Properties of Phase Transition of Traffic Flow on Urban Expressway Systems with Ramps and Accessory Roads

Transcription

1 Commun. Theor. Phys. 56 (2011) Vol. 56, No. 5, November 15, 2011 Properties of Phase Transition of Traffic Flow on Urban Expressway Systems with Ramps and Accessory Roads MEI Chao-Qun (Ö ) 1, and LIU Ye-Jin ( â ) 2 1 School of Statistics, Capital University of Economics and Business, Beijing , China 2 School of Urban Economics and Public Affair, Capital University of Economics and Business, Beijing , China (Received January 28, 2011; revised manuscript received April 13, 2011) Abstract In this paper, we develop a cellular automaton model to describe the phase transition of traffic flow on urban expressway systems with on-off-ramps and accessory roads. The lane changing rules are given in detailed, the numerical results show that the main road and the accessory road both produce phase transitions. These phase transitions will often be influenced by the number of lanes, lane changing, the ramp flow, the input flow rate, and the geometry structure. PACS numbers: a, Vn, Ak Key words: traffic flow, ramp, phase transition 1 Introduction On-off-ramps and urban expressway often produce very congested traffic, which has attracted scholars develop many methods and models to describe the formation mechanisms, the evolution, and propagation properties of these complex phenomena. [1 23] Among the above studies, Kerner [1 5] presented three phase traffic theory based on many empirical observations and found that onramp will often produce general pattern (GP) and synchronized flow pattern (SP). Helbing et al. [6 7] adopted the gas-kinetic model to study ramp and found that onramp will produce different congestion patterns when the main road traffic flow is not uniform flow. Tang et al. [8] presented a new macro model for traffic flow on a highway with ramps, which can reproduce some of the complex phenomena induced by ramps. The traffic phenomena resulted by the off-ramp of I-800 freeway in Oakland and the fifth off-ramp of the state freeway in Origine and California demonstrated that the traffic flow near the off-ramp are relevant to the saturation and the output rate of the off-ramp. [9 10] Diedrich et al. [13] used two lane-changing rules to study a ring road with ramps and obtained the fundamental diagram of ramp. Campari et al. [16] further studied a two-lane traffic system with ramps and obtained some similar results. In the real traffic system, the urban expressway often has ramps, but the ramp length and the distance between two ramps in the urban expressway are often shorter than the ones in freeway, which shows that there are more distinct interactions among ramps, main road and accessory road, so these interactions will often produce more complex phase transitions. In this paper, we investigate the phase transitions of the urban expressway with ramps and find that the phase transitions are related to the number of lanes, lane changing, and ramp flow, the input rate of the whole system and the geometry of the expressway. 2 Model In this paper, we adopt the following urban expressway system to study various phase transitions resulted by on-off-ramp. (a) Iteration rules Suppose the time step is 1 second, a cell is occupied by one vehicle or empty. v max is the vehicle s maximum velocity, v n and x n are the n-th vehicle s velocity and position, d n = x n 1 x n 1 is the n-th vehicle s gap. At the time step t t + 1, the update rules are as follows: (i) acceleration, v n min(v max, v n + 1); (ii) deceleration, v n min(v max, d n ); (iii) randomization, v n max(v n 1, 0) with probability p; (iv) position update, x n x n + v n. For simplicity, we here suppose the maximum velocities of the main road (lane 1 and 2) are v max, and the maximum velocity of the accessory road (lane 3 and 4), the ramp and the weaving section are v max. In addition, we assume that the off-ramp of the main road lies in the cell c 0 in the accessory road (see Fig. 1), the leading vehicle on the off-ramp is l c, and the leading vehicle on lane 3 before c 0 is l 3. If the vehicles l c and l 3 can simultaneously Supported by grants from the Humanities and Social Sciences Foundation of Ministry of Education of China under Grant No. 09YJC790193, Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality the Research Funds of Capital University of Economics and Business under Grant No , and the National Natural Science Foundation of China under Grant No Correspondence author, meichaoqun@126.com c 2011 Chinese Physical Society and IOP Publishing Ltd

