Dynamics of Motorized Vehicle Flow under Mixed Traffic Circumstance

Transcription

1 Commun. Theor. Phys. 55 (2011) Vol. 55, No. 4, April 15, 2011 Dynamics of Motorized Vehicle Flow under Mixed Traffic Circumstance GUO Hong-Wei (À å), GAO Zi-You (Ô Ð), ZHAO Xiao-Mei ( Ö), and XIE Dong-Fan ( ü ) Institute of System Science, School of Traffic and Transportation, Beijing Jiaotong University, Beijing , China (Received June 30, 2010; revised manuscript received October 27, 2010) Abstract To study the dynamics of mixed traffic flow consisting of motorized and non-motorized vehicles, a carfollowing model based on the principle of collision free and cautious driving is proposed. Lateral friction and overlapping driving are introduced to describe the interactions between motorized vehicles and non-motorized vehicles. By numerical simulations, the flux-density relation, the temporal-spatial dynamics, and the velocity evolution are investigated in detail. The results indicate non-motorized vehicles have a significant impact on the motorized vehicle flow and cause the maximum flux to decline by about 13%. Non-motorized vehicles can decrease the motorized vehicle velocity and cause velocity oscillation when the motorized vehicle density is low. Moreover, non-motorized vehicles show a significant damping effect on the oscillating velocity when the density is medium and high, and such an effect weakens as motorized vehicle density increases. The results also stress the necessity for separating motorized vehicles from non-motorized vehicles. PACS numbers: a, Vn, Fh Key words: car-following mode, mixed traffic flow, velocity oscillation, damping effect 1 Introduction The phenomenon of traffic flow has attracted the interest of many researchers, and several researches have been done on the modeling of traffic flow. [1 2] Different traffic problems have been studied from the perspectives of statistical physics and nonlinear dynamics. [3 6] So far, most literature has focused on the homogeneous traffic flow composed of motorized (m-) vehicles such as cars and buses. [7 10] In the traffic system, there are also nonmotorized (nm-) vehicles that are usually used as a kind of green traffic mode, such as bicycles and tricycles. Such nm-vehicles play an important role in developing countries because of their economy and flexibility. Therefore, researchers turn their attention to the traffic characteristics of nm-vehicles. Jiang et al. [11] proposed a stochastic multi-value cellular automata model for bicycle flow, and Li et al. [12] proposed a multi-valued cellular automata model to study the mixed nm-vehicle flow. On the other hand, the wide use of nm-vehicles may cause traffic addicents and capacity drop. [10] In urban roads without isolation facilities, nm-vehicles may occupy m-vehicle lanes because of the blockages in nm-vehicle lanes, especially for the positions near a bus stop or curb parking area. The situation in which m-vehicles and nmvehicles share the m-vehicle lanes becomes a typical traffic phenomenon called mixed traffic (also including mixture of m-vehicles and pedestrians). The characteristics of mixed traffic flow are complicated and they attract the attention of engineers as well as physicists. Xie et al. [13] proposed a new two-dimensional car-following model to depict the features of mixed traffic flow at an unsignalized intersection. Yang et al. [4] proposed a road capacity model for mixed traffic flow at a curbside bus stop based on the gap acceptance theory and queuing theory. Zhao et al. [15] described mixed traffic flow by combining the NaSch model and the Burger cellular automata (BCA) model, and investigated the mixed traffic system near a bus stop. Oketch [16] proposed a microscopic model to depict mixed traffic flow by combining the car-following model and lateral movement. In Ref. [17], the cellular automata model is extended to study the heterogeneous traffic observed in developing countries including car, bus, two-wheeler and three-wheeler. The oscillatory patterns in intersecting flows of pedestrian and vehicle traffic are investigated in Ref. [18]. Jiang et al. [19] studied intersecting pedestrian and vehicle flows as an example for inefficient emergent oscillations. The literature indicates that the research on dynamics of m-vehicle flow under the mixed traffic circumstance is limited compared with the research on the features of mixed traffic. Thus, to study such dynamical features is crucial for evaluating the influence of mixed traffic and traffic management. And this is the motivation of this paper. This paper is concerned about the dynamics of the m-vehicle flow under the influence of nm-vehicles. With this aim, a car-following model based on the Gipps model Supported by the National Basic Research Program of China under Grant No. 2006CB705500, and the National Natural Science Foundation of China under Grant Nos and hwguo.bjtu@gmail.com c 2011 Chinese Physical Society and IOP Publishing Ltd

