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Lane-Change-Aware Connected Automated Vehicle Trajectory
Optimization at a Signalized Intersection with Multi-Lane Roads
Handong Yao, Xiaopeng Li∗
Department of Civil and Environmental Engineering, University of South Florida, 4202 E Fowler Avenue,
ENC 3300, Tampa, FL 33620
Trajectory smoothing is an effective concept to control connected automated vehicles (CAVs)
in mixed traffic to reduce traffic oscillations and improve overall traffic performance. How-
ever, smoother trajectories often lead to greater gaps between vehicles, which may incentivize
human driven vehicles (HVs) from adjacent lanes to make lane changes to cut in. Such cut-in
lane changes may compromise the expected performance from CAV trajectory smoothing.
To address this issue, this paper builds a mixed traffic trajectory smoothing framework at a
signalized intersection with multi-lane roads and then proposes a decentralized lane-change-
aware CAV trajectory optimization model under the framework. Squared acceleration and
traffic mobility are considered as a joint objective. A commercial solver is utilized to solve
the proposed optimization problem. Numerical experiments are conducted to study the
performance of the proposed model under different scenarios. First, the results show that
the HV lane changes would cause reduction of half or more expected benefits of trajectory
smoothing along a two-lane segment adjacent to a signalized intersection. Then, we find that
the proposed model yields extra benefits in the system joint objective (10-30%), riding com-
fort (10-30%), travel time (2-5%), fuel consumption (2-10%) and safety (2-15%) compared
with the trajectory optimization model without a lane-change-aware mechanism when CAV
market penetration rate is not high. Sensitivity analyses on road segment lengths, signal cy-
cle lengths, traffic saturation rates and through-vehicle rates show that the proposed model
yields better system performance under most scenarios by restraining lane changes, e.g.,
20% extra benefit at a short road segment length, 30% extra benefit at a long signal cycle
length, 20% extra benefit at a high traffic saturation rate, and 20% extra benefit at a high
through-vehicle rate. Finally, the proposed model is verified to be effective at a signalized
intersection with three-or-more-lane roads.
Keywords: Mixed traffic, Connected automated vehicle, Decentralized control, Trajectory
smoothing, Lane change, Signalized intersections.
∗Corresponding author
Email address: xiaopengli@usf.edu (Xiaopeng Li)
1
1. Introduction
The United States Department of Transportation (USDOT) emphasized the crucial role
of connected automated vehicle (CAV) technologies that enable information sharing among
individual automated (or autonomous) vehicles to increase safety, efficiency, and reliability
of the transportation system (USDOT, 2018). Due to different communications and coop-
eration behavior among CAVs, the Society of Automotive Engineers (SAE) defined classes
of cooperation (SAE, 2020) as follows:
1) Class A: Status sharing. CAVs have full authority to decide actions based on the
perception information about the traffic environment and the status of participants.
2) Class B: Intent sharing. CAVs have full authority to decide actions based on the
information about the planned future actions of participants.
3) Class C: Agreement seeking. CAVs have full authority to decide actions with a se-
quence of collaborative messages among specific CAVs intended to influence local planning
of related actions.
4) Class D: Prescriptive. CAVs have full authority to decide actions except for very
specific circumstances in which they are designed to accept and adhere to a prescriptive
communication.
In this paper, we focus on a connected mixed traffic environment that CAVs are consid-
ered as in cooperation class B or above and human driven vehicles (HVs) have the commu-
nication capabilities (e.g., dedicated short range communication [DSRC] systems). Thus,
CAVs and HVs can share their status (e.g., locations and speeds) and intents (e.g., estimated
departure times) with other participants (e.g., road side units, traffic lights, CAVs and HVs),
and CAVs can decide their actions based on the information about the planned future actions
of participants. The state-of-art studies in such mixed traffic environment have shown that
the interactions between CAVs and HVs affect traffic performance in mobility, fuel efficiency
and safety via car following behavior and lane changing behavior.
In single-lane mixed traffic, only car following behavior exists. CAVs are capable of
communicating with surroundings (e.g., vehicles and infrastructures) and precisely control
their speeds to improve roadway capacity and fuel efficiency on highway without experiencing
lane change (LC) impacts (Aria et al., 2016; Chen et al., 2017; Liu et al., 2018; Ghiasi et al.,
2019; Chen et al., 2020). Studies have proved that CAV trajectory smoothing has the
potential to reduce travel time/delay and fuel consumption, and to enhance riding comfort
and safety by releasing traffic oscillations with smooth trajectories at intersections with a
single-lane road (Treiber et al., 2014; Ma et al., 2017; Taylor and Zhou, 2017; Wei et al.,
2017; Slade et al., 2020). These benefits are found out to increase with growing CAV market
penetration rates (MPRs) in mixed traffic (Wan et al., 2016; Jiang et al., 2017; Yang et al.,
2017; He and Wu, 2018; Zhao et al., 2018; Pourmehrab et al., 2020a,b; Yao and Li, 2020).
In multi-lane mixed traffic, both car following behavior and lane changing behavior exist.
A number of models and strategies have been proposed to investigate CAV LC related
policies and strategies, such as CAV lane assignment (Bevly et al., 2016; Chen et al., 2017;
Ghiasi et al., 2019) and CAV lane changing control (You et al., 2015; Bevly et al., 2016; Yu
et al., 2018). In this paper, only HV LCs are considered because all CAVs are assumed to
2
follow the planned trajectories and will not conduct LCs. Ioannou et al. (2005) demonstrated
that a high probability of HV cut-in LCs from adjacent lanes exists due to the large gaps
created by adaptive cruise control deceleration behavior. Besides, Milanés and Shladover
(2016) demonstrated that HV cut-in LCs drastically increase traffic oscillation in cooperative
adaptive cruise control systems equipped platoons. With larger gaps generated by trajectory
smoothing at signalized intersections, the probability of HV cut-in LCs is supposed to be
higher. To date, only a few studies have investigated the HV cut-in LC effects on CAV
trajectory smoothing at signalized intersections (Xia et al., 2013; Wan et al., 2016; Yang
et al., 2017). Xia et al. (2013) found that fuel/emission benefits from trajectory smoothing
at signalized intersections are weakened since HVs have more flexibility to change lanes
with larger gaps. Yang et al. (2017) concluded that trajectory smoothing at signalized
intersections has negative impact on overall system performance due to the LCs around
the controlled CAVs when the CAV MPR is low. While the existing studies confirmed the
negative impacts of HV cut-in LCs on the system performance of trajectory smoothing in
mixed traffic, none of them proposed a detailed mixed traffic trajectory smoothing framework
for the signalized intersections with multi-lane roads, and none of them addressed the HV
LCs induced by trajectory smoothing.
Motivated by the above research gaps, this paper extends an existing trajectory optimiza-
tion model (TO) at a signalized intersection with a single-lane road to a prescriptive mixed
traffic trajectory smoothing framework at a signalized intersection with multi-lane roads,
including a trajectory planning module that is running on CAVs and a trajectory control
module that is running on both CAVs and HVs. Then, this paper proposes a decentral-
ized lane-change-aware CAV trajectory optimization model (LCTO) under the framework
to address the HV LCs induced by TO. A joint objective is considered in LCTO with riding
comfort and mobility. Nonlinear LC restraining constraints, considering both discretionary
LCs and mandatory LCs, are linearized based on the solution from TO. Additionally, dis-
crete models are reformulated from the continuous LCTO and TO to find exact optimal
solutions utilizing commercial solvers (e.g., Gurobi). A set of numerical experiments is then
conducted to verify the effectiveness of LCTO and to compare it with TO. First, we find
that the expected benefits of trajectory smoothing are overestimated without considering
LCs, and the actual benefits in TO with LCs are decreased half or more compared with
the expected benefits without LCs. Then, sensitivity analysis results on CAV market pen-
etration at a signalized intersection with two-lane roads show that LCTO yields significant
extra benefits in the system joint objective (10-30%), riding comfort (10-30%), and safety
(2-15%), and also yields a few extra benefits in the system travel time (2-5%) and fuel con-
sumption (2-10%) by restraining HV LCs created in TO when the CAV MPR is not high.
Other sensitivity analysis results on road segment lengths, signal cycle lengths, traffic satu-
ration rates and through-vehicle rates show that TO is recommended with the same system
performance as LCTO but higher computational efficiency when either the road segment
length is relatively long or the through-vehicle rate is relatively low, and LCTO is suggested
in other traffic scenarios with significant extra benefits in the system joint objective (e.g., at
most 20%, 30%, 20% and 30% extra benefits regarding road segment lengths, signal cycle
lengths, traffic saturation rates and through-vehicle rates, respectively). Finally, additional
3
experiments verify that LCTO can be extended to a signalized intersection with three-or-
more-lane roads that has a similar system performance as LCTO at a signalized intersection
with two-lane roads.
The contributions of this paper include the following aspects.
1) This paper extends an existing trajectory optimization model at a signalized intersec-
tion with a single-lane road to a prescriptive mixed traffic trajectory smoothing framework
at a signalized intersection with multi-lane roads.
2) To address the negative effects of HV cut-in LCs, this paper first proposes a decentral-
ized lane-change-aware CAV trajectory optimization model in mixed traffic at a signalized
intersection with multi-lane roads.
