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Traffic Flow Model for Staggered Intersection without Signal Lamp
JOURNAL OF TRANSPORTATION SYSTEMS ENGINEERING AND INFORMATION TECHNOLOGY Volume 12, Issue 5, October 2012 Online English edition of the Chinese language journal
Cite this article as: J Transpn Sys Eng & IT, 2012, 12(5), 8289.
RESEARCH PAPER
Traffic Flow Model for Staggered Intersection without Signal Lamp LIU Xiaoming1,*, ZHENG Shuhui2 1 Department of Automation, North China University of Technology, Beijing 100144, China 2 Department of Software Engineering, Beijing Information Technology College, Beijing 100070, China
Abstract: Traffic characteristics investigation of staggered intersection lays the foundation of scientific and reasonable traffic control strategies. This paper first proposes the cellular automaton rules on the basis of the traffic conflict process analysis. Then with these rules, the traffic flow evolution under different parameters is presented by numerical simulations. The relationship between the arterial road traffic flow density and the average speed is explored. It is revealed that low traffic density on the main road of staggered intersections may lead to traffic congestions on the approach of main road and the road between two T-type intersections, and the average speed of traffic flow on each T-type import of main road was affected by the traffic flow changes of the other T-type approaches. Moreover, the traffic congestion on intersection without signal lamp is presented as a periodically queuing-dissipates process. The above methods and results provide meaningful guidance for traffic management and control implementation of staggered intersections without signal lamp. Key Words: traffic engineering; traffic flow; intersection without signal lamp; traffic conflict; staggered intersection
1
Introduction
The staggered intersection is a special type of intersection, which usually consists of 2 to 3 T-type intersections with certain distances between them. Although the staggered intersection is discouraged in urban traffic planning, it can still be found in the existing urban road traffic network due to some realistic factors, such as geographical space, traffic demand, etc. The particular structure of the staggered intersection makes its internal traffic characteristics rather different from an ordinary intersection. Undoubtedly, it is necessary to analyze the traffic characteristics of this kind of intersection, and then take some reasonable traffic management measures and control strategies accordingly. As for this question, Chen et al.[1] conducted a preliminary study on the staggered intersection from the perspective of traffic signal control system, traffic management, sign and marking settings, etc. Du et al.[2] discussed the relationship between distances of T-type intersections and the signal control modes, then the distances threshold values of the T-type intersections under different signal control modes were put forward. In Ref. [3], two possible signal timing methods for staggered
intersections are presented in consideration of its particular geometry and topological structure. As to the traffic management of staggered intersections, a linear control method for signal phases design is improved in Ref. [4] using the principle of road network signal linear controls. Based on the traffic characteristics analysis, the signal control methods of different distances and the signal control steps for shorter events were investigated in Ref. [5]. It can be found from the above studies that the traffic characteristics and control methods of staggered intersections have attracted the general attention of researchers and some achievements have been obtained. However, it is worth mentioning that the studies on staggered intersections have been more focused on how to design the reasonable control strategies rather than control its internal traffic characteristics. In the real traffic environment, because of fewer lanes and the lower traffic flow on branches, the signal control systems are not applied in a lot of staggered intersections, and the traffic flows are often in the form of self-organization. Further research is required to study how to improve road capacity and other aspects, such as whether the
Received date: May 11, 2012; Revised date: Jul 3, 2012; Accepted date: Jul 9, 2012 *Corresponding author. E-mail: [email protected] Copyright © 2012, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1570-6672(11)60225-0
LIU Xiaoming et al. / J Transpn Sys Eng & IT, 2012, 12(5), 8289
intersections’ traffic organization is able to meet the need and in what situation the signal control should be used and so on. Therefore, it is necessary to study traffic characteristics of staggered intersections from their form structures, traffic flow compositions, and other aspects; the results will be meaningful to guide the traffic organization of staggered intersections. The cellular automaton is a more effective tool in the study of urban traffic flow characteristics for its simple, parallel calculations and easily realized rules[6–8]. Based on this, the cellular automaton was adopted as a research tool in this paper. At first, cellular automata behavior rules were built corresponding to the typical processes behaviors of different vehicles at staggered intersections without signal lamps. Then, based on the above rules, the traffic flow evolution process and traffic characteristics under different parameters were analyzed in order to study the traffic operation mechanism.
2
Problem description and modeling
Figure 1, from a real traffic viewpoint, is the sketch of a typical staggered trickle intersection without signal lamp. Two T-type intersections without signal lamp are included in the staggered intersection. The branches of the T-type intersections A and B are ordinary branches, and the main road and branches are bidirectional two lanes; traffic flow composition is numbered in the figure as ķ to ļ. In this paper, the traffic flow is designed to contain only cars and every car occupies one lattice point. According to the structure, the relevant region of the staggered intersection is divided into two types. One is the road section of the region, which is presented as a dotted area, as shown in Fig. 1, and the vehicles in this area will execute car-following actions. The other is the junction of the region, which is presented as a black area in Fig. 1, and this area, composed of 2×2 grid points, corresponding to the four dark, marked grid points in figures 2 and 3, is the critical region of vehicles changing direction after driving into the two T-type intersections. The directions of vehicles on A and B are shown in figures 2 and 3, respectively. In this model, the road sections and the grid points constituting critical regions are collectively referred to as critical regions and critical grids, respectively. According to the different location of vehicles in the staggered intersection, vehicle behaviors can be divided into different types such as: ķ the car-following behaviors in road sections and junctions, ĸ the conflicts between traffic flows 1 and 2 in critical regions, as shown in Fig. 2(a), Ĺ the conflicts between traffic flows 1 and 3 in critical regions, as shown in Fig. 2(d), ĺ the conflicts between traffic flows 2 and 3 in critical regions, as shown in Fig. 2(b), Ļ the conflicts between traffic flows 4 and 5 in critical regions, as shown in Fig. 3(c), ļ the conflicts between traffic flows 4 and 6 in critical regions, as shown in Fig. 3(a), and Ľ the conflicts between traffic flows 5 and 6 in critical regions (Fig. 3(d)).
