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Multi-objective Optimization Method of Signal Timing for the Non-motorized Transport at Intersection
JOURNAL OF TRANSPORTATION SYSTEMS ENGINEERING AND INFORMATION TECHNOLOGY Volume 11, Issue 2, April 2011 Online English edition of the Chinese language journal RESEARCH PAPER
Cite this article as: J Transpn Sys Eng & IT, 2011, 11(2), 106−111.
Multi-objective Optimization Method of Signal Timing for the Non-motorized Transport at Intersection CHEN Xiaohong1,2, QIAN Dalin1,2, SHI Donghua1,2 1 School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China 2 MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology，Beijing Jiaotong University, Beijing 100044,China
Abstract:
Considering the coordination between vehicle and non-motorized, different traffic composition have different signal
control operates at the urban road intersection. In view of the multi-objective property of urban traffic signal control, the paper presents a multi-objective signal timing optimization model with vehicle volume and the non-motorized transport number as the inputs and traveler delay, stops and traffic capacity as the optimization objectives, and saturation degree is taken as restrictions. Furthermore, the method can adjust the weight of performance indexes according to mixed traffic composition. The multi-objective model is solved by genetic algorithm. The result shows that the model can give better signal timing than Webster method on the trade-off among various objectives and improve the traffic congestion effectively under mixed traffic condition. Key Words:
urban traffic; signal control intersection; signal timing parameter; multi-objective optimization model; non-motorized
transport; mixed traffic composition
1
Introduction
As the deepening of sustainable development of urban traffic, non-motorized transport has become the important parts of city urban traffic system. Traffic accident frequently observed at the intersection of roads considered the bottleneck of city road traffic. Traffic signal control is the basic guarantee to reduce the conflict between non-motorized and vehicle, and improve coordinated operation efficiency between nonmotorized and vehicle, and ensure traveler’s safety. The optimization of the signal timing parameter is the heart of signal control at intersection. From now on, most scholars have taken vehicles benefits performance indexes as optimal objective, without considering non-motorized benefits and environmental benefits [1-3]. In this paper, we have introduced a multi-objective optimization model of signal timing at intersection under mixed traffic condition to improve road utilization, traveler time efficiency and environmental benefits.
2
Multi-objective optimization model of signal timing at intersection
2.1 Symbolic account The following notations will be used in this paper: C: Cycle time (s); xi : Effective green time for phase i (s); λi = xi C : Split or proportion of the cycle that is effectively green for phase i; qij : Volume of vehicle for phase i lane group j (veh/h); sij : Saturation flow rate for lane group j (veh/h); yij = qij Sij : Flow ratio for phase i lane group j;
{ }
y i = max y ij : Flow ratio for critical phase i ; j
{ }
Y = ∑ max yij : Summation of flows ratios for all critical i
j
phase i. i=1, 2, ..., n, n is the phase number, j=1, 2, ..., m, m is the lane groups number for the phase. 2.2 Control performance indexes selection The main goal of city traffic control system is to make operation of all kinds of traffic flow orderly, efficiently and safely. So, the control target has been gradually developed from a single object to the efficient, safe, environmental protection and other multiple goals, and makes corresponding
Received date: Nov 08, 2010; Revised date: Dec 13, 2010; Accepted date: Jan 14, 2011 *Corresponding author. E-mail: [email protected] Copyright © 2010, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1570-6672(10)60118-3
CHEN Xiaohong et al. / J Transpn Sys Eng & IT, 2011, 11(2), 106−111
adjustments of each target’s importance according to the change of traffic status[4,5]. Based on the investigation and analysis, signal control traffic performance indexes are as follows: vehicle delay time, bicycle delay time, pedestrian delay time, queue length, traffic capacity, stops, fuel consumption, pollutant emission and so on[4]. In a word, all traffic performance indexes can be divided into three types, the road utilization, traveler time efficiency and environmental benefits. With the non-motorized transport advocacy, the proportion of pedestrians and bicycle in transportation manner is increasing, and its benefit indexes have certain effects on optimizing signal timing parameters. Yang Xiaoguang (2009) selected the vehicle delay time, pedestrian delay time and stops as cycle time optimized objectives to coordinate the vehicle benefits, pedestrian benefits and environmental benefits[4]. Considering the coordination between vehicle, bicycle and pedestrians, the paper considers vehicle delay, bicycle delay, pedestrian delay, stops and traffic capacity as the traffic control performance indexes to optimize signal timing parameters, from the aspects of road utilization rates, traveler time benefits and environmental benefits. The vehicle delay, bicycle delay and pedestrians delay belong to the category of bicycler benefit. The traffic capacity reflects road utilization and stops can be used for evaluating the effects of traffic on environmental benefits. Webster delay model [6] is the foundation of vehicle average delay, and applies to unsaturated state. The average control delay per vehicle for phase i is given by equation (1). D is control delay per vehicle, n
∑D q
i i
D=
Di = ∑ j
i =1 n
∑q
C (1 − λi )
(
2 1 − yij
2
)
i
i =1
+∑ j
yij 2 2λi qij (λi − yij )
(1)
where Di is control delay per vehicle for phase i (s/veh). Bicycle transport and pedestrian traffic are known collectively as non-motorized transport. This paper converts the volume of bicycle to volume of pedestrian, and uses the pedestrian average delay calculation formula to calculate non-motorized delay. Generally, conversion rate is 2.0 [7]. Research indicates that the average delay of pedestrians at signalized intersection crossings is not constrained by capacity, even when the pedestrian flow rates reach 2 p/s. The average delay per pedestrian for a crosswalk is given by equation (2).
