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

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Introduction Traffic Flow Models Microscopic – ODE Macroscopic – PDE Macroscopic ODE Models?

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

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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

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Constant Model Red Light: Green Light: u 0 > u 0 - v 0 : linear growth u 0 = u 0 - v 0 : equilibrium u 0 < u 0 - v 0 : linear decay RL = GL = 20; u 0 =1 v 0 =2 v 0 =1 v 0 =3

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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

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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

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M-C Model Red: Green: y > J J = 800 J = 900 u 0 = 10, τ = 10 RL = GL = 25

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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, u 0 = τ = 10 (J/4u 0 τ = 2) exists when J = 900, u 0 = τ =10 (J/4u 0 τ = 2.25)

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

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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

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Case-by-Case Model Cruising speed = c Max outflow v max = (1 vehicle)/ (time for it to exit) = = c / l L l

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Case-by-Case Model: Three cases 1. No waiting line When Call N = (no waiting line volume) 2. Maximum waiting line When Call J = (jam volume) 3. Some waiting line When N < y < J y 0 N J No waiting line Some waiting line Max waiting line

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Case-by-Case Model: u & v y N 0 J InflowOutflow u = 0 v = v max u = min (u 0, v max ) u = u 0 v = min (u 0, v max ) No waiting line Some waiting line Max waiting line

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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

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Case-by-Case Model: Plot RL = GL = 20; L = 600; l = 6; c = 30 J = 100; v max = 5 u 0 = 2 u 0 = 4 u 0 = 6

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Case-by-Case Model: Analysis No Congestion Cyclical Congestion Constant Congestion/ “Crawling” u 0 = 2 u 0 = 4 u 0 = 6

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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

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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

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

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

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DD Model: Equations Red Light: if y < J if Green Light: if y < J if

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DD Model: Plot RL = GL = 20; J = 50; v max = 5 u 0 = 5 u 0 = 20

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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; u 0 = 4 v max = 5 RL = GL = 20 Case-by-Case Model Density-Dependent Model

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DD Model: Analysis Equilibrium range of link volume Independent of initial volume on link y 0 = 100 y 0 = 50 y 0 = 0 J = 100; u 0 = 4; v max = 5; RL=GL=20

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

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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:

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

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

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DD Model: Solutions

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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

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DD Model Application: Light Synchronization 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 1Link 2 Light 1 Light 2

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Two Lights: Equations L1 & L2 are red: L1 is red & L2 is green: L1 is green & L2 is red: L1 & L2 are green:

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Two Lights: Plot u 0 = 5 J 1 = J 2 = 100 φ = 0 φ = π φ = π/2 φ = 3π/2

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Three Lights in Phase All red: ; ; All green:

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Three Lights: Plot Link 1 (RL/GL) Link 2 (RL/GL) Link 3 (RL/GL) u 0 = 6, v max = 5, J 1 = J 2 = J 3 = 100 and RL = GL = 20

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Three Lights in Phase Delay Effect Smoothing Effect Nested equilibrium ranges

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

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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)

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

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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

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DD Application 2: Traffic Junction

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Traffic Junction: Equations Light12 is green, Light34 is red: Light12 is red, Light34 is green:

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

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

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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

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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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