2.1
Car-following behaviors rules in the road section and junctions The case of car-following in the lane is shown in Fig. 4. The rules of NS model are adopted for the car-following vehicles in the lane as follows: Acceleration: vc=min(vc+1, vmax). Deceleration: vc=min(vc, (gap–1)/T). Randomization deceleration: with probability pj, to meet vc=min(vc–1, 0). where vmax is the vehicle maximum speed, T is the time step of the simulation unit, taken as a unit of time (similarly hereinafter), pj is the probability of randomization deceleration, and gap is the normal distance between the target vehicle and the preceding vehicle. Because of the existence of the conflict point, when there is no vehicle between the target vehicle and the critical lattice, the value of gap is divided into the following situations: (1) If the critical lattice is occupied by one vehicle, the value of gap is the distance between the target vehicle and the critical lattice.
Fig. 1 Staggered Intersection
(a)
(c)
(b)
(d) Fig. 2 Location of conflict points for intersection A
(a)
(b)
(c) (d) Fig. 3 Location of conflict points for intersection B
LIU Xiaoming et al. / J Transpn Sys Eng & IT, 2012, 12(5), 8289
(a)
(b3)
(b2)
(b1)
(c2) (c1) Fig. 4 Analysis of intersection A car-following Table 1 Driving rules for traffic flow ķ
NO.
Vehicle from traffic flow ķ
Vehicle from traffic flow Ĺ
Vehicle from traffic flow ĸ
Driving rules for traffic flow ķ
1
isac true ° ®acleft true °havetcw false d ¯
isbc true ® ¯bcleft true
vca (t ) 1 gapca
x aa (t 1) x d , p1 0.5 ® ¯v aa (t 1) 0, p1 0.5
2
isac true ° ®acleft true °havetcw false d ¯
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
x aa (t 1) x d , p1 0.75 ® ¯v aa (t 1) 0, p1 0.25
3
isac true ° ®acleft true °havetcw false d ¯
isbc false ° ®v ba (t ) 1 gap bd °bcleft false ¯
4
isac true ° ®acleft true °havetcw false d ¯
isbc false ° ®vba (t ) 1 t gap bd °bcleft false ¯
(2) If the critical lattice is unoccupied, the value of gap is determined by the vehicle driving direction. Traffic flows ķ, ĸ, and Ĺ are described as examples in Fig. 4. In traffic flow ķ, which is on a branch, the vehicle will slow down before entering the junction, so the value of gap is the distance between the target vehicle and the critical lattice, a, regardless of whether a is occupied or not, as shown in Fig. 4(a). In traffic flow ĸ, if there is a car driving into the critical lattice in its conflict directions in the next moment, the value of gap is the distance between the target vehicle and the critical lattice, as shown in Fig. 4(b2). If there is no car driving into the critical lattice in its conflict directions in the next moment, the value of gap is the distance between the target vehicle and the preceding vehicle and the critical lattice, as shown in Fig. 4(b3). In traffic flow Ĺ, the value of gap deterministic process is similar to that in the traffic flow ĸ, as shown in Figs. 4(c1) and 4(c2). 2.2 Processing rules for conflicts of junctions As shown in Fig. 2(b), the critical lattice b is not only tied to traffic flows ĸ and Ĺ, but also related to traffic flow ķ. When a is occupied by the traffic flow ķ at the moment t, the vehicle decision from traffic flows ķ and ĸ was affected by them. In comparison, the traffic conflict between ĸ and Ĺ is more complex. Thus, in this paper, the conflicts between ĸ and Ĺ are taken as examples to describe the processing rules
xaa (t 1) x aa (t 1) ® ¯v aa (t 1)
xd
x d , p1 0, p1
0 .5 0 .5
for conflicts of junctions. And the rules that form situations (2), (3), (5), (6), and (7) are similar to this one. Due to space limitations, the main behavior rules of different direction traffic flows, under the conflict between ĸ and Ĺ, are just listed as follows (part of the secondary rules are not listed, but can deduced from the main behavior rules): In the Table 1, isac means lattice point a is occupied by the vehicle from ķ at the moment t, isbc means lattice point c is occupied by the vehicle from Ĺ at the moment t, acleft means the vehicle from ķ turns left, bcleft means the vehicle from Ĺ turns left, havetcwd(t) is the sign that the critical lattice d is occupied by vehicles at the moment t, havetcwc(t) is the sign that the critical lattice c is occupied by vehicles at the moment t, vba(t) is the speed of Ĺ at the moment t, vca(t) is the speed of ĸ at the moment t, gapbd is the distance between the vehicle in Ĺ and the lattice point d, gapca is the distance between the vehicle in ĸ and the lattice point a, xd is the location of the critical lattice d, xaa(t+1) is the location of the vehicle from ķ at the moment t+1, p1 is the probability of the critical lattice d occupied by the vehicle from ķ. It is necessary to note that because of the influence of the traffic flows changes at staggered intersections, traffic characteristics are mainly analyzed in the following way, where the probability of behavior rules are chosen with the same value in Tables 1–3. The following provides an
LIU Xiaoming et al. / J Transpn Sys Eng & IT, 2012, 12(5), 8289
explanation for why p1=0.75 in Table 1: in the case of vehicles coming from traffic flows ķ, ĸ, and Ĺ simultaneously and avoiding the interlock phenomenon, first, if the probability of c occupied by ĸ is p=0.5, Ĺ cannot move on when c is unoccupied by ĸ, and ķ can occupy d, so the probability of d occupied by ķ is 0.5; secondly, if the probability of c unoccupied by ĸ is p=0.5, the probability of d occupied by ķ and c occupied by Ĺ is 0.5, respectively. Thus, the probability of d occupied by ķ is p1=0.5+0.58×0.5=0.75. In the same way, the probability of c occupied by Ĺ is 0.75. In
this paper, the first computing method was adopted, thus p1=0.75. The driving rules for ĸ are shown in Table 2. In the table, xaa(t) is the location of the vehicle from ĸ at the moment t, xc is the location of the critical lattice c, p2 is the selection probability of behavior rules. The driving rules for Ĺ are shown in Table 3. In the table, xba(t) is the location of the vehicle from Ĺ at the moment t, xd is the location of the critical lattice d, p3 is the selection probability of behavior rules.
