Analysis of Phase Transition in Traffic Flow based on a New Model of Driving Decision

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1 Commun. Theor. Phys. 56 (2011) Vol. 56, No. 1, July 15, 2011 Analysis of Phase Transition in Traffic Flow based on a New Model of Driving Decision PENG Yu ( Ý), 1 SHANG Hua-Yan (Ù), 2, and LU Hua-Pu (öþ ) 3 1 Programming Development Department, Shougang Corporation, Beijing , China 2 Information College, Capital University of Economics and Business, Beijing , China 3 Institute of Transportation Engineering, Tsinghua University, Beijing , China (Received November 1, 2010; revised manuscript received January 24, 2011) Abstract Different driving decisions will cause different processes of phase transition in traffic flow. To reveal the inner mechanism, this paper built a new cellular automaton (CA) model, based on the driving decision (DD). In the DD model, a driver s decision is divided into three stages: decision-making, action, and result. The acceleration is taken as a decision variable and three core factors, i.e. distance between adjacent vehicles, their own velocity, and the preceding vehicle s velocity, are considered. Simulation results show that the DD model can simulate the synchronized flow effectively and describe the phase transition in traffic flow well. Further analyses illustrate that various density will cause the phase transition and the random probability will impact the process. Compared with the traditional NaSch model, the DD model considered the preceding vehicle s velocity, the deceleration limitation, and a safe distance, so it can depict closer to the driver preferences on pursuing safety, stability and fuel-saving and has strong theoretical innovation for future studies. PACS numbers: a, Vn Key words: driving decision, cellular automaton, phase transition, traffic flow 1 Introduction Traffic flow theory aims at making models to explain the real traffic. In 1992, Nagel and Schreckenberg [1] introduced the well-known NaSch model, which made the cellular automaton (CA) model become a well-established method of traffic flow modeling. In the NaSch model, the roadway is represented by a uniform cell lattice in which each cell belongs to a discrete set of states. And the state of the cells is updated at discrete time steps with a set of update rules. Although the NaSch model is very simple, it can easily reproduce some real traffic phenomena, such as the occurrence of phantom traffic jams and the realistic flow-density relation (fundamental diagram). Based on the NaSch model, many other CA models were developed. For instance, in Fukui and Ishibashi s model (FI model), [2] a vehicle can accelerate to the maximum speed in one time step and the randomization only works on those vehicles with the desired speed. But the cruise control models [3 5] have no randomization on those vehicles with desired speed at all. The slow-to-start models [6 11] can clearly simulate the hysteresis phenomenon, for a stationary vehicle has greater randomization than a moving one. In the brake light models, [12 15] a vehicle will deliver its deceleration behaviors via a brake light and its follower will get larger randomization. Kerner presented the three-phase traffic flow models [16 17] and divided the traffic flow into such three phases as the free flow, the synchronized flow, and the jam flow. These models have a certain synchronization scope, so a preceding vehicle can affect its follower within a certain distance. [16 19] And some other models considered the velocity limitation. [20 26] All these models [1 26] focused on the themes including rules of random probability, generation of meta-stability and hysteresis, roles of the preceding vehicle, restriction of acceleration and deceleration. The four-step update rules in the NaSch model were widely used in these models and their key issues were to update the vehicles locations and velocities. A vehicle s velocity is mainly affected by the randomization and the distance between itself and its preceding vehicle. Only some models considered the influence of the preceding vehicle s speed. [20 24] Studies deep into the process of a driver s driving decision were limited. However, as we all know, a vehicle s velocity depends on the driving decision, including acceleration, deceleration and uniform movement. It shows that the subject of the driver s decision is just the acceleration. Therefore, it is very necessary to take the acceleration as a decision variable. Furthermore, the vehicle velocity is not only influenced by the distance between a vehicle and the preceding vehicle, but also by the random probability. Thus, Supported by the Program for National High-Tech Research and Development Program of China under Grant No 2007AA11Z233, National Key Technology R & D Program under Grant No. 2009BAG13A06, and China Postdoctoral Science Foundation Funded Project under Grant No Corresponding author, shanghuayan@126.com c 2011 Chinese Physical Society and IOP Publishing Ltd

