J. Cent. South Univ. Technol. (2011) 18: 1744-1749

DOI: 10.1007/s11771-011-0897-x

Cellular automation model for analyzing capacity of branch road section

WANG Ying-zi(王英姿)^{1, 2}, LONG Dong-fang(龙东方)³, SHI Feng(史峰)¹

1. School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China;

2. College of Civil Engineering, Hunan University, Changsha 410082, China;

3. Changsha Urban Construction Technology School, Changsha 410216, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2011

Abstract: In order to reduce the traffic pressure of urban arterial road with the rational utilization of the branch road, the vehicle meeting behavior on the branch road without divided lane was described, and the cellular automation (CA) model was put forward by introducing meeting behavior to reflect the relation between safe meeting speed and road width. The numerical simulation results depict several relation curves between road section capacity, speed and road width under different directional distributions of traffic flow, as well as the curves between the major and minor direction saturation flow, speed and road width. These relation characteristics indicate that except the one-way road section capacity and speed remaining unchanged, other road section capacities and speeds under different directional distributions increase with the increase of road width. On narrow road, the two-way traffic capacity and speed are less than those of one-way traffic; on wide road, the two-way traffic capacity doubles that of one-way traffic, but their speeds are almost the same. As the directional distribution moves to an even distribution of 50/50, the major direction saturation flows and speeds as well as the minor direction speeds tend to decease, while the minor direction saturation flow tends to increase.

Key words: urban traffic; road section capacity; safe meeting speed; cellular automation model

1 Introduction

Advanced urban traffic microcirculation system can serve to assign the traffic demand, alleviate the traffic pressure, and improve the vehicle speed, which is an integral part of urban road network. The branch road takes a great proportion of the whole road network with advantages of high density, long path length, and ability of distributing arterial traffic stress [1]. DREZNER and WESOLOWSKY [2] argued the benefits and defects of one-way links, and obtained the optimum configuration of one-way and two-way routes by minimizing the total transportation time in a shortest-path-selecting network. MENG et al [3] formulated the optimal contraflow lane configuration as a bi-level programming model considering the total travel time and microscopic traffic simulation. Aiming at reducing the saturation of arterial roads and branch roads, the optimization of one-way traffic organization was established with the consideration of the maximum saturation restriction [4]. SHI et al [5] discussed the bi-level model about the reconstruction of the branch road network as well as its algorithm to lower the saturation of arterial roads and reconstruction expense.

The above researches were mostly based on the traffic assignment approach, in which the capacity estimation of the branch road was similar to that of the arterial road. Due to the limited width and no physical separation between opposing traffic flows on the branch road, the interference between vehicles oncoming in opposite directions is serious. Thus, the capacity estimation of the branch road is significantly different from that of the arterial road with divided lane. Only if the capacity problem of the branch road is solved thoroughly, the relevant works can be performed effectively, such as the reconstructing optimization, and one-way traffic configuration.

This complex traffic flow with serious interference between opposing vehicles can be simulated by cellular automation (CA) model properly. The previous employment of CA traffic model was mainly about the single-lane and pure traffic flow, which discussed vehicle acceleration, deceleration, and randomly deceleration rules to model the real traffic behavior rationally [6-9]. Then, the CA traffic model was developed for multi-lane and mixed traffic flow. Through analyzing the interaction mechanism between different types of vehicles, a series of mixed traffic flow models were put forward. Based on the analysis of arriving, accelerating and overtaking characteristics of different types of vehicles, DEY et al [10] explained the impact of mixed traffic on capacity and speed of two-lane highway with CA model and evaluated the effect of different directional splits in capacity for an individual category of vehicles. XIE et al [11] addressed the interaction between the motor and non-motor vehicles at intersection. MENG et al [12] and LAN et al [13-14] proposed CA models for mixed traffic consisting of autocycles and cars considering the interactive effectiveness. YUAN et al [15] explored a two-lane traffic system with a mixture of buses and cars to propose an optimal number of bus stop. JIANG et al [16] studied the highway capacity with CA model based on three phase traffic theory, and proposed a new design of road section upstream of on-ramp to enhance the capacity of on-ramp system by homogenizing the traffic flow.

