Optimizing traffic flow on highway with three consecutive on-ramps

2012 Fifth International Joint Conference on Computational Sciences and Optimization Optimizing traffic flow on highway with three consecutive on-ramps Lan Lin, Rui Jiang, Mao-Bin Hu, Qing-Song Wu School of Engineering Science University of Science and Technology of China, Hefei 230026, China Email: tini@mail.ustc.edu.cn, rjiang@ustc.edu.cn Abstract In this paper, we study traffic flow on a highway with three consecutive on-ramps by using the Nagel- Schreckenberg model, which is a generalization of our previous work concerning with highway with two ramps. We still focus on how to improve the system capacity by assigning traffic demand to the three ramps. It is shown that when all ramps are in free flow and the main road upstream of the ramps is in congestion, assigning a highest proportion of the demand to the upstream on-ramp, and lowest proportion of the demand to the downstream on-ramp could improve the overall flow. This is explained through studying the spatiotemporal patterns and analytical investigations. The conclusion could be generalized to highway with more on-ramps, i.e., the more downstream an on-ramp is, the lower the traffic demand should be assigned. Keywords-traffic flow; Nagel-Schreckenberg model; on ramps; optimal assignment; I. INTRODUCTION In recent years the physics of traffic congestion has been an active subject of research. Traffic congestion is usually caused by traffic bottleneck which is a section of road with the capacity below neighboring sections. Among the various types of bottlenecks, on-ramps are an important composition of traffic systems and have been widely studied. Lee et al. [1,2] and Helbing et al. [3,4] found that different types of congested traffic states may occur, depending on different inflows on the main road and the on-ramp. The interactions between on-ramp and main road have been studied by Jiang et.al. They simulated the on-ramp using the Nagel-Schreckenberg (NaSch) model, in which it is assumed that the on-ramp connects the main road only on one cell [5]. In previous works [6-10], it has been pointed out that consecutive on-ramps might have an important impact on traffic flow. As explained by Kerner and Klenov [6], mainline congestion at a downstream on-ramp can affect the flow at an upstream on-ramp. Pica Ciamarra [7] performed simulations of an extended NaSch model for one-lane and two-lane roads to determine how to control each on-ramp inflow to better exploit the capacity of the road. Nevertheless, the onramps are modeled in a way of insertion probability, which is not realistic. Davis [8] has analyzed how to apportion demand between two on-ramps to reduce congestion and found that a higher proportion of the demand should be assigned to upstream on-ramp, which is consistent with remarks by Zhang and Recker [9] in a related work regarding metering rates for multiple on-ramps. It has been advised that downstream ramps should be restricted more strongly that upstream ramps. A similar conclusion is drawn by Wang et al.[10], in which it is shown that assigning a higher proportion of the demand to the upstream on-ramp could improve the overall flow. The optimal assignment is valid only in one of the regions in which both on-ramps are in free flow and the main road upstream of the on-ramps is in congestion. In this paper we extend previous work in Ref.[10] to three-on-ramp system. By using deterministic NaSch model, we investigate how to apportion a given demand (incoming flow) to three consecutive on-ramps. We study the throughput by varying the proportion of demand at each ramp. As in Ref.