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Computer-Aided Civil and Infrastructure Engineering 32 (2017) 412–428
Two-Dimensional Simulation of Turning Behavior in
Potential Conflict Area of Mixed-Flow Intersections
Zian Ma
Department of Traffic Engineering & Key Laboratory of Road and Traffic Engineering, Ministry of Education, Tongji
University, Shanghai, China and Jiangsu Province Collaborative Innovation Center of Modern Urban Traffic
Technologies, Nanjing, China
Jianbo Xie, Xiao Qi, Yiming Xu & Jian Sun*
Department of Traffic Engineering & Key Laboratory of Road and Traffic Engineering, Ministry of Education, Tongji
University, Shanghai, China
Abstract: The potential conflict area of intersection is
the space where conflicting traffic flows pass through in
the same signal phase. At this area, turning vehicles in-
teract with most traffic flows, which introduce complex
features including variation of trajectories and shared-
priority phenomenon. The traditional one-dimensional
simulation oversimplifies these features with lane-based
assumption. This study integrates the modified social
force model with behavior decision and movement con-
straints to reproduce the two-dimensional turning pro-
cess. The method is framed into a three-layered math-
ematical model. First, the decision layer dynamically
makes decision for turning patterns. Then the operation
layer uses the modified social force model to initially
generate vehicle movements. Finally, the constraint layer
modifies the vehicular motion with vehicle dynamics con-
straints, boundary of intersection and the collision avoid-
ance rule. The proposed model is validated using tra-
jectories of left-turn vehicles at a real-world mixed-flow
intersection with nonprotected signal phases, resulting in
a more realistic simulation than previous methods. The
distributions of decision points and travel time in simu-
lation are compared with the empirical data in statistics.
Moreover, the spatial distribution of simulated trajecto-
ries is also satisfactory.
1 INTRODUCTION
The potential conflict area of an intersection is defined
as the space where conflicting traffic flows go through
∗To whom correspondence should be addressed. E-mail: sunjian@
tongji.edu.cn.
the intersection in the same signal phase, represented
as shadow zones in Figure 1. Because the turning
vehicles interact with most traffic flows and are limited
by priority constraints, the characteristics of turning
traffic flow are complex in the potential conflict area.
Present studies have demonstrated three most signifi-
cant characteristics of turning flow. First and foremost,
trajectories of turning traffic are of great variation
(Alhajyaseen et al., 2013). Furthermore, turning traffic
flow shares the priority when interacting with other
flows (Troutbeck and Kako, 1999). At last, decision of
drivers is affected by other conflicting traffic flows (Sun
and Kondyli, 2010; Pascucci et al., 2015).
Two kinds of approaches are used to simulate the
turning behavior presently. The first approach models
car-following, lane-changing and crossing decision be-
havior of vehicles in their framework. The underlying
foundation of these approaches is lane-based assump-
tion. More specifically, they only calculate longitudinal
acceleration (or deceleration) of vehicles with limited
and discrete lane-changing action. Therefore, they
can be defined as a kind of one-dimensional models
as they fail to describe two-dimensional variation of
vehicle trajectories. The other kind of approach is
two-dimensional models, such as social force model.
These models can calculate speed and angular velocity
of vehicular movement simultaneously. However, they
are mostly used to simulate behavior of pedestrians and
nonmotorized vehicles, but are rarely used to model
motorized vehicles’ turning behaviors.
The challenges of applying two-dimensional model
on motorized vehicles lie in two aspects, the first
one is that vehicles’ turning decision and actions
C2017 Computer-Aided Civil and Infrastructure Engineering.
DOI: 10.1111/mice.12266
Two-dimensional simulation of turning behavior in potential conflict area of mixed-flow intersections 413
Fig. 1. Schematic map of potential conflict area at a mixed-flow intersection.
(Ma et al., 2017). Because there are no clear lane
markings within intersections, it is difficult to determine
the interacting objects, potential conflict points and
time gap between turning vehicles and straight-going
ones. Turning decision is hard to be modeled with-
out quantifying these indicators. Second, speed and
angular velocity of vehicles should be simultaneously
adjusted under constraints of vehicular dynamics, which
introduces another challenging problem.
To fill these gaps, a two-dimensional simulation
model is proposed to appropriately describe vehicles’
turning behaviors. The novel model modifies the classi-
cal social force model by better reflecting the interaction
among different traffic flows. Three main contributions
of this article are as follows. First, turning patterns
and decision model of turning vehicle are analyzed.
Typical turning process is classified into two patterns.
A logit model is used to determine the turning pattern
by identifying the critical time of implementing turning
movement. The second contribution is extending and
modifying types and expression of forces in the classical
social force model. The proposed model considers
the time headway or anticipated Post-Encroachment
Time (PET) between turning vehicles and interacting
participants. Both repulsive and attractive effects
among turning vehicles are developed. Finally, the
parameters of the novel model are calibrated with
empirical data. The model is evaluated and validated
from the aspects of the accuracy of turning decision
point and travel time and the comparison of all vehicle
trajectories.
The rest of this article is organized as follows.
Section 2 presents literature review about current turn-
ing models. Section 3 introduces the proposed simula-
tion model for turning behavior, and various modules
of model are described in detail by taking left-turn ve-
hicles as examples. Results of case study are analyzed
in Section 4. Section 5 discusses conclusion of this study
and direction of future work.
2 LITERATURE REVIEW
The turning process of drivers is consisting of two parts,
the tactical decision and operational adjustment. The
tactical level decides the desired trajectory and when to
implement turning movement. At the operational level,
turning vehicles implement movements with parame-
ters such as acceleration and turning angle, which are
dynamically influenced by the traffic environment. This
section reviews related research from the two aspects
above, respectively.
2.1 Decision-making behavior in potential conflict area
Decision-making behavior is widely studied includ-
ing gap-acceptance decision and desired trajectory
414 Ma et al.
selection of turning vehicles. In the field of one-
dimensional simulation, decision-making behaviors
consider the gap-acceptance decision as the core (PTV,
2011). And turning vehicles make decision by checking
size of acceptable gap while waiting for crossing. This
model is inefficient for interaction under mixed traffic
flow (Wu and Brilon, 2002). Limited by lane-based
assumption, the decision-making behavior only changes
acceleration in predefined lanes instead of adjusting the
trajectory. For desired trajectory decision, Zeng et al.