2 946 Communications in Theoretical Physics Vol. 56 arrive at c 0 in one time step, we should calculate the time t c and t 3 that they arrive at c 0, i.e. c 0 x lc t c = min(v max, c 0 x lc 1, v lc + 1), (1) c 0 x l3 t 3 = min(v max, c 0 x l3 1, v l3 + 1), (2) where x is the vehicle s position and v is the vehicle s velocity. Based on Eq. (1) and Eq. (2), we have: (i) when t c < t 3, l c directly enters lane 3 at its current velocity and l 3 will decelerate. (ii) when t c > t 3, l 3 goes forward at its current velocity and l c will decelerate; (iii) when t c = t 3, l c first occupies the cell c 0 if c 0 x lc < c 0 x l3 otherwise l 3 first occupies the cell c 0 ; if l 3 changes lane at this time, they will pass the cell c 0 at the same time; l c has the priority to occupy c 0 if c 0 x lc = c 0 x l3. (b) Initialization Set the most left cell of each lane as the cell 1, i.e. x = 1, the entrances of the lane 1 and 2 (lane 3 and 4) includes v max (v max ) cells, the vehicles can enter the main (accessory) road from the cell 1, 2,..., v max (1, 2,..., v max ). xi is the last vehicle on the lane i, if the entrance is empty, a vehicle will be injected into the cell min[x i v max (v max), v max (v max)] at the probability α i (i = 1, 2, 3, 4) at its maximum velocity, otherwise it is deleted. Set β 1, β 2 are respectively the output proportions from the lane 1 and 2, and β 3, β 4 are respectively the input proportions from the lane 3 and 4. We in this paper adopt the open boundary. Set the on-ramp length as L r and the off-ramp length as L c. In addition, we suppose that the vehicles that enter the main road should change lane before the cell m 2. (c) Lane-changing rules In this paper, we consider the following three kinds of lane-changing rules: (i) Lane-changing happens in the merging regions. When the vehicles on the lane 2 should enter the merging region [m 1, m 2 ] or the vehicles in the region [m 1, m 2 ] want to go into the lane 2 (see Fig. 1), lane-changing will happen. As for the above lane-changing, its condition can be reduced as follows: or g i + g + i + 1 M, g i min{g + i, max(0, M 1)}, g + i g i or g + i 1, (3a) g i + g + i + 1 < M, g + i g i or g + i 1. (3b) (ii) Lane-changing is prepared for ramp. In the region [c 1, c 2 ] (see Fig. 1), the vehicles that should go out from the main road will first change from the lane 1 to 2; in the region [c 3, c 4 ] (see Fig. 1), the vehicles that should enter the main road will change from the lane 4 to 3. As for the above lane-changing, its condition can be reduced as follows: g i min{v i + 1, M }, g i g + i or g + i 1. (4) (iii) Lane-changing is induced by the ideal velocity. As for the lane-changing, its condition can be reduced as follows: g i min{v i + 1, M }, g i < min{v i + 1, M} and g + i min{v t + 1, M}, (5) then the vehicle changes lane with probability p 1. Where g i is the gap of vehicles i and i 1, v i is the velocity of vehicle i, g i /g+ i (v i /v+ i ) is the gap (velocity) of vehicle i and the nearest back/front vehicle on the neighbor lane, M is the maximum velocity on the main road, M is the maximum velocity on the accessory road and the weaving section. Setting α 1 = α 4 = 0, and p 1 = 0 on lane 2 and lane 3, we get single-lane condition. Fig. 1 Scheme of urban expressway systems with on-off ramps and accessory roads. 3 Simulation We respectively choose 100 cells at the upstream of the cell m 1 on lane 2, the cell C 4 + 1, the cells m on lane 2 and the cell C 0 on lane 3. Denote L m = m 2 m 1 + 1, choose L r = 3, L c = 3; c 1 = m 1 40, c 2 = m 1 v max, c 3 = 300 L r 40, c 4 = 300 L r v max (Fig. 1). Combing with observation in reality, we assume that even though the lane-changing condition (iii) is satisfied, p 1 = 0 for vehicles in [m 1, m 2 ] and [a 1 20, a 2 ] of lane 1 and [c 0 v max, c 0 1] of lane 4, p 1 = 1 for vehicles in [m 1, m 2 ] and [a 1 20, a 2 ] of lane 2 and [c 0 v max, c 0 1] of lane 3; p 1 = 0.3 in other regions, and the random probability of decrease p = 0.1; v max = 3, v max = 2. We investigate the effects of the output and input proportions on the phase transition under nine cases (see Table 1), where cases 1 3 and cases 4 9 are respectively single-lane and double-lane, and cases 2 4 to simulate on-ramp, cases 8 9 to simulate off-ramp, case 1 and cases 5 7 to simulate both on-ramp and off-ramp. The phase diagram (α 2, α 3 ) under the nine cases is shown in In this paper, single-lane denotes that the main and accessory roads are single lane and double-lane means that both are double lanes.

