Coupled Map Traffic Flow Simulator Based on Optimal Velocity Functions

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1 Coupled Map Traffic Flow Simulator Based on Optimal Velocity Functions Shin-ichi Tadaki 1,, Macoto Kikuchi 2,, Yuki Sugiyama 3,, and Satoshi Yukawa 4, 1 Department of Information Science, Saga University, Saga 84, JAPAN 2 Department of Physics, Osaka University, Toyonaka 56, JAPAN 3 Division of Mathematical Science, City College of Mie, Mie 514-1, JAPAN 4 Department of Applied Physics, University of Tokyo, Bunkyo 113, JAPAN Abstract. A coupled map traffic flow model is introduced based on optimal velocity functions. The model can be simulated under open boundary conditions. It is able to bear a variety of noises as well. Noises in velocity induce increase of average car density to critical one, where power law behaviors appear in temporal spectra of density fluctuations. By introducing a traffic blockade, a hysteresis loop is reproduced in the headway-velocity plane even in open boundary systems. The exponents in temporal spectra of density fluctuations are independent on the noise level in the cases with a traffic blockade. 1 Introduction Car-following type models describe the behaviors of cars with the systems of differential equations, with each equation describing the motions of one car. Each driver controls the acceleration of his or her car under the stimuli of the headway distance or relative velocity to the preceding car. In traditional carfollowing models, the delay in response to the traffic stimuli is introduced by hand[1]. The delay in acceleration will be naturally introduced, if the motions of the cars are described by second order differential equations of the position of cars. Bando et al. proposed a new car-following model called the Optimal Velocity (OV) model [2 7]. In their model a car controls the acceleration to tune the velocity to the optimal one, which depends on the headway distance x. [ V optimal ( x) dx dt ]. (1) d 2 x dt 2 = α The optimal velocity function V optimal ( x) is a sigmoidal one, for example, [ tanh V optimal ( x) = v max 2 tadaki@ai.is.saga-u.ac.jp kikuchi@phys.sci.osaka-u.ac.jp genbey@eken.phys.nagoya-u.ac.jp yukawa@ap.t.u-tokyo.ac.jp ( 2 x x neutral x width ) + c bias ], (2)

2 where parameters, v max, x neutral, x width and c bias, can be determined from observed data[4]. Bando et al. observed trajectories in the headway-velocity plane and discussed the instability of traffic congestion with linear stability analyses and computer simulations for periodic boundary systems. The model also reproduced some observed features of real traffic flow[4]. Exact solutions for some limiting cases were also found[5, 6]. Since each car controls its acceleration under the stimuli of the headway distance to the preceding car, the system of the differential equations, Eq. (1), requires the fixed sequence of cars. Due to this restriction, simulations and analytic studies were limited to periodic boundary cases. Yukawa and Kikuchi proposed an alternative approach to model traffic flow using a coupled map idea [8 1]. In contrast to the above-mentioned car-following type modeling using differential equations, it is a discrete-time model, so that this approach is highly suitable for computer simulations. In this paper, we propose a new traffic simulator by combining these two ideas, that is, the OV model and the coupled-map based modeling. 2 Coupled Map Simulator For making the OV model applicable to several realistic traffic situations, for example, open boundary systems and multi-lane roads, we transform it to a coupled-map form by time discretization. We define the coupled map traffic flow model with optimal velocity functions as follows: The position x and velocity v of a car at t + t are given with x(t + t) x(t) = v(t) t, (3) v(t + t) v(t) =α (V optimal ( x) v(t)) t, (4) where α is a response factor. It is stressed that the time discretization is not employed for numerical integrations but for taking various realistic situations into account. Actually, the time step t is taken as a fixed parameter here. To avoid rear-end collisions and backward motions, cars are forced to stop if the headway distance are less than the minimum value x min. One of the most important improvements brought by translation from the original OV model to the coupled map form is that the model allows us to rearrange the sequences of cars. Therefore the new model can be simulated under open boundary conditions, where new cars are created and added to the tail of the car-sequence and cars are deleted from the car-sequence when they reach the end of the system. Moreover we can construct multi-lane systems using the present model. Another advantage of the coupled-map model is to allow various types of noises to be introduced to the model. For introducing noises in velocities, for example, Eq. (4) is replaced with v(t + t) =[v(t)+α (v optimal v(t)) t](1+f noise ξ), (5) where ξ [.5,.5] is a uniform random variable and f noise is the noise level.

