Variational Formulation of the LWR PDE. Anthony D. Patire

Transcription

1 Variaional Formulaion of he LWR PDE Anhony D. Paire

2

3 Wha swrong wih hese people?

4 Wha I did: x ISolved ρ + Q ( ρ ) = 0 for hree cases 1. Traffic signal 2. Sinusoidal iniial condiion 3. Acual daa using Daganzo s variaional formulaion Published in 2005 Obviaes need o explicily calculae shocks Reduces PDE o a leas-cos problem Obains exac soluions on a laice

5 Case 1: Traffic Signal The raffic signal is represened as a boundary condiion a he op (x=0) Red Ligh q( x, ) = 0 x= 0 q Green Ligh ( x, ) q x=00 cap

6 Case 2: Sinusoidal IC ρ = αρ cap If α < 1, hen a less dense zone moves forward wih raffic If α > 1 hen a speed disurbance moves backwards

7 Case 2: Sinusoidal IC ρ = αρ cap If α < 1, hen a less dense zone moves forward wih raffic If α > 1 hen a speed disurbance moves backwards

8 Case 3: Real Daa

9 How I did i: Ouline: 1. Define Moskowiz Surface 2. Daganzo s Formulaion 3. Special resuls for TFD 4. Recap Numerical Resuls

10 Time Space Diagram x Assign labels o each vehicle rajecory

11 Define N(x,) x Define N(x,) he cumulaive number of vehicles o pass poin x by ime

12 Define N(x,) x Define N(x,) he cumulaive number of vehicles o pass poin x by ime

13 Moskowiz Surface N(x,) x Vehicle rajecories are hus encoded in he values of N(x,)

14 Moskowiz Surface N(x,) N(x,) x

15 Moskowiz Surface N(x,) Traffic 30 Signal Consan Flow Space [m] Time [s]

16 Smoohed Moskowiz Surface N( x, ) Traffic 30 Signal x N( x, ) = ρ( x, ) Space [m] 0 60 Time [s] 120 N( x, ) = q( x, ) 180

17 LWR Formulaion Daganzo Formulaion ρ + x Q( ρ) = 0 x N = x N Q ( ρ) ρ N = Q x N Given I.C., B.C. Given I.C., B.C. Find ρ ( x, ) Find N ( x, )

18 N = inf{ N + Δ ( P ) P P } P D(P( P ) x poin N D ( P 1 ) Boundary Condiion pah ial Condii ion Ini P 1 P 2 N P Boundary Condiion N D ( P 2 ) To calculae he heigh of he surface a an inerior poin, we need only calculae Δ(P), he maximum possible change in N, for each valid pah o ha poin.

19 Flow measured by moving observer x v = L T L 6 # = # + # Tq o = Lρ + TQ(ρ) T q o = Q ( ρ ) vρ

20 Flow measured by moving observer x v = L T L 6 # = # + # Tq o = Lρ + TQ(ρ) T q o = Q ( ρ ) vρ

21 How o calculae Δ(P) ) Q ( ρ ) Fundamenal Diagram R ( v ) Maximum Passing Rae R( v ) v ρ cap ρ jam ρ v R( v) = sup{ Q( ρ) vρ} ρ q o Δ(P) = P D R ( v) d Inegrae over duraion of pah

22 Q ( ρ ) Wha if is riangular? Q ( ρ ) Fundamenal Diagram R ( v ) v f ρ cap v w ρ jam ρ w r max Maximum Passing Rae R ( w) = r R( ) = v f v f max 0 v Then here are only wo characerisic speeds a which informaion propagaes. We need only check wo pahs!

23 Q ( ρ ) Wha if is riangular? Iniial Con ndiion x N D ( P 1 ) w Boundary Condiion P Δ(P ( P 1 ) = rmax ( P D ) 1 v f N P P2 Δ( P 2 ) = 0 Boundary Condiion N D(P D P ( 2 ) To calculae he heigh of he surface a an inerior poin, we need only calculae Δ(P) for wo pahs, one a speed w and one a speed. v f

24 Define a Laice x w Solve by finding change in N from laice poin o v laice poin. f In general, he laice poins won end up in convenien locaions in x and. So, we choose a subse of he laice.

25 Define a Laice x Δ n = 3 Calculae N( x, ) on a subse of he laice. Δx w v f m = 2 I is bes o choose m and n o be small inegers, and You choose w, v f, and Δ. Then Δx = nwδ In his case, m = 2 and n = 3 The soluion calculaed on his laice will be exac. mn = v f w

26 Define a Laice x Δ Calculae N( x, ) on a subse of he laice. Δx w P 1 P 2 P 3 v P f 4 For his laice, one mus calculae four possible values for each poin N P (insead of wo) and ake he minimum. This is he price of omiing 6/7 of he laice inersecion poins from he compuaion.

27 Case 1: Traffic Signal The raffic signal is represened as a boundary condiion a he op (x=0) Red Ligh q( x, ) = 0 x= 0 q Green Ligh ( x, ) q x=00 cap

28 Case 2: Sinusoidal IC ρ = αρ cap If α < 1, hen a less dense zone moves forward wih raffic If α > 1 hen a speed disurbance moves backwards

29 Case 2: Sinusoidal IC ρ = αρ cap If α < 1, hen a less dense zone moves forward wih raffic If α > 1 hen a speed disurbance moves backwards

30 Case 3: Real Daa

31 Case 4: Real Daa Lane Changes

32 Wha I did: x ISolved ρ + Q ( ρ ) = 0 for hree cases 1. Traffic signal 2. Sinusoidal iniial condiion 3. Acual daa using Daganzo s variaional formulaion Published in 2005 Obviaes need o explicily calculae shocks Reduces PDE o a leas-cos problem Obains exac soluions on a laice

33 Remarks Wihou bounds on acceleraion, he LWR PDE underesimaes he velociy oscillaions observed in real raffic The LWR PDE solver may be useful as a diagnosic ool o uncover inefficiencies in raffic flow C. F. Daganzo. On he variaional heory of raffic flow: well-posedness, dualiy, and applicaions. Neworks and Heerogenous media, Vol. 1, No. 4: , 2006.