Modeling and Control of an Isolated Intersection via Hybrid Petri Nets

Transcription

1 Journal of Traffc and Logstcs Engneerng, Vol, 1, No. 1 June 2013 Modelng and Control of an Isolated Intersecton va Hybrd Petr Nets Bassem Sammoud Laboratore de Recherche en Automatque (LA.R.A.), ENIT, BP 37, 1002 Tuns Belvédère Laboratore Systèmes et Transport (SET), UTBM, Ste de Belfort, Belfort Cedex, France Emal: bassem.sammoud@utbm.fr Mahjoub Drd and Abdallah El Moudn Laboratore Systèmes et Transport (SET), UTBM, Ste de Belfort, Belfort Cedex, France Emal: mahjoub.drd@utbm.fr Mohamed Benrejeb Laboratore de Recherche en Automatque (LA.R.A.), ENIT, BP 37, 1002 Tuns Belvédère Abstract Ths paper focuses on the use of Hybrd Petr Nets (HPNs) to model and control the traffc of an solated ntersecton of four phases. It captures mportant aspects of the flow dynamcs n urban networks. In ths paper, we show how ths model can be used to descrbe the traffc functonng and to obtan control strateges that mprove the traffc flow at ntersectons. The proposed approach can be generalzed to control several connected ntersectons n a dstrbuted way. Index Terms transportaton, Hybrd Petr Net, traffc sgnal control, modelng, sgnalzed ntersecton. I. INTRODUCTION Wth the growng number of vehcles, the traffc congeston and transportaton delay on urban arterals are ncreasng worldwde; hence, t s mperatve to mprove the safety and effcency of transportaton. The control strateges consst of changng the ntersecton's stage specfcaton, the relatve green duraton of each stage, the ntersecton's cycle tme, and the offset between cycles for successve ntersectons, whle requrng the well-planned synchronzaton, schedule and control to acheve satsfactory performance. In ths work, traffc sgnal control systems are consdered to be an hybrd system, ncludng both contnuous tme and dscrete event components. There s a vast amount of lterature focused on fndng the modelng of an ntersecton [1]-[3]. For the traffc flow model based on a dscrete Petr Net (PN), Wang et al. proposed a Petr Net (PN) model that ncludes both traffc sgnal control logc and traffc flow [4]. In [5], Lst had descrbed how PN can be used to model traffc control stuatons n urban networks. On the other sde, the model based on a contnuous Petr net s sutable for macroscopc descrpton, [6] and [7]. And recently, hybrd systems have receved much attenton [8] and [9] and have been appled for the modelng and control of transport systems [10]. In general, the traffc control s characterzed by a contnuous conduct of the road [6] and behavor dscrete at road crossng [1], [11]. And, these two behavors are respectvely modeled, by a Contnuous Varable speed Petr Net (CVPN) and Tmed Petr Net (TPN). In order to complete these prevous works, we present, n ths paper, a new approach to model and to smulate the traffc sgnal control based on a Hybrd Petr Net HPNs method. Ths paper s organzed as follows: after the ntroducton of secton 1, we present, n secton 2, HPN. Then, we gve, n secton 3, a descrpton of the proposed model and the control strategy of traffc lght by usng HPN. Secton 4 descrbes the case study and reports the results of some smulatons performed under dfferent traffc scenaros. II. HYBRID PETRI NETS In the present work, a sgnalzed ntersecton wth ts nput and output flows s consdered to be an hybrd system thereby contans contnuous places and transtons (C-places and C-transtons) and dscrete places and transtons (D-places and D-transtons). The reader can fnd a more detaled presentaton of HPNs n [9]. Defnton: A marked autonomous hybrd Petr net s a sextuple R= <P, T, Pre, Post, m 0, h> fulfllng the followng condtons P = {P 1, P 2,..., P n } s a fnte, not empty, set of places; T = {T 1, T 2,..., T m } s a fnte, not empty, set of transtons; P T = Ø,.e. the sets P and T are dsjonted; h: P T {D, C}, called hybrd functon, ndcates for every node whether t s a dscrete node (sets P D and T D ) or a contnuous one (sets P C and T C ); Pre: P T Q + or N s the nput ncdence applcaton; Manuscrpt receved October 11, 2012; revsed January 13, Engneerng and Technology Publshng do: /jtle

