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A new car following model: comprehensive optimal velocity model
Jun fang Tian , Bin jia , Xin gang Li
MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, Beijing 100044, P.R.China
E-mail: [email protected]
Contributions
Results
We present a new car-following model, i.e.
comprehensive optimal velocity model (COVM), whose
optimal velocity function not only depends on the
following distance of the preceding vehicle, but also
depends on the velocity difference with the preceding
vehicle. Simulation results show that COVM is an
improvement over the previous ones theoretically. The
unrealistically high deceleration, which appears in OVM
and FVMD, will not appear in COVM. Furthermore, the
accident in the urgent braking case, which can not be
avoid in OVM and FVDM , can be avoid in COVM.
Model
In the real traffic, when the driver adjusts his speed, he
will consider the following conditions: the space headway
with his leader, the velocity difference with his leader and
so on. So, we believe the optimal velocity function is not
only just a function of the following distance, but also
should be determined by the speed difference, i.e.
Vop=V(Δxn(t),Δvn(t)). So, the dynamic behavior of
vehicles could be modeled by the following equation:
During the simulation, the value of L and c3 is selected as L = 6, c3 =0.35, other parameters are selected
in Table 1. Firstly, we simulate the vehicles’ behaviors under the decelerating case that a freely moving car
from a large distance reaches a standing car. The decelerating case is carried out as follows: The leading car
stops all the time, the follower moves with the speed 14.5m/s, the headway between them is 150m at the
initial time t = 0s. Simulation results are shown in Fig.1. One can see that the maximum deceleration rate are
6.51, 7.30, 2.98m/s2 in OVM, FVDM and COVM respectively. Since empirical deceleration should not be
larger than 3m/s2[2], the deceleration in OVM and FVDM is unrealistic high. One also could note that the
space headway is lower than 5m in OVM, so crash occurs. As to the COVM, neither high deceleration nor
crash occurs.
Secondly, we simulate the vehicles' behaviors under the urgent braking case referred to[4]. The urgent
braking case is carried out in the following. Two successive cars move with the same speed 14.5m/s at the
initial time t=0s and the space headway is 20m. The leading car decelerates suddenly with the deceleration
-8m/s2 until it stops completely. The leading car remains standing for several seconds before accelerating
back to its original speed. The simulation results are shown in Fig.2(a), which exhibits the variation of space
headway of the following car. One can see that the space headways are lower than 5m in OVM and FVDM,
this indicates that the leader and the follower collides in OVM and FVDM. But because the follower could
adjust his speed timely, so the space headway is always larger than 5 m in the COVM, i.e. the COVM avoids
the accident successfully.
For simplicity, we take:
 is the reaction coefficient to the relative velocity,
<1, so we get:
0<
Taking    , we could get the new model:
Both  and  are sensitivity. Because the optimal
velocity function in the new model is more comprehensive
than those in the existing models, so we call our new
model as comprehensive optimal velocity model (COVM).
V1(xn(t ))is selected as follows:
Lc is the length of the vehicle, which can be taken as
5m in simulations, v1 = 6.75m/s, v2 = 7.91m/s, c1 =
0.13m/s, c2 = 1.57m/s.
V2(vn(t ))is selected as below:
Fig. 1 The simulations in the OVM, FVDM and COVM under the
case that a freely moving car approaching a standing car, (a)
represents the acceleration of the follower, and (b) represents
the headway distance of the follower.
Fig. 2. The simulations in
the OVM, FVDM and COVM
under an urgent case, (a)
represents the headway
distance of the follower.
Thirdly, the delay time of car motion t and kinematic wave speed cj at jam density are examined in
COVM. We carried out the simulation as that in the reference[3]. First a traffic signal is yellow and all
vehicles are waiting with headway 7.4m , at which the optimal velocity is zero. Then at time t=0s, the signal
changes to green and cars begin to move. The simulation results are shown in Fig.3(a) and Table 1. From
Table 1, one can see that the values of and
t cj are 1.28s and 20.81m/s in COVM. As Bando et al. pointed out,
the observed isof
t the order of 1s[5], and Del Castillo and Benitez indicated that cj ranges between 17 and
23 km/h[6]. So, the COVM is successful in anticipating the two parameters.
Fig.3(b) shows the variation of acceleration under the case that two successive car initially at rest, and the
leading car is unobstructed. At t=0s, they begin to start up according to the OVM, FVDM and COVM. One
can see that the maximum value of the leading car's acceleration in COVM is not greater than that in the other
two models. As for the following car, the car in COVM accelerates more quickly than others, so the delay
time of COVM is shorter than others.
From above simulations, one can see that the COVM describes the traffic dynamics most exactly, which
verifies that the improvement in COVM is reasonable and realistic.
Where, L and c3 are constants. The simulation results
will show that this form is reasonable and realistic.
Model
δt (s)
cj(km/h)
OVM(κ=0.85 s－1, λ=0 s－1)
1.6
16.65
GFM(κ=0.41 s－1, λ=0.5 s－1)
2.2
12.11
FVDM(κ=0.41
s－1,
s－1)
1.4
19.03
CVOM(κ=0.41 s－1, λ=0.5 s－1)
1.28
20.81
Table 1
λ=0.5
the values of δt and cj
References
1. M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y.
Sugiyama, Phys. Rev. E. 51:1035-1042, 1995
2. D. Helbing and B. Tilch, Phys.Rev. E. 58:133-138, 1998
3. R. Jiang, Q. S. Wu, and Z. J. Zhu, Phys. Rev. E.
64:017101, 2001
4. X. M. Zhao and Z. Y. Gao, Eur. Phys. J. B 47:145-150,
2005
5. M. Bando, K. Hasebe, K. Nakanishi, and A. Nakayama,
Phys. Rev. E 58:5429-5435, 1998
6. J.M. Del Castillo and F.G. Benitez, Transp. Res., Part B:
Methodol. 29:373-406, 1995
Fig. 2. The simulations in the
OVM, FVDM and COVM under
an urgent case, (b) represents
the velocity of the leader.
Fig. 3. (a) exhibits the variation of velocity of all vehicles
starting from a traffic signal in COVM. (b) exhibits the
variation of acceleration of unobstructed leading car and its
following car both initially at rest in OVM, FVDM, and COVM.
Conclusions
In this paper, we present a new car-following model, i.e. comprehensive optimal velocity model (COVM),
in which the optimal velocity function not only depends on the following distance of the preceding vehicle,
but also depends on the velocity difference with the preceding vehicle. The simulation results show that the
unrealistically high deceleration will not appear in COVM, and the accident in the urgent braking case can be
avoid in COVM.
Funding
This work is financially supported by 973 Program (2006CB705500), the National Natural Science
Foundation of China (70501004 and 70701004), Program for New Century Excellent Talents in University
(NCET-07-0057), and the Natural Science Foundation of Beijing (9093020).