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Macroscopic ODE Models of Traffic Flow

Macroscopic ODE Models of Traffic Flow. Zhengyi Zhou 04/01/2010. Introduction. Traffic Flow Models Microscopic – ODE Macroscopic – PDE Macroscopic ODE Models?. Basics. Total Link Volume = y. Red light:

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Macroscopic ODE Models of Traffic Flow

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  1. Macroscopic ODE Models of Traffic Flow Zhengyi Zhou 04/01/2010

  2. Introduction • Traffic Flow Models • Microscopic – ODE • Macroscopic – PDE • Macroscopic ODE Models?

  3. Basics Total Link Volume = y • Red light: Green light: • Goal: find y(t) • MATLAB ODE numerical solver “ode15s”

  4. Outline • Constant Model • McCartney & Carey’s Model • Case-by-Case Model • Density-Dependent Model • Applied to a sequence of lights • Applied to a traffic junction

  5. Constant Model • Red Light: • Green Light: u0 > u0 - v0: linear growth u0 = u0 - v0 : equilibrium u0 < u0 - v0 : linear decay RL = GL = 20; u0=1 v0=2 v0=1 v0=3

  6. Outline • Constant Model • McCartney & Carey’s Model • Case-by-Case Model • Density-Dependent Model • Applied to a sequence of lights • Applied to a traffic junction

  7. McCartney & Carey’s Model: Intro • McCartney & Carey (1999) • Logistic outflow • , when v = 0, when y > J v = outflow; y = link vol J = jam vol;tau = trip time

  8. M-C Model • Red: Green: y > J J = 800 J = 900 u0 = 10, τ = 10 RL = GL = 25

  9. M-C Model: Equilibrium • Green Light equilibrium: Or: Green Light equilibrium exists when • System equilibrium • Equilibrium range • Exists when eg: does not exist when J = 800, u0 = τ = 10 (J/4u0τ = 2) exists when J = 900, u0 = τ =10 (J/4u0τ = 2.25)

  10. M-C Model: Features • Predict congestion • If congested: onset time of congestion • If not: equilibrium range of link volume • No mechanism to un-jam

  11. Outline • Constant Model • McCartney & Carey’s Model • Case-by-Case Model • Density-Dependent Model • Applied to a sequence of lights • Applied to a traffic junction

  12. Case-by-Case Model l • Cruising speed = c • Max outflow vmax = (1 vehicle)/ (time for it to exit) = = c / l L

  13. Case-by-Case Model: Three cases Max waiting line J • No waiting line • When • Call N = (no waiting line volume) • Maximum waiting line • When • Call J = (jam volume) • Some waiting line • When N < y < J Some waiting line N No waiting line 0 y

  14. Case-by-Case Model: u & v Inflow Outflow Max waiting line v = vmax u = 0 J u = min (u0, vmax) v = vmax Some waiting line N No waiting line v = min (u0, vmax) u = u0 0 y

  15. Case-by-Case Model: Equations • Red Light: if y N if N < y < J if y = J • Green Light: if y N if N < y < J if y = J

  16. Case-by-Case Model: Plot RL = GL = 20; L = 600; l = 6; c = 30  J = 100; vmax = 5 u0 = 2 u0 = 4 u0 = 6

  17. Case-by-Case Model: Analysis Constant Congestion/ “Crawling” Cyclical Congestion u0 = 2 u0 = 4 u0 = 6 No Congestion

  18. Case-by-Case Model: Features • All features of M-C Model • 3 congestion levels • Specific time periods of congestion • No permanent congestion • Disadvantage: discrete cases • Critical link vol (N or J) for behavioral changes

  19. Outline • Constant Model • McCartney & Carey’s Model • Case-by-Case Model • Density-Dependent Model • Applied to a sequence of lights • Applied to a traffic junction

  20. Density-Dependent Model: Intro • Drivers continuously & spontaneously adjust to existing traffic on link • Inflow and outflow are both density-dependent