2 Journal of Traffc and Logstcs Engneerng, Vol, 1, No. 1 June 2013 Post: P T Q + or N s the output ncdence applcaton; m 0 : P R + or N s the ntal markng. In the defntons of Pre, Post, and m 0, N corresponds to the case where P P D, and Q + or R + corresponds to the case where P P C. Pre and Post functons must satsfy the followng crteron: f P and T j are such that P P D and T j T C, then Pre(P,T j ) = Post(P,T j ) must be verfed. An unmarked hybrd PN s a 5-uple Q = <P, T, Pre, Post, h>,.e., n addton to the structure, the nature of each node (dscrete or contnuous) s specfed n Q. In a contnuous PN, the enablng degree of transton Tj for markng m, s defned as: mp ( ) enab( T, m) mn j P T j Pr e ( P, Tj ) A dscrete transton n a hybrd PN s enabled f each place P n Tj meets the condton (as for a dscrete PN) m( P) Pr e( P, T ) j Hybrd Petr nets (HPNs) were ntally successfully appled for modelng, performance evaluaton and the manufacturng systems desgn and more recently they had also been successfully used for studyng transportaton systems, [12]. Graphcally for HPN, the arrows represent the arcs, dscrete places are represented by crcles and dscrete transtons are represented by black boxes, whereas contnuous places are represented by double crcles and contnuous transtons are represented by whte boxes. III. MODELING A SIGNALIZED TRAFFIC INTERSECTION BY HPNS A. Studed Intersecton Descrpton The studed confguraton s related to a crossroad that has an ntersecton of four drectons, wth two lnks, one nput lnk formed of 2 lanes wth movements to turn left and to go straght or rght, and one output lnk. An example of ths sgnalzed ntersecton s gven n Fg. 1. (1) (2) We consder here what we call a hgh type of ntersectons, where all traffc flow conflcts are separated by sgnal phasng. In ths knd of ntersecton, there s four phases as shown n Fg.2. Fgure 2. Four phases ntersecton In order to prevent the conflcts, we can use the noton of compatble streams and ncompatble streams as presented n [13]. Streams, whch can smultaneously get the rght-of-way, are called compatble streams for example streams 1 and 7. On the other hand, when trajectores of two traffc streams do cross, the streams are n conflct, for example, streams 1 and 11. In our case, the studed ntersecton can be parttoned nto four Compatble Stream Groups (CSG) as shown n Fg 2. CS 1: stream 1 and 7; CS 2: stream 2 and 8; CS 3: stream 5 and 11; CS 4: stream 4 and 10 B. Analytc Model of Road Secton In ths part, we present the model by CVPN to model the flow of traffc on the ways of the crossroad. To model the problem, we dvde each secton nto N segments [X, X +1 ] of length (Fg.3). We assume that for each segment the characterstc varables of flow are supposed to depend only on the tme and not on the poston, lke n [10]. Fgure 3. The generc road secton Fgure 1. A four phases sgnalzed ntersecton The followng net of Fg 4 s composed of two contnuous places (P j and P j+1 ). The markng (m ) of place P descrbes the number of vehcles whch enter the ntersecton and the markng (m ), such that m = c m, descrbe the avalable places on the th stream, and the markngs a j and a j+1 descrbe the number of possble smultaneous frngs for transtons T and T +1. Moreover, a dscrete PN s used to represent the traffc lght. 10