ri C (1 − λi ) 2(ri − p i )
2
Pi =
(2)
where Pi is average pedestrian delay (s); ri is pedestrian saturation flow rate for the subject walkway (ped/s),
ri = K lt qh ; K is crosswalk width (m); l is effective crosswalk width (m); t qh is single pedestrian critical gap (s); pi = pi '+2bi is non-motorized volume on the subject walkway (ped/s); pi ' is pedestrian volume on the subject walkway for phase i (ped/s); bi is bicycle flow rate on the subject walkway for phase i (bicycles/s). The average stops per vehicle for phase i are given by equation (3)[6]. H is the average stops per vehicle at intersection, n
∑H q
i i
i =1 n
H=
∑q
i
i =1
H i = ∑ 0.9 × j
C (1 − λi ) 1 − yij
(3)
The traffic capacity for phase i is computed using equation (4)[6]. Q is the intersection traffic capacity, n
Q = ∑ Qi i =1
⎛x ⎞ Qi = ∑ sij ⎜ i ⎟ ⎝C⎠ j
(4)
2.3 The importance degree of each performance index It is shown that different traffic compositions have different signal control operates at the urban road intersection. According to traffic demand of reducing vehicle delay and stops in non-peak time, and improving traffic capacity in peak time, we can obtain the rule that the volume of vehicle is inversely proportional to importance degree of (weight coefficient) vehicle delay and stops, and is proportional to traffic capacity at intersection. This problem has attracted much attention due to its various applications. In the literature there are kinds of methods that can be used to deal with the importance degree of traffic control performance indexes [1,9,10]. The formulation of weight coefficient includes the following equations[9,10]: (5) ki1 = 2 (1.0 − Y ) 7 si
1.0 − Y 0.9
(6)
C 3600
(7)
ki 2 = 7 si × ki 3 = 2Y ×
where ki1 is weight coefficient of vehicle delay for phase i; ki2 is weight coefficient of stops for phase i; ki3 is weight coefficient of traffic capacity for phase i. In this paper, we have proposed a new method to calculate the weight coefficients of vehicle delay time and non-motorized delay time under mixed traffic condition. The importance degree of delay time is not only inversely proportional to the total traffic rate of intersection but also depends on the volume of each phase. Now, the formulations
CHEN Xiaohong et al. / J Transpn Sys Eng & IT, 2011, 11(2), 106−111
of the weight coefficients of vehicle delay time and non-motorized delay time are built by equation (8) that modifies equation (5). ki11 = 2
7
yi2 , ki12 = 2 y i + zi
si (1 − Y )
7
ri (1 − Y )
zi2 y i + zi
(8)
where, z i : Flows ratios of non-motorized for phase i, zi = max zij j
{ }
zij : Flows ratios of non-motorized for phase i lane group j , zij = pij rij , it reflects the congested degree of non-motorized traffic flow on walkway . pij : Non-motorized volume on the j-th(subjective) walkway for phase i. As non-motorized volume is less, the weight coefficient of non-motorized delay approximates to zero, that is without regarding to non-motorized transport benefits. 2.4 Multi-objective optimization model of signal timing In this section, considering the road utilization, traveler time efficiency and environmental benefits, we have built a signal timing optimization model of traffic control performance indexes under mixed traffic environment. This model takes saturation as the constraint. Then, we have
model, and the solution is just efficient solutions. First, we calculate the weight coefficient of all kinds of traffic performance indexes and transform the multi-objective programming model into nonlinear programming problem. Then, we chose genetic algorithm to solve the nonlinear programming model (11). Generally, three algorithms based on genetic algorithm, refuse method, repair method and penalty function were designed to solve the difficulty of constraints [11-13]. By using the corresponding fitness function, the constrained optimization problem (11) is equivalent to unconstrained optimization problem. The fitness function given by equation (12)[11], for any B is a parameter converging to zero and A is a penalty coefficient 2
F = f + A∑ fconi + B
where, A=10000000, B=0.0000001. Then, by (11), we define as following: n
fcon1 = C − ∑ xi i =1
minF (C , xi ) = min[D (C , xi ), H (C , xi ), Q(C , xi )]
∑x
s.t.
