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1 Sholars Journal of Engneerng an Tehnology (SJET) Sh. J. Eng. Teh., 204; 2(4B): Sholars Aaem an Sentf Publsher (An Internatonal Publsher for Aaem an Sentf Resoures) ISSN X (Onlne) ISSN (Prn Researh Artle Hybr Moelng of the Urban Traff Flow Douou GAYE*, Roger Mareln FAYE, Benjamn MAMPASSI Laboratore e Tratement e l Informaton, Eole Supéreure Polytehnque,Unversté Chekh Anta Dop e Dakar, BP 5085 Dakar-Fann, Senegal *Corresponng author Douou GAYE Emal: Abstrat: Nowaays, the evelopment of tools for moelng an smulatng roa traff flow beomes more an more a neessty. Ths s reflete by the evelopment of many roa traff flow moels that an reproue some traff phenomena. The objetve of ths paper s to present a hybr moel base on two moels evelope nepenently. The hybr sheme s base on the ouplng sheme evelope by Bourrel. Couple moels are the marosop LWR moel (base on ts resoluton by GODUNOV s metho) an a mrosop ar followng moel presente n ths paper. The propose moel s valate by smulaton. Keywors: Traff moelng, Marosop moel, Mrosop moel, Hybr moel. INTRODUCTION For a long tme, urban transportaton n sub- Saharan Afra, has been plae among the non-prorty setors an beome n a reent years a major onern for authortes to ontrol an evelop the setor. Thus, the provson of tools aapte to the ontet an allowng the moelng traff flow phenomena s a neessty. Ths moelng, whh onssts to a esrpton of the traff evoluton over tme an spae, an help to unerstan these phenomena. There are several approahes to moel the traff flow. These approahes an be lassfe nto two major famles whh are: the marosop approah an the mrosop approah []. The mrosop approah fouses on the nteratons between the vehles onsere nvually. The moels resultng from ths approah, epltly represent the ynam states of vehles [2]. In ontrast, the marosop approah esrbes the traff flow n a omprehensve manner [3]. Eah of these types of approahes s aapte to spef stuatons. For eample, marosop approahes fal n esrbng transtonal phases n traff. The mrosop moels allow a lear vew of these transtonal phases, but beome qukly omplate when a large network s onsere. In ths ase the omputatonal tme s very long ompare to marosop moels. The am of ths paper s to propose a moel of urban transport system, whh an represent sngular phenomena whh may be the orgn of sturbanes n the network. It s well known that the marosop LWR moel s known for ts ablty to esrbe the overall traff ynams [4]. But ths moel oes not hghlght several types of sngulartes whh have an mpat on the traff. Thus the evelopment of a hybr moel wll frst allow the overall traff ynams to be represente by the LWR moel by solvng t wth the unov s metho, an sngular elements to be represente by usng a mrosop traff flow moel that wll be presente. In the rest of ths paper, the LWR moel an ts resoluton by the unov s metho wll be presente, after, the mrosop moel wll be presente. The ouplng sheme whh s base on the sheme propose by Bourrel [5] wth some mofatons wll be esrbe. To onlue, a presentaton of some smulaton results wll be mae. THE LWR MODEL In ths moel the traff s represente as a ontnuous flu haraterze by average quanttes epenng on tme an spae. These quanttes are the flow q (,, the onentraton or ensty k(, an the spee flow v (,. 566