Table 2 Driving rules for traffic flow ĸ No.
Vehicle from traffic flow ķ
Vehicle from traffic flowĹ
1
Vehicle from traffic flowĸ
Driving rules for traffic flow ĸ
vca (t ) 1 gapca
Vehicles from traffic flow ĸ execute the NS car-following rules
2
isac true ° ®acleft true °havetcw false d ¯
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
xca (t 1) xc , p2 0.5 ® ¯vca (t 1) xc 1, p2 0.5
3
isac true ° ®acleft true °havetcw false d ¯
bcleft
vca (t ) 1 t gapca
Vehicles from traffic flow ĸ execute the NS car-following rules
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
NScar following, p2 0.5 ° ® xca (t 1) max(xca (t ), xc 1), ° p 0.5 ¯ 2
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
NScar following, p2 0.5 ° ® xca (t 1) max(xca (t ), xc 1), ° p 0.5 ¯ 2
4
5
acleft
false
isac true ° ®acleft true °havetcw true d ¯
false
Table 3 Driving rules for traffic flow Ĺ No.
1
2
Vehicle from traffic flow ķ isac true ° ®acleft true °havetcw false d ¯ isac true ° ®acleft true °havetcw false d ¯
3
Vehicle from traffic flow Ĺ
Vehicle from traffic flow ĸ
isbc true ® ¯bcleft true
vca (t ) 1 gap ca
x ba (t 1) ® ¯ v ba (t 1)
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
xba (t 1) xc , p3 0.25 ® ¯vba (t 1) 0, p3 0.75
true isbc ® true ¯ bcleft
havetcwc
true
Driving rules for traffic flow Ĺ xc , p3 0, p 3
vba (t 1)
0.5 0 .5
0
isbc false ° ®vba (t ) 1 gapbd °bcleft false ¯
Vehicles from traffic flow Ĺ execute the NS car-following rules
4
isac true ° ®acleft true °havetcw false d ¯
isbc false ° ®vba (t ) 1 t gap bd °bcleft false ¯
NScar following , p3 0.5 ® ¯ xba (t 1) xd 1, p3 0.5
5
isac false ° ®or °acleft false ¯
isbc false ° ®vba (t ) 1 t gap bd °bcleft false ¯
Vehicles from traffic flow Ĺ execute the NS car-following rules
6
isac false ° ®or °acleft false ¯
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
7
isac false ° ®or °acleft false ¯
isbc true ® ¯bcleft true
vca (t ) 1 gapca
8
havetcwd
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
9
havetcwd
isbc true ® ¯bcleft true
vca (t ) 1 gap ca
true
true
x ba (t 1) ® ¯v ba (t 1)
xc , p3 0, p 3
xba (t 1)
0 .5 0 .5
xc
NScar following, p3 0.5 ® ¯ xba (t 1) xd 1, p3 0.5
xba (t 1)
xc
LIU Xiaoming et al. / J Transpn Sys Eng & IT, 2012, 12(5), 8289
Table 4 Meaning of the parameters parameter
Meaning The vehicles’ generation probability of the main road, left side The vehicles’ generation probability of the main road, right side The vehicles’ generation probability of T-type, branch A The vehicles’ generation probability of T-type, branch B The left-turning vehicles’ generation probability of the left side on main road of A The left-turning vehicles’ generation probability of the right side on the main road of B The left-turning vehicles’ generation probability of T-type, branch A The left-turning vehicles’ generation probability of T-type, branch B
P1 P2 P3 P4 Left1 Left2 Left3 Left4
3
Simulation and analysis
Assuming that the main road and branches of staggered intersections are divided into different numbers of lattice points, the point size is 3.75 m. The road between two T-type intersections is divided into 40 points (L1=40), which represents the actual road length to be 150 m, and the road of every T-type branch is divided into 20 (L2=20, L3=20), which represents the actual road length as 75 m. To keep it simple, the vehicles are set as small cars and every point can be occupied by one car. At any moment, the point is either empty or occupied by one car. The maximum speed of the car is vmax=3, corresponding to the actual speed of about 40 km/h. The open boundary condition is used in this system, and the car is started and is driven from both sides of the main road or the end of the branch. In the chain of discrete points, there is no car in the initial moment on the driveway; if the points on both the sides of the main road or the end of the branch are empty, the vehicle whose speed is vi=3 is generated with the probability Pi, which is defined in Table 4. Then, the vehicle is driven off the system as probability 1 at the point Li+1, in which Li is the location of the export lattice point. The probability of left-turning vehicle in the importing lane is Lefti, which is defined in Table 4. The vehicles’ speed updates and position distributions obey the rules of the above model, in which the randomization deceleration probability of the NS model was set as pj=0.25. The vehicle density is defined as t=Nt/N at the moment t, the average speed is _ 1 Nt vt ¦ vi (t ) Nt i 1 _
the traffic flow is q Ut vt , where Nt is the total of traffic flow at the moment t, N is the number of lattice points, vi(t) is the speed of the i vehicle at the moment t. The evolution time of each simulation is set as 3000 steps, and the data obtained from the last 1,000 steps is taken for
numerical calculation to eliminate the initial state influence. In order to eliminate the impact of randomness on the results, every type of simulation under setting conditions is done 10 times and the result is obtained from the average of the 10 samples. During the simulation process, the vehicle will perform the traffic model rules of the cellular automata described in the previous section, based on the traffic condition changes. First, the parameters P1, P3, P4, Left1, Left2, Left3 and Left4 were set as the following. When P2 changed, the changing conditions of v1 (the average speed of T-type intersection A), v2 (e.g. the average speed of the road between T-type intersections A and B), v3 (the average speed of T-type intersection B), and v4 (e.g. the average speed of the main road) are analyzed. Some values of these parameters are then set as P1=0.5, P3=0.3, P4=0.5, Left1=0.2, Left2=0.2, Left3=0.2 and Left4=0.2, while the value of P2 is changed from 0.1 to 0.8, and the time step was 0.1. With the above parameters, the variation