2 178 Communications in Theoretical Physics Vol. 56 the study on the effects of both the driving decision and the random probability are very important. In this paper, a driver s decision will be divided into three stages: decision-making, action, and result. A new cellular automaton (CA) model based on the driving decision (DD) is built, and the effects of the density and the random probability on the phase transition in traffic flow are analyzed. Compared with the NaSch model, the key issue of the DD model is to calculate the acceleration. It considers the preceding vehicle s velocity, the deceleration limitation, and a safe distance, so it can depict closer to the driver preferences on pursuing safety, stability, and fuel-saving. This paper is organized as follows: the models used in the simulations are introduced in Sec. 2. In Sec. 3, the simulation results are discussed. Finally, the conclusions are summarized. 2 Models 2.1 Assumptions of Driving Behavior In the formulations of this study, the following assumptions are made: (i) The driving behavior can be divided into three stages, i.e. decision-making, action, and result, among which the decision-making stage is the key. At this stage, a driver decides what to do at the current time: to accelerate, to decelerate, or to keep uniform. His decision may be influenced by the factors such as the distance between himself and his preceding vehicle, his own velocity, the preceding vehicle s velocity, and other uncertainties. At the action stage, random probability happens. And at the result stage, he updates his location. (ii) A driver cannot directly obtain the preceding vehicle s velocity. But he can estimate it according to the distance s change and his own velocity. This estimation is not instant but lagged, since the current estimation is conducted previously. When the driver responses to the estimated velocity, he should always prepare for the worst condition because he fears the preceding vehicle s sudden deceleration with the greatest deceleration. In addition, his decision-making space is a bounded set, namely, the acceleration has the upper and lower bounds. (iii) The maximum acceleration is assumed to be 1 and the maximum deceleration to be 2. The maximum velocity is assumed to be 5 and the minimum velocity to be 0. These assumptions consider the preceding vehicle s velocity and the deceleration limitation, so it is more in line with the reality than the NaSch model, which thinks the preceding vehicle is always stationary. In addition, a vehicle in the NaSch model has no deceleration limitation and can decelerate from any velocity to zero. The 2 nd assumption is reasonable for the lagged estimation, since if there is no lag in the high-density traffic flow, a large number of queued vehicles possibly move with the same velocities at the same time (see Fig. 1). Fig. 1 Effect of the synchronous estimation of the preceding vehicle s velocity, where the number on each cell indicates the velocity. 2.2 Forward Motions Similar with the NaSch model, each road is divided into a one-dimensional array of L cells in the DD model. Each cell may either be occupied by at most one vehicle or be empty. Suppose that s t n and vt n denote the position and velocity of the n-th vehicle at time t, respectively. x t n is the displacement of the n-th vehicle at time t. N is the total number of all the vehicles on the road and vehicles from upstream to downstream successively number N, N 1,...,3, 2, 1. Assume that each vehicle can move with an integer velocity vn t {0, 1, 2, 3, 4, 5}. Note that the DD model is slightly different from the NaSch model. The latter has four-step update rules (acceleration, deceleration, randomization, motion) and velocity vn