2 Douou GAYE et al., Sh. J. Eng. Teh., 204; 2(4B): The funamental uaton of the moel s the onservaton uaton (uaton ), whh was frst use for moelng traff flow n 955, by Lghthll an Whtham, an nepenently by Rhars n 956 [6, 7]. Ths uaton epresses the fat that the number of vehles n a roa seton at a tme t + t s ual to the number of vehles n ths seton at tme t, to whh we a the number of vehles entere urng t, mnus the number of vehles ete urng t. k(, q(, 0 t () The flow spee s efne as the rato of flow on the ensty. q(, v(, = (2) k(, The LWR moel s supplemente by an ulbrum funamental relatonshp whh vares aorng to parameters of stue network. Usng the marosop efnton of flow spee, t s possble to erve an ulbrum relatonshp between flow an ensty. Ths relatonshp s eue from epermental observatons, an s represente by a agram alle funamental agram [8] (Fg. ). Fg. : Eample of parabol funamental agram The man parameters of a funamental agram are: mamum ensty, enote K ma, the free spee, enote Vl gven by the slope at the orgn of the funamental agram, the rtal onentraton K between the flu an saturate traff ontons, the mamum flow Q ma or apaty of the seton stue, an rtal spee V or spee of vehles at rtal onentraton. In the LWR moel, we assume that the system s always n ulbrum. Hene, the spee s funton of the onentraton. v(, = V (k(, ) (3) RESOLUTION OF THE LWR MODEL BY GODUNOV S APPROACH In the unov s sheme, eah roa seton s ve nto ells of length (Fg. 4). We enote by the length of a ell whose nterfaes are represente by ponts an, respetvely nput an output of the ell note by: C =[, ] (4) The uaton () apple to the ell C leas to the followng one: t k(, q(, 0 t After ntegraton, we have: k(, q(, q(, 0 The average ensty k ( (5) (6) k ( of ell C s ntroue: k (, (7) We enote by Q ( q(, the flow on the nterfae appromaton of the frst orer tme ervatve, the onservaton uaton n the ell C (uaton 5) leas to the followng uaton: at tme t. Conserng the Euler k ( t k Q ( Q 0 t (8) The parameters of the numeral sheme (uaton 8) are efne by supply an eman funtons (Fg. 3). 567

3 Douou GAYE et al., Sh. J. Eng. Teh., 204; 2(4B): Fg. 2: Subvson of a seton of roa to ells The supply funton enote by S ( k ) s the mamum number of ars that an enter the ell urng the tme nterval t. In flu traff stuaton t an be observe that ths funton s ual to the mamum rate that an enter the ell, whle n ongeste traff stuaton, ths funton s gven by an ulbrum relatonshp Q ( k) eue from the funamental agram. Qma S ( k) Q ( k) f k K (9) f k K The eman funton enote D ( k ), s the mamum number of ars wshng to go out urng the same nterval of tme. In stuatons of flu traff, ths funton s gven by an ulbrum relaton Q ( k ), erve from the funamental agram, whle n ongeste traff stuaton, ths funton s ual to the mamum rate that an leave the ell. Q ( k) ( k) Qma f f k K D (0) k K Fg. 3: Supply agram at left an eman agram at rght From the etermnaton of the mamum number of vehles that an enter a ell C an the mamum number of vehles that an leave the upstream ell for the nterval tme t, we an alulate the average flow Q ( t to the pont separatng the two C ells. Ths flow s the mnmum between the supply of ownstream ell C an the eman of the upstream ell C at ths pont urng ths nterval tme. Gven the uaton 8, we have: Q ( t t mn( D( k ), S( k )) t k ( t k ( Q ( t t Q ( t t ) () 568

4 Douou GAYE et al., Sh. J. Eng. Teh., 204; 2(4B): Hene, from the ntal ontons of eah ell, we an etermne the evoluton of traff flow by suessve tme steps. For the stablty of ths sheme, the tme step an the length of a ell shoul be hosen so that the ompute soluton to an nterfae oes not nterfere wth the rest. So eah ell obeys to the followng CFL onton (Courant Frerhs-Lewy) t wth tme. V l (2) the length of a ell, The resoluton algorthm s gven below: t the observaton Fg. 4: Algorthm for solvng the LWR moel by unov s metho The unov s sheme allows a smple moelng of the traff flow. It s also a moel well sute to urban envronments. However the unov s metho oes not reproue the transton phases relate to nents or vehle s startng at the en of a re lght, ether n aeleraton or eeleraton. For eample, n the ase of a spatal sontnuty (0 n Fg. 5) t an be note that, for the same spee, traff ontons are transferre from the ulbrum state orresponng to ths flow on the upstream funamental agram to that orresponng to the same flow on the ownstream agram. The spee s sontnuous at ths pont. 569