trends of v1, v2, v3, and v4 are shown in Fig. 5, in which the y-axis means average speed, the 1–8 on the x-axis means P2 ranges from 0.1 to 0.8. It can be seen in the graph that with the increase of P2, v1 is basically not changed and remains at about 0.5; v2 reaches the maximum when P2=0.1, then v2 presents relatively large fluctuation states when the value of v2 is closer to 2; v3 shows a sharp decrease when P2 ranges from 0.1 to 0.3, and v3 remains stable at about 0.06 after if P2>0.3, when there is a serious congestion phenomenon. v4 also shows a sharp decrease when P2 ranges from 0.1 to 0.3, while v4 remains stable at a smaller value when P2>0.3. In the further analysis, when P1=0.1 and other parameters are invariable; P2 is also changed from 0.1 to 0.8, the time step is 0.1, the variation trends of v1, v2, v3, and v4 are in Fig. 6, in which the y-axis represents the average speed, the 1–8 on the x-axis means P2 ranges from 0.1 to 0.8. It can be found from the graph, that compared with P1=0.5, v1 increases from
3.0 v2v2 v3v3
2.5
v1v1 v4v4
2.0 1.5 1.0
0.5 0.0 1
2
3
4
5
Fig. 5 Average speed trends when
6
7
p1=0.5
8
LIU Xiaoming et al. / J Transpn Sys Eng & IT, 2012, 12(5), 8289 3.0
2.5
2.0 v2v2 1.5
v3v3 v1v1
1.0
v4v4
0.5
0.0 1
2
3
4
5
6
7
8
9
Fig. 6 Average speed trends when p1=0.1 3.0 2.5
pp1=0.5 1=0.5
pp1=0.1 1=0.1
2.0 1.5 1.0 0.5
Through the single time step’s average speed and the result of space–time diagram, the reason for the fluctuation of v2 and the sharp decrease of v3 are given below. As the vehicle generation probability on the right side of intersection B increases, the conflicts among traffic flows ĺ, Ļ, and ļ also have increased, and the block number between the left-turning vehicles on ĺ and the vehicles on Ļ and ļ has increased; then, the left-turning vehicle queuing on ĺ has started due to vehicles behind. However, once there is an opportunity to plug a car in the above queuing, the vehicle team dissipates. Finally, combined with the increase in traffic density, v2 presented a relatively large fluctuation state and a sharp decrease. After the vehicle queue dissipated, driving into intersection A, the left-turning of the vehicles at the branch of intersection B could not be realized due to the smaller time headway. So, the vehicles just waited on the plug point for an opportunity to plug and block the vehicles driving to B on the road between intersections A and B. Then, a vehicle queue is generated on the road until the opportunity for the straight vehicles appears, after the left-turning vehicles on intersection B dissipate. Because the queuing phenomenon, brought about by the circulation mentioned above, v2 and v3 present fluctuation states.
0.0 1
2
3
4
5
6
7
8
Fig. 7 v3 between P1=0.5 and P1=0.1
0.5 to about 2.5, in general; v1 remains relatively stable with the change of P2, compared with P1=0.5, when v2 increases on the whole and is maintained at about 2.3. But it also to be noted that the volatility of v2 is no longer obvious than when P1=0.1, and v3 shows a sharp decrease when P2 ranges from 0.1 to 0.4; then the trend is no longer apparent and remains stable at about 0.06, corresponding to the same P2, compared with P1=0.5, when P1=0.1 (Fig. 7), the value of v3 increases in different degrees, especially in P2 ranges from 0.2 to 0.4. The generation of this phenomenon is due to the conflicts between the traffic flow of T-type intersection A and the left-turning traffic flow of T-type intersection B, which affect the traffic flow of B to a certain degree; and when the traffic flow of A increases, this conflict effect will further change. In addition, as can be seen in Fig. 6, v4 also presents a sharp decrease when P2 is changed from 0.1 to 0.4, and v4 remains stable at a smaller value when P2>0.4. Under the condition of P1=0.5, P3=0.3, P4=0.5, Left1=0.2, Left2=0.2, Left3=0.2, Left4=0.2, v2 is studied by each time step when P2=0.1 and P2=0.5, as shown in Figs. 8(a) and 8(b). Comparing the two figures, when P2=0.1, v2 fluctuates, but mainly when v3 ranges from 2 to 3, when P2=0.5, the parameter (v3) value fluctuation is very clear, the value between 0.02 and 3 is available, traffic jams frequently occurred, and it appears periodically, as shown in Fig. 8(b).
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1
50 99 148 197 246 295 344 393 442 491 540 589 638 687 736 785 834 883 932 981
(a) 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1
73
145 217 289 361 433 505 577 649 721 793 865 937 1009 1081
(b) Fig. 8 Change of v2: (a) P2=0.1; (b): P2=0.5
LIU Xiaoming et al. / J Transpn Sys Eng & IT, 2012, 12(5), 8289
4
Conclusions
In this paper, the traffic flow model for staggered intersection without signal lamp is established by taking cellular automata as a tool. The conflict processing rules among traffic flows from different directions are defined. On this basis, the basic traffic characteristics of single lane staggered intersections without signal lamp are further studied by using numerical simulation analysis. Results show that a smaller traffic flow density on the main road of staggered intersections can result in traffic jams on the approach to the main road and the section between the two T-type intersections, and the average speed of traffic flows on each T-type import of the main road is affected by the traffic flow changes of the other T-type import. Moreover, the traffic jam at the intersection without signal lamp will be presented as a periodically queuing-dissipation process. Overall, it is significant to study staggered intersections traffic characteristics from the structures, compositions, and directions of traffic flows and other aspects, and the method and results of this paper will provide the guidance and reference for the development of staggered intersections’ signal control strategies.