3 Journal of Traffc and Logstcs Engneerng, Vol, 1, No. 1 June 2013 When markng m reaches value 1, we pass to a second phase (case 2), labelled lnear phase and called so because the evoluton s lnear, s gven by Fgure 4. A hybrd PN model for a road ntersecton It should be noted that the sum of the markngs of places P and P (.e. m + m ) s an nvarant markng. It s the same for P j places (.e. m j ), whch remans constant over tme. c represents the capacty of each way. We have m ( t) m ' ( t) c t 0 m ( t) a t 0 j j The evoluton steps of CVPN [Davd et al 1992] can allow us to wrte the followng mathematcal expressons defnng the speed crossng q (t) and q +1 (t) related to the transtons T and T +1 respectvely q ( t) qmax mn( a j, m' ( t)) qmax mn( a j, c m ( t)) q 1( t) qmax 1 mn( a j1, m ( t)) We defne the q max as the maxmal frng frequency of transton T. And the markng m of place P s gven by the followng equaton (3) (4) dm () t q( t) q 1( t) (5) dt The markng of a place s represented by a contnuous postve real number. However, those of a dscrete place are non-negatve nteger. The crossng of a dscrete transton may affect the markng of contnuous PN and vce versa. The evoluton of the CVPN s also gven by the system of dfferental equatons (see the system of equaton (5) and (6) or (7) or (8)). We observed that there are several phases n ths evoluton (we call a phase, one perod of tme durng whch the expresson speeds accordng to markngs remans unchanged). We pass from one phase to the followng, when the markng of a place P crosses the threshold m =1, n a drecton or n the other. We can dstngush 3 cases accordng to the speed crossng q (t) and q +1 (t). A frst case, called exponental phase, wll last as long as a <c -m and a >m f a c m and a m then (5) wth q() t qmax q1( t) qmax 1m ( t) (6) f a c m and a m then (5) wth q() t q q () t q max 1 max1 (7) And a second exponental phase (case 3) whch happens when c-m=1 f a c m and a m then (5) wth q ( t) q ( c m ( t)) max q () t q 1 max 1 The resoluton of dfferental equatons, assocated wth each case, depends on the ntal condtons, on ntal tme t 0 and ntal condton m 0, leads to defnng the followng markngs equatons Case1: a c m and a m m t q m q e max 0 max qmax 1 ( tt0 ) () qmax 1 qmax 1 Case 2: a c m and a m m ( t) ( q q ) t m 0 max max 1 Case3: a c m and a m q c q m () t q m max max 1 max q c q 0 max max 1 qmax e qmax ( tt0 ) (8) (9) (10) (11) C. The Proposed HPNs Model In ths secton, the Hybrd PN of the ntersecton, gven n Fg.1 s descrbed wth the am to provde an example of a HPN model of an urban transportaton system. For ths optc, we use the model presented n Fg 4, and the overall proposed HPN model s gven n Fg 5, and t wll be generalzed later for a common confguraton. The proposed HPN model s descrbed as follow 1, 2, 4,5,7,8,10,11 P 1 : vehcles arrvng on the th stream, P 2 : vehcles present on the secton of the th stream, P 3 : avalable places on the th stream, P 3, P 6, P 9, P 12 : the output lnks, a j : smultaneous frng parameters of transtons T 1 and T 2. (a j = 2 véhcles and a j+1 = 1 véhcle). The traffc lghts n our dscret PN model are presented by places and transton where P j : green perod for streams {,j}, 11

4 Journal of Traffc and Logstcs Engneerng, Vol, 1, No. 1 June 2013 T j : green phase (wth d j. duraton) ends for streams {, j}. In ths paper, we smulate the evoluton of traffc n an solated ntersecton where the traffc lghts swtch from the usual order (fxed sgnal-tmng). We defne four compatble phases as a set of permtted conflcts: {1, 7}, {2, 8}, {5, 11} and{4, 10}. The consdered HPN model s gven n Fg.5. In the am to advance some deas leadng to a control strategy for urban traffc networks, we propose, a new heurstc approach to control the consdered model. Our control strategy s based on the green lght duraton, whch s determned on the estmated number of vehcles enterng the road durng a cycle. We defne a cycle as a sum of green lght and red lght duratons. We take the maxmal number n each par of compatble streams wth movement to turn left and we add a value of t d. as shown n equaton (12) (, j) (1,7),(4,10) (12) d max( Nbv, Nbv ) t t j j c d (13) Nbv s the number of vehcles n lnk, t d the startng tme of the vehcle at the ntersecton and t c the crossng tme. As an example, we assume that the numbers of vehcles n streams 7 and 1 are respectvely 5 and 8. So the green duraton s computed by d = 8 + t d. Snce the maxmal frng frequency to the rate of output s 1 veh/s. So, let consder t d =t c =1s, as well for the flows 4 and 10. In the other sde, as cted prevously, the arrval flow of vehcles for {2, 