Department of Mobility Systems Engineering
TUM School of Engineering and Design
Technical University of Munich

Calibration of Car Following Modelwith Genetic Algorithm and Particle
Swarm Optimization methods

A thesis presented in part fulfilment of the requirements of the Degree of Bachelor of
Science in Environmental Engineering at the Department of Civil and Environmental
Engineering, Technical University of Munich.
Supervisor

M.Sc. Qinglong Lu
Univ.-Prof. Dr. Constantinos Antoniou
Chair of Transportation Systems Engineering

Submitted by

He, Yuting
03716058
Arcisstrasse 21
80333 München

Submitted on

München, 16.10.2022

Abstract
Car following behaviour is one of the main behaviours in microscopic traffic flows and istherefore an important part of traffic flow modelling. Various car-following models havebeen proposed based on respective following principles, e.g., keeping desired spacegap, resulting in different sets of model parameters. Moreover, traffic flows usually illustrate distinct characteristics under very different traffic scenarios. Car-following models
thus should be specifically tuned to fit the traffic flow in concern, in particular the following
principle and model parameters, such that they can be used to simulate the traffic flow
with high accuracy and assist the analysis and evaluation of traffic management andcontrol measures. Considering the complicated vehicle interactions around the highway
entrances and exits areas, the thesis is to apply the genetic algorithm (GA) and particle
swarm optimization (PSO) algorithm to calibrate car-following models for the highway
entrances and exits traffic flows using a number of trajectory collections from Germany
highway entrances and exists. Moreover, the heterogeneity in car-following behavioursof vehicle combination (i.e., car following car, car following truck, truck following car, truck
following truck) is also specifically considered in the thesis. Wiedemann model is selected for calibration due to its superiority in theoretical logic and easy application in
popular traffic simulators, such as VISSIM and SUMO. Furthermore, fundamental diagrams are extracted from the selected trajectories to validate the nonnegligible differences in the traffic flow at highway entrances and exits from urban traffic and freeway
traffic. The experiment results show that PSO generally outperformed GA in terms of
convergence speed and convergence results in the devised car-following calibration procedure. Importantly, the calibrated Wiedemann models can replicate the respective traffic with a relative percentage error of less than 10%.
Keywords: car-following model, Wiedemann model, genetic algorithm, particle swarm
optimization, highway entrance/exit

Calibration of Car Following Model with Genetic Algorithm and Particle Swarm Optimization
methods

Table of Contents

1. Introduction ........................................................................................................ 1
1.1. Background and Motivation ........................................................................... 1
1.2. Research Questions and Objectives ............................................................. 2
1.3. Contribution ................................................................................................... 2
1.4. Thesis Structure ............................................................................................ 3
2. Literature review ................................................................................................. 4
2.1. Car-Following Models.................................................................................... 4
2.2. Comparison of Different Car-Following Models ............................................. 5
2.2.1.

General Motors model and Gazis-Herman-Rothery model................................................ 5

2.2.2.

Gipps model ...................................................................................................................... 6

2.2.3.

Intelligent Driver model ...................................................................................................... 7

2.2.4.

Optimal Velocity model ...................................................................................................... 7

2.2.5.

Wiedemann model............................................................................................................. 8

2.3. Calibration of Car-Following Model................................................................ 9
2.3.1.

Mathematical Principles................................................................................................... 10

2.3.2.

Non-linear Optimizations ................................................................................................. 11

2.3.3.

The current state of calibration of Wiedemann models.................................................... 14

2.4. Literature Gap ............................................................................................. 14
3. Methodology ..................................................................................................... 16
3.1. Software and Tools ..................................................................................... 16
3.2. Operating procedures.................................................................................. 16
3.3. Car-following model..................................................................................... 17
3.4. Calibration Methods .................................................................................... 21
3.4.1.

GA ................................................................................................................................... 22

3.4.2.

PSO ................................................................................................................................. 24

4. Data Description ............................................................................................... 26
4.1. Data Collection ............................................................................................ 26
4.2. Data pre-processing .................................................................................... 27
4.2.1.

Detecting and treating outlier ........................................................................................... 28

4.2.2.

Data aggregation ............................................................................................................. 28

4.3. Exploratory data analysis ............................................................................ 28
4.3.1.

Variables distributions of car ........................................................................................... 29

4.3.2.