5 Douou GAYE et al., Sh. J. Eng. Teh., 204; 2(4B): Fg. 5: Case of spatal sontnuty: veloty hange at the pont of sontnuty THE MICROSCOPIC CAR FOLLOWING MODEL In ths moel, a vehle at a poston at tme t s haraterze by ts spee v (, an ts aeleraton a (,. Lets onserng a vehle (follower) at poston behn a vehle at tme t. To represent the ynamal behavor of the rver/vehle par at tme t t, the aeleraton of ths vehle s efne as a funton of the relatve spee an spang between vehles. a f ( v v (, ( )) t (3) Fg. 6: Vehles n a seton roa Let s take t T. The aton of the rver at the t T s ontrol levers of hs vehle at tme proportonal to the spang between the poston of the vehle ( ) an the poston of the vehle at tme t (Fg. 7). Fg. 7: Graphal bas sheme of rver-vehle par The moel of the rver s behavor nlues the followng funtons: Deteton of the fferene between the atual an esre spang, Deson-makng an Aton on the ontrol levers.. Hene, a graphal moel of the rver an be represente at Fg. 8, n whh the aton of the rver s lnke to the spang between the vehle at 570

6 Douou GAYE et al., Sh. J. Eng. Teh., 204; 2(4B): an the vehle at poston at tme t, poston by the onstant parameter g (uaton 4). g( ( )) (4) ( ) ( t where g s a reaton parameter an ( ) the thrust or brakng relate to rver s behavor. In orer to take nto aount the phenomena of aeleraton an eeleraton, t s neessary to antpate a slowown or aeleraton zone. Ths antpaton wll help to prevent the too fast avanng of vehles. Thus, n aton to the elay tme orresponng to the reaton tme of rvers, the ntegraton of a verter funton for the problem of antpaton, offers a new ynam relatonshp between ( ) an the effetve aton ( ), ( ) t represente by uaton 5. T e( ) t where Fg. 8: Base moel of rver ( ) ( t T ) e( ) ( ) ( t T ) T (5) t T s the response tme of the verter. The aeleraton are relate to the effetve aton by the followng relaton: a e( ) a ( (6) t t Where s a onstant parameter an the response tme of the motor. From Equatons 5 an 6, we eue the aeleraton, represente by uaton 7. a T g g v ( ( v ( t T ) v ( t T )) ( ( t T ) ( t )) T Fnally the ar followng moel s represente by the followng system 8. a( v v ( a t ( v t T g ( v ( t T ) v g ( t T )) ( ( t T ) ( t T )) (7) (8) In ths moel, the aeleraton epens not only on the spang between the followng ar an ts leaer an the fferene between the two spees, but also on the nformaton on the spee of the followng ar at tme t. The ulbrum s haraterze by the stuaton at whh vehles travel at the same spee. At ths ulbrum we must have: v v V v ( 0 t S ( 57

7 Douou GAYE et al., Sh. J. Eng. Teh., 204; 2(4B): Then we have: T k v ( ( v ( t T ) v v ( a ( 0 t ( v ( V t k ( t T )) ( ( t T ) ( t T )) 0 (9) From uaton 9, we get the followng relaton: S V. g (20) To etermnate the parameter g from uaton 20, we use the efnton of the funamental agram. Ths agram proves the funamental ulbrum relatonshp between onentraton an spee. The onentraton k s relate to S by the uaton 2: S (2) k The alulate mamum spee shoul not be greater than that allowe at the fe free mamum onentraton K. l ma For eample, wth the ulbrum relatonshp gven above, we an efne: Smn : The mnmum ulbrum stane between two vehles, K ma S : The mnmum ulbrum stane for whh vehles run at full spee. ma K l ma Then we must have: g. S S (22) mn ( g. ma The fel observatons show that aeleraton an eeleraton are not symmetr problems. The aeleraton response tme s greater than the response tme of eeleraton. Thus we efne two response tmes an 2 suh as 2. An the term of the aeleraton or eeleraton s gven by the uaton 23. a a v v 2 T k ( v ( t T ) v T k ( v ( t T ) v 2 k ( t T )) ( ( t T ) k ( t T )) ( ( t T ) 2 ( t T )) ( t T )) Deeleraton Aeleraton (23) COUPLING THE TWO MODELS In orer to take nto aount transtonal phases not taken by the LWR moel, a hybr moel s presente here. Ths moel s use to represent some part of network by a mrosop moel whle the other parts of the network are represente by the LWR moel. The ouplng sheme we use must allow orret transmsson of nformaton from one moel to another Let s onser a roa seton wth a gven length. Applyng a hybr moel on ths roa seton onsst of vng the roa seton nto areas where, ether the mrosop moel or the LWR moel are apple. In the eample n Fg. 9, the roa s ve nto three zones, an the LWR moel wth ts resoluton by the 572

