m/s2and
p|Δa|decreases from 4.91 to 4.85 m/s2when Lincreases from 150 to 200 m under high demand in the

1250 G. WU AND R. JIANG
Tab le 9 . Impact of Lon performance of the two strategies under low traffic demand.
Two-stage Joint optimization
LL
Measure 150 200 250 300 150 200 250 300
TT (s) 11.75 15.02 18.40 21.74 14.78 18.59 21.43 23.85
(25.78%) (23.78%) (16.49%) (9.70%)
FC (mL) 24.54 29.88 34.25 38.67 14.93 20.29 26.67 33.34
(−39.15%) (−32.10%) (−22.11%) (−13.78%)
MC in China (10−2)28.08 34.75 40.75 46.78 23.06 30.16 37.30 44.33
(−17.88%) (−13.20%) (−8.47%) (−5.24%)
MC in USA (10−2$) 8.66 10.96 13.25 15.53 9.73 12.34 14.47 16.37
(12.35%) (12.66%) (9.21%) (5.43%)
p|a|(m/s2)89.17 91.37 97.76 99.58 15.71 27.30 46.82 67.12
(−82.38%) (−70.12%) (−52.11%) (−32.60%)
p|Δa|(m/s2) 10.38 10.53 10.94 11.07 2.82 4.51 7.10 8.92
(−72.86%) (−57.21%) (−35.11%) (−19.44%)
Note: Bracket indicates the change ratio of proposed strategy compared with benchmark strategy.
Table 10. Impact of Lon performance of the two strategies under high traffic demand.
Two-stage Joint optimization
Measure LL
150 200 250 300 150 200 250 300
TT (s) 14.46 17.88 21.66 24.48 16.57 20.50 23.55 26.39
(14.65%) (14.66%) (8.73%) (7.79%)
FC (mL) 20.06 25.18 29.71 34.73 17.09 21.56 26.56 32.29
(−14.81%) (−14.39%) (−10.60%) (−7.03%)
MC in China (10−2)26.75 33.37 39.81 45.89 26.12 32.63 38.87 45.50
(−2.35%) (−2.21%) (−2.37%) (−0.84%)
MC in USA (10−2$) 9.91 12.29 14.83 16.85 10.93 13.55 15.70 17.78
(10.29%) (10.32%) (5.91%) (5.56%)
p|a|(m/s2)54.12 59.14 61.93 69.41 29.60 29.56 37.33 49.99
(−45.31%) (−50.01%) (−39.73%) (−27.98%)
p|Δa|(m/s2) 7.30 7.77 8.02 8.80 4.91 4.85 5.80 7.22
(−32.73%) (−37.60%) (−27.68%) (−17.92%)
Note: Bracket indicates the change ratio of proposed strategy compared with benchmark strategy.
joint optimization strategy. This is because, the increase of Lmakes the trajectory smoother, and this
positive effect exceeds the negative effect of increasing travel distance. Moreover, with the increase
of L, although the absolute value of most percent difference between the two strategies becomes
smaller, there are some opposite cases. For example, the percent difference of MC in USA slightly
increases from 12.35% to 12.66% when Lincreases from 150 to 200 m under the low demand. Under
the high demand, when Lincreases from 150 to 200 m, the percent difference of TT slightly increases
from 14.65% to 14.66%.
6. Conclusion
This paper proposes a discrete-time MILP model to jointly optimize the AIM and TSD, aiming to
improve traffic efficiency, fuel economy and driving comfort simultaneously. In this model, collision
avoidance (at divergence points, converge points and conflict points), acceleration and jerk constraints
are considered. To more reasonably describe the vehicle movement in ZONE 2, the vehicle width is
considered and vehicle trajectory within ZONE 2 is treated as a channel. Travel time, fuel consumption,

TRANSPORTMETRICA B: TRANSPORT DYNAMICS 1251
monetary cost, and driving comfort are selected as measures to evaluate the performance of pro-
posed model. Sensitivity analysis has been performed with respect to the left-turn ratio, the weighted
coefficient of TSD, and the control zone length.