In this work, the relation between the road width and safe meeting speed was discussed incorporating with characteristics of the narrow width and no divided lane on the branch road, and then a CA model of traffic flow considering meeting behavior was put forward to obtain the relations between the branch road section capacity, speed and road width. Finally, a comparison was made to present the characteristics of each direction saturation flow and speed.

2 Relation between branch road section width and safe meeting speed

The narrow width, no divided lane and frequent vehicle meeting behavior on the branch road lead to the drop of driving speed and road section capacity. Generally, before meeting vehicle, driver needs to observe the oncoming vehicle and traffic condition to choose the proper meeting place. When meeting vehicle, driver should reduce the speed and maintain enough lateral clearance so as to ensure the meeting safety. The minimum safe lateral clearance is related to the vehicle speed. If the speed is faster, the lateral clearance will be larger.

In general, the more skillful a driver is, the less safe lateral clearance and higher safe meeting speed are preferred. On wide road, driver can meet others with sufficient lateral clearance and fast speed. When the road is narrow, driver has to choose a lower meeting speed.

Figure 1 shows the vehicle meeting process explicitly. During this process, the opposing vehicles A and B run to A′ and B′, respectively. Note d_AB as the minimum safe lateral clearance between vehicles A and B; l_A and l_B are the widths of vehicle A and B, respectively; d_A and d_B are the lateral clearances of A and B from their outside wheels to each nearest roadside. L denotes the road width, L=d_A+l_A+d_AB+l_B+d_B. v_A and v_B are the speeds of A and B before meeting, and v_safe is the safe meeting speed.

Fig.1 Meeting representation of two motor vehicles

The minimum safe lateral clearance needed in vehicle meeting is related to the running speed. Generally, when the speed is up to 40-60 km/h, the lateral clearance between cocurrent vehicles is 1-1.4 m, between opposite vehicles is 1.2-1.4 m, and the lateral clearance between vehicle and its nearest lane edge is 0.5-0.8 m. When the speed is about 30 km/h, the minimum lateral clearance between vehicles is 0.75 m, and the lateral clearance between vehicle and lane edge is 0.6 m. Concluded from the research and practical investigation [17], the details about meeting speed and minimum safe lateral clearance are listed in Table 1.

Table 1 Relation between safe meeting speed and safe lateral clearance

Suppose that the limited speed is 50 km/h and the average vehicle width is 1.8 m. From the data in Table 1, a fitting curve of the safe meeting speed and branch road section width is obtained in Fig.2.

Fig.2 Fitting curve of safe meeting speed and road section width

The curve in Fig.2 follows an upward trend, in which the safe meeting speed increases from 0 km/h and levels off at 50 km/h with road section width ranging from 4.6 m to 6.6 m. When the width decreases gradually, the safe meeting speed declines drastically. The deficient road breath had more significant influence on car speed than sufficient road breath, and WANG and GUO [18] also mention that the decrease of car speed resulted from one meter deduction in road breath was considerably larger than the increase of car speed caused by one meter increase in road breath.

3 Decision-making for vehicle meeting

In response to the characteristic of narrow width, vehicle overtaking is assumed to be forbidden on the branch road section. The meeting behavior under two-way traffic organization is mainly analyzed.

Through observing the speed and position of the approaching vehicle, driver slows down in advance. The advance time should be enough to ensure that the speeds of two pre-meeting vehicles decline to a safe level. The meeting begins when two vehicle heads reach each other and ends when one vehicle tail reaches the other tail. Both speeds of two vehicles maintain at the safe level during the whole process. After the meeting, two vehicles recover to the normal state.

Set v(i) and v(j) to be the speeds of vehicle i and its nearest opposing vehicle j. The space headway between vehicle i and j is d_head(i), so the pre-meeting time of two vehicles with their current speed unchanged is T(i)= . If the two vehicles speed down with decelerations of a_reduce(i) and a_reduce(j), the following requirement should be satisfied:

It the pre-meeting time  for decelerating is long enough, it can be ensured that vehicle slows down to the safe speed in accordance with the above deceleration requirement. The stage of pre-meeting deceleration is described by 0≤T(i)≤ In general, the deceleration of vehicle i is

To describe the meeting process explicitly, the space range between vehicle i and its nearest opposing vehicle can be extended from  to -2l_car, in which l_car is the standard car length. This means the space between two opposing vehicles is not determined by the vehicle head but by the rear. In this way, the meeting period was denoted as -2l_car≤≤0.