Variables distributions of truck ......................................................................................... 31

Calibration of Car Following Model with Genetic Algorithm and Particle Swarm Optimization
methods

4.4. Fundamental Diagram of Traffic Flow.......................................................... 34
4.5. Selection of Following Pairs ........................................................................ 35
5. Results and Discussion ................................................................................... 38
5.1. Experimental Design ................................................................................... 38
5.1.1.

Initialization of models ......................................................................................................38

5.1.2.

Goodness-of-Fit Function Definition.................................................................................40

5.2. Optimization Convergence Analysis ............................................................ 41
5.2.1.

Convergence of GA .........................................................................................................41

5.2.2.

Convergence of PSO .......................................................................................................44

5.2.3.

Comparison of GA and PSO ............................................................................................47

5.3. Optimization Performance Analysis ............................................................. 47
5.3.1.

Calibrated parameter values of W-99 model ....................................................................47

5.3.2.

Estimation performance evaluation ..................................................................................49

5.3.3.

Simulation by calibrated model ........................................................................................51

6. Conclusion and Outlook .................................................................................. 55
6.1. Conclusion .................................................................................................. 55
6.2. Outlook ....................................................................................................... 56

Calibration of Car Following Model with Genetic Algorithm and Particle Swarm Optimization
methods

List of Figures
Figure 1 Thesis framework ............................................................................................ 3
Figure 2.1 Typical illustration for car-following .............................................................. 4
Figure 2.2 Schematic representation of thresholds and regimes in Wiedemann model . 8
Figure 2.3 Examples of objectie function: (a) smooth function with a single global
minimum, (b) smooth function with no definie minimum, and (c) objective function witha rugged fitting landscape (Treiber & Kesting, 2013). .................................................. 11................................................................................................................................... 16
Figure 3.1 Thesis methodology ................................................................................... 16
Figure 3.2 Thresholds and driving regimes in W-99 model (Zhu et al., 2018) .............. 17
Figure 3.2 Calculation process in Wiedemann 99 model for acceleraton of following
vehicle (Zhu et al., 2018)............................................................................................. 20
Figure 3.3 General framework of calibration............................................................... 21
Figure 3.4 Flow chart of GA ........................................................................................ 22
Figure 3.5 An example of the mechanism of fitness proportionate selection (roulette
wheel selection) .......................................................................................................... 23
Figure 3.6 Concept of double-point crossover ............................................................. 23
Figure 3.5 Flow chart: Steps of PSO algorithm. .......................................................... 24
Figure 4.1 Flow chart of data processing and analysis ................................................ 26
Figure 4.1 Overview image of one recording from the exiD dataset (Moers et al., 2022)................................................................................................................................... 27
Figure 4.2 Distribution and kernel density estimation of velocity and acceleration of car:(a) acceleration distribution, (b) kernel density estimation for acceleration, (c) velocity
distribution, and (d) kernel density estimation for velocity............................................ 29
Figure 4.3 Distribution and kernel density estimation of distance headway and time
headway of the following car: (a) distance headway distribution, (b) kernel density
estimation for distance headway, (c) time headway distribution, and (d) kernel density
estimation for time headway........................................................................................ 30
Figure 4.4 Distribution and kernel density estimation of velocity and acceleration oftruck: (a) velocity distribution, (b) kernel density estimation for velocity, (c) acceleration
distribution, and (d) kernel density estimation for accelration. ..................................... 32

Calibration of Car Following Model with Genetic Algorithm and Particle Swarm Optimization
methods