3 3 Simulations and Results In this paper we simulate systems with open boundary conditions. The schematic view of the systems is shown in Fig. 1. The course consists of three parts: The middle part of length L observe is used for the observation of the car density. Common parameters used in these simulations are given in Table. 1. All parameters except t are compatible with those in Ref. [4], which are determined so as to fit to the real data observed in Chuo Expressway in Japan[11]. Table 1. Common Parameters. There values, except t, correspond to those in Ref. [4]. parameter value (unit) x neutral 25. m x width 23.3 m v max 33.6 m/sec α 2. sec 1 c bias.913 t.1 sec 7.2 m x min Injection Ejection L 1 L observe L 2 Fig. 1. Schematical view of simulated systems. The system has no car at the initial state of simulations. From the left end, namely the upstream of the system, a new car is injected with zero speed at every second if the position of the car at the tailend of the car-sequence is larger than x min. This injection method corresponds to the situation that there is a virtual traffic jam with infinite number of cars in the upstream of the left end of the system and a new injected cars is one escaping from the virtual traffic jam[12]. We first observe trajectories in the headway-velocity plane. After relaxation of 2, second, a test car is injected from the left end of the system. The values of the headway distance x and velocity v of the test car are recorded for 1, seconds (Fig. 2). The newly injected car only accelerates to follow up the preceding car in the absence of noise. With noises in velocity, the velocity of the test car fluctuates around the maximum velocity v max. Traffic congestion is considered to emerge, because the amplitude of velocity fluctuations around v max is larger than f noise v max.

4 5 v (m/sec) v (m/sec) v (m/sec) f noise = f noise = f noise = Fig. 2. Trajectories in the headway-distance versus velocity plane. The system configuration is L 1 = L 2 =1, m. Noises in velocity are f noise =.,.5 and.1. We observe the density of cars to investigate the emergence of traffic jam. Observations are done in the middle part with L observe = 1km length. The noise in velocity enhances the amplitude of the density fluctuation and the average density (Fig. 3), where the average density increases as ρ + ρ 1 fnoise 2. In the high noise level region, the average density reaches to almost.27m 1, which corresponds to almost 37m of the average headway distance and thus allows most of cars run with the maximum velocity. Namely, the high level noise enhances the formation of the maximum throughput flow. The time sequences of the density ρ(t) are observed over T =1, seconds, and the temporal spectra of ρ(t), T 1 1 I(k) = ρ(t)e 2πikt/T, (6) T t= are calculated. In the higher noise level region than f noise.2, the temporal spectra obey the power law, I(k) k β, in the wide range of frequency, This corresponds to the formation of the maximum throughput flow. In the low noise level region, the temporal spectra are almost white. The values of the effective exponent β are summarized in Fig. 3. In the low noise level region, the exponent β increases with the noise level until it has a plateau above the noise level f noise.2. Next we introduce a traffic blockade [13] by restricting the maximum velocity, V optimal (t) V optimal ( x)/2, in the right half region of the road segment L 2. Then the cars suddenly decelerate to reduce their velocity at the beginning point of that region. The traffic blockade produces the wave of traffic jam, which propagates to the upstream. The values of the headway distance and velocity of the test car are recorded during 1, second (Fig. 4). The hysteresis loop in the headway-velocity plane, similar to one observed in the periodic boundary system of the original OV model[2, 3], is reproduced in the absence of the noise. The test car produces the hysteresis loop with fluctuations in the presence of noise. The density fluctuations and its temporal spectra are observed also for the systems with traffic blockade. The temporal spectra again obey the power law

