-
MICROSCOPIC MODELLING OF THE
AREA-BASED TRAFFIC FLOW
NIKHIL CHANDRA SARKAR
MASTER OF SCIENCE IN MATHEMATICS
Submitted in fulfilment of the requirements for the degree
of
Doctor of Philosophy
School of Civil Engineering and Built Environment
Faculty of Science and Engineering
Queensland University of Technology
2019
-
Microscopic Modelling of the Area-Based Traffic Flow i
Keywords
Angle-based discretization, area-based traffic, car-following
model, confusion
matrix, heterogeneous traffic, lane-changing model, macroscopic
characteristics,
microscopic modelling, modified IDM, multinomial logit model,
multinomial probit
model, ROC space, simulation, visual perception area.
-
ii Microscopic Modelling of the Area-Based Traffic Flow
Abstract
Area-based (i.e., non-lane based) heterogeneous traffic (as in
developing
countries) differs significantly from lane-based homogeneous
traffic (as in developed
countries). In area-based traffic, drivers generally ignore the
lane markings and
perceive the entire road space while progressing longitudinally.
Traditional car-
following and lane-changing models are not directly applicable
to modelling such
driving behaviour.
This research aimed to microscopically model the dynamic of the
subject
vehicle in area-based traffic flow. The modelling was conducted
in two steps. In Step
1, discrete choice-based modelling was conducted to identify the
area-based
movement direction of the subject vehicle. In Step 2, a
vehicle-following behaviour
model was developed to simulate the next position of the subject
vehicle (along the
direction of a selected alternative, as modelled in Step 1 of
this modelling).
In Step 1, the choice space of the subject vehicle was divided
into a number of
realistic radial cones considering the possible movement
directions of the subject
vehicle in the next simulation time step. These radial cones
formed the alternatives
for the decision of the subject vehicle. The attributes of the
alternatives were defined
in terms of angular deviation from the direction of the flow,
spacing and relative
speed. A multinomial logit model was used for the alternative
selection of the subject
vehicle. This model was calibrated and validated using real
trajectory data from
India. The alternative selection for the subject vehicle as a
car was correctly captured
approximately 84% to 86% for each alternative prediction. The
alternative selection
for subject vehicle as a motorcycle was approximately 87% to 97%
for each
alternative predication. A receiver operating characteristics
graph was used to
illustrate the performance of the multinomial logit model and
the results
demonstrated the reliability of the proposed modelling framework
for area-based
traffic flow using a discrete choice approach.
In Step 2, a modified intelligent driver model (MIDM) was
proposed and tested
on the data. Here, the spacing and relative speed parameters of
the intelligent driver
model (IDM) were modified considering the vector projection of
the relative position
-
Microscopic Modelling of the Area-Based Traffic Flow iii
and the relative vehicle speed along the direction of the
selected alternative (in Step
1). For the MIDM, a subject vehicle had a choice space in front
from the visual
perception area consistent with the first step of this
modelling. The acceleration
behaviour of a subject vehicle was simulated along the direction
of an alternative
selected from the first step. The performance of the entire
modelling was tested on
randomly selected vehicles from the replications of the model on
the dataset.
The macroscopic validation of the model was performed to ensure
the
robustness of the model. The deterministic parameters for MNL
model and the
empirical distribution of the MIDM model parameters were
utilized to stochastically
simulate vehicle trajectories with the initial and boundary
conditions determined by
the real dataset. The comparison of the macroscopic properties
between the
simulated and real dataset provides promising results for the
simulation applicability
of the proposed modelling framework.
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iv Microscopic Modelling of the Area-Based Traffic Flow
Table of Contents
Keywords
...................................................................................................................................
i
Abstract
.....................................................................................................................................
ii
Table of Contents
.....................................................................................................................
iv
List of Figures
.........................................................................................................................
vii
List of
Tables............................................................................................................................
xi
List of
Abbreviations...............................................................................................................
xii
Statement of Original Authorship
..........................................................................................
xiv
Acknowledgements
.................................................................................................................
xv
Chapter 1: Introduction
......................................................................................
1
1.1 Background
.....................................................................................................................
1
1.2 Definitions
......................................................................................................................
3
1.3 Problem Statement
..........................................................................................................
4
1.4 Research Questions
.........................................................................................................
6
1.5 Scope
...............................................................................................................................
7
1.6 Hypotheses
......................................................................................................................
8
1.7 Aim and Objective
..........................................................................................................
9
1.8 Significance
..................................................................................................................
10
1.9 Thesis Outline
...............................................................................................................
10
Chapter 2: Literature Review
...........................................................................
13
2.1 Advancements in Traffic Flow Theory
.........................................................................
13 2.1.1 Car-following (CF) Models
................................................................................
13 2.1.1.1 Safe distance CF models
..................................................................................
15 2.1.1.2 Stimulus-response models
...............................................................................
16 2.1.1.3 Psychophysical CF models
..............................................................................
24 2.1.1.4 Fuzzy logic-based CF models
..........................................................................
25 2.1.2 Lane-changing Models
.......................................................................................
26
2.2 Gianluca Antonini’s Pedestrian
Modelling...................................................................
28
2.3 Iztok Lebar Bajec’s Birds Flocking
Model...................................................................
29
2.4 Modelling Explicitly on Area-Based Traffic
................................................................
31
2.5 Microscopic Traffic Software Packages
.......................................................................
43 2.5.1 CF and LC Logics in Aimsun
.............................................................................
43 2.5.2 CF and LC Logics in VISSIM
............................................................................
44
2.6 Model Calibration and Validation in Literature
........................................................... 45
2.7 Summary
.......................................................................................................................
52
Chapter 3: Vehicle Trajectory Data
.................................................................
55
3.1 Description of Dataset
..................................................................................................
55
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Microscopic Modelling of the Area-Based Traffic Flow v
3.2 Data
Exploration...........................................................................................................
57 3.2.1 Characteristics of Vehicle Trajectories
.............................................................. 57
3.2.2 Macroscopic Characteristics
..............................................................................
58 3.2.3 Following Behaviour Data Extraction
...............................................................
62
3.3 Data Partition
................................................................................................................
68
3.4 Strength and Limitation of Data
...................................................................................
68
Chapter 4: Theoretical Model Development
................................................... 71
4.1 Behavioural Framework
...............................................................................................
71
4.2 Discrete Choice Theory in Transportation
...................................................................
72
4.3 Modelling Framework
..................................................................................................
74 4.3.1 Area-Selection
....................................................................................................
74 4.3.1.1 Defining the
alternatives..................................................................................
75 4.3.1.2 Defining the attributes of an alternative
.......................................................... 78
4.3.1.3 Modelling the selection of an alternative
........................................................ 82 4.3.2
Vehicle Movement
.............................................................................................
84
4.4 Model Calibration Framework
.....................................................................................
87
4.5 Area-Selection Model Development for Calibration
................................................... 88
4.6 Area-Selection Model Selection
...................................................................................
89
4.7 Indicators of Performance Measures During Calibration of
Area-Selection Model .... 90
4.8 Vehicle Movement Model Calibration
Procedure........................................................
92
4.9 Model Validation Framework
......................................................................................
93
Chapter 5: Model Calibration for Car
.............................................................
95
5.1 Data Build-Up for NLOGIT
.........................................................................................
95
5.2 MNL Model Selection for Cars
....................................................................................
97
5.3 Calibration Results For Selected MNL Model
.............................................................
98
5.4 Performance During MNL Model Calibration
.............................................................
99
5.5 Performance of MNL Model on Validation Data
...................................................... 102
5.6 MIDM Calibration for Car
.........................................................................................
105
Chapter 6: Model Calibration for Motorcycle
.............................................. 107
6.1 Data Build-Up for NLOGIT
.......................................................................................
107
6.2 MNL Model Selection for Motorcycle
.......................................................................
108
6.3 Calibration Results for Selected MNL Model
............................................................
109
6.4 Performance during MNL Model Calibration
............................................................
110
6.5 Performance of MNL Model on Validation Data
...................................................... 113
6.6 MIDM Calibration for
Motorcycle.............................................................................
114
Chapter 7: Model Application: Simulating area-based traffic
.................... 117
7.1 Initial and Boundary Conditions for the Simulation
.................................................. 117
7.2 Model Parameters for the Simulation
.........................................................................
117
7.3 Performance of the Model
..........................................................................................
119
7.4 Macroscopic Properties of the Model
........................................................................
121
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vi Microscopic Modelling of the Area-Based Traffic Flow
Chapter 8: Conclusions and Future
Research............................................... 127
8.1 Conclusions
.................................................................................................................
127
8.2 Future Research Directions
.........................................................................................
132
Bibliography
...........................................................................................................
135
Appendices
..............................................................................................................