VI

The distribution of time headway and distance headway between a truck, as a vehicle
with following behaviour, and the vehicle in front of it is shown in Figure 4.5. ............. 32
In graphical terms, the distance headway and the time headway have coherence...... 32
Figure 4.5 Distribution and kernel density estimation of distance headway and time
headway of the following truck: (a) distance headway distribution, (b) kernel density
estimation for distance headway, (c) time headway distribution, and (d) kernel density
estimation for time headway. ...................................................................................... 33
Figure 4.6 Fundamental diagram of Data collection site 01 in exiD dataset ................ 35
Figure 4.7 Flowchart of following-pair selection: by truck follows car........................... 36
Figure 5.1 Flowchart of Wiedemann 99 model calibration by GA and PSO ................. 38
Figure 5.2 Convergence of GA calibrator: by following-pair type with car-truck, 𝑛𝑃𝑜𝑝 =10, 𝑚𝑎𝑥𝐼𝑡𝑒𝑟 = 50, and 𝑝𝑀𝑢𝑡𝑎𝑡𝑖𝑜𝑛 = 0.004 ................................................................. 41
Figure 5.3 The optimization process by GA-calibrator in different following-pair type: (a)car-car, (b) car-truck, (c) truck-car, and (d) truck-truck. ............................................... 42
Figure 5.4 The influence of different mutation rates on the convergence of the GA
calibrator under identical conditions: following-pair type car-truck, 𝑛𝑃𝑜𝑝 = 10,
𝑚𝑎𝑥𝐼𝑡𝑒𝑟 = 50. ............................................................................................................. 43
Figure 5.5 Convergence of PSO calibrator: by following-pair type with truck-car,𝑛𝑃𝑜𝑝 = 10, 𝑚𝑎𝑥𝐼𝑡𝑒𝑟 = 40, 𝑤 = 0.8, 𝑐1 = 1, and 𝑐2 = 1 ............................................... 44
Figure 5.6 The optimization process by PSO-calibrator in different following-pair type:(a) car-car, (b) car-truck, (c) truck-car, and (d) truck-truck........................................... 45
Figure 5.7 The influence of different inertia weight parameter on the convergence of the
PSO calibrator under identical conditions: following-pair type car-truck, 𝑛𝑃𝑜𝑝 = 10,
𝑚𝑎𝑥𝐼𝑡𝑒𝑟 = 40, 𝑐1 = 𝑐2 = 1.49 ..................................................................................... 46
Figure 5.8 The influence of different cognitive and social coefficient on the convergenceof the PSO calibrator under identical conditions: following-pair type car-truck, 𝑛𝑃𝑜𝑝 =10, 𝑚𝑎𝑥𝐼𝑡𝑒𝑟 = 40, 𝑤 = 0.3 .......................................................................................... 46
Figure 5.9 Relative error of velocity by GA calibrator in different following-pair types: (a)car-car, (b) car-truck, (c) truck-car, and (d) truck-truck. ............................................... 50
Figure 5.10 Relative error of velocity by PSO calibrator in different following-pair types:(a) car-car, (b) car-truck, (c) truck-car, and (d) truck-truck........................................... 51

Calibration of Car Following Model with Genetic Algorithm and Particle Swarm Optimization
methods

VIIFigure 5.11 Simulation of single vehicle by GA-calibrated W-99 model of different
following-pair in accelration and velocity: (a) car-car, (b) car-truck, (c) truck-car, and (d)truck-truck ................................................................................................................... 52
Figure 5.12 45-degree-line of simulated velocity by GA calibrated W-99 model and theobserved velocity of different following-pair in accelration and velocity: (a) car-car, (b)car-truck, (c) truck-car, and (d) truck-truck .................................................................. 54

Calibration of Car Following Model with Genetic Algorithm and Particle Swarm Optimization
methods

VIII

List of Tables
Table 2.1 Classification of car following models based on their logic (Ahmed et al.,2021) ............................................................................................................................ 5
Table 3.3 Wiedemann parameters value (Zhu et al., 2018)......................................... 19
Table 4.2 Description of the features of cars ............................................................... 31
Table 4.3 Discription of the features of trucks ............................................................. 33
Table 4.4 Summary of extraction criteria for car-following behaviour .......................... 36
Table 4.5 Amount of different car-following type in dataset ......................................... 37
Table 5.1 The default value and bound of parameters in W-99 model ........................ 39
Table 5.2 Initial parameter values of PSO calibrator ................................................... 40
Table 5.3 Initial parameter values of PSO calibrator ................................................... 40
Table 5.1 Calibrated parameters of W-99 model by GA calibrator .............................. 47
Table 5.2 Calibrated parameters of W-99 model by PSO calibrator ........................... 48

Calibration of Car Following Model with Genetic Algorithm and Particle Swarm Optimization
methods