Technology Support Program Research of China (No. 2011BAH16B01-05).
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Cite this article as: J Transpn Sys Eng & IT, 2012, 12(5), 8289.
RESEARCH PAPER
Traffic Flow Model for Staggered Intersection without Signal Lamp LIU Xiaoming1,*, ZHENG Shuhui2 1 Department of Automation, North China University of Technology, Beijing 100144, China 2 Department of Software Engineering, Beijing Information Technology College, Beijing 100070, China
Abstract: Traffic characteristics investigation of staggered intersection lays the foundation of scientific and reasonable traffic control strategies. This paper first proposes the cellular automaton rules on the basis of the traffic conflict process analysis. Then with these rules, the traffic flow evolution under different parameters is presented by numerical simulations. The relationship between the arterial road traffic flow density and the average speed is explored. It is revealed that low traffic density on the main road of staggered intersections may lead to traffic congestions on the approach of main road and the road between two T-type intersections, and the average speed of traffic flow on each T-type import of main road was affected by the traffic flow changes of the other T-type approaches. Moreover, the traffic congestion on intersection without signal lamp is presented as a periodically queuing-dissipates process. The above methods and results provide meaningful guidance for traffic management and control implementation of staggered intersections without signal lamp. Key Words: traffic engineering; traffic flow; intersection without signal lamp; traffic conflict; staggered intersection
1
Introduction
The staggered intersection is a special type of intersection, which usually consists of 2 to 3 T-type intersections with certain distances between them. Although the staggered intersection is discouraged in urban traffic planning, it can still be found in the existing urban road traffic network due to some realistic factors, such as geographical space, traffic demand, etc. The particular structure of the staggered intersection makes its internal traffic characteristics rather different from an ordinary intersection. Undoubtedly, it is necessary to analyze the traffic characteristics of this kind of intersection, and then take some reasonable traffic management measures and control strategies accordingly. As for this question, Chen et al.[1] conducted a preliminary study on the staggered intersection from the perspective of traffic signal control system, traffic management, sign and marking settings, etc. Du et al.[2] discussed the relationship between distances of T-type intersections and the signal control modes, then the distances threshold values of the T-type intersections under different signal control modes were put forward. In Ref. [3], two possible signal timing methods for staggered
intersections are presented in consideration of its particular geometry and topological structure. As to the traffic management of staggered intersections, a linear control method for signal phases design is improved in Ref. [4] using the principle of road network signal linear controls. Based on the traffic characteristics analysis, the signal control methods of different distances and the signal control steps for shorter events were investigated in Ref. [5]. It can be found from the above studies that the traffic characteristics and control methods of staggered intersections have attracted the general attention of researchers and some achievements have been obtained. However, it is worth mentioning that the studies on staggered intersections have been more focused on how to design the reasonable control strategies rather than control its internal traffic characteristics. In the real traffic environment, because of fewer lanes and the lower traffic flow on branches, the signal control systems are not applied in a lot of staggered intersections, and the traffic flows are often in the form of self-organization. Further research is required to study how to improve road capacity and other aspects, such as whether the
Received date: May 11, 2012; Revised date: Jul 3, 2012; Accepted date: Jul 9, 2012 *Corresponding author. E-mail: [email protected] Copyright © 2012, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1570-6672(11)60225-0
LIU Xiaoming et al. / J Transpn Sys Eng & IT, 2012, 12(5), 8289
intersections’ traffic organization is able to meet the need and in what situation the signal control should be used and so on. Therefore, it is necessary to study traffic characteristics of staggered intersections from their form structures, traffic flow compositions, and other aspects; the results will be meaningful to guide the traffic organization of staggered intersections. The cellular automaton is a more effective tool in the study of urban traffic flow characteristics for its simple, parallel calculations and easily realized rules[6–8]. Based on this, the cellular automaton was adopted as a research tool in this paper. At first, cellular automata behavior rules were built corresponding to the typical processes behaviors of different vehicles at staggered intersections without signal lamps. Then, based on the above rules, the traffic flow evolution process and traffic characteristics under different parameters were analyzed in order to study the traffic operation mechanism.
2
Problem description and modeling
Figure 1, from a real traffic viewpoint, is the sketch of a typical staggered trickle intersection without signal lamp. Two T-type intersections without signal lamp are included in the staggered intersection. The branches of the T-type intersections A and B are ordinary branches, and the main road and branches are bidirectional two lanes; traffic flow composition is numbered in the figure as ķ to ļ. In this paper, the traffic flow is designed to contain only cars and every car occupies one lattice point. According to the structure, the relevant region of the staggered intersection is divided into two types. One is the road section of the region, which is presented as a dotted area, as shown in Fig. 1, and the vehicles in this area will execute car-following actions. The other is the junction of the region, which is presented as a black area in Fig. 1, and this area, composed of 2×2 grid points, corresponding to the four dark, marked grid points in figures 2 and 3, is the critical region of vehicles changing direction after driving into the two T-type intersections. The directions of vehicles on A and B are shown in figures 2 and 3, respectively. In this model, the road sections and the grid points constituting critical regions are collectively referred to as critical regions and critical grids, respectively. According to the different location of vehicles in the staggered intersection, vehicle behaviors can be divided into different types such as: ķ the car-following behaviors in road sections and junctions, ĸ the conflicts between traffic flows 1 and 2 in critical regions, as shown in Fig. 2(a), Ĺ the conflicts between traffic flows 1 and 3 in critical regions, as shown in Fig. 2(d), ĺ the conflicts between traffic flows 2 and 3 in critical regions, as shown in Fig. 2(b), Ļ the conflicts between traffic flows 4 and 5 in critical regions, as shown in Fig. 3(c), ļ the conflicts between traffic flows 4 and 6 in critical regions, as shown in Fig. 3(a), and Ľ the conflicts between traffic flows 5 and 6 in critical regions (Fig. 3(d)).