4 Cellular automation mode of branch road capacity

Based on the above traffic characteristic analysis on the branch road and cellular automation theory, the cellular mode of the branch road capacity is built, in which the time, space and speed are integrally discrete. By the central road line, the road section is divided into two cells widthwise which are empty or occupied by vehicle. Each vehicle occupies one cell on the right side. The speed is one of 0, 1, 2, …, v_max.

The cell automation model contains the vehicle following and vehicle meeting rules. The former obeys the NS model [19] for following behavior but changes the deceleration order that vehicle decreases speed regularly after decelerating randomly, since this altered order is more reasonable and its corresponding maximum flow conforms to the real observed data which exceeds the flow of NS rule by 40% under the same condition [20]. Like the vehicle following rule, the rule of meeting behavior includes updating vehicle speed and vehicle position. At the pre-meeting stage, the speed is updated for the meeting safety by vehicle deceleration; at the on-meeting stage, the safe meeting speed keeps unchanged; with the updating of vehicle speed, the vehicle position should be updated correspondingly.

During the simulating process, the vehicle meeting rule is prior to the vehicle following rule. This means that vehicle chooses the meeting rule rather than the following rule when staying at the pre-meeting or on-meeting stage, otherwise vehicle takes the following rule. Consequently, there is only deceleration but no acceleration in the meeting rule, and the acceleration is just implemented in the vehicle following rule.

Set x(i, t), v(i, t) as the position and speed of vehicle i at time t and both are vectors. If they are negative, their absolute value represents the vehicle position and speed in the negative direction of road section (the positive and negative directions of road section are defined in advance). d_head(i) is the space headway between vehicle i and its nearest preceding vehicle. v_max is the maximum driving speed. p is the random decelerating probability. sgn(?) is a sign function, whose value is 1 if the variant is positive, otherwise it is -1.

During the period from t to t+1, the evolvement of cellular automation model conforms to the vehicle following and meeting rules as below.

1) Vehicle following rule

(1) Updating vehicle speed

Step 1: Accelerating: v(i, t+1/3)←min{v(i, t)+ sgn(v(i, t)), v_max} vehicle running at its maximum speed;

Step 2: Decelerating randomly: v(i, t+2/3)←v(i, t+1/3)-sgn(v(i, t+1/3)) with probability of p determined by various undefined factors;

Step 3: Decelerating: v(i, t+1)←min{sgn(v(i, t+2/3))×d_head(i), v(i, t+2/3)} to void conflict with preceding vehicle.

(2) Updating vehicle position: x(i, t+1)←x(i, t)+v(i, t+1).

2) Vehicle meeting rule

(1) Vehicle pre-meeting: if 0≤T(i)≤ v(i, t+1)← min{v(i,t)-a_reduce(i)?sgn(v(i, t)), v_safe};

(2) Vehicle meeting: if -2l_car≤≤0, v(i, t+1)←v_safe;

(3) Updating vehicle position: x(i, t+1)←x(i, t)+v(i, t+1).

5 Computer simulation and result analysis

5.1 Simulating condition

Each cell length corresponds to 5.6 m (the length covered by a vehicle in congested queue). The total road length has 100 cells, which is actually 560 m. Each time unit is equal to 2 s, and the total time contains 1 800 units, namely 1 h. Each speed unit equals 10 km/h, and the increase by a speed unit during a time unit is equivalent to speed up at acceleration of 1.4 m/s². The maximum vehicle speed is 50 km/h, so v_max=5 speed unit. The pre-meeting time for decelerating 12 s.

After updating vehicle condition in each step, new motor vehicles are added in simulation with probabilities of p_in,  at the road entrances, respectively. At the road exits, each coming vehicle is tested whether the running distance exceeds the road length. If it is satisfied, the vehicle will finish its trip and be deleted from the simulation.

The probabilities of p_in,  should be large enough to make sure that all input vehicles can finish their trips in the simulation. In this way, the road section capacity can be obtained by multiple simulations.

An unbalanced bidirectional flow often occurs on the branch road. The ratio of the two direction flows is defined as the directional distribution (or directional split) of traffic flow, such as 50/50, 60/40, 75/25, especially, the distribution of 100/0 represents the one-way traffic. The direction with large flow is set as the major direction; the other with small flow is the minor direction.