i
+L=C
i
i = 1,2,L, n
(9)
30 ≤ C ≤ 160 α ≤ 0.9 5 where,
D ( C , x ) : The delay time of traveler (s); H (C , x ) : The stops per vehicle; Q(C , x ) : Traffic capacity at intersection (veh/h); α = max{yi λi } : Intersection saturation, and then the i
following relation hold:
α ≤ 0.95 ⇔ xi ≥ YC 0.95
(10)
L : The lost time per cycle at intersection (s)； n : The numbers of phase. Generally, it is difficult that traffic capacity reaches the saturated flow rate, that is Q < S . Thus, by combining (1)-(4), and (6)-(10), the model can be written as
(
)
minF (C , xi ) = min ∑ ki11Di + ki12 Pi + ki2 H i + ki3 (Si − Qi ) i
s.t.
i
∑x + L = C i
i
i = 1,2,L, n
(11)
30 ≤ C ≤ 160 xi ≥ YC 0.95 where, k i11 , ki12 , k i2 , ki3 are from equations (8), (6), (7).
3
Genetic algorithm It is difficult to solve the multi-objective programming
(12)
i =1
⎧0 fcon2 = ⎨ ⎩ C − 160
C ≤ 160 C > 160
⎧0 fcon3 = ⎨ ⎩ 30 − C
C ≥ 30 C < 30
xi ≥ YC 0.95 ⎧0 fcon4 = ⎨ x YC xi < YC 0.95 − 0 . 95 ⎩ i Remark: The fitness function evaluates the quality of the chromosome. Specifically, we define the fitness of a chromosome as the value of the objective function of the corresponding basic feasible solution corrected by a penalty term if the chromosome is non-feasible. In order to associate chromosomes with solutions of (11), we encode each chromosome as an n string of integers whose components are the indices of the basic variables. The factors and their levels are as follows: Crossover probability ( pc = 0.5 ); mutation probability ( pm = 0.05 ); and population size ( N = 20 ). As a result, each factor combination provides a configuration of the algorithm. This loop uses a little logic algorithm to ensure that if the user does not press the STOP button after a reasonable amount of time (101 iterations), and the loop stops anyway.
4
Numerical results
We have presented our computational results in this section. The purpose of the numerical experiment is to illustrate the characteristics of the multi-objective optimization model of signal timing at intersection. The numerical results are based on Jiaoda east road-Xueyuan south road signal control intersection (be called SIDAOKOU for short), which is
CHEN Xiaohong et al. / J Transpn Sys Eng & IT, 2011, 11(2), 106−111
pedestrian flow and then carries on calculations, and the conversion coefficient can be regarded as 2.0[7].The saturation flow rate of crosswalks in each entrance direction is 2ped/s, and the saturation flow rate of vehicle lanes is about 1800pcu/h. Through the analysis of vehicle flow and non-motorized transport flow at the intersection, the paper determines the importance degree of road utilization rate, traveler benefits and environmental benefits. The weight coefficients of vehicle average delay, non-motorized transport average delay, vehicle drops and traffic capacity are calculated in Table 2. Table 1 and Table 2 indicate that when vehicle flows vary slightly, the total flow ratios at the intersection are almost equal during peak time and non-peak time. During peak time, vehicle flow ratio and non-motorized transport flow ratio in east-west direction are approximate, so the average delay weight coefficients of both are basically the same; whereas vehicle flow ratio in south-north direction is two times larger than non-motorized transport flow ratio, and the proportion of vehicle average delay in this direction is larger than that of non-motorized transport average delay. In non-peak time, the vehicle flow is obviously higher than non-motorized transport flow, so the weight coefficient of vehicle average delay is larger than that of non-motorized transport average delay; furthermore, the proportion of vehicle drops and traffic capacity at the intersection is improved compared with that during peak time.
applied the multi-objective optimization model to set signal timing parameters of this intersection. The infrastructure and geometric configuration of this intersection are as shown in Fig. 1.
Fig. 1 The plan of SIDAOKOU
Through practical investigation, we find that this intersection is under mixed traffic condition with vehicle, bicycle and pedestrians. Table 1 presents data on flows in each direction at the intersection, and it is demonstrated that the flow of vehicles during peak time is equal to that during non-peak time, but the flow of bicycles is large during peak time. Considering the similarity crossing the street between pedestrians and bicycle, the paper converts the bicycle flow to
Table 1 Traffic flow of Sidaokou intersection Traffic
Entrance
Vehicle flow
Pedestrian
Bicycle
Vehicle flow
Non-motorized
Total flow ratio at
condition
direction
（veh/h）
flow(ped/h)
flow(bic/h)
ratio yi
transport flow ratio zi
the intersection Y
0.369 3
0.354 7
0.45
0.207 2
0.371
0.191 9
0.394
0.124 7
East entrance
1 108
167
1 277
West entrance
775
312
847
South entrance
810
179
746
North entrance
490
161
396
East entrance
1 113
213
691
Non-peak
West entrance
890
281
542
time
South entrance
710
176
449
North entrance
552
171
400
Peak time
0.819 3
0.765 4
Table 2 Importance of objectives Environmental
Traveler benefits Traffic condition
Peak time Non-peak time
Entrance phase
benefits
vehicle average delay
non-motorized transport
/(s/veh)
average delay /(s/ped)
East-west phase
0.066 3
0.069 3
0.181 8
South-north phase
0.100 8
0.026
0.195 8
vehicle drops
East-west phase
0.111 7
0.033 9
0.253 9
South-north phase
0.127 3
0.015 5
0.23 6
Combining with the related data shown in Tables 1 and 2, the paper applies the proposed signal control multi-objective optimization model to conduct timing parameters optimization
Road utilization rate traffic capacity/ (veh/h) 0.032 4 0.037 2
design at Sidaokou intersection; comparing with Webster method[14] and the practical survey data, results are shown in Table 3. The east-west phase is the first signal phase, the
CHEN Xiaohong et al. / J Transpn Sys Eng & IT, 2011, 11(2), 106−111
south-north phase is the second signal phase, and the lost time is 10s. It can be seen from the analysis that the timing scheme achieved by the paper proposed timing parameter optimization
algorithm in the aspect of vehicles optimization index is better than the Webster method.