This study can be extended in several aspects in the future work. Firstly, the intersection we studied
is rather simple. Performance of the model needs to be tested at more complex intersections with more
approach lane and different lane allocation (He et al. 2018; Levin, Rey, and Schwartz 2019). Secondly, in
ZONE 2, vehicles heading to the same direction cruise with the same constant speed. One can expect
more benefits if this restriction can be relaxed. The lane-changing behaviour in the approach lanes
also needs to be considered. Finally, perfect communication reception is too idealistic. Relaxing this
assumption, vehicle information uncertainty should be considered in modelling (Vitale, Kolios, and
Ellinas 2022).
Acknowledgements
The authors thank Dr Xiaopeng Li and Dr Yiheng Feng for their helpful discussion.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Funding
The work was supported by the National Natural Science Foundation of China [grant nos. 71931002, 72288101].
ORCID
Guohong Wu http://orcid.org/0000-0002-3038-5509
References
Ahn, K., H. Rakha, A. Trani, and M. Van Aerde. 2002. “Estimating Vehicle Fuel Consumption and Emissions
Based on Instantaneous Speed and Acceleration Levels.” Journal of Transportation Engineering 128 (2): 182– 190.
doi:10.1061/(ASCE)0733-947X(2002)128:2(182).
Altan, O. D., G. Y. Wu, M. J. Barth, K. Boriboonsomsin, and J. A. Stark. 2017. “GlidePath: Eco-Friendly Automated
Approach and Departure at Signalized Intersections.” IEEE Transactions on Intelligent Vehicles 2 (4): 266– 277.
doi:10.1109/TIV.2017.2767289.
Bashiri, M., and C. H. Fleming. 2017. “A Platoon-Based Intersection Management System for Autonomous Vehicles.” In
2017 IEEE Intelligent Vehicles Symposium (IV) 667–672. CA, USA.
Berkovitz, L. D. 1974.Optimal Control Theory. New York: Springer Science & Business Medi.
Carlino, D., S. D. Boyles, and P. Stone. 2013. “Auction-based Autonomous Intersection Management.” In 16th International
IEEE Conference on Intelligent Transportation Systems (ITSC 2013) 529–534. Netherlands.
Chai, L. G., B. G. Cai, S. G. Wei, W. Jian, and H. S. Wang. 2018. “Connected and Autonomous Vehicles Coordinat-
ing Approach at Intersection Based on Space-Time Slot.” Transportmetrica A-Transport Science 14 (10): 929–951.
doi:10.1080/23249935.2018.1452308.
Chen, X. D., M. Li, X. Lin, Y. F. Yin, and F. He. 2021. “Rhythmic Control of Automated Traffic-Part I:Concept and Properties
at Isolated Intersections.” Transportation Science 55 (5): 969–987. doi:10.1287/trsc.2021.1060.
Denney, R. W., E. Curtis, and P. Olson. 2012. “The National Traffic Signal Report Card.” Ite Journal-Institute of Transportation
Engineers 82 (6): 22–26.
Dresner, K., and P. Stone. 2004. “Multiagent Traffic Management: A Reservation-Based Intersection Control Mechanism.”
In Proc. 3rd Int. Joint Conf. Auton. Agents Multiagent Syst., 530–537. New York, USA.
Dresner, Kurt, and Peter Stone. 2006. “Human-usable and emergency vehicle-aware control policies for autonomous inter-
section management.” In Fourth International Workshop on Agents in Traffic and Transportation (ATT), 17–25. Hakodate,
Japan.
Eastmoney. 2022. “Regional Gasoline Prices.” https://data.eastmoney.com/cjsj/yjtz/default.html.
Fayazi, S. A., and A. Vahidi. 2018. “Mixed-Integer Linear Programming for Optimal Scheduling of Autonomous Vehicle
Intersection Crossing.” IEEE Transactions on Intelligent Vehicles 3 (3): 287–299. doi:10.1109/TIV.2018.2843163.
Feng, Yiheng, Chunhui Yu, and Henry X. Liu. 2018. “Spatiotemporal Intersection Control in a Connected and Automated
Vehicle Environment.” Transportation Research Part C: Emerging Technologies 89: 364–383. doi:10.1016/j.trc.2018.02.
001.