2.1
Car-following behaviors rules in the road section and junctions The case of car-following in the lane is shown in Fig. 4. The rules of NS model are adopted for the car-following vehicles in the lane as follows: Acceleration: vc=min(vc+1, vmax). Deceleration: vc=min(vc, (gap–1)/T). Randomization deceleration: with probability pj, to meet vc=min(vc–1, 0). where vmax is the vehicle maximum speed, T is the time step of the simulation unit, taken as a unit of time (similarly hereinafter), pj is the probability of randomization deceleration, and gap is the normal distance between the target vehicle and the preceding vehicle. Because of the existence of the conflict point, when there is no vehicle between the target vehicle and the critical lattice, the value of gap is divided into the following situations: (1) If the critical lattice is occupied by one vehicle, the value of gap is the distance between the target vehicle and the critical lattice.
Fig. 1 Staggered Intersection
(a)
(c)
(b)
(d) Fig. 2 Location of conflict points for intersection A
(a)
(b)
(c) (d) Fig. 3 Location of conflict points for intersection B
LIU Xiaoming et al. / J Transpn Sys Eng & IT, 2012, 12(5), 8289
(a)
(b3)
(b2)
(b1)
(c2) (c1) Fig. 4 Analysis of intersection A car-following Table 1 Driving rules for traffic flow ķ
NO.
Vehicle from traffic flow ķ
Vehicle from traffic flow Ĺ
Vehicle from traffic flow ĸ
Driving rules for traffic flow ķ
1
isac true ° ®acleft true °havetcw false d ¯
isbc true ® ¯bcleft true
vca (t ) 1 gapca
x aa (t 1) x d , p1 0.5 ® ¯v aa (t 1) 0, p1 0.5
2
isac true ° ®acleft true °havetcw false d ¯
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
x aa (t 1) x d , p1 0.75 ® ¯v aa (t 1) 0, p1 0.25
3
isac true ° ®acleft true °havetcw false d ¯
isbc false ° ®v ba (t ) 1 gap bd °bcleft false ¯
4
isac true ° ®acleft true °havetcw false d ¯
isbc false ° ®vba (t ) 1 t gap bd °bcleft false ¯
(2) If the critical lattice is unoccupied, the value of gap is determined by the vehicle driving direction. Traffic flows ķ, ĸ, and Ĺ are described as examples in Fig. 4. In traffic flow ķ, which is on a branch, the vehicle will slow down before entering the junction, so the value of gap is the distance between the target vehicle and the critical lattice, a, regardless of whether a is occupied or not, as shown in Fig. 4(a). In traffic flow ĸ, if there is a car driving into the critical lattice in its conflict directions in the next moment, the value of gap is the distance between the target vehicle and the critical lattice, as shown in Fig. 4(b2). If there is no car driving into the critical lattice in its conflict directions in the next moment, the value of gap is the distance between the target vehicle and the preceding vehicle and the critical lattice, as shown in Fig. 4(b3). In traffic flow Ĺ, the value of gap deterministic process is similar to that in the traffic flow ĸ, as shown in Figs. 4(c1) and 4(c2). 2.2 Processing rules for conflicts of junctions As shown in Fig. 2(b), the critical lattice b is not only tied to traffic flows ĸ and Ĺ, but also related to traffic flow ķ. When a is occupied by the traffic flow ķ at the moment t, the vehicle decision from traffic flows ķ and ĸ was affected by them. In comparison, the traffic conflict between ĸ and Ĺ is more complex. Thus, in this paper, the conflicts between ĸ and Ĺ are taken as examples to describe the processing rules
xaa (t 1) x aa (t 1) ® ¯v aa (t 1)
xd
x d , p1 0, p1
0 .5 0 .5
for conflicts of junctions. And the rules that form situations (2), (3), (5), (6), and (7) are similar to this one. Due to space limitations, the main behavior rules of different direction traffic flows, under the conflict between ĸ and Ĺ, are just listed as follows (part of the secondary rules are not listed, but can deduced from the main behavior rules): In the Table 1, isac means lattice point a is occupied by the vehicle from ķ at the moment t, isbc means lattice point c is occupied by the vehicle from Ĺ at the moment t, acleft means the vehicle from ķ turns left, bcleft means the vehicle from Ĺ turns left, havetcwd(t) is the sign that the critical lattice d is occupied by vehicles at the moment t, havetcwc(t) is the sign that the critical lattice c is occupied by vehicles at the moment t, vba(t) is the speed of Ĺ at the moment t, vca(t) is the speed of ĸ at the moment t, gapbd is the distance between the vehicle in Ĺ and the lattice point d, gapca is the distance between the vehicle in ĸ and the lattice point a, xd is the location of the critical lattice d, xaa(t+1) is the location of the vehicle from ķ at the moment t+1, p1 is the probability of the critical lattice d occupied by the vehicle from ķ. It is necessary to note that because of the influence of the traffic flows changes at staggered intersections, traffic characteristics are mainly analyzed in the following way, where the probability of behavior rules are chosen with the same value in Tables 1–3. The following provides an
LIU Xiaoming et al. / J Transpn Sys Eng & IT, 2012, 12(5), 8289
explanation for why p1=0.75 in Table 1: in the case of vehicles coming from traffic flows ķ, ĸ, and Ĺ simultaneously and avoiding the interlock phenomenon, first, if the probability of c occupied by ĸ is p=0.5, Ĺ cannot move on when c is unoccupied by ĸ, and ķ can occupy d, so the probability of d occupied by ķ is 0.5; secondly, if the probability of c unoccupied by ĸ is p=0.5, the probability of d occupied by ķ and c occupied by Ĺ is 0.5, respectively. Thus, the probability of d occupied by ķ is p1=0.5+0.58×0.5=0.75. In the same way, the probability of c occupied by Ĺ is 0.75. In
this paper, the first computing method was adopted, thus p1=0.75. The driving rules for ĸ are shown in Table 2. In the table, xaa(t) is the location of the vehicle from ĸ at the moment t, xc is the location of the critical lattice c, p2 is the selection probability of behavior rules. The driving rules for Ĺ are shown in Table 3. In the table, xba(t) is the location of the vehicle from Ĺ at the moment t, xd is the location of the critical lattice d, p3 is the selection probability of behavior rules.