3) CAV market penetration analyses are conducted to verify that the expected benefits
of trajectory smoothing are impaired by HV LCs, and LCTO is capable of improving the
system riding comfort, travel time, fuel consumption and safety by restraining the HV LCs
when the CAV MPR is not high.
4) Sensitivity analyses on road segment lengths, signal cycle lengths, traffic saturation
rates and through-vehicle rates imply that LCTO is effective in most traffic situations except
those when the road segment length is too long and the through-vehicle rate is too low.
The organization of this paper is as follows. Section 2 describes the framework of CAV
trajectory smoothing in mixed traffic at a signalized intersection with multi-lane roads.
Section 3 formulates LCTO in continuous and discrete formats, respectively. Section 4
conducts numerical experiments to study the performance of LCTO under different traffic
settings. Section 5 discusses the limitations and extensions of LCTO. Finally, Section 6
concludes this manuscript and briefly discusses future research directions.
2. Extension of the Existing Trajectory Optimization Model in a Multi-lane
Context
This section describes the framework of trajectory smoothing with signal phase and
timing (SPaT) in mixed traffic. The focus of this framework is on a signalized intersection
with a fixed-time signal setting, formed by a two-lane road (note that the two-lane setting
is for conciseness of the presentation and it can be easily adapted to multi-lane instances,
see Section 4.4), as illustrated by Figure 1. For convenience of readers, the key notation is
listed in Table 1.
4
Table 1: Notation list.
Notation Definition
N:= [1,2,...,N]Set of vehicles, where the number of vehicles is N.
NCAV := [1,2,...,NCAV]Set of CAVs, where the number of CAVs is NCAV .
NHV := [1,2,...,NHV]Set of HVs, where the number of HVs is NHV .
M P R Market penetration rate, and MPR =NCAV/N.
N0Set of leading vehicles.
kIndex of lane k∈ K := [1, K], where Kis the number of lanes.
tIndex of time t∈ T := [0, T ], where Tis the maximum simulation time.
nIndex of a vehicle, n∈ N .
lveh Length of a vehicle.
sCAV
0Minimum spacing of a CAV.
sHV
0Minimum spacing of an HV.
τCAV Minimum time gap of a CAV.
τHV Minimum time gap of an HV.
LLocation of stop-line, i.e., road segment length.
Llimit Location of forbidding LCs.
LMLocation of allowing mandatory LCs.
X:={Xn}n∈N Set of vehicle tra jectories.
Xn={xn(t)}t∈T Trajectory of vehicle n∈ N .
xn(t)Location of vehicle n∈ N at time t∈[0, T ].
t−
nTime point of vehicle n∈ N at location 0, i.e., arrival time.
t+
nTime point of vehicle n∈ N at location L, i.e., departure time.
vn(t)Speed of vehicle n∈ N at time t∈[0, T ], and vn(t) = ˙xn(t).
vSpeed limit (i.e., maximum allowed speed).
an(t)Acceleration of vehicle n∈ N at time t∈[0, T ], and an(t) = ¨xn(t).
aMaximum acceleration.
aMinimum acceleration (or maximum deceleration with a negative sign).
∆aChanging threshold.
on(t)Current lane of vehicle n∈ N at time t∈[0, T ].
o∗
n(t)Target lane of vehicle n∈ N at time t∈[0, T ].
enDesired movement of vehicle n∈ N .
e
OnDesired lane set of vehicle n∈ N .
fsTraffic saturation rate.
GEffective green length.
REffective red length.
CCycle length, and C=G+R.
5
Figure 1: An illustration of the CAV trajectory optimization at a signalized intersection with a two-lane
road.
Consider a time horizon of T:= [0, T ], where Tis the time window for CAV trajectory
control. Consider a two-lane road segment approaching a signalized intersection with a
control zone of length L. Let Llimit denote the location after which LCs are prohibited. Let
LMdenote the location after which mandatory LCs are activated. Let K:= [1, K]denote
the set of lanes, where Kis the number of lanes. Let k= 1 denote the leftmost lane (i.e.,
lane 1), and the index increases rightwards, thus indicating k=Kas the rightmost lane
(i.e., lane K). Note that K= 2 in the two-lane setting.
Let N:= 1,2, . . . , N denote the set of vehicles. NHV := 1,2,..., NHV and NCAV :=
NHV + 1, NHV + 2,..., N HV +NCAV =Ndenote the sets of HVs (i.e., the blue-dashed
curves in Figure 1) and CAVs (i.e., the black-solid curves in Figure 1), respectively. NHV
and NCAV are the numbers of HVs and CAVs, respectively. Note that CAVs are assumed to
be equipped with cooperation class B or above systems, and HVs are assumed to be equipped
with connected vehicle communication systems (e.g., DSRC systems). Let Xn={xn(t)}t∈T
denote the trajectory of vehicle n∈ N , where xn(t)is the location of vehicle nat time t.
Let vn(t) = ˙xn(t)∈[0, v], where vdenotes the speed limit. And let an(t) = ¨xn(t)∈[a, a]
denote the acceleration of vehicle n, where aand adenote the minimum and maximum
acceleration rates respectively.
Let endenote the desired turning movement of vehicle n, where en= 0 for through
vehicles, en= 1 for left-turn vehicles, and en= 2 for right-turn vehicles. Assign lane 1
to through and left-turn vehicles and lane 2 to through and right-turn vehicles. Thus, the
6
desired lane set e
On(en)of vehicle nis formulated as follows,
e
On(en) =
{1,2},if en= 0;
{1},if en= 1;
{2},otherwise.
(1)
Assume the control zone is reserved for smoothing longitudinal CAV trajectories while
CAV lateral LCs, if any, are completed upstream before arriving the control zone. Whereas
HVs are randomly distributed in all lanes. Thus, in the control zone, CAVs will not make
LCs, and only HVs will make LCs.
Additionally, let on(t)denote the current lane of vehicle nat time t. In the two-lane
setting, at time t∈ T , if a vehicle is located in lane 1, then its target lane (the lane to which
vehicle nintends to change to) is lane 2. Otherwise, its target lane is lane 1. Thus, the
target lane o∗
n(t)of vehicle nat time tcan be calculated as
o∗
n(t) = (on(t)+1,if on(t) = 1,
on(t)−1,if on(t) = 2.(2)
Further, assume the intersection signal control follows a fixed signal timing plan with
an effective green time Gand an effective red time R, i.e., with a signal cycle length of
C=G+R(note that the yellow time is split and included in Gand R). Assume that the
intersection is equipped with road side units that allow the SPaT information to be shared
with all vehicles in the control zone.
Let t−
nand v−
ndenote the arrival time and arrival speed of vehicle nrespectively, which
are predetermined by the upstream traffic. To capture general stochastic vehicle arrival
patterns with different traffic saturation rates (fs) and green/cycle ratios (G/C), assume
the arrival time of vehicle n∈ N \ N 0follows,
t−
n=
t−
l−
n+τCAV +sCAV
0+lveh
v−
n×1 + ξn×C
fsG−1,if n∈ N CAV,
t−
l−
n+τHV +sHV
0+lveh
v−
n×1 + ξn×C
fsG−1,otherwise, (3)
where ξnis an uniform random number over [0,2] and fs∈(0, C/G]is the traffic saturation
rate. l−
nis the initial preceding vehicle of vehicle n.τCAV and sCAV
0are the minimum time
gap and minimum spacing of a CAV, respectively. τHV and sHV
0are the minimum time gap
and minimum spacing of an HV, respectively. lveh is the uniform vehicle length of a vehicle
for all CAVs and HVs. Thus, τCAV +sCAV
0+lveh
v−
nand τHV +sHV
0+lveh
v−
nare the safety time gaps
that ensure CAVs and HVs arrive without any collision. Additionally, N0is the set of leading
vehicles (e.g., vehicles that do not have preceding vehicles) and the arrival time t−
l−
n∈N 0of
a leading vehicle is randomly selected. Each time, the arrival speed of vehicle nfollows an
uniform random number over [v/2, v]to allow stochasticity in arrival speeds.
7
If vehicle nviolates the safety spacing, the arrival times and speeds are changed to keep
safety in the following ways. If sn,l−
n(t−
n)< sCAV
0+τCAVv−
n,∀n∈ N CAV \ N 0, we update
v−
n=v−
l−
n, t−
n=t−
l−
n+τCAV +sCAV
0+lveh
v−
n
;(4)
if sn,l−
n(t−
n)< sHV
0+lveh +τHVv−
n,∀n∈ N HV \ N 0, we update
v−
n=v−
l−
n, t−
n=t−
l−
n+τHV +sHV
0+lveh
v−
n
,(5)
where sn,m(t) = xm(t)−xn(t)−lveh is the spacing gap between vehicles nand mat time
t∈ T .
Figure 2: Flow chart of the trajectory smoothing framework in mixed traffic at a signalized intersection with
multi-lane roads.
Figure 2 plots the flow chart of the trajectory smoothing framework in mixed traffic at a
signalized intersection with multi-lane roads. First, we check the arrival times of all vehicles
8
to decide their arrival sequence. Then, a trajectory planning module is designed to optimize
CAV trajectories within a rolling-time window. Last, a real-time trajectory control module
is used to run on both CAVs and HVs at each time window. The detailed operations of these
two modules are described in the following subsections.