(2014) use logit model to select desired destinations
and entering points at crosswalks and adjust direction
to avoid conflicts. Anvari et al. (2015) validate social
force model further with empirical data, and the model
is modified with obstacle avoidance strategy and path
planning optimization method. Sch ¨
onauer et al. (2012)
use potential field method to find the path to the des-
tination and handle vehicular conflicts by game theory.
From agent-based approach, Sun and Wu (2014) de-
velop a Behavior Library and select a suitable behavior
at each step considering surrounding environment and
individual features. However, the studies for desired
trajectory decision only use path planning to achieve
the variation of the trajectory, which does not combine
with long-term behavior choice like gap-acceptance
model. The innovation of this article is selection
of turning patterns. The proposed decision model
classifies turning patterns into two kinds, two-stage
pattern (straight and turning) and one-stage pattern
(directly turning), to explain the formation of path
diversity.
2.2 Microscopic simulation model of traffic flows
in potential conflict area of an intersection
At the tactical level, many microscopic simulation the-
ories are able to describe the behavior of pedestrians
and vehicles. Among them, the typical theories include
lattice gas model (Helbing et al., 2003), cellular au-
tomata (CA) (Hafstein et al., 2004; Jiang and Wu, 2006;
Chu, 2009; Zhang and Chang, 2011), potential field
method (Okazaki, 1979), social force model (Helbing
and Moln´
ar, 1995; Huynh et al., 2013; Zeng et al.,2014;
Anvari et al., 2015). CA and lattice gas models belong
to discrete model. These models conduct simulation
by setting the running rules in discrete road space and
time. The rules in corresponding road structure can
determine the destination, describe the movement of
the traffic flow and avoid the collision. Chai and Wong
(2015) integrate CA with fuzzy logit model to simulate
decision-making process of driver behavior. Crociani
and L¨
ammel (2016) use CA to simulate multidesti-
nation pedestrian flow and design a learning cycle to
optimize the planning rule. But its speed is discrete,
and it is difficult to explain the continuous movement
behavior of the actual traffic participants (Toledo,
2007; Zeng et al., 2014). The social force model and the
potential field method belong to the continuum model.
According to Coulomb’s law, Okazak (1979) establishes
the magnetic force model for pedestrian simulation.
The model considered pedestrians as magnetic objects.
It can well describe the reaction of pedestrians to obsta-
cles. But the load parameters of each pedestrian cannot
be measured, and cannot reflect the characteristics of
the group. Helbing and Moln´
ar (1995) put forward a
social force model, which can effectively describe the
behavior of pedestrians for the first time. The model
compared pedestrians with a particle of Newton’s sec-
ond law. Huynh et al. (2013) use the social force model
to simulate the nonmotorized vehicle, and got satisfied
results. Huang et al. (2012) apply the social force model
to the vehicle simulation under the condition of mixed
traffic flow. And Anvari et al. (2015) combine the path
planning with the social force model and the avoidance
of conflict strategy. The application of social force
model is gradually expanded from the pedestrian only
to mixed traffic environment. However, the existing
studies are mostly based on the social force model for
pedestrians, and describe the behavior of motorized
vehicle by parameter calibration (Huang et al., 2012;
Pascucci et al., 2015; Anvari et al., 2015). On one hand,
the motorized vehicles, which have poor flexibility, are
subject to vehicle dynamic. Constraints on vehicular
movements are needed. On the other hand, speed of
vehicles varies in large range. Therefore, the forces for
vehicles should have different expression from those
for pedestrians to ensure the simulation accuracy.
3 SIMULATION MODEL
3.1 Framework
The classical social force model has certain applica-
bility in pedestrian simulation, but it cannot describe
decision-making behavior, which is important in vehic-
ular simulation under mixed traffic flow. To fill the gap,
this study proposed a modified social force simulation
model, which takes interactions between different
traffic flows into consideration. The framework of
the model has three layers, namely decision layer,
operation layer, and constraint layer (See Figure 2).
Decision layer handles turning patterns selection
and recognizes current stage of the selected pattern
(please see Section 3.2 in detail). Under different stages,
corresponding destinations are obtained, which influ-
ences the direction of driving force in operation layer
(please see Section 3.3 in detail). Operation layer
Two-dimensional simulation of turning behavior in potential conflict area of mixed-flow intersections 415
Fig. 2. Concept framework turning vehicles simulation model.
uses modified social force model to calculate vehicular
velocity and driving direction from current position
to the selected destination. Constraint layer includes
constraints of vehicle dynamic and collision-avoidance
strategy to make the simulation closer to reality (please
see Section 3.4 in detail). The constraints include min-
imum turning radius, acceleration, and boundary con-
straints. The following sections will discuss the three lay-
ers in detail. For convenience, this research chooses left-
turning vehicles at a two-phase intersection as an exam-
ple. The conflict flows include: motorized straight-going
vehicles, nonmotorized straight-going vehicles, and
pedestrian. The behavior is more complex than right-
turn vehicles at potential conflict area of intersections.
3.2 Decision layer
The proposed decision-making model aims at deciding
whether to slow down or to pass the intersection step
by step until the result is true. In most existing models,
turning vehicles make decisions before passing the
stop line. However, the process of turning is not like
that in reality. Based on the field observation, left-turn
vehicles are not necessary to stop and decide at the
stop line. Instead, the vehicle passes the stop line and
slows down when waiting for an acceptable gap. To
reflect this phenomenon, the decision model must
make suitable crossing decision at every step instead
of at a fixed decision point. Therefore, two turning
patterns can be observed. One pattern is the one-stage
pattern, turning directly (it will be further explained
in Section 4.2.1), and the other one is the two-stage
pattern, going straight and then turning. In one-stage
pattern, left-turn vehicle implements turning behavior
once passing the stop line, represented by the green line
Fig. 3. Two turning patterns: one-stage and two-stage
turning.
in Figure 3. In two-stage pattern, left-turn vehicle slows
down and goes straight after passing the stop line, and
then implements turning behavior when the situation
is suitable, represented by the blue line in Figure 3.
The two patterns determine the desired direction of the
vehicle at the current simulation step, which impact the
driven force in the operation layer.