143
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Microscopic Modelling of the Area-Based Traffic Flow vii
List of Figures
Figure 1 A schematic diagram for traffic flow differentiating
(a) lane-based,
from (b) area-based (a rectangular block represents a vehicle
and an
arrow indicates flow direction).
.....................................................................
1
Figure 2 A real scenario for traffic flow differentiating (a)
lane-based, from (b)
area-based.
.....................................................................................................
2
Figure 3 Outline of the thesis.
....................................................................................
12
Figure 4 Schematic diagram and notations used in CF models.
................................ 14
Figure 5 Schematic and simplified representation of the regimes
of the
Wiedemann’s (1974) CF model (Kesting and Treiber (2013)).
.................. 24
Figure 6 Lane-changing model framework (modified; Ahmed (1999)).
................... 27
Figure 7 Schematic diagram of lane changing process.
............................................. 27
Figure 8 Choice set of Gianluca Antonini’s dicrete choice model
(source:
Antonini et al.
(2006))..................................................................................
29
Figure 9 The perception model used in fuzzy animate (source:
Lebar Bajec
(2005))..........................................................................................................
30
Figure 10 An illustration of inhomogeneous CA modelling for car
and
motorcycle (Lan and Chang (2005)).
........................................................... 33
Figure 11 Basic structure of CA model used by Mallikarjuna and
Rao (2009)......... 34
Figure 12 Modified cell structure in the CA-based heterogeneous
traffic flow
model at two different occupancy levels (Mallikarjuna and
Rao
(2011))..........................................................................................................
35
Figure 13 Schematic diagrams of motorcycle following
relationships: (a)
longitudinal headway and (b) oblique and lateral headway (T.-C.
Lee
et al. (2009)).
................................................................................................
36
Figure 14 Schematic diagram of path choice decision: (1) lateral
clearance
beside preceding vehicle, (2) lateral distance to
ready-to-overtake
position, and (3) interaction from vehicle behind or beside
(T.-C. Lee
et al. (2009)).
................................................................................................
37
Figure 15 Concept of strip (Mathew et al. (2013)).
................................................... 38
Figure 16 Model structure (Choudhury and Islam (2016)).
....................................... 39
Figure 17 Illustration of the dynamical model for staggered CF
behaviour. The
lateral separation distance between vehicle (𝑛) and vehicle (𝑛 −
1) is defined as 𝑏𝑛 (Jin et al. (2012)).
..................................................................
41
Figure 18 The overview of calibration method for traffic
simulation models. .......... 48
Figure 19 Schematic diagram of data collection site in Chennai,
India
(Kanagaraj et al. (2015)).
.............................................................................
56
Figure 20 Data collection site in Chennai, India (Kanagaraj et
al. (2015)). .............. 56
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viii Microscopic Modelling of the Area-Based Traffic Flow
Figure 21 Distribution of vehicles in the
dataset........................................................
57
Figure 22 Vehicle trajectories for a short period of time.
.......................................... 58
Figure 23 Time-space diagram for vehicle trajectories.
............................................. 59
Figure 24 Empirical flow-density relations for dataset.
............................................. 60
Figure 25 Empirical speed-density relationship for dataset.
...................................... 61
Figure 26 Empirical speed-flow relationship for dataset.
.......................................... 62
Figure 27 Dynamic bounds for vehicle following behaviour in
area-based
traffic.
...........................................................................................................
63
Figure 28 Car to Car following behaviour in heterogeneous
traffic. ......................... 64
Figure 29 Motorcycle to Motorcycle following behaviour in
heterogeneous
traffic.
...........................................................................................................
64
Figure 30 Bus to Bus following behaviour in heterogeneous
traffic. ........................ 65
Figure 31 Auto-rickshaw to Auto-rickshaw following behaviour
in
heterogeneous traffic.
...................................................................................
65
Figure 32 Subject vehicle to first, second, third and group of
leaders in vehicle
following behaviour.
....................................................................................
66
Figure 33 Angle deviation calculation for vehicle (S).
.............................................. 67
Figure 34 Deviation of the movement direction of area-based
heterogeneous
traffic.
...........................................................................................................
68
Figure 35 The conceptual behavioural framework for modelling
area-based
traffic flow.
...................................................................................................
72
Figure 36 An illustration for defining the finite number (N) of
alternatives in the choice space of subject vehicle (S).
....................................................... 75
Figure 37 The choice set of a subject vehicle as a car and
defined angular
bounds for three alternatives based on “macular” peripheral
vision
from −9° to +9°.
.........................................................................................
76
Figure 38 The choice set of a subject vehicle as a motorcycle
and defined
angular bounds for five alternatives based on angle deviation
observed
from aforementioned data.
...........................................................................
77
Figure 39 Self-explanatory illustration of the dynamic of choice
space for a
subject vehicle.
.............................................................................................
78
Figure 40 Illustration of physical barrier for lateral movement
of subject
vehicle (S).
...................................................................................................
80
Figure 41 (a) Angle deviations for a subject vehicle, (b)
spacing for a subject
vehicle, (c) relative speed for a subject vehicle.
.......................................... 82
Figure 42 Calibration framework of the model.
......................................................... 88
Figure 43 Validation framework of the model.
.......................................................... 94
Figure 44 A ROC graph illustrates the performance of the MNL
model for car
in different time steps during calibration.
.................................................. 102
Figure 45 Distribution of observations in the two datasets.
..................................... 104
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Microscopic Modelling of the Area-Based Traffic Flow ix
Figure 46 Distribution of calibrated parameters of the MIDM from
130 car
trajectories.
.................................................................................................
106
Figure 47 Distribution of RMSE of the MIDM from 130 car
trajectories. .............. 106
Figure 48 A ROC graph illustrates the performance of the MNL
model for
motorcycle during calibration.
...................................................................
112
Figure 49 Distribution of calibrated parameters of the MIDM from
40
motorcycle trajectories.
..............................................................................
115
Figure 50 Distribution of RMSE of the MIDM from 40 motorcycle
trajectories. ... 116
Figure 51 Flow chart of the model simulation and validation.
................................ 119
Figure 52 Distribution of errors from five replications of the
model for Car-I. ...... 120
Figure 53 Box and whisker plots for three cars randomly selected
from five
replications of the model.
...........................................................................
121
Figure 54 Box and whisker plots for three motorcycles randomly
selected from
five replications of the model.
...................................................................
121
Figure 55 Observed vehicle trajectories in a 60 (s) x 100 (m)
time-space region. .. 122
Figure 56 Simulated vehicle trajectories in a 60 (s) x 100 (m)
time-space
region.
........................................................................................................
123
Figure 57 Illustration of total number of vehicles in defined
time-space. ............... 124
Figure 58 a) Illustration of time series of the total time taken
(TTT in seconds)
for vehicles defined in time-space windows; and b) The
comparison of
the observed (x-axis) and simulated (y-axis) TTT using
45-degree
line..............................................................................................................
125
Figure 59 a) Illustration of time series of the total distance
travelled (TDT in
meters) for vehicles defined in the time-space windows; and b)
The
comparison of the observed (x-axis) and simulated (y-axis) TDT
using
45-degree
line.............................................................................................
126
Figure 60 Normal quantile-quantile (Q-Q) plots for estimated
parameters of the
MIDM.
.......................................................................................................
146
Figure 61 Gamma quantile-quantile (Q-Q) plots for estimated
parameters of the
MIDM.
.......................................................................................................
147
Figure 62 Weibull quantile-quantile (Q-Q) plots for estimated
parameters of
the MIDM.
.................................................................................................
147
Figure 63 Kernel density curves for different smoothing
functions for
maximum acceleration.
..............................................................................
150
Figure 64 Kernel density curves for different smoothing
functions for safety
time headway.
............................................................................................
150
Figure 65 Kernel density curves for different smoothing
functions for linear
jam distance.
..............................................................................................
151
Figure 66 Kernel density curves for different smoothing
functions for non-
linear jam distance.
....................................................................................
151
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x Microscopic Modelling of the Area-Based Traffic Flow
Figure 67 The goodness of fit measured by kernel cdf with
empirical cdf of
maximum acceleration.
..............................................................................
152
Figure 68 The goodness of fit measured by kernel cdf with
empirical cdf of
safety time headway.
..................................................................................
153
Figure 69 The goodness of fit measured by kernel cdf with
empirical cdf of
linear jam distance.
....................................................................................
153
Figure 70 Simulated and observed trajectory for a Car-I
......................................... 155
Figure 71 Simulated and observed trajectory for a Car-II
....................................... 156
Figure 72 Simulated and observed trajectory for a Car-III
...................................... 156
Figure 73 Simulated and observed trajectory for a Car-IV
...................................... 157
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Microscopic Modelling of the Area-Based Traffic Flow xi
List of Tables
Table 1 Example of IDM model parameters (modified) used by
Treiber et al.