2.1 Trajectory planning module
A decentralized control based CAV trajectory optimization model (TO) has been studied
in our previous work (Yao and Li, 2020) considering a signalized intersection with a single-
lane road, and it can be extended to TO at a signalized intersection with multi-lane roads
without considering possible conflicting HV LCs in a similar decentralized control scheme
to reduce travel time and to improve riding comfort simultaneously. Thus, the continuous
TO in the trajectory planning module is formulated as NCAV sub-problems with each CAV
as an individual sub-problem. For each CAV n∈ N CAV,
CTO: min
Xn
Jn(Xn):=ωmt+
n−t−
n+ωaZt+
n
t−
n
an(t)2dt, (6)
where t+
nis the departure time of vehicle n.ωmand ωaare the weights of mobility and
squared acceleration, respectively. Then, the continuous TO is subject to the following
constraints.
•Arrival constraints:
xn(t−
n) = 0.(7)
vn(t−
n) = v−
n.(8)
•Departure constraints:
xnt+
n=L. (9)
t+
n≤t+
n≤t+
n,(10)
where the lower bound of departure time t+
ncan be calculated by
t+
n=G(max{t−
n+(v−v−
n)2
2av +L
v, t+
ln(t−
n)+τCAV +sCAV
0+lveh
v}).(11)
G(t):=(t, if mod (t, C)∈[0, G],
t
C×C, otherwise.(12)
When vehicle narrives at time t−
n, the preceding vehicle ln(t−
n)at time t−
nis considered as
the vehicle n’s preceding vehicle until vehicle ndeparts. To let Equation (11) satisfy leading
vehicles, let t+
ln0(t−
n)=−∞,∀n0∈ N 0. With Equation (11), t+
n=t−
n+(v−v−
n)2
2av +L
vif the
trajectory of the preceding vehicle does not affect the subject vehicle. Otherwise, vehicle n
9
should keep a safety spacing to the preceding vehicle, and thus t+
n=t+
ln(t−
n)+τCAV +sCAV
0+lveh
v.
With Equation (12), the lower bound of departure time is definitely in green time.
To maintain a reasonable throughput, the upper bound of departure time t+
nis set as
t+
n= min ($t+
n
C%×C+G, t+
n+τCAV +sCAV
0+lveh
v).(13)
•Speed constraints:
0≤vn(t)≤v, ∀t∈t−
n, t+
n.(14)
•Acceleration constraints:
a≤an(t)≤a, ∀t∈t−
n, t+
n.(15)
•Safety constraints:
xln(t−
n)(t)−xn(t)≥sCAV
0+lveh +τCAVvn(t),∀t∈t−
n, t+
n.(16)
Because vln0(t)=vand xln0(t)= +∞,∀n0∈ N 0∩ N CAV, Equation (16) is also valid for
leading CAVs. Equation (16) is set to ensure safety between vehicle nand its preceding
vehicle.
Although we can easily solve the continuous TO by commercial solvers (e.g., Gurobi)
and smooth trajectory for each CAV, the planned trajectories are optimized based on the
TO considering a single-lane road. At a signalized intersection with multi-lane roads, CAVs
might not follow the planned trajectories due to HV LCs. Thus, the following trajectory
control module is activated to control vehicles in real-time.
2.2 Trajectory control module
The trajectory control module is run on each vehicle to control vehicle’s trajectory. If a
HV arrives, the trajectory planning module will not work and the HV will follow predefined
HV car following and lane changing models. If a CAV arrives, it will follow the planned
trajectory and, if necessary, regulate its trajectory with a safety feature.
(1) HV car following model
HVs (i.e., n∈ N HV) follow Gipps’ car following model described in Treiber and Kesting
(2013).
vnt+τHV=
FGipps
n,ln(t)vn(t), vln(t)(t), sn,ln(t)(t),
if mod (t, C)∈[0, G],
min nFGipps
n,ln(t)vn(t), vln(t)(t), sn,ln(t)(t), F Gipps
n,ln(t)vn(t),0, L +sHV
0−xn(t)o,
otherwise.
(17)
10
FGipps
n,ln(t)vn(t), vln(t)(t), sn,ln(t)(t)= min vn(t) + aτHV, v, vsafe
HV vln(t)(t), sn,ln(t)(t),(18)
where vsafe
HV vln(t)(t), sn,ln(t)(t)=aτ HV +q(aτHV)2+vln(t)(t)2+ 2a×sn,ln(t)(t)−sHV
0is
the safe speed of an HV. Further, set vln(t)=vand xln(t)= +∞,∀n0∈ N 0to make
Equation (18) compatible with a leading vehicle.
(2) HV lane changing model
The HV lane changing model is described in Treiber and Kesting (2013) involving safety
and incentive checks. The safety checks require two components: safety check 1 with respect
to the preceding vehicle on the target lane (Equation (19)) and safety check 2 with respect
to the following vehicle on the target lane (Equation (20)).
sn,ˆ
ln(t)(t)> sHV
0+τHVvn(t) + max (0,vˆ
ln(t)(t)2−vn(t)2
2a),(19)
sˆ
fn(t),n(t)> sHV
0+τHVvˆ
fn(t)(t) + max n0,vn(t)2−vˆ
fn(t)(t)2
2ao,if ˆ
fn(t)∈ N HV ,
sˆ
fn(t),n(t)> sCAV
0+τCAVvˆ
fn(t)(t) + max n0,vn(t)2−vˆ
fn(t)(t)2
2ao,otherwise,(20)
ˆ
fn(t)is the nearest following vehicle of vehicle nin the target lane at time t. To make
Equation (20) compatible to the last vehicle in a traffic stream, let vˆ
fN(t)= 0 and xˆ
fN(t)=
−∞. Safety check 1 (i.e., Equation (19)) passes when the spacing gap between vehicles n
and ˆ
ln(t)is greater than the safety spacing, and safety check 2 (i.e., Equation (20)) passes
when the spacing gap between vehicles nand ˆ
fn(t)is greater than the safety spacing.
The incentive check is formulated as
ean,ˆ
ln(t)(t)−ean,ln(t)(t)>∆a, (21)
where ˆ
ln(t)is the preceding vehicle of vehicle nin the target lane at time t,ean,ln(t)(t) =
FGipps
n,ln(t)(·)is the acceleration of vehicle nfollowing its current preceding vehicle ln(t). And
ean,ˆ
ln(t)(t) = FGipps
n,ˆ
ln(t)(·)is the acceleration of vehicle nfollowing its target preceding vehicle
ˆ
ln(t). Incentive check passes when ean,ˆ
ln(t)(t)is great than ean,ln(t)(t)by a changing threshold
of ∆a. This makes sure that an HV changes its lane only when there exists a significant
mobility advantage, i.e., the increases of acceleration after the LC greater than ∆a.
Then, we consider two types of LCs: 1) discretionary LCs that are optional for a vehicle
to improve its mobility and is not on the critical path for the vehicle to reach its destination,
and 2) mandatory LCs that are required for a vehicle to reach its desired destination (i.e.,
e
On). A conditional changing threshold ∆ais used to represent these two types of LCs.
Before entering LM, i.e., xn(t)∈0, LM, only the discretionary LCs are considered
and ∆ais set as a small constant value (e.g., 0.1m
/s2in Treiber and Kesting (2013)). The
discretionary LCs will occur when both safety and incentive checks pass. Between the
locations LMand Llimit, i.e., xn(t)∈LM, Llimit,∆ais formulated as follows,
11
∆a=
+∞,if on(t)∈e
Onand o∗
n(t)/∈e
On,
−∞,if on(t)/∈e
Onand o∗
n(t)∈e
On,
0.1,otherwise.
(22)
When the current lane is in the desired lane set and the target lane is not in the desired
lane set (i.e., ∆a= +∞), Equation (21) will not pass and thus neither mandatory nor
discretionary LCs will not occur. When the current lane is not in the desired lane set and
the target lane is in the desired lane set (i.e., ∆a=−∞ ), Equation (21) will always pass and
thus mandatory LCs are considered by only checking Equations (19) and (20). Otherwise,
the discretionary LCs are considered by checking both incentive and safety checks.
For simplicity, we assume that a LC completes immediately once it starts. Further, if a
vehicle cannot change lane before Llimit, the vehicle will pass the intersection in the current
lane and try to make LCs after passing the intersection.
(3) CAV car following model
Although the proposed TO will smooth trajectories, it does not guarantee that the
planned trajectory is safe. For example, in Figure 1, CAV n’s planned trajectory will
intersect with HV m’s trajectory when HV mcuts in front of the CAV n. Therefore, there
is a need for a safety feature to ensure the avoidance of collision. The safety feature is
considered to follow a CAV car following model (e.g., Gipps’ car following model with sCAV
0
and τCAV).
vnt+τCAV=
FCAV
n,ln(t)vn(t), vln(t)(t), sn,ln(t)(t),
if mod (t, C)∈[0, G],
min nFCAV
n,ln(t)vn(t), vln(t)(t), sn,ln(t)(t), F CAV
n,ln(t)vn(t),0, L +sCAV
0−xn(t)o,
otherwise.