3 No. 5 Communications in Theoretical Physics 947 Fig. 2. The absolute value of the flow (velocity) difference q( v) on the main (accessory) road before and after the phase transition α 2 (α 3 ) is shown in Fig. 3. The maximum flow and its corresponding velocity during the phase transition are shown in Fig. 4. From the three figures, we have: (i) The boundary of each region illustrates that the interactions between the main and accessory roads will produce phase transition and increase with the slope of the boundary (see Fig. 2). The curves q and v can reflect the properties of the phase transition by increasing α 2 (α 3 ) when fixing α 3 (α 2 ) (see Fig. 3). (ii) First we consider single-lane. As to case 1, the area I is large (see Fig. 2), the maximum flow and its corresponding velocity are high (see Fig. 4). When 0 α 2 (α 3 ) 0.4, q, v on the main road and accessory road are all small and the flow and velocity is still high after transition, which shows continuous phase transition F S on roads; at the same time, the gradient of the boundary between area I and II is mild, which shows that the main road have weak influences on the accessory road. When 0.4 < α 2 < 0.6, the boundary line turns acclivitous, which shows that the main road has obvious effects on the accessory road, increasing α 2 makes the phase transition on the accessory road ahead of time; When α 2 > 0.4, q, v become bigger (the maximum q is about 0.02 and v is about 0.4), and the phase transition F S is discontinuous. When 0 < α 3 < 0.43, the boundary of area I and IV is acclivitous, which shows that the accessory road affect the main road even if the main road input rate is very small; When α 2 > 0.4, the main road shows a similar but lighter phase transition to the accessory road. In this case, the in-vehicles and out-vehicles conflict on the weaving section, but the serious congestion does not happen. The reason is the out-vehicles vacate space for in-vehicles on main road, which help the in-vehicles enter into main road successfully; only the out-vehicles received some hindrances which causing discontinuous phase transition F S on main road. The enough long weaving section become glacis anyhow. Table 1 Nine cases of simulation conditions. Case 1: β 2 = 0.5, β 3 = 0.5 Case 2: β 2 = 0, β 3 = 1, L m = 2 Case 3: β 2 = 0, β 3 = 1 Case 4: β 1 = 0, β 2 = 0, β 3 = 1, β 4 = 0 Case 5: β 1 = 0, β 2 = 1, β 3 = 1, β 4 = 0 Case 6: β 1 = 0, β 2 = 0.5, β 3 = 0.5, β 4 = 0 Case 7: β 1 = 0.25, β 2 = 0.25, β 3 = 0.25, β 4 = 0.25 Case 8: β 1 = 0, β 2 = 0.5, β 3 = 0, β 4 = 0 Case 9: β 1 = 0, β 2 = 1, β 3 = 0, β 4 = 0 Fig. 2 The phase diagram (α 2, α 3) under the nine cases, where lane 2 and lane 3 are free flow in I, lane 2 is free flow and lane 3 is congested flow in II, lane 3 is free flow and lane 2 is congested flow in IV, lane 2 and lane3 are congested flow in III.