Table 3 Compare the result with Webster method at Sidaokou intersection Traffic
Green time/s
Vehicle average
Non-motorized transport
phase2
delay /(s/veh)
average delay /(s/ped)
24
20
16
0.824 2
55
26
17
0.818 1
3 762
38
35
31
17
0.876 5
3 675
87
45
32
27
16
0.800 0
3 815
85
41
34
40
15
0.900 0
3 755
Computing method
Cycle/s
phase1
Proposed method
71
37
Webster method
110
45
Practical data
83
Non-peak
Proposed method
time
Webster method
condition Peak time
Through calculations, the saturation degree is 0.856 3 during peak time and 0.855 9 during non-peak time, which illustrates that the utilization rate of road or intersection is guaranteed very well, with no congestion. At the same time, the road resources can be made full use of. During peak time, vehicle flow ratio and non-motorized transport flow ratio are almost the same, Better than the Webster method, the vehicle average delay is reduced by 23%, and the non-motorized transport average delay is reduced by 5.8%; whereas drops is increased by 3.3%, and the traffic capacity is reduced by 1.1%. During non-peak time, vehicle flow remains the same, whereas non-motorized transport flow drops, and the proportion of non-motorized transport average delay drops. The calculations indicate that the timing parameters tend to increase vehicle benefits. Compared with the Webster method, vehicle delay is reduced by 32.5%, vehicles drops is reduced by 11.1%, the traffic capacity of the intersection is increased by 1.6%, whereas non-motorized transport average delay is increased by 6.7%.
5
Conclusions
The paper mainly aims at vehicle and non-motorized transport at mixed signal control intersection, in view of the conjunct benefits of non-motorized transport and vehicle, regarding vehicle delay, non-motorized transport delay, vehicle stops and traffic capacity of intersection as control performance indexes; regarding cycle length and split as decision variables, then building multi-objective optimization model of signal timing at the intersection. Given weight coefficient expressions of each performance index, the paper discusses coordination and rationality of right-of-way distribution between vehicle and non-motorized transport at signal control intersection under the different mixed traffic compositions. With different mixed traffic compositions, the signal timing parameters differ. The numerical example indicates that under the unsaturated condition, the proposed timing algorithm is feasible and effective, compared it with Webster method, each performance index can get corresponding improvement. The proposed algorithm opens up new research thinking for ensuring traveler benefit and
Drops
Traffic capacity(pcu/h) 3 717
safety of non-motorized transport, coordinating vehicle and non-motorized transport travel, and reducing carbon emission. But it is still not perfect, especially the processing method of benefit performance between bicycle traffic and pedestrian traffic of non-motorized transport needs to strengthen, so as to ensure the accuracy and rationality of timing parameters further.
Acknowledgements This research was funded by the National Science Foundation of China (No. 70972041), Ph.D. Programs Foundation of Ministry of Education of China (No. 20100009110010) and Excellent Doctoral Technology Innovation Foundation of Beijing Jiaotong University (No. 141079522).
References [1] Cao C T, Xu J M. Multi-object traffic signal control method for single intersection. Computer Engineering and Application, 2010, 46(16): 20–22. [2] Lu K. Signal control strategies for intersection under different traffic flow. Journal of Highway and Transportation Research and Development, 2010, 46(16): 20–22. [3] Wang W, Yang Z S, et al. Forming opportunity decision of traffic congestion and its corresponding control strategy for urban expressway. Traffic and Computer, 2007, 25(3): 1–5. [4] Ma Y Y, Yang X G, Zeng Y. Multi-objective cycle length optimization
model
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solution.
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Uniersersity (Natural &Science), 2009, 37(6): 761–765. [5] Jessica Anderson. Tessa sayers and michael bell, the objectives of traffic signal control. Traffic Engineering Control, 1998(3): 167. [6] Yang P K, Wu B. Traffic management and control．Beijing： China Communications Press, 2003. [7] Li W Y, Chen X W, Wang Q, et al. Algorithm of mid-block street crossing capacity and cross-street green time. Journal of Wuhan University of Technology(Transportation Science & Engineering),2006, 30(5): 751–754. [8] Feng S M, Pei Y L. Research on delay of pedestrian crossing.
