Modelling traffic flow with constant speed using the Galerkin finite element method

Transcription

1 Modelling raffic flow wih consan speed using he Galerin finie elemen mehod Wesley Ceulemans, Magd A. Wahab, Kur De Prof and Geer Wes Absrac A macroscopic level, raffic can be described as a coninuum flow. Lighhill Wiham and Richards (LWR) have developed a raffic flow model based on he fluid dynamics coninuiy equaion, which is nown as he firs order LWR raffic flow model. The resuling firs order parial differenial equaion (PDE) can be analyically solved for some special cases, given iniial and boundary condiions, and numerically using for eample he finie elemen mehod (FEM). This paper maes use of he Galerin FEM o solve he LWR model wih consan speed. The road is divided ino a number of road segmens (elemens) using he Galerin FEM. Each elemen consiss of wo nodes. Each node has one degree of freedom (d.o.f.), namely he raffic densiy. The FEM provides a soluion for he degrees of freedom, i.e. raffic densiies of each node. The resuling simulaneous equaions are solved a differen ime seps using he Euler bacward ime-inegraion algorihm. In Belgium and also in he Neherlands, here is a special echnique ha can be used in order o preven raffic jams and increasing safey in siuaions wih high volume of cars on he roads, i.e. bloc driving. I is a echnique where cars drive in groups by order of he police when he roads are crowded. In his paper bloc driving is used as a pracical eample of he LWR model wih consan speed. Thereby, i is simulaed using he Galerin FEM and he resuls are compared wih he analyical soluion. The FEM gives good resuls providing ha: he road segmens and ime seps are small enough. A road wih lengh 5 m, consan speed of 25 m/s, segmen lengh of m and ime seps of s gives good resuls for he sudied case. A poins of raffic densiy rae disconinuiies, depending on he road segmen size and ime sep size, he Galerin FEM is accurae and requires reasonable compuaional effor. From he research wor carried ou in his paper, i is found ha he Galerin FEM is suiable for modelling raffic flow a macroscopic level. The elemen size and ime sep size are imporan parameers in deermining he convergence of he soluion in case of disconinuiies in raffic densiy rae. Alhough his paper considers he case of consan speed, he echnique can be eended in he fuure o include he case of non-consan speed, i.e. speed as a funcion of raffic densiy. Inde Terms Macroscopic raffic flow, Galerin finie elemen mehod, LWR model, bloc driving. Manuscrip received March 23, 29. This wor was suppored by Hassel Universiy (Transporaion Research Insiue, Faculy of Applied Economics) and XIOS Universiy College Limburg (Deparmen of Indusrial Sciences and Technology). Wesley Ceulemans ( wesley.ceulemans@ios.be), Magd A. Wahab (corresponding auhor, phone: ; fa: ; magd.wahab@ios.be) and Kur De Prof ( ur.deprof@ios.be) are wih he Deparmen of Indusrial Sciences and Technology, XIOS Universiy College Limburg, Universiaire Campus Agoralaan Gebouw H, B-359 Diepenbee, Belgium. Geer Wes ( geer.wes@uhassel.be) is wih he Transporaion Research Insiue, Faculy of Applied Economics, Hassel Universiy, Weenschapspar 5 bus 6, B-359 Diepenbee, Belgium. I. INTRODUCTION There has been a lile bi research done in he lieraure concerned wih macroscopic raffic flow modelling using he Galerin finie elemen mehod (FEM). In [], [2], and [3] a Galerin FEM ype is used o solve he macroscopic Lighhill Wiham and Richards (LWR) [4], [5] raffic flow model in conjuncion wih Greenshields flow-densiy relaionship [6]. A wavele-galerin FEM is used in [7] o solve he macroscopic non-consan speed LWR raffic model. A disconinuous Galerin FEM is used in [8] for solving red-and-green ligh models for he raffic flow. This paper presens a numerical soluion (using a Galerin FEM [9]) of he LWR raffic flow model wih consan speed. The validaion of resuls is done by using he analyical mehod of characerisics []. The firs secion describes he mos imporan pars of he macroscopic raffic flow heory. The LWR model is presened in deails. A descripion of he analyical soluion (using he mehod of characerisics) and he numerical soluion (using he Galerin FEM in combinaion wih he Euler bacward ime-inegraion algorihm [9]) can be found in he ne secions. The las secion provides a numerical eample, namely a bloc driving simulaion. II. TRAFFIC FLOW THEORY There are several raffic flow models, which can be mosly divided in 4 caegories: macroscopic, mesoscopic, microscopic and submicroscopic (lised wih growing level of deails). The firs order LWR raffic flow model is a macroscopic coninuum model. A. Macroscopic raffic flow parameers The mos imporan macroscopic raffic flow parameers are: Densiy (): epressed in vehicles per ilomere (veh/m) Flow (q): epressed in vehicles per hour (veh/h) Speed (u): epressed in ilomeres per hour (m/h) B. Fundamenal relaion The unique relaion beween he hree macroscopic raffic flow parameers densiy, flow and speed is: q = u () C. Fundamenal diagrams Beside he fundamenal relaion, here are also eperimenal relaions beween he raffic flow parameers.

