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TRAFFIC PERFORMANCE MODELS I: TRAFFIC FLOW THEORY AND SIMULATION APPROACHES

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time-mean speed ut: average speed of vehicles crossing a point of the roadway ... time-mean speed space-mean speed (typically ut = 1.06 1.12us) ... – PowerPoint PPT presentation

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Title: TRAFFIC PERFORMANCE MODELS I: TRAFFIC FLOW THEORY AND SIMULATION APPROACHES


1
TRAFFIC PERFORMANCE MODELS I TRAFFIC FLOW
THEORY AND SIMULATION APPROACHES
  • Haris N. Koutsopoulos
  • Northeastern University
  • M.I.T. Summer Professional Program 1.10s
  • Modeling and Simulation for Dynamic
    Transportation Management Systems
  •  July/August 2003
  •  

2
Outline
  • Basic Elements
  • Macroscopic models
  • Traffic Stream Models
  • Continuum Models
  • Microscopic models
  • Car Following Models
  • Lane Changing
  • Traffic Simulation Models

3
Basic Elements of Traffic Flow Theory
  • flow (q) number of vehicles crossing a point
    per unit of time
  • speed (u)
  • time-mean speed ut average speed of vehicles
    crossing a point of the roadway
  • space-mean speed us average speed of vehicles
    over a roadway segment at any given time, or
    based on average time it takes to cross the
    segment
  • concentration
  • density (k) number of vehicles per unit length
    of roadway
  • occupancy (o) percent of time a point of the
    road is occupied
  • headway
  • space headway (s) distance between two
    consecutive vehicles
  • time headway (h) time between passage of two
    consecutive vehicles

4
Measurements
  • Fixed Point
  • Fixed Time (aerial photograph)
  • Moving observer

5
Time-Space Diagram Analysis at a Fixed Position
6
Time-Space Diagram Analysis at a Fixed Time
7
Measurements
  • Flow
  • Density

8
Measurements
  • Time-mean speed

At a fixed location, time T ui instantaneous
speed
  • Space-mean speed

Observation at fixed location over time T
Instantaneous photograph, speeds over section of
length L
  • time-mean speed ? space-mean speed (typically
    ut 1.06 1.12us)

9
Fundamental Relationship
  • q us ? k
  • where,
  • q flow
  • us space-mean speed
  • k density

10
Traffic Stream Models
  • Objective
  • Provide fundamental relationships among
    macroscopic traffic stream characteristics for
    uninterrupted flow conditions
  • speed-density
  • flow-density
  • speed-flow

11
Field Data (Holland Tunnel, Eddie 63)
12
Relationships
  • Structure
  • Single regime same functional form under all
    traffic conditions
  • Multiple regime different models for different
    traffic conditions

13
Examples
  • Single regime models
  • Greenshields model

uf, kj, parameters to be calibrated
14
Examples
  • Single regime models
  • Greenbergs model ?
  • Underwoods model ?

uf, kj, km, parameters to be calibrated
15
Examples
  • Multiple regime models
  • Edies model
  • Uncongested region
  • Underwoods model
  • Congested region
  • Greenbergs model
  • Discontinuity

16
Speed-Density Relationships
17
Flow-Density Relationships
18
Speed-Flow Relationships
19
Empirical Results
  • Hall et. al. (1992)

20
Speed-Flow Relationship
Source HCM 2000
21
Continuum Flow (Kinetic) Models
  • Conservation of flow
  • density at time t k
  • density at time tdt kdk
  • conservation of flow qdt kdx (q dq)?dt
    (kdk)?dx

22
Continuum Flow (Kinetic) Models, contd
  • First order models (Lighthill and Whitham,
    Richards, LWR)

Conservation of flow
Basic equation of flow
Equilibrium speed (e.g. Greenshields
speed-density model)
q flow u space-mean speed k density g(x,t) gen
eration rate
23
Solution of First Order Models
  • Problem
  • Given a set of initial and boundary conditions,
  • e. g. k0 k(x0, 0)
  • Find k(x,t)
  • Solution
  • Exact solution is based on the method of
    characteristics (waves) lines with the same
    density


24
Solution of First Order Models
  • Properties
  • characteristics (waves) are straight lines along
    which the density is constant (equal to the
    density at the initial point they emanate from)
  • a disturbance at a point propagates along the
    characteristic through that point
  • a disturbance travels at a speed given by the
    slope of the characteristic