(23)
FCAV
n,ln(t)vn(t), vln(t)(t), sn,ln(t)(t)= min vn(t) + aτCAV, v, vsafe
CAV vln(t)(t), sn,ln(t)(t),(24)
where vsafe
CAV vln(t)(t), sn,ln(t)(t)=aτ CAV +q(aτ CAV)2+vln(t)(t)2+ 2a×sn,ln(t)(t)−sCAV
0
is the safety speed of a CAV. Also, the CAV car following model can be applied in a no
control scenario without any trajectory optimization.
As soon as the planned trajectory becomes safe again, the CAV will switch back to follow
the planned trajectory. See Figure 1, CAV nwill first follow a planned trajectory generated
from TO. However, due to the cut-in LC of HV m∈ N HV, CAV nwill abandon the planned
trajectory and then control its trajectory with the CAV car following model to ensure safety.
Although it ensures that two consecutive vehicles do not collide, the planned trajectory of
CAV nis affected by HV m’s LCs and thus may weaken the expected system performance
of TO with extra “stop-and-go” traffic.
12
3. Lane-Change-Aware Trajectory Optimization Model
To address the above problem existing in TO, this section proposes a decentralized
trajectory optimization model with a lane-change-aware mechanism (LCTO) to replace TO
in the above trajectory planning module. A discrete model is also formulated to find an
exact optimal solution and to make LCTO implementable in numerical experiments.
3.1 Continuous model
The continuous LCTO is formulated based on the continuous TO considering additional
constraints to restrain HV cut-in LCs. For each CAV n∈ N CAV,
CLCTO: min
Xn
Jn(Xn):=ωmt+
n−t−
n+ωaZt+
n
t−
n
an(t)2dt, (25)
subject to the same constraints in the continuous TO (i.e., Equations (7) - (16)) and extra
•LC restraining constraints:
1−βm(t)(t)×xm(t)(t)−xn(t)−sCAV
0+lveh−τCAVvn(t)≤0,∀t∈t−
n, t+
n,
(26)
where βm(t)(t)is an indicator whether HV m(t) = ˆ
ln(t)∈ N HV has a potential to make LCs
at time t∈t−
n, t+
nand is formulated as follows.
βm(t)(t) =
0,if eam(t),ˆ
lm(t)(t)(t)−eam,lm(t)(t)(t)>∆a
or sm(t),ˆ
lm(t)(t)(t)> sHV
0+τHVvm(t)(t),
1,otherwise.
(27)
When xn(t)∈0, Llimit, we have
∆a=
+∞,if on(t)∈e
Onand o∗
n(t)/∈e
On
and xn[i]≥LM,
−∞,if on(t)/∈e
Onand o∗
n(t)∈e
On
and xn[i]≥LM,
0.1,otherwise.
(28)
If βm(t)(t) = 1, HV m(t)does not have a potential to make LCs and Equation (26)
will always be 0. For the discretionary LC restraining, βm(t)(t)=1indicating that either
safety check 1 (Equation (19)) or incentive check (Equation (21)) does not pass. For the
mandatory LC restraining, βm(t)(t)=1indicating that safety check 1 (Equation (19)) does
not pass. If βm(t)(t) = 0, HV m(t)has a potential to make LCs and Equation (26) will only
be determined by the safety check 2 (i.e., Equation (20)).
Because m(t)is time-variant, Equations (26) - (28) are nonlinear. To linearize LC re-
straining constraints, Pn(t):=m0|xn(t)≤xm0(t)≤xln(t)(t),∀om0(t)6=on(t),∀n∈ N CAV, is
defined as a preceding vehicle set that includes vehicles in adjacent lanes located between
the preceding vehicle ln(t)and CAV nat time t.
13
Remark 1: If the optimal trajectory xTO
n(t)of CAV nat time t∈t−
n, t+
nin TO and the
optimal solution x∗
n(t)in LCTO exist, then x∗
n(t)≥xTO
n(t).
Proof: Without LC restraining constraints, see Figure 3, TO has the optimal solution
xTO
n(t),∀t∈t−
n, t+
n(i.e., the black-dotted curve). With LC restraining constraints,
if there is no potential for HVs on the adjacent lane to make LCs in front of CAV n,
then x∗
n(t) = xTO
n(t)(i.e., black-solid curve); otherwise, the trajectory of CAV nshould
be closer to HV m0(i.e., the grey-dashed curve in Figure 3 (b)) to restrain LCs, i.e.,
x∗
n(t)> xTO
n(t). Thus, x∗
n(t)≥xTO
n(t). This completes the proof.
(a) Without cut-in LCs. (b) With cut-in LCs.
Figure 3: Illustrations of remark 1.
Then, Pn(t):=nm0|xTO
n(t)≤xm0(t)≤xln(t−
n)(t),∀om0(t)6=on(t)oand we can find the first
vehicle that has potential to make LCs in Pn(t), i.e., pn(t) = argmaxk∈Pn(t)(xk(t)|βk(t) = 0).
Remark 2: If LCs of vehicle pn(t)are restrained, then the other vehicles in Pn(t)cannot
change lanes in front of CAV n.
Proof: Let p0
n(t)be the nearest following vehicle of vehicle pn(t), then xpn(t)(t)−xp0
n(t)(t)>
lveh +sHV
0+τHVvp0
n(t)(t). If vehicle pn(t)cannot make LCs in front of CAV n, we have
xpn(t)(t)−xn(t)≤lveh +sCAV
0+τCAVvn(t). Combined these two equations, we have
xp0
n(t)(t)< xpn(t)(t)−lveh +sHV
0+τHVvp0
n(t)(t)≤xn(t) + lveh +sCAV
0+τCAVvn(t)−
lveh +sHV
0+τHVvp0
n(t)(t), i.e., xp0
n(t)(t)−xn(t)< lveh +sCAV
0+τCAVvn(t). Thus, this
completes the proof.
14
With Remarks 1 and 2, we only need to consider vehicle pn(t)at time tin Equations
(26) and (27), and to reformulated them as follows,
1−βpn(t)(t)×xpn(t)(t)−xn(t)−sCAV
0+lveh−τCAVvn(t)≤0,∀t∈t−
n, t+
n.(29)
βpn(t)(t) =
0,if eapn(t),ˆ
lpn(t)(t)(t)−eapn(t),lpn(t)(t)(t)>∆a
or spn(t),ˆ
lpn(t)(t)(t)> sHV
0+τHVvpn(t)(t),
1,otherwise.
(30)
where ∆afollows Equation (28). Because pn(t)is known at any time t,βpn(t)(t)and xpn(t)(t)
are known and thus Equation (29) is linear.
Figure 4: Flow chart of the mixed traffic LCTO framework at a signalized intersection with multi-lane roads.
15
Figure 4 plots the flow chart of the mixed traffic LCTO framework at a signalized in-
tersection with multi-lane roads. The motions of downstream vehicles are rarely affected
by upstream traffic, and thus, the sub-problems will be solved sequentially from the lead
CAV to the last CAV. In each sub-problem, LCTO will be activated when CAV marrives
the control zone. The locations, speeds and occupied lanes of preceding vehicles are known
(CAVs follow the planned trajectories generated in LCTO, and HVs follow the HV car fol-
lowing model in Equations (17) and (18) and the lane changing model in Equations (19) -
(21). Then, we can estimate the departure time of vehicle m, and construct LC restraining
constraints with the solution from TO and obtain CAV m’s trajectory by solving LCTO.
Note that LCTO only works within the approaching road segment. Once a CAV passes the
intersection, it follows the CAV car following model (i.e., Equations (23) and (24)).
3.2 Discrete model
Since the continuous LCTO model is difficult to be solved as the continuous time points
where decision variables dwell lead to infinite dimensionality, this section reformulates the
continuous LCTO as a discrete model to find the exact optimal solutions in numerical
experiments.
Let I:= [0,1,2, . . . , I ]denote the set of discrete-time points with a discrete-time interval
∆tand a maximum discrete-time I=dT/∆te. Let i∈ I denote the index of a discrete-
time point. Let i−
n:= t−
n/∆tdenote the discretized arrival time of vehicle n∈ N . Let
i+
n:= t+
n/∆tdenote the discretized departure time of vehicle n. Let xn[i],vn[i]and an[i]
denote the location, speed and acceleration of vehicle nat discrete-time point i, respectively.
We also have xn[i] = xn[i−
n] + Pi
j=i−
n+1 vn[j]∆t, where xn[i−
n] = v−
n×(i−
n∆t−t−
n)and
i+
n=lt+
n/∆tm. Let VI:=VI
ndenote the set of discretized speeds, where VI
n={vn[i]}i∈I is
a sequence of vehicle n’s discretized speeds.
