Acta Mech Sin (2008) 24:399–407
DOI 10.1007/s10409-008-0163-0
RESEARCH PAPER
A car-following model with the anticipation effect of potential
lane changing
Tieqiao Tang · Haijun Huang · S. C. Wong ·
Rui Jiang
Received: 6 November 2007 / Revised: 19 March 2008 / Accepted: 31 March 2008 / Published online: 1 July 2008
© Springer-Verlag 2008
Abstract In this paper, a new car-following model is
presented, taking into account the anticipation of potential
lane changing by the leading vehicle. The stability condition of the model is obtained by using the linear stability
theory. The modified Korteweg–de Vries (KdV) equation
is constructed and solved, and three types of traffic flow in
the headway-sensitivity space, namely stable, metastable and
unstable ones, are classified. Both the analytical and simulation results show that anxiety about lane changing does
indeed have an influence on driving behavior and that a consideration of lane changing probability in the car-following
model could stabilize traffic flows. The quantitative relationship between stability improvement and lane changing probability is also investigated.
The project supported by the National Natural Science Foundation of
China (70701002, 70521001), the National Basic Research Program
of China (2006CB705503), and the Research Grants Council of the
Hong Kong Special Administrative Region (HKU7187/05E).
T. Tang
School of Transportation Science and Engineering,
Beihang University, Beijing 100083, China
e-mail: [email protected]
T. Tang · H. Huang (B)
School of Economics and Management, Beihang University,
Beijing 100083, China
e-mail: [email protected]
S. C. Wong
Department of Civil Engineering, The University of Hong Kong,
Pokfulam Road, Hong Kong, China
R. Jiang
School of Engineering Science,
University of Science and Technology of China,
Hefei 230026, China
Keywords Car-following model · Lane changing
probability · Stability
1 Introduction
To date, complex traffic phenomena have been analyzed by
different models, and some important results have been
obtained [1]. Among all of these models, the cellular automaton and car-following models are the most frequently adopted
to capture the basic characteristics of vehicle motion. In the
car-following models, velocity, headway, and relative velocity are often considered to be the main variables that determine the vehicle acceleration [2–13]. Recently, researchers
have begun to involve additional factors in the car-following
models for enhancing their abilities of describing real traffic [14–23]. All of these researches were subject to singlelane traffic flow. Kurata and Nagatani [24] and Nagai et al.
[25] used the optimal velocity model to study lane changing
behavior in a two-lane traffic system with given lane changing
rules. Tang et al. [26–31] proposed several two-lane traffic
flow models and revealed such complex phenomena as triangular shock, soliton waves, and oscillating waves. Tang et al.
[32] presented an extended speed-gradient model for twolane traffic flow by incorporating lane changing volume in
the flow conservation and dynamics equations. These studies, however, do not give consideration to the anticipation
effect of potential lane changing on the dynamic equation of
a car-following model.
In this paper, we present a new car-following model taking
into account the anticipation effect of the lane changing probability of the leading vehicle on the car-following behavior of
the following vehicle. The stability condition of the model
is derived by using linear stability theory. Three types of
traffic flow, stable, metastable and unstable, are classified by
123
400
T. Tang et al.
analyzing the neutral stability curves and the coexisting
curves that are given by the solution of the modified KdV
equation. Numerical results show that a consideration of lane
changing probability can improve the stability of traffic flow.
Our new model could also effectively reproduce the spatialtime evolution of headway, traffic flow, and potential lane
changing flow that result from lane changing probability.
2 The car-following model
In general, the single-lane car-following model can be written
as follows [1]:
d2 x n
= f (vn , xn , vn ),
(1)
dt 2
where f (·) is the stimulus function, xn and vn are the position
and velocity of vehicle n, respectively, xn = xn+1 − xn is
the headway, and vn = vn+1 − vn is the relative velocity.
