An Optimization Approach for Intersection Signal Timing Based on Multi-Objective Particle Swarm Optimization

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1 A Approach for Itersecto Sgal Tmg Based o Mult-Objectve Partcle Swarm Hao Pag Feg Che Departmet of Automato Uversty of Scece ad Techology of Cha Hefe, Ahu, , Cha shamrock@mal.ustc.edu.c chefeg@ustc.edu.c Abstract Itersecto sgal tmg s oe of the key techques tellget trasportato system (ITS). Both the average delay ad stop frequecy are mportat dces for evaluatg the level of servce (LOS) for sgalzed tersectos. Tradtoal sgal tmg models ether optmze oly oe of them or deal wth them as a sgle objectve usg weghted average methods. I ths paper, a Mult-Objectve Partcle Swarm (MOPSO) method s proposed to optmze the both evaluato dces sychroously. A well-dstrbuted set of Pareto optmal solutos s obtaed, ad the most satsfed soluto s selected by the mult-objectve decso-maker module. The expermetal results dcate ths optmal method s steady ad effectve. Keywords average delay, average stop frequecy, multobjectve, Partcle Swarm I. INTRODUCTION Wth the urba populato ad cars creasg, traffc cogesto has become more ad more serous. Itersecto s a mportat compoet of the urba trasportato etworks; meawhle, t s also the major occurrg place of traffc cogesto. Traffc sgal cotrol ca provde for the orderly movemet of traffc flow, ad reduce the frequecy ad severty of traffc jams at the tersecto. The ma approach of sgal cotrol s to mplemet the tmg model for a sgalzed tersecto. Sgal tmg model ca geerate the optmal sgal tmg pla by calculatg approprate cycle legth ad gree splt of each phase, whch s crucal for sgal cotrol ad mproves the sgalzed tersecto s level of servce (LOS) effectvely. Therefore, lots of scholars devoted themselves to desgg a reasoable tmg model. Referece [1] gave a formula of the optmal cycle legth for mmzg vehcle average delay, ad the gree-lght tme was determed accordg to the traffc flow rato. But stop frequecy per vehcle was ot cosdered ths model. I [2], o the bass of Webster s model, the stoppg compesato coeffcet was added to modfy the formula for calculatg the optmal cycle legth, but t was dffcult to be acqured ths coeffcet precsely. Referece [3] trasformed the average delay, average stops ad traffc capacty to a sgle objectve by calculatg ther weghted sum, ad adopted Tabu search algorthm to fd approxmate soluto. However, the rato for the weghted coeffcet of average delay ad average stop frequecy was costat her model, whch meat the two parameters had a lear correlato. That was cosstet wth the actual codtos. Average cotrol delay ad average stop frequecy are mportat evaluato dces of traffc sgal pla. They are useful for the determato of the cycle legth ad the gree splts. Referece [4] aalyzed the correlato coeffcet betwee the two dces ad other traffc parameters, ad suggested the relatoshp betwee these two dces was ether lear or absolutely mootoe, but a grey correlato. It meas whe oe of the two dces s creased, the other mght be creased or decreased. Therefore, t s sutable to establsh two objectves optmal model so as to optmze both of them smultaeously. Partcle Swarm (PSO) s a fast optmzato algorthm based o swarm tellgece. Its heret characterstcs, such as mplct parallelsm, ca mprove the effcecy of mult-objectve optmal problem. I ths study, a mult-objectve optmzato algorthm based o PSO s proposed to optmze average delay ad average stop frequecy smultaeously. Ths algorthm calculates the Pareto soluto aggregate of the cycle legth ad gree-lght tme of each phase, the most satsfed soluto s provded by the decso-maker module. The expermetal results demostrate ths algorthm effectve. II. BASIC THEORY A. Model of Objectve Fuctos I sgal tmg models, average cotrol delay fuctos are essetal for evaluatg the traffc codtos of a sgalzed tersecto. These fuctos drectly relate wth the LOS of the whole tersecto. The basc defto of average cotrol delay s the travel tme loss caused by traffc frcto resstace ad sgal cotrol [5]. It s also related to other /08 /$ IEEE CIS 2008

