Algebraic Methods for Traffic Flow Densities Estimation

Transcription

1 BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume No Sofa 0 Prn ISSN: -970; Onlne ISSN: -08 DOI: 0.78/ca Algebrac Mehods for Traffc Flow Denses Esmaon H. Abouaïssa V. Iordanova Unversé Llle Nord France F Llle France U-Aros LGIA (EA. 96) Technoparc Fuura F-600 Béhune France CETE Nord-Pcarde Déparemen Transpor Moblés Groupe Sysèmes de Transpor rue de Bruxelles BP Llle France Emals: hassane.abouassa@unv-aros.fr Volna.Iordanova@developpemen-durable.gouv.fr Absrac: An esmaon approach ha allows recoverng of he raffc sae s proposed n hs paper. The mehod used s based on numercal dfferenaon whch does no need any negraon of dfferenal equaons and urns ou o be que robus wh respec o perurbaons and measuremens noses. Numercal smulaons carred-ou by usng he so-called Cell Transmsson Model (CTM) demonsrae he relevance of he proposed on-lne esmaon scheme. Keywords: Traffc flow cell ransmsson model algebrac mehods sae esmaon.. Inroducon Dynamc raffc managemen and conrol sysems are becomng he mos common soluons o allevae he daly problem of congeson. Indeed several sudes have confrmed he effcency of such sysems n mprovng raffc flow neworks and ensurng safe dsplacemen of goods and people. They also conrbue o polluon reducon. Traffc conrol n freeway neworks consss of he use of several acons and measuremens such as dynamc speed lms roue gudance ramp meerng [6 ]. 5

2 Mos of hese sraeges are based on he use of a macroscopc model. Neverheless he performance of any sraeges requres accurae nformaon abou he raffc sae. Such nformaon s ofen provded by a se of loop deecors. As saed n [] he loop deecor daa s frequenly ncomplee or conans bad samples. In mos cases he avalable sensors are fauly. Therefore he use of onlne sae esmaon schemes becomes necessary n order o provde he whole raffc nformaon needed. I s mporan o underlne ha he works devoed o raffc sae esmaon are few. In he frame of survellance sysems [5] and [] have developed a mehod based on me seres of speed and flow daa from a se of sensors n order o generae esmaes of vehcles accouns. These crude esmaes are hen flered usng a Kalman fler. Ths mehod has wo major drawbacks. Frsly requres large daa sorage and becomes hen mpraccable. Secondly he use of a Kalman fler s mosly adaped for lnear models whle he measuremens are a nonlnear funcon of he sae varables. [8] and [9] have consdered he problem of processng daa a a fxed spaal locaon o produce esmaes of spaal average quanes. As saed n [] wh good nal condons Nah s mehod shows he ably o esmae he densy closely n homogeneous suaons [0]. In [] an esmaon scheme was proposed based on a nonlnear swchng model. In [5] M u ñ o z e al. have used a sem-auomaed mehod based on he leassquares echnques. In [] K o h a n has nroduced a robus sldng mode observer n order o predc he freeway raffc saes such as raffc densy and velocy. Such a mehod hough robus suffers of he chaerng phenomena. Noce ha he wdely used raffc sae esmaons mehods are semmng from Kalman flerng echnques. In hs conex [] for example has proposed an Exended Kalman fler usng second order models (see also [] for a general approach o real-me freeway esmaon of boh sae and paramerc esmaons). See also e.g. [6] as a comparson sudy of several flers confguraons for freeway raffc sae esmaon). However alhough such mehods are more adaped when he se of he measuremen and model unceranes are assumed o be whe nose wh normal dsrbuons n pracce hs assumpon s vald only for raffc measuremens and he unceranes nvolved n he model equaons such as dsurbances and modelng errors canno be smply regarded as whe nose. Moreover several dffcules are sll perssen wh respec o unng (gan schedule) numercal analyss (Rcca s equaon where he precse sascs of he nose has o be que accuraely known) and sensvy o perurbaons. Oher works gven n [5] have proposed an esmaon scheme based on a parcle fler. Such a mehod was formulaed whn a Bayesan recursve framework where he raffc sae s modeled as a hybrd sochasc sysem (see e.g. [7]). The man objecve of hs paper s o deal wh raffc sae esmaon usng he recen advances on he algebrac mehods n order o provde a real-me sae esmaon of he raffc densy of freeway secons. Such an esmaon mehod whch s of algebrac characer was developed by M. Fless e al. (see [9]) and dffers from oher approaches of hs ype because nvolves he use of dfferenal Traffc densy (or occupancy) represens he basc sae varable of he mos avalable raffc flow models. 6