t is the decision-making variable. But the DD model simulates based on three-step update rules (decision-making, action, result), in which not the velocity vn t but the acceleration a n is taken as the decision-making variable. Here, a n represents the acceleration of the n-th vehicle. Let a n = { 2, 1, 0, 1} and a min = 2 hold. d n denotes the distance between the n-th vehicle and its preceding vehicle, d brake (v) denotes the braking distance, i.e. the minimum required distance for a vehicle to slow down from v to 0. And d safe (v) is the safe distance, i.e. the minimum required distance for a vehicle to drive at the velocity v behind its preceding vehicle. Two formulas can be obtained: d safe (v) = d brake (v) + v, (1) d brake (v) = max(v + a min, 0) 1 + max(v + 2a min, 0) (2) Thus we have the relationships among v t n, d brake (v), and d safe (v), where v t n {0, 1, 2, 3, 4, 5}, d safe (6) = + (see Table 1). The correspondence of a n, d brake (v), and d safe (v) is shown in Fig. 2. Table 1 d safe (v). The correspondence of v t n, d brake (v), and v n d brake (v) d safe (v)

3 No. 1 Communications in Theoretical Physics 179 Fig. 2 The correspondence of a n, d brake (v), and d safe (v). The DD model can be described as follows: Step 1 Decision-making. Calculate the acceleration a t n = f(d n, vn, t vn+1): t 1, if D n [d safe (vn t + 1), + ), 0, if D n [d safe (vn t a n = ), d safe(vn t + 1)), 1, if D n [d safe (vn t 1), d safe (vn)), t (3) 2, if D n [d brake, d safe (vn t 1)), where D n is a new parameter and defined as a dynamic distance, D n = d n + max(0, vn+1 t + a min ). max(0, vn+1 t + a min ) denotes the minimum displacement of the (n+1)-th vehicle and a min = 2. It can be seen that D n considers the combined effects of the preceding vehicle s velocity and the minimum displacement. Step 2 Action. Randomization with probability p: Step 3 Result. Each vehicle moves forward: a n = max(a min, a n 1), (4) v t+1 n = max(0, v t n + a n ). (5) s t+1 n = s t n + v t+1 n. (6) Step 1 performs the driver s reaction and is the core of the DD model. For example, D n [d safe (vn t + 1), + ] shows that D n can satisfy the safe distance with the velocity vn t + 1, so this vehicle may accelerate at time t + 1. Here, a n = 1. Step 1 can be summarized into a programming problem: maxa n d safe (vn t + a n ) D n, D n = d n + max(0, vn+1 t + a min) a n = 2, 1, 0, 1, vn t 0, d n 0, (7) where D n = d n if a n (, 1]. We can find that if a n (, 1], then D n = d n. Here Step 1 has the equivalently combined effects of acceleration and deceleration in the NaSch model. In addition, the greatest difference between the DD model and the NaSch model is that the former considers the deceleration limitation (i.e. the acceleration s lower bound). Since there is a minimum acceleration, there should also be a minimum displacement max(0, vn+1 t + a min), which differs from the constant zero in the NaSch model. Then, the dynamic distance D n = d n + max (0, vn+1 t + a min ) can be computed. Similarly, if a n (, + ), the effect of Step 1 is equivalent to the combined effects of acceleration and deceleration in the FI model. It is important to note that the randomization in the DD model will affect on the acceleration a n instead of directly on the velocity v n. Especially when a n = a min, a vehicle has reached the maximum deceleration limitation and the randomization has no effect at all. From the above modeling process, we can find that the DD model fully considers the preceding vehicle s velocity, the deceleration limitation, and a safe distance. Moreover, it has a stricter limitation on the safe distance than the NaSch model. Thus, a driver in the DD model prefers to keep a proper distance with the preceding vehicle rather than pursue a maximum speed. 3 Simulations Results and Discussions The simulations are carried out under cycle boundary condition. To compare with the NaSch model, the model parameters are set as follows: the road is equally divided into L = 1000 cells; each cell may be either empty or occupied by one vehicle with an integer velocity v n between 0 and v max = 5; each time step is one second. The initial state is set to be congested. The first 2000 time steps are discarded to avoid the transient behavior. 3.1 Fundamental Diagram Figure 3 exhibits the fundamental diagram for different random probability p in DD model. With the increase of p, the peak of the flow gradually decreases. Fig. 3 The fundamental diagram in the DD model. Figure 4 compares the DD model with the NaSch model. It shows that traffic flux q varies with the density in the DD model and its general trend is similar with