2 u u f u c q q c u u f u c Free-flow raffic Free-flow raffic Free-flow raffic Jammed raffic c Capaciy-flow raffic Jammed raffic Figure : The fundamenal u- diagram according o Greenshields c Capaciy-flow raffic Jammed raffic j Figure 2: The fundamenal q- diagram according o Greenshields Capaciy-flow raffic q c q Figure 3: The fundamenal u-q diagram according o Greenshields These relaionships are called fundamenal diagrams. They are obained from measuremens. The fundamenal diagrams are: Speed-densiy relaionship: u = uf ( ) (2) j Speed-flow relaionship: q = uf ( ) (3) j Flow-densiy relaionship: u q = j u ( ) (4) u f Fig., 2, and 3 give graphical overviews of he fundamenal diagrams according o Greenshields [6]. Traffic can have differen regimes (characerized by j variables relaed o he raffic sae): Free-flow raffic is characerized by a low densiy (high speed), which resuls in a free-flow speed u f. Mosly u f is he maimum allowed speed. Capaciy-flow raffic is characerized by a maimum flow which is called he capaciy flow q c. Jammed raffic is characerized by a maimum densiy (low or no speed) called he jam densiy j. In pracice, ransiions occur in ime from one regime o anoher. D. Firs order LWR model Lighhill, Wiham and Richards considered ha raffic was an inviscid bu compressible fluid (fluid-dynamic model). Densiies, speed values and flows were defined as coninuous variables in each poin in ime and space (coninuum, macroscopic model). The firs order parial differenial equaion (PDE) from his model is: q + = (5) Crucial o he approach of Lighhill, Wiham and Richards was he fundamenal hypohesis (see secion II.B), i.e. flow is a funcion of densiy and speed: ( u) + = (6) Lighhill and Wiham assumed ha he fundamenal hypohesis holds a all raffic densiies, no jus for ligh-densiy raffic bu also for congesed raffic condiions. Using he fundamenal diagrams (see secion II.C) o relae he wo dependen variables in he lef-hand side of (5) (densiy and flow q) o one anoher, i is possible o solve he parial differenial equaion, given iniial and boundary condiions. Equaion (6) can be made simpler by assuming a consan speed. In his paper a consan speed is assumed, i.e. u = u : ( u ) + = (7) Therefore, (7) becomes + u = (8) I is worh menioning ha Lighhill, Wiham and Richards noed ha because of he coninuiy assumpion, he heory only holds for a large number of vehicles (long crowded roads). III. ANALYTICAL SOLUTION: METHOD OF CHARACTERISTICS Equaion (8) is a firs order parial differenial equaion (more specific: he firs order wave equaion wih speed u ). The mehod of characerisics [] can be used o find a soluion for he iniial boundary value problem. An iniial boundary value problem assumes (beside he differenial equaion) wo era equaions (coninuous or disconinuous): Iniial values: he densiy values a ime = ( (,) = f( )) Boundary values: he densiy values a disance = ( (, ) = g ( ))

3 The general form of he soluion from he firs order parial differenial equaion (8) wih consan speed u, densiy, iniial condiion (,) = f( ) and boundary condiion (, ) = g ( ) is: f( u ) u (,) = g ( ) u u IV. NUMERICAL SOLUTION: GALERKIN FEM Applicaion of he Galerin FEM o (8) gives: δ w ( + u ) dv = V (9) () Galerin ses he weigh funcion δ w equal o he shape funcion vecor [ N ] : V N ( + u ) dv = [ ] () where V is he elemen volume. For a linear homogeneous elemen dv can be replaced by d and he inegraion can be done over : N ( + u ) d = [ ] (2) A. Single elemen marices This secion gives he derivaion of he marices of a single elemen wih lengh L and wo nodal densiies and 2. Fig. 4 gives a graphical