2 traffc parameters such as cycle legth, gree splts, ad saturato. I the case of a usaturated traffc stuato, the delay formula s expressed as (1): 2 2 C(1 λ) X c (2+ 5 λ ) d = X (1) 2(1 λx) 2 q(1 X) q Where d (sec/pcu) s average cotrol delay per vehcle o the partcular lae group of the -th phase; (pcu meas the passeger car ut.) C (sec) s the cycle legth; q (pcu/sec) s the flow rate o the partcular lae group of the -th phase; λ s the proporto of effectve gree respect to cycle legth of the -th phase; (.e. g /C ad g (sec) s effectve gree tme of the - th phase.) ad X s the degree of saturato of the -th phase of the tersecto [6]. The frst term of (1) represets the delay whe the traffc s assumed to be arrvg uformly. The secod term of the equato deotes the expereced delay due to vehcles arrvg radomly. The thrd term of the equato s a emprcal correcto term to gve a closer ft for all values of traffc flow. Normally, the last term s qute small compared to the whole delay ad s frequetly eglected for actual calculato [6]. The average cotrol delay the whole cycle legth should equal to the value of weghted average of each phase ad be expressed as (2): d d q / q 1 1 = (2) Where s the umber of phase oe cycle [7]. The average stop frequecy per vehcle s also a mportat parameter for judgg the LOS of a sgalzed tersecto. It represets the umber of stops whe oe vehcle passes through the sgalzed tersecto. The average stop frequecy per vehcle of the -th phase s expressed as (3): 1 λ h = 0.9 (3) 1 y Where h meas average stop frequecy per vehcle of the -th phase; ad y s the flow rato of the partcular lae group of the -th phase. It represets the rato of actual flow rate to saturato flow rate [7]. Smlar to the average delay, the average stop frequecy the whole cycle legth should equal the value of weghted average of each phase ad be expressed as (4): h hq / q 1 1 1/3 = (4) Where s also the umber of phase oe cycle [7]. As a result, the optmal tmg model take to cosderato both average cotrol delay ad average stop frequecy s descrbed as follows: m y = d C, λ, h C, λ (5) { ( ) ( ) } B. Mult-Objectve Problem If there are two ad more objectves to be optmzed smultaeously, there s o loger a sgle optmal soluto but a whole set of possble solutos of equvalet qualty [8]. Geerally, the defto of mmum mult-objectve problem wth decso varables x ad m objectves y s expressed as follows: m y = f ( x) = { f1( x), f2( x),, fm ( x) } (6) Where x = ( x1, x2,, x ); y = ( y1, y2,, y ); x S = { x gj ( x) 0, j = 1,2,, p} ; Ad where x s called decso vector; y s called objectve vector; S s the feasble soluto rego, ad g j represets the j-th costrat of ths problem [8]. 1) DEFINITION 1: A decso vector u S s referred to domate a decso vector v S oly f: 1,..., { } f( u) f( v) Ad j { 1,..., } f ( u) < f ( v) [8]. Based o ths defto, Pareto optmal solutos ca be defed as follows: 2) DEFINITION 2: Let u S be a arbtrary decso vector. (a) The decso vector u s referred to be o-domated regardg a set S' S oly f there s o vector S ' whch domates u ; Usually, u domates v ca be wrtte as u (b) The decso vector u s called Pareto optmal soluto oly f u s o-domated regardg the whole feasble soluto space S. The key to solvg mult-objectve problem s to fd the Pareto optmal solutos from the feasble soluto rego, ad decso makers ca choose the most satsfed soluto or a set of optmal solutos from them [8]. C. Partcle Swarm The Partcle Swarm (PSO) s a parallel stochastc search algorthm frst proposed by Keedy ad Eberhart. It s a populato-based algorthm ad ts procedure s smpler tha Geetc Algorthm s. Ths algorthm smulates socal behavor such as flyg brd flock searchg of food. The behavor of each partcle s affected by the behavors of eghborhoods ad the whole swarm [9]. Ths algorthm s talzed wth a swarm of radom soluto at the feasble soluto rego. The dvdual called partcle flows through the soluto space by followg the curret best oe. At each terato, the posto of each partcle s updated by a ew velocty calculated through (7) ad (8) whch s based o ts precous velocty, the posto of the best soluto so far has bee acheved by the partcle tself (pbest), v