(2000)
...........................................................................................................
22
Table 2 Comparative overview for different CF models with
calibration
parameters
....................................................................................................
23
Table 3 A brief overview of the calibration of CF models
formulated based on
trajectory
data...............................................................................................
50
Table 4 Statistics for the leaders in vehicle following
behaviour .............................. 66
Table 5 Number of parameters to be estimated in different models
.......................... 89
Table 6 A sample of choice data format in NLOGIT for individual
car.................... 96
Table 7 Different log-likelihood values of MNL models for car
............................... 97
Table 8 Estimated parameters of the selected MNL model for cars
.......................... 98
Table 9 Illustration of TP, FP, TN and FN for each alternative
during model
calibration on
car........................................................................................
100
Table 10 Accuracy of prediction for each alternative during
calibration on car ..... 100
Table 11 The summary of the performance measured of the MNL
model for
car in three different time steps during calibration
.................................... 101
Table 12 The performance of MNL model for car on validation data
..................... 104
Table 13 Comparison of actual observation in the two datasets
.............................. 105
Table 14 A sample of choice data format in NLOGIT for individual
motorcycle .. 107
Table 15 Different log-likelihood values of the MNL models for
vehicle type
“Motorcycle”..............................................................................................
108
Table 16 Estimated parameters of the selected MNL model for
motorcycles ......... 109
Table 17 Illustration of TP, FP, TN and FN for each alternative
during model
calibration
..................................................................................................
111
Table 18 Summary of the performance measured of the MNL model
for
motorcycles during calibration
..................................................................
112
Table 19 The performance of MNL model for motorcycle for the
validation
data
.............................................................................................................
114
Table 20 List of K-S Test on Estimated Parameters of the MIDM
......................... 148
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xii Microscopic Modelling of the Area-Based Traffic Flow
List of Abbreviations
ASC alternative specific constant
CA cellular automata
CDF cumulative probability density function
CF car-following
DLC discretionary lane-change
FN false negative
FP false positive
FVD full velocity difference
GF generalized force
GM general motor
IDM intelligent driver model
LC lane-changing
LCV light commercial vehicle
LR log-likelihood ratio
MIDM modified intelligent driver model
MLC mandatory lane-change
MNL multinomial logit
MNP multinomial probit
OV optimal velocity
PCE passenger car equivalent
RMSE root mean square error
ROC receiver operating characteristics
TN true negative
TP true positive
-
Microscopic Modelling of the Area-Based Traffic Flow xiii
TTT total time taken
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xiv Microscopic Modelling of the Area-Based Traffic Flow
Statement of Original Authorship
The work contained in this thesis has not been previously
submitted to meet
requirements for an award at this or any other higher education
institution. To the
best of my knowledge and belief, the thesis contains no material
previously
published or written by another person except where due
reference is made.
Signature:
Date:
Nikhil Chandra Sarkar
May 2019
QUT Verified Signature
-
Microscopic Modelling of the Area-Based Traffic Flow xv
Acknowledgements
I wish to express my gratitude to principal supervisor Dr Ashish
Bhaskar,
associate supervisor Dr Marc Miska and external supervisor Dr
Zuduo Zheng for
their mentoring, advice, cooperation and guidelines to complete
this thesis.
I also extend my deepest thanks to Prof Richard Brown for his
valuable
suggestions and advice during my research journey.
I would like to thank my fellow friends, fellow students and
administrative
staff at school of civil engineering and built environment and
the school of
mathematical sciences at Queensland University of
Technology.
I am grateful to Queensland University of Technology for the
financial support
to complete this research. The financial support from the
International Postgraduate
Research Sponsorship (IPRS), QUT Postgraduate Research Award
(QUTPRA) and
QUT Excellence Top-up Scholarship are acknowledged.
I would also like to thank professional editor, Kylie Morris,
who
provided copyediting and proofreading services, according to
university-endorsed
guidelines and the Australian Standards for editing research
theses.
I am grateful to my parents, brothers, sisters, uncles, and
other family members
for their endless support and encouragement.
Finally, I express my deepest love and affection to my wife,
Runa Kundu, and
my adorable daughter, Win Sarkar, for their endless love, care,
and support while
I completed this thesis.
-
Chapter 1: Introduction 1
Chapter 1: Introduction
This chapter provides the background for different types of
traffic flow and
modelling (Section 1.1). Section 1.2 presents some basic
definitions related to this
thesis. A detailed problem statement, specific to car-following
models, lane-changing
models, and area-based models is presented separately in the
problem statement
section (Section 1.3). The research questions, scope,
hypotheses, aims, and
objectives, and thesis significance are presented in Sections
1.4-1.8. Finally, the
thesis outline is provided in Section 1.9.
1.1 BACKGROUND
Due to high population density and limited road capacity, the
level of congestion on
the road is generally high for many countries in the world. A
high percentage of
passenger cars on urban roads can be considered homogeneous with
respect to the
vehicle type (herein referred to as homogeneous, Figure 1(a) and
Figure 2 (b). With
respect to vehicle type, heterogeneous traffic (herein referred
to as heterogeneous) is
composed of multiple vehicle types with different behavioural
and mechanical
characteristics. This includes passenger cars, motorcycles,
auto-rickshaws, heavy
vehicles such as buses and trucks and light commercial vehicles
(LCV). In
developing countries (such as India) traffic is heterogeneous
and drivers generally
ignore lane markings. Here, drivers perceive the entire road
space for their
movement and this traffic movement is referred to as area-based
traffic (Figure 1 (b)
and Figure 2 (b)). The resulting traffic flow is significantly
different from lane-based
traffic where drivers maintain lane marking rules while
driving.
(a) Lane-based traffic flow (b) Area-based traffic flow
Figure 1 A schematic diagram for traffic flow differentiating
(a) lane-based, from (b) area-
based (a rectangular block represents a vehicle and an arrow
indicates flow direction).
-
2 Chapter 1: Introduction
(a) Lane-based traffic flow in Washington; (b) Area-based
traffic flow in New Delhi
(Photos’ source: Internet)
Figure 2 A real scenario for traffic flow differentiating (a)
lane-based, from (b) area-based.
Modelling the behaviour of traffic on the road has long been an
area of
research (Greenshields, Channing, and Miller (1935), Pipes
(1953)). In the literature,
traffic flow modelling is generally classified into two major
types (see van
Wageningen-Kessels, Van Lint, Vuik, and Hoogendoorn (2015) for
mesoscopic
modelling based on gas-kinetic theory), such as:
a) Macroscopic: Macroscopic modelling is based on the analogy
of
hydrodynamic continuity equation (Lighthill and Whitham (1955)
and
Richards (1956)) from an aggregated point of view. The traffic
flow state is
characterised by aggregated macroscopic variables, such as
density, flow, and
speed. The first major step in the macroscopic modelling of
traffic was
undertaken by Lighthill and Whitham (1955). Richards (1956)
modified the
model by introducing “shock-waves on the highway” into the
model, and
thereafter this model has widely been known as the LWR model and
is
considered the basis for macroscopic modelling of traffic flow.
Other
researchers, such as Payne (1971), Liu, Lyrintzis, and
Michalopoulos (1998),
and Zhang (1998) derived similar but higher order models for
homogeneous
traffic flow based on fluid-dynamic equations. Wong and Wong
(2002)
developed a multi-class traffic flow model as an extension of
the LWR model
with heterogeneous drivers based on their choice of speeds. The
model
considers the distribution of these drivers characterised by
their choice of
speeds in a traffic stream.
b) Microscopic: Microscopic modelling is developed from a
disaggregated point
of view, which describes the dynamic of an individual vehicle
composing the
-
Chapter 1: Introduction 3
traffic stream. The dynamic variables of the models represent
microscopic
properties; for example, the position and velocity of the
vehicles.
Microscopic modelling was developed to advance the traffic flow
theory and
describes two primary individual driving behaviours on roads;
that is, car-
following (CF) (see Saifuzzaman and Zheng (2014) and Brackstone
and
McDonald (1999) for a review) and lane-changing (He et al.) (see
Zheng
(2014) for a review). Microscopic modelling has gained the
interest of
researchers and numerous mathematical models have been developed
to
advance the microscopic traffic flow theory.
Several simulation packages, such as, DynaMIT, VISSIM, Aimsun,
AVENUE,
MITSIMLab, SUMO, METANET, and so forth have been developed (see
Jaume
Barceló (2010) for details about these packages). These micro-
simulation tools are
widely used to explicitly analyse the driving behaviours in
controlled environments,
test the performance of the traffic management strategies, and
to understand the
underlying behaviour of traffic congestion, and so forth in
lane-based traffic. Traffic
flow characteristics in area-based heterogeneous conditions vary
substantially from
lane-based traffic. In area-based traffic, the flow becomes more
complex in nature
and challenging to model due to the unregulated vehicle-dynamic,
often lateral
separation and movement, and interactions among different
vehicles over time.