For CAV n∈ N CAV, the discrete LCTO is then formulated as follows,
DLCTO: min
VI
n
JnVI
n:=ωmL−xnhi+
ni+ωa
i+
n
X
i=i−
nvn[i+ 1] −vn[i]
∆t2
∆t, (31)
where xnhi+
ni=xn[i−
n] + Pi+
n
j=i−
n+1 vn[j]∆tand i+
n=t+
n/∆t. If L−xnhi+
nibecomes smaller,
the departure time t+
nwill be closer to the i+
nand thus it will decrease the travel time of
CAV n. Further, the discrete LCTO is subject to
•Arrival constraint:
vn[i−
n] = v−
n.(32)
•Departure constraints:
L≤xni+
n=xn[i−
n] +
i+
n
X
j=i−
n+1
vn[j]∆t≤L+v×i+
n∆t−t+
n.(33)
16
xnhi+
ni=xn[i−
n] +
i+
n
X
j=i−
n+1
vn[j]∆t≤L, (34)
•Speed constraints:
0≤vn[i]≤v, ∀i∈i−
n, i+
n.(35)
•Acceleration constraints:
a≤an[i] = vn[i+ 1] −vn[i]
∆t≤a, ∀i∈i−
n, i+
n.(36)
•Safety constraints:
xln[i−
n][i]−xn[i]
=xln[i−
n][i]−
xn[i−
n] +
i
X
j=i−
n+1
vn[j]∆t
≥sCAV
0+lveh +vn[i]τCAV,∀i∈i−
n, i+
n,(37)
•LC restraining constraints:
1−βpn[i][i]×xpn[i][i]−xn[i]−sCAV
0+lveh−τCAVvn[i]≤0,∀i∈i−
n, i+
n.(38)
βpn[i][i] =
0,if eapn[i],ˆ
lpn[i][i][i]−eapn[i],lpn[i][i][i]>∆a
or spn[i],ˆ
lpn[i][i][i]> sHV
0+τHVvpn[i][i],
1,otherwise.
(39)
When xn[i]∈0, Llimit, we have
∆a=
+∞,if on[i]∈e
Onand o∗
n[i]/∈e
On
and xn[i]≥LM,
−∞,if on[i]/∈e
Onand o∗
n[i]∈e
On
and xn[i]≥LM,
0.1,otherwise.
(40)
Equation (33) then is reformulated as a general linear function
A0[i] + A1[i]xn[i−
n] + A1[i]∆t
i
X
j=i−
n+1
vn[j] + A2[i]vn[i]≤0,∀i∈i−
n,i+
n,(41)
where A0[i] = (1 −βpn[i][i]) ×(xpn[i][i]−(sCAV
0+lveh)),A1[i] = −(1 −βpn[i][i]), and A2[i] =
(1 −βpn[i][i]) ×(−τCAV).
17
Now, LCTO is formulated as a quadratic optimization problem with linear constraints.
Thus, we can easily solve it using commercial solvers (e.g., Gurobi) to find the optimal
solution in each sub-problem and plan all trajectories from the lead CAV to the last CAV
following the logic described in Figure 4.
To be compared with the discrete LCTO, a discrete TO is also formulated for CAV
n∈ N CAV,
DTO: min
VI
n
JnVI
n:=
N
X
n=1
ωmL−xnhi+
ni+ωa
i+
n
X
i=i−
nvn[i+ 1] −vn[i]
∆t2
∆t
,(42)
subject to Equations (32) - (37).
4. Numerical Experiments
This section conducts a set of numerical experiments to test the performance of TO
and LCTO. The numerical experiments are conducted using Matlab on a PC with 3.6
GHz Intel Core i7 CPU, 16 GB RAM. The parameter settings are described in Subsection
4.1. Subsection 4.2 analyzes the effects of CAV market penetration on TO and LCTO at a
signalized intersection with two-lane roads. Subsection 4.3 investigates the system sensitivity
performance of TO and LCTO with varying road segment lengths, signal cycle lengths, traffic
saturation rates and through-vehicle rates at a signalized intersection with two-lane roads.
Subsection 4.4 extends LCTO at a signalized intersection with three-or-more-lane roads.
4.1. Parameter settings
Parameters are set as following default values. The simulation time is T= 800 s and the
time interval is ∆t= 0.1 s. The road segment length is L= 500 m and the number of lanes is
K= 2. The number of vehicles in each lane is 150 and vehicles are evenly distributed across
lanes, and thus the total number of vehicles is N=N×K= 300 veh. The vehicle length
is lveh = 4 m. The speed limit is v= 16 m
/s. The maximum acceleration is a= 2 m
/s2and
the minimum acceleration is a=−2m
/s2. LCs are forbidden after location Llimit =L+v2
/2a,
and mandatory LCs occur after location LM=Llimit
/2. The minimum spacing and time gap
of a CAV are sCAV
0= 1 m and τCAV = 0.7 s respectively. The minimum spacing and time
gap of an HV are sHV
0= 4 m and τHV = 1 s respectively. The signal cycle length is C= 60 s
and the effective green length is G=C/2. The through-vehicle rate, left-vehicle rate and
right-vehicle rate are T= 0.6,L= 0.2and R= 0.2, respectively. The CAV MPR in lane 1
is M P R1= 50% and the CAV MPR in lane 2 is M P R2= 50%. To better reflect the reality,
we divide the traffic into two parts according to Sun et al. (2015): the queuing period (e.g.,
the first 2/3 vehicles in each lane) when “stop-and-go” traffic occurs seriously at high traffic
saturation rates; and the dissipation period (e.g., the last 1/3 vehicles in each lane) when
the “stop-and-go” traffic is dissipating gradually at low traffic saturation rates (set as 0.5 in
this study). In the queuing period, the traffic saturation rate in lane 1 is fs
1= 1 and the
traffic saturation rate in lane 2 is fs
2= 1. Further, the arrival times and arrival speeds are
18
generated from Equations (3) - (5). Figure 5 depicts an example of the arrival patterns with
the queuing and dissipation periods. Due to a higher traffic saturation rate in the queuing
period, the slope in the queuing period is smaller than that in the dissipation period; i.e.,
vehicles arrive with smaller time gaps in the queuing period than those in the dissipation
period.
Figure 5: An example of the arrival patterns.
In the numerical experiments, the following indicators are adopted to measure the system
performance.
•Riding comfort: The summation of squared acceleration is used to measure the system
riding comfort.
SA =PN
n=1 (an[i])2∆t
N,(43)
where SA is the average system squared acceleration. Lower value of the system squared
acceleration means less sharp acceleration, and thus yields more riding comfort.
•Travel time: The summation of all vehicles’ travel time during road segment Lis used
to measure the system travel time.
T T =PN
n=1(t+
n−t−
n)
N,(44)
where T T is the average system travel time. Lower value of the system travel time means
better traffic mobility.
•Fuel consumption: The summation of the instantaneous fuel consumption function
(i.e., the VT-micro model used in Ma et al. (2017)) is used to measure the system fuel
consumption.
F C =PN
n=1 exp nP3
j1=0 P3
j2=0 κj1j2(vn[i])j1(an[i])j2o∆t
N,(45)
19
where F C is the average system fuel consumption. j1and j2are the power indexes. κj1j2
is a constant coefficient. See Table 2 for the value of the coefficients (Zegeye et al., 2013).
Lower value of the system fuel consumption means better fuel efficiency.
Table 2: Coefficients for fuel consumption. Note that the unit of fuel consumption, speed and acceleration
are in l
/s,m
/s, and m
/s2, respectively.
κj1j2j2= 0 j2= 1 j2= 2 j2= 3
j1= 0 -7.537 0.4438 0.1716 -0.0420
j1= 1 0.0973 0.0518 0.0029 -0.0071
j1= 2 -0.0030 -7.42E-4 1.09E-4 1.16E-4
j1= 3 5.3E-5 6E-6 -1E-5 -6E-6
•Safety: The system safety is measured by a safety surrogate (i.e., the inverse time-to-
collision) described in Gettman and Head (1840) and Ma et al. (2017).
SS =PN
n=1 max n0,vn[i]−vln[i][i]
xln[i][i]−xn[i]−lveh o∆t
N,(46)
where SS is the average system safety surrogate. Equation (46) is compatible with leading
vehicles because of vln0[i]=vand xln0[i]=∞,∀n0∈ N 0. Lower value of the system safety
surrogate means less potential of collision, and thus yields safer riding experience.
Further, four scenarios are conducted in the numerical experiments for comparison.
1) CF (benchmark): HVs follow Gipps’ car following and lane changing models; CAVs
follow the CAV car following model.
2) TO: HVs follow Gipps’ car following and lane changing models; CAVs follow the
planned trajectories generated in TO model and the CAV car following model.
3) LCTO: HVs follow Gipps’ car following and lane changing models; CAVs follow the
planned trajectories generated in LCTO model and the CAV car following model.
4) TOWOLC: HVs follow Gipps’ car following model without considering the lane chang-
ing behavior; CAVs follow the planned trajectories generated in TO model.
To select suitable weights of mobility and squared acceleration, we compare LCTO with
CF by setting the weight of squared acceleration ωa= 1 and varying the weight of mobility
ωm∈[0,2] with an increment of 0.1. The result is shown in Figure 6. The black-solid curve
depicts the benefit of the system riding comfort in LCTO. The red-dashed curve depicts the
benefit of the system travel time in LCTO. We find that the benefit of the system riding
comfort decreases in a small range (from 94% to 91%) as the weight of mobility grows, and it
becomes stable when ωm≥0.5. Additionally, the benefit of the system travel time increases
as the weight of mobility grows. When ωm≤0.2, LCTO focuses more on the system riding
comfort and it may sacrifice the system travel time to increase the system riding comfort.