Note that in Eq. (1), the acceleration of a vehicle is completely determined by velocity vn , headway xn , and relative
velocity vn . Recently, researchers have also considered the
multiple headways of vehicles in their car-following models,
as follows [14–18]:
d2 x n
= f (vn , xn , xn+1 , . . . , xn+m , vn ),
(2)
dt 2
where xn+i = xn+i+1 − xn+i . Zhao and Gao [19] found
that a collision will occur under certain given conditions
when this model is used to describe traffic flow. They then
presented an improved model by considering the effect of
the acceleration of the leading vehicle on the car-following
behavior of the following vehicle, i.e.,
d2 x n
= f (vn , xn , vn , d2 xn+1 /dt 2 ).
(3)
dt 2
To enhance further the stability of traffic flow, Wang et al.
[20] proposed a multiple velocity difference model, as follows:
d2 x n
= f (vn , xn , vn , vn+1 , . . . , vn+k ),
(4)
dt 2
where vn+i = vn+i+1 − vn+i . Numerical results have
shown that models (2)–(4) can improve the stability of traffic
flow in comparison with model (1). In addition, Ge et al. [18]
declared that drivers do not necessarily consider the effects
of an arbitrary number of vehicles ahead of them, but they
do consider those of the three that are ahead of them.
All of the above car-following models are suitable only to
describe single-lane traffic flow. Tang et al. [27–29] recently
developed a two-lane car-following model involving lateral
distance, as follows:
d2 xl,nl
= f (vl,nl , xl,nl , vl,nl , l,nl ),
dt 2
123
(5)
Fig. 1 Scheme of car-following
where l,nl is the lateral distance of vehicle nl on lane l,
away from the leading vehicle in the neighboring lane. They
showed that the consideration of lateral distance can improve
the stability of traffic flow compared with no lane changing. Frequent lane changing results in an oscillating wave,
whereas a soliton wave occurs for infrequent lane changing
[27–29].
The model (5), however, does not take into account the
anticipation effect of the potential lane changing on the acceleration of the following vehicle. Our observations and survey
show that the leading vehicle will change lanes at some probability, regardless of whether or not there is an opportunity
to do so. Hence, during the car-following process, the driver
of the following vehicle has to consider the possibility that
the leading vehicle will change lanes even when it does not
actually happen. Referring to Fig. 1, we incorporate the lane
changing probability ε(xn+1 (t)) of the leading vehicle, the
headway xn+1 (t) = xn+2 (t) − xn+1 (t), and the relative
velocity ṽn (t) = vn+2 (t) − vn (t) into Eq. (1). The new
car-following model is as follows:
d2 x n
= f (vn , xn , xn+1 , vn , ṽn , ε(xn+1 (t))). (6)
dt 2
Applying a similar method to those that were used in
[13,14], we have
dxn (t + τ )
dt
= V (xn (t), xn+1 (t), vn (t), ṽn (t), ε(xn+1 (t))),
(7)
where V (xn (t), xn+1 (t), vn (t), ṽn (t), ε(xn+1 (t)))
is the optimal velocity of vehicle n at time t, and τ is the delay
time. Note that the optimal velocity used herein is different
from that one formulated in Bando et al. [2,3].
Expanding the right-hand side of Eq. (6) in the Taylor
series and neglecting the higher order terms, we obtain
1
d2 xn (t)
= (V (xn (t), xn+1 (t), vn (t), ṽn (t), εn+1 )
dt 2
τ
−dxn (t)/dt),
(8)
A car-following model with the anticipation effect of potential lane changing
where εn+1 = ε(xn+1 (t)) for short. With ṽn (t) =
vn+1 (t) + vn (t), the optimal velocity can be rewritten as
V (xn (t), xn+1 (t), vn (t), ṽn (t), εn+1 )
= V (xn (t), xn+1 (t), vn (t), vn+1 (t)
+vn (t), εn+1 ).