However, due to a lack of models for area-based traffic flow
(see Section 2.2), the
simulation packages for lane-based traffic are generally not
applicable for such
traffic conditions.
1.2 DEFINITIONS
In the absence of any clear definition, the following
definitions related to
traffic flow modelling are used in this thesis:
• Homogeneous traffic flow: The flow of vehicles is composed of
identical
vehicles.
• Heterogeneous traffic flow: The flow of vehicles is composed
of different
types of vehicles.
• Lane-based flow: Here, vehicles follow lane discipline and
each vehicle
should align itself with the centre of the lane for lateral
movement. The driver
-
4 Chapter 1: Introduction
crosses the lane-marking only for overtaking, lane-changing, and
merging
manoeuvres.
• Area-based (or non-lane based) flow: Vehicles do not follow
the lane-based
discipline and instead look for an available gap in both lateral
and
longitudinal directions at all times.
• Acceleration models (or CF models): These models describe the
longitudinal
movement of the vehicle in the traffic stream.
• LC models: These models describe the lateral movement of the
vehicle in
different traffic conditions.
• Decision models: Different discrete-choice situations, such as
entering a
priority road, are defined by decision models.
1.3 PROBLEM STATEMENT
Most research has focussed on lane-based traffic conditions, and
as such,
modelling the vehicle dynamics in area-based traffic conditions
has been overlooked.
Area-based movements and the heterogeneity of the vehicles on
the road make
it possible for drivers to use the entire road space along
lateral and longitudinal
directions, which differs from the driving behaviour of
lane-based homogeneous
traffic. In area-based traffic, a subject vehicle is not
necessarily influenced by a
single vehicle, but by the vehicles in its visual perception
area. The continuous lateral
movement in area-based traffic cannot be directly modelled by
existing LC models.
Additionally, the driving behaviour of the subject vehicle that
models the
longitudinal movement in lane-based traffic through existing CF
models may not be
suitable in area-based traffic. To address these issues related
to area-based traffic,
several area-based microscopic models and limited macroscopic
model have been
developed.
The leading-edge area-based traffic flow models can be grouped
into:
• Coordinate based simulation framework (e.g., Benekohal and
Treiterer
(1988); Hossain and McDonald (1998));
-
Chapter 1: Introduction 5
• Cellular automata (Antonini, Bierlaire, & Weber) based
models (e.g., Lan and
Chang (2005); Mathew, Gundaliya, and Dhingra (2006);
Mallikarjuna and
Rao (2009));
• Agent-based integrated simulation model (e.g., T.-C. Lee,
Polak, and Bell
(2009));
• Staggered CF model based on lateral separation (e.g., Jin,
Wang, Xu, and
Huang (2012) );
• Strip-based framework (e.g., Mathew, Munigety, and Bajpai
(2013));
• Random utility and stimulus-response based framework (e.g.,
Choudhury and
Islam (2016)); and
• Porous flow based macroscopic modelling (e.g. Nair,
Mahmassani, and
Miller-Hooks (2011)).
The coordinate-based framework defines the vehicle position in a
coordinate
system and uses existing CF and LC behaviour models in
area-based traffic. CA
model results are dependent on rules that should incorporate
different area-based
driving behaviours. The agent-based integrated simulation model
captures the
characteristic movement of motorcycles. The characteristic
dynamics of motorcycles
include travelling alongside another vehicle, swerving,
tailgating, oblique following,
shorter headway, and virtual lane-based movement. The staggered
CF model
considers the effects of lateral separation in area-based
traffic and captures staggered
following behaviour. The strip-based approach is a
simplification of lane-based
simulation, with virtual lanes termed as strips. The porous flow
approach is
analogous to granular flow through a porous medium that relies
on different pore
sizes and pore space distributions are complicated for
area-based traffic. In area-
based traffic, the drivers look for the suitable gaps in
different situations to make
manoeuvres like merging and crossing. The gap requirements
basically vary from
one vehicle class to other. In addition, because of the
area-based movement for
mixed traffic, it is difficult for the drivers to perceive the
time gap. It can be
concluded that research on area-based modelling is still in the
early stages and there
are significant avenues for model development and more realistic
representations of
driving behaviour.
-
6 Chapter 1: Introduction
The driving behaviour of area-based flow involves frequent
lateral movements
while progressing longitudinally. The lateral movement of
individual vehicles
basically generate the angular deviation from the direction of
the flow. This is
analogous to the pedestrian walking behaviour, though the
manoeuvre is limited by
the mechanical characteristics of the vehicle. In literature,
the discrete choice
framework is proposed by Antonini et al. (2006) to model the
pedestrian dynamics.
Motivated by which, this research aims to develop a parsimonious
modelling
framework to capture the driving behaviour of area-based traffic
flow. We propose a
two-step hierarchical modelling framework where a discrete
choice framework in
step-1 is developed to identify the direction in which the
vehicle can move. The
second step of vehicle movement specify how far the vehicle will
move along the
direction selected in step-1. The combined effect of these two
steps defines the
trajectory of the vehicle.
Discrete choice modelling theory has long been utilised in
number of
engineering and behavioural applications. The uniqueness of its
applications in
literature is the way the alternatives and their attributes are
defined. The novelty of
our research compared to the discrete choice framework proposed
by Antonini et al.
(2006) includes:
a) The logical procedure to define the number of alternatives
and the attributes
of the alternatives for direction selection in area-based
traffic (in step -1); and
b) Modelling the vehicular movement in step-2.
The goal is to develop a robust, realistic, and easy to
implement simulation
framework for modelling the driving behaviour of area-based
heterogeneous traffic.
1.4 RESEARCH QUESTIONS
Modelling area-based traffic behaviours is a challenging task
due to its
complex driving manoeuvres. To develop a theoretical model for
area-based traffic
flow, this research addresses the following key research
questions:
• What are the limitations of existing traffic flow models for
area-based traffic
flow conditions? This question is partially addressed by the
research problem.
See Chapter 2 for details about the strengths and limitations of
existing area-
based traffic flow models.
-
Chapter 1: Introduction 7
• How can the area-based dynamics of traffic be modelled
(conceptually and
mathematically)? See Chapter 4 for details about the methodology
of the
model development.
o What are the key parameters that influence area-based traffic
flow?
This question is addressed in Chapter 3.
o How can the lateral and longitudinal movements of different
types of
vehicles be incorporated into the model? This question is
addressed in
Chapter 4.
o How can we calibrate the aforementioned model? This question
is
addressed in Chapter 5 and Chapter 6 regarding model
calibration.
o How can we validate the aforementioned model? This question
is
addressed in Chapter 7 regarding model validation.
1.5 SCOPE
This research focuses on microscopic modelling of the
area-based
uninterrupted traffic flow, where the breakdown of the traffic
is due only to internal
friction between the vehicles. Consideration of external
influences, such as traffic
signals, pedestrians, on-street parked vehicles, road geometry,
etc., was deemed to be
outside the scope of this research.
The model can be furthered extended to interrupted traffic flow,
such as
signals, by incorporating stop-go conditions into the model.
Before developing the
methodological framework for area-based traffic, the main
challenges behind the
microscopic modelling of area-based heterogeneous traffic are
first demonstrated.
The main challenges are described below:
• Mechanical variation of heterogeneous traffic: Significant
differences have
been observed in the mechanical behaviour of heterogeneous
traffic
resulting in significant variations in the driving manoeuvres of
these
vehicles.
• Area-based dynamic: Apart from the mechanical variation and
heterogeneity
in driving manoeuvrability, vehicles do not maintain proper lane
discipline
due to the off-centre position on the lane. Several different
vehicles can
occupy a single lane. Even if there are lane-markings, in
congested driving
-
8 Chapter 1: Introduction
situations, vehicles also frequently position themselves in
between other
vehicles to share the space and thus occupy multiple lanes. This
creates
difficulty in identifying some core elements for the model
formulation, such
as identification of the proper leader, the lane changing
manoeuvre, the
position of vehicles, and so on. In area-based situation, the
vehicle can stay
anywhere on the road without maintaining the lane mark. Thus,
lateral
movement behaviour can be modelled as a continuous process
rather than
discrete.
• Physical shape variation: The vehicles in heterogeneous
traffic do not have
the uniform physical shape and dimensions of homogeneous
traffic. Based
on the physical dimension, vehicles occupy the space on the road
and this
varies from vehicle type to type.
• Data noise: In contrast to homogeneous traffic, heterogenous
traffic causes
difficulties for image processing software to maintain accuracy
for lateral
position. Thus, this becomes a matter of concern for microscopic
modelling
of area-based traffic.