represenaion of a single raffic flow elemen. The raffic densiy in he elemen as a funcion of he nodal densiies, degree of freedom (d.o.f.), is given by = N + N2 2 = [ N N2] (3) 2 The differeniaion of he densiy wih respec o he ime is given by 2 = N + N2 = [ N N2] (4) 2 The differeniaion of he densiy wih respec o he posiion is given by N N2 N N2 = + 2 = (5) 2 The shape funcions N and N2 are [ ] [ 2] N = N N = (6) L L 2 Subsiuing (3), (4), (5) in (2) gives: N ( [ 2] N N ) d N2 2 (7) N N N 2 + ( u ) d = N2 2 Simplifying he noaion of (7) gives: ([ ] [ ] { T } T N N N ) d + ( u [ N ] { }) = d (8) Afer derivaion and inegraion of he shape funcions, (8) becomes: L 2 u { } 6 + = 2 2 (9) Because of he ime-dependency, here is a need for a ime-inegraion algorihm. Applicaion of he Euler bacward ime-inegraion algorihm [9] + = resuls in: L 2 L u 2 Re-wriing (2) gives: where + { } { } + { } + = + [ ] + [ ] { } = [ ] { } (2) A B A (2) L 2 = 6 2 (22) B = 2 (23) [ A] u [ ] + = { } = 2 { } B. General soluion for m ime seps and n elemens BC L IC n + (24) (25) L m Figure 5: Problem definiion for n finie elemens and m ime seps Figure 4: A single finie elemen

4 Given a road, which is divided in n elemens wih uniform lengh L, iniial condiions, a = s, o n+ (Fig. 5) and boundary condiions, a = m, o m (Fig. 5) hen, he densiies a all imes and all posiions can be calculaed from: where + { } = [ ] + [ ] [ ] { } [ A] A B A (26) 2 4 L = (27) v B = (28) 2 [ ] V. SIMULATION OF BLOCK DRIVING By choosing a pracical raffic eample, bloc driving, he Galerin FEM soluion mehod is esed on accuracy, convergence and compuaional needs. Bloc driving is a echnique where cars drive in groups by order of he police, in order o preven jams. When he raffic densiy of he road is high, a car of he police drives wih a consan speed in he cenre of he road. The vehicles drive behind he police car wih he same consan speed. Because of his, acceleraion and braing (he harmonica effec) are avoided. The chance of accidens reduces and raffic jams become shorer in disance. In his secion bloc driving is simulaed wih he Galerin FEM. I should be noed ha all densiies are epressed in vehicles per meer (veh/m), disances in meers (m) and imes in seconds (s). Figure 6: Boundary condiions a he beginning of he road bloc driving A. Deails of he simulaion The simulaion is described in 5 ime inervals (Fig. 6): Case : no vehicles eners/on he road during 24 seconds. Case 2: 6 vehicles ener he road (a vehicle per second driving a 25 m/s) during 6 seconds. This case is responsible for he creaion of he firs raffic wave. Case 3: no vehicles ener he road. The 6 vehicles drive on he road and leave he road whereas he road becomes clear. The ime beween enering and leaving he road from one car is 2 s (ravelling 5 m wih a speed of 25 m/s). This case simulaes he evoluion of he firs raffic wave. Case 4: repeaing case 2. This case is responsible for he creaion of raffic wave 2. Case 5: repeaing case 3. This case simulaes he evoluion of raffic wave 2. B. Mahemaical ranslaion The LWR model wih consan speed is given by (8). All simulaions are done from s o 84 s and he road lengh L is 5 m. The consan speed u is 25 m/s. Case conains an iniial condiion. The densiy a = s is given by: (,) = 5 (29) Cases, 2, 3, 4, and 5 conain boundary condiions. The densiy a = m (he beginning of he road) is given by: < 3 (, ) = 3 < 54 (3).4 54 < 6 6 < 84 Fig. 6 gives a graphical represenaion of he boundary condiions. C. Analyical soluion using he mehod of characerisics The analyical soluion wih he mehod of characerisics (see secion III) is given by he following