3 ad the posto at whch the best soluto so far has bee acheved by the global swarm (gbest): + 1 = * + + (7) ( ) ( () ()) ( ) () () () pt ( 1) pt ( ) V( t 1) V t w V t cu pbest t p t c v gbest t p t = + + (8) Where w s a weght determg the proporto of the partcle s prevous preserved; c 1 ad c 2 are two postve accelerato costats, u ad v are two uform radom sequeces produced from U (0, 1) [9]. Ftess values obtaed from the objectve fuctos drve the partcles to fly through the soluto space ad are attracted to both ther persoal best soluto ad the best posto foud by the global swarm. Fally, they coverge to the optmal soluto [9]. III. TIMING MODEL BASED ON MOPSO A. MOPSO Algorthm Process Mult-objectve partcle swarm optmzato (MOPSO) uses PSO algorthm to optmze each objectve the feasble soluto rego. Ths optmzato approach s effectve to solve mult-objectve optmzato problem ad easy to be mplemeted. Stregth Pareto Evolutoary Algorthm (SPEA) s a classcal method amog all major mult-objectve EAs whch s based o the Pareto-optmalty ad domace [8]. Ths study proposes the MOPSO algorthm based o SPEA to optmze average delay ad average stop frequecy smultaeously. The detaled procedure s descrbed as follows: Step 1: Italze the partcle swarm P. I ths model, each partcle represets a sgal tmg pla the same traffc codto. The dmeso of the partcle equals to the umber of phases o ths pla, ad ts posto represets the vector of the gree-lght tme of each phase. The talzed posto of each partcle s geerated radomly the feasble soluto rego ad the talzed speed s set to zero. The talzato of pbest ad gbest of each partcle s set to tself. The value rage of the posto ( Xm, Xmax ) ad the maxmum speed V max should be talzed to avod the dvduals flowg out of the soluto space. A empty exteral set of o-domated dvduals P ' s also created. Step 2: Update the posto ad speed of each partcle by usg (7) ad (8). Step 3: Calculate the objectve fucto values of each partcle. I ths model, average delay ad average stops are calculated wth (2) ad (4). Step 4: Update the pbest of each partcle. For each dvdual the swarm, the pbest s replaced by the curret posto f the curret posto domates the pbest of ths partcle. Otherwse, the pbest s ot updated. Step 5: Fd all o-domated members of P at ths terato, ad copy them to P '. Step 6: Remove solutos wth P ' whch are domated by ay other member of P '. Step 7: If the umber of stored o-domated solutos exceeds a gve maxmum N, reduce the populato sze of P ' to N by meas of clusterg procedure. Ths teratve procedure clusters partcles whch have the mmum dstace P '. Step 8: For each partcle, select a soluto radomly from P ' ad compare to the gbest of ths partcle. If the soluto domates gbest, the replaces t; else do ot update the gbest of ths partcle. I order to accelerate the covergece speed of ths algorthm, the value of gbest should ot vary frequetly. That makes the partcle search o a steady approach. A varato coeffcet of the partcle s gbest s also troduced to avod the algorthm droppg to local optmum [10]. Step 9: If maxmum umber of geeratos s reached, the stop; else go to Step 2. B. Desg of Mult-Objectve Decso-Maker Mult-Objectve Decso-maker s a mechasm that makes sure the most sutable soluto for the practcal stuato be selected from the set of o-domated solutos. By usg MOPSO algorthm, a set of Pareto optmal solutos are geerated. However, sgle tersecto usually uses oly oe sgal tmg pla. The mult-objectve decso-maker helps to choose the most sutable tmg pla accordg to the evromet ad dyamc traffc codtos. Whe the saturato degree of the tersecto s closed to 1, the decso-maker may choose the