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Journal of Harerbin Institute of Technology, 2007, 39(4): 613–616. [9] Gu H Z, Wang W. A global optimization simulated annealing algorithm for intersection signal timing. ournal of Southeast University,1998, 28(3): 68–72. [10] Yan Y X, Li W Q. Ant colony optimization for signalized intersection. Journal of Highway and Transportation Research and Development, 2006, 23(11): 116–125.
42(22): 46-49. [12] Cao W, Zhang N Z. Approach for solving nonlinear equation group based on genetic algorithm. Computer Era, 2009, (9): 26–31. [13] Liang X M, Zhu C, Yan D H. Novel genetic algorithm based on species
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[11] Wei L Y, Chai Y T, Zhao M. A hybrid genetic algorithm for
[14] Li H Q. Study on the optimization methods of signal timing
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Cite this article as: J Transpn Sys Eng & IT, 2011, 11(2), 106−111.
Multi-objective Optimization Method of Signal Timing for the Non-motorized Transport at Intersection CHEN Xiaohong1,2, QIAN Dalin1,2, SHI Donghua1,2 1 School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China 2 MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology，Beijing Jiaotong University, Beijing 100044,China
Abstract:
Considering the coordination between vehicle and non-motorized, different traffic composition have different signal
control operates at the urban road intersection. In view of the multi-objective property of urban traffic signal control, the paper presents a multi-objective signal timing optimization model with vehicle volume and the non-motorized transport number as the inputs and traveler delay, stops and traffic capacity as the optimization objectives, and saturation degree is taken as restrictions. Furthermore, the method can adjust the weight of performance indexes according to mixed traffic composition. The multi-objective model is solved by genetic algorithm. The result shows that the model can give better signal timing than Webster method on the trade-off among various objectives and improve the traffic congestion effectively under mixed traffic condition. Key Words:
urban traffic; signal control intersection; signal timing parameter; multi-objective optimization model; non-motorized
transport; mixed traffic composition
1
Introduction
As the deepening of sustainable development of urban traffic, non-motorized transport has become the important parts of city urban traffic system. Traffic accident frequently observed at the intersection of roads considered the bottleneck of city road traffic. Traffic signal control is the basic guarantee to reduce the conflict between non-motorized and vehicle, and improve coordinated operation efficiency between nonmotorized and vehicle, and ensure traveler’s safety. The optimization of the signal timing parameter is the heart of signal control at intersection. From now on, most scholars have taken vehicles benefits performance indexes as optimal objective, without considering non-motorized benefits and environmental benefits [1-3]. In this paper, we have introduced a multi-objective optimization model of signal timing at intersection under mixed traffic condition to improve road utilization, traveler time efficiency and environmental benefits.
2
Multi-objective optimization model of signal timing at intersection
2.1 Symbolic account The following notations will be used in this paper: C: Cycle time (s); xi : Effective green time for phase i (s); λi = xi C : Split or proportion of the cycle that is effectively green for phase i; qij : Volume of vehicle for phase i lane group j (veh/h); sij : Saturation flow rate for lane group j (veh/h); yij = qij Sij : Flow ratio for phase i lane group j;
{ }
y i = max y ij : Flow ratio for critical phase i ; j
{ }
Y = ∑ max yij : Summation of flows ratios for all critical i
j
phase i. i=1, 2, ..., n, n is the phase number, j=1, 2, ..., m, m is the lane groups number for the phase. 2.2 Control performance indexes selection The main goal of city traffic control system is to make operation of all kinds of traffic flow orderly, efficiently and safely. So, the control target has been gradually developed from a single object to the efficient, safe, environmental protection and other multiple goals, and makes corresponding
Received date: Nov 08, 2010; Revised date: Dec 13, 2010; Accepted date: Jan 14, 2011 *Corresponding author. E-mail: [email protected] Copyright © 2010, China Association for Science and Technology. Electronic version published by Elsevier Limited. All rights reserved. DOI: 10.1016/S1570-6672(10)60118-3
CHEN Xiaohong et al. / J Transpn Sys Eng & IT, 2011, 11(2), 106−111
adjustments of each target’s importance according to the change of traffic status[4,5]. Based on the investigation and analysis, signal control traffic performance indexes are as follows: vehicle delay time, bicycle delay time, pedestrian delay time, queue length, traffic capacity, stops, fuel consumption, pollutant emission and so on[4]. In a word, all traffic performance indexes can be divided into three types, the road utilization, traveler time efficiency and environmental benefits. With the non-motorized transport advocacy, the proportion of pedestrians and bicycle in transportation manner is increasing, and its benefit indexes have certain effects on optimizing signal timing parameters. Yang Xiaoguang (2009) selected the vehicle delay time, pedestrian delay time and stops as cycle time optimized objectives to coordinate the vehicle benefits, pedestrian benefits and environmental benefits[4]. Considering the coordination between vehicle, bicycle and pedestrians, the paper considers vehicle delay, bicycle delay, pedestrian delay, stops and traffic capacity as the traffic control performance indexes to optimize signal timing parameters, from the aspects of road utilization rates, traveler time benefits and environmental benefits. The vehicle delay, bicycle delay and pedestrians delay belong to the category of bicycler benefit. The traffic capacity reflects road utilization and stops can be used for evaluating the effects of traffic on environmental benefits. Webster delay model [6] is the foundation of vehicle average delay, and applies to unsaturated state. The average control delay per vehicle for phase i is given by equation (1). D is control delay per vehicle, n