8 Douou GAYE et al., Sh. J. Eng. Teh., 204; 2(4B): unov s sheme s apple n two zones separate by a zone n whh the ar followng moel s apple. At nterfaes we efne transtonal ells. These ells are vrtual an allow nformaton ehanges between the two moels at the nterfaes. The nterest of these transton ells s to allow satsfatory bounary ontons to eah moel, an also enable a graual transfer of nformaton from one moel to another. Fg. 9: Couplng sheme Let a ertan nstant t where the state of the system s known, an t an t m the temporal sretsaton steps apple respetvely to the marosop moel an the mrosop moel. The ouple moels are srete tme an the problem s to etermne the state of the system at t t. By mposng m t N t, we wll have nstants where the two moels know the state of the system smultaneously, wth N a postve nteger. Thus nformaton s ehange at the nterfaes at eah tme step of the marosop moel. The prnple of ths sheme s ve nto four stages: Step : We alulate for the unov s moel, the upstream eman of the mrosop area urng a tme step, an the ownstream supply Step 2: After we translate these onstrants nto vehular onstrants for ell transton through the generaton of vehles at the nterfae Marosop/Mrosop an etermnaton of the trajetory of the frst vehle at the nterfae Mrosop/Marosop. Step 3: we evolve the whole vehle of the mrosop, base on the ar followng moel, from t to t t per suessve tme step t m, takng nto aount the onstrants at nterfaes. Step 4: we eue from these trajetores, the ensty of these ells from t to t t. These flows are use as bounary ontons for the marosop moel an allow the alulaton of ts state at t t. At the nterfaes of the mrosop area, we mpose the eman an supply of marosop moel as bounary ontons. Thus, the mrosop moel shoul prove to the marosop moel, ownstream supply at nterfae Marosop/Mosop, an upstream eman of the nterfae Mrosop/Marosop. Conversely, the marosop moel must prove to the mrosop moel, the generaton tmes of vehles at the nterfae Marosop/Mrosop, an the trajetory of the frst vehle at upstream nterfae Mrosop/Marosop. Flows are alulate retrospetvely: the growth of vehles from t to t t gves the supply an eman of transton ells. The generaton of vehles at the nterfae Marosop/Mrosop must satsfy both the eman on the marosop moel an ownstream traff ontons. It s assume that the generaton of vehles at the entrane of eah seton wll be mae unformly. We efne: The mnmum tme nterval between two generatons of vehle at the nterfae Marosop/MrosopCI, as the nverse of the eman mpose by the marosop moel. CI D( k) Nsas: entry transton sas. Ths sas s esgne to materalze at the en of tme step, the presene of a porton of t vehle not suffent to generate a vehle 573