1252 G. WU AND R. JIANG
Guo, Yi, Jiaqi Ma, Chenfeng Xiong, Xiaopeng Li, Fang Zhou, and Wei Hao. 2019. “Joint Optimization of Vehicle Trajectories
and Intersection Controllers with Connected Automated Vehicles: Combined Dynamic Programming and Shooting
Heuristic Approach.” Transportation Research Part C: Emerging Technologies 98: 54–72. doi:10.1016/j.trc.2018.11.010.
He, Z. B., L. Zheng, L. L. Lu, and W. Guan. 2018. “Erasing Lane Changes from Roads: A Design of Future Road Intersections.”
IEEE Transactions on Intelligent Vehicles 3 (2): 173–184. doi:10.1109/TIV.2018.2804164.
Insurance_Information. 2021. “Latest National Average Salary in 2021.” Accessed January 1 2022. http://www.baoxianzx.
com/shebao/54590.html.
Jiang,H.F.,J.Hu,S.An,M.Wang,andB.B.Park.2017. “Eco Approaching at an Isolated Signalized Intersection Under
Partially Connected and Automated Vehicles Environment.” Transportation Research Part C-Emerging Technologies 79:
290–307. doi:10.1016/j.trc.2017.04.001.
Jiang, S. M., T. L. Pan, R. X. Zhong, C. Chen, X. A. Li, and S. M. Wang. 2022. “Coordination of Mixed Platoons
and Eco-Driving Strategy for a Signal-Free Intersection.” IEEE Transactions on Intelligent Transportation Systems.
doi:10.1109/Tits.2022.3211934.
Kamal, M. A. S., J. Imura, T. Hayakawa, A. Ohata, and K. Aihara. 2015. “A Vehicle-Intersection Coordination Scheme for
Smooth Flows of Traffic Without Using Traffic Lights.” IEEE Transactions on Intelligent Transportation Systems 16 (3):
1136–1147. doi:10.1109/TITS.2014.2354380.
Levin, M. W., and D. Rey. 2017. “Conflict-point Formulation of Intersection Control for Autonomous Vehicles.” Transporta-
tion Research Part C-Emerging Technologies 85: 528–547. doi:10.1016/j.trc.2017.09.025.
Levin, M. W., D. Rey, and A. Schwartz. 2019. “Max-pressure Control of Dynamic Lane Reversal and Autonomous Intersection
Management.” Transportmetrica B-Transport Dynamics 7 (1): 1693–1718. doi:10.1080/21680566.2019.1691675.
Li, Li, and Xiaopeng Li. 2019. “Parsimonious Trajectory Design of Connected Automated Traffic.” Transportation Research
Part B: Methodological 119: 1–21. doi:10.1016/j.trb.2018.11.006.
Li,G.F.,S.L.Li,S.Li,andX.D.Qu.2021. “Continuous Decision-Making for Autonomous Driving at Intersections Using Deep
Deterministic Policy Gradient.” IET Intelligent Transport Systems. doi:10.1049/itr2.12107.
Li,L.,andF.Y.Wang.2006. “Cooperative Driving at Blind Crossings Using Intervehicle Communication.” IEEE Transactions
on Vehicular Technology 55 (6): 1712–1724. doi:10.1109/TVT.2006.878730.
Liang, X., S. I. Guler, and V. V. Gayah. 2018. “Signal Timing Optimization with Connected Vehicle Technology: Platooning
to Improve Computational Efficiency.” Transportation Research Record 2672 (18): 81–92. doi:10.1177/0361198118786
842.
Lin,Q.F.,S.B.E.B.Li,S.B.Xu,X.J.Du,D.G.Yang,andK.Q.Li.2021. “Eco-Driving Operation of Connected Vehicle With
V2I Communication Among Multiple Signalized Intersections.” IEEE Intelligent Transportation Systems Magazine 13 (1):
107–119. doi:10.1109/MITS.2020.3014113.
Liu, M. Q., S. Hoogendoorn, and M. Wang. 2020. “Receding Horizon Cooperative Platoon Trajectory Planning on Corridors
with Dynamic Traffic Signal.” Transportation Research Record 2674 (12): 324–338. doi:10.1177/0361198120954869.
