SECOND ORDER TRAFFIC FLOW MODELING: SUPPLY- DEMAND ANALYSIS OF THE INHOMOGENEOUS RIEMANN PROBLEM AND OF BOUNDARY CONDITIONS

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1 Advancd OR and AI Mhods in Tanspoaion SECOND ORDER TRAFFIC FLOW MODELING: SUPPLY- DEMAND ANALYSIS OF THE INHOMOGENEOUS RIEMANN PROBLEM AND OF BOUNDARY CONDITIONS Jan-Paick LEBACQUE 1, Habib HAJ SALEM 2, Sai MAMMAR 3 Absac. Rcny Aw, Rasc and Zhang inoducd a scond od od (ARZ) ha dos no xhibi h usua dawbacks of his faiy of ods, i.. ngaiv vociis and/o dnsiis. In his pap w anayz h inhoogous Riann pob fo his od, which is shown o b uivan o h inhoogous Riann pob fo a ad fis od od wih odifid uiibiu fow dnsiy aionship. Th bounday condiions fo h ARZ od a dducd. Thy can b xpssd in s of an upsa dand and downsa suppy. 1. Inoducion Th Payn-Wiha scond od affic fow od [1], a hypboic sys wih axaion, has had uch succss wih appicaions. Nvhss his od xhibis known icincis. Noaby i has bn shown ha his od can xhibi ngaiv spds [2], a vioaion of h anisoopic chaac of h affic, du o h fac ha infoaion can av fas han cas in his od and hus ovak cas. In od o ovco his icincy, Aw and Rasc [3], and indpndny Zhang [4] poposd a nw scond od od cad ARZ in h pap, basd on vy diffn aionas. Th objc of his pap is o anayz h inhoogous Riann pob fo his od. Th ais a h foowing: Div coc fouas fo h Godunov sch appid o h discizaion of his od, Dfin pop bounday condiions, Dfin consisn inscion ods. 1 bacu@ins.f 2 haj-sa@ins.f 3 INRETS/GRETIA, 2 av du G Ma-Joinvi F Acui Fanc, saa@ins.f

2 Scond od affic fow oding: suppy-dand anaysis 19 In h cas of h fis od LWR (Lighhi-Whiha-Richads) affic fow od [5], [6] his poga has bn caid ou succssfuy [7], [8], [9], and i has bn shown ha h concp of oca suppy and dand of affic is h fundana hoica oo fo his poga. In his pap w gnaiz hs idas o h AWR od. Th ouin of h pap is h foowing. Fis w dscib h AWR od and ca so basic facs concning wavs in h AWR od. Thn w poin ou h invaianc popis of h aiv spd (a Lax-Riann invaian of h od). I foows ha h inhoognous Riann pob fo h AWR od can b sovd as an inhoognous Riann pob fo a LWR od wih suiaby odifid fundana diaga and ighhand-sid condiions. Thus w can ca upon h souion of h inhoognous Riann pob fo h LWR od [7] o sov h pob fo h ARZ od. Finay w sa h suing bounday condiions fo h ARZ od on a ink. Ths bounday condiions a xpssd in s of suiab odifid suppy and dand funcions. 2. Th ARZ od Th ARZ od can b sad as: (consvaiv fo.), wih x, : h posiion and i ( v ) = + x (1a) y + x 1 p = y τ, x : h paia divaiv wih spc o i and spac vaiabs (x, : h dnsiy a i and ocaion x v (x,: h spd a i and ocaion x ( x, ( x, v( x, = : h fow a i and ocaion x y( x, ( x, ( v( x, V ( ( x,, x) ) = ( x, Q ( ( x,, x) = : h aiv fow, i.. h diffnc bwn h acua fow and h uiibiu fow, Q (, x) V (, x) : h uiibiu fow (a ocaion x), i. h fundana diaga, : h uiibiu spd (a ocaion x), p( x, ( x, ( v( x, V ( ( x,, x) ) = v( x, y( x, = : h fux of aiv fow, τ : a axaion i consan. L us aso in h aiv spd I (x, a i and ocaion x : = ( x, v( x, V ( ( x,, x) (1b) I (2) No ha y = I and ha p = v y= I. Th aiv spd I saisfis h foowing uaion, as a su of (1a) and (1b):