(9)
Suppose that the above optimal velocity can be formulated
as a linear combination of the perceived headway-induced
optimal velocity and the perceived relative velocity. It follows:
V (xn (t), xn+1 (t), vn (t), vn+1 (t) + vn (t), εn+1 )
= V (xn (t) + εn+1 xn+1 (t)) + λ(vn (t)
+ εn+1 vn+1 (t)),
(10)
where λ is a coefficient that represents the weight of the relative velocity. In Eq. (10), the perceived headway by vehicle n
is in fact the expected headway that is caused by considering
the lane changing probability, i.e.,
xn (t) + εn+1 xn+1 (t)
= εn+1 (xn (t) + xn+1 (t)) + (1 − εn+1 )xn (t),
where xn (t) + xn+1 (t) = xn+2 (t) − xn (t) is the space
between vehicles n + 2 and n when vehicle n + 1 is anticipated to disappear with probability εn+1 , whereas xn (t) =
xn+1 (t) − xn (t) is the space between vehicles n + 1 and n
when vehicle n + 1 is anticipated to remain in the lane with
probability 1−εn+1 . Similarly, the perceived relative velocity
of vehicle n is in fact the expected relative velocity, i.e.,
vn (t) + εn+1 vn+1 (t)
= εn+1 (vn (t) + vn+1 (t)) + (1 − εn+1 )vn (t),
where vn (t) + vn+1 (t) = vn+2 (t) − vn (t) is the velocity
difference between vehicles n + 2 and n.
Note that the second term of the right-hand side in Eq. (10)
can be rewritten as
λ(vn (t) + εn+1 vn+1 (t))
= λ(1−εn+1 )vn (t) + λεn+1 (vn (t)+vn+1 (t)).
(11)
Clearly, λ(1 − εn+1 ) > λεn+1 when εn+1 < 0.5. This is usually the case. In other words, the driver of following vehicle
n always pays greater attention to the relative speed of the
leading vehicle, rather than to that of a vehicle that is further
downstream.
Substituting Eq. (10) into Eq. (8), we obtain the new carfollowing model, as follows:
d2 xn (t)
= α(V (xn (t) + εn+1 xn+1 (t)) − vn )
dt 2
+ κ(vn + εn+1 vn+1 (t)),
(12)
401
where α = 1/τ represents the sensitivity coefficient of a
driver to the difference between the optimal velocity and
the current velocities, and κ = λ/τ is the sensitivity coefficient to the perceived relative velocity. Obviously, Eq. (12)
becomes the standard optimal velocity (OV) model when
λ = 0 and ε = 0, and the full velocity difference (FVD)
model when λ > 0 and ε = 0. We call Eq. (12) with λ = 0
and ε > 0 the extended OV model.
To perform stability analyses of the model, we discretize
the continuous model using the asymmetric forward difference, and rewrite Eq. (12) as follows:
xn (t + 2τ )
= xn (t + τ ) + τ (V (xn+1 (t) + εn+2 xn+2 (t))
−V (xn (t) + εn+1 xn+1 (t))) + λ(xn+1 (t + τ )
−xn+1 (t) − xn (t + τ ) + xn (t))
+λεn+2 (xn+2 (t + τ ) − xn+2 (t)
−xn (t +τ )+xn (t))+λ(εn+2 − εn+1 )(xn+1 (t + τ )
−xn+1 (t) − xn (t + τ ) + xn (t)),
(13)
where εn+2 = ε(xn+2 (t)). Recall that, due to a consideration of the lane changing factor, the actual headway of
vehicle n should be replaced by its expected headway xn +
εn+1 xn+1 . Thus, the headway-induced optimal velocity
V (xn (t) + εn+1 xn+1 (t)) in Eq. (12) can be written as
V (x̄n ), where x̄n = xn + εn+1 xn+1 . In this paper, we
adopt the following form for the optimal velocity function
V (x̄n ) =
vmax
(tanh(x̄n − h c ) + tanh(h c )),
2
(14)
where h c is the safety distance and vmax is the maximum
speed. Note that this headway-induced optimal velocity has
the following properties:
1. it is a monotonically increasing function of headway x̄n
and bounded by the maximal velocity vmax ;
2. it has a turning point at x̄n = h c , i.e.,
d2 V (x̄n ) V (h c ) =
= 0.