Liu, C. L., C. W. Lin, S. Shiraishi, and M. Tomizuka. 2018. “Distributed Conflict Resolution for Connected Autonomous
Vehicles.” IEEE Transactions on Intelligent Vehicles 3 (1): 18–29. doi:10.1109/TIV.2017.2788209.
Liu, M. Q., J. Zhao, S. P. Hoogendoorn, and M. Wang. 2021. “An Optimal Control Approach of Integrating
Traffic Signals and Cooperative Vehicle Trajectories at Intersections.” Transportmetrica B-Transport Dynamics.
doi:10.1080/21680566.2021.1991505.
Liu, Meiqi, Jing Zhao, Serge Hoogendoorn, and Meng Wang. 2022. “A Single-Layer Approach for Joint Optimization of
Traffic Signals and Cooperative Vehicle Trajectories at Isolated Intersections.” Transportation Research Part C: Emerging
Technologies 134: 103459. doi:10.1016/j.trc.2021.103459.
Ma,J.Q.,X.P.Li,F.Zhou,J.Hu,andB.B.Park.2017. “Parsimonious Shooting Heuristic for Trajectory Design of Connected
Automated Traffic Part II: Computational Issues and Optimization.” Transportation Research Part B-Methodological 95:
421–441. doi:10.1016/j.trb.2016.06.010.
Malikopoulos, Andreas A, Logan Beaver, and Ioannis Vasileios J Automatica Chremos. 2021. “Optimal Time Trajectory and
Coordination for Connected and Automated Vehicles.” Automatica 125: 109469. doi:10.1016/j.automatica.2020.109
469.
Malikopoulos, A. A., C. G. Cassandras, and Y. J. Zhang. 2018. “A Decentralized Energy-Optimal Control Framework for Con-
nected Automated Vehicles at Signal-free Intersections.” Automatica 93: 244–256. doi:10.1016/j.automatica.2018.03.
056.
Medina, A. I. M., N. van de Wouw, and H. Nijmeijer. 2018. “Cooperative Intersection Control Based on Virtual Platooning.”
IEEE Transactions on Intelligent Transportation Systems 19 (6): 1727–1740. doi:10.1109/TITS.2017.2735628.
Mirheli, A., M. Tajalli, L. Hajibabai, and A. Hajbabaie. 2019. “A Consensus-Based Distributed Trajectory Control in a Signal-
free Intersection.” Transportation Research Part C-Emerging Technologies 100: 161–176. doi:10.1016/j.trc.2019.01.004.
Niroumand, Ramin, Mehrdad Tajalli, Leila Hajibabai, and Ali Hajbabaie. 2020. “Joint Optimization of Vehicle-Group Tra-
jectory and Signal Timing: Introducing the White Phase for Mixed-Autonomy Traffic Stream.” Transportation Research
Part C: Emerging Technologies 116: 102659. doi:10.1016/j.trc.2020.102659.
Ntousakis, Ioannis A., Ioannis K. Nikolos, and Markos Papageorgiou. 2016. “Optimal Vehicle Trajectory Planning in the
Context of Cooperative Merging on Highways.” Transportation Research Part C: Emerging Technologies 71: 464–488.
doi:10.1016/j.trc.2016.08.007.

TRANSPORTMETRICA B: TRANSPORT DYNAMICS 1253
Numbeo. 2022. “Cost of Living in United States.” Accessed January 1 2022. https://www.numbeo.com/cost-of-living/coun
try_result.jsp?country =United+States.
Sun, W. L., J. F. Zheng, and H. X. Liu. 2018. “A Capacity Maximization Scheme for Intersection Management with Automated
Vehicles.” Transportation Research Part C-Emerging Technologies 94: 19–31. doi:10.1016/j.trc.2017.12.006.
Tachet, R., P. Santi, S. Sobolevsky, L. I. Reyes-Castro, E. Frazzoli, D. Helbing, and C. Ratti. 2016. “Revisiting Street Intersections
Using Slot-Based Systems.” Plos One 11 (3): 149607. doi:10.1371/journal.pone.0149607.
Timmerman, R. W., and M. A. A. Boon. 2021. “Platoon Forming Algorithms for Intelligent Street Intersections.” Transport-
metrica A-Transport Science 17 (3): 278–307. doi:10.1080/23249935.2019.1692962.
