Urban Werea Traffic Signal Timing Optimization based on Sa-PSO Chaojun DONG

Insititute of Information, Wuyi University,

Jiangmen 529020,China cjun-dong@163.com

Shiqing Huang Insititute of Information,

Wuyi University, Jiangmen 529020,China

huangshiqing_2005@163.com

Xiankun LIU Insititute of Information,

Wuyi University, Jiangmen 529020,China

lxklml@163.com

Abstract— Urban traffic problem was an important factor that affects the development and restricts the economic construction of cities. It’s a complex system in a random way so it was necessary to optimize traffic control signals to cope with so much urban traffic problems. A simulated annealing-particle swarm optimization (Sa-PSO) algorithm was developed which bases on particle swarm optimization (PSO) and metropolwas rule. It effectively shows in dealing with the optimization of urban traffic signal timing. Simulation was carried out in a nine-intersection werea network and the result shows that the use of Sa-PSO method can reduce 41.0% of the average delay per vehicles, and 30.6% of the average stop rates comparing with fixed time plans.

Keywords-PSO, simulated annealing, Sa-PSO, Traffic Signal Optimization

I. INTRODUCTION It was important to develop an efficient traffic signal

control method with intelligent technologies for efficient urban traffic control. The idea of simulated annealing algorithm (SA) was presented by Metropolwas in 1953, and was used in compounding optimization by Kirkpatrick in 1983. SA algorithm was a random optimization algorithm which based on the MenteCarlo iteration solution, the basic standpoint of which was based on the similarity between physical solid substances annealing process and general combinatorial optimization problem. It accepted the current optimal solution at a probability after searching, which called Metropolwas law. Particle swarm optimization (PSO), a evolvement-computation technology based on swarm intelligence which was rawased by Kennedy and Eberhart who were aroused by the research results about artificial life in 1995. The basic idea of PSO was every particle always follows the two best positions—the best position in the whole swarm and itself in iteration computation [1], so it converges very fast, but there were several shortcomings in PSO: 1.Getting in a locally optimal point easily; 2.Difficult to process the constraints of the optimization problem. Combining the ideology of simulated annealing and PSO, constructing a new multiple algorithm (Sa-PSO) could be a global optimal method and it has been proved to be a effective way to deal with optimal problem through simulation.

II. SA-PSO ALGORITHMS At the beginning, the individual best point and the global

best point were accepted by the Metropolwas rule, which meaned both of them accepted the hypo-best point with

certain probability, the aim function was allowed to become worse at a certain extent, the acceptance rule was decided by the coefficient T, T was the anneal temperature. Along with T descending, the searching region would be around the best point, the accepting probability of the algorithm to the hypo-best point became small, when the T descended to the lower limit, the probability of the accepting algorithm to the hypo-best point became zero, the algorithm degenerated to the basic PSO algorithm. According to the relation between the annealing temperature and the inertial factor, a mapping relationship was built, the inertial factor varied with the temperature, and the searching speed could be increased.

A. Basic PSO Algorithm There were m particles in a swarm which was in a space

of D dimensions, the ith particle’s position in the space was:

iX = ),,,( 21 iDii xxx , mi ,,2,1= , which was a latent solution. The i th particle’s flit speed was:

iv = ),,,( 21 iDii vvv , mi ,,2,1= , until now the best position of the i th particle was:

iP = ),,,( 21 iDii ppp , mi ,,2,1= , the best position in the whole swarm for now was:

gP = ),,,( 21 gDgg ppp , the PSO algorithm was shown:

ididid vxx += α (1)

)()( 2211 idgdidididid xpcxpcwvv −+−+= γγ (2)

Where, mi ,,2,1= , Dd ,,2,1= , 1c and 2c were the study coefficients of cognitive learning factor and society cognitive learning factor, and both were positive

constants. The relative value of 1c and 2c expresses the relative importance-degree of ip and gP with

evolvement. 1γ and 2γ were both random numbers between

0 and 1, ],[ maxmax vvvid −∈ , maxv was decided by the user, α and ω were, respectively, constraint factor and inertial factor ( w >0).

B. Metropolwas Acceptance probability law The accepted probability function was given by the extent Boltzman-Gibbs dwastributing:

( )[ ] ( )1 11 1 hP h f T −= − − Δ (3)

2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.257

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2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.257

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Where ( )i if f X PΔ = − ; ( )if X was the ith particle

solution, iP was the hwastorical best solution, T was the

anneal temperature; h was real number that regulate the

relation between difference value and temperature, when

1h → , then:

( )expp f T= −Δ (4)

Thwas was the general SA algorithm accept probability

function, it was the special one of function (5), and the Sa-

PSO algorithm also use it.