equaion: u 5 f( u ) u 24 g ( u).4 24 < u 3 g ( u) (,) = (3) 3 < u 54 g ( u).4 54 < u 6 g ( u) 6 < u 84 g ( u) D. Numerical soluion wih he Galerin FEM The densiy is calculaed for all imes ( o 84 s) and disances ( o 5 m) by using (26) and applying he iniial and boundary condiions. The convergence sudy on he densiy is done wih simulaions wih differen elemen sizes and ime seps. The densiy versus ime a disance 2 m is fully analysed by using convergence parameers and CPU calculaion imes. The evoluion of densiy in disance (a disances m, 25 m and 5 m) and ime (a imes 57 s, 67 s and 77 s) is

5 invesigaed wih he use of a simulaion wih elemen size m and ime seps of s. ) Convergence sudy on densiy The analyical soluion resuls in disconinuous raffic bloc waves. The numerical soluion resuls in coninuous rounded raffic waves. The disconinuiies become approimaed by a coninuous funcion using he Galerin FEM. The CPU imes of he numerical simulaions are presened in Table I. Table I: CPU imes for he numerical simulaions of bloc driving Δ (s) Δ (m) Simulaion ime (s) The accuracy parameers (mean error, sandard deviaion of he error and oal error) of he numerical simulaions are presened in Table II. The higher he oal error, mean error or sandard deviaion (sd), he less accurae he resuls. The error resuls are calculaed via he difference beween analyical and numerical values. Table II: Accuracy parameers for he densiy versus ime a disance 2 m Δ (s) Δ (m) Mean error a 2 m (veh/m) Sd error a 2 m (veh/m) Toal error a 2 m (veh/m) Figure 7: Convergence of densiy as funcion of Δ bloc driving Figure 8: Convergence of densiy as funcion of Δ bloc driving Fig. 7 gives an overview of he densiy versus ime a disance 2 m for simulaions wih = s and = m, 25 m, and m. Simulaions wih = s and = m or m gives good resuls (boh a small mean error of.2 veh/m wih a small sandard deviaion of.46 veh/m and.45 veh/m), bu compuaionally hese simulaions are very inensive (high CPU ime of s and s). This simulaions are no presened in Fig. 7 because of he same graphical resuls as wih = s and = m. Simulaions wih = s and = m gives good resuls (a small mean error of.22 veh/m wih a small sandard deviaion of.46 veh/m). These simulaions are compuaionally fas (low CPU ime of 2.67 s). Simulaions wih = s and = 25 m gives also good resuls (a small mean error of.23 veh/m wih a small sandard deviaion of.48 veh/m) bu hose are less accurae hen simulaions wih = m. Simulaions wih = s and = m gives bad resuls (a large mean error of.57 veh/m wih a large sandard deviaion of.73 veh/m). The effecs of a larger segmen lengh are: Broader raffic wave Lower maimum densiies Earlier sar of raffic wave Fig. 8 gives an overview of he densiy versus ime a disance 2 m for simulaions wih = m and = s, s, and 3 s. Simulaions wih = m and =. s gives good resuls, bu compuaionally hese simulaions are very inensive (more han s CPU ime). This simulaion is no presened in Fig. 8 because of he same graphical resuls as wih = m and = s. Simulaions wih = m and = s gives good resuls (a small mean error of.22 veh/m wih a small sandard deviaion of.46 veh/m) and hey are compuaionally no inensive (low CPU ime of 2.76 s). Simulaions wih = m and = s or 3 s gives bad resuls (boh a large mean error of.64 veh/m and.72 veh/m wih a large sandard deviaion of.85 veh/m and.98 veh/m). The effecs of a larger ime sep size are: Broader raffic wave Lower maimum densiies Laer sar of raffic wave