[10], the optimal assignment could be reached in the region that the main road upstream of the ramps is in congestion while all ramps are in free flow. In this region, the overall flow could be improved by assigning a highest proportion of the demand to the upstream on-ramp, and lowest proportion of the demand to the downstream onramp. Through analysis on spatiotemporal diagrams of the main road, it is shown that optimal assignment happens in a critical state of the system, when a random walking shock appears on the main road section between the ramps. Any deviation from the critical state decreases the overall flow of the system. We have carried out theoretical analysis, which agrees very well with the simulation results. The paper is structured as follows. We start with a short review of the NaSch model, and of the rules used to model on-ramps. In section III, the simulation results are presented and discussed. Then, conclusion is given in section IV. II. MODEL In our work, we use the deterministic NaSch model [11] to simulate the forward motion of vehicles. The parallel update rules of the model are as follows: (R1) acceleration: v i min(v max,v i +1); (R2) slowing down: v i min(d i,v i ); (R3) car motion: car i moves v i cells. Here v i and d i are the velocity of car i and the number of empty cells in front of car i, respectively. We use an updating time step of 1 s and a maximum velocity v max =5. Fig.1 presents the sketch of the three-on-ramp system. The main road and the ramps are single lane. We assumed 978-0-7695-4690-2/12 $26.00 2012 IEEE DOI 10.1109/CSO.2012.118 514

Figure 1. Sketch of the three-on-ramp system. that the three on-ramps connect to the main road at one lattice C 0,E 0 and F 0, respectively. We denote the main road upstream of C 0, the first on-ramp, the main road between C 0 and E 0, the second on-ramp, the main road between E 0 and F 0, the third on-ramp and the main road downstream of F 0 as roads A, B, C, D, E, F and G, respectively. In the scenario, we denote the leading cars on roads A, B, C, D, E, F, G as A lead,b lead,c lead,d lead,e lead,f lead,g lead and the last cars on road C, E, G as C last,e last,g last, respectively. To implement the update, we calculate the time needed to arrive at C 0 for cars A lead and B lead, denoted as t a and t b. t c,t d,t e,t f are defined similarly. x C0 x Alead t a = (1) min(v max,x Clast x Alead 1,v Alead +1) x C0 x Blead t b = (2) min(v max,x Clast x Blead 1,v Blead +1) x E0 x Clead t c = (3) min(v max,x Elast t x Clead 1,v Clead +1) x E0 x Dlead t d = (4) min(v max,x Elast x Dlead 1,v Dlead +1) x G0 x Elead t e = (5) min(v max,x Glast x Elead 1,v Elead +1) x G0 x Flead t f = (6) min(v max,x Glast x Flead 1,v Flead +1) Specifically, for cars A lead and B lead, in one time step, we compare the values of t a and t b.ift a > 1 or t b > 1, the updates of cars on road A and road B are not affected by each other; otherwise, i.e. t a 1 and t b 1, four cases are distinguished: (i) if t a <t b, car A lead has the priority to occupy or pass lattice C 0, and becomes the new last car on road C, while the new velocity of car B lead will be confined by the new last car on road C; (ii) if t a >t b, the situation is similar, just the roles of roads A and B interchange; (iii) if t a = t b, for cars A lead and B lead with different distances to lattice C 0, it is reasonable to endow the the Figure 2. Dependence of flow rates J A,J B and J C on α 2 in the case of α 3 = α 4 =0and α 1 =1. car nearer to the lattice C 0 with priority. Then the problem becomes the same as the case t a >t b or t a <t b ; (iv) if t a = t b and the distances to C 0 are the same for cars A lead and B lead, car A lead is provided with priority to update first because it is on the main road. The situation turns to be the same as t a <t b. For cars C lead,d lead,e lead and F lead, the evolution is the same except that the roles of cars A lead and B lead are replaced by the roles of cars C lead,d lead,e lead and F lead, respectively. The open boundary conditions are employed as follows. At the entrance of the main road and on-ramps, after the evolution of all cars is completed in one time step, we check the positions of the last cars on roads A(B,D,E), which are denoted as x Alast (x Blast,x Dlast,x Flast ). If x Alast (x Blast,x Dlast,x Flast ) >v max, then a new car with velocity v max is inserted with probability α 1 (α 2,α 3,α 4 ) at the lattice min[x Alast (x Blast,x Dlast,x Flast ) v max,v max ]. Near the exit of the road G, the leading car is removed if x Glead >L G (L G denotes the position of the right most cell on road G) and the following car becomes the new leading car and it moves without hindrance. III. SIMULATION RESULTS In this section, the simulation results are presented and discussed. In the simulations, roads A-G are divided into 100 v max cells and configured as initially empty. We gather data for 100000 time steps after ignoring initial 40,000 time steps. We denote the flow rate on roads A-G as J A,J B,J C,J D,J E,J F and J G, respectively. Ref.