3 geomery. As menoned n [] he mehod s appled o oban an esmae of he dervave from any sgnal hus avodng relance on he sysem model a leas n he esmaon of saes. The proposed approach assumes ha he so-called Cell Transmsson Model (CTM) whch was developed by D a g a n z o [] models he suded sysem. The paper s organzed as follows. Secon descrbes he prncple of CTM. Secon recalls a shor background of he proposed algebrac mehod and s applcaon for densy esmaon. Some numercal smulaons are provded n Secon. Fnally he las secon presens some conclusons and nex sep nvesgaons.. Cell Transmsson Model prncples Macroscopc raffc flow models are based on he conservaon of he vehcles law ha reads: ρ () + q x = 0 where x s he longudnal poson along he freeway s he me; ρ ( x ) and q( x ) are respecvely he vehcle densy a poson x and me and he raffc flow n vehcles per hour (veh/h). In hs paper he macroscopc cell-ransmsson raffc model was seleced due o s analycal smplcy and ably o reproduce mporan raffc behavoral phenomena such as he backward propagaon of a congeson wave. Neverheless whle CTM uses he cell occupancy we use for smplcy he cell densy as sae varables and accepaons followng [] for nonunform cell lenghs. We also consder space dscree modelng. Consder he followng smple secon depced n Fg.. Fg.. Example of a freeway secon For space dscree represenaon defne he raffc densy ρ as he number of vehcles n segmen a me dvded by he segmen lengh L. The raffc volume q n veh/h s defned as he number of vehcles enerng segmen ; q s he number of vehcles leavng segmen. I should be emphaszed ha he proposed mehod does no use any probablsc or opmzaon echnques [8]. 7

4 The equaon of he nonlnear model for each freeway segmen s as follows: () ρ () = ( q () q () + a r () bp () ) L where a and b are bnary varables whch ndcae respecvely he presence or he absence of an on-ramp r ( ) and an off-ramp p ( ). The model parameers nclude v w Q max and ρ j whch are depced n he followng fundamenal dagram (Fg. ). 8 Fg.. Fundamenal dagram They can be unform over all cells or allowed o vary from a cell o a cell. The free-flow speed v s he average speed a whch he vehcles ravel down he hghway under uncongesed (low densy) condons. w s he average speed a whch congeson waves propagae upsream whn congesed (hgh densy) regons of he hghway. Q s he maxmum flow rae and ρ j s he jam densy. max ρ c he crcal densy s he densy a whch he free-flow curve Q ( ρ ) nersecs he congeson curve Q ( j ) = vρ ρ = w ρ ρ. The congeson saus of cell s deermned by comparng he cell densy wh crcal densy: f ρ < ρ c he cell has a free-flow saus oherwse ρ ρ c and he cell s sad o have a congesed saus. Three dfferen ypes of nercell connecons are allowed: smple connecon merge and dverge. In hs paper we wll focus on he frs one. As descrbed n [] q ( ) he flow enerng cell from he manlne s deermned by akng he mnmum of wo quanes: () q ( ) = mn ( S ( ) R( ) ) where S ( ) mn ( v Qmax ) by cell under free-flow condons. () = ρ s he maxmum flow ha can be suppled R = mn w ρj ρ Qmax s he maxmum flow ha can be receved by cell under congesed condons. Accordng o he saus of each cell several modes can be depced whch leads o he so-called Swchng-Mode Model (SMM) [5]. Indeed SMM s a hybrd model whch swches beween a se of lnear dfferenal equaons dependng on he congeson saus of he cells and he values of he manlne

