The LWR Model in Lagrangian coordinaes ACI-NIM: Mah Models on Traffic Flow Ludovic Leclercq, LICIT (ENTPE/INRETS) Jorge Laval, Georgiaech Universiy 31 ocober 2007 INRETS
Ouline Lagrangian resoluion of he homogeneous LWR model Exension o heerogeneous flow Numerical examples 2
The Homogeneous case
The LWR model Variables: k, densiy; v, speed; q=kv, flow Conservaion equaion: k + kv = 0 x q q m a: Flow/densiy relaionship Q Fundamenal diagram (FD): v = V ( k) or q = kv ( k) = Q( k) v m -w k c k m k The model can be synhesized as a scalar hyperbolic equaion: k + Q( k) = 0 x The Godunov scheme is classically used for numerical resoluion 4
The cumulaive coun funcion N(x,) N(x,) represens he cumulaive number of vehicles ha cross locaion x by ime k=- x N and q= N The conservaion equaion reduces o: x, N=,x N (exisence of N) The LWR model can hen be expressed as: N = Q( N) Hamilon-Jacobi equaion x (x,) Eulerian coordinaes (N,) Lagrangian coordinaes 5
The LWR Model in (N,) coordinaes as a conservaion law Variables: s=1/k, spacing; N(x,), cumulaive coun funcion The conservaion equaion becomes : s + v = 0 N v v m b: Speed/spacing relaionship V * FD can be expressed as : v = V s = V s * (1/ ) ( ) wk m s m =1/k m s c =1/k c s The model reduces o a scalar hyperbolic equaion: s + V * ( s) = 0 x 6
The LWR model in (N,) coordinaes as a variaional principle Le X(n,) be he inverse of N(x,) (X is obained by solving for x in n=n(x,)) X(n,) represens he rajecory of vehicle n X * X X verifies: = V (Hamilon-Jacobi equaion) N The model soluions in X also saisfy a leas-cos pah problem 7
Definiion of he leas-cos pah problem in (N,) Valid Pahs are locaions in (N,) where s is consan ( o characerisics in (x,)) Pah Slopes passing raes Pah Coss inercep c V*(s) c n s min s 0 c Passing rae X(n,) s c 8
Numerical resoluion using he variaional principle When he FD is riangular, only wo pahs have o be considered: free-flow pah (slope: 0 ; cos: v m ) congesed pah (slope: wk x ; cos: -w) V*(s) v m wk x 0 n -w s min s n n- n X(n,) X(n- n,) 0 wk x X(n,+ ) (, ) m (, ) X n + v X ( n, + ) = min X n n w wih = n / wk x Iniial nodes + This scheme is exac as each node is linked by a valid pah 9
Advanages of (N,) coordinaes (1) when n=1 he oupus are vehicle rajecories (carfollowing model) vehicle characerisics can hen be easily incorporaed: desinaion own characerisics (desired speed, size, reacion ime) when n>1 mesoscopic models n<1 moving boundary condiions 10
Advanages of (N,) coordinaes (2) v q wk x v m w n Lagrangian approach s x Eulerian approach k n A recangular laice only requires = n/wk x A recangular laice requires v m /w o be an ineger 11
Muliclasse in Lagrangian Framework
Principle Define a specific FD for each Lagrangian cell ( n=1) Each vehicle ype i is defined by hree parameers: The free-flow speed v m,i The jam densiy k x,i ( inverse of vehicle size) The wave-speed w i n a: Grid in (n,) plane n+ n n n- n Cell i+1 Cell i Cell i-1 n Flow [veh/s] v m,i w i + Densiy [veh/m] k x,i 13
Numerical resoluion using he variaional principle Free-flow pah: only he cos (v m,i ) is modified Congesed pah: he slope (w i k x,i ) and he cos (-w i ) are modified n n X(n,) 0 X(n,+ ) Slope modificaions change he srucure of he nework n-1 X(n-1,) w i k x,i wk x This poin is no on he nework grid => he value of X is unknown Iniial nodes + 14
X value esimaion soluion 1 Assume ha he spacing is uniform beween n and n-1 a ime and esimae he X value: ( ) ( 1 α ) (, ) α ( 1, ) wih α, X = X n + X n = w k i i i i m i The numerical scheme does no remain exac 15
X value esimaion soluion 2 Sore he X values a previous ime seps and look for he ime when he congesed pah join a nework node n n n-1 X(n,) X(n-1,) w i k x,i 0 X(n,+ ) This is possible if for each vehicle: w i is he same he raio k x,i /max(k x,i ) is an ineger Iniial nodes + Under hese assumpions, he numerical scheme is exac 16
Numerical examples
Case of sudy One-lane road Cars and rucks wih specific FDs A consan flow rae a he 0.7 enry (1080 veh/h) 0.6 Composiion: 0-150s: 90% cars, 10% rucks 150s-350s: 60% cars, 40% rucks A raffic signal (cycle=90s, green ime=60s) Flow [veh/s] 0.8 0.5 0.4 0.3 0.2 0.1 u=20 m/s u=12 m/s Trucks Cars 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Densiy [veh/m] 18
Variaional scheme 800 (Cars = 90% ; Trucks = 10%) (Cars = 60% ; Trucks = 40%) 700 600 500 Space [m] 400 300 200 100 0 0 50 100 150 200 250 300 Time [s] 19
Simulaion Wihou lane-changing: Coupled wih a lane-changing model: 20
Conclusion on Lagrangian approach oupu are vehicle rajecories. This makes i easier o incorporae exensions such as: origin/desinaion differen vehicle ypes moving bolenecks Allow a direc exension o heerogenous flow: Parsimonious and easy o calibrae (FD by vehicle ype) Exac under lile resricive assumpions (riangular FD, consan w, ineger raio beween jam densiies) Fully compaible wih exising exension of he LWR model and especially lane-changing one (Laval and Leclercq, 2007) 21
Thank you for your aenion 22