∑D q
i i
D=
Di = ∑ j
i =1 n
∑q
C (1 − λi )
(
2 1 − yij
2
)
i
i =1
+∑ j
yij 2 2λi qij (λi − yij )
(1)
where Di is control delay per vehicle for phase i (s/veh). Bicycle transport and pedestrian traffic are known collectively as non-motorized transport. This paper converts the volume of bicycle to volume of pedestrian, and uses the pedestrian average delay calculation formula to calculate non-motorized delay. Generally, conversion rate is 2.0 [7]. Research indicates that the average delay of pedestrians at signalized intersection crossings is not constrained by capacity, even when the pedestrian flow rates reach 2 p/s. The average delay per pedestrian for a crosswalk is given by equation (2).
ri C (1 − λi ) 2(ri − p i )
2
Pi =
(2)
where Pi is average pedestrian delay (s); ri is pedestrian saturation flow rate for the subject walkway (ped/s),
ri = K lt qh ; K is crosswalk width (m); l is effective crosswalk width (m); t qh is single pedestrian critical gap (s); pi = pi '+2bi is non-motorized volume on the subject walkway (ped/s); pi ' is pedestrian volume on the subject walkway for phase i (ped/s); bi is bicycle flow rate on the subject walkway for phase i (bicycles/s). The average stops per vehicle for phase i are given by equation (3)[6]. H is the average stops per vehicle at intersection, n
∑H q
i i
i =1 n
H=
∑q
i
i =1
H i = ∑ 0.9 × j
C (1 − λi ) 1 − yij
(3)
The traffic capacity for phase i is computed using equation (4)[6]. Q is the intersection traffic capacity, n
Q = ∑ Qi i =1
⎛x ⎞ Qi = ∑ sij ⎜ i ⎟ ⎝C⎠ j
(4)
2.3 The importance degree of each performance index It is shown that different traffic compositions have different signal control operates at the urban road intersection. According to traffic demand of reducing vehicle delay and stops in non-peak time, and improving traffic capacity in peak time, we can obtain the rule that the volume of vehicle is inversely proportional to importance degree of (weight coefficient) vehicle delay and stops, and is proportional to traffic capacity at intersection. This problem has attracted much attention due to its various applications. In the literature there are kinds of methods that can be used to deal with the importance degree of traffic control performance indexes [1,9,10]. The formulation of weight coefficient includes the following equations[9,10]: (5) ki1 = 2 (1.0 − Y ) 7 si
1.0 − Y 0.9
(6)
C 3600
(7)
ki 2 = 7 si × ki 3 = 2Y ×
where ki1 is weight coefficient of vehicle delay for phase i; ki2 is weight coefficient of stops for phase i; ki3 is weight coefficient of traffic capacity for phase i. In this paper, we have proposed a new method to calculate the weight coefficients of vehicle delay time and non-motorized delay time under mixed traffic condition. The importance degree of delay time is not only inversely proportional to the total traffic rate of intersection but also depends on the volume of each phase. Now, the formulations
CHEN Xiaohong et al. / J Transpn Sys Eng & IT, 2011, 11(2), 106−111
of the weight coefficients of vehicle delay time and non-motorized delay time are built by equation (8) that modifies equation (5). ki11 = 2
7
yi2 , ki12 = 2 y i + zi
si (1 − Y )
7
ri (1 − Y )
zi2 y i + zi
(8)
where, z i : Flows ratios of non-motorized for phase i, zi = max zij j
{ }
zij : Flows ratios of non-motorized for phase i lane group j , zij = pij rij , it reflects the congested degree of non-motorized traffic flow on walkway . pij : Non-motorized volume on the j-th(subjective) walkway for phase i. As non-motorized volume is less, the weight coefficient of non-motorized delay approximates to zero, that is without regarding to non-motorized transport benefits. 2.4 Multi-objective optimization model of signal timing In this section, considering the road utilization, traveler time efficiency and environmental benefits, we have built a signal timing optimization model of traffic control performance indexes under mixed traffic environment. This model takes saturation as the constraint. Then, we have
model, and the solution is just efficient solutions. First, we calculate the weight coefficient of all kinds of traffic performance indexes and transform the multi-objective programming model into nonlinear programming problem. Then, we chose genetic algorithm to solve the nonlinear programming model (11). Generally, three algorithms based on genetic algorithm, refuse method, repair method and penalty function were designed to solve the difficulty of constraints [11-13]. By using the corresponding fitness function, the constrained optimization problem (11) is equivalent to unconstrained optimization problem. The fitness function given by equation (12)[11], for any B is a parameter converging to zero and A is a penalty coefficient 2
F = f + A∑ fconi + B
where, A=10000000, B=0.0000001. Then, by (11), we define as following: n
fcon1 = C − ∑ xi i =1
minF (C , xi ) = min[D (C , xi ), H (C , xi ), Q(C , xi )]
∑x
s.t.