soluto whch optmzes the average delay. Whe the tersecto closes to resdetal areas, t may select the soluto whch optmzes the average stop frequecy to avod polluto creasg due to vehcles start ad brake. The mult-objectve decso-maker s desged as follows: Step 1: Desg a fuzzy matrx about the proporto of each dex cosdered the decso. The dmeso of ths matrx s usually set to 3, ad ts elemets ca be chaged frequetly. Step 2: Accordg to the actual stuato about ths tersecto, determe whch s more mportat betwee average delay ad average stop frequecy. The select a elemet form the fuzzy matrx to represet the rato of weght betwee cotrol delay ad stop frequecy. Step 3: Modfy the value of objectves by usg the elemet selected at Step 2. The sort the members wth the set of o-domated solutos by the value of the objectve whch s more mportat. Step 4: Offer the most fttg soluto. Because the set of o-domated solutos has bee modfed by cosderg the effect of actual stuato, ad also bee sorted by the value of the key objectve. So the frst soluto wll be the best sgal tmg pla that meets the optmzato of the both dces ad the practcal traffc codto about ths tersecto. IV. EXPERIMENTS AND DISCUSSION I order to verfy the property of ths traffc tmg model based o MOPSO, we select two tersectos dfferet areas for our expermets. 1) INTERSECTION 1:

4 The frst tersecto called Bahuaj s Hefe, Ahu provce, Cha. It s a crossroad ad locates ear the people s lvg rego. The flow rate of each approach s usually ot hgh, ad the umber of phase s set to 4. The traffc data from 13:30 to 14:30 at May 23, 2003 are lsted Table 1: TABLE I. TRAFFIC DATA ON BAIHUAJIN Lae Drecto Actual Flow Rate Saturato Flow East straght East left-tur South straght South left-tur West straght West left-tur North straght North left-tur ) INTERSECTION 2: The secod tersecto called Logsha Road locates Aqg, Ahu provce, Cha. Ths tersecto s also a crossroad, ad ts locato s o a slope top. The capacty of each lae s lower tha the frst tersecto because of the arrow wdth of laes ad the exstece of the slope. However, the flow rate of each lae s ofte hgh because ths tersecto s ear the cty ceter. So the umber of phase s usually set to 2. The traffc data from 8:00 to 12:00 are lsted Table 2: TABLE II. TRAFFIC DATA ON LONGSHAN ROAD Lae Drecto Actual Flow Rate Saturato Flow East straght East left-tur South straght South left-tur West straght West left-tur North straght North left-tur Accordg to the actual stuato, some costrats have to be descrbed before the expermets: 0.5 X 0.95; 60 C 180; 10 g 45; L = 3 (sec); Where L s the lost tme of each phase a cycle legth. I these two expermets, we compare curret traffc sgal pla, Webster traffc model [1], ad tmg model based o PSO for sgle objectve wth MOPSO so as to verfy the effcecy of our algorthm. Each method has operated for 50 tmes. Each method of SPSO ad MOPSO terates tmes ad the populato sze s set to 50. The results are show Table 3 ad Table 4: TABLE III. COMPARISON OF ALGORITHM RESULTS ABOUT BAIHUAJIN INTERSECTION Methods Delay (sec/pcu) Stops (um/pcu) Curret sgal pla Webster SPSO (average delay) SPSO (average stops) MOPSO (decso 1) MOPSO (decso 2) MOPSO (decso 3) TABLE IV. COMPARISON OF ALGORITHM RESULTS ABOUT LONGSHAN ROAD INTERSECTION Methods Delay (sec/pcu) Stops (um/pcu) Curret sgal pla Webster SPSO (average delay) SPSO (average stops) MOPSO (decso 1) MOPSO (decso 2) MOPSO (decso 3) I Table 3 ad Table 4, SPSO (average delay) tres to optmze the average cotrol delay whle SPSO (average stops) tres to optmze the average stop frequecy. The three expermets of MOPSO use the same optmzato algorthm, but ther decso-makers choose the dfferet rato of weght betwee average delay ad average stop frequecy. MOPSO (decso 1) thks optmzg average delay s much more mportat tha optmzg average stop frequecy, so t chooses the bggest rato of weght betwee average delay ad stop frequecy. O the cotrary, MOPSO (decso 2) selects the smallest