IX

1. Introduction
1.1. Background and Motivation
The simulation of traffic flows presents an essential focus in the field of traffic control andplanning. The traffic flow simulation reproduces the behaviour and characteristics of anatural traffic flow and provides a foundation for the implementation of traffic flow regulation and traffic planning.
Car-following models are an essential component of traffic flow simulation. It simulates
the following behaviour of vehicles in their driving direction: the following vehicle adjusts
its velocity and acceleration to the preceding vehicle's driving behaviour (Olstam &Tapani, 2004). The car-following model defines the interaction of drivers in the same laneof the traffic flow by defining the response of the target vehicle to the driving behaviourof the vehicle in front of it in the same lane with it (Brackstone & McDonald, 1999). Since
the publication of the first car-following model (Pipes, 1953), various car-following models have been developed in accordance with different logical foundations, like stimulusbased models (Gazis et al., 1961), safety distance models (Gipps, 1981), optimal speed
models (Bando et al., 1995), psycho-physical models (Wiedemann & Reiter, 1992), etc.
Among these models, the Wiedemann model has the most well-established theoretical
logic and has been proven as capable of simulating microscopic traffic flows under diverse traffic scenarios. Therefore, several mainstream traffic flow simulation software
nowadays utilises the Wiedemann model as the foundation for car-following simulations.
The behavioural characteristics of microscopic traffic flows are susceptible to the influence of external factors. The typology of roadways, the composition type of microscopic
traffic flows, the countries or regions situated, and time-of-day contribute to variations intraffic flow characteristics. By varying the parameter set settings of the model, the model
is enabled to simulate traffic flows with different characteristics. To improve the simulation accuracy of the model, the parameter set of the model needs to be adjusted according to the traffic scenario being simulated.
The process of adjusting the parameter set of models by using observed data to improve
the accuracy of the simulation is known as calibration. To reflect actual driver behaviourin the simulation, before a car-following model is used for microscopic traffic flow simulations, the parameters in the model should be calibrated. The calibration of car-following
models usually uses real-world traffic trajectory data recorded in the traffic scenario withsimilar characteristics.

Introduction1For the acquisition of real-world traffic trajectory data, several researchers have collected
the data by installing sensors and cameras on vehicles (Pourabdollah et al., 2017; Zhu
et al., 2018). However, as drivers differ individually in their driving styles and conventions,
extrapolating the overall condition from the sample situation could easily be biased, especially with the insufficient sample. Therefore, some studies calibrated the car-following
models by using datasets that record the overall traffic flow trajectory (He et al., 2015;
Liu et al., 2021).
The interaction of vehicles surrounding highway entrances and exits is complicated. To
ensure the safety of vehicles during travel, the simulation of traffic flow in this scenario
is meaningful. The Wiedemann model was selected as the calibration model because of
the complete logical basis and the possibility of direct application in mainstream microscopic traffic flow simulators such as SUMO and VISSIM. In this thesis, a recently published dataset recording traffic flow variations in the surroundings of entrances and exits
in German highways was utilized for the calibration. Considering the influence of theheterogeneity of vehicle types on driving behaviour, in this study different types of following pairs are distinguished and calibrated separately. Because of the multi-parameter
property of the Wiedemann model, the calibration was conducted by using genetic algorithm (GA) and particle swarm optimization (PSO) methods.

1.2. Research Questions and Objectives
The main objective of this study was to calibrate the car-following model of entrance/exitarea in German highway and to obtain separate sets of parameters for different types ofvehicles. Based on the above objectives, the following research questions are proposed:
1) What is the difference of highway entrance/exit traffic flow from the other traffic
scenarios?
2) What is the difference between the following behaviour of different vehicle types?
3) Are the simulations using the calibrated model resembling real traffic situations?

1.3. Contribution
The main contributions of this thesis are threefold:
1) A newly published dataset is utilized to analyse the following behaviour of vehicles in a highway entrance/exit scenario and the car-following model is calibrated
by using this dataset.

Introduction22) The behavioural differences between vehicle types were considered. The carfollowing data was filtered by different vehicle types and the respective parameter sets are calibrated separately.
3) The characteristics of the traffic trajectory data from the highway entrance/exitarea were analysed utilizing the traffic flow fundamental diagram, which confirmed the non-negligible differences of traffic flow around the highway entrance/exit compared to the regular highway sections.

1.4. Thesis Structure
Chapter 1 of the thesis presents the background and motivation for the study, describes
the study objectives, and provides an overview of the thesis in general. Chapter 2 reviews the previous literature in general and presents the current literature gaps in termsof the car-following model and the calibration methods. Chapter 3 focuses on the methodology used in this study. Chapter 4 introduces the dataset. In this chapter, traffic flow
data is processed and analysed. Chapter 5 presents the results of the study as well as
the analysis and discussion of the results. Chapter 6 provides an overall review andsummary of the thesis and discusses the problems from the study as well as an outlookof possible further work. Figure 1 shows the framework of the thesis structure.