[5] has analyzed the simple case that there are no ramp. The problem reduces to a deterministic NaSch model problem in open boundary conditions, and the flow rates depends on α 1 (the injection rate of road A). When α 1 is small, the flow rate equals to α 1.Whenα 1 =1, the density on the main road is ρ = 1/(1 + v max ). As a result the 515

Figure 3. Phase diagram of the three-on-ramp system at different values of α 2 with the parameter α 1 =1.(a)α 2 =0,(b)α 2 =0.1, (c)α 2 =0.2, (d) α 2 =0.3. maximum flow is J max = v max ρ = v max /(1 + v max )= 5/6. In another case that there are only one on-ramp, the plot of J C versus α 2 is shown in Fig.2, in which α 1 =1and α 3 = α 4 =0. One can see that when α 2 =0, J C reaches the maximum flow 5/6. When0 <α 2 < 0.2, roadais in congestion and on-ramp B is in free flow, and the curve of J C can be approximately described by a linear function with slope value 7/6 J C =5/6 7/6 α 2 (7) When α 2 > 0.2, traffic flows on both roads A and B are in congestion. Being the sum of J A and J B, J C turns into a constant. Since α 3 = α 4 =0, the flow on road G is equal to J C. Now we study the situation α 2,α 3,α 4 > 0. This paper considers the case α 1 = 1, and situation α 1 < 1 will be reported elsewhere. In Fig.3, the phase diagram in the (α 3,α 4 ) space is shown. Road A is always congested in all regions. Four regions are categorized in Fig.3(b) and Fig.3(c). In region I, the traffic flows on ramps B, D and F are free; in region II, it is still free flow on ramp D and F but becomes congested on ramp B; in region III, flows on ramps B and D are in congestion, while ramp F remains free; in region IV, flows are congested on all ramps. In the special case α 2 =0(Fig.3(a)), the system reduces to two-ramp system. As a result, in region I, the traffic flows on ramps D and F are free; In region III, flow on ramp D is in congestion, while ramp F remains free; In region IV, flows are congested on both ramps D and F. From Fig.3, we can also see the influence of α 2 on the phase diagram. Regions III and IV always remain invariant with respect to α 2. However, in the range 0 <α 2 < 0.2, region I shrinks and region II expands with the increase of α 2.Whenα 2 =0, since there is no flow on ramp B, region II disappears and regions I, III, IV transit into regions I, III, IV (Fig.3(a)). When α 2 exceeds 0.2, region I disappears (Fig.3(d)) and the phase diagram becomes independent of α 2. Next we discuss the four regions of the phases diagram. In region I, α 2,α 3 and α 4 are small, thus the flow rates on the ramps are equal to the injection rates. Suppose J 0 = J B + J D + J F = α 2 + α 3 + α 4, which denotes the total on-ramp demand. Through analysis on spatiotemporal diagrams of the main road, it is found that optimal apportionment of the fixed total on-ramp demand happens in the critical state as shown 516

Figure 4. Typical spatiotemporal patterns on road C (a) and road E (b) at the critical state which corresponds to the optimal assignment in region I. below. Fig.4 shows the typical spatiotemporal patterns on roads C and E at the optimal assignment proportion which corresponds to the maximum value of J G. One can see that, a shock exists on both roads C and E, which performs a random walk typical for a one-dimensional driven diffusive system at the transition from low density to high density. Any deviation from the critical state will decrease the overall flow of the whole system. To find the exact optimal apportionment of the fixed total on-ramp demand through an analytical investigation, we adopt Eq.(7) to calculate the value of J G, by varying the proportion of demand at each on-ramp. With the change of ramp injection rate, the flow J G can be calculated in three cases as follows: (i) When both roads C and road E are free, J C can be calculated by J C = 5 6 7 6 α 2 (8) Thus, J G = J C + J D + J F = 5 6 7 6 α 2 + α 3 + α 4 (9) (ii) When road C is congested and road E is free, J G is J G = J E + J F = 5 6 7 6 α 3 + α 4 (10) (iii) When road E is congested (no matter road C is free or congested), J G is J G = 5 6 7 6 α 4 (11) Consequently, if the injection rates reach the optimal assignment proportion, J Gmax is achieved when Eqs.