 
25
Solution of First Order Models
26
Comments
  • Shockwave speed

27
Comments
  • If (A, B) (2, 1) w21 is positive (shockwave
    moves forward)
  • If (A, B) (3, 2) w32 is negative (shockwave
    moves backward)

28
Example Traffic Lights
time
29
Shockwaves at Traffic Lights (red)
Stopping waves
uw
A
qA
Flow (q)
B
D
qD 0
kj
kA
Density (k)
30
Shockwaves at Traffic Lights (green)
Starting waves
uw
C
qB qmax
Flow (q)
B
D
kj
kC
31
Discussion on First Order Models
  • Issues
  • driver differences
  • motion through a shock
  • stability
  • Implications
  • light traffic
  • desired speed
  • platoon formation
  • restricted traffic
  • instantaneous change of speed
  • stop and go instabilities

32
High Order Continuum Flow Models
  • Attempt to represent the acceleration of the
    traffic stream
  • Acceleration f(traffic ahead, equilibrium
    speed, reaction time,)

33
High Order Continuum Flow Models
  • Example Paynes model

conservation of flow
basic equation of flow
dynamic speed equation
  • where,
  • T reaction time, relaxation parameter
  • ? anticipation parameter
  • f(k) equilibrium speed (speed-density model)
  • Rate of change of speed is a function of
  • Convection (tendency to travel upstream with
    current speed)
  • Relaxation to equilibrium
  • Anticipation (impact off density ahead)

34
Microscopic Models
  • Car-following
  • Lane changing
  • Gap acceptance

35
Car-Following Models
Common Model Response(t) sensitivity?stimulus(t
-T) T reaction time Stimulus ?v (relative
speed), ?x (relative distance) Sensitivity
function of ?x, speed, traffic conditions Response
acceleration, speed
36
Car-Following Models
GM models (Herman, Gazis)
acceleration of vehicle n at
time t ?, l, m parameters
37
Car-Following Models
  • Distance-based models (Newell)
  • Stimulus is a function of relative distance
  • Psycho-physical spacing model (Wiedemann, 1974)
  • reaction based on relative speed-relative
    distance combinations
  • General acceleration models
  • Ahmed, Ben-Akiva, Koutsopoulos (2002)
  • Multiple regimes of traffic
  • Free flowing
  • Car-following
  • Time headway threshold and reaction time
    distributions

38
General Acceleration, contd
  • Ahmed, Ben-Akiva, Koutsopoulos (2002)
  • Car following
  • Free flowing
  • Distributed headway threshold and reaction time

39
Car-following and Traffic Stream Models
mgt
  • Are car-following and traffic stream models
    consistent?
  • Under steady-state, single lane conditions
    integration of car-following models results in
    traffic stream models

40
Flow Models Derived from Car-Following Models
41
Lane Changing Models
  • Mandatory and discretionary lane-changing
  • mandatory getting off the current lane in order
    to continue on the desired path (e.g. exiting),
    or to avoid lane closure
  • discretionary attempting to achieve desired
    speed, avoid following trucks, avoid merging
    traffic, etc.

42
Lane-changing, contd
  • Ahmed, Ben-Akiva, Koutsopoulos (1999)
  • Mandatory and discretionary
  • 3 levels
  • Decision to consider lane change
  • Target lane choice for DLC
  • Random utility framework
  • Gap acceptance
  • Lead and lag gaps
  • Forced merging

43
Dimensions of Lane Changing
44
Gap Acceptance
  • Critical Gap
  • if available gap lt critical gap reject the gap
  • if available gap gt critical gap accept the gap
  • Lead and lag gap
  • Critical gap is a function of
  • Relative speed
  • First gap
  • Number of gaps rejected
  • Remaining length (mandatory lane changing)
  • Other opportunities
  • Traffic conditions

The Critical Gap is Function of explanatory
variables Gng (t) expXng (t)?g ?ng (t)
45
Limitations of Existing Models
  • Independent behaviors
  • Acceleration and lane-changing
  • Mandatory and discretionary lane changing
  • Reactive
  • No anticipation
  • Myopic
  • No planning

46
Other Traffic Performance Models
  • Delay models
  • Queuing theory (deterministic and stochastic
    approaches)
  • Empirical models
  • Simulation models
  • Synthesis of traffic theory models, delay models,
    traffic dynamics representation