U.S.E.P.A. 2018. “Sources of Greenhouse Gas Emission.” Accessed February 13 2020. https://www.epa.gov/ghgemissions/
sources-greenhouse-gas-emissions#transportation.
Vitale, C., P. Kolios, and G. Ellinas. 2022. “Autonomous Intersection Crossing with Vehicle Location Uncertainty.” IEEE
Transactions on Intelligent Transportation Systems. doi:10.1109/TITS.2022.3152006.
Wan, N. F., A. Vahidi, and A. Luckow. 2016. “Optimal Speed Advisory for Connected Vehicles in Arterial Roads and the
Impact on Mixed Traffic.” Transportation Research Part C-Emerging Technologies 69: 548–563. doi:10.1016/j.trc.2016.01.
011.
Wei, Yuguang, Cafer Avcı, Jiangtao Liu, Baloka Belezamo, Nizamettin Aydın, Pengfei Li, and Xuesong Zhou. 2017. “Dynamic
Programming-based Multi-Vehicle Longitudinal Trajectory Optimization with Simplified Car Following Models.” Trans-
portation Research Part B: Methodological 106: 102–129. doi:10.1016/j.trb.2017.10.012.
Wu, Y. Y., H. P. Chen, and F. Zhu. 2019. “DCL-AIM: Decentralized Coordination Learning of Autonomous Intersection Man-
agement for Connected and Automated Vehicles.” Transportation Research Part C-Emerging Technologies 103: 246–260.
doi:10.1016/j.trc.2019.04.012.
Wu, W., Y. Liu, W. Hao, G. A. Giannopoulos, and Y. J. Byon. 2022. “Autonomous Intersection Management with Pedestrians
Crossing.” Transportation Research Part C-Emerging Technologies 135. doi:10.1016/j.trc.2021.103521.
Xu,B.,S.E.Li,Y.G.Bian,S.Li,X.J.Ban,J.Q.Wang,andK.Q.Li.2018. “Distributed Conflict-free Cooperation for Multiple
Connected Vehicles at Unsignalized Intersections.” Transportation Research Part C-Emerging Technologies 93: 322–334.
doi:10.1016/j.trc.2018.06.004.
Xu, H., W. Xiao, C. G. Cassandras, Y. Zhang, and L. Li. 2022. “A General Framework for Decentralized Safe Optimal Con-
trol of Connected and Automated Vehicles in Multi-Lane Signal-Free Intersections.” IEEE Transactions on Intelligent
Transportation Systems. doi:10.1109/TITS.2022.3151080.
Xu, H. L., Y. Zhang, L. Li, and W. X. Li. 2020. “Cooperative Driving at Unsignalized Intersections Using Tree Search.” IEEE
Transactions on Intelligent Transportation Systems 21 (11): 4563–4571. doi:10.1109/TITS.2019.2940641.
Yao, Handong, Jianxun Cui, Xiaopeng Li, Yu Wang, and Shi An. 2018. “A Trajectory Smoothing Method at Signalized Inter-
section Based on Individualized Variable Speed Limits with Location Optimization.” Transportation Research Part D:
Transport and Environment 62: 456–73. doi:10.1016/j.trd.2018.03.010.
Yu, Chunhui, Yiheng Feng, Henry X. Liu, Wanjing Ma, and Xiaoguang Yang. 2018. “Integrated Optimization of Traffic
Signals and Vehicle Trajectories at Isolated Urban Intersections.” Transportation Research Part B: Methodological 112:
89–112. doi:10.1016/j.trb.2018.04.007.
Yu,C.H.,Y.H.Feng,H.X.Liu,W.J.Ma,andX.G.Yang.2019. “Corridor Level Cooperative Trajectory Optimiza-
tion with Connected and Automated Vehicles.” Transportation Research Part C-Emerging Technologies 105: 405–421.
doi:10.1016/j.trc.2019.06.002.
Zhao, W. M., R. H. Liu, and D. Ngoduy. 2021. “A Bilevel Programming Model for Autonomous Intersection Control and
Trajectory Planning.” Transportmetrica A-Transport Science 17 (1): 34–58. doi:10.1080/23249935.2018.1563921.