3 11 J.-P. Lbacu a. I& 1 = I + v xi = I (3) τ which xpsss ha h aiv spd I dcass xponniay aong ajcois. In h s of h pap w sha focus on h od wihou axaion, i.. h aiv spd I is consan aong ajcois, and (1b) ads: y + x p = (4) Th fundana diaga and h uiibiu spd dnsiy aionship a dpicd bow: Figu 1 : uiibiu aionships 3. Wavs Th aia of his scion is fuy dscibd in a copanion pap. L us ca h ain sus. Th sys of hypboic consvaion aws (1a), (4) can b xpssd as: wih U and F ( U) ( U ) = U + xf (5) ( ) Q + y = = y y ( Q ( ) ) + y Th ignvaus of A( U ) = F( U ) a: λ ( v) = v + V '( ) and λ v) = v, (6) 1 2 In h (,v) pan, wo affic sas U and U a conncd by a 1-wav if and ony if hi aiv spd is idnica, i.. I U = I U v V = v V (7) ( ) ( ) ( ) ( ) Fuh, wo sas U and U conncd by a 1-wav and such ha U is upsa of U a conncd by a 1-shockwav if v v o and hy a conncd by a 1-afacion wav shockwav if v o. (Lax nopic condiion). v

4 Scond od affic fow oding: suppy-dand anaysis 111 Two affic sas U and U a conncd by a 2-wav if hi aiv spd is idnica, v= v (8) 2-wavs a conac disconinuiis which popaga a h spd of h affic fow. 4. Dynaics of h aiv spd I a. Foowing fo (3), if h axaion pocss is ngcd, I & = I + v I = (9) and I is consan aong vhic ajcois (ins x ( = v( x(, & ). I us b nod ha (9) hods ony wh h a no disconinuiis of spd and/o dnsiy. b. Foowing h sus of scion 3, I is consvd hough 1-wavs. c. L us consid a fundana diaga disconinuous wih spc o posiion x, a say x = x. Fo any givn uaniy B dpnding picwis coninuousy on h posiion, h sybo [B] dnos h diffnc bwn h vaus of B on h f and igh sid of h disconinuiy: [ ] B( x + ) B( ) B = x. Th vociy of popagaion of h disconinuiy a x is. I foows fo h Rankin- Hugonio foua ha: = [ p] [ y] Fo [] = i foows ha h affic fow is consvd hough h disconinuiy. Thus and [I] = : I is consvd a x. = x [ ] [ ] [ p ] = ( x )[ I] = Suay: I is consvd aong vhic ajcois, hough 1-wavs and acoss fixd disconinuiis. 5. Inhoognous Riann pob a. Th inhoognous Riann is ind as foows: Th pob is ind on a who in Th iniia daau ( x) = U( x, is consan on ach haf-in x <, x > : = U U ( x) = U if x < ( x) = U if x > (1) Th fundana diaga is assud picwis coninuous wih spc o h spac vaiab x, wih a sing disconinuiy a x = : x) = Q, ( ) if x < x) = Q ( ) if x > Q and Q, V V x) = V, ( ) if x < x) = V ( ) if x >,

5 112 J.-P. Lbacu a. Th souion is assud sf-siia. Th souion of his pob is ind by scos in which h affic sa is consan and unifo, spaad by wavs. U uas U in sco (T). As is shown in h copanion pap, in h hoognous cas h a wo wavs, a sow 1-wav and a fas 2-wav. In h inhoognous cas h 1-wav ay spi ino wo 1-wavs, on wih ngaiv spd and on wih posiiv spd. Th phnonon is siia in h LWR od, xcp ha in h a, ony on yp of wavs occu. b. In viw of h sus of scion 4, h aiv spd on h f-hand sid of h disconinuiy, I( U ) I =, is popagad acoss h disconinuiy. L us in h affic sa U by h foowing condiions: ( U ) = I v = v( U ) = v v( U ) I = I, = By sf-siiaiy of h souion of h Riann pob, h affic sa U popagas a spd v aong a 2-wav (W ) and is givn by: v = v = V 1, ( v I ) (11) Figu 2: odifid fundana diaga, suppy and dand