Thus, the benefit of the system travel time is negative. We also find that the benefit of the
system travel time becomes stable when ωm≥1. Therefore, we select ωm= 1 to ensure
20
positive and stable benefits of the system riding comfort and travel time. After the selection
of weights, we will conduct a set of numerical experiments to investigate the performance of
TO and LCTO.
Figure 6: Selection of weights.
4.2. CAV market penetration analysis
This subsection investigates CAV market penetration impacts on TO and LCTO at a
signalized intersection with two-lane roads. We assume that M P R =MP R1=M P R2and
MPR is set to vary from 10% to 100% with an increment of 10%. The other parameters are
fixed at their default values, respectively.
Figures 7 and 8 plot examples of trajectories in TO and LCTO respectively. When the
CAV MPR is low, see Figure 7 (a) and (b), a number of HVs change lanes in front of CAVs
due to the large gaps induced by TO. When the CAV MPR grows to a medium level, see
Figure 7 (c) and (d), only a few HVs change lanes in front of CAVs. When the CAV MPR
grows to a high level, see Figure 7 (e) and (f), almost all vehicles are CAVs and there is less
probability for HVs to make LCs. We also find these LCs might cause extra “stop-and-go”
traffic and “spillback” traffic. Further, in Figure 8, we find less LCs and less “stop-and-go”
traffic in LCTO compared with those in TO. Therefore, we can say that LCTO generates
smoother trajectories by restraining HV LCs than TO.
Then, we plot the benefits of the system performance with varying CAV MPRs in Figure
9. The system joint objective, LC frequency per vehicle, the system riding comfort, the
system travel time, the system fuel consumption and the system safety are considered. The
gray-dotted curve denotes the benefits of the system performance in TOWOLC compared
with those in CF. The black-solid curve denotes the benefits of the system performance in
TO compared with those in CF. The red-dashed curve denotes the benefits of the system
performance in LCTO compared with those in CF. The yellow bars denote the extra benefits
of the system performance in LCTO compared with those in TO. The black and gray bars
denote the LC frequencies per vehicle in TO and LCTO, respectively.
21
(a) Lane 1 with 20% MPR. (b) Lane 2 with 20% MPR.
(c) Lane 1 with 50% MPR. (d) Lane 2 with 50% MPR.
(e) Lane 1 with 80% MPR. (f) Lane 2 with 80% MPR.
Figure 7: Examples of trajectory results in TO.
22
(a) Lane 1 with 20% MPR. (b) Lane 2 with 20% MPR.
(c) Lane 1 with 50% MPR. (d) Lane 2 with 50% MPR.
(e) Lane 1 with 80% MPR. (f) Lane 2 with 80% MPR.
Figure 8: Examples of trajectory results in LCTO.
23
(a) Joint objective. (b) LC frequency per vehicle.
(c) Riding comfort. (d) Travel time.
(e) Fuel consumption. (f) Safety.
Figure 9: Benefits of the system performance in TO and LCTO with varying CAV MPRs.
See Figure 9 (a), we find that the benefits of the system joint objective in both TO and
LCTO increase with the increasing CAV MPRs. The benefits are about 30% even at a low
CAV MPR (e.g., 10%), and reach to 90% after the CAV MPR exceeds 80%. The benefits
of TO and LCTO are relatively low due to the predominance of HVs in mixed traffic when
the CAV MPR is low. From Figure 9 (b), we find that the LC frequencies per vehicle in
both TO and LCTO are higher at a low CAV MPR than those at a high CAV MPR. This
is consistent with Yang et al. (2017), because there are less gaps for HVs to change lanes
when more CAVs follow the planned trajectories at a higher CAV MPR. Additionally, note
24
that TOWOLC provides the upper bound of the expected benefits of trajectory smoothing
without considering LCs. The expected benefits in TOWOLC could be significant when the
CAV MPR is relatively low (e.g., 50% in the system joint objective, riding comfort, and
safety, 30% in the system fuel consumption, as well as 10% in the system travel time when
the CAV MPR is only 10%). However, the presence of LCs will significantly impair the
expected benefits. Comparing TOWOLC with TO, we see that the expected benefits are
decreased half or more due to actual HV LCs. Therefore, LCTO is necessary to restrain the
HV LCs and improve the system performance.
Compared LCTO with TO, as shown in Figure 9 (a), the extra benefit of the system
objective in LCTO increases first at low CAV MPRs and then decreases at high CAV MPRs.
The LC frequencies per vehicle in Figure 9 (b) explain this observation: 1) Although more
LCs occur at a low CAV MPR, the effects of both TO and LCTO are weak due to the
predominance of HVs and thus LCTO yields less extra benefits; 2) When the CAV MPR
grows, there still exist more LCs occurred in TO than in LCTO, and both TO and LCTO
yield significant benefits, and thus LCTO yields more extra benefits by restraining these
LCs; 3) When the CAV MPR further grows, more CAVs follow the planned trajectories
with less gaps, and less or no LCs are induced by TO, and thus LCTO yields less extra
benefits.
Additionally, Figure 9 (c) - (f) show that the benefits of the system riding comfort, travel
time, fuel consumption and safety in TO and LCTO increase as the CAV MPR increases,
and the extra benefits of the system riding comfort, travel time, fuel consumption and
safety in LCTO show similar trends as the system joint objective. Because the system joint
objective integrates the system riding comfort and mobility (i.e., travel time surrogate), thus
the benefits of the system riding comfort and travel time increase accordingly. However, the
benefit of the system travel time is negative when the CAV MPR is lower than 40%. This
is because that HVs change lanes with a purpose of acquiring mobility advantages, and thus
the system travel time in CF might be smaller than that in TO/LCTO when the CAV
MPR is low (Li et al., 2020). Also, the benefits of the system travel time in both TO and
LCTO are a little higher at 10% CAV MPR than those at 20% CAV MPR. This is probably
because the benefits of both TO and LCTO are relatively insignificant at a extremely low
CAV MPR and HVs change lanes at a purpose of obtaining mobility advantage, and thus
the system travel time is higher at a lower CAV MPR. When the CAV MPR is high, there
are more CAVs in the system and they will follow the planned trajectories to improve the
system travel time. Also, we find that the system fuel consumption and safety are related
to accelerations in Equation (45). Thus, the system fuel consumption and safety are also
improved by optimizing the system joint objective. Note that the extra benefit of the system
fuel consumption in LCTO is marginal. Because the system fuel consumption is not only
related to accelerations but also to speeds, and thus we might need to use a fuel consumption
model instead of the squared acceleration in the objective function for yielding more benefits
in the system fuel consumption.
Further, the solution time per vehicle in TO is less than 1 second, and LCTO costs twice
or more than TO. This is because that TO is nested in LCTO with Equations (38) - (41).
Overall, TO and LCTO are able to be implemented in real-time applications with an edge
25
computing framework that distributes the computation to each individual vehicle on board
computers. In implementation, it is expected that LCTO yields more benefits in the system
joint objective, riding comfort, travel time, fuel consumption and safety when the CAV MPR
is lower than 70%. TO is suggested to be used for a higher computational efficiency with a
similar system performance in LCTO when the CAV MPR is higher than 70%.
4.3. Sensitivity analysis on other parameters
In this subsection, the system sensitivity analyses of TO and LCTO are investigated
with varying parameters. Figures 10 - 13 plot the benefits of the system joint objective in
TO and LCTO with varying road segment lengths, signal cycle lengths, traffic saturation
rates and through-vehicle rates, respectively. The black-solid curve denotes the benefit of
TO compared with CF, the red-dashed curve denotes the benefit of LCTO compared with
CF. The yellow bars show the extra benefits of LCTO compared with TO. The black and
gray bars show the LC frequencies per vehicle in TO and LCTO, respectively.
1) Road segment length
We vary the road segment length range between 300 m and 1100 m with an increment
of 200 m while fixing the other parameters at their default values. Compared with CF,
as shown in Figure 10 (a), the benefit of TO first grows with the increasing road segment
length, and then drops a little as the road segment length further increases. From Figure
10 (b), we find that the reason is probably that there is more “stop-and-go” traffic when
the road segment is shorter, and the LC frequency increases with more room when the road
segment length increases from a relatively short initial length. With a further increase of
the road segment length, the “stop-and-go” traffic and the LC frequency per vehicle are
less, and thus TO yields less benefits. Compared with TO, the extra benefit of LCTO first
increases when the road segment length increases at low road segment lengths and then
decreases with the further increasing road segment length. This is because that there exist
less LCs in TO when the road segment length is extremely short, and LCTO yields less extra
benefits. As the road segment length further increases, there are less “stop-and-go” traffic
and less LCs. Thus, LCTO has a similar performance as TO when the road segment length
is long. Overall, LCTO is better than TO when the road segment length is not relatively
long (10-20% extra benefits).
26
(a) Joint objective. (b) LC frequency per vehicle.
Figure 10: Benefits of the system performance in TO and LCTO with varying road segment lengths.