5, 8, 11} s more mportant than the ones wth movements to turn left, so the green duraton s computed as follows: we dvde the number of vehcles enterng n the ntersecton by the number of roads (.e. equaton (13)). Fgure 5. The proposed hybrd PN ntersecton wth IL: the number of nput lnks, Nbv k : the number of vehcles n each lnk, t d : the startng tme of the vehcle at the ntersecton, t c : the crossng tme. Stream Nbv TABLE I. SIMULATION PARAMETERS 1 Sm 1 Sm 2 Sm 3 (case a) Sm 3 (case b) Sm 4 d j TE1 d j TE2 d j TE3 d j TE4 d j TE6 IV. SIMULATION AND RESULTS As follows, we suppose that the arrval rate of vehcles s Nb_Vehcles dvded randomly n dfferent stream (see Table I). wth Nbv: arrval rate of vehcles of stream, TE: evacuaton tme of all vehcles for each flow, d j : green lght duraton. Durng the smulatons, we consdered that the maxmal frng frequency for T 1 s equal to 2veh/s; on the other sde, the rate of output (for T 2 ) s 1veh/s and we consdered ntersecton currently ruled by a fxed sgnal-tmng plan

5 Journal of Traffc and Logstcs Engneerng, Vol, 1, No. 1 June 2013 At frst step, we suppose that Nb_Vehcles=110 and the delay d j assocated to the dscrete transton (green lght duraton), s the same and fxed to 5 seconds (Sm1). The evoluton of markngs s shown n Fg 6. Fgure 9. Markng evoluton (Sm 3 (case b)). Fgure 6. Markng evoluton (Sm 1). In the second step, we do not consder a same delay d j for all the streams, and we dstngush three types of smulaton (Sm 2, Sm 3 cases a and b). In the frst tme, we start by ncreasng the green lght duraton,.e. 7 seconds for left turn movements (streams 1/7 and streams 4/10) and 10 seconds for the streams to go straght or rght (streams 2/8 and streams 5/11). The evoluton of markng s shown n Fg 7. We can see that the total evacuaton tme s reduced from 99 to 85 s. In a second tme, and referrng to a real ntersecton, we can notce that the arrval rate of vehcles n the streams {1, 4, 7, 10}, are less than {2, 5, 8, 11}. For ths reason, n the smulaton (Sm 3) we start by fxng the green duraton for the last streams to 15 n case a and then to 20 seconds n case b. The other streams are left wth duraton of 10 seconds. The behavor of markng for these two cases s presented n Fg 8 and Fg 9. The smulaton (Sm 3) shows clearly that despte ths duraton seems to be drectly related to the tme needed to avod all the ways; we can notce that t can not be always the case, as shown n Sm 3 (case a and case b). In the last step, and for the last smulaton, we use the proposed heurstc approach to optmze the green tme duraton. Ths approach s based on the estmated number of vehcles whch enter the crossroad durng a cycle (equaton 10 and 11). We obtan the markng evoluton presented n Fg 10. We can clearly see that the total evacuaton tme s greatly reduced, snce t passes to 74s. Fgure 10. Markng evoluton (Sm 4). Fgure 7. Markng evoluton (Sm 2). In order to prove the robustness and the effcency of our approach, we realzed other smulaton by changng parameter Nb_Vehcles. These smulatons are reported n Table II. We can see that our approach provdes an nterestng result. In Fg. 11, we report the results of all these approaches for dfferent flows capactes and we can see that for all the consdered cases our heurstc provdes the best results (Sm 4). Fgure 8. Markng evoluton (Sm 3 (case a)). V. CONCLUSION AND FUTURE WORK Ths paper has presented a new hybrd model of sgnalzed ntersecton. Ths model based on HPNs descrbes the behavor of the traffc wth reference to a four phase sgnalzed ntersectons. We partonned ntersecton streams nto several Compatbles Streams Group (CSG) wth a smultaneous movement through the ntersecton. Some expermental results about a consdered case study were dscussed n order to descrbe the dynamcs of the model and the effcency of the proposed 13

