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Modelling of Traffic Flow and Related Transport Problems. Andreas Schadschneider Institute for Theoretical Physics University of Cologne Germany. www.thp.uni-koeln.de/~as. www.thp.uni-koeln.de/ant-traffic. Overview.
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Modelling of Traffic Flow and Related Transport Problems Andreas Schadschneider Institute for Theoretical Physics University of Cologne Germany www.thp.uni-koeln.de/~as www.thp.uni-koeln.de/ant-traffic
Overview General topic: Application of nonequilibrium physics to various transport processes/phenomena • Highway traffic • Traffic on ant trails • Pedestrian dynamics • Intracellular transport Topics: • basic phenomena • modelling approaches • theoretical analysis • physics Aspects:
Introduction • Traffic = macroscopic system of interacting particles • Nonequilibrium physics: • Driven systems far from equilibrium • Various approaches: • hydrodynamic • gas-kinetic • car-following • cellular automata
Cellular Automata Cellular automata (CA) are discrete in space time state variable (e.g. occupancy, velocity) Advantage: very efficient implementation for large-scale computer simulations often: stochastic dynamics
Asymmetric Simple Exclusion Process q q • Asymmetric Simple Exclusion Process (ASEP): • directed motion • exclusion (1 particle per site) • stochastic dynamics Caricature of traffic: “Mother of all traffic models” For applications: different modifications necessary
Update scheme In which order are the sites or particles updated ? • random-sequential: site or particles are picked randomly at each step (= standard update for ASEP; continuous time dynamics) • parallel (synchronous): allparticles or sites are updated at the same time • ordered-sequential: update in a fixed order (e.g. from left to right) • shuffled: at each timestep all particles are updated in random order
ASEP ASEP = “Ising” model of nonequilibrium physics • simple • exactly solvable • many applications • Applications: • Protein synthesis • Surface growth • Traffic • Boundary induced phase transitions
Periodic boundary conditions fundamental diagram • no or short-range correlations
Influence of Boundary Conditions • open boundaries: density not conserved! exactly solvable for all parameter values! Derrida, Evans, Hakim, Pasquier 1993 Schütz, Domany 1993
Phase Diagram Maximal current phase J=J(p) Low-density phase J=J(p,) 2.order transitions 1.order transition High-density phase J=J(p,)
Spontaneous Jam Formation space time jam velocity: -15 km/h (universal!) Phantom jams, start-stop-waves interesting collective phenomena
Fundamental Diagram Relation: current (flow) $ density free flow congested flow (jams) more detailed features?
Cellular Automata Models • Discrete in • Space • Time • State variables (velocity) velocity dynamics: Nagel – Schreckenberg (1992)
Update Rules • Rules (Nagel, Schreckenberg 1992) • Acceleration: vj! min (vj + 1, vmax) • Braking: vj! min ( vj , dj) • Randomization: vj ! vj – 1 (with probability p) • Motion: xj! xj + vj (dj = # empty cells in front of car j)
Example Configuration at time t: Acceleration (vmax = 2): Braking: Randomization (p = 1/3): Motion (state at time t+1):
Interpretation of the Rules • Acceleration: Drivers want to move as fast as possible (or allowed) • Braking: no accidents • Randomization: • a) overreactions at braking • b) delayed acceleration • c) psychological effects (fluctuations in driving) • d) road conditions • 4) Driving: Motion of cars
Realistic Parameter Values • Standard choice: vmax=5, p=0.5 • Free velocity: 120 km/h 4.5 cells/timestep • Space discretization: 1 cell 7.5 m • 1 timestep 1 sec • Reasonable: order of reaction time (smallest relevant timescale)
Discrete vs. Continuum Models • Simulation of continuum models: • Discretisation (x, t) of space and time necessary • Accurate results: x, t ! 0 • Cellular automata: discreteness already taken into account in definition of model
Simulation of NaSch Model Simulation • Reproduces structure of traffic on highways • - Fundamental diagram • - Spontaneous jam formation • Minimal model: all 4 rules are needed • Order of rules important • Simple as traffic model, but rather complex as stochastic model
Analytical Methods 1 1 2 2 3 3 4 4 d1=1 d2=0 d3=2 2 3 1 4 • Mean-field: P(1,…,L)¼ P(1) P(L) • Cluster approximation: • P(1,…,L)¼ P(1,2) P(2,3) P(L) • Car-oriented mean-field (COMF): • P(d1,…,dL)¼ P(d1) P(dL) with dj = headway of car j (gap to car ahead)
Fundamental Diagram (vmax=1) vmax=1: NaSch = ASEP with parallel dynamics • Particle-hole symmetry • Mean-field theory underestimates flow: particle-hole attraction
Paradisical States (AS/Schreckenberg 1998) • ASEP with random-sequential update: no correlations (mean-field exact!) • ASEP with parallel update: correlations, mean-field not exact, but 2-cluster approximation and COMF • Origin of correlations? (can not be reached by dynamics!) Garden of Eden state (GoE) in reduced configuration space without GoE states: Mean-field exact! => correlations in parallel update due to GoE states not true forvmax>1 !!!