Table 2 Driving rules for traffic flow ĸ No.
Vehicle from traffic flow ķ
Vehicle from traffic flowĹ
1
Vehicle from traffic flowĸ
Driving rules for traffic flow ĸ
vca (t ) 1 gapca
Vehicles from traffic flow ĸ execute the NS car-following rules
2
isac true ° ®acleft true °havetcw false d ¯
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
xca (t 1) xc , p2 0.5 ® ¯vca (t 1) xc 1, p2 0.5
3
isac true ° ®acleft true °havetcw false d ¯
bcleft
vca (t ) 1 t gapca
Vehicles from traffic flow ĸ execute the NS car-following rules
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
NScar following, p2 0.5 ° ® xca (t 1) max(xca (t ), xc 1), ° p 0.5 ¯ 2
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
NScar following, p2 0.5 ° ® xca (t 1) max(xca (t ), xc 1), ° p 0.5 ¯ 2
4
5
acleft
false
isac true ° ®acleft true °havetcw true d ¯
false
Table 3 Driving rules for traffic flow Ĺ No.
1
2
Vehicle from traffic flow ķ isac true ° ®acleft true °havetcw false d ¯ isac true ° ®acleft true °havetcw false d ¯
3
Vehicle from traffic flow Ĺ
Vehicle from traffic flow ĸ
isbc true ® ¯bcleft true
vca (t ) 1 gap ca
x ba (t 1) ® ¯ v ba (t 1)
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
xba (t 1) xc , p3 0.25 ® ¯vba (t 1) 0, p3 0.75
true isbc ® true ¯ bcleft
havetcwc
true
Driving rules for traffic flow Ĺ xc , p3 0, p 3
vba (t 1)
0.5 0 .5
0
isbc false ° ®vba (t ) 1 gapbd °bcleft false ¯
Vehicles from traffic flow Ĺ execute the NS car-following rules
4
isac true ° ®acleft true °havetcw false d ¯
isbc false ° ®vba (t ) 1 t gap bd °bcleft false ¯
NScar following , p3 0.5 ® ¯ xba (t 1) xd 1, p3 0.5
5
isac false ° ®or °acleft false ¯
isbc false ° ®vba (t ) 1 t gap bd °bcleft false ¯
Vehicles from traffic flow Ĺ execute the NS car-following rules
6
isac false ° ®or °acleft false ¯
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
7
isac false ° ®or °acleft false ¯
isbc true ® ¯bcleft true
vca (t ) 1 gapca
8
havetcwd
isbc true ® ¯bcleft true
vca (t ) 1 t gapca
9
havetcwd
isbc true ® ¯bcleft true
vca (t ) 1 gap ca
true
true
x ba (t 1) ® ¯v ba (t 1)
xc , p3 0, p 3
xba (t 1)
0 .5 0 .5
xc
NScar following, p3 0.5 ® ¯ xba (t 1) xd 1, p3 0.5
xba (t 1)
xc
LIU Xiaoming et al. / J Transpn Sys Eng & IT, 2012, 12(5), 8289
Table 4 Meaning of the parameters parameter
Meaning The vehicles’ generation probability of the main road, left side The vehicles’ generation probability of the main road, right side The vehicles’ generation probability of T-type, branch A The vehicles’ generation probability of T-type, branch B The left-turning vehicles’ generation probability of the left side on main road of A The left-turning vehicles’ generation probability of the right side on the main road of B The left-turning vehicles’ generation probability of T-type, branch A The left-turning vehicles’ generation probability of T-type, branch B
P1 P2 P3 P4 Left1 Left2 Left3 Left4
3
Simulation and analysis
Assuming that the main road and branches of staggered intersections are divided into different numbers of lattice points, the point size is 3.75 m. The road between two T-type intersections is divided into 40 points (L1=40), which represents the actual road length to be 150 m, and the road of every T-type branch is divided into 20 (L2=20, L3=20), which represents the actual road length as 75 m. To keep it simple, the vehicles are set as small cars and every point can be occupied by one car. At any moment, the point is either empty or occupied by one car. The maximum speed of the car is vmax=3, corresponding to the actual speed of about 40 km/h. The open boundary condition is used in this system, and the car is started and is driven from both sides of the main road or the end of the branch. In the chain of discrete points, there is no car in the initial moment on the driveway; if the points on both the sides of the main road or the end of the branch are empty, the vehicle whose speed is vi=3 is generated with the probability Pi, which is defined in Table 4. Then, the vehicle is driven off the system as probability 1 at the point Li+1, in which Li is the location of the export lattice point. The probability of left-turning vehicle in the importing lane is Lefti, which is defined in Table 4. The vehicles’ speed updates and position distributions obey the rules of the above model, in which the randomization deceleration probability of the NS model was set as pj=0.25. The vehicle density is defined as t=Nt/N at the moment t, the average speed is _ 1 Nt vt ¦ vi (t ) Nt i 1 _
the traffic flow is q Ut vt , where Nt is the total of traffic flow at the moment t, N is the number of lattice points, vi(t) is the speed of the i vehicle at the moment t. The evolution time of each simulation is set as 3000 steps, and the data obtained from the last 1,000 steps is taken for
numerical calculation to eliminate the initial state influence. In order to eliminate the impact of randomness on the results, every type of simulation under setting conditions is done 10 times and the result is obtained from the average of the 10 samples. During the simulation process, the vehicle will perform the traffic model rules of the cellular automata described in the previous section, based on the traffic condition changes. First, the parameters P1, P3, P4, Left1, Left2, Left3 and Left4 were set as the following. When P2 changed, the changing conditions of v1 (the average speed of T-type intersection A), v2 (e.g. the average speed of the road between T-type intersections A and B), v3 (the average speed of T-type intersection B), and v4 (e.g. the average speed of the main road) are analyzed. Some values of these parameters are then set as P1=0.5, P3=0.3, P4=0.5, Left1=0.2, Left2=0.2, Left3=0.2 and Left4=0.2, while the value of P2 is changed from 0.1 to 0.8, and the time step was 0.1. With the above parameters, the variation trends of v1, v2, v3, and v4 are shown in Fig. 5, in which the y-axis means average speed, the 1–8 on the x-axis means P2 ranges from 0.1 to 0.8. It can be seen in the graph that with the increase of P2, v1 is basically not changed and remains at about 0.5; v2 reaches the maximum when P2=0.1, then v2 presents relatively large fluctuation states when the value of v2 is closer to 2; v3 shows a sharp decrease when P2 ranges from 0.1 to 0.3, and v3 remains stable at about 0.06 after if P2>0.3, when there is a serious congestion phenomenon. v4 also shows a sharp decrease when P2 ranges from 0.1 to 0.3, while v4 remains stable at a smaller value when P2>0.3. In the further analysis, when P1=0.1 and other parameters are invariable; P2 is also changed from 0.1 to 0.8, the time step is 0.1, the variation trends of v1, v2, v3, and v4 are in Fig. 6, in which the y-axis represents the average speed, the 1–8 on the x-axis means P2 ranges from 0.1 to 0.8. It can be found from the graph, that compared with P1=0.5, v1 increases from