6 Journal of Traffc and Logstcs Engneerng, Vol, 1, No. 1 June 2013 heurstc. The researches contnue by a development of an algorthm to decde the optmal traffc sgnal plan by the mnmzaton of the overall evacuaton tme. Our future work wll focus also on how several neghbor ntersectons wll be modeled and controlled together. REFERENCES [1] R. Davd and H. Alla, Du Grafcet Aux Réseaux De Petr, Pars, Edtons Hermès, [2] J. Wang, C. Jn, and Y. Deng, Performance analyss of traffc networks based on stochastc tmed Petr net models, n Proc. IEEE Engneerng of Complex Computer Systems, 1999, pp [3] L. G. Zhang, Z. L. L, and Y. Z. Chen, Hybrd Petr net modelng of traffc flow and sgnal control, n Proc. Seventh Internatonal Conference on Machne Learnng and Cybernetcs, Kunmng, July 2008, pp [4] H. Wang, G. F. Lst, and F. DCesare, Modelng and evaluaton of traffc sgnal control usng tmed petr nets, n Proc. IEEE Systems, Man and Cybernetcs, 1993, pp [5] G. F. Lst and M. Cetn, Modelng traffc sgnal control usng petr nets, IEEE Trans. Intellgent Transportaton Systems, vol. 5, pp , [6] C. Tolba, D. Lefebvre, P. Thomas, and A. ElMoudn, Contnuous petr nets models for the analyss of traffc urban networks, n Proc. IEEE Systems, Man, and Cybernetcs Conference, Tucson, Arzona, USA, 2001, pp [7] J. Julvez and R. Boel, Modelng and controllng traffc behavor wth contnuous Petr nets. n Proc. 16 th Trennal World Congress of the Internatonal Federaton of Automatc Control, [8] J. Le Bal, H. Alla, and R. Davd, Hybrd petr nets, n Proc. European Control Conference, Grenoble, France, 1991, pp [9] R. Davd and H. Alla, Dscrete, Contnuous and Hybrd Petr Nets, Lbrary of Congress Control Number: , Berln Hedelberg: Sprnger-Verlag, [10] A. D Febrarro, D. Gglo, and N. Sacco, Urban traffc control structure based on hybrd petr nets, IEEE Transactons on Intellgent Transportaton Systems, vol. 5, no 4, pp [11] C. Tolba, D. Lefebvre, P. Thomas, A. EL Moudn, Performance evaluaton of the traffc control n a sngle crossroad by Petr nets, n Proc. 9 th IEEE Internatonal Conference on Emergng Technologes and Factory Automaton, Lsbon, Portugal, 2003, vol. 2, pp [12] M. K. Jbra and M. Ahmed, Computer smulaton: a hybrd model for traffc Sgnal Optmzaton, Journal of Informaton Processng Systems, vol.7, no.1, March [13] F. Yan, M. Drd, and A. EL Moudn, A schedulng approach for ntellgent vehcle sequencng problem at mult-ntersecton networks, Internatonal Journal of Operaton Research, vol. 1, pp, Bassem SAMMOUD was born n Tunsa n He obtaned the Master of Automatc Control and Sgnal Processng from the Ecole Natonale d Ingneurs de Tuns n he s currently a PhD student n Automatc Control and Sgnal Processng n LARA-ENIT. Hs current research nterests are the traffc control. and optmzaton methods for dscrete events systems and operatonal research domans. Mahjoub DRIDI was born n Tunsa, n He receved n 2001 the M.Sc. degree n control and computer scences from de Unversty of Llle 1, Vlleneuve d'ascq, France. He receved hs Ph.D. degree n 2004 from the Ecole Centrale de Llle, France. He s currently an assstant Professor at Unverstéde Technologes de Belfort Montbélard n Systèmes Et Transport Laboratory. Hs research s related to traffc control, producton plannng, computer scence, optmzaton methods for dscrete events systems and operatonal research. Abdellah EL MOUDNI Receved a PH.D degree n Automatc Control n 1985 from Llle Unversty, Llle, France. He s currently a Professor at UTBM, where he s n charge of courses n automatc control. Hs research nterests nclude transportaton systems, nonlnear control theory, dynamc systems, control of dscrete event systems and sngular perturbaton methods n dscrete tme. Mohamed BENREJEB has obtaned hs Dploma of Ingéneur IDN (French Grande Ecole ) n 1973, a Master degree of Automatc Control n 1974, a Ph.D. n Automatc Control of the Unversty of Llle n 1976 and the DSc of the same Unversty n He s currently a full professor at the Ecole Natonale d Ingéneurs de Tuns and an nvted Professor at the Ecole Centrale de Llle. Hs research nterests are n the area of analyss and synthess of complex systems based on classcal and non conventonal approaches and recently n dscrete event systems doman. 14