A MODIFIED CELLULAR AUTOMATON MODEL FOR RING ROAD TRAFFIC WITH VELOCITY GUIDANCE

International Journal of Modern Physics C Vol. 20, No. 5 (2009) 711 719 c World Scientific Publishing Company A MODIFIED CELLULAR AUTOMATON MODEL FOR RING ROAD TRAFFIC WITH VELOCITY GUIDANCE C. Q. MEI,, H. J. HUANG, and T. Q. TANG School of Economics and Management Beijing University of Aeronautics and Astronautics Beijing 100191, China School of Statistics, Capital University of Economics and Business Beijing 100070, China School of Transportation Science and Engineering Beijing University of Aeronautics and Astronautics Beijing 100191, China haijunhuang@buaa.edu.cn Received 26 September 2008 Accepted 19 January 2009 We present a modified cellular automaton model to study the traffic flow on a signal controlled ring road with velocity guidance. The velocity guidance is such a strategy that when vehicles approach the traffic light, suggested velocities are provided for avoiding the vehicles sharp brakes in front of red light. Simulation results show that this strategy may significantly reduce the vehicles stopping rate and the effect size is dependent upon the traffic density, the detector position, the signal s cycle time and the obedience rate of vehicles to the guidance. Keywords: Traffic flow; velocity guidance; traffic light; cellular automaton model. PACS Nos.: 89.40.-a, 45.70.Vn, 64.60.Ak, 02.60.Cb. 1. Introduction Various traffic models have been developed to study the complex traffic phenomena in recent years. 1 10 The cellular automata (CA) model has become a wellestablished method to analyze, understand and even forecast the behavior of real road traffic, because the automata s evolution rules are simple, straightforward to understand, computationally efficient and sufficient to emulate much of the behavior of observed traffic flow. 11 17 Real-time feedback strategy and adaptive cruise control (ACC) system have been adopted in driving control and some important results have been obtained. 8 25 An effective real-time feedback strategy can guide drivers to improve urban traffic. 18 20 In order to maintain a safe distance and to improve the stability of traffic flow, the ACC system adjusts the vehicles speeds according to some criteria. 4,21 24 711

712 C. Q. Mei, H. J. Huang & T. Q. Tang Traffic is generally controlled by signal lights. The green and red lights often produce start-waves and stop-waves on traffic flow. These two waves make vehicles release much more exhaust gas than the stable state of flow in which vehicles run at invariable velocity. Keeping invariable velocity for vehicles can greatly reduce traffic emission. Brockfeld et al. 12 found that under the synchronized strategy of flow, the traffic on a road controlled by several lights can be reduced to the ring road traffic with one light. They pointed out that the most remarkable benefit of green wave strategy is that a street with total length L consisting of N street segments (the length of each street segment is D), can be reduced to one street intersected only once. In the light of this finding, we use the CA model to study the traffic flow on a ring road with one traffic light. The classical CA model is modified to incorporate such a velocity guidance strategy that, when vehicles approach the traffic light, suggested velocities are provided for avoiding their sharp brakes when the signal light turns red. Through comparing numerical results obtained by the classical model and the modified model with various model parameters, we find that the velocity guidance strategy may significantly reduce the vehicles stopping rate (the emissions as a result) and the effect size is dependent upon the initial traffic density, the detector position, the signal s cycle time and the obedience rate of vehicles to the guidance. 2. The CA Model and Velocity Guidance The classical CA models include those proposed by Wolfram 25 and Nagel and Schreckenberg. 26 They found that the start-wave and stop-wave of traffic flow often appear in the congested region. In this paper, we extend the classical CA model through incorporating a velocity guidance strategy into the NS model 26 and use it to study the traffic flow on a ring road with one traffic light. Figure 1 shows the scheme of ring road traffic with one traffic light, where the ring is unfolded as a straight line. Let the timestep be 1 s. We first recall the update rules of a CA model without velocity guidance strategy (Model A). It follows: Acceleration: ν i min(ν i + 1, ν max ). Braking due to traffic light: Case 1. Traffic light is red, ν i min(ν i, d i, s i ); Case 2. Traffic light is green, ν i min(ν i, d i ). Randomization: ν i max(ν i 1, 0) with probability p. Movement: x i x i + ν i. Traffic light Velocity detector Traffic light car i car i-1 D 0 m 0 s i Fig. 1. Schematic illustration of traffic on an unfolded ring road with one traffic light.