Zhao, Weiming, Dong Ngoduy, Simon Shepherd, Ronghui Liu, and Markos Papageorgiou. 2018. “A Platoon Based Cooper-
ative Eco-Driving Model for Mixed Automated and Human-Driven Vehicles at a Signalised Intersection.” Transportation
Research Part C: Emerging Technologies 95: 802–821. doi:10.1016/j.trc.2018.05.025.
Appendix
Take a left-turn vehicle on arm 1 as an example, we explain how to calculate xλ+
iand xλ−
iof turning vehicles with the
consideration of vehicle size.
Step 1: determining nλ−
i(purple node in Figure A1)/nλ+
i(red node in Figure A1) which are first and last node of cell λ
passed through by vehicle i.
As shown in Figure A1,NODEλ
idenotes node set of cell λalong the channel of vehicle i. The node set NODEλ
iconsists
of two types of nodes. CROSSλ
iis the first type of node in set NODEλ
i, which indicates the cross nodes of trajectory channel
boundaries of vehicle iand the side of cell λ(the green and red nodes in Figure A1).VERTEXλ
iis the second type of node
in set NODEλ
i, which indicates the vertex of the cell λwithin the channel (the purple node in Figure A1).
We denote DDnas the deviation degree of node nfrom Nstart, which is calculated by Equation (A1). We find nλ−
iand
nλ+
iby calculating DDnof all the nodes in set NODEλ
i. As shown in Equations (A2) and (A3), the nodes corresponding to
the minimum and maximum value of DDnare defined as nλ−
i(purple node in Figure A1)andnλ+
i(red node in Figure A1),

1254 G. WU AND R. JIANG
Figure A1. Illustration of xλ−
iand xλ+
i.
respectively.
DDn=|xn−xstart |+|yn−ystart|,∀n∈NODEλ
i(A1)
nλ−
i=arg min(DDn),∀n∈NODEλ
i(A2)
nλ+
i=arg max(DDn),∀n∈NODEλ
i(A3)
where (xn,yn)and (xstart,ystart )are the coordinates of node nand Nstart .
Step 2: determining ¯
nλ−
i/¯
nλ+
iand θ−/θ+
without loss of generality, ¯
nrepresents projection point of node non the trajectory centre line. Because ¯
nis on
trajectory centre line, the coordinate of ¯
n,(x¯
n,y¯
n), satisfies the parametric equation
x¯
n=x0+acentercosθ,θ∈0, π
2(A4)
y¯
n=y0+bcentersinθ,θ∈0, π
2(A5)
where (x0,y0)and (x¯
n,y¯
n)denote the coordinate of the ellipse arc centre N0and ¯
n, respectively. θis eccentric angle
of ¯
n,acenter and bcenter represent the semi-major axis and the semi-minor axis of the vehicle i’s trajectory centre line,
respectively.
The tangent slope of trajectory centre line at ¯
nis calculated by equation (A6)
K¯
n=−
bcenter cos θ
acenter sin θ,θ∈0, π
2(A6)
Moreover, coordinates of nand ¯
nsatisfies Equation (A7)
y¯
n−yn
x¯
n−xn
×K¯
n=−1, θ−∈0, π
2(A7)

TRANSPORTMETRICA B: TRANSPORT DYNAMICS 1255
¯
nλ−
iand ¯
nλ+
iare denoted by the two yellow nodes in Figure A1, which are the corresponding projections of nλ−
iand
nλ+
ion the trajectory centre line, respectively. θ−(θ+)is the eccentric angles of ¯
nλ−
i(¯
nλ+
i), which can be obtained by
solving (A4)-(A7).
Step 3: determining xλ−
i/xλ+
i
θ−(θ+)is obtained in step 2. According to arc length calculation formula, xλ−
i(thick green line in Figure A1)and
xλ+
i(thick blue line and green line in Figure A1) are calculated by equation (A8) and (A9).
xλ−
i=θ−
0bcenter2(cos(θ ))2+acenter 2(sin(θ ))2dθ
(A8)
xλ+
i=d+θ+
0bcenter2(cos(θ ))2+acenter 2(sin(θ ))2dθ
(A9)
where dis the sum of car length lcar and jam gap sjam.