The probability to implement turning behavior Ptis
influenced by many factors related to the left-turn ve-
hicle, motorized straight-going vehicles, nonmotorized
straight-going vehicles, pedestrians, and so on. Because
416 Ma et al.
Fig. 4. Influential factors for turning decision.
the logit model is widely used for binary choice (Cassidy
et al., 1995; Devarasetty et al., 2012; Zhang et al., 2016),
especially for the crossing behavior of left-turn vehicles,
this study selects logit model to predict whether the
vehicle begins to implement turning at the current
simulation step t. Considering features of different
traffic participants, influencing factors of the utility V
to implement turning behavior include: velocity of left-
turning vehicle vα, time gap between the left-turning
vehicle and motorized through vehicle T, the number
of nonmotorized vehicles Nnand the number of pedes-
trians Npin potential conflict area (See Figure 4), as
shown in Equation (1). According to discrete choice
model, the probability Ptof choosing left-turning
behavior can be expressed as Equation (2).
V=β0+β1vα+β2T+β3Nn+β4Np(1)
Pt=exp (V)
1+exp (V)(2)
3.3 Operation layer
When the turning pattern and desired turning point is
determined, the approximate trajectory of left-turn ve-
hicle is determined. The operation layer model then cal-
culates turning motion parameters to implement micro-
scopic adjustment of trajectory based on surrounding
traffic environment. The parameters include velocity
and direction of movement. In microscopic simulation
area, social force model has natural advantage in de-
scribing continuous traffic flows. But as it is discussed in
Section 2.2, classic social force model has its limitation
in microscopic vehicle simulation. Therefore, the modi-
Fig. 5. Social force of pedestrians at crosswalk.
fied social force model is established, which can describe
the behavior of left-turning vehicles more accurately.
Classic social force model has been used to describe the
influence of pedestrians, obstacles, boundary, destina-
tion, and so on. Helbing and Moln´
ar (1995) proposed
a social force model of pedestrians based on hydrody-
namic equations. The social force includes driving force
caused by destination, repulsive force caused by other
pedestrians, boundary or obstacles, and disturbing
behavior. For example, force on pedestrian at an in-
tersection is shown in Figure 5 (Zeng et al., 2014). The
social force model can be expressed as Equation (3).
−−→
F(t)=−→
FD+−→
FB+−→
FV+−→
FP+−→
FS(3)
where −−→
F(t) is social force on pedestrians, −→
FDis driving
force, −→
FBis force of boundary, −→
FVis repulsive force of
vehicles, −→
FPis force of pedestrians, −→
FSis driving force of
traffic light.
Similar with the model for pedestrian, the social force
on turning vehicle includes: driving force −→
FD, force of
boundary −→
FB, force caused by turning vehicle ahead
−−−−→
Ffollow
αfand −→
Fαf, force of motorized through vehicles
−→
Fαm, force of nonmotorized through vehicles −→
Fαn, force
of pedestrians −→
Fαp(see Figure 6). The number of forces
is more than the model of pedestrian, and there are
both repulsive force and attractive effects in force from
preceding left-turn vehicles. It can be expressed as in
Two-dimensional simulation of turning behavior in potential conflict area of mixed-flow intersections 417
Fig. 6. Social force of turning vehicles at shared space.
Equation (4).
−−→
F(t)=−→
FD+−→
FB+−−−−→
Ffollow
αf+−→
Fαf+−→
Fαm+−→
Fαn+−→
Fαp
(4)
Pedestrian has very good flexibility in movement.
Velocity of pedestrian is stable and difference between
pedestrians is limited. So the social force model has
been widely used in the field of pedestrian simula-
tion. However, there are essential differences when
it comes to behavior of turning vehicles. The velocity
of motorized vehicle is unstable and changeable and
the influence of velocity cannot be ignored to avoid
collision. There are two kinds of trajectories and two
situations of driving force; it will be discussed in more
detail in Section 3.3.2.1.
In classic social force model (Johansson et al.,
2007), the value of social force mainly has negative
correlation with distance. Social force −−→
FαUbetween the
simulated pedestrian αand an interacting object Ucan
be expressed as in Equations (5) and (6).
−−→
FαU=AαUexp−dαU
BαU−→
nαUfαU(5)
fαU=λ+(1−λ)1+cos (ϕαU)
2(6)
In Equation (5), diis distance between the two
agents. Because each agent occupies a circular or
elliptical space, diis calculated by subtracting the sum
of radii of two agents from the distance between their
centers. −→
nαUis the unit vector pointing from the Uto
Fig. 7. Boundary of vehicles.
α,fαUexpresses the anisotropic character of the inter-
action and AαU,BαUare parameters to be estimated.
In Equation (6), fαUis related to the angle difference
ϕαUbetween the direction of an interacting object and
the direction of movement of the simulated vehicle
(please see Figure 7), the parameter λvaring from 0 to
1 determines the extent affected by ϕαU.
Equation (5) is suitable to pedestrian, which has
low velocity, small velocity change, and great velocity
controllability (large acceleration and deceleration).
But it is not suitable to motorized vehicle, which has
high velocity and relatively large velocity change: the
influence of vehicles with different velocities is different
when the distances are the same.
In view of this, this research redefined the social
force model. The core of social force model is to use
risk compensation principle to ensure safety. Time
distance has more advantages over distance to measure
the safety of vehicles. Therefore, the proposed model
takes time distance tαUas the independent variable
of social force. The basic turning vehicle social force
model can be expressed as in Equation (7). Because the
modified model is used for mixed flow, −−→
FαUrepresents
social force from other agents U, which can be the front
turning vehicle f, motorized vehicles m, nonmotorized
vehicles n, and pedestrians p. Each force is discussed
in Section 3.3.2. In Equation (8), considering the
movement constraints of vehicle and blind area, the
parameter qdetermines the influencing field by angle.
If the absolute value of angle between the direction
of the interacting agent and the direction of vehicular
movement is less than 60◦, the simulated vehicle is
impacted by the social force. Otherwise, the interacting
agent does not impact the simulated vehicle.