5 .27 road length=21m road length=11m 1. Total Road Length 11 m Total Road Length 21 m.6 β average density fnoise fnoise.3.4 Fig. 3. The average density (left) and the exponents (right) of the temporal spectra of the density fluctuations. 5 v (m/sec) 5 v (m/sec) 5 v (m/sec) fnoise=. 5 1 fnoise= fnoise= Fig. 4. Trajectories in the headway-distance versus velocity plane with traffic blockades. The system configuration is L1 = L2 = 1, m. Noises in velocity are fnoise =.,.5 and.1. with the exponents ( ), which are much larger than those in the system without traffic blockade and seem not to depend on the noise level. 4 Summary and Discussion A coupled map traffic simulation model was introduced by discretizing the Optimal Velocity model introduced by Bando et al. It is capable of rearrangement of the car sequence and introduction of a variety of noises. It, therefore, can be extended to an open boundary system and a multi-lane road system. We investigated open boundary systems with noises in velocity in this paper. The trajectories in the headway-velocity plane were observed. A injected car just accelerates to catch up the preceding car in open boundary systems without any causes of traffic jam. The noises in velocity produce the fluctuation around the maximum velocity. The increase of the noise level enhances the average density. The traffic flow organizes to one with the maximum throughput as a consequence. The temporal density fluctuation shows f β behavior at the same time. In the low noise level region, the formation of the flow with the maximum throughput is not sufficient to give the f β behavior in broad frequency region.

6 The formation of f β behavior also depends on the value of the response factor α in Eq. (1). In the original OV model, the car sequences become unstable when α/2 <V optimal (b)[2, 3]. The critical value b c for the headway distance is 32.3m for the parameter in Table 1, which give the optimal velocity V optimal (b c )= 24.7m/sec =.74 v max. The high level noise exceeding the stable region is considered to make the car sequence unstable to form traffic congestion in the present model. So the balance of α and the noise level seems to be a key to distinguish the two regions shown in Fig. 3. Further investigations are required to characterize these two regions. When the traffic blockade is introduced, traffic jams are created at the blockade and propagate to the upstream of the road. The hysteresis loops in the headway-velocity plane are observed. The power law behaviors are also observed in the system with traffic blockade. The exponents of the spectra are larger than those in the system without blockade and almost independent of the noise level. Finally let us summarize the advantages of our coupled map traffic simulator. Each car responds to the stimuli of the preceding car individually. The individuality of cars can be introduced through a variety of parameters and forms of optimal velocity functions. Flexible sequences of cars enabled the model applicable to multi-lane roads and traffic systems with junctions. The model can include several types of noises as well. References 1. Leutzbach, W.: Introduction to the Theory of Traffic Flow, (Springer-Verlag, Berlin, 1988). 2. Bando, M., Hasebe, K., Nakayama, A., Shibata A. and Sugiyama, Y.: Jap. J. Ind. Appl. Math. 11 (1994) Bando, M., Hasebe, K., Nakayama, A., Shibata A. and Sugiyama, Y.: Phys. Rev. E51 (1995) Bando, M., Hasebe, K., Nakanishi, K., Nakayama, A., Shibata, A. and Sugiyama, Y.: J. Phys. I France 5 (1995) Sugiyama, Y. and Yamada, H.: Phys. Rev. E55 (1997) Nakanishi, K., Itoh, K., Igarashi Y., and Bando, M.: Phys. Rev. E55 (1997) Sugiyama, Y : in this proceedings 8. Yukawa, S. and Kikuchi, M.: J. Phys. Soc. Japan 64 (1995) Yukawa, S. and Kikuchi, M.: J. Phys. Soc. Japan 65 (1996) Yukawa, S. and Kikuchi, M.: in this proceedings. 11. Koshi, M., Iwasaki, M. and Ohkura, I.: Some Findings and an Overview on Vehicular Flow Characteristics, V. F. Hurdle et al. (eds.) Proc. 8th Intl. Symp. on Transp. and Traffic Theory, Nagel, K. and Paczuski, M.: Phys. Rev. E51 (1995) Yukawa,S., Kikuchi, M. and Tadaki, S.: J. Phys. Soc. Japan 63 (1994) 369.