3.0 v2v2 v3v3
2.5
v1v1 v4v4
2.0 1.5 1.0
0.5 0.0 1
2
3
4
5
Fig. 5 Average speed trends when
6
7
p1=0.5
8
LIU Xiaoming et al. / J Transpn Sys Eng & IT, 2012, 12(5), 8289 3.0
2.5
2.0 v2v2 1.5
v3v3 v1v1
1.0
v4v4
0.5
0.0 1
2
3
4
5
6
7
8
9
Fig. 6 Average speed trends when p1=0.1 3.0 2.5
pp1=0.5 1=0.5
pp1=0.1 1=0.1
2.0 1.5 1.0 0.5
Through the single time step’s average speed and the result of space–time diagram, the reason for the fluctuation of v2 and the sharp decrease of v3 are given below. As the vehicle generation probability on the right side of intersection B increases, the conflicts among traffic flows ĺ, Ļ, and ļ also have increased, and the block number between the left-turning vehicles on ĺ and the vehicles on Ļ and ļ has increased; then, the left-turning vehicle queuing on ĺ has started due to vehicles behind. However, once there is an opportunity to plug a car in the above queuing, the vehicle team dissipates. Finally, combined with the increase in traffic density, v2 presented a relatively large fluctuation state and a sharp decrease. After the vehicle queue dissipated, driving into intersection A, the left-turning of the vehicles at the branch of intersection B could not be realized due to the smaller time headway. So, the vehicles just waited on the plug point for an opportunity to plug and block the vehicles driving to B on the road between intersections A and B. Then, a vehicle queue is generated on the road until the opportunity for the straight vehicles appears, after the left-turning vehicles on intersection B dissipate. Because the queuing phenomenon, brought about by the circulation mentioned above, v2 and v3 present fluctuation states.
0.0 1
2
3
4
5
6
7
8
Fig. 7 v3 between P1=0.5 and P1=0.1
0.5 to about 2.5, in general; v1 remains relatively stable with the change of P2, compared with P1=0.5, when v2 increases on the whole and is maintained at about 2.3. But it also to be noted that the volatility of v2 is no longer obvious than when P1=0.1, and v3 shows a sharp decrease when P2 ranges from 0.1 to 0.4; then the trend is no longer apparent and remains stable at about 0.06, corresponding to the same P2, compared with P1=0.5, when P1=0.1 (Fig. 7), the value of v3 increases in different degrees, especially in P2 ranges from 0.2 to 0.4. The generation of this phenomenon is due to the conflicts between the traffic flow of T-type intersection A and the left-turning traffic flow of T-type intersection B, which affect the traffic flow of B to a certain degree; and when the traffic flow of A increases, this conflict effect will further change. In addition, as can be seen in Fig. 6, v4 also presents a sharp decrease when P2 is changed from 0.1 to 0.4, and v4 remains stable at a smaller value when P2>0.4. Under the condition of P1=0.5, P3=0.3, P4=0.5, Left1=0.2, Left2=0.2, Left3=0.2, Left4=0.2, v2 is studied by each time step when P2=0.1 and P2=0.5, as shown in Figs. 8(a) and 8(b). Comparing the two figures, when P2=0.1, v2 fluctuates, but mainly when v3 ranges from 2 to 3, when P2=0.5, the parameter (v3) value fluctuation is very clear, the value between 0.02 and 3 is available, traffic jams frequently occurred, and it appears periodically, as shown in Fig. 8(b).
3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1
50 99 148 197 246 295 344 393 442 491 540 589 638 687 736 785 834 883 932 981
(a) 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1
73
145 217 289 361 433 505 577 649 721 793 865 937 1009 1081
(b) Fig. 8 Change of v2: (a) P2=0.1; (b): P2=0.5
LIU Xiaoming et al. / J Transpn Sys Eng & IT, 2012, 12(5), 8289
4
Conclusions
In this paper, the traffic flow model for staggered intersection without signal lamp is established by taking cellular automata as a tool. The conflict processing rules among traffic flows from different directions are defined. On this basis, the basic traffic characteristics of single lane staggered intersections without signal lamp are further studied by using numerical simulation analysis. Results show that a smaller traffic flow density on the main road of staggered intersections can result in traffic jams on the approach to the main road and the section between the two T-type intersections, and the average speed of traffic flows on each T-type import of the main road is affected by the traffic flow changes of the other T-type import. Moreover, the traffic jam at the intersection without signal lamp will be presented as a periodically queuing-dissipation process. Overall, it is significant to study staggered intersections traffic characteristics from the structures, compositions, and directions of traffic flows and other aspects, and the method and results of this paper will provide the guidance and reference for the development of staggered intersections’ signal control strategies.
Technology Support Program Research of China (No. 2011BAH16B01-05).
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This research was funded by the Beijing Outstanding Talents Training Individual Projects of China (No. 2010D00500200003) and the National Science and
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