−−→
FαU=AαUexp −tαU
BαU−−→
nαUλ+(1−λ)1+cos (ϕαU)
2q
(7)
418 Ma et al.
q=1,if |ϕαU|≤60◦
0,if |ϕαU|>60◦(8)
3.3.1 Geometry size of moving objects. The geometry
size model of moving objects need to be determined
first. In social force model of moving object (Helbing
et al., 2000), the distance rαbetween boundary and
center of the object is related to the angle ϕαUbetween
the direction of an interacting object to the moving
direction of the simulated vehicle. According to the
size of motorized vehicle, nonmotorized vehicle and
pedestrian, motorized and nonmotorized vehicles are
considered as elliptical objects, and pedestrians consid-
ered as circular object whose radius is rp. The shape of
a vehicle’s boundary is an ellipse (see Figure 7).
And the distance between boundary and center of
vehicle rαUcan be calculated by Equation (9).
rα=w
1−ε2cos (ϕαU),ε =√l2−w2
l(9)
where rαis distance between boundary and center of
vehicle, ϕaU is the angle, lis semimajor axis of ellipse, w
is semiminor axis of ellipse.
3.3.2 Modified social force model. As is discussed
before, the social force of left-turning vehicle consists
of six kinds of forces. According to kinematics law
of physics, when a vehicle with the velocity of (−−→
V(t))
moves to next position in ttime, the velocity of
the vehicle at t+tmoment can be expressed as in
Equation (10).
−−−−−−−→
V(t+t)=−−→
V(t)+−−→
F(t)×t(10)
Supposing the position of vehicle at tmoment is −−→
P(t),
the position of vehicle at t+tmoment −−−−−−→
P(t+t) can
be expressed as in Equation (11).
−−−−−−−→
P(t+t)=−−→
P(t)+−−−−−−−→
V(t+t)+−−→
V(t)
2×t(11)
This research will describe the forces, which left-
turning vehicle meets in turning process in Equation
(4) in the following sections.
3.3.2.1 Driving force. The driving force is the force
to drive the turning vehicle to destination. It can be
expressed as in Equation (12).
−→
FD=vd
α−→
eα−−→
v0
τ(12)
where vd
αis value of expected velocity, −→
eαis direction of
expected velocity, v0is velocity of turning vehicle, τis
buffer time of acceleration.
The key is to determine desired direction. Two
turning patterns are discussed in the decision layer.
In the straight-going stage of two-stage pattern, the
desired destination is in front of the vehicle (See
Figure 8a). In left-turning stage of one-stage pattern,
the desired destination is the point at outgoing lanes of
the intersection (See Figure 8b).
The desired velocity is determined by probability dis-
tribution. The range of motorized vehicle’s velocity is
large; and the situation of potential conflict area is com-
plex. It is hard to find desired velocity without influence
of other traffic flows. So, in this research, the velocity
of left-turning vehicle when entering the intersection
is measured as expected velocity−→
vd. The velocity is
collected from samples corresponding to the normal
distribution which can be expressed as Equation (13).
f(vd=x)=1
√2πσ exp −(x−μ)2
2σ2(13)
where f(vd=x) is probability density function of nor-
mal distribution, vdis value of expected velocity, μis
mean value of expected velocity, σis standard deviation
of expected velocity.
3.3.2.2 Force from left-turning vehicles ahead. Left-
turning vehicles ahead have both positive and negative
effect on left-turning vehicle. Positive effect is the
attractive force −−−−−→
Ffollow
αf,which makes the vehicle follow
the vehicle ahead. Negative effect is the repulsive force
−→
Fαf, which makes the vehicle keep a certain distance
from the vehicle ahead (See Figure 9a).
The force from left-turning vehicles ahead can be
expressed as in Equation (14).
−−→
Fsoc
αf=−−−−→
Ffollow
αf+−→
Fαf(14)
When the left-turning vehicle is in straight-going
stage and is waiting for a chance of turning, the vehicle
slows down to the tail of the queue and is mainly
controlled by the repulsive force. So the attractive force
−−−−→
Ffollow
αfcan be expressed as in Equation (15).
−−−−→
Ffollow
αf=⎧
⎪
⎪
⎨
⎪
⎪
⎩
v
τexp tαf−Ts
Bfollow−→
nαf,
left −turning stage
0,straight −going stage
(15)
where vis velocity difference between left-turning
vehicle and vehicle ahead, Tsis safe headway (for ex-
ample, Ts=1.5s), τis buffer time for car following,
Two-dimensional simulation of turning behavior in potential conflict area of mixed-flow intersections 419
Fig. 8. Driving force from destination on left-turning vehicle.
Fig. 9. Social force from other flows.
tαfis headway, −→
nαfis unit vector of following direction,
Bfollowis parameter to be estimated.
The repulsive force −→
Fαfcan be calculated as in
Equations (7) and (8) by changing the subscript U
to t. Because there is no potential conflict point, the
critical conditions with highest risk are the front vehicle
immediately stops. Therefore, time distance tαfis
defined as headway to measure the highest risk. The
time distance for interacting with the turning vehicle
ahead can be expressed as in Equation (16).
420 Ma et al.
tαf=dαf−rαf×cos ϕαf
|vα|
rαf=rα+rf
(16)
where dαfis distance between center of left-turning ve-
hicle and the vehicle ahead, vαis velocity of left-turning
vehicle, rαis distance between boundary and center of
left-turning vehicle, rfis distance between boundary
and center of left-turning vehicle ahead.
3.3.2.3 Force from motorized straight-going vehicles.
In typical intersection, the first conflict flow of left-
turning vehicle is motorized straight-going vehicle.
Repulsive force from motorized through vehicles −→
Fαmis
determined by the time difference tαmbetween the left-
turning vehicle αand the straight-going vehicle m.Be-
cause their directions are parallel, no obvious potential
conflict point is detected as shown in Figure 9b. Because
the condition with the highest risk is that the left-turning
vehicle immediately moves into the way of the straight-
going vehicle, the time distance is determined by the
headway to measure the proximity of the two vehicles in
the worst condition. The force from motorized straight-
going vehicles −→
Fαmcan be calculated by Equations (7)
and (8) with the parameter mreplacing U.
tαm=dαm−rαm×cos(ϕαm)
|vm|
rαm=rα+rm
(17)
The time distance tαmcan be expressed as in Equa-
tion (17), where dαmis the distance between centers
of left-turning vehicle and motorized through vehicle,
|vm|is velocity of motorized straight-going vehicle, rαis
distance between boundary and center of left-turning
vehicle, rmis distance between boundary and center of
motorized straight-going vehicle.