dx̄n2 x̄n =h c
(15)
As explained in [18], the reason to choose Eq. (14) is
that it has a turning point at x̄n = h c , which is important
for deriving the Berger, KdV, and MKdV equations from
Eq. (12). For mathematical convenience, we assume in the
linear and nonlinear stability analyses that ε(xn+1 (t)) is
continuous and differentiable. In Sect. 5, however, we use
a non-smooth function ε(xn+1 (t)) in the simulation for
demonstrating that the results are highly consistent with those
obtained from the stability analyses.
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402
T. Tang et al.
3 Linear stability analysis
We first investigate the stability of a uniform traffic flow. The
uniform traffic flow is defined as a state in which all vehicles
move with constant headway h and optimal velocity V (h).
Clearly, the steady-state solution of the model (12) is
xn,0 (t) = hn + V (h)t, with h = L/N ,
(16)
where N is the total number of vehicles on the road, L is the
length of the road, and xn,0 (t) is the position of vehicle n in
a steady state.
Let yn (t) be a small perturbation of the steady-state solution xn,0 (t), then the perturbed solution is xn (t) = xn,0 (t) +
yn (t). Correspondingly, the headway can be expressed as
xn (t) = h + yn (t). Let the lane changing probability be
εn+1 = ε(h + yn+1 (t)) = ε(h) + ε (h)yn+1 (t). Substituting it into Eq. (13), linearizing the equation and neglecting
the higher order terms (yn )m with m ≥ 2, we obtain
yn (t + 2τ )
= yn (t +τ )+τ V (h̄)(yn+1 (t)−yn (t)+ε0 (yn+2 (t)
−yn+1 (t))) + λ(yn+1 (t + τ ) − yn+1 (t)
−yn (t + τ ) + yn (t)) + λε0 (yn+2 (t + τ )
−yn+2 (t) − yn+1 (t + τ ) + yn+1 (t)),
(17)
where V (h̄) = dV (x̄)/dx̄ with x̄ = h̄ = h + ε0 h, and
ε0 = ε(h).
Expand yn in the Fourier modes, that is, yn (t) =
A exp(ikn + zt). Then, Eq. (9) can be rewritten as
(ezτ − 1)(ezτ − λ(eik − 1) − λε0 e2ik + λε0 eik )
−τ V (h̄)(eik − 1)(1 + ε0 eik ) = 0.
(18)
Solving Eq. (18) with respect to z, we find that the leading
term of z is the order of ik. As z → 0 when ik → ∞, we can
express the long-wave solution of z as z = z 1 (ik)+ z 2 (ik)2 +
. . .. Inserting this into Eq. (18) and neglecting the second and
higher order terms, we obtain two roots of z, as follows:
z 1 = (1 + ε0 )V (h̄),
z 2 = −1.5τ (1 + ε0 )2 (V (h̄))2 + 0.5V (h̄)
(19)
+1.5ε0 V (h̄) + λ(1 + ε0 )2 V (h̄).
Clearly, the flow becomes unstable if z 2 < 0, and stable if
z 2 > 0. Thus, the demarcation point between stable and
unstable conditions (hereafter called the “neutral stability
point”) is
α=
3(1 + ε0 )2 V (h̄)
1
=
.
τ
(1 + 3ε0 ) + 2λ(1 + ε0 )2
(20)
Thus, an unstable flow will evolve from a small perturbation
in the uniform flow, if the following condition holds
α<
3(1 + ε0 )2 V (h̄)
.