5.2 Result analysis

The relation between road section capacity and width is shown in Fig.3 (The unit of road capacity is represented as passenger car units per hour (pcu/h)). When the directional splits are 75/25, 60/40 and 50/50, the corresponding road section capacities rise steeply with the width increasing. When the directional distribution is 100/0, the road section capacity keeps unchanged due to not considering the effect of road width in vehicle speed. When the road section is wide enough, the meeting behavior influences the vehicle speed rarely. Consequently, the capacities under distributions of 75/25, 60/40 and 50/50 tend to be four thirds, five thirds and six thirds of the capacity under 100/0. When the road is extremely narrow, the one-way capacity is larger than two-way capacity. Specially, the one-way capacity is close to the two-way capacity with a road width of about 5.0 m.

Fig.3 Relation curve between road section capacity and width under different directional splits

The relation between road section speed and width is shown in Fig.4. Except that the speed under 100/0 maintains constant, the speeds under other distributions go up gradually with width increasing. For a certain road width, the speeds under distributions of 75/25, 60/40 and 50/50 rank in a descending order. The one-way traffic speed is the largest on narrow road, but the smallest on wide road which is close to the speed under an even directional flow distribution.

Fig.4 Relation curve between road section speed and width under different directional splits

In order to compare each direction traffic flow characteristics under different directional splits, Figs.5-8 present the relation curves between major and minor direction saturation flow, speed and road width.

Figures 5 and 6 show that the major direction saturation flows and speeds under distributions of 75/25, 60/40 and 50/50 rank in a descending order. With road widening, the corresponding flows and speeds follow an upward trend and incline to the flow and speed of one-way traffic.

Fig.5 Relation curve between major direction saturation flow and road width under different directional splits

Fig.6 Relation curve between major direction speed and road width under different directional splits

Figures 7 and 8 illustrate that the minor direction saturation flows and speeds under distributions of 75/25, 60/40 and 50/50 rank in an ascending order, while the speeds of 75/25, 60/40 and 50/50 rank in a descending order. With road widening, the corresponding saturation flows and speeds increase moderately and the spaces between each curve show a divergent trend.

Fig.7 Relation curve between minor direction saturation flow and road width under different directional splits

Fig.8 Relation curve between minor direction speed and road width under different directional splits

The comparison between Fig.6 and Fig.8 indicates that under the same road width and directional distribution, the speed of minor direction is often higher than that of the major direction, but lower than that on extremely narrow road. The reason is that the meeting behavior has a significant effect on the vehicle speed. On very wide road, the considerably high density of major direction saturation flow has serious influence on the regular running of the vehicles when the vehicles meet the other direction vehicles, thus their speeds decrease correspondingly. On very narrow road, the more meeting events of the minor direction vehicles cause the speed to decrease more than that of the major direction.

6 Conclusions

1) The relations between capacity, speed, and width under different directional distributions on the branch road section with no divided lane as well as the relation between the major and minor direction saturation flow, speed, and width are obtained.

2) When the road width locates around a certain fixed value (about 5.0 m), the one-way capacity is close to the two-way capacity; on wide road, the more even the directional distribution is, the larger capacity the road section has; on narrow road, the situation is totally different with more even distribution but less road section capacity. The two-way traffic capacity is twice as large as that of one-way traffic when the road is significantly wide.

3) Except the minor direction saturation flows under distributions of 75/25, 60/40 and 50/50 rank in an ascending order, the major and minor direction saturation flows and speeds under the above distributions rank in a descending order. Each direction saturation flow and speed increases with the increase of road width. Only on the extremely narrow road, the minor direction speed is lower than that of the major direction.

References

[1] SHI Feng, HUANG En-hou, WANG Ying-zi. Study on the functional characteristics of urban transportation micro-circulation system [J]. Urban Studies, 2008, 15(3): 34-36. (in Chinese)

[2] DREZNER Z, WESOLOWSKY G O. Selecting an optimum configuration of one-way and two-way routes [J]. Transportation Science, 1997, 31(4): 386-394.

[3] MENG Q, KHOO H L, CHEU R L. Microscopic traffic simulation model-based optimization approach for the contraflow lane configuration problem [J]. Journal of Transportation Engineering, 2008, 134(1): 41-49.