2 720 Communications in Theoretical Physics Vol. 55 framework [20] is proposed. The collision avoidance and cautious driving are considered as the principle of m- vehicles driving under the mixed traffic circumstance in the new model. The lateral friction and the overlapping driving are adopted to highlight the impact of nm-vehicles on m-vehicle flow. The flux-density relation and the velocity revolution of m-vehicle flow are investigated by numerical simulations. The results show different dynamical features of m-vehicle flow under the influence of nmvehicles and emphasize the necessity of setting facilities to segregate nm-vehicles from m-vehicle. This paper is organized as follows. In Sec. 2, a car-following model under the mixed traffic circumstance is proposed. Based on the new model, numerical simulations are performed and results are given in Sec. 3. Finally, Section 4 presents the conclusions. 2 Model 2.1 Gipps Model The dynamics of m-vehicle flow is investigated through developing a specific car-following model framework proposed by Gipps. [20] There are three basic assumptions: the vehicles are collision free; the vehicles do not exceed the safe velocity v safe ; the acceleration a and the deceleration b are bounded. It is assumed that one vehicle with velocity v f follows another vehicle (with velocity v l ) within a gap x. When the leader suddenly decelerates and stops eventually, the follower responds this stimulus τ s later and then brakes to aviod a rear-end collision. The safe criterion for the follower is defined as τv f + d(v f ) x + d(v l ), (1) where d( ) is the braking distance with maximum deceleration. Let b (b > 0) be the maximum deceleration and the breaking distance is given by d(v) = v 2 /2b. Considering the acceleration characteristics of m-vehicles, the safe velocity for the follower can be written as v safe = min(v f + 2.5aτ(1 v f /v max ) ( v f /v max ), bτ + b 2 τ 2 + vl 2 + 2b x 2bv fθ), (2) where θ is a possible additional delay to present a safety margin. Note that in the right hand side of Eq. (2), the first term represents the velocity boundary and characteristic of free acceleration; the second term represents the collision freeness behaviour. The main reason for adopting such a framework is that the parameters used in the model correspond to obvious characteristics of drivers and vehicles. Therefore, most of these parameters can be assigned values without calibration and validation. Additionally, the collision freeness is in accordance with the driving behaviour of m-vehicles in mixed traffic circumstance because m-vehicle drivers are cautious enough to respond any unexpectedness caused by nm-vehicles (e.g. suddenly appear in the middle of lane). 2.2 New Model for Mixed Traffic We consider a two-lane road system consisting of an m-vehicle lane and an nm-vehicle lane. For simplicity, there are two kinds of elements in such a system: m- vehicles (representing cars) and nm-vehicles (representing bicycles). M-vehicles should travel in the m-vehicle lane, and nm-vehicles should travel in the nm-vehicle lane in principle. In reality, some nm-vehicles often enter and occupy the m-vehicle lane for some reason. Figure 1 shows a schematic illustration of the model for the mixed traffic circumstance. As schematized in Fig. 1, the interactions between the two types of vehicles depend on the lateral offsets of nmvehicles in the m-vehicle lane (see L off in Fig. 1). Firstly, if L off > 1 (generally, the width of m-vehicle lane is 3.8 m and the width of m-vehicle is 1.8 m) the nm-vehicles will hinder the m-vehicles behind them. Then the m-vehicle drivers have to slow down and follow the nm-vehicles cautiously. In this situation, which is called overlapping driving (OD) situation, the m-vehicle drivers also travel at a safe velocity so as to stop completely before colliding with the leading nm-vehicles. The OD situation is similar to general car-following process. According to Eq. (1), the safe velocity for the m-vehicles travelling behind the nm-vehicles is defined as v safe bτ + b 2 τ 2 + b(vl 2/ b + 2 x), (3) where b ( b > 0) is the deceleration of nm-vehicles. Fig. 1 Schematic illustration of mixed traffic flow. Secondly, if L off is not significant (L off 1), the following m-vehicles can overtake the nm-vehicles by using the escape route (see W er in Fig. 1), instead of following behind the leading nm-vehicles. Thus unlike conventional car-following behaviour, the follower s velocity needs to be adjusted to the overtaking velocity (v over ) rather than being reduced to zero. Here, the overtaking velocity reflects the lateral friction effects between the two types of vehicles. This situation is defined as the lateral friction (LF) situation, where the m-vehicles decelerate (or accelerate) to overtake the leading nm-vehicles. Such LF situation reflects the cautious psychology of m-vehicle drivers under

