Formation of Synchronized Flow at the Upper Stream of Bottleneck in Optimal Velocity Model

ELSEVIER

Copyright © IFAC Control in Transportation Systems, Tokyo, Japan, 2003

IFAC PUBLICATIONS www.elsevier.com/localc/ifac

FORMATION OF SYNCHRONIZED FLOW AT THE UPPER STREAM OF BOTTLENECK IN OPTIMAL VELOCITY MODEL M. Kikuchi· Y. Sugiyama·· S. Tadaki ••• S. Yukawa····

• Cybermedia Center, Osaka University, Toyonaka 560-8531, Japan •• Graduate School of Information Science, Nagoya University, Nagoya, 464-8601, Japan ••• Computer and Network Center, Saga University, Saga 840-8502, Japan •••• Department of Applied Physics, University of Tokyo, Bunkyo 113-8656, Japan

Abstract: The synchronized flow is observed during congestion at the upper stream of a bottleneck. One of the main features of the synchronized flow is the complex behavior in the fundamental diagram (the density-flow relation). Computer simulations based on the coupled-map optimal velocity model show that the synchronized flow is observed even in one-lane expressways. The model also reproduces the other features of the congested two-lane traffic at the upper stream of a bottleneck. Copyright © 2003 IFAC Keywords: synchronized flow, coupled map, optimal velocity, bottleneck

1. INTRODUCTION

model is an introduction of an optimal velocity (OY) function.

The study of the traffic flow has been activated since early 1990s in the physical point of view. Some simple models have been proposed and been studied with analytic methods and computer simulations in the context of nonequilibrium statistical physics and pattern formation. Since late 1990s, those researches have been improved based on observational data (Helbing et al., 2000; Chowdhury et al., 2000; Helbing, 2001; Nagatani, 2002).

In the OY model, each car controls its acceleration d 2 x/de to tune its speed dx/dt to the optimal (safety) velocity Voptimal, which depends on the headway distance ~x to its preceding car.

ddt2 2x

=

0:

[

Voptimal

dX]

(~x) - dt

'

(1)

where 0: is the susceptibility and corresponds to the inverse of the delayed response time. The OY function should be a sigmoidal function of the headway distance.

One of modeling methods for the traffic flow is a car-following model (Leutzbach, 1988). The method of car-following traffic flow models has been greatly improved by introducing the new model called Optimal Velocity (OY) Model (Bando et al., 1995a). The central concept of the OY

The OY model has a very simple form of secondorder differential equations. We can obtain an exact solution for a sequence of cars in a circuit (periodic boundaries), if we employ a step function as

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the OV function (Sugiyama and Yamada, 1997). By analytic discussions with the OV model with periodic boundaries, independent of the detail form of the OV function, the emergence of the traffic jam can be understood as the instability of the uniform flow (Bando et al., 1995a). Namely the uniform flow with an average headway b is unstable if V:ptimal(b) > 0/2, where I denotes a derivative (Bando et al., 1995a). For realistic simulations, we employ the following OV function: Voptimal

d) + ],(2)

Vmax [ tanh (boX (box) = -2~

We can introduce effects of noises into the CMOV model. Noises in car motions are very complex mixture of human, machine and road condition factors. In the simulations, we just install a simple form of noise by replacing Eq. (4) with

Vk(t

The parameter f denote the noise level and (-0.5,0.5) is a uniform random number.

w Vmax

c

value 25. 23.3 33.6 0.913

Fig. 1. The configuration of the one-lane simulation system.

(unit) m m m/sec

We construct a simulation system where it has one lane with the injection and ejection flow (Fig. 1). A bottleneck structure is introduced at the end of the system, where all cars reduce their desired speed V max in the OV function. Without bottleneck, we can not observe the traffic jam. The injection and ejection methods employed in this simulation are very simple. A car is injected with a probability Pinjection, if there is a space larger than boXmin, which corresponds to the minimum value of headway, from the left edge of the road. This injection method corresponds the virtual queue with infinite length at the injection point. If a car can not find the preceding car within the system, the headway is set to be infinite. If such a car reaches the right edge (the ejection point) of the system, the car just disappears from the system.

In the CMOV model, the speed Vk and the position Xk of k-th car is updated according to the following rules. Values of speed and position for all cars are updated pararellly.

At the upper stream of the bottleneck, the synchronized flow is known to be observed (Kerner and Rehborn, 1996; Lee et al., 2000; Tadaki et al., 2002b; Tadaki et al., 2002a) during congestion. The congested traffic flow on the fast and slow lanes synchronize each other. Namely the difference of the average speed of the flow both on two lanes remains small. In the fundamental diagram (the density-flow relation), the synchronized flow appears as a scattered data in the right side of the peak (Fig. 2).

+ bot) = Vk(t) + 0 [V (boXk(t» - Vk(t)] bot, (3)

+ bot) = Xk(t) + vk(t)bot,

E

L

By discretizing the OV model In time, we define the Coupled Map Optimal Velocity (CMOV) model (Tadaki et al., 1998; Tadaki et al., 1999; Uchida and Tadaki, 2001). The car sequences in the CMOV model can be modified as a result of the discretization. The advantages of the CMOV model include that the model can be applied to the system with injection and ejection flow and we can construct two-lane expressway systems with the model.

