Enhancing highway capacity by lane expansion and traffic light regulation

Transcription

1 Enhancing highway capacity by lane expansion and traffic light regulation Rui Jiang, Mao-Bin Hu, Qing-Song Wu, Bin Jia, and Ruili Wang Abstract This paper studies the traffic flow in a cellular automaton (CA) model with an on-ramp in the open boundary conditions. This CA model is in the framework of three phase traffic theory and can reproduce the first order phase transition from the free flow to synchronized flow [R.Jiang and Q.S.Wu, Eur. Phys. J.B 46, 581 (2005)]. It is shown that when some special homogeneous boundary conditions are used, the capacity of the on-ramp system is greatly enhanced because the strong interactions of main road vehicles and on-ramp vehicles, which occur under inhomogeneous boundary conditions, are avoided. Based on this finding, we propose a new design of road section upstream of on-ramp, i.e., the main road firstly expands into several branches and then the branches merge together. With the help of traffic light control, the inhomogeneous vehicles are homogenized. Accordingly, the capacity is enhanced. Keywords: Traffic flow; highway capacity; on-ramp I. INTRODUCTION Traffic flow research has a quite long history (see, e.g., [1], [2], [3], [4], [5]). It has been shown that freeway traffic is a complex dynamic process, which shows different kinds of transitions between different traffic patterns. Also, numerous empirical observations show that the onset of congestion in an initial free flow is associated with an abrupt decrease in vehicle speed (breakdown phenomenon) and this breakdown is in general accompanied by a drop in highway capacity [6], [7], [8], [9], [10], [11], [12]. Nevertheless, both from operation and from design perspectives, the determination of a road s capacity is one of the most important applications. To explain the capacity drop, different traffic theories and models have been developed. Generally, these models and theories can be categorized into two types: the fundamental diagram approach and the three-phase traffic theory. A. Fundamental Diagram Approach Generally speaking, when traffic density is low, the flow rate increases with the increase of density. It reaches the maximum at medium densities and subsequently decreases with further increase in density. The curve of the average flow rate as a function of the density in the flow density plane is called the fundamental diagram. As pointed out by Kerner et al. [13], the empirical fundamental diagram was the reason R.Jiang, M.B.Hu, and Q.S.Wu are with School of Engineering Science, University of Science and technology of China, Hefei , China rjiang@ustc.edu.cn B.Jia is with School of Traffic and Transportation, Institute of System Science, Beijing Jiaotong University, Beijing China R.Wang is with Institute of Information Sciences and Technology, Massey University, New Zealand why already the first traffic flow models were based upon the postulate that hypothetical spatially homogeneous and time-independent solutions, where all vehicles have the same distances to their neighbors and move with the same constant speed, exist in the flow density plane. The congested patterns from fundamental diagram approach are due to the instability of steady states of the fundamental diagram within some range of vehicle densities. When perturbed, they decay into a single or a sequence of wide moving jams. Specifically, in the fundamental diagram approach, free flow is metastable if the density is equal to or higher than ρ out, the density of the outflow from a jam [14]. The capacity drop is associated with the transition from metastable free flow to jam (F J) caused by a local perturbation. The critical amplitude of the local perturbation needed for the transition decreases as traffic density increase. It reaches a maximum at ρ = ρ out and becomes zero at some critical density ρ c2 > ρ out. Therefore, the probability that the F J transition (i.e., the capacity drop) occurs in a given time interval should decrease with density. B. Three-phase Traffic Theory Recently, based on a series of empirical observations, Kerner [5] found out that in congested traffic two different traffic phases should be distinguished: synchronized flow and wide moving jam. Therefore, there are three traffic phases: (1) free flow, (2) synchronized flow, (3) wide moving jam. A wide moving jam is a localized structure moving upstream and limited by two fronts where the vehicle speed changes sharply. A wide moving jam propagates through either free or synchronized flows and through any bottlenecks keeping the velocity of its downstream front. In contrast, the downstream front of synchronized flow is usually fixed at the bottleneck. Wide moving jams do not emerge spontaneously in free flow. Instead, there is a sequence of two first order phase transitions: first the transition from free flow to synchronized flow occurs (F S), and later and usually at a different location moving jams emerge in the synchronized flow (S J). It is pointed out by Kerner [5] that the dynamical behavior near an on-ramp differs significantly in the fundamental diagram approach and in other empirical observations. For example, in the diagram of congested states based on the fundamental diagram approach, at the highest values of onramp flow rate, the homogeneous congested traffic (HCT) occurs and no jam appears. This is in contradiction with