3 No. 4 Communications in Theoretical Physics 721 the influence of nm-vehicles. The overtaking velocity is in inverse proportion to L off. In this situation, the breaking distance of the follower is given by d(v) = (v 2 vover)/2b 2 and the safe velocity for overtaking is defined as v safe bτ + b 2 τ 2 + vover 2 + b(vl 2/ b + 2 x) (4) with v over = 3.96L 2 off 7.97L off , 0 < L off 1. (5) We also propose a simplified nm-vehicle model. The nm-vehicles move at a constant velocity v b in the nmvehicle lane. Some nm-vehicles would move into the m- vehicle lane at time step t in two cases. In the first case, a certain number of nm-vehicles enter the m-vehicle lane with probability p a and L off (L off 1), resulting in the LF situation. The nm-vehicles in the LF situation will move back to the nm-vehicle lane at next time step since the LF situation is a short time effect. In the second case, also at time step t, some of the nm-vehicles in the LF situation may laterally move further with probability p b and safe criterion x and cause the OD situation (L off > 1). Such a safe criterion indicates that nm-vehicles cannot drive between two m-vehicles that travel at close distance. It needs to be explained that nm-vehicle drivers can only drive in front of m-vehicles for a short time because the nm-vehicle drivers themselves may feel dangerous or the following m-vehicles may sound horn to force them to go away. Therefore, we use T to represent the duration time of the OD situation. The model is defined with continuous state variables x, v and discrete time step t. In each time step, every m-vehicle updates its position after calculating v safe according to v des = min(v safe, v t + a t, v max ), (6) v t+ t = max(v des µa t, 0), (7) x t+ t = x t + v t+ t t, (8) where v des is the desired velocity, v t+ t is the velocity at time step t + t, x t and x t+ t are the positions at time step t and t + t respectively, µ is a random number uniformly distributed in the interval [0,1]. The update rules can be called the continuous Nagel- Schreckenberg model. [21] The main reason for adopting these rules is that they make the calculation of the update rules easier. More importantly, such rules can describe the driving behaviour accurately. Firstly, Eq. (6) represents that vehicles accelerate gradually and travel at some safe velocity to avoid collisions. Secondly, Eq. (7) introduces the random deceleration to reflect the uncertainties in driving behaviour. The random deceleration is more practical especially under mixed traffic circumstance. Additionally, the concept of the safety distance proposed by the Gipps model is appropriate for describing driving behaviour of m-vehicles influenced by nm-vehicles. A little deficiency to be noted is that it seems unreasonable to use a driving strategy that requires anticipating the unlikely event of having to brake to a complete stop using maximum deceleration at every time step. In this paper, a safety margin term is ignored for it can lead to longer headway and higher velocity. Nevertheless, the concentration is about the interaction mechanism between m-vehicles and nm-vehicles, so such a deficiency can be neglected. 3 Simulation In this section, the dynamics of m-vehicle flow under the influence of nm-vehicles is investigated through numerical simulations. A system of length L = 1000 is simulated under periodic boundaries. For this purpose, a fixed set of parameters is used, namely: a = 2, b = 6, v max = 12, v b = 4, x = 6, l m = 7, and l mn = 3, where l m is the length of m-vehicles and l mn is the length of nmvehicles. The duration parameter T is 3. M-vehicles and nm-vehicles are initialized randomly with initial velocity v 0 = 0 in respective lanes. For simplicity, nm-vehicles are initialized with the same amount as m-vehicles. The density is defined as ρ = N m l m /L with the number of m-vehicles N m, and the flux is defined as q = ρ v with average velocity v. The simulation time step t is 0.5 s. The simulations last for 8800 time steps and the first 4000 time steps are discarded to avoid transient behaviour. 