Fundamental Diagram (vmax>1) • No particle-hole symmetry
Phase Transition? • Are free-flow and jammed branch in the NaSch model separated by a phase transition? No! Only crossover!! Exception: deterministic limit (p=0) 2nd order transition at
Modelling of Traffic Flow and Related Transport Problems Lecture II Andreas Schadschneider Institute for Theoretical Physics University of Cologne Germany www.thp.uni-koeln.de/~as www.thp.uni-koeln.de/ant-traffic
Nagel-Schreckenberg Model velocity • Acceleration • Braking • Randomization • Motion vmax=1: NaSch = ASEP with parallel dynamics vmax>1: realistic behaviour (spontaneous jams, fundamental diagram)
Fundamental Diagram II more detailed features? high-flow states free flow congested flow (jams)
Metastable States • Empirical results: Existence of • metastable high-flow states • hysteresis
VDR Model • Modified NaSch model: • VDR model (velocity-dependent randomization) • Step 0: determine randomization p=p(v(t)) • p0 if v = 0 • p(v) = with p0 > p • p if v > 0 • Slow-to-start rule Simulation
Jam Structure NaSch model VDR model VDR-model: phase separation Jam stabilized by Jout < Jmax
Fundamental Diagram III Even more detailed features? non-unique flow-density relation
Synchronized Flow • New phase of traffic flow (Kerner – Rehborn 1996) • States of • high density and relatively large flow • velocity smaller than in free flow • small variance of velocity (bunching) • similar velocities on different lanes (synchronization) • time series of flow looks „irregular“ • no functional relation between flow and density • typically observed close to ramps
3-Phase Theory free flow (wide) jams synchronized traffic 3 phases
Cross-Correlations Cross-correlation function: ccJ() /h (t) J(t+) i - h(t) ihJ(t+)i free flow,jam: synchronized traffic: free flow jam synchro Objective criterion for classification of traffic phases
Time Headway synchronized traffic • free flow density-dependent many short headways!!!
Brake-light model • Nagel-Schreckenberg model • acceleration (up to maximal velocity) • braking (avoidance of accidents) • randomization (“dawdle”) • motion • plus: • slow-to-start rule • velocity anticipation • brake lights • interaction horizon • smaller cells • … Brake-light model (Knospe-Santen-Schadschneider -Schreckenberg 2000) good agreement with single-vehicle data
Fundamental Diagram IV • Empirical results • Monte Carlo simulations
Highway Networks • Autobahn network • of North-Rhine-Westfalia • (18 million inhabitants) • length: 2500 km • 67 intersections (“nodes”) • 830 on-/off-ramps • (“sources/sinks”)
Data Collection • online-data from • 3500 inductive loops • only main highways are densely equipped with detectors • almost no data directly • from on-/off-ramps
Online Simulation • State of full network through simulation based on available data • “interpolation” based on online data: online simulation (available at www.autobahn.nrw.de) classification into 4 states
Traffic Forecasting forecast for 14:56 actual state at 14:54 state at 13:51
2-Lane Traffic • Rules for lane changes (symmetrical or asymmetrical) • Incentive Criterion: Situation on other lane is better • Safety Criterion: Avoid accidents due to lane changes
Defects Locally increased randomization: pdef > p shock Ramps have similar effect! Defect position
City Traffic • BML model: only crossings • Even timesteps: " move • Odd timesteps: ! move • Motion deterministic ! 2 phases: Low densities: hvi > 0 High densities: hvi = 0 Phase transition due to gridlocks
More realistic model • Combination of BML and NaSch models • Influence of signal periods, • Signal strategy (red wave etc), … Chowdhury, Schadschneider 1999
Summary • Cellular automata are able to reproduce many aspects of • highway traffic (despite their simplicity): • Spontaneous jam formation • Metastability, hysteresis • Existence of 3 phases (novel correlations) • Simulations of networks faster than real-time possible • Online simulation • Forecasting