4 948 Communications in Theoretical Physics Vol. 56 For case 2, we suppose the entrance to main road can hold one vehicle, as the in-vehicles should change to main road before cell m, so L m = 2. Compare to case 1, area I is smaller and the others are bigger, and from q and v (Fig. 3), we find that the degree of phase transition in case 2 is stronger than case 1. It is not difficult to see that the phase transition F S is discontinuous on accessory road even if α 2 = 0, which shows that though the road construct goes against the accessory road (only one lattice at the on-ramp entrance to main road), the intensive desire on lane changing meliorate the deficiency of hardware; at the same time, the strike that accessory to main road is not so big. Fig. 3 The absolute value of the flow (velocity) difference q( v) on the main (accessory) road before and after the phase transition α 2(α 3), where M denotes the main road and A represents the accessory road. Now change L m = 2 to L m = 20, we get case 3. There is markedly bigger area III and IV than in case 2 and 1, and remarkable slope of boundary line between I, IV (see Fig. 3); It is found that confirming the boundary of free flow and congestion is difficult as the tiny disturbance of accessory road may bring severe impact to the main road. From Fig. 3 we see, v > 2 and q < 0.4 when α 3 0.2, especially when α 3 0.5, v > 2.8, and q < 0.07 show the main road transits from the free flow with high velocity (above 2.8) to very serious jam sharply, there is transition F J, the space-time plot reproduce this discontinuous transition (Figs. 5(a), 5(b)). The above results show that the enough long weaving section causes more disturbances on main road; these disturbances cause intense phase transition. The influence of main road to accessory road is slighter: the area I and IV are bigger than that in case 2 and somewhere in case 1 (Fig. 2), the change of flow and velocity are comparatively calm (Fig. 3), which is owing to that the strong lane-changing desire can be realized under the help of the enough long weaving section. So the phase transition on accessory road is mostly induced by itself, i.e. the increasing input rate. Figure 5 also shows that the congestion formation on main road is F J, but the congestion fading is J S F as the decreasing of in-vehicles, and there is more congested traffic in synchronized phase along with the lighter fall of α 3 (Figs. 5(c), 5(e)). (iii) As to double-lane, first we compare case 4 with case 3, the main road and the accessory road get melioration, especially on main road: the area I and IV become larger, the area I and II become larger remarkably (see Fig. 2); transition on main road happened at larger α 3, q, and v are smaller (Fig. 3), the maximum flow on transition are higher and the corresponding velocities are lower (Fig. 4).The small q and v on accessory road show the gently phase transition on the accessory road.

5 No. 5 Communications in Theoretical Physics 949 Combing the results of case 3 and case 4, we conclude that lane changing conduces the improvement of traffic; we also have gotten the space-time plots when α 1 = 0.11, α 2 = 0.11, α 3 = 0.45, α 4 = 0.45 (Fig. 6(b)) and α 1 = 0.7, α 2 = 0.11, α 3 = 0.45, α 3 = 0.45 (Fig. 6(c)), which shows that when vehicles on lane 2 have difficulty to change to lane 1(for there are too much vehicles on lane 1), then the congestion on lane 2 appears ahead of schedule even when α 2 is small. Through the simulation process, we have also found that the traffic never gets into F J even with the big enough input rate (Figs. 6(a) and 6(d)). Fig. 4 Maximum flows and the corresponding velocities on main roads and accessory roads. (a) and (c): main roads, (b) and (d): accessory roads. Fig. 5 Congestion formulation and fading away near the on-off ramp on cellular [71,420] of the single-lane main road, fixing on-ramp at cellular 300 and off-ramp at cellular 320, β 2 = 0, β 3 = 1, α 2 = 0.1, α 3 = 0.46 when t [0, 20000] and α 2 = 0.1, α 3 = 0.05 when t [20001, 30000]. (a) t [5001, 5500]; (b) t [19501, 