(1 + 3ε0 ) + 2λ(1 + ε0 )2
123
(21)
Fig. 2 Phase diagram in the headway-sensitivity space. Each model
has a pair of curves that have the same maximal sensitivity. The upper
curve is called the “coexisting curve” and the lower the “neutral curve”.
For each pair of curves, the space is divided into three regions: the stable
region above the coexisting curve, the metastable region between the
neutral and coexisting curves, and the unstable region below the neutral
curve. The curves given by the full velocity difference model are very
close to those by the extended OV model and thus are not depicted in
the figure
Figure 2 shows the neutral stability curves in the
parameterized space (x, α), based on Eq. (20), where the
sensitivity parameter is α = 1/τ . For comparison purposes,
the curves given by the optimal velocity (OV) model and the
extended OV model are also depicted in the figure. Note from
Eq. (14), that V (h̄) reaches the maximal value of 0.5vmax
at the turning point of h̄ = h c , and thus the critical points
(h c , αc ) for these neutral stability curves exist. One curve
corresponds to a specific car-following model that is characterized by the value of (ε0 , λ). Figure 2 shows that with the
same value of h c , the value of αc given by our new model
is the lowest. Therefore, the model proposed in this study
pulls the neutral stability curve down, which enlarges the
stability region for uniform flow in the parameterized space
(x, α).
We then compute the values of αc against different ε0 values, by changing ε0 from 0 to 1/3. The results are shown in
Fig. 3. It can be seen that αc gets down when ε0 increases.
This implies that the stability of traffic flow can be improved
if the following vehicle pays greater attention to the possible lane changing behavior of the leading vehicle. We do
not investigate the situations with 1/3 < ε0 < 1, because
in these situations the leading vehicle may actually change
lane, which renders the anticipation assumption invalid and
is beyond the scope of this study.
A car-following model with the anticipation effect of potential lane changing
403
A42 =
(1 + ε0 )2 (1 + ε0 ) V (h c ),
6
1 + 15ε0 5b4 τ 3
−
V (h c )
8
24
4b + 6b2 τ + 4b3 τ 2
−λ
24
28b + 18b2 τ + 4b3 τ 2
,
−λε0
24
1 + 3ε0
V (h c )
(1 + ε0 )2
.
=
6
2
A51 =
A52
Fig. 3 The αc value against lane changing probability for 0 < ε0 ≤ 1/3
4 Nonlinear analysis
To further study the anticipation effect of lane changing
probability on the stability of traffic flow, we also conduct a
nonlinear analysis for investigating the slowly varying behavior of long waves in the stable and unstable regions. We introduce slow scales for space variable n and time variable t and
define slow variables X and T as follows:
X = θ (n + bt) and T = θ 3 t, 0 < θ 1,
(22)
(23)
where R(X, T ) is a function to be determined.