A Modified CA Model for Ring Road Traffic with Velocity Guidance 713 In the above, x i is the position of vehicle i, d i = x i 1 x i l is the number of empty cells to the nearest vehicle ahead, l is the vehicle s length, s i = D 0 x i 1 is the distance to the traffic light ahead, D 0 is the position of the traffic light, p is the probability of the randomization. In our study, the length of a single cell is set to be 2.5 m, one vehicle is 3 cells long, and the maximal velocity is ν max = 6 cell/s (54 km/h). Now, we present the extended CA model (Model B) in which a velocity guidance strategy is incorporated. The road is divided into L cells, 1 m 0 < L. Once x i D 0 + m 0 and x i < D 0 + m 0 + l which means vehicle i arrives at the detector, a suggested velocity ν sugg can then be computed according to the vehicle s current velocity and the information of the next traffic light for avoiding from meeting the next red light. This suggested velocity is provided to the driver. The driver can adjust his velocity to ν sugg and cross the next traffic light with this velocity. For a velocity ν {1, 2,..., ν max }, ν sugg is computed as follows: { ( ν sugg = max ν 0 t + ν i ν + s ) } i s a + l mod T < αt, (1) a ν where t is the time from receiving the guidance message to starting velocity adjustment, a is the acceleration, ν i ν /a is the time required by driver for adjusting his velocity from ν i to ν, s a = (ν i + ν) ν i ν /2a is the distance that vehicle moves during the velocity adjustment period, T is the cycle time of traffic light, α is the cycle s split, x mod y = x y int(x/y). Thus, ν sugg is the optimal velocity for avoiding red light signal and crossing the next traffic light as quickly as possible. For simplicity, we suppose a = 1 in this study. The update rules of Model B are as follows: Adjustment: min(ν i + 1, ν max ), if ν i < ν sugg ν i max(ν i 1, 0), if ν i > ν sugg ν sugg, otherwise Braking due to traffic light: Case 1. The traffic light is red, ν i min(ν i, d i, s i ); Case 2. The traffic light is green, ν i min(ν i, d i ). Randomization: ν i max(ν i 1, 0) with probability p. Movement: x i x i + ν i. Suppose that before the simulation, N vehicles are uniformly distributed on the ring road. Since one vehicle occupies three cells, the initial traffic density is ρ = 3N L. (2) In this paper, we only focus our attention on the deterministic case, i.e. p = 0 and α = 0.5 for simplicity.

714 C. Q. Mei, H. J. Huang & T. Q. Tang 3. Simulation Results We now investigate the effect of velocity guidance strategy on traffic flow and the vehicles stopping rate. At every timestep of the simulation, we compute the stopping rate of vehicles as follows: r 0 = N 0 N, (3) where N 0 is the number of stopped vehicles. Traffic flow is computed as follows: q = ρ N i=n i=1 ν i. (4) The numerical results are shown in Fig. 2 when T = 60, L = 300 and m 0 = 9. Note that all the simulation results shown in Figs. 2 to 6 are obtained after 50 000 iterations, and the flow q and the ratio r 0 in Figs. 3 to 6 are the averaged ones on the base of the last 10 000 timesteps. From Fig. 2, we have: (a) When the density is low (0 ρ 0.18), r 0 is approximately equal to zero if the velocity guidance is considered, otherwise it is very high and produces very serious oscillating waves; and the velocity guidance has little effect on q [see Fig. 2(a)]; (b) When 0.18 < ρ 0.698, the oscillating amplitude of r 0 in model B is obviously less than that in model A, but the velocity guidance reduces q slightly [see Fig. 2(b)]. (c) When the density is high (0.698 < ρ 1), the velocity guidance has little effect on q but cuts down on r 0 obviously [see in Fig. 2(c)]. For demonstrating the effect that the velocity guidance has on traffic flow, Fig. 3 depicts the flow and the stopping rate against density generated by both models, in which L = 300, m 0 = 9 and T = 60. From this figure, we have: (a) Whether the velocity guidance is used or not, (i) the flow increases with the density when the density is very low; (ii) as the density is between the down critical value and the up critical value, the flow reaches at a constant capacity; (iii) when the density is higher than the up critical value, the flow decrease with the density. These results are consistent with the findings in Ref. 10. This shows that the velocity guidance strategy can generate reasonable outputs. (b) When the density is low or high, the velocity guidance has little effect on the flow since there is little difference between the ν sugg in model B and the velocity in model A. Here the velocity in either model A or model B is referred to as the average one. When the density is between the two critical values, the flow by model B is smaller a bit than that by model A. This is because, without the velocity guidance, all vehicles can move at relatively high velocities during green signal, which might produce high flow through making up for the deficiency of low flow during red signal.

