Modelling Traffic Flow with Constant Speed using the Galerkin Finite Element Method

Transcription

1 Proceedings of he World Congress on Engineering 29 Vol II WCE 29, Jly - 3, 29, London, U.K. Modelling Traffic Flow wih Consan Speed sing he Galerin Finie Elemen Mehod Wesley Celemans, Magd A. Wahab, Kr De Prof and Geer Wes Absrac A macroscopic level, raffic can be described as a coninm flow. Lighhill Wiham and Richards (LWR) have developed a raffic flow model based on he flid dynamics coniniy eqaion, which is nown as he firs order LWR raffic flow model. The resling firs order parial differenial eqaion (PDE) can be analyically solved for some special cases, given iniial and bondary condiions, and nmerically sing for eample he finie elemen mehod (FEM). This paper maes se of he Galerin FEM o solve he LWR model wih consan speed. The road is divided ino a nmber of road segmens (elemens) sing 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 solion for he degrees of freedom, i.e. raffic densiies of each node. The resling simlaneos eqaions are solved a differen ime seps sing he Eler bacward ime-inegraion algorihm. In Belgim and also in he Neherlands, here is a special echniqe ha can be sed in order o preven raffic jams and increasing safey in siaions wih high volme of cars on he roads, i.e. bloc driving. I is a echniqe where cars drive in grops by order of he police when he roads are crowded. In his paper bloc driving is sed as a pracical eample of he LWR model wih consan speed. Thereby, i is simlaed sing he Galerin FEM and he resls are compared wih he analyical solion. The FEM gives good resls providing ha: he road segmens and ime seps are small enogh. A road wih lengh 5 m, consan speed of 25 m/s, segmen lengh of m and ime seps of s gives good resls for he sdied case. A poins of raffic densiy rae disconiniies, depending on he road segmen size and ime sep size, he Galerin FEM is accrae and reqires reasonable compaional effor. From he research wor carried o in his paper, i is fond ha he Galerin FEM is siable for modelling raffic flow a macroscopic level. The elemen size and ime sep size are imporan parameers in deermining he convergence of he solion in case of disconiniies in raffic densiy rae. Alhogh his paper considers he case of consan speed, he echniqe can be eended in he fre o inclde he case of non-consan speed, i.e. speed as a fncion of raffic densiy. Inde Terms Macroscopic raffic flow, Galerin finie elemen mehod, LWR model, bloc driving. Manscrip received March 23, 29. This wor was sppored by Hassel Universiy (Transporaion Research Insie, Facly of Applied Economics) and XIOS Universiy College Limbrg (Deparmen of Indsrial Sciences and Technology). Wesley Celemans ( wesley.celemans@ios.be), Magd A. Wahab (corresponding ahor, phone: ; fa: ; magd.wahab@ios.be) and Kr De Prof ( r.deprof@ios.be) are wih he Deparmen of Indsrial Sciences and Technology, XIOS Universiy College Limbrg, Universiaire Camps Agoralaan Gebow H, B-359 Diepenbee, Belgim. Geer Wes ( geer.wes@hassel.be) is wih he Transporaion Research Insie, Facly of Applied Economics, Hassel Universiy, Weenschapspar 5 bs 6, B-359 Diepenbee, Belgim. I. INTRODUCTION There has been a lile bi research done in he lierare concerned wih macroscopic raffic flow modelling sing he Galerin finie elemen mehod (FEM). In [], [2], and [3] a Galerin FEM ype is sed o solve he macroscopic Lighhill Wiham and Richards (LWR) [4], [5] raffic flow model in conjncion wih Greenshields flow-densiy relaionship [6]. A wavele-galerin FEM is sed in [7] o solve he macroscopic non-consan speed LWR raffic model. A disconinos Galerin FEM is sed in [8] for solving red-and-green ligh models for he raffic flow. This paper presens a nmerical solion (sing a Galerin FEM [9]) of he LWR raffic flow model wih consan speed. The validaion of resls is done by sing 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 solion (sing he mehod of characerisics) and he nmerical solion (sing he Galerin FEM in combinaion wih he Eler bacward ime-inegraion algorihm [9]) can be fond in he ne secions. The las secion provides a nmerical eample, namely a bloc driving simlaion. II. TRAFFIC FLOW THEORY There are several raffic flow models, which can be mosly divided in 4 caegories: macroscopic, mesoscopic, microscopic and sbmicroscopic (lised wih growing level of deails). The firs order LWR raffic flow model is a macroscopic coninm 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 hor (veh/h) Speed (): epressed in ilomeres per hor (m/h) B. Fndamenal relaion The niqe relaion beween he hree macroscopic raffic flow parameers densiy, flow and speed is: q = () C. Fndamenal diagrams Beside he fndamenal relaion, here are also eperimenal relaions beween he raffic flow parameers. ISBN: WCE 29