Figure 1 Thesis framework

Introduction

3

2. Literature Review
In this chapter, the previous research relevant to the topic of this thesis is reviewed andsummarised. Section 2.1 introduces the origins and development of the car-following
model. In section 2.2 different car-following models are classified and briefly introduced.
Sections 2.3 explains the reasons for calibrating the car-following model and the methods commonly used. These two methods are used in this thesis to calibrate the carfollowing model. Section 2.4 describes the current research gap.

2.1. Car-Following Models
Car-following models describe the following behaviour of drivers towards the vehicle infront of them in the traffic flow. Based on the rules of safe driving in traffic regulations,the theory of car-following was first proposed by Pipes in 1953 (Pipes, 1953). Using the
car-following model, the driving behaviour of individual drivers can be simulated, whichplays an important role in the study of microscopic traffic flows. The purpose of car-following model is to describe the influence of the leading vehicle on the driving behaviourof the vehicle behind during driving process from the individual level. The car-following
process is one of the main processes in modern traffic flow theory. With the rise anddevelopment of autonomous driving, the importance of this type of model has increased
further in recent years.

Figure 2.1 Typical illustration for car-following

Figure 2.1 illustrates a typical vehicle following scenario. The car-following behaviour
studies the longitudinal travel process of the vehicle. As the following car, the driving
behaviour of the yellow vehicle is modulated by the behaviour of the blue vehicle aheadin the same lane (Olstam & Tapani, 2004).

Literature Review

4Depending on the regulation logic used, car-following models were classified into fivecategories. Table 1 shows a summary comparison of the categories. A brief introduction
and comparison between these five types of models is given in the next subsection.
Table 2.1 Classification of car following models based on their logic (Ahmed et al., 2021)

Representative
Model

Model Class

Model logic and Assumption

Stimulus based models

The stimulus-response function between the leader andGazis-Hermanfollower vehicle: the acceleration of the follower is proporRothery (GHR)
tional to its speed, the speed difference between follower
model
and leader, and the space headway.

Safety distance models

The follower always keeps a safe distance from the frontvehicle.

Desired measure models

An ideal situation represented by certain measures (e.g.,following distance and following speed) is assumed. The Intelligent driver
following driver is constantly attempting to eliminate themodel (IDM)
difference between the ideal and actual situation.

Optimal velocity models

The optimal safe velocity of following vehicle depends onthe distance between follower and leader and the accelerOptimal velocity
ation of the following vehicle is determined based on themodel (OVM)
difference between the actual speed and the optimal
speed(Lazar et al., 2016).

Psycho-physical models

Thresholds are used to regulate the driving behaviour offollowing vehicle. The driver reacts to the vehicle when a Wiedemann modelthreshold value for relative speed or spacing is reached.

Gipps model

2.2. Comparison of Different Car-Following Models
2.2.1. General Motors model and Gazis-Herman-Rothery model
The General Motors (GM) model is initial developed by General Motors research lab in1958 and it was expanded and improved in 1961 (Gazis et al., 1961). The GM model
might be the most widely known car-following model, and it provides the basis for allother car-following models with varying sensitivity parameters (Koutsopoulos & Farah,2012). The original GM model is a linear model with a constant sensitive parameter andhas a very simple structure, where the acceleration of the following vehicle is linearly
related to the relative speed between it and the leading vehicles (Ahmed et al., 2021).
One of the reasons for the deviation between the basic GM model's simulation of thefollowing vehicle's acceleration and the actual situation is that the impact of the spacing
between the front and following vehicles on the stimulus is not considered. Based on
this, an extended GM model was proposed by Gazis et al. (1961), which is also known
as the Gazis-Herman-Rothery (GHR) model. In GHR model the acceleration equation isextended to incorporate traffic density as a sensitive term and proposes different sets ofLiterature Review