  21. DD Model: u & v • Inflow: ↓ linearly as link volume ↑ • Outflow: ↑ linearly as link volume ↑

  22. DD Model: Equations • Red Light: if y < J if • Green Light: if y < J if

  23. DD Model: Plot RL = GL = 20; J = 50; vmax = 5 u0 = 5 u0 = 20

  24. DD Model vs. Case-by-Case Model • DD Model • Superior: model driver’s behaviors better • Constant adjustment  less likely to jam • Fewer cars get through Same parameter values: J = 100; u0 = 4 vmax = 5 RL = GL = 20 Case-by-Case Model Density-Dependent Model

  25. DD Model: Analysis • Equilibrium range of link volume • Independent of initial volume on link J = 100; u0 = 4; vmax = 5; RL=GL=20 y0 = 100 y0 = 50 y0 = 0

  26. DD Model: Analysis Red:if y < J; if Green: if y < J; if Non-dimensionalization Equilibria: Red:  stable Green: stable Red: Green: where r =

  27. DD Model: Rate of approach • Switch to approach 2 stable equilibria  stable equilibrium range • Approach at the same rate? • If yes, center of equilibrium range = weighted average of 2 equilibrium points • Numerical simulations:

  28. DD Model: Rate of Approach • Center lower; approach to green equilibrium is faster • RL/GL ↑, center↓ Weighted average of equilibriums

  29. DD Model: Solutions • Solve ODEs by discretization • Red: • ……………………………(1) • Green: • ……………….(2) UB LB

  30. DD Model: Solutions

  31. Outline • Constant Model • McCartney & Carey’s Model • Case-by-Case Model • Density-Dependent Model • Applied to a sequence of lights • Applied to a traffic junction

  32. DD Model Application: Light Synchronization Light 1 Light 2 • Outflow of Link 1 = Inflow of Link 2 • Optimal synchronization for smoothest flow • Light 1: red if sin(t) > 0; green if sin(t) < 0 Light 2: red if sin(t+φ) > 0; green if sin(t+φ) <0 φ : phase difference, 0 ≤ φ < 2π Link 1 Link 2

  33. Two Lights: Equations • L1 & L2 are red: • L1 is red & L2 is green: • L1 is green & L2 is red: • L1 & L2 are green:

  34. φ = 0 φ = π/2 Two Lights: Plot φ = π φ = 3π/2 u0 = 5 J1 = J2 = 100

  35. Three Lights in Phase • All red: ; ; • All green:

  36. Three Lights: Plot u0 = 6, vmax = 5, J1 = J2 = J3 = 100 and RL = GL = 20 Link 1 (RL/GL) Link 2 (RL/GL) Link 3 (RL/GL)

  37. Three Lights in Phase • Delay Effect • Smoothing Effect • Nested equilibrium ranges

  38. Three Lights in Phase • Independent of initial link volumes Link 1 (RL/GL) Link 2 (RL/GL) Link 3 (RL/GL)

  39. Three Lights in Phase • Independent of jam vol (link length) on different links Link 1 (RL/GL) Link 2 (RL/GL) Link 3 (RL/GL)

  40. Three Lights in Phase • Non-Dimensionalization • Red: • Green: Integrating Factor = • Later link’s y = integral of previous link’s y • Smoothing

  41. Outline • Constant Model • McCartney & Carey’s Model • Case-by-Case Model • Density-Dependent Model • Applied to a sequence of lights • Applied to a traffic junction

  42. DD Application 2: Traffic Junction

  43. Traffic Junction: Equations • Light12 is green, Light34 is red: • Light12 is red, Light34 is green:

  44. Traffic Junction: Plot1 α = β = 0.9 Link 1 Link 3 Link 2 Link 4 u0 = 6, vmax = 5, J = 100, RL = GL = 20

  45. Traffic Junction: Plot2 α = β = 0.6 Link 1 Link 3 Link 2 Link 4

  46. Conclusions & Further Research Summary • Case-by-Case Model • Density-Dependent Model • Applied to a sequence of lights and a junction Further Research • Different RL/GL in DD equilibrium range analysis • Traffic junction with fewer simplifying assumptions • Compare with macroscopic PDE models • Delay differential equations

  47. References & Acknowledgements • McCartney, M. and Carey, M. “Modeling Traffic Flow: Solving and Interpreting Differential Equations”, Teaching Mathematics and Its Applications 18, no. 3 (1999): 118-119. • MATLAB • Professor Gallegos, Buckmire, Cowieson & Lawrence • Math Department • Friends

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