5 boundary daa (see e.g. [5] for more explanaons and dscussons abou he SMM). As saed n [5] he SMM s a modfed verson of CTM and can be obaned by expressng he ner-cellular q as an explc funcon of cell densy q = vρ or q w( ρ j ρ) = or as a consan q = Qmax. Accordng o hese expressons a cell s called congesed-c f s no able o accep he flow delvered by s upsream neghborng cell oherwse s consdered as Free-F. In he general case he denses vecor gves he sae of he sysem: ρ= [ ρ ρ ]. N Noce ha he loop deecors locaed upsream and downsream of he suded freeway segmen gve he measuremens of he raffc flow denoed as q e and q s respecvely. In he sequel we consder ha wo adjacen cells can be n one of he followng modes: Free-Flow (FF) Congesed-Free (FC) and Congesed-Congesed (CC). The man objecve hen s o desgn an esmaor for hese dfferen suaons usng he new algebrac mehods of denfcaon. The followng secon recalls a shor background of hs approach.. Esmaon mehod.. Background on numercal dfferenaon Numercal dfferenaon s based on he algebrac seng sared by F l e s s [7] F l e s s e al. [-6] and provdes a powerful ool for he esmaon of dervaves of a nosy sgnal. In hs secon we jus skech he prncple of he mehod for more deals and neresng dscussons and comparsons he reader mgh refer o [8 ]. ( Consder a real-valued sgnal () ) y = y ( 0 ) whch for sake of = 0! smplcy s assumed o be analyc around = 0 and nroduce s runcaed Taylor expanson: N ( () ) N + y = y ( 0 ) + O ( ). = 0! Approxmae y () n he nerval ( 0 ε )0< ε β by s runcaed Taylor expanson N ( ) y N = y ( 0 ) = 0! of degree N. The usual rules of symbolc calculus n Schwarz s dsrbuons heory yeld N+ N ( N ) y = y 0 δ y δ N 9

6 where δ s he Drac measure a 0. Mulply boh sdes by ( ) and usng he () ( ) rules δ = 0 δ = δ leads o a rangular sysem of lnear equaons from whch dervaves y ( ) ( 0 ) can be obaned ( N) ( N+ (5) ) N ( N ) ( N y ( 0 )... ) N = δ y + +δy ( 0 ). ( N )! I means ha hose quanes are lnearly denfable []. The me dervave of yn ( ) he Drac measures and s dervaves are removed by negrang wh respec o me boh sdes of equaon (5) a leas v mes v > N v ( N + ) N v... v N v : τ y d d dτ = N ( N = y( 0 ) y ) ( 0 ). N! v N! v! Remark. These eraed negrals are low pass flers whch aenuae he noses whch are vewed as hghly flucuang or oscllaory phenomena (see [7] for more deals). Remark. An excellen esmaed value whch s derved va eraed me negrals may be obaned by ulzng que shor me wndows. The above formulae may easly be exended o sldng me wndows n order o oban real mes esmaes []. Remark. The same calculaons can be acheved usng he operaonal calculus (see e.g. [8 7])... Traffc denses esmaon n a FF mode In he case of a Free-Flow mode all he cells { N} are able o accep he raffc flow comng from s upsream neghborng cell. As demonsraed n [5] he observably resuls were derved usng sandard lnear sysems echnques. In he case of FF mode he suded freeway segmen s observable from he downsream measuremens of q s. Consder for smplcy s sake he followng freeway secon wh hree cells (Fg. ). In he free-flow mode he densy evoluon s Fg.. A freeway secon dvded no hree cells We consder n hs paper ha all cells are wh he same lengh L. 0