3.1 Flux-Density Relation Firstly, the influence of the LF situation is evaluated through the increased p a (from 0.05 to 0.4) and the constant p b (p b = 0.05). As shown in Fig. 2. The dotted curve in the fundamental diagrams represents the fluxdensity relation without influence; and the solid curves with different symbols represent flux-density relation influenced by nm-vehicles. It can be seen that there are obvious drops in the maximum flux of the m-vehicle flow (from 0.58 to 0.49) with a further increase of p a. The fundamental diagrams show the free flow phase and the congested phase. The free flow phase distributes at low density (ρ < 0.40) where the m-vehicle flux increases with increasing density. The phase transitions occur at a critical density that increases from 0.45 to 0.56 as p b increases. In the congested phase, the flux begins to decline with the increase of density. At high density (ρ > 0.60), the differences in flux become smaller and the fundamental diagrams match together. As shown in Fig. 2, nm-vehicles have a significant impact on the m-vehicle flux. In the LF situation, m-vehicle drivers need to slow down and adjust to safe overtaking velocity so that the flux decreases. The results in such situation conform to the cautious driving behaviour in mixed traffic circumstance.

4 722 Communications in Theoretical Physics Vol. 55 higher if not influenced, drops and leads to the flux reduction. Thirdly, the duration parameter T increases from 1 to 5 in order to evaluate the effect of the duration of the OD situation. As shown in Fig. 4, the longer the duration is, the more significantly the flux reduces. With the same p a and p b, the maximum flux declines by 9% due to the increase of duration from 1 to 5. Fig. 2 Fundamental diagrams of m-vehicle flow under LF situation. Secondly, the influence of the OD situation is evaluated by using the constant p a (p a = 0.1) and the increased p b (from 0.05 to 0.4). At a time step, the maximum proportion of the m-vehicles being influenced is 10% and the maximum proportion of the m-vehicles in the OD situation is from 0.5% to 4%. The fundamental diagrams are shown in Fig. 3. Fig. 4 Fundamental diagrams of m-vehicle flow for various T. The results of the above three scenarios directly reflect that the nm-vehicles in the m-vehicle lane have a significant influence on the m-vehicle flux. The nm-vehicles can influence the phase transition of m-vehicle flow and reduce the critical flux. When the density of m-vehicle flow is high (ρ > 0.60), the influence of nm-vehicles is weakened remarkably because m-vehicles are as slow as nm-vehicles under such condition. Fig. 3 Fundamental diagrams of m-vehicle flow under OD situation. The free flow phase and the congested phase in the OD situation are similar with those in the LF situation. The phase transitions occur at critical densities ranging from 0.49 to For comparison, the maximum flux with p a = 0.1 and p b = 0.4 is approximately equal to the maximum flux with p a = 0.4 and p b = It means 4% of the m-vehicles in the OD situation and 38% of the m- vehicles in the LF situation will lead to the same results. Therefore, the results show that the flux reduction in the OD situation is more significant. Here, the optimal velocity function of the new model can be derived analytically as V opt = min{(1/ρ 1/ρ max )/τ, v max }. [21] The optimal velocity depends on the change of m-vehicle density or headway. The influence of nm-vehicles can be considered as an increase of system density so that the optimal velocity decreases. Meanwhile, the velocity, which will be 3.2 Time-Space Diagram In this section, the temporal and spatial dynamics of m-vehicle flow are presented by the time-space diagrams. In order to examine various features of m-vehicle flow, three influential degrees are considered: no influence with p a = 0 and p b = 0; weak influence with p a = 0.1 and p b = 0.1; and strong influence with p a = 0.3 and p b = 0.3. Fistly, we draw the time-space diagrams for the free flow phase (ρ = 0.40). Figure 5(a) shows the stable state of the pure m-vehicle flow without external influence. As shown in Figs. 5(b) and 5(c), the stable state is interrupted by nm-vehicles. In Fig. 5(b), the m-vehicle flow is interrupted and local clusters appear. The influence of nm-vehicles can make m-vehicles slow down so that the headways distribute unevenly. Figure 5(c) shows more obvious local clusters than Fig. 5(b). There are also more strip blanks in Fig. 5(c), which reflects that the m-vehicle flow is seriously interrupted by nm-vehicles when more m-vehicles travel in the OD situation.