47
Simulation Models
  • Definition
  • the process of designing a model of a real
    system and conducting experiments with this model
    for the purpose either of understanding the
    behavior of the system or of evaluating various
    strategies (within the limits imposed by a
    criterion or set of criteria) for the operation
    of the system (Shannon, 1975)
  • Approaches
  • Discrete event
  • synchronous
  • Asynchronous
  • Continuous time
  • Hybrid

48
Simulation Models
  • Functionality and level of detail
  • Network representation
  • Flow representation
  • Traffic dynamics
  • Support of control strategies
  • Surveillance
  • Travel behavior/demand
  • Overall structure
  • Event-based
  • Time-based
  • Output
  • Measures of effectiveness (MOEs)

49
Level of Detail
  • Based on their flow and traffic dynamics
    representation traffic simulation models are
    characterized as
  • Macroscopic
  • Fluid representation of flow
  • Time and space discretization
  • Mesoscopic
  • Individual vehicle representation
  • Continuous space
  • Usually discrete time
  • Microscopic
  • Individual vehicle representation
  • Traffic dynamics through vehicle interactions and
    movements
  • Nanoscopic
  • Many common elements with microscopic
  • Detailed representation of vehicle dynamics

50
Overall StructureTime- vs. Event-Based Simulation
  • Time-based models advance the clock at fixed
    intervals ?t
  • ?t may be different for different processes
  • choice of ?t important for
  • Efficiency
  • Accuracy
  • Event-based models maintain an event list. First
    event in the list is processed next
  • sequencing of events may be difficult
  • less control over efficiency

51
Macroscopic Model Characteristics
  • In effect numerical solutions to continuum flow
    models
  • Usually deterministic
  • Common for evaluation of freeway corridor
    operations
  • Traffic dynamics
  • queuing theory
  • kinetic theory
  • simple input/output
  • simple continuum
  • high order continuum
  • Basic approach
  • numerical solution of kinetic equations
  • discretization of space and time

52
General Approach
  • define a grid of points (n, j) in the time-space
    domain
  • replace derivatives by finite differences
  • Forward, Backward, Central
  • Lax-Friedrichs scheme
  • Codunovs finite difference approximation

53
Macroscopic Model Example
  • Numerical solution based on Laxs method

?j coefficient for converting flow to density
(e.g. ?t/?xj)
54
Paynes model (High Order)
55
Macroscopic Models Examples
56
NETCELL (Daganzo)
  • Characteristics
  • Dynamic evolution of multi-commodity traffic over
    large scale freeway networks
  • Links divided into cells traversed in one
    simulation step (under free flow speed)
  • Consistent with hydrodynamic theory of flow (cell
    transmission model)
  • Time-based, with an event list
  • Inputs
  • network geometry
  • incidents
  • O-D flows
  • routing information

57
NETCELL, contd
  • Principles (example)

58
NETCELL, contd
cell i1
cell i-1
ni(t1) ni(t) yi(t) - yi1(t)  yi(t) min
ni-1(t), Qi(t), Ni(t) - ni(t)
  • l length of cell
  • ni(t) number of vehicles in cell i at time t
  • Ni(t) max number of vehicles in cell i
  • Qi(t) capacity flow into i for time interval t,
    and
  • yi(t) number of vehicles that flow from cell i-1
    to cell i in (t, t1)
  • Note results are independent of the processing
    order of cells

59
Macroscopic Models Comments
  • Debate first order vs. high order models
  • theoretical considerations
  • empirical evidence
  • Recent developments include
  • effects due to lane drops
  • freeway-to-freeway interactions
  • bus effects
  • Importance of method for numerical solution
  • Numerical solution requires appropriate
    discretization of time and space

60
Macroscopic Models Comments
  • Advantages
  • data requirements
  • execution speed
  • Usual output aggregate measures of performance
  • queue lengths
  • speed contours
  • total delay
  • total travel time
  • fuel/pollution statistics

61
Mesoscopic Traffic Simulation Models
  • Flow Representation
  • individual vehicles or groups of vehicles with
    similar characteristics (packets)
  • Traffic Dynamics
  • fluid approximation
  • queuing theory
  • Network Representation
  • integrated networks or corridors
  • link-based
  • lane-based
  • Traffic Control
  • aggregate by equivalent capacities
  • Detailed
  • Structure
  • time-based