7 (6) = and assume ha L ρ = qe ρ v L ρ = ρ v ρ v L ρ = ρ v ρ v. y = ρ s an oupu varable. Some Le qs vρ smple manpulaons allow us o express he mssng denses values ( ρ and ρ ) as a funcon of y y and y (7) ρ () = y L v ρ () = y + y v v L L v v ρ () = y + + y + y. v v v v v.. Traffc denses esmaon n CF mode In hs case all he cells { N} are no able o accep he raffc flow comng from s upsream neghborng cell. Neverheless he compuaon of he observably marx allows concludng ha he freeway secon s observable from he downsream measuremens of flow q e. We assume hen ha he measured varable s y= ρ. The raffc model n he CC mode s gven as: (8) () ( j ) ( j ) () ( j ) ( j ) () ( j ) s. L ρ = w ρ ρ w ρ ρ L ρ = w ρ ρ w ρ ρ L ρ = w ρ ρ q where w = represen he wave speeds. The same calculaons performed above allow us o express he varables ρ and ρ as follows: (9) ρ () = y L w w ρ = + ρ +ρ () y y j j w w w L L w w ρ () = y + + y + y ρ j +ρ j. w w w w w w

8 .. Traffc denses esmaon n CF mode We assume ha he frs segmen s congesed whle he second and he hrd are free. The raffc model descrbng hs suaon s: L ρ () = w ( ρ j ρ) w ( ρj ρ ) (0) L ρ () = vρ vρ L ρ () = v ρ q s. The observably marx allows confrmng ha n CF mode s observable from boh upsream and downsream measuremens. Assumng ha y = ρ we oban he same resuls as n equaon (7).. Tme dervave esmaon For he generaon of me dervaves of he measured oupus y consder a h order approxmaon of a smooh sgnal y(). Thus s no necessary o desgn he dervave esmaor from a specfc dynamc model of he raffc flow. d y ( ) () = 0. d Rewrng expresson () n he operaonal doman we ge [7] () s y ( s ) s y ( 0) s y ( 0) sy ( 0) y ( 0) = 0. In order o elmnae he nal condons we ake successve dervaves wh respec o he operaonal varable s unl he number of hree s obaned: () ( s y) d = 0 ds d d d d y () s s s s y ( s) ds + 6 ds + 7 ds ds = 0. Mulplyng he above equaon () by s yelds: d y d y d y d y s s + 96s + s y ds ds ds ds s = 0. Accordng o he equvalence: s (5) Dervave sad algebrac : ( n ) n s : Mulple negrals. 0 0 d ds

9 n d ds n d d n n n n ( ) and s =. n Equaons (5) can be expressed n he me doman as follows: (6) d ( ) ( y () ) 6 y ( ) + 7( y( ) ) 96( y () ) + ( y() ) = 0. d For smplcy we have used here he followng noaon: ( j ) σ σ k j k (7) y =... σ y σ dσ... dσ. ( ) 0 0 j j j These expressons yeld afer some algebrac manpulaons he approxmaons of he frs and second order me dervaves of y ( ) : (8) dy d e ( y ) ( y ) ( y ) y = = (9) where d y d e ( y ) ( y ) ( y ) [ y ] e = = [ ] e s he esmaon value. Noce ha expresson (9) for he second order me dervave esmae requres he oucome of he evaluaon of he frs dervave esmae. Ths s n complee agreemen wh he announced rangular srucure of he generang sysem of equaons. The above formulas are vald for > 0. Snce (8) and (9) provde an approxmaed value of he frs and second dervaves hese are only vald durng a perod of me. Thus he sae esmaon mus be calculaed perodcally as follows: (0) = y 96 ( ) y + + ( ) dy d ( ) + 7 y + y e =

10 () where 0 ( ) = y 96 ( ) y + + ( ) d y = d ( ( ) y) ( ) [ y ] e > s he esmaon perod. e 5. Smulaon resuls In order o llusrae he relevance of he proposed approach consder he freeway secon depced n Fg.. For he smulaon purpose we consder he followng raffc demand n vehcles per hours (Fg. ). Fg.. Traffc demand Table summarzes he used model parameers. Table. Model parameers Parameers Free-flow speed (m/s) Wave speed (m/s) Crcal densy (veh/m) Jam densy (veh/m) Segmen Segmen Segmen Fgs 5 and 6 show he raffc denses me evoluon when all he cells are n he free-flow suaon. A very shor me s needed for he developed esmaor n order o reach he smulaed raffc denses.