5 No. 4 Communications in Theoretical Physics 723 Fig. 5 Time-space diagrams of m-vehicle flow for ρ = (a) pa = 0, pb = 0; (b) pa = 0.1, pb = 0.1; (c) pa = 0.3, pb = 0.3. Secondly, the time-space diagrams for the congestion phase are shown in Fig. 6. The m-vehicle flow shows a stopand -go wave and an obvious wide strip jam, as shown in Fig. 6(a). Under the influence of nm-vehicles, the wide strip jams become narrow and change into local jams. The velocity reduction of m-vehicle flow weakens the oscillation of the headways of m-vehicles and causes such a phenomenon. The wide strip jams dissipate and change into the local jams that are represented by the strip blanks in Figs. 6(b) and 6(c). Here, the influence of nm-vehicles can be considered as damping effect that will lead to the decrease in the oscillation intensity of m-vehicle velocity. Moreover, because of the safe criterion for the OD situation, the m-vehicles in the OD situation will reduce or even disappear as the density increases. Therefore, the strip blanks in Fig. 6(c) are obviously fewer than those in Fig. 5(c) with the same pa and pb. Fig. 6 Time-space diagrams of m-vehicle flow for ρ = (a) pa = 0, pb = 0; (b) pa = 0.1, pb = 0.1; (c) pa = 0.3, pb = 0.3. From the time-space diagrams, we can see the different effects caused by nm-vehicles. In the free flow phase, the influence of nm-vehicles leads to headway oscillation and local jams. Furthermore, in the congested phase, the nmvehicles show damping effect on the oscillating headways of m-vehicles and turn wide jams into local jams. 3.3 Temporal Distribution of Velocity The time evolution of the velocity distribution is investigated in this section. In order to present the velocity distribution intuitionally, we take the first m-vehicle as an example and draw the velocity evolution curve in 200 s for the free flow phase (ρ = 0.40) and the congestion phase (ρ = 0.60) respectively. In addition, the same influential degrees mentioned in Subsec. 3.2 are adopted to distinguish the overall coherence of the results. Figure 7(a) shows the velocity distribution of the free flow phase. The m-vehicle can travel at a uniformly high velocity if not influenced (see the solid curve in Fig. 7(a)). If the m-vehicle travels under the influence of nm-vehicle, its velocity de- creases and oscillates obviously (see the dotted curve and the dashed curve in Fig. 7(a)). With increasing pa and pb, the velocity still shows oscillation distribution but the maximum velocity decreases obviously. The average velocities for the three influential degrees are 11.5, 7.8, and 4.6. The velocity distribution of the congestion phase is shown in Fig. 7(b). Oscillating velocity in the stop-and-go wave can be seen if the m-vehicle travels without any influence. Then, the velocity distribution shows damping oscillation and the velocity ranges decrease with increasing pa and pb. As shown in Fig. 7(b), the velocity oscillation ranges from 12 to 1 with pa = 0.1 and pb = 0.1 and from 8.5 to 2 with pa = 0.3 and pb = 0.3. Therefore, at a high m-vehicle density, the influence of nm-vehicle can be considered as damping effect in oscillation system. The temporal velocity distributions also verify the results in the time-space diagrams. In order to adapt the varying velocity caused by the nm-vehicles, the m-vehicle drivers have to slow down and speed up frequently; and pay more attention to deal with the unexpected at any