In this study, the first two challenges—mechanical variation and
area-based
dynamic—were considered in the microscopic modelling; while the
last two—
physical shape variation and data noise—were deemed to be
outside the scope of this
research. This research used data from an urban midblock road
section in Chennai,
India collected by Kanagaraj, Asaithambi, Toledo, and Lee (2015)
and it was
assumed that the data were accurate.
1.6 HYPOTHESES
The following hypotheses were developed and tested using a real
traffic dataset
for area-based traffic flow:
A. Existing traffic flow models for lane-based traffic are not
directly
applicable for area-based traffic.
B. Frequent lateral movement causes the vehicles to maintain
less CF
behaviours.
C. In area-based traffic flow, it is assumed that the subject
vehicle is
influenced by the potential individual vehicle in choice space
from a
-
Chapter 1: Introduction 9
perceived visual area in front of the decision maker (i.e.,
subject vehicle).
The potential individual vehicle in choice space from a
perceived visual
area of a subject vehicle represents the potential leader.
D. The driving behaviour of area-based traffic very often
involves lateral
movements, ignores lane markings, and perceives the entire road
space
while progressing longitudinally. The lateral movements of the
individual
vehicle in area-based traffic basically generate angular
deviation from the
direction of the flow and a discrete choice modelling framework
can be
applied to model the area-based movement direction in choice
space from
the visual perception area of a subject vehicle.
E. A vehicle-following model can be developed to simulate
acceleration
along the direction of selected alternatives from choice space
for a subject
vehicle at the next time step given the dynamics (acceleration
and
position) of all vehicles up to the current time step.
1.7 AIM AND OBJECTIVE
This research aims to enhance the knowledge on modelling
area-based traffic
flow. Specifically, the objectives include:
• The development of a theoretical model to capture the
area-based driving
behaviour of heterogeneous traffic. This objective is addressed
through the
research question “How can the area-based dynamics of traffic be
modelled
(conceptually and mathematically)?”. (see Chapter 4 for details
about the
model’s development).
• To perform calibration of the model with the goal of studying
the robustness
of the model with respect to its parameters. Specifically, by
estimating the
parameters of the model for which the model is accurate and
reliable. This
objective is addressed by the research question “How can we
calibrate the
aforementioned model?” (see Chapter 5 and Chapter 6 for details
about the
calibration and performance measures of the model during
calibration).
• Testing and validation of the model through a case study of a
real road
segment. This objective is addressed by the research question
“How can we
-
10 Chapter 1: Introduction
validate the aforementioned model?”. (see Chapter 7 for details
about model
testing and validation).
1.8 SIGNIFICANCE
The significance of this research can be categorised in two
different aspects:
scientific and practical contributions.
• Scientific contributions: This research aims to develop a
leading-edge and
robust theoretical model based on the driving behaviour of
area-based traffic
to advance traffic flow theory. A framework is proposed for
microscopically
modelling the aggregated lateral and longitudinal driving
behaviour in area-
based traffic. Specifically:
o The proposal of a discrete choice-based model to capture the
direction
of the movement of vehicles in area-based traffic.
o The existing IDM model is modified to incorporate the
area-based
movement of the vehicle along the selected direction.
• Practical contributions: The proposed methodology is seen as a
building
block towards the development of a micro-simulation model for
area-based
traffic flow. The simulation model could be utilised for
detailed traffic impact
analysis, decision support systems, explicitly analysing driving
behaviours to
understand the underlying behaviour of traffic congestion, and
so on, in area-
based traffic. The proposed framework can also act as a research
tool for
further enhancement and the associated analytical techniques can
be used in
the planning, design, and operation of the transportation
system.
1.9 THESIS OUTLINE
The remainder of this thesis is designed as shown in Figure 3.
Chapter 2
describes the leading-edge literature regarding area-based
traffic flow modelling,
including CF and LC models, pedestrian modelling, bird flocking
modelling,
microscopic traffic simulators, and a brief overview of the
calibration and validation
methods for the traffic flow models used. This chapter addresses
the research
question “What are the limitations of existing traffic flow
models on area-based
traffic flow conditions?”.
-
Chapter 1: Introduction 11
Chapter 3 provides a description of the vehicle trajectory
dataset for area-based
traffic, with data exploration based on the characteristics of
the trajectories,
macroscopic characteristics, and vehicle following behaviour
data extraction
presented in separate sections. This chapter addresses the
research question “Which
key parameters influence area-based traffic flow?”.
Chapter 4 defines the comprehensive driving behavioural and
methodological
framework for area-based traffic. The discrete choice theory in
transportation, a
detailed two step modelling framework, model calibration
framework, area-selection
model development and selection, the indicators of the
performance measures during
calibration, the vehicle movement model calibration procedure,
and the entire model
validation framework are presented in separate sections. This
chapter addresses the
research question “How can the lateral and longitudinal movement
of different types
of vehicles be incorporated into the model?” and develops a
theoretical model to
capture the area-based driving behaviour of heterogeneous
traffic.
Chapter 5 presents the calibration results from the area-section
and the vehicle
movement steps of the proposed modelling for subject vehicles as
cars
independently. This chapter presents a detailed description of
the data preparation
for the MNL model calibration for the subject vehicle as a car,
the results from the
MNL model selection, the estimation results for the selected MNL
model, the
performance measured during calibration of the MNL model, and
the estimation
results from the MIDM model in separate sections. This chapter
partially addresses
the research question “How can we calibrate the aforementioned
model?” and
estimates the parameters of the model for which the model is
accurate and reliable.
Chapter 6 presents the calibration results from the area-section
and vehicle
movement steps of the proposed modelling for subject vehicles as
motorcycles
independently. This chapter provides a detailed description of
data preparation for
MNL model calibration for the subject vehicle as a motorcycle,
the results from the
MNL model selection for motorcycles, the estimation results for
the selected MNL
model for motorcycles, the performance measured during
calibration of MNL model,
and the estimation results from the MIDM model for motorcycles
in separate
sections. This chapter also partially addresses the research
question “How can we
calibrate the aforementioned model?” and estimates the
parameters of the model for
which the model is accurate and reliable.
-
12 Chapter 1: Introduction
Chapter 7 presents the results from the validation of the entire
model. The
results from the comparison of the performance of the model, the
simulated
macroscopic characteristics, and the statistical measures of the
model validity are
presented in separate sections. This chapter addresses the
research question “How
can we validate the aforementioned model?” and tests and
validates the model using
a case study using a real road segment.
The conclusions and the future research directions of the
proposed model for
area-based heterogeneous traffic flow are presented in Chapter
8.
Figure 3 Outline of the thesis.
-
Chapter 2: Literature Review 13
Chapter 2: Literature Review
This chapter reviews the literature on the following topics:
advancements in
traffic flow theory (Section 2.1) describes car-following and
lane-changing models,
Section 2.2 discusses Gianluca Antonini’s pedestrian modelling
of pedestrian
dynamic behaviours and develops the conceptual modelling
framework for this
study. Section 2.3 examines Iztok Lebar Bajec’s birds flocking
model for the fuzzy
based simulation of birds flocking. Section 2.4 presents the
leading-edge area-based
traffic flow models and their limitations. Commercial
microscopic traffic simulators
are presented in Section 2.5. A brief overview of the
calibration and validation
methods for traffic flow models in the literature is discussed
in Section 2.6. Finally,
a summary of the comprehensive review of advancements in
car-following, lane-
changing, and area-based traffic flow modelling are presented in
Section 2.7.
2.1 ADVANCEMENTS IN TRAFFIC FLOW THEORY
Traffic flow theory has been studied and developed since 1935,
when
Greenshields et al. (1935) measured observed traffic flow
characteristics using
photographic measurement techniques for the first time. Since
then, numerous traffic
flow models have been developed to advance the theory in the
category of
microscopic simulation. Microscopic traffic models describe
individual driving
behaviours, such as car-following (CF) and lane-changing (He et
al.). These two
basic driving behaviours have been studied over the years
(Chandler, Herman, and
Montroll (1958) for the CF model; Gipps (1986) for the LC
decision model) to
develop various micro-simulation techniques to capture real
traffic scenarios. The
detailed modellings of these behaviours of individual vehicles
are discussed in the
following sub-sections.
2.1.1 Car-following (CF) Models
The longitudinal movement of vehicles following one another in
the same lane
of the traffic stream is demonstrated by the principle of
Newtonian physics (equation
of motion), which has been studied for almost sixty-five years
(Pipes (1953)). In
recent years, a detailed understanding of this key process has
become more important
in traffic simulation, capacity analysis, level-of-service, and
traffic safety research. A
-
14 Chapter 2: Literature Review
schematic diagram of two vehicles traveling in same lane and
some relevant
notations describing CF models are shown in Figure 4. The lead
vehicle is denoted
by "L” with length 𝑙𝐿, a subject vehicle is denoted by "S" and
the time of observation
of vehicle is denoted by the variable "𝑡". These notations are
used consistently
throughout this thesis, due to their simplicity and
consistency.