5parameters to decide the acceleration and deceleration. The GHR model also consider
the non-linear behaviour in the sensitivity of the distance between the following and theleading vehicle and sensitivity of the relative speed of the following vehicle.
A large amount of work on the calibration of the GHR model was published between the
1960s and 1990s (Brackstone & McDonald, 1999). However, models based on GM
model are currently used less frequently because of their own limitations in behaviour.
Drivers respond to arbitrarily any small change in stimulus, e.g., the relative speed of thefollowing vehicle to the leading vehicle is still influenced by its leading vehicle even atvery large relative distances. When the relative speed is zero, the follower's response
disappears. In case that the magnitude of fluctuations of acceleration increases, GHR
model creates a larger headway than reality. Moreover, microscopic analysis from Rockwell and Treiterer confirms that the car-following behaviour is likely to vary with traffic
and flow conditions (Rockwell et al., 1968), which is not considered in GM-based models.
2.2.2. Gipps model
Gipps model is the first general model based on safety distance logic. Unlike the GMbased models that use the stimulus-response equations, in the safety-distance model asafe following distance is always maintained between the following vehicle and the leading vehicle. Within a safe distance, a collision would be unavoidable if the driver of thelead vehicle acted “unpredictably” (Brackstone & McDonald, 1999). The basic rule ofsafety distance was quoted by Pipes in 1953 from the safe driving regulations of the
California Motor Vehicle Code: "A good rule for following another vehicle at a safe distance is to allow yourself the length of a car (about fifteen feet) for every ten miles perhour you are traveling" (Pipes, 1953).
As an important development of safe-distance model, the Gipps model was proposed by
Gipps in 1981. The Gipps model records the car-following behaviour when leading vehicle exists and the free-flowing behaviour of the vehicle when no vehicle is ahead. The
following driver plans his or her speed for the following instant (i.e., after the time delay)
to make sure he or she can stop safely even the leading vehicle break suddenly. In casethat there is no vehicle in front of the driver, the driving speed can be obtained from theempirical formulations which combines two conditions: (a) the speed never exceeds thedriver’s desired speed; and (b) the acceleration decreases as the speed increases andbecomes zero until the desired speed is reached (Gipps, 1981).
Gipps model has been extended, modified, and calibrated by various researchers and isused in a variety of micro-simulation tools, such as AIMSUN, DRACULA, and SISTM for
its simple calibration requirements (Barceló & Casas, 2005). However, one weakness of

Literature Review

6the safe-distance model such as Gipps model is that the drivers’ perception is not considered. Any minor changes might end in the reaction of the following driver (Lazar et al.,2016).
2.2.3. Intelligent Driver model
The intelligent driver model (IDM) is one of the most popular models using desired
measures (Treiber et al., 2000). The IDM considers both the desired speed and the desired space headway. When the front vehicle is far away, the IDM performs as a freeflow model where the acceleration is controlled by the desired speed of the driver. The
desired space headway in IDM is dependent on several factors: speed, speed difference,
the maximum acceleration, a comfortable deceleration, the minimum spacing at thestandstill situation, and the desired time headway. The IDM introduces a maximum acceleration and a comfortable deceleration rate, which prevents the model from producing
unrealistically high acceleration or deceleration. This is a significant advantage of the
IDM over earlier car-following models such as the GHR and Gipps models (Saifuzzaman& Zheng, 2014). The IDM was later extended by Treiber et al. in 2003. This extended
model captures the effects of driver adaptation to the surrounding environment by using
a memory function (Treiber & Helbing, 2003). This extension was based on the observation that after spending time in congested traffic, most drivers will adjust their drivingstyle.
The main difficulty for models with desired measures such as IDM is that most parameters are inherently unobservable, which makes their estimation more challenging. As aresult, many such models are not empirically estimated using real traffic data.
2.2.4. Optimal Velocity model
The instability of traffic flow can lead to congestion on highways and expressways. To
describe this erratic behaviour, the optimal speed model is proposed (Bando et al., 1995).
The OVM uses the equation of motion of each vehicle to describe the dynamic processof traffic congestion. Bando makes three assumptions about this model, including: (a)each driver responds to the stimulus of the vehicle in front of him with a legal speed; (b)each driver observes the motion of the vehicle in front of him and controls his acceleration/deceleration accordingly to maintain safe speed; (c) no response time delay is considered. In the OVM model, the driver does not overtake the vehicle in front of him under
any circumstances. This model ensures that erratic traffic flows generate congestion rather than traffic accidents.
Like the GHR model, OVM has the advantage that the model structure is simple andeasy to calibrate using numerical methods. Another advantage of the OVM is that it isLiterature Review