Xk(t

~

3. SYNCHRONIZED FLOW IN ONE-LANE EXPRESSWAYS

2. COUPLED MAP OPTIMAL VELOCITY MODEL

Vk(t

(5)

C

Table 1. The parameter in the optimal velocity function. d

[V (boXk(t» - Vk(t)] bot]

x(l+fO.

where parameters V max , d, wand c can be obtained by observation of car-following behavior. Parameters in the optimal velocity functio:l (Table 1) used in the following simulations are compatible values with those obtained by observations (Bando et al., 1995b).

parameter

+ bot) = [Vk(t) + 0

(4)

where boXk(t) is the headway distance to the preceding (k -l-th) car. The time step bot is fixed one. In the following simulation, the time step is fixed as bot = O.lsec, which corresponds the human reaction time for traffic stimuli.

The synchronized flow seems not to be proper to two-lane road systems. During congestion, it may be difficult for cars to find a chance to change

348

200 q (1/5min)

200 q (1/5min) o

slow lane x

150

150

100

100

50

50 o

o

o

o

~

0

0

fast lane

0 ·0

50

100

150

0.fE"'------r---......------. 0

50

p(1/Km)

100

150

p(1/Km)

Fig. 2. Fundamental diagram (density-flow relation) observed at the upper stream of Nihonzaka Tunnel in Tomei Expressway, August 1996). 8000 1....

8500 -"

9000

-'-'

formation of weak congestion in the vicinity of the beginning point of the bottleneck. Figure 4 shows the space-time diagram of the car density at the upper stream of a bottleneck.

9500 (m) -"

1900

1950

100 2000 (sec)

Fig. 3. The space-time diagram of car orbits at the upper stream of a bottleneck in an onelane simulation system. Each line represents the orbit of a car. Dense lines shows the emergence of traffic jam.

E

2. 50 a.

lanes. Therefore two lanes are almost decoupled during congestion. So the important point of the synchronized flow will not be the synchronization between lanes. The growing fluctuation at the upper stream of bottleneck seems to be the essential point of the synchronized flow.

Fig. 4. The space-time diagram of the density at the upper stream of a bottleneck in an onelane simulation system.

In other words, the synchronized flow will be observed even in one-lane expressways. The complex behavior in the fundamental diagram corresponding to the synchronized flow should be understood as a transient phenomenon under the growing fluctuation.

In the vicinity of the bottleneck, the flow runs, as an average, with low speed without oscillation. The fluctuation in the density grows with the distance from the bottleneck. The instability included in the low speed flow near the bottleneck effects the scattered data in the fundamental diagram.

Figure 3 shows the space-time diagram of car orbits at the upper stream of a bottleneck. The simulated system is 10,000m long. The desired speed V max distributes ±20% around the average. In the last 1,000m segment, the maximum speed is reduced to 50% as a bottleneck. We can see the

Figure 5 is the fundamental diagram corresponding to the simulation results shown in Fig 4. Data are observed at 5,000m, 6,000m and 7,000m points from the injection. The data are scattered widely

349

200

D

-~100 --_. . ---

_om ____

~ D

•

c

.

-...

100

: 5000m

--..,

150

7000m

(.)oonr:~

oSlow lane

D

D

~....,

50

_OODaPJoa -CJ~ ~Q2 D _ l< 0

~

1t~

.".."

50

"'. "'.-

..

0/0

'>oil'

..

o o

o+------::r:'-------:-! o 50 100

o-f"---.,....--.,....--.,....---, o 100 200 300 400

P (1/Km)

q (1/5min)

Fig. 5. The fundamental diagram corresponding to the simulation in Fig. 4. The observation points are 5,000m, 6,000m and 7,000m from the injection point.

Fig. 7. Lane occupation ratio observed at the upper stream of Nihonzaka Tunnel in Tomei Expressway, August 1996.

near the bottleneck at the right side of the peak. Namely, we can reproduced the synchronized flow on one-lane expressway at the upper stream of a bottleneck.

exceeds the one on the slow lane (Fig. 7), on the contrary to the traffic regulation in expressways. The reverse lane usage phenomena usually happen except midnights and early mornings in the observation data of Fig. 7. During the free-flow states, the car densities on both lanes are almost the same values. The excess of the average speed on the fast lane causes the excess of the flow. During congestion, the average speeds on both lanes take almost the same values. The excess of the flow on the fast lane comes from the excess of the density on the fast lane.

4. TWO-LANE SYSTEM

The coupled-map optimal velocity model can be applied to two-lane expressway systems. The schematic view of the system is shown in Fig. 6. L Injection

100 aslow lane

Fig. 6. The configuration of the two-lane simulation system. The detail features of lane-changing behavior are not clearly understood. So we introduce a set of reasonable and simple lane-changing rules. If the headway to the preceding car is not large enough to run with the desired speed, the car hopes to change lane. There must be a enough room in the target lane for safe lane-change. If these two conditions are fulfilled, the car moves to the target lane with some probability.