Figure 4 Schematic diagram and notations used in CF models.
The definitions of the variables illustrated in Figure 4 are
presented below:
𝑥𝐿(𝑡) is the position of the lead vehicle L at time 𝑡
𝑥𝑆(𝑡) is the position of the subject vehicle S at time 𝑡
�̇� 𝐿(𝑡) is the speed of the lead vehicle at time 𝑡
�̇� 𝑆(𝑡) is the speed of the subject vehicles at time 𝑡
�̈� 𝑆(𝑡) is the acceleration or deceleration of the subject
vehicle at time 𝑡
𝑙𝐿 is the length of the lead vehicle
[𝑥𝐿(𝑡) − 𝑥𝑆(𝑡)] is the spacing (front bumper to front bumper)
between a leader
and a follower at time 𝑡
[𝑥𝐿(𝑡) − 𝑥𝑆(𝑡) − 𝑙𝐿] is the separation between the front bumper
of subject
vehicle to the rear bumper of lead vehicle at time 𝑡
The basic theory of the CF model can be expressed as the
response of the
subject vehicle influenced by the perceived stimulus resulting
from the driving
-
Chapter 2: Literature Review 15
behaviour of the lead vehicle. The spacing, separation, speed,
and relative speed can
be considered elements of a stimulus function. In addition, the
sensitivity parameter
can have several functional forms in different CF models of
different researchers,
such as constant in Chandler et al. (1958) model, step
functioning in Herman,
Montroll, Potts, and Rothery (1959) model, and reciprocal
spacing in Gazis, Herman,
and Potts (1959) model. CF models are broadly discussed in the
following categories
by Siuhi (2009):
• safe distance CF models;
• stimulus-response CF models;
• psychophysical CF models;
• fuzzy logic-based CF models.
2.1.1.1 Safe distance CF models
The minimum safe distance model was developed by Pipes (1953),
inspired by
the vehicle separation code that states that “A good rule for
the following another
vehicle at a safe distance is to allow yourself the length of a
car (about fifteen feet,
4.5 m) for every ten miles per hour (16 km/h) you are
traveling”. The model always
maintains the minimum safe time headway distance as a linear
function of speed.
The model has shortcomings due to the lack of other important
variables, such as
relative speed. Relative speed may influence how drivers always
maintain a
minimum safe following distance during a CF situation. As a
result, the driver
response behaviour of acceleration and deceleration may be
affected. Gipps (1981)
developed a model to overcome the shortcomings of Pipes (1953)
model by
incorporating the thresholds for driver performance and safe
speed with respect to
the lead vehicle. Gipps (1981) suggested that drivers should
maintain the speed limit
and the vehicles should perform acceleration and deceleration
capacities accordingly.
The model uses the form depicted in Equation (1):
[𝑥𝐿(𝑡) − 𝑥𝑆(𝑡) − 𝑙𝐿] ≥[�̇� 𝐿(t)]
2
2𝑏𝐿+ [�̇� 𝑆(t)]
𝜏
2+ [�̇� 𝑆(𝑡 + 𝜏)]𝜏 −
[�̇� 𝑆(𝑡+𝜏)]2
2𝑏𝑆 (1)
where 𝜏 is the reaction time of the driver.
𝑏𝑖 is the most severe braking of the vehicle 𝑖 and (𝑏𝑖 < 0);
𝑖 = 1, 2, … , 𝑛.
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16 Chapter 2: Literature Review
The model was validated using real trajectory data from a three
lane divided
highway. The results replicated reasonable driving behaviour and
propagated the
disturbance of traffic flow, though the model assumed the same
driver response time
lags and ignored vehicles composition.
2.1.1.2 Stimulus-response models
The General Motors Research Laboratories are part of General
Motors (GM),
who first developed the CF models (Chandler et al. (1958). The
researchers
associated with GM developed a numbers of CF models that
described the individual
response behaviour of a subject vehicle. Basically, each driver
of a vehicle in CF
models maintains the following relationship:
𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒 = 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 (𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑖𝑡𝑦, 𝑠𝑡𝑖𝑚𝑢𝑙𝑢𝑠) (2)
GM models define stimulus as the relative speed between a
subject vehicle and a
lead vehicle. The response is then taken as acceleration or
deceleration according to
the positive or negative relative speed of the subject-lead
vehicles, respectively.
When a negative relative speed is generated in the traffic
stream, the deceleration is
taken as response of a subject vehicle. In contrast, when a
positive relative speed is
generated in the traffic flow, the acceleration is triggered as
a response of a subject
vehicle. Moreover, the magnitude of the response depends on the
sensitivity
parameter, which can have several functional forms.
The first version of the stimulus-response based linear CF model
was
developed by Chandler et al. (1958). The response is considered
to be directly
proportional to the relative speed of the subject-lead vehicle,
which is described in
Equation (3):
�̈� 𝑆(𝑡 + 𝜏) = 𝜆(�̇�𝐿(𝑡) − �̇�𝑆(𝑡)) (3)
where, �̈�𝑆(𝑡 + 𝜏) represents the response parameter, such as
the acceleration or
deceleration of the subject vehicle at time 𝑡 + 𝜏, 𝜏 is the
reaction time, �̇�𝐿(𝑡)
represents the speed of the lead vehicle L at time 𝑡, �̇�𝑆(𝑡)
represents the speed of the
subject vehicle 𝑆 at time 𝑡, and 𝜆 is the constant sensitivity
parameter.
Driving behaviour in the form of acceleration, no response, or
deceleration of
this model depends on the positive, zero, or negative value of
relative speed,
respectively. The model is considered a constant sensitivity
parameter; however,
-
Chapter 2: Literature Review 17
field experiments show significant variation in sensitivity
values. Moreover, the
sensitivity parameter appears to closely depend on the distance
between the subject
and lead vehicles spacing. In addition, the constant sensitivity
theory does not show
the maximum flow characteristics at an optimum density and does
not yield flow of
zero when the density is zero.
Gazis et al. (1959) developed a CF model by addressing the
shortcomings of
Chandler et al. (1958) linear CF model by incorporating spacing
between the subject
and lead vehicles in the sensitivity parameter. The sensitivity
(𝜆) is no longer taken
as constant as in previous model (Chandler et al. (1958)) but is
inversely proportional
to spacing. Then model becomes as follows:
�̈�𝑆(𝑡 + 𝜏) = 𝜆(�̇�𝑆(𝑡)(�̇�𝐿(𝑡)−�̇� 𝑆(𝑡))
𝑥𝐿(𝑡)−𝑥𝑆(𝑡) (4)
Gazis et al. (1959) investigated some of the properties of the
steady-state flow of
traffic based on CF theory of vehicle interaction. The
experimental measurement
indicated that an optimum density exists for which the traffic
flow is a maximum.
Gazis, Herman, and Rothery (1961) developed a generalised
non-linear CF
model based on stimulus-response framework. The model is
mathematically defined
in Equation (5):
�̈�𝑆(𝑡 + 𝜏) =𝜆(�̇� 𝑆(𝑡))
𝛽(�̇� 𝐿(𝑡)−�̇� 𝑆(𝑡))
(𝑥𝐿(𝑡)−𝑥𝑆(𝑡))𝛾 (5)
The model parameters can be easily estimated from the vehicle
trajectory data. The
macroscopic fundamental relationship can be obtained from this
model by setting
𝛽 = 0 and 𝛾 = 2. However, the model is highly sensitive to the
relative speed of the
subject vehicle. When the relative speed becomes zero, any
arbitrary value of
spacing is acceptable which is unrealistic for traffic
safety.
Edie (1961) developed a modified CF model to address the
shortcomings of
Chandler et al. (1958) in low density traffic. In extremely low
density traffic there is
no interaction between two vehicles. Moreover, when the density
approaches zero,
the speed becomes infinite and speed and density relation can
then be derived. Edie
(1961) argued that the sensitivity parameter of a driver is
closely related to the speed
of his /her vehicle. When the speed is very high, greater
sensitivity is required for
safe driving. The Edie’s model (1961) then becomes
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18 Chapter 2: Literature Review
�̈�𝑆(𝑡 + 𝜏) =𝜆(�̇�𝑆(𝑡))(�̇�𝐿(𝑡)−�̇�𝑆(𝑡))
(𝑥𝐿(𝑡)−𝑥𝑆(𝑡))2
(6)
CF behaviour was advanced by Gipps (1981), in which the model
basically
consists of two core components; acceleration and deceleration.