9 Douou GAYE et al., Sh. J. Eng. Teh., 204; 2(4B): an to reue the osllatons n the hybr moel. To etermne the generaton tmes of vehles at the nterfae Marosop/Mrosop, we alulate the frst moment of generaton, to omplete the porton of vehle alreay present n the ell, gven by (- Nsas) CI. The prete generaton tmes of other vehles are alulate usng tme spang ual to CI. The reaton of a vehle wll be mae only, f the spang between the last vehle an the entrane of the mrosop area s more than or ual to the ulbrum spang orresponng to the spee of the last vehle, f ths spee s less than or ual to the spee alulate from the eman at the entrane of the mrosop area. The trajetory of a vehle omng out from the mrosop area must meet ownstream traff ontons. We also etermne the et tmes of vehles. We efne NsasS as the output transton sas. SIMULATIONS WITH THE HYBRID MODEL PROPOSED In ths seton we wll suss fferent senaros to stuy the behavor of the moel, espeally wth regar to the sprea of nformaton epenng on fferent parameters. In smulaton, a 6km streth of roa s use. A tme step t 5s s hosen. The seton of roa 88, 2m s ve nto ells of length. The ells are numbere from the upstream to ownstream; the ells 58 an 59, loate between 4km an 4.400km are represente by the mrosop moel. Senaro : Propagaton of ongeston to upstream For ths purpose, we take an ulbrum onton where eman at the network entry s set to 0.467veh/s an the supply at the network output s set to veh/s. The ntal onentraton of ells s set to 50veh/km. Through the results of presente n fgure 0, there s a ongeston propagatng upstream. In ths fgure we have represente the evoluton of the traff flow on ths streth of roa n the plane (,. Fg. 0: Propagaton of ongeston to upstream: plane (, Senaro 2: Reuton of ongeston Ths tme we are n an ulbrum stuaton where eman at the nput of the network s fe to veh/s an supply at the output of the network s set to veh/s. The ntal onentraton of ells s set at 260veh/km. 574

10 Douou GAYE et al., Sh. J. Eng. Teh., 204; 2(4B): Fg. : Reuton of ongeston n the hybr moel Through the results presente n fgure, the formaton of a range whh propagates through the mrosop moel an be observe wthout storton. In ths fgure we have represente the evoluton of the traff flow on the streth of roa n the plane(,. Sénaro 3: Reuton of apaty The eman at the nput of the network s set to veh/s an the supply at the output s set to veh/s. The ntal onentraton of ells s set at 50veh/km. Fg. 2: Results of the smulaton wth the hybr moel n the ase of a reue apaty Through the results presente n fgure 2, we observe that the hybr moel proves an aeptable representaton of the traff, wth a representaton of the traff evoluton at the sontnuty pont, keepng the flow onservaton an onserng the spee as a bas parameter at the sontnuty pont. The smulaton allowe to observe the evoluton of the rse n ongeston to upstream that s observable n realty. CONCLUSION In ths paper we presente a hybr traff flow moelng base on the marosop LWR moel an a mrosop traff flow moel evelope nepenently. The nterest of ouplng the two moels s to represent a large network wth a marosop frst orer moel, an usng a mrosop moel to represent the sngular elements. The results show that wth ths hybr moel t s able to obtan aeptable results for a representaton of the fferent haratersts of traff. Ths hybr moel an also represent phenomena that an be observe at 575

11 Douou GAYE et al., Sh. J. Eng. Teh., 204; 2(4B): the ponts of spatal sontnutes, an that are not represente by the marosop moels. REFERENCES. El Hmam MS ; Contrbuton à la moélsaton et à la smulaton hybre u flu e traf. Ph.D. thess, Unversté Ártos, Deember Ma HD; Sur la apaté opératonnelle es moèles affetaton ynamque u traf et la onvergene es algorthmes équlbrage. Ph.D. thess, Eole Natonale es Pont et Chaussées Frane, Deember Chabaut N; Calbraton et valaton un moèle marosopque éoulement u traf nterurban: Vers un albrage en lgne pour une utlsaton en temps réel, V.A. Transports, ENTPE/INRETS, Frane, Ska M, Rau C, Hermener V ; Smulaton e la ynamque u système e éplaements urbans : une plate-forme e moélsaton, Laboratore D eonome Des Transports, UMR 5593 CNRS, Bourrel E ; Moélsaton ynamque e l éoulement u traf router : u marosopque au mrosopque, Ph.D. thess, Insttut natonal es senes applquées e Lyon, Frane, Deember Lghthll MJ, Wtham GB; On knemat waves. II. A theory of traff flow on long rowe roas, Proeeng of the Royal Soety of Lonon Seres A, 955; 229(78): Kumar N; Towars pratal mplementaton of omputatonal soluton of the knemat-wave moel for smulatng traff-flow senaros, Ph.D. thess, Offe of Grauate Stues of Teas AM Unversty, Lelerq L ; Moélsaton ynamque u traf et applatons à l estmaton u brut router, Ph.D. thess, Insttut natonal es senes applquées e Lyon, Frane, Otober,