50

0t----:-:!-::---=-=--~r::_--__:_J o 100 200 300 400 q (1/5min)

Usually all cars are usually asked to run on the slow lane in expressways. The fast lane should be used only for overtaking. The lane-changing rules should be asymmetric between lanes. Therefore some rules should be added for cars on the fast lane for going back to the slow lane.

Fig. 8. Lane occupation ratio observed at the upper stream in the simulation. The simulation system can reproduce the same feature depending on the parameter set (Fig. 8). Through the simulation we find the importance

If the total flow is large enough, we can observe the reverse lane usage, where the flow on the fast lane

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of the acceleration on the fast lane for overtaking. In the model, cars on the fast lane change their desired speed V max to fv max where f 2: 1 is a overtaking factor. The reverse lane usage is difficult to be reproduced, if the overtaking factor f = 1. The overtaking factor induces the excess of the flow during the free-flow states.

ACKNOWLEDGEMENTS A part of this work is financially supported by Grant-in-Aid No. 15607014 from t.linistry of Education, Science. Sports and Culture, Japan.

REFERENCES

The current simulation model can not reproduce the difference of the density well on lanes during congestion. The observational data suggests that the OV function should be modified when a car moves to the fast lane.

Bando, M.. K. Hasebe, A. Nakayama. A. Shibata and Y. Sugiyama (1995a). Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E 51,1035-1042. Bando, M., K. Hasebe. K. Nakanishi, A. Nakayama, A. Shibata and Y. Sugiyama (1995b). Phenomenological study of dynamical model of traffic flow. J. Phys. I France 5, 1389-1399. Chowdhury, D., L. Santen and A. Schadschneider (2000). Statistical physics of vehicular traffic and some related systems. Physics Reports 329, 199-329. Helbing, D. (2001). Traffic and related selfdriven many-particle systems. Rev. Mod. Phys. 73, 1067-1141. Helbing, D., Herrmann, H. J., Schreckenberg, M. and Wolf, D. E., Eds.) (2000). Traffic and Granular Flow '99. Springer-Verlag. Berlin. Kerner, B. S. and H. Rehborn (1996). Experimental properties of complexity in traffic flow. Phys. Rev. E 53, R4275-R4278. Lee, H. Y., H.-W. Lee and D. Kim (2000). Phase diagram of congested traffic flow: An empirical study. Phys. Rev. E 62, 4727-4741. Leutzbach, W. (1988). Introductoin to the theroy of traffic flow. Springer-Verlag. Berlin. Nagatani, T. (2002). The physics of traffic jam. Rep. Prog. Phys 65, 1331-1386. Sugiyama, Y. and H. Yamada (1997). Simple and exactly solvable model for queue dynamics. Phys. Rev. E 55, 7749-7752. Tadaki, S., K. Nishinari, M. Kikuchi, Y. Yugiyama and S. Yukawa (2002a). Analysis of congested flow at the upper stream of a tunnel. Physica A 315,156-162. Tadaki, S., K. Nishinari, M. Kikuchi, Y. Yugiyama and S. Yukawa (2002b). Observation of congested two-lane traffic caused by a tunnel. J. Phys. Soc. Jpn. 71, 2326-2334. Tadaki, S., M. Kikuchi, Y. Sugiyama and S. Yukawa (1998). Coupled map traffic flow simulator based on optimal velocity functions. J. Phys. Soc, Jpn. 67, 2270-2276. Tadaki, S., M. Kikuchi, Y. Sugiyama and S. Yukawa (1999). Noise induced congested traffic flow in coupled map optimal velocity model. J. Phys. Sac, Jpn. 68, 3110-3114. Uchida, T. and S. Tadaki (2001). Congested flow induced by noise in headway measurements. J. Phys. Sac, Jpn. 70, 1842-1848.

5. SUMMARY The traffic flow have attract researchers from a variety of research fields. The physical research interests have been activated since early 1990s. One of the physical traffic flow models is the optimal velocity model. The optimal velocity (OV) model enables us to understand the traffic phenomena as many-body physical processes. The emergence of the traffic jam can be described as the result of the instability in the uniform traffic flow. The coupled-map optimal velocity (CMOV) model was introduced as a discretized version of the OV model. By the discretization in time, the CMOV model becomes suitable for computer simulations of various configurations. The model can be applied to open-boundary systems with two lanes. In an one-lane-expressway model, we studied the emergence of the synchronized flow at the upper stream of a tunnel. In the vicinity of the entrance of the tunnel. we observed the low-speed flow. The fluctuation included in the low-speed flow grow according to the upper stream. The complex behavior appearing in the fundamental diagram is the result of the instability of the low-speed flow at the upper stream of a bottleneck. The synchronized flow has been observed in twolane expressways. The observational data in onelane expressways, however, have never been analyzed in the context of the synchronized flow. In an two-lane-expressway model, we studied the reverse lane usage phenomenon. The reverse lane usage phenomena have been observed during congestion in the traffic data at the upper stream of a bottleneck. The simulation with the CMOV model can reproduce the fundamental features of the reverse lane usage. By introducing the overtaking factor, the flow on the fast lane exceeds the flow on the slow lane.

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