62
Mesoscopic Simulation Models Examples
  • Microscopic version has also been developed

63
Microscopic Traffic Simulation Models
  • Detailed
  • Synthesis of models
  • Driving behavior
  • Travel behavior
  • Control and routing strategies
  • Driver classes
  • Integrated networks (freeways, urban streets)
  • Usually time-based (e.g. 0.1 sec. time step)
  • Stochastic

64
Microscopic Traffic SimulationApplication Needs
65
Microscopic Traffic SimulationFunctionality
Needs
66
Examples of Microscopic Simulation Models
  • AIMSUN 2 Universitat Politècnica de Catalunya,
    Spain
  • ANATOLL ISIS and Centre dEtudes Techniques de
    lEquipement France
  • ARTEMiS University of New South Wales, School of
    Civil Engineering Australia
  • ARTIST Bosch Germany
  • CASIMIR Institut National de Recherche sur
    les Transports et la Sécurité France
  • CORSIM Federal Highway Administration
    USA
  • DRACULA Institute for Transport Studies,
    University of Leeds UK
  • FLEXSYT II Ministry of Transport
    Netherlands
  • FREEVU University of Waterloo, Department
    of Civil Engineering Canada
  • FRESIM Federal Highway Administration
    USA
  • HUTSIM Helsinki University of
    Technology Finland
  • INTEGRATION Queens University, Transportation
    Research Group Canada
  • MELROSE Mitsubishi Electric Corporation
    Japan
  • MICROSIM Centre of parallel computing (ZPR),
    University of Cologne Germany
  • MICSTRAN National Research Institute of Police
    Science Japan
  • MITSIMLab Massachusetts Institute of
    Technology USA
  • NEMIS Mizar Automazione, Turin
    Italy

67
Examples of Microscopic Simulation Models, contd
  • PADSIM Nottingham Trent University NTU UK
  • PARAMICS The Edinburgh Parallel Computing Centre
    and Quadstone UK
  • PHAROS Institute for simulation and
    training USA
  • PLANSIM-T Centre of parallel computing (ZPR),
    University of Cologne Germany
  • SIGSIM University of Newcastle UK
  • SIMDAC ONERA Centre d'Etudes et de Recherche
    de Toulouse France
  • SIMNET Technical University Berlin
    Germany
  • SISTM Transport Research Laboratory,
    Crowthorne UK
  • SITRA-B ONERA Centre d'Etudes et de Recherche
    de Toulouse France
  • TRANSIMS Los Alamos National Laboratory USA
  • THOREAU The MITRE Corporation USA
  • VISSIM PTV System Software and Consulting GMBH
    Germany

A detailed description of microscopic models can
be found in www.its.leeds.ac.uk/smartest
68
TRANSIMS Microsimulation
intersection with multipleturn buffers (not
internallydivided into grid cells)
single-cell vehicle
multiple-cell vehicle
7.5 meter ? 1 lane cellularautomaton grid cells
69
TRANSIMS Microsimulation, contd
  • For all particles i simultaneously
  • IF ( vi gt gapi ? deceleration)
  • ELSE IF (vi lt vmax ? acceleration)
  • ELSE (vi vmax and vi lt gapi ? free-flow)

70
Nanoscopic Simulation Models
  • General characteristics
  • detailed driver behavior
  • detailed vehicle dynamics model
  • detailed vehicle-drive interactions

71
Applications of Traffic Simulation Models
  • Evaluation at the planning and policy level
  • Evaluation at the operational level
  • traffic control
  • intersection/urban street operations
  • freeway corridors
  • ITS
  • Automated Highway Systems (AHS)
  • Public transportation
  • Design
  • Optimization

72
Applications of Traffic Simulation Models
  • Real time decision support systems
  • Prediction
  • Strategy design/evaluation
  • route guidance
  • traffic control
  • Research and development
  • New concepts and algorithms
  • Role
  • used independently
  • elements of larger systems
  • supply representation
  • dynamic network loading models (DNL)

73
Applications of Traffic Simulation Models
 
74
Summary
  • Integrated networks
  • Traffic dynamics
  • queue build-up and spillbacks
  • Travel behavior and demand
  • Driver characteristics and vehicle classes
  • Dynamic Traffic Management
  • Trends
  • Hybrid models

75
General Comments
  • Model complexity
  • inputs
  • Calibration
  • Validation
  • program is a close approximation to reality
  • Estimation and calibration
  • Output analysis
  • large amounts of output
  • interpretation of results
  • Sources of error
  • functional
  • distributional
  • independence
  • aggregation
  • boundary effects
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