i
+L=C
i
i = 1,2,L, n
(9)
30 ≤ C ≤ 160 α ≤ 0.9 5 where,
D ( C , x ) : The delay time of traveler (s); H (C , x ) : The stops per vehicle; Q(C , x ) : Traffic capacity at intersection (veh/h); α = max{yi λi } : Intersection saturation, and then the i
following relation hold:
α ≤ 0.95 ⇔ xi ≥ YC 0.95
(10)
L : The lost time per cycle at intersection (s)； n : The numbers of phase. Generally, it is difficult that traffic capacity reaches the saturated flow rate, that is Q < S . Thus, by combining (1)-(4), and (6)-(10), the model can be written as
(
)
minF (C , xi ) = min ∑ ki11Di + ki12 Pi + ki2 H i + ki3 (Si − Qi ) i
s.t.
i
∑x + L = C i
i
i = 1,2,L, n
(11)
30 ≤ C ≤ 160 xi ≥ YC 0.95 where, k i11 , ki12 , k i2 , ki3 are from equations (8), (6), (7).
3
Genetic algorithm It is difficult to solve the multi-objective programming
(12)
i =1
⎧0 fcon2 = ⎨ ⎩ C − 160
C ≤ 160 C > 160
⎧0 fcon3 = ⎨ ⎩ 30 − C
C ≥ 30 C < 30
xi ≥ YC 0.95 ⎧0 fcon4 = ⎨ x YC xi < YC 0.95 − 0 . 95 ⎩ i Remark: The fitness function evaluates the quality of the chromosome. Specifically, we define the fitness of a chromosome as the value of the objective function of the corresponding basic feasible solution corrected by a penalty term if the chromosome is non-feasible. In order to associate chromosomes with solutions of (11), we encode each chromosome as an n string of integers whose components are the indices of the basic variables. The factors and their levels are as follows: Crossover probability ( pc = 0.5 ); mutation probability ( pm = 0.05 ); and population size ( N = 20 ). As a result, each factor combination provides a configuration of the algorithm. This loop uses a little logic algorithm to ensure that if the user does not press the STOP button after a reasonable amount of time (101 iterations), and the loop stops anyway.
4
Numerical results
We have presented our computational results in this section. The purpose of the numerical experiment is to illustrate the characteristics of the multi-objective optimization model of signal timing at intersection. The numerical results are based on Jiaoda east road-Xueyuan south road signal control intersection (be called SIDAOKOU for short), which is
CHEN Xiaohong et al. / J Transpn Sys Eng & IT, 2011, 11(2), 106−111
pedestrian flow and then carries on calculations, and the conversion coefficient can be regarded as 2.0[7].The saturation flow rate of crosswalks in each entrance direction is 2ped/s, and the saturation flow rate of vehicle lanes is about 1800pcu/h. Through the analysis of vehicle flow and non-motorized transport flow at the intersection, the paper determines the importance degree of road utilization rate, traveler benefits and environmental benefits. The weight coefficients of vehicle average delay, non-motorized transport average delay, vehicle drops and traffic capacity are calculated in Table 2. Table 1 and Table 2 indicate that when vehicle flows vary slightly, the total flow ratios at the intersection are almost equal during peak time and non-peak time. During peak time, vehicle flow ratio and non-motorized transport flow ratio in east-west direction are approximate, so the average delay weight coefficients of both are basically the same; whereas vehicle flow ratio in south-north direction is two times larger than non-motorized transport flow ratio, and the proportion of vehicle average delay in this direction is larger than that of non-motorized transport average delay. In non-peak time, the vehicle flow is obviously higher than non-motorized transport flow, so the weight coefficient of vehicle average delay is larger than that of non-motorized transport average delay; furthermore, the proportion of vehicle drops and traffic capacity at the intersection is improved compared with that during peak time.
applied the multi-objective optimization model to set signal timing parameters of this intersection. The infrastructure and geometric configuration of this intersection are as shown in Fig. 1.