As 0 < θ 1, the lane changing probability ε(xn+1 (t))
is approximated by
εn+1 = ε(xn+1 (t))
(24)
Denote ε(h c ) as ε0 . Substituting Eqs. (22)–(24) into Eq. (13),
expanding θ by the Taylor series to the fifth order, and neglecting the terms that are multiplied by ε (h c )θ m , m ≥ 3, we
obtain the following nonlinear partial differential equation:
θ 2 (b − (1 + ε0 )V (h c ))∂ X R + θ 3 A3 ∂ X2 R
+θ (∂T R +
4
A41 ∂ X3 R
− A42 ∂ X R )
+θ (3bτ ∂ X ∂T R + A51 ∂ X4 R − A52 ∂ X2 R 3 ) = 0,
(25)
where
dV (x) V (h c ) =
,
dx x=h c
d3 V (x) ,
V (h c ) =
dx 3 x=h c
1 + 3ε0 3b2 τ
−
V (h c ) − λ(1 + ε0 )b,
A3 =
2
2
A41
θ 4 (∂T R − m 1 ∂ X3 R + m 2 ∂ X R 3 )
+ θ 5 (m 3 ∂ X2 R + m 4 ∂ X2 R 3 − m 5 ∂ X4 R) = 0,
(27)
m1 = −
V (h c )
6
7
(1 + ε0 )(1 + 3ε0 + 2λ(1 + ε0 )2 )2 − (1 + 7ε0 )
9
−λ(3(1 + ε0 ) + (1 + ε0 )2 (1 + 3ε0 + 2λ(1 + ε0 )2 ))
−λε0 (9(1 + ε0 ) + (1 + ε0 )2 (1 + 3ε0 + 2λ(1 + ε0 )2 )) ,
(1 + ε0 )3 V (h c ),
6
1 + 3ε0 + 2λ(1 + ε0 )2 V (h c ),
2
V (h c )
1 + 3ε0
(1 + ε0 )2
,
m4 =
6
2
m3 =
m5 = −
5(1 + ε0 )(1 + 3ε0 + 2λ(1 + ε0 ))3 V (h c )
216
1 + 15ε0 V (h c ) + λ(1 + ε0 )
24
36+18(1+3ε0 +2λ(1+ε0 ))+4(1+ε0 )(1+3ε0 +2λ(1+ε0 ))2
×
72
+
3
5
(26)
where τc = ((1 + 3ε0 ) + 2λ(1 + ε0 )2 )/(3(1 + ε0 )2 V (h c )).
Then, Eq. (25) can be rewritten as
m2 = −
= ε(h c + θ R(θ (n + 1 + bt), θ 3 t))
= ε(h c ) + ε (h c )(θ R + θ 2 ∂ X R).
τ/τc = 1 + θ 2 ,
where
where b is a constant to be determined later. Let
xn (t) = h c + θ R(X, T ),
Taking b = (1+ε0 )V (h c ), we can eliminate the second order
terms of θ from Eq. (25). We consider the neighborhood of
the critical point τc , such that
7b3 τ 2 −(1 + 7ε0 )V (h c )−λ(3b + 3b2 τ )−λε0 (9b + 3b2 τ )
,
=
6
+ λε0 (1 + ε0 )
252+54(1+3ε0 +2λ(1+ε0 ))+4(1+ε0 )(1+3ε0 +2λ(1+ε0 ))2
×
.
72
To derive the regularized equation, we make a transformation for Eq. (27), as follows:
T̂ = m 1 T,
R=
m1
R̂.
m2
(28)
123
404
T. Tang et al.
Then, Eq. (27) can be rewritten as the following regularized
equation.
∂T̂ R̂ = ∂ X3 R̂ − ∂ X R̂ 3
1
m1m4 4
m 3 ∂ X2 R̂ +
−θ
∂ X R̂ − m 5 ∂ X2 R̂ 3 .
m1
m2
(29)
Equation (29) is simply the modified KdV equation after
ignoring the perturbed term O(θ ). The kink solution of this
equation is
√
c
R̂0 (X, T̂ ) = c tanh
(X − c T̂ ) ,
(30)
2
the stable region above the coexisting curve, the metastable
region between the neutral and coexisting curves, and the
unstable region below the neutral curve. It can be seen that
both the neutral and the coexisting curves decrease when the
lane changing probability increases from 0 (the OV model) to
a positive value (the extended OV model) and further decline
when the relative velocity coefficient increases from 0 (the
extended OV model) to a positive value (the proposed model).