2 Proceedings of he World Congress on Engineering 29 Vol II WCE 29, Jly - 3, 29, London, U.K. f c q q c f c Free-flow raffic Free-flow raffic Free-flow raffic Jammed raffic c Capaciy-flow raffic These relaionships are called fndamenal diagrams. They are obained from measremens. The fndamenal diagrams are: Speed-densiy relaionship: = f ( ) (2) Speed-flow relaionship: q = f ( ) Flow-densiy relaionship: q = j ( ) j f j Jammed raffic Figre : The fndamenal - diagram according o Greenshields c j Capaciy-flow raffic Jammed raffic Figre 2: The fndamenal q- diagram according o Greenshields j Capaciy-flow raffic Figre 3: The fndamenal -q diagram according o Greenshields q c q (3) (4) Fig., 2, and 3 give graphical overviews of he fndamenal diagrams according o Greenshields [6]. Traffic can have differen regimes (characerized by variables relaed o he raffic sae): Free-flow raffic is characerized by a low densiy (high speed), which resls in a free-flow speed f. Mosly f is he maimm allowed speed. Capaciy-flow raffic is characerized by a maimm flow which is called he capaciy flow q c. Jammed raffic is characerized by a maimm densiy (low or no speed) called he jam densiy j. In pracice, ransiions occr in ime from one regime o anoher. D. Firs order LWR model Lighhill, Wiham and Richards considered ha raffic was an inviscid b compressible flid (flid-dynamic model). Densiies, speed vales and flows were defined as coninos variables in each poin in ime and space (coninm, macroscopic model). The firs order parial differenial eqaion (PDE) from his model is: q + = (5) Crcial o he approach of Lighhill, Wiham and Richards was he fndamenal hypohesis (see secion II.B), i.e. flow is a fncion of densiy and speed: ( ) + = (6) Lighhill and Wiham assmed ha he fndamenal hypohesis holds a all raffic densiies, no js for ligh-densiy raffic b also for congesed raffic condiions. Using he fndamenal 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 eqaion, given iniial and bondary condiions. Eqaion (6) can be made simpler by assming a consan speed. In his paper a consan speed is assmed, i.e. = : ( ) + = (7) Therefore, (7) becomes + = (8) I is worh menioning ha Lighhill, Wiham and Richards noed ha becase of he coniniy assmpion, he heory only holds for a large nmber of vehicles (long crowded roads). III. ANALYTICAL SOLUTION: METHOD OF CHARACTERISTICS Eqaion (8) is a firs order parial differenial eqaion (more specific: he firs order wave eqaion wih speed ). The mehod of characerisics [] can be sed o find a solion for he iniial bondary vale problem. An iniial bondary vale problem assmes (beside he differenial eqaion) wo era eqaions (coninos or disconinos): Iniial vales: he densiy vales a ime ISBN: WCE 29

3 Proceedings of he World Congress on Engineering 29 Vol II WCE 29, Jly - 3, 29, London, U.K. = ( (,) = f( )) Bondary vales: he densiy vales a disance = ( (, ) = g( )) The general form of he solion from he firs order parial differenial eqaion (8) wih consan speed, densiy, iniial condiion (,) = f( ) and bondary condiion (, ) = g( ) is: f ( ) (, ) = g( ) (9) 2 L Figre 4: A single finie elemen IV. NUMERICAL SOLUTION: GALERKIN FEM Applicaion of he Galerin FEM o (8) gives: δ w ( + ) dv = V () Galerin ses he weigh fncion δ w eqal o he shape fncion vecor[ N ] : V N ( + ) dv = [ ] () where V is he elemen volme. For a linear homogeneos elemen dv can be replaced by d and he inegraion can be done over : N ( + ) 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 fncion 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 = N + N = [ N N ] 2 (4) The differeniaion of he densiy wih respec o he posiion is given by N N2 N N2 = + 2 = 2 The shape fncions N and N2 are = = L [ N] [ N N ] L (5) (6) ISBN: WCE 29