3.3.2.4 Force from nonmotorized vehicles. After
interacting with motorized vehicles, the direction of
velocity of left-turning vehicle has changed greatly.
Obvious potential collision points can be found when
the turning vehicle interacts with nonmotorized ve-
hicles as shown in Figure 9c. Therefore, the widely
accepted measurement of safety, PET, is used to define
time distance, which reflects the proximity between
two agents with potential conflicts. The repulsive force
from nonmotorized vehicle −→
Fαnis determined by the
changing the subscript Uto n. The time distance tαncan
be expressed as in Equation (18).
tαn=|tn−tα|
tn=dn
−→
vn
cos (ϕn)
tα=dα
−→
vα
cos (ϕαn)
(18)
In Equation (18), tnor tαis the component of
time distance between the potential conflict point and
nonmotorized or turning vehicle in the direction of −→
Fαn,
dnor dαis distance between potential conflict point and
nonmotorized vehicle or the turning vehicle, −→
vnor −→
vα
is velocity of the nonmotorized vehicle or the turning
vehicle, ϕnis the difference between the direction of
the turning vehicle and the moving direction of the
nonmotorized vehicle.
3.3.2.5 Force from pedestrians.
FB=
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
0,if vehicle αis within the virtual boundary
Ar
Bexp r
Br
B−→
nαB,if vehicle αis outside of the inside range
AR
Bexp R
BR
B−→
nαB,if vehicle αis outside of the outer range
(19)
Left-turning vehicle has conflict with pedestrians
near crosswalk. Only when a potential conflict point is
detected, pedestrian has an influence on left-turning ve-
hicle. And the conflicting situation (shown in Figure 9d)
is similar with that in Figure 9c. In this situation, the
repulsive force from pedestrians (
Fp) is also determined
by the expected PET tαp. The force from pedestrians
(
Fp) can be expressed as Equations (7) and (8), where
the parameter Uis changed to p. The expected PET
tαcan be calculated by changing the subscript nto pin
Equation (18).
3.3.2.6 Force from virtual boundary. Although no
real boundary can be observed within intersection,
turning flow uses a certain space to pass the intersection
because of safety and comfort. Therefore, virtual
boundaries are potential conflict area of intersections.
When the left-turning vehicle is within the virtual
boundary, it is not affected. Otherwise, a repulsive
force drives the vehicle to go back (see Figure 10). The
repulsive force from virtual boundary
FBcan be ex-
pressed as in Equation (19), where Ar
B,Br
B,AR
B,and BR
B
are parameters to be estimated, (−→
nαB) is unit vector of
(
FB), ris distance between vehicle and the inside range,
Ris distance between vehicle and the outer range.
Two-dimensional simulation of turning behavior in potential conflict area of mixed-flow intersections 421
Fig. 10. Force from virtual boundary.
3.4 Constraint layer
To avoid unrealistic phenomenon in critical situation,
such as unfitness of the vehicle dynamics, the research
adds constraint layer to modify the results from the
two layers above. The constraint layer establishes
constraints and collision-avoidance model. Constraints
include turning radius, acceleration, and boundaries.
3.4.1 Constraint of turning radius. Because of the
limitation of motorized vehicle itself, the turning radius
of a vehicle is always greater than the minimum turning
radius in running process. In the work by Fitzpatrick
and Schneider (2005), the turning radius has linear
correlation with the turning speed. However, according
to the basic knowledge of kinematics, as the turning
speed increases, the increment of radius become in-
creasingly large. Therefore, the Equation (20) models
the relationship between the minimum turning radius
and turning speed as an exponential function.
Rmin
R0=exp v
v0(20)
where R0and v0are parameters to be estimated, vis
velocity of vehicle.
In running process of vehicle, the turning radius Rα
of every point in simulation trajectory should satisfy
Rα≥Rmin. If the turning radius Rαin simulation com-
puting is smaller than Rmin , this constraint considers
that Rα=Rmin.
3.4.2 Constraint of acceleration. The traffic condition
at potential conflict area of intersections is relatively
complex. So the acceleration of vehicle should be within
a certain range for safety when the vehicle passes in-
tersection. Different vehicles have different maximum
accelerations a+
max and maximum decelerations a−
max
. The constraint of acceleration in simulation can be
expressed as in Equation (21), where ais acceleration
of a vehicle.
−a−
max <
=a<
=a+
max (21)
3.4.3 Constraint of boundaries. As discussed before,
virtual boundaries exist at potential conflict area of
intersection in simulation. But only a small number of
vehicles run out of the boundaries in real situation. The
effect of the virtual boundary is hard to be calibrated.
Therefore, the force of boundary is strengthened
in simulation. The proposed model adds a certain
value to parameters to strengthen the constraint of
virtual boundary. The constraint of boundaries can be
expressed as in Equation (22).
AR
B=AR
BC+ε1,BR
B=BR
BC+ε2
Ar
B=Ar
BC+ε3,Br
B=Br
BC+ε4
(22)
where AR
B,BR
B,Ar
B,and Br
Bare estimated parameters,
AR
BC,BR
BC,Ar
BC,and Br
BCare parameters before esti-
mate, ε1,ε
2,ε
3,and ε4are parameters to be estimated.
3.4.4 Collision-avoidance control tactics. Social force
model cannot describe the sudden deceleration behav-
ior when the vehicle meets serious conflict. Therefore,
collision-avoidance model is included in constraint
layer. According to severity of potential conflicts, the
model decides whether to take collision-avoidance mea-
sures. The collision-avoidance control is determined by
time distance tabetween potential conflict point and
vehicle and time distance tvbetween potential conflict
point and conflict vehicle. If 0 <ta−tv<TTC
min
(TTC
min is the minimum of nonserious conflict TTC,
usually take 2 seconds as TTC
min), the through vehi-
cle will pass the potential conflict point earlier than
left-turning vehicle. The left-turning vehicle will take
collision-avoidance control tactics and decelerate.
Suppose the deceleration is a,ashould satisfy the
Equation (23). The equation is used to ensure that the
left-turning vehicle takes the minimum deceleration
ain situation that the difference of time distance to
conflict point between two flows is larger than T.