2 empirical observations where wide moving jams always emerge spontaneously in general pattern (GP). Moreover, at the low values of on-ramp flow rate, jams always emerge as a single moving local cluster (MLC) or triggered stop-and-go traffic (TSG) in the fundamental diagram approach. However, no jam appears spontaneously in synchronized pattern (SP) in empirical observations. To overcome the problem in the fundamental diagram approach, Kerner and his colleagues developed a three-phase traffic theory [5]. They postulate that the steady states (homogeneous and stationary states, time-independent solutions in which all vehicles move with the same constant speed) of synchronized flow cover a two-dimensional region in the flow-density plane, i.e., there is no fundamental diagram of traffic flow. In 2002, Kerner and Klenov [15] developed a first microscopic model of the three-phase traffic theory, which can reproduce empirical spatiotemporal patterns. However, this model is relatively complex. Later some new microscopic models based on three phase traffic theory have been developed, such as Kerner-Klenov-Wolf (KKW) cellular automaton (CA) model [10], [13], the CA model introduced by Lee et al., which considers the mechanical restriction versus human overreaction [16], and the model proposed by Davis [17]. As in empirical observations, these models exhibit the sequence of F S J transitions leading to wide moving jam emergence in free flow; in addition, the models show all types of congested patterns found in empirical observations. Recently, Jiang and Wu [18], [19], [20] presented a CA model based on the comfortable driving model of Knospe et al. [21]. It meets the fundamental hypothesis of three phase traffic theory that some steady-state model solutions cover a two-dimensional region in the flow-density plane and it can reproduce the synchronized flow quite satisfactorily. Moreover, the spatial-temporal patterns at an isolated onramp are also investigated and it is shown that the patterns are qualitatively consistent with empirical observations. The first order phase transition from free flow to synchronized flow is also reproduced. In this paper, using the Jiang-Wu model, we study the traffic flow on highway with an on-ramp in the open boundary conditions. It is shown that the capacity is different when different boundary conditions are adopted. Based on this, we propose a new design of road section upstream of on-ramp which can enhance the capacity of the on-ramp system. The paper is organized as follows. In section 2, the Jiang- Wu model is briefly reviewed and the setup of the on-ramp is presented. In section 3, the simulation results are presented and the new design is introduced. The conclusions are given in section IV. II. MODEL, BOUNDARY CONDITIONS AND SETUP OF ON-RAMP In this section, we briefly recall the Jiang-Wu model and present the on-ramp setup. The Jiang-Wu model is a CA model based on three-phase theory. In CA models, a road is divided into cells which can be either empty or occupied. The vehicles have integer velocity v = 0, 1, 2,, v max and their velocities and positions are usually updated in parallel. Compared with other models, the CA models evolution rules are simple, straightforward to understand, computationally efficient and sufficient to emulate much of the behavior of observed traffic flow, therefore, using CA has become a wellestablished method to model, analyze, understand, and even forecast the behavior of real road traffic. The first CA models for freeway traffic go back to Cremer and Ludwig [22] and to Nagel and Schreckenberg [23]. Since then, there has been an overwhelming number of proposals and publications in this field [1], [2], [24]. For example, in the Fukui-Ishibashi model [25], the vehicle has unlimited acceleration capability; in the cruise control model, the self-organized criticality is found [26]; in the Velocity Dependent Randomization (VDR) model [27], the hysteresis phenomenon is reproduced; in the discrete optimal velocity model [28], the coherent moving state is in good agreement with real data; in the velocity effect model [29], the anticipation effect of vehicles are considered. These CA models belong to the fundamental diagram approach, and they fail to reproduce some empirical phenomena. In contrast, the KKW model, the Jiang-Wu model, and the model of Lee et al. are CA models based on threephase traffic theory. Therefore, in this paper, the simulations are carried out based on one of them, the Jiang-Wu model. For the sake of the completeness, we briefly recall the Jiang-Wu model. The parallel update rules of the model are as follows: 1) Determination of the randomization parameter p n (t + 1): p n (t + 1) = p(v n (t), b n+1 (t), t h,n, t s,n ) 2) Acceleration: if ((b n+1 (t) = 0or t h,n t s,n ) and (v n (t) > 0)) then v n (t + 1) = min(v n (t) + 2, v max ) else if (v n (t) = 0) then v n (t + 1) = min(v n (t) + 1, v max ) else v n (t + 1) = v n (t) 3) Braking rule: v n (t + 1) = min(d eff n, v n (t + 1)) 4) Randomization and braking: if (rand() < p n (t + 1)) then: v n (t + 1) = max(v n (t + 1) 1, 0) 5) The determination of b n (t + 1): if (v n (t + 1) < v n (t)) then: b n (t + 1) = 1 if (v n (t + 1) > v n (t)) then: b n (t + 1) = 0 if (v n (t + 1) = v n (t)) then: b n (t + 1) = b n (t) if v n (t + 1) v c and t f,n t c1, then: b n (t + 1) = 0 6) The determination of t st,n : if v n (t + 1) = 0 then: t st,n = t st,n + 1 if v n (t + 1) > 0 then: t st,n = 0 7) The determination of t f,n : if v n (t + 1) v c then: t f,n = t f,n + 1 if v n (t + 1) < v c then: t f,n = 0 8) Car motion: x n (t + 1) = x n (t) + v n (t + 1).