6 Simulaions wih segmen lenghs of m and ime seps of s give bes resuls complying wih compuaion power and accuracy. This resuls in he following convergence crieria: L 5 s 2) Evoluion of densiy in disance and ime Fig. 9 gives a graphical represenaion of he evoluion of he densiy in disance a = m, 25 m, and 5 m. The movemen of raffic waves and 2 is clearly visible in Fig. 9. The analyical and numerical resuls are he same for = m because of he boundary condiion. Since raffic waves are ravelling wih a consan speed of 25 m/s, he maimum densiy a disance 25 m and ime 37 s (poin A) becomes a disance 5 m a ime 47 s (poin B). Fig. gives a graphical represenaion of he evoluion of he densiy in ime a = 57 s, 67 s, and 77 s. This is a graphical represenaion of he movemen from raffic wave 2 over he road wih lengh 5 m. Since he raffic wave is ravelling wih a consan speed of 25 m/s, a densiy a ime 57 s and disance m (poin A) becomes a ime 67 s a disance 25 m (poin B) and a 77 s a disance 5 m (poin C). Figure 9: Evoluion of densiy in disance bloc driving VI. CONCLUSIONS AND FUTURE WORK The Galerin FEM can be used o solve he firs-order macroscopic LWR raffic flow model wih consan speed. The densiy, flow and speed values are calculaed in each poin on he road, a any ime. The resuls of he Galerin finie elemen analysis are compared wih ha of he analyical mehod of characerisics. By he use of an analyical and numerical echnique, bloc driving is simulaed. A simulaion wih road lengh 5 m, consan speed of 25 m/s, segmen lenghs of m and ime seps of s resuls in accurae and fas numerical resuls. The difficulies wih he numerical simulaions appear a he disconinuiies. This can be prevened by choosing he elemen size and ime seps small enough. Using larger segmen lenghs and/or ime seps can give inaccurae resuls. Using very small segmen lenghs and/or ime seps can resul in inensive simulaions. Fuure research will concenrae on he applicaion of he Galerin FEM o he LWR model wih non-consan speed. In such a case, he speed is a funcion of he densiy. The LWR model can also be eended o include social forces and resisances. REFERENCES [] D. E. Besos and P. G. Michalopoulos. (984, May-June). An applicaion of he finie elemen mehod in raffic signal analysis. Mechanics Research Communicaions. (3). pp [2] D. E. Besos, P. G. Michalopoulos and J. K. Lin. (985, Ocober). Analysis of raffic flow by he finie elemen mehod. Applied Mahemaical Modelling. 9(5). pp [3] I. Ouani, D.E. Besos and P.G. Michalopoulos. (986, June). Finie elemen analysis of freeway dynamics. Engineering Analysis. 3(2). pp [4] M. J. Lighhill and G. B. Whiham. (955, May). On inemaic waves: II. A heory of raffic flow on long crowded roads. Proceedings of he Royal Sociey of London (Series A). 229(78). pp [5] P. I. Richards. (956, February). Shoc waves on he Highway. Operaions Research. 4(). pp [6] B. D. Greenshields, J. R. Bibbins, W. S. Channing and H. H. Miller. (934). A sudy of raffic capaciy. Highway Research Board Proceedings. 4. pp [7] G. C. K. Wong and S. C. Wong. (2, March). A wavele-galerin mehod for he inemaic wave model of raffic flow. Communicaions in Numerical Mehods in Engineering. 6(2). pp [8] R. Liu, H. Li and Z. Wang. (2, March). The disconinuous finie elemen mehod for red-and-green ligh models for he raffic flow. Mahemaics and Compuers in Simulaion. 56(). pp [9] R. D. Coo, D. S. Malus and M. E. Plesha. Conceps and applicaions of finie elemen analysis. Canada, CA: Wiley, 989, ch. 5. [] P. DuChaeau and D. Zachmann. Applied Parial Differenial Equaions. CA: Dover publicaions, 22, ch. 7. [] F. L. Hall, H. M. Zhang, R. Kuhne and P. Michalopoulos. Traffic flow heory A Sae-of-he-Ar Repor. CA: Transporaion Research Board, 2, ch. 2 and ch. 5. Figure : Evoluion of densiy in ime bloc driving