⎧
⎪
⎪
⎨
⎪
⎪
⎩
vαtα−0.5at
α2=d1−rm−L
tα−tv=T
tv=d2
vm
(23)
where vαis velocity of left-turning vehicle, tαis time dis-
tance between potential conflict point and vehicle, ais
422 Ma et al.
Fig. 11. Layout of case intersection and traffic flow trajectory.
deceleration, d1is distance between potential conflict
point and vehicle, rmis size of through vehicle, Lis semi-
major axis of vehicle ellipse, tmis time distance between
potential conflict point and conflict vehicle, Tis dif-
ference of time distance between vehicle and conflict ve-
hicle, d2is distance between potential conflict point and
conflict vehicle, vmis velocity of through vehicle.
The collision-avoidance model for other flows is
similar to that for straight-going vehicles, so they are
not discussed any more in detail.
4 CASE STUDY
The purpose of this study is to establish a two-
dimensional simulation model, which can appropriately
describe the turning vehicles in a mixed-flow traffic
environment. Therefore, the selected intersection
for the case study should meet the following two
conditions. First, turning vehicles compete with other
traffic participants at the potential conflict area under
a nonprotected phase. Second, pedestrians and nonmo-
torized vehicles are allowed to pass in the same phase.
The Jianhe–Xianxia intersection in Shanghai is a typical
two-phase intersection where the traffic is composed
of motorized vehicles, nonmotorized vehicles and
pedestrians. The left-turning traffic flow interacts with
others frequently in this intersection. Therefore, it is
selected in this study. The left-turning vehicles are used
for analysis as they are most conflicted. The layout of
the intersection is shown in Figure 11.
4.1 Data preparation
The empirical trajectories data were extracted from
video records with a high-accuracy video-processing
software. The software has been used in the previous
studies (e.g. Malinovskiy et al., 2009; Sun et al., 2014). A
camera was set on a tall building at the northeast corner
of the intersection. The weather condition was clear.
The video recorded traffic operation from 4:00 pm to
4:50 pm, which were rush hours of the day. Left-turn
vehicles from west bound approach were studied, as
represented by the blue lines in Figure 11. The left-turn
vehicles mainly interacted with the motorized and
nonmotorized vehicles in the opposite direction, rep-
resented as green and red lines in Figure 11. In the
video, 109 effective left-turn vehicle trajectories were
observed, 326 straight-going vehicle trajectories, 442
nonmotorized vehicle trajectories, 581 bidirectional
pedestrian trajectories as well. Left-turn vehicles go
through the potential conflict area when they first inter-
act with straight-going vehicles, then with straight-going
nonmotorized vehicles and finally with pedestrians in
the crosswalk.
4.2 Model calibration and validation
The turning behavior within intersections is a complex
process with the combination of vehicular interaction
and circular motion. Several models (e.g., decision
model, social force model and collision-avoidance
model) are needed to replicate the behavior. Uncer-
tainty of each of them impacts the movement of turning
vehicles simultaneously. That is, a driver can come out
with several turning solutions even under one given
traffic environment. Therefore, aggregated indicators
are used to measure the performance of the proposed
model. It should be mentioned that few similar studies
compare the fitness of trajectories one by one.
In this study, the distribution of decision points,
travel time, and the overlapping area are used to
evaluate performance of the proposed model from
different aspects. For the three-layer framework, the
distribution of decision points verifies the accuracy of
the decision layer, whereas the travel time evaluates the
movements of vehicles. The overlapping area illustrates
the difference between the two-dimensional behavior
and the lane based behavior.
The parameters in the proposed model are calibrated
with the data from the initial 30 minutes, whereas the
data from the following 20 minutes are used to validate
the proposed model.
4.2.1 Parameters calibration of decision-making model
of left-turning vehicle. As mentioned above, turning
behavior can be divided into two patterns. One is
two-stage pattern including straight movement and
turning movement. The other is one-stage pattern
movement, which is turning directly. Accordingly, the
Two-dimensional simulation of turning behavior in potential conflict area of mixed-flow intersections 423
Fig. 12. Comparison of single left-turn vehicle trajectory.
left-turn vehicle trajectories are divided into two parts,
the straight-going stage and the left-turning stage. As
shown in Figure 12, the critical point of the transition
from the straight-going stage to the left-turning stage
is called the desired trajectory change point. For each
desired trajectory change point, according to the de-
scription of Section 3.2, the speed of the vehicle, the
time gap between the left-turning vehicle and motor-
ized through vehicle, the nonmotorized vehicle density
in the conflict zone, and the pedestrian density in the
conflict zone are estimated. A total of 2,679 valid sam-
ples were obtained. All statistics data were analyzed by
binomial logit regression. After excluding no significant
variables, the model expression is as follows:
Pt=exp (0.056 −0.183vα+0.266T)
1+exp (0.056 −0.183vα+0.266T)(24)
According to the regression results, the effects of
pedestrians in the conflicting area and density of non-
motorized vehicles are not significant. Time distance of
straight traffic T, speed of left-turning vehicles vαare
significant variables. According to the expression, the
probability of implementing turning left movement Pt
has positive correlation with acceptance gap Tand neg-
ative correlation with vehicles’ own speed vα. When T
is increased by 10%, the utility Vis increased by 2.66%.
When vαis increased by 10%, the utility Vis decreased
by 1.83%. When the influential variables are only Tand
V, the prediction accuracy is the highest, which is 77.4%.
4.2.2 Parameters calibration of social force model. In
this article, the social force model calibration contains
multiple parameters. According to the research of
Zeng et al. (2014), parameters can be classified into two
categories: one is measurable parameters; the other is
calibrated by means of parameter fitting. Therefore,
three steps are taken to calibrate the parameters. First,
for the statistical parameters that cannot be measured
from the data, such as the driver’s visual angle, authors
directly refer to the relevant studies. Second, for the pa-
rameters that can be observed from the data, such as the
expected speed, they are carried out directly via statis-
tical analysis. Third, the least-square method is used to
calibrate the parameters, which have no clear physical
meaning.
4.2.2.1 Measurable parameters. A total of five mea-
surable parameters were measured in the modified
social force model including range of view θ, expected
speed vd
α, maximum acceleration a+
max, maximum de-
celeration a−
max, minimum turning radius Rmin.The
vehicle angle range is 120°. As mentioned before, the
expected velocity distribution is consistent with the
normal distribution. The average speed is 14.67 km/h,
the standard deviation is 5.8 km/h.