C. Annealing scheme Acceptance probability was controlled by the annealing

temperature. Annealing scheme used the formula of a fast annealing (VFSA) algorithm which shown blow:

( ) ( )1/0 exp NT k T ck= − (5)

Where 0T was the initialized temperature, k was annealing-time; c was constant defined; N was number of syllogwasm coefficient. The function (7) also could be shown as below:

( ) 1/

0

NkT k T α= (6) Where α was rate of cooling 0.7 1.0α≤ ≤ .The 1 / N in the function (8) was replaced by 1.5 or 1.0 in real use.

D. Inertial Factor Function The inertial factor in the function (1) was used to control the effect between the previous step speed and the current particle speed, the bigger inertial factor w can enhance the global searching ability [8], and the smaller inertial factor w could enhance the local searching ability. In the simulated annealing algorithm, the temperature T has the similitude ability, it could enhance the global searching ability at higher temperature and enhance local searching ability at lower temperature. A new mapping function was presented to describe relation between the two factors. It was used to enhance the global searching ability at the beginning and improve the searching speed at end of algorithm. The function was shown below:

( )00

0

(1 )*

T T kw w

Tβ−

= − (7)

Where β was a adjusted factor which needs to be

adjusted when it was in a real problem, 0w was the basic PSO initialization inertial factor.

E. The astringency of Sa-PSO algorithm Assume the

gP and iP keep unchanged in the evolution,

when ( )22 1 4 2w wφ+ − − < , the particles ( iX ) of PSO

algorithm converge to the weight center of ,g iP P , which

means 1 2i g

i

p pX

φ φ

φ

+→ , ( )1 2 1 1 1 2 2 2, ,c r c rφ φ φ φ φ= + = =

. But in real use, the ,g iP P keep on changing in the

evolution, it could also be proved that if the algorithm fits the condition above, the astringency of PSO algorithm can be sure.

The Sa-PSO algorithm’s configuration was similitude to the PSO, the different between two algorithms was at the beginning of algorithm, with the temperature cooling down, the Sa-PSO degenerate to PSO, so the algorithm content the condition above, the astringency of Sa-PSO would be sure.

F. Steps of Algorithm The function of (3) and (4) was the basic PSO algorithm, the steps of Sa-PSO shown below:

Step1: To initialize the coefficient, it includes the annealing

temperature T and 1 2w c c、 、 .To initialize the particle

swarm, it includes the particle random position and the first

speed. Evaluate every particles adaptive value ( )if X ;

Step2: For each particle, the adaptive value ( )if X was

compwered with one of the hwastorical best position iP , if

the adaptive value was better than one of iP . Then, iX was

consider as the best position iP , otherwwase, using the

accept-probability law function (6) to decide if thwas point

be accepted;

Setp3: For each particle, the best point iP itself was

compwered with the whole best point gP , if iP was better

than gP , then reset gP , otherwwase, the global best point

was accepted according to the probability function

( )exp /i gp P P T= − , T was the annealing temperature;

Step4: Change the position and speed of each particle

following the function (3) and (4), several times later, adjust

the temperature T following the function (8) and the inertial

weight w following the function (9);

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Step5: If it haven’t got to the stop condition (the condition

was the best adaptive value or setting a maximum iteration

times in generally), back to step2, else the algorithm stops;

G. 2.5 Algorithm Simulation The general Griewank function was test function. It was a classical optimized test function which often be used to compwere the ability with different algorithm.

To compwere the algorithm ability, three algorithms were used here: The first one was the basic standard particle swarm optimization (Std-PSO) algorithm; The second was particle swarm optimization with the inertial factor reducing linearly (Iw-PSO); The third was Sa-PSO presented in thwas paper. The general Griewank function shown below:

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1 1

1min ( ) cos 14000

ii

i i

xf X xi= =

⎛ ⎞= + +⎜ ⎟⎝ ⎠

∑ ∏

[ ]600, 600ix ∈ − (8) Fig.1 shows the Griewank function in 3 dimensions

coordinate. The function has the global best point ( )* 0, , 0X = , in which the function value was 0. The

Griewank function was similar to the Sphere function with adding the nowase part, with dimension increasing, thwas nowase part tends to 0, so the function has more local best points when the dimension was lower, and it’s hard to convergence to the global best point. The experiment parameters were the number of particles was 30, run randomly 20 times , the results were shown in Table1

Simulation results show: Both in high dimension and in low dimension, the Sa-PSO gets the better solution than the other algorithm and has powerful optimization ability.

III. TRAFFIC SIMULATION RESEARCH The object to be simulated was a nine-intersection werea in a city as shown in Figure 2.Suppose the east-west direction route was the main road, and the north-south direction route was the subsidiary. Every sections was double-lane, and every intersection was a four-phase control intersection, which means there were east-west Straight, east-west left, north-south straight and north-south left. No control to the right turn of the west and the east.