[4] SHI Feng, HUANG En-hou, CHEN Qun, WANG Ying-zi. Optimization of one-way traffic organization of urban micro- circulation transportation network [J]. Journal of Transportation Systems Engineering and Information Technology, 2009, 9(4): 30-35. (in Chinese)

[5] SHI Feng, HUANG En-hou, CHEN Qun, WANG Ying-zi. Bi-level programming model for reconstruction of urban branch road network [J]. Journal of Central South University of Technology, 2009, 16(1): 172-176.

[6] SCHRECKENBERG M, SCHADSCHNEIDER A, NAGEL K, ITO N. Discrete stochastic models for traffic flow [J]. Physical Review E, 1995, 51(4): 2939-2949.

[7] FUKUI M, OIKAWA H, ISHIBASHI Y. Flow of cars crossing with unequal velocities in a two-dimensional cellular automation model [J]. Journal of the Physical Society of Japan, 1996, 65(8): 2514-2517.

[8] KNOSPE W, SANTEN L, SCHADSCHNEIDER A, SCHRECKENBERG M. Towards a realistic microscopic description of highway traffic [J]. Journal of Physics A: Mathematical and General, 2000, 33(48): 477-485.

[9] LI Xiao-bai, WU Qing-song, JIANG Rui. Cellular automaton model considering the velocity effect of a car on the successive car [J]. Physical Review E, 2001, 64(6): 66-128.

[10] DEY P P, CHANDRA S, GANGOPADHYAY S. Simulation of mixed traffic flow on two-lane roads [J]. Journal of Transportation Engineering, 2008, 134(9): 361-369.

[11] XIE Dong-fan, GAO Zi-you, ZHAO Xiao-mei, LI Ke-ping. Characteristics of mixed traffic flow with non-motorized vehicles and motorized vehicles at an unsignalized intersection [J]. Physica A, 2009, 388(10): 2041-2050.

[12] MENG Jian-ping, DAI Shi-qiang, DONG Li-yun, ZHANG Jie-fang. Cellular automation model for mixed traffic flow with motorcycles [J]. Physica A, 2007, 380: 470-480.

[13] LAN L W, CHANG C W. Motorbike’s moving behavior in mixed traffic: particle-hopping model with cellular automata [J]. Journal of the Eastern Asia Society for Transportation Study, 2003, 5: 23-37.

[14] LAN L W, CHIOU Y C, LIN Z S, HSU C C. Cellular automation simulations for mixed traffic with erratic motorcycles’ behaviors [J]. Physica A, 2010, 389(10): 2077-2089.

[15] YUAN Yao-ming, JIANG Rui, WU Qing-song, WANG Rui-li. Traffic behavior in a two-lane system consisting of a mixture of buses and cars [J]. International Journal of Modern Physics C, 2007, 18(12): 1925-1938.

[16] JIANG Rui, HU Mao-bin, JIA Bin, WANG Rui-li, WU Qing-song. Enhancing highway capacity by homogenizing traffic flow [J]. Transportmetrica, 2008, 4(1): 51-61.

[17] LI Dong-jiang, YU Hai-ming. The skill and taboo of vehicle driving [M]. Beijing: China Machine Press, 2004: 38-42. (in Chinese)

[18] WANG Wei, GUO Xiu-cheng. Traffic engineering [M]. Nanjing: Southeast University Press, 2000: 152-157. (in Chinese)

[19] ZANG Xiao-dong, TANG Yuan-hong, ZHANG Xian-jun, HU Xiu-lian. Former analysis of side interference for mixed traffic [J]. Technology & Economy in Areas of Communications, 2000, 2(3): 54-55. (in Chinese)

[20] JIA Bin, GAO Zi-you, LI Ke-ping, LI Xin-gang. Models and simulations of traffic system based on the theory of cellular automation [M]. Beijing: Science Press, 2007: 70-73. (in Chinese)

(Edited by YANG Bing)

Foundation item: Project(71171200) supported by the National Natural Science Foundation of China

Received date: 2010-09-16; Accepted date: 2011-02-18

Corresponding author: SHI Feng, Professor, PhD; Tel: +86-731-82656451; E-mail: shifeng@csu.edu.cn