The maximum acceleration and deceleration is
extracted from field observation. The maximum accel-
eration is 1.71 m/s2, the minimum deceleration is –2.71
m/s2. The research sets the maximum acceleration as
a+
max=2m/s
2, the minimum deceleration as a−
max =
–3 m/s2.
Minimum turning radius is the minimum radius
of the motorized vehicle. Due to the interference of
motorized vehicles in the intersection potential conflict
area, the minimum turning radius cannot be directly
observed. The minimum turning radius Rmin expression
is shown in Equation (25). The parameter R0is 0.75 m,
whereas the parameter v0is 8.74 m/s.
Rmin
0.75 =exp v
8.74(25)
4.2.2.2 Nonmeasurable parameters. The remaining
parameters, such as τ, Ar
B,Br
B,AR
B,BR
B,A
v,Bv,
Asoc
lv,Bsoc
lv,τ
,Bfollow
lv,Anv,Bnv,A
p,Bp, are difficult
to be observed directly after estimation of measurable
parameters. For these parameters, this study used
nonlinear least-squares calibration and found the
optimal estimation value when the object function Y(u)
is minimum. The calibration results are shown in
Table 1.
Y(u)=
m
i=1f2
xi (u)+f2
yi (u)
fxi (u)=axti−Fx(ti,u)
fyi (u)=ayti−Fy(ti,u)
(26)
424 Ma et al.
Table 1
Results of social force model calibration
Parameter Equation Estimation Description
τ(12) 6.22 Buffer time of driving force
τ(15) 19.45 Car following force
Bfollow(15) 29.9
Aαf(7) 0.13 Repulsive force of front left-turn vehicle
Bαf(7) 3.13
Aαm(7) 0.26 Repulsive force of opposite straight-going vehicle
Bαm(7) 19.82
Aαn(7) 0.14 Repulsive force of opposite nonmotorized vehicle
Bαn(7) 3.13
Aαp(7) 0.07 Repulsive force of pedestrians
Bαp(7) 14.94
Ar
B(19) 0.09 Inner boundary repulsive force
Br
B(19) 1.34
AR
B(19) 0.40 Outer boundary repulsive force
BR
B(19) 19.90
Fig. 13. Distribution of social force.
In Equation (26), Y(u) is the residual sum of squares,
mis the size of samples, fxi(u) and fyi(u)arethe
residuals of the ith sample at Xand Ycoordinates,
respectively, axtiis the component of acceleration in X
coordinate at sampling time ti,aytiis that in Ycoordi-
nate and Fx(ti,u) and Fy(ti,u) are the components in X
and Ycoordinate of the social force at ti, respectively.
The above calibration results show that estimation
value of acceleration buffer time τis 6.22 seconds. But
the influence of the following force is weak, which is
mainly caused by few car-following samples in data
set. The weight of force from straight-going vehicles is
highest and that from pedestrians is lowest. The virtual
boundary force calibration results show that the effect
is relatively weak. To analyze the accuracy of the cali-
bration results, a sample of left-turn tracks was selected.
The simulated trajectory and observed trajectory are
shown in Figure 12, and these two trajectories show
the same two-stage pattern and the decision points
are close considering the size of a vehicle. Besides, the
social force of the left-turn vehicle at each moment is
calculated and shown in Figure 13. The characteristics
of the modified social force model are expressed in
three aspects.
In aspect of weight of the social force, the driving
force −→
FDis the highest, the force from motorized
Two-dimensional simulation of turning behavior in potential conflict area of mixed-flow intersections 425
vehicle −→
Fαmand the preceding left-turn vehicle −→
Fαfis
the second, force from pedestrians −→
Fαpand nonmotor-
ized vehicles −→
Fαnis the third. Because of the number
of pedestrian, pedestrian force −→
Fαpis higher than that
of nonmotorized vehicle −→
Fαn. The size of the virtual
boundary force −→
FBis lowest.
In aspect of time sequence and lasting time, driving
force −→
FD, following force −−−−→
Ffollow
αfand force from the
front left-turn vehicle −→
Fαtare always existing. The force
from straight-going motorized vehicles −→
Fαmis present
in the previous stage. The force from nonmotorized
vehicles −→
Fαnappears at the same time as straight-going
vehicles −→
Fαm, but last longer than straight-going vehi-
cles. Pedestrians’ force −→
Fαpappears at the later stage,
which is more in line with the reality of the impact of
traffic on the left-turn vehicles.
From the variation of social force, the change of
each social force reflects innovation of the modified
social force model. For example, the change of the
driving force −→
FDreflects the turning vehicle moves from
straight-going stage to turning stage. The change of
opposite direction straight-going vehicles’ force −→
Fαm
is caused by disappearance of interaction with the
front straight-going vehicle or a new one entering the
intersection. The change of pedestrian and nonmo-
torized vehicles force −→
Fαn,−→
Fαpcan be considered as
appearance of new conflicts between pedestrians and
turning vehicle or disappearance of current conflict.
4.3 Simulation result analysis of left-turning vehicle
The performance of the proposed model is evaluated
from the aspects of distribution of decision points, distri-
bution of travel time, and spatial distribution of trajec-
tories. As mentioned above, the three indicators are se-
lected to comprehensively reflect the model’s ability of
reproducing the details in turning process. The data in
the last 20 minutes of the video are used for validation.
4.3.1 Decision points. The distributions of the decision
points in empirical and simulation data are shown in
Figure 14.
The differences of Xand Ycoordinate between
the observed and simulated decision points are calcu-
lated. The average of differences of Xcoordinate is
0.53 m with the standard deviation 0.81 m, whereas
the average of difference of Ycoordinate is 0.54 m
with the standard deviation 0.78 m. Paired sample ttests
are used to compare the two sets of data, because they
are correlated. The tvalue and pvalue of difference
of Xcoordinate are –4.01 and 0.00, respectively, which
rejects the null hypothesis that the average of the two
Fig. 14. Distribution of decision points.
data sets are equal in statistics. However, considering
the size of vehicles, the difference is acceptable. The
tand pvalues of difference of Ycoordinate are 0.41
and 0.68, respectively, which means the difference of Y
coordinate is not statistically difference.