4 180 Communications in Theoretical Physics Vol. 56 that in the NaSch model. However, the peak of the curve in the DD model is low while that in the NaSch model is high and sharp. Further comparisons are as follows: Fig. 4 Fundamental diagrams in the NaSch model and in the DD model (p = 0.2). (i) When ρ [0, 0.1] [0.4, 1], the NaSch model s curve almost completely overlap the DD model s curve, which shows the two models are almost equivalent under the freeflow or the high-density state. Imaginably, the distances between vehicles are large and the deceleration limitation does not play a role under the free-flow state. And under the high-density state, vehicles average velocity is less than 2, so even if a vehicle instantaneously decelerates to zero, it still can meet the deceleration limitation. (ii) When ρ (0.1, 0.4), the flow in the DD model is slightly lower than that in the NaSch model and its curve is relatively smooth. Two interactions result in this phenomenon. On the one hand, the flow reduces due to the limitation of the safe distance. The drivers in the DD model will drive more carefully to ensure adequate deceleration distance. They pursue smooth and safe driving instead of a maximum speed. Therefore, under the same space distance, the vehicles in the DD model will be slower than those in the NaSch model. On the other hand, the effects of the preceding vehicle can help increase the traffic flow. Since the downstream vehicle s velocity and the dynamic space distance are considered, the upstream vehicle in the DD model will probably be faster than that in the NaSch model if the downstream vehicle is fast. So the effects of the safe distance and the preceding vehicle in the DD model will offset each other. These combined effects will lead to a little reduction of flux q. Vehicles detailed motions are displayed in Fig. 5. In Fig. 5(a), a vehicle with velocity of 5 (in the black lattice of the first line) has 5 lattices distance from a stationary vehicle (in the black lattice of the second line) at time t. According to the NaSch model, it should brake harry to 0 at time t + 2 (in the black lattice of the third line). But according to the DD model, it may decelerate to 2. Obviously, a vehicle in the DD model decelerates gradually with the distance s reduction and a hurry-braking phenomenon seldom happens. In Fig. 5(b), a vehicle with velocity of 5 has 4 lattices distance from its preceding vehicle at time t, whose velocity is 3. According to the NaSch model, it should accelerate to 4 at time t + 1 and decelerate to 3 at t + 2. But according to the DD model, it can maintain the steady speed of 3. It does not rush to accelerate when the distance increases, on the contrary, it prefers to maintain a constant velocity within a certain range of gaps. These characteristics reveal that the drivers in the DD model will prefer the safety (maintaining a safe distance), the smooth (changing the velocity smoothly), and fuel-saving (braking seldom). Fig. 5 Vehicles motions in the NaSch model and in the DD model. 3.2 Analysis of Phase Transition: Density s Impacts This section will discuss the density s impacts on the phase transition. An interesting result is that the DD model can simulate the synchronized flow phenomenon but the NaSch model cannot. Figure 6 explores the traffic state s transition with the density when the random probability is low (p = 0.1). As is customary, traffic propagates in the direction of vertical axis, whereas the horizontal axis is for time. From Fig. 6, we have: (i) When ρ = 0.05, traffic flow is in the free-flow state