4 Proceedings of he World Congress on Engineering 29 Vol II WCE 29, Jly - 3, 29, London, U.K. Sbsiing (3), (4), (5) in (2) gives: N ( [ ] N N ) d N2 2 (7) N N N + ( ) d = N2 2 Simplifying he noaion of (7) gives: ([ ] [ ] { T } T N N N ) d + ( [ N ] {}) = d (8) Afer derivaion and inegraion of he shape fncions, (8) becomes: L 2 {} 6 + = 2 (9) Becase of he ime-dependency, here is a need for a ime-inegraion algorihm. Applicaion of he Eler bacward ime-inegraion algorihm [9] +Δ = resls in: Δ L 2 +Δ L 2 { } { } 6 6 Δ Δ (2) +Δ + { } = 2 Re-wriing (2) gives: +Δ [ A] + [ B] { } = [ A] { } (2) where L 2 [ A] = 6 Δ (22) [ B ] = 2 (23) +Δ +Δ { } = +Δ 2 (24) = 2 { } B. General solion for m ime seps and n elemens BC m Δ L IC n + Figre 5: Problem definiion for n finie elemens and m ime seps (25) Given a road, which is divided in n elemens wih niform lengh L, iniial condiions, a (Fig. 5) and bondary condiions, a = s, = m, o n+ m o (Fig. 5) hen, he densiies a all imes and all posiions can be calclaed from: +Δ { } = [ A] + [ B] [ A] { } (26) where 2 4 L [ A] = (27) 6 Δ 4 v [ B ] = (28) 2 V. SIMULATION OF BLOCK DRIVING By choosing a pracical raffic eample, bloc driving, he Galerin FEM solion mehod is esed on accracy, convergence and compaional needs. Bloc driving is a echniqe where cars drive in grops 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. Becase of his, acceleraion and braing (he harmonica effec) are avoided. The chance of accidens redces and raffic jams become shorer in disance. In his secion bloc driving is simlaed wih he Galerin FEM. I shold be noed ha all densiies are epressed in vehicles per meer (veh/m), disances in meers (m) and imes in seconds (s). Densiy (veh/m) Case Case 2 Case 4 Case 3 Case Time (s) Figre 6: Bondary condiions a he beginning of he road bloc driving Δ ISBN: WCE 29

5 Proceedings of he World Congress on Engineering 29 Vol II WCE 29, Jly - 3, 29, London, U.K. A. Deails of he simlaion The simlaion is described in 5 ime inervals (Fig. 6): Case : no vehicles eners/on he road dring 24 seconds. Case 2: 6 vehicles ener he road (a vehicle per second driving a 25 m/s) dring 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 simlaes he evolion 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 simlaes he evolion of raffic wave 2. B. Mahemaical ranslaion The LWR model wih consan speed is given by (8). All simlaions are done from s o 84 s and he road lengh L is 5 m. The consan speed 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 bondary 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 bondary condiions. C. Analyical solion sing he mehod of characerisics The analyical solion wih he mehod of characerisics (see secion III) is given by he following eqaion: 5 f( ) 24 g( ).4 24 < 3 g( ) (, ) = (3) 3< 54 g( ).4 54 < 6 g( ) 6 < 84 g( ) D. Nmerical solion wih he Galerin FEM The densiy is calclaed for all imes ( o 84 s) and disances ( o 5 m) by sing (26) and applying he iniial and bondary condiions. The convergence sdy on he densiy is done wih simlaions wih differen elemen sizes Δ and ime seps Δ. The densiy verss ime a disance 2 m is flly analysed by sing convergence parameers and CPU calclaion imes. The evolion of densiy in disance (a disances m, 25 m and 5 m) and ime (a imes 57 s, 67 s and 77 s) is invesigaed wih he se of a simlaion wih elemen size m and ime seps of s. ) Convergence sdy on densiy The analyical solion resls in disconinos raffic bloc waves. The nmerical solion resls in coninos ronded raffic waves. The disconiniies become approimaed by a coninos fncion sing he Galerin FEM. The CPU imes of he nmerical simlaions are presened in Table I. Table I: CPU imes for he nmerical simlaions of bloc driving Δ (s) Δ (m) Simlaion ime (s) The accracy parameers (mean error, sandard deviaion of he error and oal error) of he nmerical simlaions are presened in Table II. The higher he oal error, mean error or sandard deviaion (sd), he less accrae he resls. The error resls are calclaed via he difference beween analyical and nmerical vales. Table II: Accracy parameers for he densiy verss 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) Densiy (veh/m) Traffic wave Traffic wave Time (s) Analyical a 2 m FEM (Δ = s, Δ = m) a 2 m FEM (Δ = s, Δ = 25 m) a 2 m FEM (Δ = s, Δ = m) a 2 m Figre 7: Convergence of densiy as fncion of Δ bloc driving ISBN: WCE 29