4.3.2 Travel time. Figure 15a shows the actual time
distribution of vehicles passing the intersection. The
mean is 15.77s seconds and the standard deviation is
4.78 seconds. Figure 15b shows the simulated time
distribution of vehicles through the intersection, the
mean is 16.18 seconds, standard deviation 5.42 seconds.
Because the sample size is large and the samples do
not show obvious normal distribution, we use z-test
to check consistency of the two sets. The statistic of
z-test is 0.32, which is smaller than the threshold of
95% confidence coefficient, 1.96. Hence, it is indicated
that the average travel time of simulation and empirical
data does not show significant difference.
4.3.3 Trajectory analysis. Distribution of frequency
difference is analyzed to verify the accuracy of simu-
lation trajectories. The frequency of each cell denotes
the times that actual or simulated vehicles pass through
the cell. As shown in Figure 16, distribution of dif-
ference between actual frequency faand simulated
frequency fsdemonstrates the variation of the two sets
of trajectories.
The simulated trajectories are of great variation,
which is also observed in field data. It reflects the
dynamic interaction and diversification of turning
behaviors of left-turn vehicles. The overall distribution
of the simulated trajectory is in a reasonable range of
distribution and consistent with the distribution of the
actual trajectory.
426 Ma et al.
Fig. 15. Comparison of travel time of left-turn vehicles to pass through intersection.
Fig. 16. Comparing actual and simulation trajectories with
distribution of frequency difference.
In the area between the start point and Section 1,
turning vehicles are mainly at straight-going stage. In
simulation, the vehicles move either strictly straight
or toward the destination, whereas the actual drivers
sometimes adjust the driving direction slightly to keep
more safe distance from the opposing vehicles. The
largest difference of frequency reaches about 18, which
is affected by uncertainty of the driving habit. In
the area between Sections 1 and 2, the difference of
frequency is almost between –5 and 5. Considering the
sample size is 71, the difference is acceptable. In the
area between Section 2 and destination, the absolute
value of difference is more than 5 in a fraction of cells.
The reason is that the attractive force from destination
may directly move the vehicles to the destination when
vehicles are near and point to the destination.
To test the similarity degree between simulation tra-
jectory and actual trajectory, this study uses coverage
rate, which is obtained by dividing area covered by
simulated trajectories to that of empirical trajectories.
The coverage rate is denoted as Pcand can be expressed
by Equation (27).
Pc=SC
Sr×100% (27)
Where SCis overlapped area of actual trajectory and
simulation trajectory, the unit is m2,Sris the area cov-
ered by the actual trajectory, the unit is m2.
According to the distribution range of the real trajec-
tory, the area of the potential conflict area is 1,000 m2
(x:1540 m; y:1050 m). For the convenience
of calculation, the potential conflict area is divided
into 0.5 m ×0.5 m small cells. The coverage area
is estimated by the number of small squares, which
are covered by the simulation trajectory and the real
trajectory. The coverage area of real trajectory is 332
m2, whereas that of simulation trajectory is 290.25 m2.
The coverage rate is 87.5%, which shows that the sim-
ulation trajectory coverage is of great coincidence with
real trajectory. The overlapped and nonoverlapped
parts are shown in Figure 17a. The difference between
the simulated trajectory and the real trajectory is
mainly on the trajectory boundary. The main reason is
that the real left-turn vehicles may move out of turning
boundary (such as bottom left corner).
Overlapped areas of commercial simulation model
VISSIM and real turning trajectories, are shown in the
Figure 17b. The coverage rate is only 9.7%, while the
VISSIM simulation covers an area of 32.25 m2.
Two-dimensional simulation of turning behavior in potential conflict area of mixed-flow intersections 427
Fig. 17. Coverage area of left-turn trajectories.
This is because VISSIM is a one-dimensional simula-
tion model. Its turning trajectory is two desired turning
paths (this study intersection has two exit lanes).
Finally, there is a need to point out that although
the modified social force model with collision-avoidance
constraint is developed, accidents still can be observed
especially when interacting with pedestrians. The driv-
ing force from the destination improves the speed of
the simulated vehicles when no obstacles are observed.
When an object suddenly moves into the intersection,
the high speed leads to an unavoidable accident. The
range of perception and how to handle the interaction in
the decision layer need to be further analyzed in future.
5 CONCLUSIONS
The behaviors of turning vehicles at potential conflict
area of mixed-flow intersections are particularly com-
plex because of interactions between turning vehicles
and conflicting flows. The proposed method breaks the
lane-based assumption for the one-dimensional simula-
tion method. And the main conclusions are as follows.
The proposed model includes three layers: decision
layer, operation layer, and constraint layer, which are
highly related. The direction of driving force in opera-
tion layer is determined by the decision made in deci-
sion layer. The constraints in constraint layer have influ-
ence on the social force in operation layer. Based on this
framework, the proposed model not only describes the
decision-making behavior of vehicles reasonably, but
also reproduces the entire turning process of vehicles.
The validation of the proposed model indicates that
the distributions of the desired turning points and
travel time do not have significant difference with the
empirical data in statistics. In addition, the distribution
of frequency difference is basically less than 10 out of
71 samples and the coverage rate between simulated
trajectory and real trajectory is 87.5%. It should be
emphasized that the coverage rate is the most obvious
indicator showing the improvement of performance
from one-dimensional to two-dimensional simulation.
Therefore, the proposed model reproduces the turning
process more appropriately in potential conflict area at
intersections.
The proposed research is a preliminary exploration
of two-dimensional model to describe behavior of
turning vehicles at potential conflict area of mixed-flow
intersections. There are still some differences between
simulated and real trajectories. The main reason is the
randomness in modeling the desired turning points. An-
other reason is the limited influence from constraints on
constraint layer. There are also some serious conflicts
between vehicles and pedestrians. Therefore, the accu-
racy of decision-making results, effects of constraints
and safety of operation needs to be improved in future.
ACKNOLEGEMENTS
The authors would like to thank the Natural Science
Foundation of China (51422812), and the Shanghai
Science and technology project of international coop-
eration (16510711400) for supporting this research. We
are also thankful to the reviewers to provide detailed
comments which are helpful in improving this article.
428 Ma et al.
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