The desired speed of
a subject vehicle is given by:
𝑉𝑆𝑓(𝑡 + 𝜏) = 𝑉𝑆(𝑡) + 2.5 𝑎𝑆
𝑚𝑎𝑥𝜏 (1 −𝑉𝑆(𝑡)
𝑉𝑆𝑑 ) √0.025 +
𝑉𝑆(𝑡)
𝑉𝑆𝑑 (7)
where:
𝑉𝑆𝑓(𝑡 + 𝜏) is the obtained desired speed of subject vehicle S at
time 𝑡 + 𝜏;
𝑉𝑆(𝑡) is the speed of subject vehicle S at time 𝑡 for the
current section;
𝑎𝑆𝑚𝑎𝑥 is the maximum acceleration for subject vehicle S;
𝜏 is the reaction time.
In a congested traffic flow situation, the maximum speed of a
subject vehicle is given
by:
𝑉𝑆𝑐(𝑡 + 𝜏) = 𝑑𝑆
𝑚𝑎𝑥𝜏
+√(𝑑𝑆𝑚𝑎𝑥)2𝜏2 − 𝑑𝑆
𝑚𝑎𝑥 [2 {𝑥𝐿(𝑡) − 𝑙𝐿 − 𝑥𝑛−1(𝑡) − 𝑉𝑆(𝑡)𝜏 −𝑉𝐿
2(𝑡)
𝑑𝐿𝑑 }]
(8)
where: 𝑑𝑆𝑚𝑎𝑥(< 0) is the maximum deceleration desired by
subject vehicle S;
𝑉𝑆𝑐(𝑡 + 𝜏) is the obtained speed of the subject vehicle S at 𝑡 +
𝜏;
𝑥𝑆(𝑡) is the position of subject vehicle S at time 𝑡;
𝑥𝐿(𝑡) is the position of lead vehicle L at time 𝑡;
𝑙𝐿 is the effective length of lead vehicle L at time 𝑡;
𝑑𝐿𝑑 is an estimation of lead vehicle L desired deceleration.
The final speed for subject vehicle S during the time interval
(𝑡, 𝑡 + 𝜏) is defined as:
𝑉𝑆(𝑡 + 𝜏) = min{𝑉𝑆𝑓(𝑡 + 𝜏), 𝑉𝑆
𝑐(𝑡 + 𝜏)} (9)
The position of subject vehicle S is updated in Equation
(10):
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Chapter 2: Literature Review 19
𝑥𝑆(𝑡 + 𝜏) = 𝑥𝑆(𝑡) + 𝑉𝑆(𝑡 + 𝜏)𝜏 (10)
The estimation of deceleration for the lead vehicle (𝑑𝐿) is a
function of a parameter 𝜆
defined per vehicle type named the “sensitivity factor” and the
model then becomes:
𝑑𝐿𝑑 = 𝑑𝐿 ∗ 𝜆 (11)
When 𝜆 < 1, the vehicle underestimates the deceleration of
the leader and as a result
the vehicle becomes more aggressive, decreasing the gap with its
follower. When
𝜆 > 1 , the vehicle overestimates the deceleration of the
leader and as a result the
vehicle becomes more careful, increasing the gap with its
follower.
Bando, Hasebe, Nakayama, Shibata, and Sugiyama (1995) introduced
a time-
continuous optimal velocity (OV) model which is defined by the
acceleration
equation, as shown in Equation (12)
�̈�𝑆(𝑡) = 𝜆{𝑉(Δ𝑥(𝑡)) − �̇�𝑆(𝑡)} (12)
where, Δ𝑥(𝑡) = 𝑥𝐿(𝑡) − 𝑥𝑆(𝑡) is the spacing in between the
subject and lead
vehicles, 𝜆 is the constant sensitivity parameter, which can be
written in another
form, such as 𝜆 =1
𝜏 , where 𝜏 is the reaction time; and 𝑉 is the optimal
velocity
function (OVF), which decides the safety speed for the headway.
The subject vehicle
controls the acceleration or deceleration according to the
difference between the
optimal and his/her own speed. The following OVF is considered
to calibrate the OV
model by Helbing and Tilch (1998)
𝑉(Δ𝑥(𝑡)) = 𝑉1 + 𝑉2 tanh[𝐶1(Δ𝑥(𝑡) − 𝑙𝐿) − 𝐶2] (13)
where 𝑙𝐿 is the length of the lead vehicle; 𝑉1, 𝑉2 , 𝐶1 and 𝐶2
are the calibration
parameters. The resulting optimal parameter values for city
traffic, in particular, road
in Stuttgart are 𝑉1 = 6.75 𝑚/𝑠, 𝑉2 = 7.91 𝑚/𝑠, 𝐶1 = 0.13 𝑚−1, 𝐶2
= 1.57.
Bando, Hasebe, Nakanishi, and Nakayama (1998) then proposed the
next
version of the OV model with explicit delay (i.e., reaction
time), as defined in
Equation (14):
�̈�𝑆(𝑡 + 𝜏) = 𝜆{𝑉(Δ𝑥(𝑡)) − �̇�𝑆(𝑡)} (14)
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20 Chapter 2: Literature Review
where 𝜏 is the explicit delay, widely known as reaction time.
Here, Bando et al.
(1998) used the parameter 𝜆 = 2.0 𝑠−1 and the OVF was defined
based on their
observed data.
𝑉(Δ𝑥(𝑡)) = 16.8 [tanh 0.0860(Δ𝑥(𝑡) − 25) + 0.913] (15)
The main shortcoming of the OV model is that it produces
unreasonable driving
behaviours, such as acceleration and deceleration, due to the
direct dependency of
the OVF on the following distance.
To control for unrealistic deceleration in the OV model, Helbing
and Tilch
(1998) developed a modified OV model that added velocity
difference, which is
known as the “generalised force” (Jin, Wang, Tao, & Li)
model, as defined in
Equation (16)
�̈�𝑆(𝑡) = 𝜆1{𝑉(Δ𝑥(𝑡)) − �̇�𝑆(𝑡)} + 𝜆2(Δ𝑥(𝑡))𝐻(−Δ𝑥(𝑡)) (16)
where 𝜆1 and 𝜆2 are the sensitivity parameters, 𝐻 is a Heaviside
function, and value
is defined by
𝐻 = {1; 𝑣𝐿 < 𝑣𝑆
0; 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (17)
Jiang, Wu, and Zhu (2001) developed the “full velocity
difference” (FVD)
model, which is essentially an extended model of the GF that
considers both negative
and positive velocity differences, as defined in Equation
(18):
�̈�𝑆(𝑡) = 𝜆1{𝑉(Δ𝑥(𝑡)) − �̇�𝑆(𝑡)} + 𝜆2(Δ𝑉(𝑡)) (18)
where Δ𝑉(𝑡) = 𝑣𝐿(𝑡) − 𝑣𝑆(𝑡) is the velocity difference of the
subject and lead
vehicles. Here, Jiang et al. (2001) used the similar OV function
of Helbing and Tilch
(1998). However, FVD also contains shortcomings, as the model
produces unrealistic
acceleration and deceleration behaviours.
Treiber, Hennecke, and Helbing (2000) proposed a novel
single-lane based CF
model, widely known as the “Intelligent Driver Model” (IDM).
Here, acceleration
(�̈�𝑆) is defined in the IDM as a function of the spacing,
speed, and relative speed of
subject vehicle, S, to the lead vehicle, L, as shown in Equation
(19):
�̈�𝑆 = 𝑎𝑆
𝑚𝑎𝑥 [1 − (𝑣𝑆
𝑣𝑆𝑑)
𝛿
− (𝑠∗(𝑣𝑆,∆𝑣)
𝑠)
2
] (19)
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Chapter 2: Literature Review 21
where 𝑎𝑆𝑚𝑎𝑥 is the maximum acceleration of a subject vehicle S,
𝑣𝑆
𝑑 is the desired
speed of a subject vehicle S, 𝑠 = ∆𝑥 − 𝑙𝐿 is the spacing between
the subject vehicle
and the leader, where 𝑙𝐿 is the length of lead vehicle, 𝑠∗ is
the desired minimum gap,
and 𝛿 is an acceleration exponent (model parameter).
The IDM equation (19) combines the acceleration strategy in
Equation (20)
𝑎𝑆
𝑓𝑟𝑒𝑒= 𝑎𝑆
𝑚𝑎𝑥 [1 − (𝑣𝑆
𝑣𝑆𝑑)
𝛿
] (20)
where 𝑎𝑆𝑓𝑟𝑒𝑒
represents the free flow acceleration of the subject vehicle
towards a
desired speed 𝑣𝑆𝑑 on a free road when the lead vehicle is far
away.
The braking declaration (strategy) induced by the lead vehicle L
is defined in
Equation (21).