3 Here x n and v n are the position and velocity of vehicle n (here vehicle n + 1 precedes vehicle n), d n is the gap of the vehicle n, b n is the status of the brake light (on(off) b n = 1(0)). The two times t h,n = d n /v n (t) and t s,n = min(σv n (t), h), where h determines the range of interaction with the brake light, are introduced to compare the time t h,n needed to reach the position of the leading vehicle with a velocity dependent interaction horizon t s,n. d eff n = d n + max(v anti gap safety, 0) is the effective distance, where v anti = min(d n+1, v n+1 ) is the expected velocity of the preceding vehicle in the next time step and gap safety controls the effectiveness of the anticipation. rand() is a random number between 0 and 1, t st,n denotes the time that the car n stops, t f,n denotes the time that car n is in the state v n v c. The randomization parameter p is defined: p(v n (t), b n+1 (t), t h,n, t s,n ) = p b : if b n+1 = 1 and t h,n < t s,n p 0 : if v n = 0 and t st,n t c p d : in all other cases Here v c, t c and t c1 are parameters. Next we present the setup of the on-ramp. We assume the main road is a single lane road and its length is L. The onramp is assume to be from L 1 to L 2 and the length of the merge section is L m (see Fig.1). In the merge region, the on-ramp vehicle changes lane to the main road, provided the following conditions are met: x n x n d > λv n, (1) x + n x n d > λv n. (2) Here superscripts + and denote the preceding vehicle and the trailing vehicle in the target lane, respectively; λ reflects the personality of the driver, d is the vehicle length. Fig. 1. The sketch of the on-ramp. The open boundary conditions are adopted as follows. In the exit, when the first vehicle goes beyond L, it is removed from the system and the second vehicle becomes the new leading vehicle and it moves without hindrance. In the entrance of the main road, if the last vehicle is beyond x in, then a new vehicle is inserted at min(x last x in, x in ) with maximum velocity v max with probability α A. Here x last is. position of the last vehicle on main road. In the entrance of the on-ramp, if the last vehicle is beyond L 1 + x in, then a new vehicle is inserted at min(x on,last x in, L 1 +x in ) with maximum velocity v max with probability α B. Here x on,last is position of the last vehicle on on-ramp. In the next section, the simulations are carried out. In the simulations, each cell corresponds to 1.5 m and a vehicle has a length of five cells. One time step corresponds to 1 s. The parameter values are t c = 7 s, t c1 = 30 s, v c = 18 (corresponding to 97.2 km/h), v max = 20 (corresponding to 108 km/h), p d = 0.1, p b = 0.94, p 0 = 0.5, h = 6 s, gap safety = 7 (corresponding to 10.5 m), σ = 1/1.5 s 2 /m, λ = 1.0 s, L = 30000, L 1 = 15000, L 2 = 16000, L m = 300 (corresponding to 45 km, 22.5 km, 24 km, and 0.45 km, respectively) unless otherwise mentioned. III. SIMULATION RESULTS In this section, the simulation results are presented. We assume initially there is no vehicle on the road. From t = 0 the vehicles enter the main road with probability α A and enter the on-ramp with probability α B. The simulation lasts for time steps (3 hours). In Fig.2, the typical traffic patterns found in empirical observation are reproduced. In Fig.2(a), the flow rates on both main road and onramp are high. As a result, the general pattern (GP) is observed. GP remains when on-ramp flow rate decreases. However, when on-ramp flow rate is small, the outflow rate of the jam is smaller than the capacity of the onramp system, the free flow will form between the location of the on-ramp and the downstream front of the jam. As a result, the dissolving general pattern (DGP) is observed (Fig.2(b)). When the on-ramp flow further decreases, the jam will not spontaneously occur in the synchronized flow and the widening synchronized flow pattern (WSP) appears (Fig.2(c)). When the on-ramp flow is very small, the main road flow is perturbed by the on-ramp flow and consequently the synchronized flow is induced. However, the on-ramp flow will not become an bottleneck to the main road flow. In this way, the alternations of free and synchronized flow (ASP) is observed (Fig.2(d)). Between the region of GP and free