19500]; (c) t [20001, 20500]; (d) t [29501, 29500]; (e) α 2 = 0.1, α 3 = 0.46 when t [0, 20000] and α 2 = 0.1, α 3 = 0.25 when t [20001, 30000], t [20001, 20500]. Comparing the results of case 5 with case 4, we find area II becomes larger, and there is little change about area I and IV, but there is a more acclivitous boundary line between I and IV in case 4 (Fig. 2). We think as no out-vehicles in case 4, vehicles on lane 2 received more hindrances causing by in-vehicles than that in case 5, for out-vehicles can change to weaving section and some space are vacated for the in-vehicles. So as for phase transition F J on lane 2, the above causations put off it in case 5 and make it more sensitive with the increasing of α 3 in case 4. As for the main

6 950 Communications in Theoretical Physics Vol. 56 roads, the values of v at α3 = 0 is not smaller than that of α3 > 0 show that phase transition on main road is more induced by itself, which rests with that out-vehicles on lane 2 will be hindered by the vehicles that have changed to the weaving section even there is no in-vehicles; q is and v is (Fig. 3), the transition is F S; The simulations also show that this synchronized phase is easy to be more congested as small disturbance, along with the velocities sharply drop to As for accessory roads, the phase transition on which become stronger than that in case 4 (see v and q in Fig. 3), because the in-vehicles conflict with the out-vehicles on the weaving section. Fig. 6 Space-time plot near the on-off ramp on cellular [71, 420] of the double-lanes main road, fixing on-ramp at cellular 300 and off-ramp at cellular 320, β1 = 0, β2 = 0, β3 = 1, β4 = 0, t [39501, 40000]. (a) α1 = 0.4, α2 = 0.4, α3 = 0.5, α4 = 0.5; (b) α1 = 0.11, α2 = 0.11, α3 = 0.45, α4 = 0.45; (c) α1 = 0.7, α2 = 0.11, α3 = 0.45, α4 = 0.45; (d) α1 = 0.6, α2 = 0.6, α3 = 0.6, α4 = 0.6. Changing β1 = 0, β2 = 1, β3 = 1, β4 = 0 to β1 = 0, β2 = 0.5, β3 = 0.5, β4 = 0, we get case 6. From Figs. 2 and 3, we find the similar results with that of case 5, but there are gentler phase transition both on main roads and accessory roads. Comparing with case 1, we find the double-lane alleviates the tension of main roads on accessory roads, especially when α2 > 0.4; At the same time, lane changing on lane 2 make the phase transition on main roads happens ahead of time. Comparing the results of case 7 (Figs. 2 4) with case 6, we get the similar trends, except that the phase transition degree of case 7 is a little more intense and phase transition on main road and accessory road both happened ahead of time, which are induced by lane changing. So lane changing is one of the main factors to induce phase transition. Fig. 7 Space-time plot near the off ramp on cellular [121, 470], which include [121, 320] on main road and [321, 323] on off-ramp lane and [324, 470] on accessory road, fixing on-ramp at cellular 300 and off-ramp at cellular 320, β1 = 0, β2 = 1, β3 = 0, β4 = 0, t [29501, 30000]. (a) Main road of double-lanes, α1 = 0.5, α2 = 0.5, α3 = 0.4, α4 = 0.4; (b) Accessory road of double-lanes, α1 = 0.5, α2 = 0.5, α3 = 0.4, α4 = 0.4; (c) Main road of single-lane, α2 = 0.5, α3 = 0.4; (d) Accessory road of single-lane, α2 = 0.5, α3 = 0.4. (iv) At last, we study the off-ramp (case 8 and case 9), the results are shown in Figs From Fig. 2, we find both congested area of main road and accessory road for the later one are bigger, there is only gradient boundary line between area I and II in case 8, but which expand to that of I and IV in case 9. The out-vehicles cause the phase transition on accessory road ahead of schedule, and the same impact will spread to the main road itself along with the increasing of proportion of out-vehicles. Except for main road of case 8, Fig. 4 shows the maximum flow and the corresponding velocities descend all along and the simulation process shows that increasing the input rate on the lane will cause sharply decrease on velocity after phase transition; As to the main road of case 8, the maximum flow ( ) and the corresponding velocity ( ) are all along at relatively high level (see Fig. 4), and v is always small (see Fig. 3). As for case 9, q and v in Fig. 2 tell us that when α3 = 0, the maximum v arises to 1.178; v rises to over 2 and the maximum flow decreases to under 0.35 when α3 > 0.3