A. Cycle and offset Optimization For real-time traffic control, in order to reduce the interference caused by traffic flow when the program changes, the changes in cycle and offsets can not be too much. Therefore, during the optimization of cycle and offset, the cycle should be optimized in a incremental form.

The offset should be optimized in the way of preacting or delaying the starting time of the coordinated phase for several seconds. So that it reduces the feasible solution space on one hand, on the other hand, it was easy to control the incremental range of the cycle and offset. Formulas shown as follows:

c c c= + Δ (9)

min maxid id idt t t− ≤ ≤ (10) Where, cΔ was the cycle incremental (or decreasing) time; formulas limit the range of the starting time of the coordinated phases.

B. Werea Traffic Control Timing Optimization steps 1. The constraints method to cycle and phase takes section 3.1, for other constraints, it uses penalty function incorporate the objective function PI. 2. Given a set of initial value, calculating the objective function initial value J0, then set a small valueεas the algorithm terminational criterion. 3. Calculate the Cycle and Offset by formula (11) and (12) while doing the optimization, then calculate it again by using the Sa-PSO, If the terminating conditions of the calculation didn’t meet, return and recalculate it after adjusting by （ 11 ） and （ 12 ） ,keep calculating Until terminating conditions was met.

C. Traffic Simulation As werea traffic control was complex system, traffic specific simulation softwwere TSWAS5.1 was used in thwas paper. TSWAS (Traffic Softwwere Integrated System) was a large-scale integrated simulation toolbox, which conswasts of the following 4 components: CORSIM, TSHELL, TRAFVU, TRAFED. It applies to the city roads and highway system which was under the signal control and it can also simulate many details in various traffic conditions. In thwas paper, Vwasual C + + was used in writing RTE (Run-Time Extension) Interface.

The simulation experiment parameters were set as: The saturation for all the lanes were set to 1800 PCU/h, the maximum queue length was 45pcu, Cycle was 40-120 seconds, maximum green time was 110 seconds, minimum green time was 10 seconds, the yellow light was 3 seconds, the red time was 2 seconds, the vehicle average starting time was 2 seconds, PSO swarm size was 50, simulation section time was 600 seconds without concerning the effect of the pedestrians and buses. In order to be more congruous with the reality, three significantly different classes traffic flow demand were set for the simulation, which were slight-demand, moderate-demand and heavy-demand. Each demand has five traffic demand data. Then simulation and calculation were carried out in the ways of Sa-PSO and

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fixed-cycle method (FIX), comparwason and analyswas were included.

D. Results and Analyswas

Simulation result of average delay and average stop rates comparwason Sa-PSO and FIX was shown in Table 2. Simulation results show: Comparing with FIX, the use of Sa-PSO method can reduce 41.0% of the average delay per vehicles, and 30.6% of the average stop rates.

IV. CONCLUSION 1. Integrating Simulated Annealing and PSO was a meaningful attempt, It improves the convergence rate and effectively overcomes the defects that PSO algorithm was easy falling into local optimal solutions and easy to diverge. 2. Urban traffic signal timing optimization based on Sa-PSO can reduce 41.0% of the average delay per vehicles, and 30.6% of the average stop rates comparing with FIX, which can improve the traffic efficiency. 3. The Sa-PSO can also be used in other fields, but the research of Sa-PSO was just at the beginning, there were still so much problems for further study.

V. ACKNOWLEDGEMENTS Thwas research was supported by GuangDong Natural Science Foundation (8152902001000014) and Guangdong College Major Research Item (05Z025).

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Fig.1 Griewank function

Table 1 Simulation result.

dimension Times

Griewank mean value 310−×

Std-

PSO

Iw-

PSO

Sa-

PSO

2 200 1.262 0.887 0.258

10 2000 12.764 8.917 6.594

20 2000 37.431 15.664 13.572

30 2000 55.322 35.007 21.361

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Fig.2 A nine-intersection werea traffic network

Table 2. Simulation result of average delay and average stop rates comparwason on Sa-PSO and FIX .

CLASS Traffic Average delay Average stop rates

Demand Sa-PSO FIX Sa-PSO FIX

Slight

Demand

600 19.3 38.2 70.1 108.7 700 20.9 40.7 72.9 111.3 800 22.8 44.4 75.5 116.2 900 24.1 47.3 80.6 120.4

Moderate

Demand

1000 28.6 52.6 84.8 125.9 1100 31.4 56.9 89.7 133.9 1200 35.5 69.1 94.3 139.5 1300 40.3 76.2 99.2 152.7 1400 45.7 83.4 105.7 160.8

Heavy

Demand

1500 52.8 93.9 118.4 172.1 1600 61.5 105.5 129.1 187.6 1700 73.2 119.8 140.3 201.1 1800 88.4 137.4 157.5 219.6 2000 134.2 198.3 196.4 245.3 2100 163.7 241.1 223.6 272.4

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