7better able to simulate following traffic in congested conditions on expressways and highways. However, as noted in some studies (Lazar et al., 2016), in some cases OVM can
produce unrealistically large accelerations.
2.2.5. Wiedemann model
The Wiedemann model is one of the most studied psycho-physical models. This type ofmodel considers the driver's behaviour as varying according to the traffic state he/she isin. Psycho-physical models use a “perceptual threshold” to define the boundary conditions for different states, which is a function of relative speed and relative distance to theleading vehicle (Wiedemann, 1974). The driver only responds to changes in spacing or
relative speed when the threshold is reached. In this way, it also records the driver's
attention to small spacing and lack of following behaviour for large spacing. In 1992,
Wiedemann and Reiter applied the psycho-physical method to a simulation system
named MISSION (Fellendorf & Vortisch, 2010). The Wiedemann model describes theinteraction of follower and leader vehicle pairs using varying thresholds in four different
regimes (Wiedemann & Reiter, 1992), including (1) free-flowing (the subject vehicle is
not influenced by the leading vehicle), (2) approaching (the subject vehicle is influenced
consciously by the slower leading vehicle), (3) following (The subject vehicle unconsciously follows the preceding vehicle and the following behaviour approaches a steadystate equilibrium), and (4) emergency (critical situations requiring braking action).

Figure 2.2 Schematic representation of thresholds and regimes in Wiedemann model

Six thresholds are defined in the relative speed and space diagram for the following pair,including: AX: the desired distance, ABX: the desired minimum following distance (the lower limit of the following regime),
 SDX: the maximum following distance (the upper limit of the following regime),
Literature Review

8 SDV: the approaching point where the following driver perceives that he or she isapproaching a slower leading vehicle,
 CLDV: the decreasing speed difference, and OPDV: the increasing speed difference.
Figure 2.2 depicts the boundary limits of the regime for different thresholds.
The concepts involved in the Wiedemann model have rarely been studied. While thewhole system is able to simulate driving behaviour very well, the calibration for individual
thresholds does not always seem to be accurate. However, many simulations haveshown that the basis on which it is built is undoubtedly the most consistent and best
describes most of the characteristics we see in daily driving behaviour (Brackstone &McDonald, 1999). Therefore, it is currently used by most micro-simulation systems, suchas VISSIM and SUMO. Since its introduction, the Wiedemann model has been continuously enhanced. In 1999, Wiedemann updated his model (Wiedemann 99, W-99) fromthe original one (Wiedemann 74, W-74) and defined some of the thresholds in a different
way to simulate freeway traffic better. For this reason, the driving behaviour in PTV VISSIM is set to W-99 for freeway driving and W-74 for urban driving.
After combing through the existing mainstream car-following models, it can be seen thatthe early models (such as the GHR and Gipps models) have a simple structure and fewer
parameters to be calibrated, which makes them easy to apply. Desired measure models,such as the IDM model, are difficult to calibrate using real traffic data as their parameters
are not directly observable. Optimal velocity models are good at simulating the dynamics
of traffic flows, such as congestion on highways, but can still produce unrealistically large
accelerations in specific cases. Psychophysiological models, such as the Wiedemannmodel, are the most complete car-following models available to describe the logic ofdriver behaviour and can describe most of the observed features of the daily driving
behaviour well, and therefore it is the dominant car-following model used in most microsimulation systems today. However, the calibration of individual thresholds in it is notalways accurate. For these reasons, we choose the Wiedemann model to calibrate andsimulate the following behaviour.

2.3. Calibration of Car-Following Model
The car-following model describes the behaviour of the vehicle in the longitudinal direction, which includes not only the driving behaviour of the driver, but also the local traffic
rules and possible restrictions. Drivers in different countries have different driving styles,
drive different types of vehicles, and are subject to different traffic regulations. Drivers'
driving behaviour and habits also respond differently on different road types. Therefore,
Literature Review

9a car-following model could not simply use the default set of parameters directly to handle a particular task. Parameters must be modified and adapted to fit the problem underconsideration. This process is known as calibration. Calibration are defined as (Treiber& Kesting, 2013): “Calibration is the estimation of parameters to maximize the model’s
descriptive power to reproduce local driver behaviour and/or collective traffic-flow characteristics. The descriptive power is specified by an objective function to be applied tothe test data.” For linear car-following models, such as the GM model, the solution canbe obtained directly by linear regression. However, for most non-linear car-following
models, after constructing the objective function using mathematical methods, the approximate minimum value of the objective function needs to be obtained by iteration. The
following discussion focuses on the mathematical principles of constructing the objectivefunction and the commonly used non-linear optimisation methods.
2.3.1. Mathematical Principles
There are two main mathematical methods to formulate the calibration problem, the leastsquared errors, and the maximum likelihood.
1)