5 No. 1 Communications in Theoretical Physics both in the DD model and in the NaSch model. (ii) When ρ = 0.25, large region of free-flow and part of jam appear in the NaSch model. However, the light synchronized flow and the heavy synchronized flow fill the whole region in the DD model, and there is almost no free flow or jam flow. Since the drivers in the DD model have more stable behaviors, they do not stop and go as in the NaSch model. (iii) When ρ = 0.35, the jam flow and the free flow are roughly half and half in the NaSch model. But in the DD model, the whole region is filled of the heavy synchronized flow and the jam flow. Since the jam flow occupies a small region and dissipates soon, there is no centralized 181 jam zone. (iv) When ρ = 0.4, the jam s region further increases but the phase composition keeps unchanged in the two models. From the above analyses, we have the process of the phase transition in the NaSch model: the free flow the jam flow and that in the DD model: the free flow the synchronized flow the jam flow. Moreover, during the process from the free flow to the synchronized flow in the DD model, the free flow first transits to the light synchronized flow, then part transits to the heavy synchronized flow, finally it disappears. Fig. 6 Time-space diagrams in the NaSch model and in the DD model. 3.3 Analysis of Phase Transition: Random Probability s Impacts In the DD model, the phase transition s process is directly related to the random probability p, as shown in Fig. 7: (i) When p = 0, ρ has two critical values, i.e and The phase transition can be divided into three steps. Step 1: When ρ [0, 0.15], all the flow is the free

6 182 Communications in Theoretical Physics Vol. 56 flow. Step 2: When ρ (0.15, 0.23], the flow is the synchronized flow. Specially, when ρ > 0.15, the free flow disappears and immediately transits to the light synchronized flow. And as ρ 0.23, the light synchronized flow is gradually replaced by the heavy one. When ρ = 0.23, all the flow is the heavy synchronized flow. Step 3: When ρ (0.23, 1], the heavy synchronized flow and the jam flow exist simultaneously. When ρ 1, more and more heavy synchronized flows will be replaced by the jam flows and eventually block the whole road (see Fig. 7(a)). (ii) When p (0, 0.1], the phase transition includes four steps: the free flow; the free flow the synchronized flow; the light synchronized flow the heavy synchronized flow; the synchronized flow the jam flow (see Fig. 7(b)). (iii) When p = 0.1, the two critical values of ρ are 0.13 and The phase transition can also be divided into three steps. Step 1: When ρ [0, 0.13], all the flow is the free flow. Step 2: When ρ (0.13, 0.25], the synchronized flow and the free flow exist simultaneously. Specially, when ρ > 0.13, the free flow does not disappear immediately, instead, it is replaced by the synchronized flow gradually. Until ρ = 0.25, the free flow disappears completely and all the flow is the heavy synchronized flow. Step 3: When ρ (0.25, 1], the synchronized flow and the jam flow exist simultaneously. These three steps can be summarized by Fig. 7(c), where the process of the free flow the synchronized flow is gradual when p = 0.1 but it is a jump one when p = 0. (iv) When p [0.1, 1], the phase transition also includes four steps: the free flow; some of the free flow the synchronized flow; the free flow the synchronized flow and meanwhile some of the free flow the jam flow; the synchronized flow the jam flow. Note that in the third step, the free flow, the synchronized flow and the jam flow exist simultaneously. Additionally, the larger p is, the smaller the synchronized flow occupies under the same density (see Fig. 7(d)). (v) When p 1, the free flow does not appear any longer. The proportion of the jam flow and the free flow is similar with the densityρ (see Fig. 7(e)). Fig. 7 Phase transition in the DD model for different p. Thus, during p: 0 1, the region for the free flow to separately exist reduces gradually and eventually disappears; but the coexistence region for the three phases increases. The free flow intrudes into the high-density region while the jam flow intrudes into the low-density region. The synchronized flow expands to the low-density region, but its proportion becomes smaller and smaller due to the extrusions of both the free flow and the jam flow, eventually it disappears. These results illustrate that the random probability is an important factor to impact the phase transition. It impacts the proportion of the traffic phases for the same density. When it is different, the critical density for the phase transition will be different too.