6 Proceedings of he World Congress on Engineering 29 Vol II WCE 29, Jly - 3, 29, London, U.K. Densiy (veh/m) Traffic wave Traffic wave 2 Simlaions wih segmen lenghs of m and ime seps of s give bes resls complying wih compaion power and accracy. This resls in he following convergence crieria: Δ L 5 Δ s Time (s) Analyical a 2 m FEM (Δ = s, Δ = m) a 2 m FEM (Δ = s, Δ = m) a 2 m FEM (Δ = 3 s, Δ = m) a 2 m Figre 8: Convergence of densiy as fncion of Δ bloc driving Fig. 7 gives an overview of he densiy verss ime a disance 2 m for simlaions wih Δ = s and Δ = m, 25 m, and m. Simlaions wih Δ = s and Δ = m or m gives good resls (boh a small mean error of.2 veh/m wih a small sandard deviaion of.46 veh/m and.45 veh/m), b compaionally hese simlaions are very inensive (high CPU ime of s and s). This simlaions are no presened in Fig. 7 becase of he same graphical resls as wih Δ = s and Δ = m. Simlaions wih Δ = s and Δ = m gives good resls (a small mean error of.22 veh/m wih a small sandard deviaion of.46 veh/m). These simlaions are compaionally fas (low CPU ime of 2.67 s). Simlaions wih Δ = s and Δ = 25 m gives also good resls (a small mean error of.23 veh/m wih a small sandard deviaion of.48 veh/m) b hose are less accrae hen simlaions wih Δ = m. Simlaions wih Δ = s and Δ = m gives bad resls (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 maimm densiies Earlier sar of raffic wave Fig. 8 gives an overview of he densiy verss ime a disance 2 m for simlaions wih Δ = m and Δ = s, s, and 3 s. Simlaions wih Δ = m and Δ =. s gives good resls, b compaionally hese simlaions are very inensive (more han s CPU ime). This simlaion is no presened in Fig. 8 becase of he same graphical resls as wih Δ = m and Δ = s. Simlaions wih Δ = m and Δ = s gives good resls (a small mean error of.22 veh/m wih a small sandard deviaion of.46 veh/m) and hey are compaionally no inensive (low CPU ime of 2.76 s). Simlaions wih Δ = m and Δ = s or 3 s gives bad resls (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 maimm densiies Laer sar of raffic wave 2) Evolion of densiy in disance and ime Fig. 9 gives a graphical represenaion of he evolion 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 nmerical resls are he same for = m becase of he bondary condiion. Since raffic waves are ravelling wih a consan speed of 25 m/s, he maimm 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 evolion 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). Densiy (veh/m) Densiy (veh/m) Traffic wave Traffic wave 2 A Time (s) FEM a m FEM a 25 m FEM a 5 m Figre 9: Evolion of densiy in disance bloc driving.4.3 A B C Disance (m) FEM a 57 s FEM a 67 s FEM a 77 s B Traffic wave 2 Figre : Evolion of densiy in ime bloc driving ISBN: WCE 29

7 Proceedings of he World Congress on Engineering 29 Vol II WCE 29, Jly - 3, 29, London, U.K. VI. CONCLUSIONS AND FUTURE WORK The Galerin FEM can be sed o solve he firs-order macroscopic LWR raffic flow model wih consan speed. The densiy, flow and speed vales are calclaed in each poin on he road, a any ime. The resls of he Galerin finie elemen analysis are compared wih ha of he analyical mehod of characerisics. By he se of an analyical and nmerical echniqe, bloc driving is simlaed. A simlaion wih road lengh 5 m, consan speed of 25 m/s, segmen lenghs of m and ime seps of s resls in accrae and fas nmerical resls. The difficlies wih he nmerical simlaions appear a he disconiniies. This can be prevened by choosing he elemen size and ime seps small enogh. Using larger segmen lenghs and/or ime seps can give inaccrae resls. Using very small segmen lenghs and/or ime seps can resl in inensive simlaions. Fre research will concenrae on he applicaion of he Galerin FEM o he LWR model wih non-consan speed. In sch a case, he speed is a fncion of he densiy. The LWR model can also be eended o inclde social forces and resisances. REFERENCES [] D. E. Besos and P. G. Michalopolos. (984, May-Jne). An applicaion of he finie elemen mehod in raffic signal analysis. Mechanics Research Commnicaions. (3). pp [2] D. E. Besos, P. G. Michalopolos and J. K. Lin. (985, Ocober). Analysis of raffic flow by he finie elemen mehod. Applied Mahemaical Modelling. 9(5). pp [3] I. Oani, D.E. Besos and P.G. Michalopolos. (986, Jne). 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, Febrary). 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 sdy 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. Commnicaions in Nmerical Mehods in Engineering. 6(2). pp [8] R. Li, H. Li and Z. Wang. (2, March). The disconinos finie elemen mehod for red-and-green ligh models for he raffic flow. Mahemaics and Compers in Simlaion. 56(). pp [9] R. D. Coo, D. S. Mals and M. E. Plesha. Conceps and applicaions of finie elemen analysis. Canada, CA: Wiley, 989, ch. 5. [] P. DChaea and D. Zachmann. Applied Parial Differenial Eqaions. CA: Dover pblicaions, 22, ch. 7. [] F. L. Hall, H. M. Zhang, R. Khne and P. Michalopolos. Traffic flow heory A Sae-of-he-Ar Repor. CA: Transporaion Research Board, 2, ch. 2 and ch. 5. ISBN: WCE 29