Hence, the stability region is enlarged, and the metastable
and unstable regions are reduced, when compared to other
models. This shows that the simultaneous consideration of
the two factors enhances the stability of traffic flow.
where c is the propagation speed of the kink wave. This speed
is determined by the O(θ ) term, which is solvable if
5 Simulation
+∞
( R̂0 , M[ R̂0 ]) ≡
d X R̂0 (X, T̂ )M[ R̂0 (X, T̂ )] = 0,
Huang [33] employed the cellular automaton model to study
lane changing behavior and found that lane changing probability increases with traffic density when traffic is light and
decreases when it is heavy. According to this and our observations in reality, we set the lane changing probability ε(x)
as follows:
⎧
0,
x ≤ x1c ,
⎪
⎪
⎪
⎪
⎪
⎪
⎪
x − x1c
⎪
⎪ 0.1
, x1c < x < x2c ,
⎨
x2c − x1c
(35)
ε(x) =
⎪
⎪
x
−
x
⎪
3c
⎪
−0.1
, x2c < x < x3c ,
⎪
⎪
⎪
x3c − x2c
⎪
⎪
⎩
0,
x ≥ x3c ,
(31)
−∞
where
M[ R̂0 ] =
1
m1
m1m4 2 3
m 3 ∂ X2 R̂ +
∂ X R̂ − m 5 ∂ X4 R̂ .
m2
After integration, we can obtain the propagation speed
c=
27(1 + 2λ)(1 + ε0 )
.
1 + 3ε0 + 26λ(1 + ε0 ) − 19λ2 (1 + ε0 )2 − 20λ3 (1 + ε0 )3
(32)
Thus, we have derived the solution of the modified KdV
equation. That is,
m1
c
(X − m 1 cT ).
tanh
(33)
R(X, T ) =
m2
2
If the optimal velocity takes the form of Eq. (14), then
V (h c ) = 1 and V (h)c = −2. The amplitude of the kink
solution is given by
1/2
m 1 c αc
−1
A=
,
(34)
m2 α
0)
with αc = (1+3ε3(1+ε
2.
0 )+2λ(1+ε0 )
The kink wave solution represents the coexisting phase,
which consists of the freely moving phase with low density
and the congested phase with high density. The headway of
the freely moving phase is x = h c + A, and that of the
congested phase is x = h c − A, by which we can depict
the coexisting curves in the parameterized space (x, α).
Figure 2 shows the coexisting curves that are generated by
the models and those that are generated by the other two
models for comparison. Each model has a pair of curves that
have the same maximal sensitivity. The upper curve is the
coexisting curve and the lower the neutral stability curve. For
each pair of curves, the space is divided into three regions:
2
123
where x1c , x2c , and x3c are three constants with x1c <
x2c < x3c . Equation (35) states that lane changing is
impossible when headway is too small and is not necessary when it is too large. Thus, lane changing probability
increases when the headway is within [x1c , x2c ] and
decreases when it is within [x2c , x3c ].
The initial values of the model parameters for simulation
are given below:
xn (0) = xn (1) = x0 , n = 0.5N , n = 0.5N + 1,
xn (0) = xn (1) = x0 + 0.1, n = 0.5N ,
(36)
xn (0) = xn (1) = x0 − 0.1, n = 0.5N + 1,
where N (= 200) is the number of vehicles and x0 (= 4.0)
is the average headway. A periodic boundary condition is
adopted in the simulation. Other parameters are
x1c = 4.0,
x2c = 10.0,
x3c = 30.0,
vmax = 2.0,
α = 2.0,
L = 800,
(37)
where L is the length of the highway. The values of above
parameters may be different from those calibrated from
A car-following model with the anticipation effect of potential lane changing
405
Fig. 4 The headway evolution
after 104 time steps and the
profile at t = 104 , a with the OV
model with λ = 0 and
ε(xn ) = 0, b with the FVD
model with λ = 0.1 and
ε(xn ) = 0, c with the
extended OV model with λ = 0
and ε(xn ), given by Eq. (35),
and d with the new model that is
proposed in this paper with
λ = 0.1 and ε(xn ), given by
Eq. (35)
Fig. 5 Flow q and potential
lane changing volume qpLC at
t = 104 , a with the OV model
with λ = 0 and ε(xn ) = 0,
b with the FVD model with
λ = 0.1 and ε(xn ) = 0, c with
the extended OV model with
λ = 0 and ε(xn ), given by
Eq. (35), and d with the new
model that is proposed in this
paper with λ = 0.1 and ε(xn ),
given by Eq. (35)
123
406
T. Tang et al.
reality. It is, however, expected that the qualitative findings
from this study would not be influenced by these values.