7 No. 1 Communications in Theoretical Physics Conclusions Considering drivers behaviors and taking the acceleration as a decision variable, three core factors affecting drivers decision-making, i.e. distance between adjacent vehicles, their own velocity, and the preceding vehicle s velocity, are analyzed. In accordance with the process of decision-making, action, and result, a cellular automaton (CA) model based on driving decision (DD) is built. The DD model has the following characteristics: (i) It builds a three-step update rules under the driver s driving decision: decision-making, action and result. Compared with the four-step update rules in the NaSch model, it can depict closer to the driver preferences on pursuing safety, stability and energy-efficiency. (ii) The DD model is novel in taking the acceleration a n as a decision variable based on the CA model. This way is much closer to the driver s actual decision-making. In addition, it can more flexibly describe a vehicle s deceleration limitation and car following behavior. Meanwhile, the effect of randomization can be explained better: randomization is not to reduce the velocity necessarily but to reduce the acceleration. (iii) a t n = f(d n, vn t 1, vn+1 t 1 ) means that a driver s driving decision is restricted to the combined effects of the distance between adjacent vehicles d n, its own velocity, and the preceding vehicle s velocity vt 1 v t 1 n n+1. (iv) The DD model proposes a min = 2 and a max = 1 based on the real experimental data. If a n (, 1], the decision-making step in the DD model is equivalent to the combined steps of acceleration and deceleration in the NaSch model. And if a n (, + ), it is equivalent to that in the FI model. Additionally, simulation results imply more findings: (i) In the low-density and high-density regions, the fundamental diagrams in both the DD model and the NaSch model change linearly. However, the DD model s curve is smooth within a certain region near the peak value. (ii) The DD model can simulate the light synchronized flow and the heavy synchronized flow. Compared with other CA models, its rules are very simple and need not specifically define some functions to describe the impacts of the synchronization region. What is more, the synchronized flow is spontaneously formed by a large number of particles after some simple interactions. (iii) Both the density ρ and the random probability p have great impacts on the phase transition in traffic flow. All these findings show that the DD model has strong theoretical innovation for future studies. However, it should be mentioned that overtaking is not involved in this work. Therefore, multi-lane DD models are our ongoing research. References [1] K. Nagel and M. Schreckenberg, J. Phys. I 2 (1992) 221. [2] M. Fukui and Y. Ishibashi, J. Phys. Soc. Jpn. 65 (1996) [3] T. Nagatani, Rep. Prog. Phys. 65 (2002) [4] P. Ramachandran, Commun. Theor. Phys. 52 (2009) 646. [5] R. Jiang and Q.S. Wu, Phys. Lett. A 359 (2006) 99. [6] S.C. Benjamin, N.F. Johnson, and P.M. Hui, J. Phys. A: Math. Gen. 29 (1996) [7] R. Barlovic, L. Santen, A. Schadschneider, and M. Schreckenberg, Eur. Phys. J. B 5 (1998) 793. [8] X.B. Li, Q.S. Wu, and R. Jiang, Phys. Rev. E 64 (2001) [9] R. Jiang and Q.S. Wu, J. Phys. A: Math. Gen. 36 (2003) 381. [10] L. Lei, L.Y. Dong, T. Song, and S.Q. Dai, Acta Phys. Sin. 55 (2006) [11] Q. Zhuang, B. Jia, and X.G. Li, Chin. Phys. B 18 (2009) [12] W. Knospe, L. Santen, A. Schadschneider, and M. Schreckenberg, J. Phys. A: Math. Gen. 33 (2000) [13] E. Brockfeld, R. Barlovic, A. Schadschneider, and M. Schreckenberg, Phys. Rev. E 64 (2001) [14] W. Knospe, L. Santen, A. Schadschneider, and M. Schreckenberg, J. Phys. A: Math. Gen. 35 (2002) [15] R. Jiang and Q.S. Wu, Eur. Phys. J. B 46 (2005) 581. [16] B.S. Kerner, S.L. Klenov, and D.E. Wolf, J. Phys. A: Math. Gen. 35 (2002) [17] B.S. Kerner, Physica A 333 (2004) 379. [18] S.D. Liu, S.Y. Shi, S.K. Liu, Z.T. Fu, F.M. Liang, and G.J. Xin, Commun. Theor. Phys. 43 (2005) 604. [19] Z.H. Peng, G. Sun, and J.Y. Zhu, Commun. Theor. Phys. 51 (2009) 145. [20] B. Jia, Z.Y. Gao, K.P. Li, and X.G. Li, Models and Simulations of Traffic System Based on the Theory of Cellular Automaton, Science Press, Beijing (2007). [21] H.Y. Lee, R. Barlovic, M. Schreckenberg, and D. Kim, Phys. Rev. Lett. 92 (2004) [22] C.Q. Mei, H.J. Huang, and T.Q. Tang, Acta Phys. Sin. 57 (2008) [23] K.P. Li, Commun. Theor. Phys. 45 (2006) 113. [24] P. Ramachandran, Commun. Theor. Phys. 52 (2009) 646. [25] Z.Q. Wei Y.G. Hong, and D.H. Wang, Physica A [26] B.S. Kerner S.L. Klenov, and D. Ewolf, J. Phys. A: Math. Gen. 35 (2002) 9971.