(9)-(11) become equal 5 6 7 6 α 2+α 3 +α 4 = 5 6 7 6 α 3+α 4 = 5 6 7 6 α 4 (12) We can obtain the optimal proportion of ramp injection rates α 2 = 169 309 J 0; α 3 = 91 309 J 0; α 4 = 49 309 J 0 (13) J G,max = 5 6 7 6 49 309 J 0 (14) which are in good agreement with the simulations. In a general case that J C does not decrease linearly with α 2, we replace Eq.(7) by a monotonically decreasing function J C = f(α 2 ). Thus, Eq.(12) becomes J G,max = f(α 2 )+α 3 + α 4 = f(α 3 )+α 4 = f(α 4 ) (15) Due to the monotonic decrease of f(x), it is easy to derive that α 2 >α 3 >α 4. This demonstrates that the upstream onramp injection rate should be higher than downstream one so as to improve the overall flow. This result could be generalized to highway with more on-ramps. In the region where the main road upstream of the on-ramps is congested and all ramps are free, the more downstream an on-ramp is, the lower the traffic demand should be assigned. In region II, both roads A and B are in congestion, while roads D and F are in free flow. As a result, the overall flow is independent of α 2, and the system reduces to the one with two on-ramps, as studied in Ref.[10]. In region III, road D has become congested, while road F is still free. In this case, variations of α 2 or α 3 do not influence the overall flow. Finally, in region IV, roads A,B,D and F are all congested, and the throughput of road G is a constant J G,max =0.6. IV. CONCLUSION In this paper we have studied how to assign a given demand among three consecutive on-ramps to optimize highway traffic flow. It is shown that in the region where the main road upstream of the on-ramps is in congestion while all ramps are in free flow, assign a highest proportion of the demand to the upstream on-ramp, and lowest proportion of the demand to the downstream on-ramp could maximize highway throughput. The conclusion could be generalized to highway with more on-ramps, i.e., the more downstream an on-ramp is, the lower the traffic demand should be assigned. 517

ACKNOWLEDGEMENTS This work is supported by the National Basic Research Program of China (No.2012CB725404), the NNSFC (No.11072239 and 71171185), and the Fundamental Research Funds for the Central Universities (No. WK2320000014). REFERENCES [1] H.Y. Lee, H.W. Lee, and D. Kim, Origin of synchronized traffic flow on highways and its dynamic phase transition, Physical Review Letters, 81: 1130-1133, 1998. [2] H.Y. Lee, H.W. Lee, and D. Kim, Dynamic states of a continuum traffic equation with on-ramp, Physical Review E, 59: 5101-5111, 1999. [3] D. Helbing and M. Treiber, Gas kinetic-based traffic model explaining observed hysteretic phase transition, Physical Review Letters, 81: 3042-3045, 1998. [4] D. Helbing, A. Hennecke, and M. Treiber, Phase diagram of traffic states in the presence of inhomogeneities, Physical Review Letters, 82: 4360-4363, 1999. [5] R. Jiang, Q.S. Wu and B.H. Wang, Cellular automata model simulating traffic interactions between on-ramp and main road,physical Review E, 66: 036104, 2002. [6] B.S. Kerner and S.L. Klenov, Microscopic theory of spatialtemporal congested traffic patterns at highway bottlenecks, Physical Review E, 68: 036130, 2003. [7] M. Pica Ciamarra, Optimizing on-ramp entries to exploit the capacity of a road,physical Review E, 72: 066102, 2005. [8] L.C. Davis, Driver choice compared to controlled diversion for a freeway double on-ramp in the framework of three-phase traffic theory,physica A, 387: 6395-6410, 2008. [9] H.M. Zhang, W.W. Recker, On optimal freeway ramp control policies for congested traffic corridors, Transportation Research Part B, 33: 417-436, 1999. [10] Q.M. Wang, R. Jiang, X.Y. Sun and B.H. Wang, Assigning on-ramp flows to maximize highway capacity, Physica A, 388: 3931-3938, 2009. [11] K. Nagel, M. Schreckenberg, A cellular automaton model for freeway traffic, Journal of Physics I France, 2: 2221-2229, 1992. 518