flow, the localized synchronized flow pattern (LSP) exists (Fig.2(e)). We investigate the capacity of the on-ramp system. Here for simplicity, we only consider the case that both main road flow and on-ramp flow are high. Our simulations show that when GP occurs, the capacity is approximately 1700 vehicles/h, which is essentially independent of α A and α B because congestion occurs on both main road and on-ramp. The contribution of the main road is 890 vehicles/h and that of the on-ramp is 810 vehicles/h. Next we adopt a different boundary condition for the onramp. A new car is inserted at x = 1 at time step 3m + 1 on main road with maximum velocity, and a new car is inserted at x = L 1 at time step 3m + 2 on on-ramp with maximum velocity. Here m is an integer 0, 1, 2,... The parameters used are L = 14400, L 2 = 400, L 1 = 50. Initially there is no vehicle on the road. Under such boundary

4 condition, the capacity is 2400 vehicles/h, much higher than 1700 vehicles/h. The contributions of both main road and the on-ramp are 1200 vehicles/h. Let us have a look at the distribution of the vehicles in the merge region. It is found that the headways of the vehicle on on-ramp to the vehicle ahead (on target lane) and vehicle back (on target lane) slightly fluctuates around 30 (corresponding to 45 m). In other words, when the first vehicle (suppose it is from main road) reaches x = x 1, the second vehicle (which is from on-ramp) reaches x x 1 30, the third vehicle (which is from main road again) reaches x x 1 60, and so on. Therefore, the on-ramp vehicles always have the chance to change lane to main road far before they reach the end of the merge region. Moreover, when the vehicle on on-ramp changes lane to main road, it does not brake and the back vehicle on target lane does not brake, either. In other words, the strong interactions of main road vehicles and on-ramp vehicles, which occur under inhomogeneous boundary conditions, are avoided. As a result, the high capacity is maintained. Fig. 2. The typical traffic patterns induced by the on-ramp. (a) GP, x in = 30, α A = 0.6, α B = 0.5; (b) DGP, x in = 30, α A = 0.6, α B = 0.06; (c) WSP, x in = 30, α A = 0.6, α B = 0.04; (d) ASP, x in = 30, α A = 0.6, α B = 0.01; (e) LSP, x in = 30, α A = 0.36, α B = The vehicles move from left to right and the time is increasing in up direction. Each point represents there is a vehicle at location x at time t. Therefore, the darker the region is, the higher the local densities are. In real traffic, the enter of vehicles is always inhomogeneous. For example, we consider the following traffic situations: the vehicles enter the main road with probability α A = 0.5 and x in = 30 and enter the on-ramp with probability α B = 0.4 and x in = 30, which implies main road inflow rate q in 1680 vehicles/h and on-ramp inflow rate q on 1390 vehicles/h. Under normal on-ramp setup as shown in Fig.1, the capacity is approximately 1700 vehicles/h as shown before. Next we propose a new design of road section upstream of on-ramp, which homogenize the vehicles and accordingly enhance the capacity. The design is shown in Fig.3. Both the main road and on-ramp expand into two roads and then the two roads merge into one again just upstream of the merge region. Note that this method is widely used at tollbooth system on highways [30], [31]. On each of the four branches, a traffic light is set. Suppose traffic light 1 is at x = x 1, then traffic light 2 is at x 2 = x 1, traffic light 3 is at x 3 = x 1 10, traffic light 4 is at x 4 = x Traffic light 1 is switched to green on time step 6m + 1 to allow one vehicle to pass, and then is switches to red. Similarly, traffic light 2 is switched to green on time step 6m + 4; traffic light 3 is switched to green on time step 6m + 2; traffic light 4 is switched to green on time step 6m + 5. We suppose that when the traffic light switches to green, the vehicle accelerates with constant acceleration a = 1 before it reaches the maximum velocity, then it moves according to Jiang-Wu model. In this way, the vehicles enter the merge region homogeneously and high capacity 2400 vehicles/h is maintained. One each branch, the flow rate is 600 vehicles/h. Figure 4 shows the spatio-temporal plot of the main road.