oe. MOPSO (decso 3) cosders both of the dces ad the tersecto s practcal evromet, ad obtas the sutable soluto by selectg the approprate elemet of that matrx. The Maxmum umber of members wth the o-domated set equals to 50 for all three MOPSO. Accordg to the expermets focused o mmzg the average delay, t s obvous that SPSO (average delay) ad MOPSO (decso 1) are better tha the curret tmg pla ad Webster method, especally whe the average delay s hgh. That the average delay from MOPSO (decso 1) s close to the oe from SPSO (average delay) dcates that the Pareto optmal solutos have a reasoable dstrbuto. The comparso betwee SPSO (average stops) ad MOPSO (decso 2) shows the same cocluso. Although average delay from MOPSO (decso 3) s feror to the result from MOPSO (decso 1) ad SPSO (average delay), the other objectve s superor to the oe from MOPSO (decso 1) ad SPSO (average delay). I Table 3, the soluto obtaed from MOPSO (decso 3) s closer to the oe from MOPSO (decso 2) tha MOPSO (decso 1). However, the soluto obtaed from MOPSO (decso 3) has the opposte stuato Table 4. Because the

5 evromet ad the traffc codtos of Bahuaj tersecto are dfferet from Logsha Road tersecto. Bahuaj tersecto s ear the lvg areas ad the flow rate s ot hgh; ths stuato, the decso-maker of MOPSO (decso 3) suggests optmzg average stop frequecy s more mportat tha optmzg average delay. O the cotrary, Logsha Road tersecto locates the cty ceter ad capacty of ts laes s low. To avod traffc cogesto ad decrease the degree of saturato, the decso-maker of MOPSO (decso 3) suggests optmzg average cotrol delay s more mportat tha optmzg average stops. The soluto of MOPSO (decso 3) s also selected from the Pareto optmal solutos, so the results from ths soluto wll be the most fttg tmg pla for fxed-cycle sgal cotrol. V. CONCLUSIONS I ths paper, a mult-objectve optmzato algorthm based o PSO (MOPSO) s proposed to optmze the average delay ad average stop frequecy smultaeously. The Pareto solutos of the cycle legth ad gree lght tme of each phase are obtaed as well. Fally, the most satsfed soluto s provded by the mult-objectve decso-maker module. The expermet results show that ths algorthm ot oly obtas better soluto tha Webster s optmal model whe mmzg average delay, buy also provdes other best solutos whch mmze average stop frequecy or optmze them both. The decso-maker s also desged to choose the sutable soluto accordg to the traffc stuato. The further study wll focus o mprovg the covergece speed of MOPSO algorthm by addg ew operators, ad researchg the teracto betwee the mult-objectve optmzato algorthm ad the mult-objectve decso- maker. REFERENCES [1] F. V. Webster, " Traffc Sgal Settgs," Road Research Techcal Paper, No. 39, Lodo: Her Majesty s Statoery Offce, 1958, pp [2] R. Akcelk, Traffc Sgals: Capacty ad Tmg Aalyss, Sydey: TRRL, [3] Y. Zhag, C. Gao, ad H. Huag, Sgal realtme tmg model ad Tabu search algorthm, J. Wuha Uv. (Nat. Sc. Ed. ) vol. 49, o. 1, Feb., pp , [4] J. Yag, D. Yag, Optmzed sgal tme model sgaled tersecto, Joural of Togj Uversty, vol. 29, o. 7, Jul., pp , [5] Trasportato Research Board, Hghway Capacty Maual, 4th ed., Washgto D C: Natoal Research Coucl, 2000, [6] D. Cheg, C. Messer, Z. Ta, J. Lu, Modfcato of Webster s mmum delay cycle legth equato based o HCM 2000, preseted at the 82 d TRB Aual Meetg, Washgto, D. C., [7] Q. Wag, X. Ta, ad S. Zhag, Sgal tmg optmzato of urba sgle-pot tersectos, Joural of Traffc ad Trasportato Egeerg, vol. 6, Ju., pp , [8] I. F. Sbalzar, S. Mǖller, ad P. Koumoutsakos, Multobjectve optmzato usg evolutoary algorthms, Proc. Ceter for Turbulece Research o Summer Program 2000, Nov. 2000, pp [9] W. Ja, Y. Xue, ad J. Qa, A mproved partcle swarm optmzato algorthm wth dsturbace, IEEE Iteratoal Coferece o Systems, Ma ad Cyberetcs, Oct [10] H. Jag, J. Zheg, L. Che, A Partcle Swarm Algorthm for Multobjectve Optmzaro Problem, Joural of Patter Recogto ad Artfcal Itellgece, vol.20, Oct., pp , 2007