11 Fg. 5. Traffc denses evoluon of he second cell: FF mode Fg. 6. Traffc denses evoluon of he frs cell: FF mode Fg. 7 confrms he relevance of he algebrac esmaon echnques for boh CC and CF modes. Fg. 7. Traffc denses evoluon of he cells: CC and CF modes 5

12 6. Concluson The work presened n hs paper demonsraes he relevance of he recenly nroduced denfcaon mehods for raffc sae esmaon. Such echnques whch are of algebrac characer do no use eher opmzaon nor probablsc or asympoc echnques and lead o robus and fas raffc esmaon. The proposed approach was successfully appled for he denses esmaon of a freeway secon modelled by usng he so-called Swchng Mode Model (SMM). I allows he esmaon of he raffc denses accordng o he mode of he freeway.e. FF CF and CC modes. Furher works wll be focused on a comparave sudy beween he proposed approach and he classcal one and he more frequenly used n he raffc area sae esmaors such as he Exended Kalman Fler (EKF). Oher research works wll be realzed n order o explo he approach proposed for dynamc raffc managemen and conrol measuremens desgn such as ramp meerng dynamc speed lms and dynamc raffc roung. Acknowledgemen: Ths paper s parly suppored by FP7 projec 6087 ACOMIN Advance compung and nnovaon. References. D a g a n z o C. F. The Cell Transmsson Model: A Dynamc Represenaon of Hghway Traffc Conssen wh Hydrodynamc Theory. Transporaon Research B Vol F l e s s M. H. S r a-r a m í r e z. An Algebrac Framework for Lnear Idenfcaon. ESAIM Conrol Opmz. Calculus Vara. Vol F l e s s M. H. S r a-r a m í r e z. Reconsruceurs d éa. C. R. Acad. Sc. Pars ser. I F l e s s M. C. J o n H. S r a-r a m í r e z. Robus Resdual Generaon for Lnear Faul Dagnoss: An Algebrac Seng wh Examples. Inerna. J. Conrol Vol Fless M. C. Jon H. Mouner. An Inroducon o Nonlnear Faul Dagnoss wh an Applcaon o a Congesed Inerne Rouer. In: Lecure Noe Advances n Communcaon and Conrol Neworks Sprnger Vol F l e s s M. C. J o n H. S r a-r a m í r e z. Closed-Loop Faul Toleran Conrol for Unceran Nonlnear Sysems n Conrol and Observer Desgn for Nonlnear Fne and Infne Dmensonal Sysems. In: Lecure Noes n Conrol Informa. Sc. T. Meurer K. Grachen E. D. Glles Eds. Vol F l e s s M. Analyse Non Sandard du Bru. C. R. Acad. Sc. Pars Ser. I Vol F l e s s M. C. J o n H. S r a-r a m í r e z. Non-Lnear Esmaon s Easy. Inernaonal Journal of Modellng Idenfcaon and Conrol Vol. 008 No G a z s D. C. C. H. K n a p p. On-Lne Esmaon of Traffc Denses from Tmes-Seres of Flow and Speed Daa. Transporaon Scence 97 No H e g y A. D. G r m o n e R. B a b u s ÿ k a B. D e S c h u e r. A Comparson of Fler Confguraons for Freeway Traffc Sae Esmaon. In: Proc. of IEEE Conf. ITSC Sepember 006 Canada.. K o s a l o s A. M. P a p a g e o r g o u. Moorway Nework Traffc Conrol Sysems 00. European Journal of Operaonal Research Vol

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