6 724 Communications in Theoretical Physics Vol. 55 time. Moreover, such mixed traffic circumstance can easily cause traffic accidents in which nm-vehicle drivers will be direct victim. In addition, frequent acceleration and deceleration also make m-vehicles consume more energy. Fig. 7 Velocity distribution of the first m-vehicle. (a) free flow (b) congestion flow. 4 Conclusion The present work analyzes the dynamical characteristics of the traffic flow under the influence of nm-vehicles. The results show that the existing nm-vehicles in the m- vehicle lane can decrease the m-vehicle flux at the range of low and medium density (ρ < 0.65). Under strong influence, both the OD situation and the LF situation make the maximum flux decline by about 13% (the three cases show the maximum fluxes reduce by 17% to 9%). The nm-vehicles can also make the m-vehicle velocity drop obviously. An interesting result is that the influence of nmvehicles can cause velocity oscillation in the free flow phase and damp the velocity oscillation in the congestion phase. Furthermore, the nm-vehicles in the m-vehicle lane would make m-vehicle drivers nervous and uncomfortable. Such a mixed traffic circumstance can easily lead to traffic accidents and more energy consumption. Therefore, good efforts should be made to avoid the conflicts between m- vehicles and nm-vehicles. It is important to install the segregated facility to separate m-vehicles from nm-vehicles on urban arterial roads, especially for the road segments with bus stops or curb parking areas. It should be pointed out that at present the interactions mechanism between m-vehicles and nm-vehicles are roughly considered as overlapping drive and lateral friction. Actually, the driving behaviour of the m-vehicle drivers in mixed traffic is very complex. For example, m-vehicles may move laterally to avoid collisions rather than lane-based driving in general model. Compared with the m-vehicle model, the nm-vehicle model is too simple. These will be studied in the future. References [1] D. Chowdhury, L. Santen, and A. Schreckenberg, Phys. Rep. 329 (2000) 199. [2] D. Helbing, Rev. Mod. Phys. 73 (2001) [3] T.Q. Tang and H.J. Huang, Commun. Theor. Phys. 53 (2010) 983. [4] T.Q. Tang, et al., Commun. Theor. Phys. 51 (2009) 71. [5] Y.F. Wei, S.L. Guo, and Y. Xue, Commun. Theor. Phys. 47 (2007) 499. [6] K.P. Li, Commun. Theor. Phys. 45 (2006) 113. [7] Z.P. Li, X.B. Gong, and Y.C. Liu, Commun. Theor. Phys. 46 (2006) 367. [8] H.X. Ge, et al., Commun. Theor. Phys. 43 (2005) 321. [9] T. Nagatani, Physica A 388 (2009) [10] S.I. Khan and P. Maini, Transp. Res. Rec (1999) 234. [11] R. Jiang, B. Jia, and Q.S. Wu, J. Phys. A 37 (2004) [12] X.G. Li, et al., Acta Phys. Sin. 57 (2007) 4777 (in Chinese). [13] D.F. Xie, Z.Y. Gao, and X.M. Zhao, Physica A 388 (2009) [14] X.B. Yang, et al., Transp. Res. Rec (2009) 18. [15] X.M. Zhao, B. Jia, and Z.Y. Gao, arxiv: v1. [16] T.G. Oketch, Transp. Res. Rec (2000) 61. [17] C. Mallikarjuna and K.R. Rao, J. Adv. Transp. 43 (2010) 321. [18] D. Helbing, R. Jiang, and M. Treiber, Phys. Rev. E 72 (2005) [19] R. Jiang, D. Helbing, P.K. Shukla, and Q.S. Wu, Physica A 368 (2006) 567. [20] P.G. Gipps, Trans. Res. B 15 (1981) 105. [21] S. Krauss, P. Wagner, and C. Gawron, Phys. Rev. E 54 (1996) 3707.