5 Fig. 3. The sketch of the new design of road section upstream of on-ramp. Here the length of the expansion region is 300 and it ends at x 0, the position of the traffic light 1 is at x 1 = x One can see that the lane expansion and the traffic lights homogenize the vehicles. In Fig.4(b), the details near the traffic lights are shown. One can see that the density is higher in the lane expansion region (upstream of the traffic lights) than the density further upstream. Downstream of the traffic lights, the homogeneous free traffic forms (on each branch). The vehicles on the four branch almost does not interact with each other when they enter the merge region area. Since usually the on-ramp flow is not so high as main road flow, we may split the main road into three roads and the on-ramp does not split. In this way, the flow rate on main road can reach 1800 vehicles/h. If we split the main road into four roads, then a higher capacity can be reached. In this case, suppose traffic light 1 is at x = x 1, then traffic light 2 is at x 2 = x , traffic light 3 is at x 3 = x 1 + 4, traffic light 4 is at x 4 = x , traffic light 5, which is on on-ramp, is at x 5 = x The traffic lights are switched to green in the following way: Traffic light 1 is on time step 7m, traffic light 2 is on time step 7m + 2, traffic light 3 is on time step 7m + 3, traffic light 4 is on time step 7m + 5, traffic light 5 is on time step 7m+6. In this way, the vehicles enter the merge region homogeneously with headway 28 and the capacity is 2570 vehicles/h. One each branch, the flow rate is 514 vehicles/h. In real traffic, the heterogeneous vehicles may bring some problems to the design. For example, some vehicles have large acceleration capability, some vehicles have large maximum velocity, some trucks are much longer. Therefore, it is important to limit the acceleration and maximum velocity of the vehicles near the merge region. Otherwise, the homogeneous boundary conditions will be violated. This, we believe, can be fulfilled in the near future due to adaptive cruise control (ACC) [32], [33]. When a long truck is detected passing the light, we may stop the next traffic light from switching to green for one time. In this way, the headway between the truck and its follower will be large enough and the follower does not need to brake. IV. CONCLUSIONS In this paper, we have studied the traffic flow in a CA model with an on-ramp in the open boundary conditions. Fig. 4. The spatio-temporal plot of the main road under new design. Note only the traffic situations on one branch are shown and those on other branch are similar. (b) shows the details near the traffic lights. The cars are moving from left to right and the time in increasing in up direction. This CA model is in the framework of three phase traffic theory, it can reproduce the synchronized flow quite satisfactorily and can reproduce the first order phase transition from the free flow to synchronized flow. The simulations show that the typical traffic patterns found in empirical observation can be reproduced under inhomogeneous boundary conditions. Furthermore, we found that when homogeneous boundary conditions are used, the capacity of the on-ramp system is greatly enhanced because the strong interactions of main road vehicles and on-ramp vehicles, which occur under inhomogeneous boundary conditions, are avoided. Next we propose a new design of road section upstream of on-ramp, i.e., to split the main road into several branches and then the branches merge together. With the help of traffic light control, the inhomogeneous vehicles are homogenized. Accordingly, the capacity increases. In our future work, empirical data need to be collected at the on-ramps to verify our simulation results, i.e., the homogeneity of traffic approaching the merge point increases merge capacity. Furthermore, in real traffic, the main road usually has two lanes or three lanes. we will apply the new design to multi-lane main road and evaluate the benefit of the new design. ACKNOWLEDGEMENT We acknowledge the support of National Basic Research Program of China (No.2006CB705500), the National Natural Science Foundation of China (NNSFC) under Key Project No and Project Nos , , , , the CAS President Foundation, and the Chinese Postdoc Research Foundation (No ). R. Wang acknowledges the support of the ASIA:NZ Foundation Higher Education Exchange Program (2005), Massey Research Fund (2005), and International Visitor Research Fund

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