𝑎𝑆
𝑏𝑟𝑎𝑘𝑒 = −𝑎𝑆𝑚𝑎𝑥 (
𝑠∗(𝑣𝑆,∆𝑣)
𝑠)
2
(21)
where 𝑎𝑆𝑏𝑟𝑎𝑘𝑒 represents the breaking deceleration of the
subject vehicle. The model
is free from producing unreasonably high acceleration. The
desired space headway in
the IDM model is dependent on the speed of subject vehicle 𝑣𝑆,
relative speed ∆𝑣,
the minimum spacing at the jam situation (standstill) 𝑠0 , the
maximum acceleration
of the subject vehicle 𝑎𝑆𝑚𝑎𝑥, a comfortable deceleration b, and
the desired time
headway 𝑇𝑑.
The desired space headway can be measured using Equation
(22)
𝑠∗(𝑣𝑆 , ∆𝑣) = 𝑠0 + 𝑣𝑆𝑇𝑑 +𝑣𝑆∆𝑣
2√𝑎𝑆𝑚𝑎𝑥𝑏
(22)
where 𝑠0 is the minimum spacing in a congested traffic
situation, 𝑇𝑑 is the desired
(safety) time headway, and 𝑏 is comfortable deceleration.
Moreover, IDM brakes
stronger than comfortable deceleration when the gap becomes too
small, which
basically makes the model collision-free. Similar to maximum
acceleration, the
addition of comfortable deceleration in Equation (22) prevents
the model from
producing unreasonably high decelerations. However, the model
still contains
shortcomings, as it ignores human reaction time. The main
challenge of the model is
measuring the desired parameters, such as desired speed,
spacing, and safety time
headway when following other vehicles due to these being
unobservable in nature. A
list of the estimated parameters for the IDM is included in
Table 1.
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22 Chapter 2: Literature Review
Table 1 Example of IDM model parameters (modified) used by
Treiber et al. (2000)
Parameters Typical value
Desired velocity (𝑣𝑆)
Safe time headway (𝑇𝑑)
Maximum acceleration (𝑎𝑆𝑚𝑎𝑥)
Desired deceleration (𝑏)
Acceleration exponent (𝛿)
Jam distance (𝑠0)
120 𝑘𝑚/ℎ𝑟
1.6 𝑠
0.73 𝑚/𝑠2
1.67 𝑚/𝑠2
4
2 𝑚
Newell (2002) proposed a lower order CF model for homogeneous
traffic
wherein one vehicle, S, (in other models say the subject
vehicle) follows its
preceding vehicle, L, (typically known as the lead vehicle) and
the time-space
trajectory of the subject vehicle is fundamentally identical to
that of the lead vehicle,
except for space and time shifts, as defined in Equation
(23):
𝑥𝑆(𝑡 + 𝛿𝑡) = 𝑥𝐿(𝑡) − 𝛿𝑥 (23)
where 𝛿𝑡 is time displacement and 𝛿𝑥 is space displacement
in-between two
consecutive vehicle trajectories. The trajectory of vehicle S
will approximately
follow the trajectory of the lead vehicle, L, as shown in
Equation (23), for some
appropriate values of 𝛿𝑡 and 𝛿𝑥. The model parameter 𝛿𝑡 can be
interpreted in
different ways, such as:
(a) the reaction time: with Newell’s model interpreted as a
time-delay differential
equation.
(b) the speed adaptation time: in an optimal velocity model.
(c) the numerical update time: in a discrete time model.
Newell (2002) suggested that the gap between two adjacent
trajectories at time
𝑡 depends on the average speed of the vehicle, and remains
relatively constant in
homogenous traffic flow. Furthermore, Newell (2002) assumed that
the values of the
pair of two parameters (𝛿𝑡, 𝛿𝑥) would vary as if they were
sampled independently
from some joint probability distribution with coefficients of
variation comparable
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Chapter 2: Literature Review 23
with 1. In addition to its simplicity (i.e., only two
parameters: 𝛿𝑡 and 𝛿𝑥 ), the model
is analogous with Chandler et al. (1958) model, with the
exception of considering
reaction time. Moreover, the model is consistent with the LWR
theory (Lighthill and
Whitham (1955); Richards (1956)), with a triangular shaped
fundamental diagram
but no shock waves. It should be noted that diverging waves may
be seen in the
interface between free and congested traffic flow. In addition,
the model is limited to
predicting the characteristics of traffic oscillations due to
the follower’s trajectory
essentially being replicated from the leader’s vehicle by
translating time and space.
A comparative overview of the different core stimulus response
CF models is
presented in Table 2.
Table 2 Comparative overview for different CF models with
calibration parameters
Parameters Name of CF models
GM
model
Eq. (5)
Edie’s
model
Eq.(6)
Gipps
model
Eq. (7)
OV
model
Eq. (13)
GF
model
Eq. (16)
FVD
model
Eq. (18)
IDM
Eq.
(19)
Newell
model
Eq.(23)
Reaction time √ √ √ √ √
Sensitivity
constant √ √ √ √ √
Speed
parameter √ √
Spacing
parameter √ √
Space
displacement
√
Desired speed √ √
Safe time
headway
√
Maximum
acceleration
√ √
Acceleration
exponent
√
Comfortable
deceleration
√ √
Minimum
spacing/jam
distance
√ √
Maximum
desired
deceleration
√
Length of lead
vehicle
√ √
(√ ∶ 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 )
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24 Chapter 2: Literature Review
2.1.1.3 Psychophysical CF models
The perception of the psychophysical CF models is almost
analogous to the
stimulus-response based CF models to model the longitudinal
movement of vehicles
in same lane. Kesting and Treiber (2013) described the
Wiedemann’s (1974) CF
model, which considers the local traffic context (i.e.,
congested or jammed traffic), as
well as action points. The model considers the perceptual
aspects of driving
behaviour using thresholds to define the different driving
regimes, such as free flow,
approaching, CF, and critical situations. Each of these regimes
is bound by the
different curves of perception-based thresholds. The perception
thresholds are
defined from the different perception areas. Figure 5 shows the
approaching
behaviour of the subject vehicle where the gap is decreasing due
to the higher
positive value of the relative speed and enters a perception
area by crossing the first
threshold (SDV). After crossing the SDV, the subject vehicle has
to reduce its speed
and passes another threshold CLDV, where it reacts and reduces
speed even further
to enter an unconscious reaction of a CF episode. Following
this, the subject vehicle
maintains the CF behaviour as long as it remains bound by the
other thresholds
OPDV, SDX, SDV and ABX. However, the model becomes sophisticated
due to
different acceleration functions, several nonlinear forms of
boundaries for different
regimes, and acceleration noise. Moreover, model calibration is
a challenging task
due to its complex nature.
Figure 5 Schematic and simplified representation of the regimes
of the Wiedemann’s (1974)
CF model (Kesting and Treiber (2013)).
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Chapter 2: Literature Review 25
The basic hypothesis of the visual angle model was first defined
by Michaels
(1963) in his perception-based CF model. In this model, Michaels
(1963) stated that
when a subject vehicle moves towards a lead vehicle, the subject
vehicle perceives
the situation from the changes in the apparent size of the lead
vehicle. In this
situation, the relative speed is perceived through the changes
in the visual angle
subtended by the lead vehicle. The challenging part of this
model is to measure an
appropriate visual angle threshold. Michaels and Cozan (1963)
measured the
threshold of the visual angle, which ranged between 0.0003 to
0.001 rad/sec, with a
mean of 0.0006 rad/sec.
Jin, Wang, and Yang (2011) introduced the concept of visual
angle in a
stimulus-response-based CF model. CF models potentially require
human perception
in decision making and response processes. The individual driver
has a perception
threshold for perceiving distance and speed in CF models. In
this model, the stimulus
parameter is defined by the rate of change of the visual angle.
However, defining the
appropriate visual angle threshold can be challenging in
different traffic conditions.
2.1.1.4 Fuzzy logic-based CF models
The deterministic approaches used in CF models have several
shortcomings in
replicating drivers’ CF behaviours. In traditional CF models,
many of the quantities,
such as relative speed, spacing, reaction time, acceleration,
deceleration, etc., are
considered deterministic; however, they are actually stochastic.
These values carry
considerable uncertainty in different traffic environments; for
example: free flow,
congested flow, and stop-go flow conditions. Capturing the
appropriate behaviour of
drivers in a traffic condition using deterministic models is not
precise due to the
imprecise and ambiguous nature of driver perceptions and
decisions in traffic
streams. Kikuchi and Chakroborty (1992) developed a fuzzy
logic-based CF model.
Certain membership functions are used to transform input factors
into linguistic
forms. For example, driver responses have the following form in
Kikuchi and
Chakroborty (1992) model:
“IF spacing is sufficient and relative speed is close to
zero
THEN the subject vehicle can accelerate.�
