TRAFFIC PERFORMANCE MODELS I: TRAFFIC FLOW THEORY AND SIMULATION APPROACHES

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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
•  

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Outline

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

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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

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Measurements

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

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Time-Space Diagram Analysis at a Fixed Position
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Time-Space Diagram Analysis at a Fixed Time
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Measurements

• Flow

• Density

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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)

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Fundamental Relationship

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

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Traffic Stream Models

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

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Field Data (Holland Tunnel, Eddie 63)
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Relationships

• Structure
• Single regime same functional form under all
  traffic conditions
• Multiple regime different models for different
  traffic conditions

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Examples

• Single regime models

• Greenshields model

uf, kj, parameters to be calibrated
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Examples

• Single regime models

• Greenbergs model ?
• Underwoods model ?

uf, kj, km, parameters to be calibrated
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Examples

• Multiple regime models

• Edies model
• Uncongested region
• Underwoods model
• Congested region
• Greenbergs model
• Discontinuity

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Speed-Density Relationships
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Flow-Density Relationships
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Speed-Flow Relationships
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Empirical Results

• Hall et. al. (1992)

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Speed-Flow Relationship
Source HCM 2000
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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

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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
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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

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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

 
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Solution of First Order Models
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Comments

• Shockwave speed

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Comments

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

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Example Traffic Lights
time
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Shockwaves at Traffic Lights (red)
Stopping waves
uw
A
qA
Flow (q)
B
D
qD 0
kj
kA
Density (k)
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Shockwaves at Traffic Lights (green)
Starting waves
uw
C
qB qmax
Flow (q)
B
D
kj
kC
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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

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High Order Continuum Flow Models

• Attempt to represent the acceleration of the
  traffic stream
• Acceleration f(traffic ahead, equilibrium
  speed, reaction time,)

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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)

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Microscopic Models

• Car-following
• Lane changing
• Gap acceptance

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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
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Car-Following Models
GM models (Herman, Gazis)
acceleration of vehicle n at
time t ?, l, m parameters
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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

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General Acceleration, contd

• Ahmed, Ben-Akiva, Koutsopoulos (2002)
• Car following

• Free flowing

• Distributed headway threshold and reaction time

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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
  

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Flow Models Derived from Car-Following Models
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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.

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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

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Dimensions of Lane Changing
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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)
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Limitations of Existing Models

• Independent behaviors
• Acceleration and lane-changing
• Mandatory and discretionary lane changing
• Reactive
• No anticipation
• Myopic
• No planning

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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

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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

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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)

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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

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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

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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

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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

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Macroscopic Model Example

• Numerical solution based on Laxs method

?j coefficient for converting flow to density
(e.g. ?t/?xj)
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Paynes model (High Order)
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Macroscopic Models Examples
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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

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NETCELL, contd

• Principles (example)

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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

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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

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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

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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

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Mesoscopic Simulation Models Examples

• Microscopic version has also been developed

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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

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Microscopic Traffic SimulationApplication Needs
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Microscopic Traffic SimulationFunctionality
Needs
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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

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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
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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
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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)

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Nanoscopic Simulation Models

• General characteristics
• detailed driver behavior
• detailed vehicle dynamics model
• detailed vehicle-drive interactions

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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

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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)

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Applications of Traffic Simulation Models
 
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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

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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