7 No. 5 Communications in Theoretical Physics 951 (Fig. 3). The reason is that the out-vehicles disturb the accessory road much, but if the proportion of out-vehicles is not so big and out-vehicles has priority to the accessory road, the bad impact that accessory road to main road is not so distinct; if the proportion of out-vehicles is big, the accessory road is very easy to disturb the traffic flow on the main road and causes the phase transition F S. We draw the space-time plot of double-lane and singlelane (Fig. 7), F J appears on the accessory road of single-lane. 4 Summary In this paper, we study the traffic flow phase transition of urban expressway systems with on-off-ramps and accessory roads by a CA model. We draw the phase paragraph and analyze the maximum flow and the corresponding velocity and the skip absolute values of velocity ( v) and flow ( q) before and after phase transition, and then conclude the impact factors and the features of phase transition, the results are as follows: (a) The more different between the proportion of invehicles and out-vehicles are, the earlier of the phase transition on the road happens. (b) The weaving section acts on easing the conflict of vehicles especially when the proportions of out-vehicles and in-vehicles are both mezzo, which make the phase transition more mild and keep the flow and velocity relatively high. The simulation shows that constructing the weaving section will greatly reduce the deterioration of traffic as well as not harm the accessory road much. (c) Lane changing can make the roads escaping from the phase transition F J as well as induce the phase transition happens ahead of schedule. The phase transition J S F always happens on main road of single-lane when decreasing the entering probability of main road. The simulation results may offer some advisements to traffic management and traffic engineering in reality. For example, prompt in-vehicles and out-vehicles change to outboard lane ahead of time, and forbid lane-changing when vehicles have arrived at the aim-lane; on rush hour, build some geometry settings for prohibition or let manager channel off to keep on the mild phase transition in order to get high flow and the moderate velocity. References [1] B.S. Kerner, Phys. Rev. E 65 (2002) [2] B.S. Kerner, Math. Comput. Model 35 (2002) 481. [3] B.S. Kerner, S.L. Klenov, and D.E. Wolf, J. Phys. A 35 (2002) [4] B.S. Kerner, H. Rehborn, M. Aleksic, and A. Haug, Trans. Res. Part C 12 (2004) 369. [5] B.S. Kerner, S.L. Klenov, A. Hiller, and H. Rehborn, Phys. Rev. E 73 (2006) [6] D. Helbing, A. Hennecke, and M. Treiber, Phys. Rev. Lett. 82 (1999) [7] D. Helbing and M. Treiber, Phys. Rev. Lett. 81 (1998) [8] T.Q. Tang, H.J. Huang, S.C. Wong, Z.Y. Gao, and Y. Zhang, Commun. Theor. Phys. 51 (2009) 71. [9] J.C. Muñoz and C.F. Daganzo, Trans. Res. Part A 36 (2002) 483. [10] M.J. Cassidy, S.B. Anani, and J. Haigwood, Trans. Res. Part A 36 (2002) 563. [11] S. Tadaki, K. Nishinari, M. Kiuchi, et al., J. Phys. Soc. Jpn. 71 (2002) [12] S. Tadaki, K. Nishinari, M. Kiuchi, et al., Physica A 315 (2002) 156. [13] G. Dirdrich, L. Santen, A. Schadschneider, et al., Int. J. Mod. Phys. C 11 (2000) 335. [14] H. Ez-Zahraouy, Z. Benrihane, and A. Benyoussef, Int. J. Mod. Phys. C 13 (2002) [15] D.W. Huang, Int. J. Mod. Phys. C 13 (2002) 739. [16] E.G. Campari and G. Levi, Eur. Phys. J. B 17 (2000) 159. [17] B. Jia, R. Jiang, and Q.S. Wu, Physica A 345 (2005) 218. [18] B. Jia, R. Jiang, and Q.S. Wu, Phys. Rev. E 69 (2004) [19] C.Q. Mei, H.J. Huang, and T.Q. Tang, Acta Phys. Sin. 58 (2009) [20] T.Q. Tang and H.J. Huang, Commun. Theor. Phys. 53 (2010) 983. [21] T.Q. Tang, C.Y. Li, H.J. Huang, and H.Y. Shang, Commun. Theor. Phys. 54 (2010) [22] R. Jiang and Q.S. Wu, Physica A 366 (2006) 523. [23] R. Jiang, Q.S. Wu, and B.H. Wang, Phys. Rev. E 66 (2002)