2) Signal cycle length
We vary the signal cycle length between 40 s and 120 s with an increment of 20 s while
fixing the other parameters at their default values. Compared with CF, as shown in Figure
11 (a), the benefit of the system joint objective in TO grows first at low signal cycle lengths
and then drops as the signal cycle length further increases. This is because a longer signal
cycle length means that more vehicles can pass the intersection per signal cycle and yield
less “stop-and-go” traffic. This is consistent with the findings in Xia et al. (2013), and thus
TO has less potential to improve the system performance. Further, when the signal cycle
length is short, the speed variation is smaller and thus the benefit in TO is less. Compared
with TO, the extra benefit of LCTO increases at low signal cycle lengths, and then decreases
with the further increasing signal cycle lengths. From Figure 11 (b), we find that the LC
frequencies per vehicle decrease as the signal cycle length increases. This is probably because
“stop-and-go” traffic occurs more at a lower signal cycle length (i.e., vehicles may need to
wait for several signal cycles in the queue before passing the intersection), and more HVs
change lanes to avoid slower preceding vehicles. By restraining LCs induced by TO, LCTO
increases the extra benefits at low signal cycle lengths. Then, with the further increasing
signal cycle length, there are less “stop-and-go” traffic and less LCs induced by TO, and thus
the extra benefit of LCTO decreases. Overall, the results suggest to use LCTO in different
signal cycle lengths to yield extra benefits (10-25%).
27
(a) Joint objective. (b) LC frequency per vehicle.
Figure 11: Benefits of the system performance in TO and LCTO with varying road segment lengths.
3) Traffic saturation rate
We vary the traffic saturation rate in the queuing period between 0.6 and 1.6 with an
increment of 0.2 and the traffic saturation rate in the dissipation period remains 0.5. The
other parameters are fixed at their default values. Compared with CF, as shown in Figure
12 (a), the benefit of TO first grows as the traffic saturation rate initially increases, and
then drops a little at higher traffic saturation rates. Because there is less “stop-and-go”
traffic and TO yields less benefits when the traffic saturation rate is lower. When the traffic
saturation rate is higher, there is less room for TO to regulate LCs. Compared with TO,
the extra benefit of LCTO increases at low traffic saturation rates, and then decreases a
little at higher traffic saturation rates. From Figure 12 (b), we find that the LC frequency
per vehicle is higher due to more room in both TO and LCTO when the traffic saturation
rate is lower. As the traffic saturation rate increases, LCTO yields more extra benefits by
restraining the LCs induced by TO. In implementation, LCTO is expected to yield extra
benefits (10-25%) across all traffic saturation rates.
(a) Joint objective. (b) LC frequency per vehicle.
Figure 12: Benefits of the system performance in TO and LCTO with varying traffic saturation rates.
4) Through-vehicle rate
28
We vary the through-vehicle rate range between 0 and 1 with an increment of 0.2 while
fixing the other parameters at their default values. We also assume L=R=(1−T)/2.
Compared with CF, as shown in Figure 13 (a), the benefit of TO drops with the increasing
through-vehicle rates. Because turn vehicles will not change lanes once they are in their
desired lanes in the mandatory LC zone, but through vehicles will change lanes with the
discretionary lane changing behavior before approaching Llimit. Thus, the LC frequency
per vehicle grows as the through-vehicle rate increases, as shown in Figure 13 (b). And
the benefit of TO decreases due to more LCs. Compared with TO, the extra benefit of
LCTO increases at low through-vehicle rates, and then decreases a little with the further
increasing through-vehicle rate. Because LCTO restrains more LCs induced by TO to yield
more benefits when the through-vehicle rate is low. When the through-vehicle rate increases
to a high level, a number of LCs that are not iduced by TO exist in both TO and LCTO,
and thus the extra benefit of LCTO decreases a little. Only considering the through-vehicle
rate, LCTO is suggested to be applied when the through-vehicle rate is not extremely low
(10-30% extra benefits).
(a) Joint objective. (b) LC frequency per vehicle.
Figure 13: Benefits of the system performance in TO and LCTO with varying through vehicle rates.
In summary, the above results suggest that using LCTO to restrain LCs induced by TO
will yield extra benefits in most traffic scenarios except those when the road segment length
is relatively long and the through-vehicle rate is relatively low.
4.4. Extension with three-or-more-lane roads
From the above analyses, we find that LCTO yields more benefits than TO at a signalized
intersection with two-lane roads. To generalize LCTO at a signalized intersection with
three-or-more-lane roads, we extend the lane changing model and conduct another set of
experiments.
Assume a multi-lane road with Klanes, as shown in Figure 14. HV nis the subject
vehicle, and vehicle lnis HV n’s preceding vehicle in the same lane. ˆ
lR
nand ˆ
lL
ndenote HV
n’s preceding vehicles in right and left adjacent lanes, respectively. ˆ
fR
nand ˆ
fL
ndenote HV
n’s following vehicles in right and left adjacent lanes, respectively. Assign lane 1 to left-turn
29
vehicles only and lane Kto right-turn vehicles only. Thus, the desired lane set e
On(en)of
vehicle nis formulated as follows,
e
On(en) =
{2, . . . , K},if en= 0;
{1},if en= 1;
{K},otherwise.
(47)
Thus, the vehicles in the leftmost and rightmost lanes follow the Equations (21) - (20).
The other vehicles in the middle lanes need to consider both adjacent lanes. Then, we need
to find the target lane that HV ndesires to change lane to. For discretionary LC vehicles
(i.e., eon= 0), the incentive check (i.e., Equation (21)) is reformulated as follows,
max nean,ˆ
lR
n(t)(t),ean,ˆ
lL
n(t)(t)o−ean,ln(t)(t)>∆a. (48)
The preceding vehicle ˆ
l∗
n(t)in the target lane o∗
n(t)is obtained by
ˆ
l∗
n(t) = (ˆ
lR
n(t),if ean,ˆ
lR
n(t)(t)>ean,ˆ
lL
n(t)(t),
ˆ
lL
n(t),otherwise.(49)
o∗
n(t) = (min {on(t)+1, K},if ean,ˆ
lR
n(t)(t)>ean,ˆ
lL
n(t)(t),
max {on(t)−1,1},otherwise.
Then, the target lane ˆ
l∗
n(t)is used to replace ˆ
ln(t)in safety checks (i.e., Equations (19)
and (20)).
Figure 14: An Illustration of LCs in a multi-lane road.
30
(a) Lane 1 in TO. (b) Lane 1 in LCTO.
(c) Lane 2 in TO. (d) Lane 2 in LCTO.
(e) Lane 3 in TO. (f) Lane 3 in LCTO.
Figure 15: Examples of trajectory results in TO and LCTO at a signalized intersection with three-lane
roads.
31
With the above equations, we can study the lane changing behavior at a signalized
intersection with three-or-more-lane roads. We will study a signalized intersection with 3,
4, and 5 lanes under different CAV MPRs, respectively. M P Rkin each lane is set from 10%
to 100% with an increment of 10%, and these values in all lanes will vary simultaneously.
The other parameters are set at their default values as in section 4.2.
Figure 15 shows examples of trajectories in TO and LCTO at a signalized intersection
with three-lane roads under 50% CAV MPR. We find that a number of HV LCs occur in
TO, and thus it causes “stop-and-go” traffic and “spillback” traffic. There, however, are less
LCs in LCTO, and it causes less “stop-and-go” traffic and no “spillback” traffic. That verifies
the effectiveness of LCTO at signalized intersections with three-lane roads.
Further, Figure 16 plots the benefits of LCTO in the system joint objective with vary-
ing CAV MPRs at a signalized intersection with three-lane, four-lane and five lane roads,
respectively. The black-solid curve, the red-dashed curve, the blue-dash-dotted curve, and
the gray-dotted curve denote the benefits with 2-lane, 3-lane, 4-lane, and 5-lane settings,
respectively. Figure 16 (a) shows the benefits of the system joint objective in LCTO com-
pared with those in CF. Since more LCs occur as the lane number increases, the benefit of
LCTO decreases as the lane number increases. From Figure 16 (b), we find that the extra
benefits in the three-or-more-lane experiments have a similar trend (i.e., increases initially
and then decreases with the increasing CAV MPR) as in the two-lane experiments. How-
ever, more extra benefits of LCTO are found with more lanes at high CAV MPRs. This is
probably because that there are more HVs and more opportunities for making LCs in the
experiments with more lanes. Overall, LCTO is capable to restrain LCs and improve the
system performance at a signalized intersection with three-or-more-lane roads.
(a) LCTO - CF. (b) LCTO-TO.
Figure 16: Benefits of the system performance in LCTO with varying CAV MPRs at a signalized intersection
with multi-lane roads.
5. Limitations and Discussions
From the above numerical experiment results, LCTO is capable to restrain HV LCs in-
duced by TO at a signalized intersection with multi-lane roads. However, the applicability of
LCTO is limited to certain scenarios (e.g., a limited lane-change-aware mechanism, isolated
32
intersections, fixed (or predictable) signal timing, and cyber-physical constraints). This sec-
tion summarizes these limitations and possible future directions to implement LCTO in real
world.