Fig. 1 The plan of SIDAOKOU
Through practical investigation, we find that this intersection is under mixed traffic condition with vehicle, bicycle and pedestrians. Table 1 presents data on flows in each direction at the intersection, and it is demonstrated that the flow of vehicles during peak time is equal to that during non-peak time, but the flow of bicycles is large during peak time. Considering the similarity crossing the street between pedestrians and bicycle, the paper converts the bicycle flow to
Table 1 Traffic flow of Sidaokou intersection Traffic
Entrance
Vehicle flow
Pedestrian
Bicycle
Vehicle flow
Non-motorized
Total flow ratio at
condition
direction
（veh/h）
flow(ped/h)
flow(bic/h)
ratio yi
transport flow ratio zi
the intersection Y
0.369 3
0.354 7
0.45
0.207 2
0.371
0.191 9
0.394
0.124 7
East entrance
1 108
167
1 277
West entrance
775
312
847
South entrance
810
179
746
North entrance
490
161
396
East entrance
1 113
213
691
Non-peak
West entrance
890
281
542
time
South entrance
710
176
449
North entrance
552
171
400
Peak time
0.819 3
0.765 4
Table 2 Importance of objectives Environmental
Traveler benefits Traffic condition
Peak time Non-peak time
Entrance phase
benefits
vehicle average delay
non-motorized transport
/(s/veh)
average delay /(s/ped)
East-west phase
0.066 3
0.069 3
0.181 8
South-north phase
0.100 8
0.026
0.195 8
vehicle drops
East-west phase
0.111 7
0.033 9
0.253 9
South-north phase
0.127 3
0.015 5
0.23 6
Combining with the related data shown in Tables 1 and 2, the paper applies the proposed signal control multi-objective optimization model to conduct timing parameters optimization
Road utilization rate traffic capacity/ (veh/h) 0.032 4 0.037 2
design at Sidaokou intersection; comparing with Webster method[14] and the practical survey data, results are shown in Table 3. The east-west phase is the first signal phase, the
CHEN Xiaohong et al. / J Transpn Sys Eng & IT, 2011, 11(2), 106−111
south-north phase is the second signal phase, and the lost time is 10s. It can be seen from the analysis that the timing scheme achieved by the paper proposed timing parameter optimization
algorithm in the aspect of vehicles optimization index is better than the Webster method.
Table 3 Compare the result with Webster method at Sidaokou intersection Traffic
Green time/s
Vehicle average
Non-motorized transport
phase2
delay /(s/veh)
average delay /(s/ped)
24
20
16
0.824 2
55
26
17
0.818 1
3 762
38
35
31
17
0.876 5
3 675
87
45
32
27
16
0.800 0
3 815
85
41
34
40
15
0.900 0
3 755
Computing method
Cycle/s
phase1
Proposed method
71
37
Webster method
110
45
Practical data
83
Non-peak
Proposed method
time
Webster method
condition Peak time
Through calculations, the saturation degree is 0.856 3 during peak time and 0.855 9 during non-peak time, which illustrates that the utilization rate of road or intersection is guaranteed very well, with no congestion. At the same time, the road resources can be made full use of. During peak time, vehicle flow ratio and non-motorized transport flow ratio are almost the same, Better than the Webster method, the vehicle average delay is reduced by 23%, and the non-motorized transport average delay is reduced by 5.8%; whereas drops is increased by 3.3%, and the traffic capacity is reduced by 1.1%. During non-peak time, vehicle flow remains the same, whereas non-motorized transport flow drops, and the proportion of non-motorized transport average delay drops. The calculations indicate that the timing parameters tend to increase vehicle benefits. Compared with the Webster method, vehicle delay is reduced by 32.5%, vehicles drops is reduced by 11.1%, the traffic capacity of the intersection is increased by 1.6%, whereas non-motorized transport average delay is increased by 6.7%.
5
Conclusions
The paper mainly aims at vehicle and non-motorized transport at mixed signal control intersection, in view of the conjunct benefits of non-motorized transport and vehicle, regarding vehicle delay, non-motorized transport delay, vehicle stops and traffic capacity of intersection as control performance indexes; regarding cycle length and split as decision variables, then building multi-objective optimization model of signal timing at the intersection. Given weight coefficient expressions of each performance index, the paper discusses coordination and rationality of right-of-way distribution between vehicle and non-motorized transport at signal control intersection under the different mixed traffic compositions. With different mixed traffic compositions, the signal timing parameters differ. The numerical example indicates that under the unsaturated condition, the proposed timing algorithm is feasible and effective, compared it with Webster method, each performance index can get corresponding improvement. The proposed algorithm opens up new research thinking for ensuring traveler benefit and
Drops
Traffic capacity(pcu/h) 3 717
safety of non-motorized transport, coordinating vehicle and non-motorized transport travel, and reducing carbon emission. But it is still not perfect, especially the processing method of benefit performance between bicycle traffic and pedestrian traffic of non-motorized transport needs to strengthen, so as to ensure the accuracy and rationality of timing parameters further.
Acknowledgements This research was funded by the National Science Foundation of China (No. 70972041), Ph.D. Programs Foundation of Ministry of Education of China (No. 20100009110010) and Excellent Doctoral Technology Innovation Foundation of Beijing Jiaotong University (No. 141079522).
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