Figure 4 depicts the headway evolution after 104 time steps
and the profile at t = 104 . From these figures, we have the
following findings.
1. In Fig. 4a–c, the stop-and-go traffic appears. It can be
seen that the headway evolution and the profile are very
similar to the solution of the mKdV equation. This is
because the initial value condition lies in the unstable
region when the OV model, the FVD model, and the
extended OV model are used to reproduce the evolution
of small perturbations (36). When small perturbations
are put into a uniform traffic flow, they are amplified
with time, and consequently, the uniform traffic flow
finally evolves into an inhomogeneous flow. The jam in
Fig. 4a is the most serious. In Fig. 4d, the perturbation
finally disappears. This verifies that the introduction of
lane changing probability can improve the stability of
traffic flow.
2. Figure 4b and c show that considering relative velocity
and lane changing probability separately cannot completely eliminate perturbation. These two factors must
be covered simultaneously in the car-following model to
enhance the stability of traffic flow.
3. The density waves in Fig. 4a–d always propagate backwards. This has been observed in reality and reported in
literature.
4. Figure 4c shows that only considering the lane changing
probability in the optimal velocity may produce some
shocks and smooth transition layers. This is very similar
to the results obtained in [34].
We now further investigate the effects of lane changing
probability on traffic flow and on so-called potential lane
changing volume. The potential lane changing volume is
defined below.
qpLC (x) = q(x)ε(x),
(38)
where ε(x) is the lane changing probability at x. Although
real lane changing is not here allowed, there is potential lane
changing volume due to the existence of lane changing probability.
Figure 5a shows that the OV model generates oscillating
waves in both q and qpLC . Figure 5b and c illustrate that the
oscillating waves still exist when the relative velocity and
the lane changing probability are considered separately in
the car-following models. The oscillating wave disappears
in Fig. 5d. This further verifies that the introduction of lane
changing probability can enhance the stability of traffic flow.
Tang et al. [30] found that frequent lane changing produces
oscillating waves; here, we show that unstable flow leads to
more frequent lane changing. This just proves such a fact that
123
unstable flow will produce frequent lane changing if real lane
changing is allowed. The potential lane changing volumes in
Fig. 5a and b are obviously larger than those in Fig. 5c and d.
Figure 5d gives the most stable traffic flow and, consequently,
the lowest potential lane changing volume. Thus, drivers do
not like to change lanes in a stable flow.
6 Summary
In this paper, we present a new car-following model taking
into account the anticipation effect of potential lane changing by the leading vehicle in the acceleration equation of
the following vehicle. The stability of traffic flow is studied
analytically by using linear and nonlinear analyses. A critical point associated with neutral stability curve exists in the
new model. The traffic behavior near this point is described
by a modified KdV equation. It is found that the consideration of lane changing probability can stabilize traffic flows.
Simulation results are largely consistent with the analytical
investigation.
It should be pointed out that in our modeling and the simulation, the leading vehicle does not actually perform a lane
changing action. The driver’s intention is only anticipated by
the driver of the following vehicle when he or she makes a
decision to accelerate or decelerate his/her vehicle. Nevertheless, we found that such anxiety does indeed influence the
driving behavior and stabilize traffic flow.
Some theoretical findings from this study can be used
to improve the lane changing rules adopted in reality. For
instance, our study shows that when the lane changing probability exceeds some value, the critical point αc will increases
with the probability, which implies that the flow stability
declines and thus lane changing should not be encouraged
furthermore. In addition, our study also shows that it is much
easier for unstable traffic flow to lead to frequent lane changing than the stable flow. So, the controller should pay much
attention to traffic when it is in unstable state.
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