As shown in Figures 8 and 15, there exist some LCs in LCTO. Figure 17 plots examples
of three possible cases of LCs in LCTO. First, Case 1 shows a type of LCs that are not
induced by TO. In Case 1, the blue-dashed curves denote the trajectories of HVs in the
current lane (i.e., Vehicle 1, 2, and 3) and the black-solid curves denote the trajectories
of CAVs in the current lane (i.e., Vehicle 4). Before CAV 4 arrives the control zone (i.e.,
Location 0), HV 2 changes its lane due to a low speed of HV 1 and a large spacing gap in
the adjacent lane. Thus, CAV 4 does not have an opportunity to restrain this type of LCs.
And we find that this type of LCs occurs more frequently at a low traffic saturation rate,
as shown by those trajectories during the dissipation periods in Figure 8 and 15. Second,
Case 2 shows a second type of LCs that occur after a CAV. In Case 2, the blue-dashed
curves denote the trajectories of HVs (i.e., Vehicles 1, 3, and 4), and the black-solid curves
denote the trajectories of CAVs (i.e., Vehicle 2). When doing LCTO for CAV 2, HV 3 is
considered as a following vehicle of CAV 2 in the same lane. Consequently, HV 3 changes
its lane due to a low speed of CAV 2 and a large spacing gap in the adjacent lane, and
then changes its lane in front of CAV 2 due to its mandatory lane changing behavior. The
second type of LCs usually occurs at the beginning of a vehicle platoon. Finally, Case 3
shows a third type of LCs that are caused by the following CAVs in adjacent lanes. In Case
3, blue-dashed curves denote the trajectories of HVs in the current lane (i.e., Vehicle 1) and
black-solid curves denote the trajectories of CAVs in the current lane (i.e., Vehicle 2). The
light-blue-dashed curves denote the trajectories of HVs in the adjacent lane (i.e., Vehicle 11)
and the grey-solid curves denote the trajectories of CAVs in the adjacent lane (i.e., Vehicle
10). In LCTO, the predefined car following models are used to predict trajectories of other
vehicles. CAV 2 does not predict any LCs in adjacent lanes and thus plans its trajectory
with a smoother trajectory. After that, CAV 10 arrives and plans its trajectory with LCTO,
and then HV 11 changes lane in front of CAV 2 due to a low speed of CAV 10 and a large
spacing gap in the adjacent lane. The third type of LCs occurs more frequently in a low
traffic saturation rate, as shown by the trajectories during the dissipation periods in Figure
8 and 15. Although the above analyses imply that LCTO has a limited lane-change-aware
mechanism to restrain all LCs, the results in the numerical experiments show that LCTO
can still significantly improve the system performances by restraining most LCs. To address
this limitation, an effective way might be using the reinforcement learning methods (e.g.,
Q-learning and the policy gradient algorithms) to build this problem as a Markov decision
process in real-time control. The goal of the reinforcement learning methods is to learn a
policy that maximizes the expected return (e.g., throughput, fuel efficiency and safety) from
the current state.
33
(a) Case 1. (b) Case 2.
(c) Case 3.
Figure 17: Examples of three possible LC cases in LCTO.
Additionally, we believe that LCTO can be extended from an isolated signalized inter-
section to a more complex geometric with multiple signalized intersections (e.g., signalized
corridors). For controlling a signalized corridor, departed vehicles might need to be con-
trolled as well. In this case, depending on the distance between two adjacent intersections
and the road segment lengths, departed vehicles of one intersection might simply be seen
as arrival vehicles by another intersection. This way, the proposed algorithm would be
adaptable to handle a signalized corridor.
Further, this study assumes a fixed signal setting at intersections due to wide and simple
implementations of fixed signal settings in real world. To extend LCTO from a fixed signal
setting to an adaptive signal control (Feng et al., 2015), we can use a bi-level collaborative
34
control framework that focuses the high-level signal control/optimization on a central con-
troller while distributing complex low-level trajectory control to individual CAVs. This way,
the extend collaborative control framework can further improve the system performance and
easily be implemented in real world.
Finally, there exist a number of cyber-physical constraints in real-world applications.
1) This study assumes all CAVs are in the same cooperation class (cooperation class B or
above defined by SAE J3216), however CAVs might have different cooperation classes in
real world due to different designs of manufactures and preferences of riders. Thus, a mixed
CAV cooperation class model needs to be extended from a mixed traffic with two types of
vehicles (e.g., CAVs in the same cooperation class and HVs) to a mixed traffic with five types
of vehicles (e.g., cooperation classes A-D CAVs and HVs). 2) This study assumes that all
CAVs will accept the planned trajectories. Note that cooperation class C equipped CAVs
have abilities to negotiate with other vehicles and accept/reject the planned trajectories.
Thus, we need to use the game theory to decide the best strategy for accepting/rejecting the
planned trajectories. 3) This study assumes that communication delays, sensing errors and
prediction errors are neglected. Actually each vehicle may need to frequently (e.g., every
20 milliseconds) adjust its direct drive-by-wire control variables (e.g., throttle and brake
levels and steering wheel angle) to ensure the actual vehicle trajectory can closely follow
the planned trajectory based on feedback controllers (e.g., the Model Predictive Control
[MPC] and the proportional-integral-derivative [PID] controllers). An objective measure of
the error will be proposed in the MPC (e.g., the weighted mean square errors of location
and speed). In the PID control, each control variable is a simple linear function of the
discrepancies of the status (e.g., locations and speeds) between the actual and the planned
trajectories. Then, in the real-time control, a series of control variables will be optimized to
minimize the expected objective error.
Overall, the proposed methodology in this study can be extended in number of ways
as briefly illustrated above to deal with more complex real-world applications. This study
plays an instrumental role and draws fundamental insights into studies focusing on trajectory
smoothing with a lane-change-aware mechanism along signalized intersections with multi-
lane roads.
6. Conclusion
This paper extends the existing CAV trajectory optimization model at a signalized in-
tersection with single-lane roads to a mixed traffic trajectory smoothing framework at a
signalized intersection with multi-lane roads. HV LCs occur frequently due to lager gaps
induced by CAV trajectory optimization and they may impair the expected system per-
formance of the CAV trajectory optimization. To address this issue, this paper proposes
a decentralized lane-change-aware CAV trajectory optimization model under the proposed
framework considering a joint objective of squared acceleration and mobility. Nonlinear lane
change restraining constraints are linearized based on the solution in TO. Then, a discrete
model is reformulated from the continuous model to find the exact optimal solution utilizing
commercial solvers (e.g., Gurobi). Numerical experiments first investigate the system per-
35
formance in CF, TO and LCTO at a signalized intersection with two-lane roads. The results
of CAV market penetration analyses show that the expected benefits of trajectory smooth-
ing without considering LCs are decreased half or more due to HV LCs in TO. The results
also show that LCTO yields extra benefits by restraining HV LCs induced by TO in the
system joint objective (about 10-30%), riding comfort (about 10-30%), travel time (about
2-5%), fuel consumption (about 2-10%) and safety (about 2-15%) when the CAV MPR is not
extremely high. Sensitivity analysis results considering road segment lengths, signal cycle
lengths, traffic saturation rates and through-vehicle rates show that LCTO has significant
effects on restraining LCs induced by TO without being aware of possible LCs, and thus
yields extra benefits in most traffic scenarios, such as 10-20% extra benefits when the road
segment length is not extremely long, 10-30% extra benefits when the signal cycle length
varies from 40 s to 120 s, 10%-25% extra benefits when the traffic saturation rate varies from
0.6 to 1.6, and 10-30% extra benefit when the through-vehicle rate is not extremely low. Fi-
nally, the extension experiments show that LCTO is also effective (similar performance as
in the 2-lane experiments) at a signalized intersection with three-or-more-lane roads.
This study reveals a significant potential of using a lane-change-aware mechanism in
CAV trajectory smoothing concepts at a signalized intersection with multi-lane roads in
transportation engineering applications. The study can be further extended in multiple
directions. 1) Although using the proposed lane-change-aware trajectory optimization can
improve the system performance, the lane-change-aware mechanism may not restrain all
redundant LCs. To yield further benefits by restraining more LCs in real-time control,
reinforcement learning methods might be helpful. 2) This study is limited at an isolated
signalized intersection. More complex geometric, such as corridors and networks, may be
investigated in the future. 3) This study considers a fixed signal setting. A cooperative
optimization of signal timing and CAV trajectories may be developed to further improve
the system performance. 4) In real-time applications, the proposed method might suffer from
certain cyber-physical constraints, such as mixed CAV cooperation classes, non-compliant
behavior due to riders’ preferences, communication delays, sensing errors, and prediction
errors due to stochastic human driving behavior. Feedback controllers (e.g., MPC and PID
controllers) could be developed to regulate CAV trajectories such that they will follow the
planned trajectories when approaching to intersections. Such control strategies may address
the implementation of trajectory smoothing concept in the real world and reduce the gaps
between the actual system performance and theoretical system performance.
CRediT Authorship Contribution Statement
Handong Yao: Conceptualization, Methodology, Software, Validation, Formal analysis,
Visualization, Writing – Original Draft. Xiaopeng Li: Supervision, Conceptualization,
Methodology, Formal analysis, Writing- Reviewing and Editing.
Acknowledgment
This research is supported by the U.S. National Science Foundation through Grants
CMMI #1558887 and CMMI #1932452.
36
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