A Modified CA Model for Ring Road Traffic with Velocity Guidance 715 (a) (b) (c) Fig. 2. Flow and stopping rate after 50 000 iterations, where the left sub-figures are the results of model A and the right for model B, (a) ρ = 0.15, (b) ρ = 0.3, (c) ρ = 0.7.

716 C. Q. Mei, H. J. Huang & T. Q. Tang Fig. 3. Flow and stopping rate against density. Fig. 4. Flow and stopping rate against density for different m 0 values. (c) The velocity guidance can remarkably cut down the stopping rate of vehicles. Next, we explore the effect that the position of the detector has on traffic flow. Figure 4 shows the results given by model B, in which L = 300 and T = 60. From this figure, we have: (a) When m 0 is very large (m 0 = 289), q and r 0 in model B are closer to those in model A. The reasons are as follows: the larger the parameter m 0 is, the shorter the distance between the detector and the next traffic light is, so the suggested velocity ν sugg of model B has little effect on traffic flow when m 0 is very large (m 0 = 289). (b) When m 0 takes 149 or 219, r 0 and q drop remarkably for densities between the two critical values. The reasons are as follows: when drivers reach the detector, the distance to the next traffic light and the time to the latest green phase produce a relatively small ν desired in order to avoid red signal and then make the average flow of the last 10 000 timesteps relatively low.

A Modified CA Model for Ring Road Traffic with Velocity Guidance 717 Fig. 5. Flow and stopping rate against cycle time. We now analyze the flow and the stopping rate against the cycle time of traffic light. The results by both models under three different densities are shown in Fig. 5. From this figure, we have: (a) The velocity guidance can always generate lower r 0 than model A; and the reduction of r 0 becomes more obvious as T increases. (b) The velocity guidance may produce relatively low flow when the cycle time T is long. The reasons are as follows: if the time when a driver passes the detector is within the green phase of the next traffic light, he or she can reach the next traffic light in the same green phase; if the time when the driver passes the detector is within the red phase, the suggested velocity ν sugg would be smaller than the current velocity so that the time of reaching the traffic light is within the green phase, which makes the flow relatively low. (c) If T is short enough, the velocity guidance has little effect on q in comparison with model A. Finally, we investigate the effect that the obedience rate of drivers to guidance, P obey, has on q and r 0 (see Fig. 6). From this figure, we have: (a) In the traffic with ρ = 0.15 or ρ = 0.7, P obey has little effect on q; but r 0 always decreases with the increase of P obey. Thus, in the situation with light or heavy traffic the velocity guidance can produce remarkable influence in decreasing the stopping rate provided more drivers obey the guidance. (b) In the middle traffic with ρ = 0.3, q decreases with the increase of P obey for 0 P obey 0.3 and approximates constant for 0.3 P obey 1. The stopping rate goes down all along.

718 C. Q. Mei, H. J. Huang & T. Q. Tang Fig. 6. Flow and stopping rate against obedience rate of drivers to the guidance. In summary, regardless of the values of parameters adopted, the velocity guidance strategy proposed in this paper may drop the average flow slightly but can significantly reduce the vehicles stopping rate. 4. Summary In this paper, we presented a modified cellular automaton model for studying the traffic flow on a signal-controlled ring road with velocity guidance. The velocity guidance is such a strategy that when vehicles approach the traffic light, suggested velocities are provided for avoiding the vehicles sharp brakes in front of red light. We conducted a lot of simulations using the classical cellular automaton model and the modified model with various model parameters. Results show that the velocity guidance strategy can significantly reduce the vehicles stopping although the average flow may be dropped slightly. This implies the traffic flow becomes more stable and the emissions will be reduced correspondently. Particularly, the effect in stopping reduction is obvious when the density is low or high and the signal cycle time is long. The strategy proposed in this paper is easier to implement than the green wave strategy. It is true when there are turning vehicles at intersections and the traffic density varies along roads in urban networks. Thus, the velocity guidance strategy is practicable in reality. In this paper, however, we have only investigated the effects that the density and the parameters m 0 and P obey have on the traffic flow and the stopping rates. Our numerical results and conclusions lie on the factors considered. Other factors, such as the distance and offset between two traffic lights, and the flow rate at the upstream of traffic lights, may influence the traffic. These will be studied in our follow-up work. Acknowledgments The work described in this paper was supported by grants from the National Basic Research Program of China (2006CB705503) and the National Natural Science Foundation of China (70521001, 70701002).

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