6 Scond od affic fow oding: suppy-dand anaysis 113 Th Riann pob can b sicd o sco (S) boundd by h haf-in x < and h 2-wav (W ), wih iniia and bounday condiions U and U. Now (4) ducs o I= I on (S), foowing h sus of scion 4 and appying h iniion of U. Thfo and (1a) ducs o: ( x) + I and = Q( x) + I v = V,, (12) + x ( x) + I ) = Q (13) (13) is a LWR-ik od, wih a odifid fundana diaga Q x) + I (s figu 2 abov, wih I > ), and h souion of (13) yids h dnsiy, h souion of (12) h spd in sco (S). c. Th hodoogy fo soving (13) has bn dvopd in [7], and is basd on h oca dand and suppy concp. Cucia fo h divaion of h fu souion a h fow a h oigin and h dnsiis and, on h f- and igh- sid of h oigin., Figu 3: Exap of a souion o h Riann pob, wih suppy < dand In od o appy h hod, i is ncssay o in h suppy and dand associad o h odifid fundana diaga (s figu 2 abov, wih I < ). Fis w in h ciica dnsiy and h axiu fow fo h odifid fundana diaga: ax,* [ +I ] fo * [ ] fo * =, ( I ) = Ag Max Q,* ( ) ( I ) = Max Q ( ) + I ci,*,* =, Thn h dand and suppy funcions a ind fo h odifid fundana diaga: I ) + I if ci,* ( I ) ( I ) if ( I ) Q,*,* I ) = fo * =, (14a) ax,* ci,*,*( I ) if ci,*( I ) I ) + I if ( I ) ax Σ,* I ) = fo * =, (14b) Q,* ci,* Th dand on h f and igh sid of h oigin foow:,,, ( I ), Σ = Σ (, I ) = (14c) and h fow a h oigin fo a is > is obaind by h usua Min foua [ Σ ] = Min, (15)

7 114 J.-P. Lbacu a. No ha h downsa suppy Σ dpnds on h upsa aiv spd I, a popy fundanay a odds wih h suppy in h LWR od. Finay, h dnsiis f and igh of h oigin a a is > a dducd:,, if = 1 Σ, = 1, (, I ) (, I ) if = if if = Σ Σ = Σ = Σ (16a) (16b) 6. Concusion: bounday condiions and Godunov sch Upsa bounday condiions fo a ink a h upsa odifid dand, foowing (14c), and h upsa aiv spd. Th upsa aiv spd and h ink dnsiy yid h ink suppy, by (11), (14b), (14c). Th ink infow is hn givn by (15). Th dnsiy a h nanc of h ink sus fo (16b). Downsa condiions fo a ink a h downsa vociy; and h downsa fundana diaga. By (11) h downsa sa U can b dducd, and fo (14b), and h downsa suppy sus. Th ink oufow is hn givn by (15); h ink dand is saighfowad. Th Godunov sch [7], is a discizaion sch fo nwok affic siuaion, which is consvaiv and basd on a picwis consan appoxiaion of h affic sa. In od o appy h sch, ony fows fo on c o h nx a uid, hy su fo (14) and (15). Rfncs [1] Payn, H.J., Mods of Fway Taffic and Cono. Siuaion Councis Poc. S. Mah. Mods Pubic Sys. 28(1), 51-61, [2] Daganzo, C.F. Rui fo scond od fuid appoxiaion o affic fow. Tanspoaion Rsach Pa B, 29B, 4, , [3] A. Aw and M. Rasc. Rsucion of Scond Od Mods of Taffic fow. SIAM J. App. Mah., 6(3): , 2. [4] Zhang, H.M. A Non-Euiibiu Taffic Mod Dvoid of Gas-Lik Bhavio. Tanspoaion Rsach Pa B, Vo 36, p , 22. [5] Lighhi M.H., Whiha G.B., On kinaic wavs II: A hoy of affic fow on ong cowdd oads. Poc. Roya Soc. (London) A 229: [6] Richads P.I., Shock-wavs on h highway. Op. Rs. 4:

8 Scond od affic fow oding: suppy-dand anaysis 115 [7] Lbacu J.P., Th Godunov sch and wha i ans fo fis od affic fow ods. In: Tanspoaion and Taffic Thoy, pocding of h 13 h ISTTT (J.B. Lso d.) , Esvi [8] Lbacu, J.P., Khoshyaan, M.M. Macoscopic fow ods. In "Tanspoaion panning: h sa of h a'' (Edios: M. Paiksson, M. Labbé), Kuw Acadic Pss. 22. [9] Lbacu J.P., Khoshyaan M.M. Fis od acoscopic affic fow ods: inscion oding, nwok oding. Accpd fo pubicaion in h Pocdings of h 16 h ISTTT (Innaiona Syposiu on Tanspoaion and Taffic Thoy). 25.