The Ordinary Least Square (OLS) method

OLS is an intuitive method in regression analysis to approximate the solution of overdetermined systems. The objective function is defined directly in terms of a sum of squared
errors (SSE), or the mean squared error (MSE) between the test data and the model
prediction. The MSE is treated as a function of the parameters while the data are fixed.
For models which comprise a linear combination of the parameters, the regression model
is linear, which is called linear least squares. For non-linear models, generally could not
find a closed-form solution. In this case, numerical algorithms are used to find the valueof the parameters that minimize the objective. By setting initial values and iteratively refining the parameters, an approximation to the minimum value can be obtained.
2)

Maximum likelihood estimation (MLE)

The MLE is a method of estimating the parameters of an assumed probability distribution.
As the MLE is explicitly based on probability, random elements with specific properties
are necessary. If the model to be calibrated is deterministic, random noise needs to be
added to it. The probability or probability density 𝑝(𝒚𝑖𝑠𝑖𝑚 |𝜷) is defined that, at time 𝑡𝑖 , themodel makes the predictions 𝒚𝑖𝑠𝑖𝑚 to a given parameter vector 𝜷 and suitable data driven
conditions for the prior stem 𝑡𝑖−1 or the initial state. The state to be predicted could be
the speed and the gap between the following-pair, but also the detector counts, travel
times, or propagation velocities of traffic waves. The likelihood function is defined as thejoint probability that the model predicts all data points. The MLE of a parameter is

Literature Review

10obtained by maximizing the likelihood function. The MLE reduces the calibration problem
to a multi-variate non-linear optimization problem.
2.3.2. Non-linear Optimizations
For the problem that minimizing non-linear functions of several variables, is usually acomplex task with no unique “one size fits all” solution. In terms of speed and robustness,the “best” solution for finding the global minimum depends on the complexity of the objective function.

Figure 2.3 Examples of objectie function: (a) smooth function with a single global minimum, (b) smooth function with nodefinie minimum, and (c) objective function with a rugged fitting landscape (Treiber & Kesting, 2013).

Figure 2.3 shows three forms of objective functions. Figure 2.3 (a) and (b) show the local
trajectory ML calibrations, and (c) shows a global OLS calibration to stationary detector
data.
The calibration methods for car-following models can be grouped into two categories.
The first category revolves around the concept of local fit. At each simulation step, theempirical vehicle state is fed into the formulated model to predict its acceleration. A costfunction is then constructed to capture the error between the simulated acceleration trajectory and the corresponding experimental data, and the parameters are adjusted
through iterations to minimize the error. This type of problem generally has a smoothobjective function.
For smooth and unimodal objective functions (Figure 2.3 (a)), which are differentiable
and have a global minimum, the global optimum solution can be found using simple deterministic methods. Commonly used methods are:

Literature Review

11 Newton’s method: This method assumes that the objective function is quadraticallydifferentiable and approximately quadratic. At each iteration, the objective function isapproximated by a quadratic function and the extreme point of this function is the next
iteration point. The advantage of Newton’s method is fast convergence.
 Gauss-Newton algorithm: This algorithm is a variation of Newton's method specificallyfor minimization of the sums of squares. The second derivatives are no longer required; however, this method is less robust compared to the Newton’s method.
 Gradient descent method: the search path proceeds always along the gradient at thelast iteration point in this method. Along this direction we can easily determinate theminimum by a linear search, which becomes the new iteration point. This method is
slower but more robust than the Newton and Gauss-Newton methods.
 Levenberg-Marquardt algorithm: This method makes a smooth transition between thegradient descent method and the Newton’s method and therefore combines the advantages of them. The transition is controlled by an adaptive “trust region”, which
prevents premature convergence of the Gauss-Newton method. Levenberg-Marquardt algorithm is the most popular method for objective functions in form of a differentiable sum of squares.
The second category of calibration is global-fit. Compared with local-fit, the global-fit calibration results better in the minimization of fitting error than a local-fit calibration (Punzo& Simonelli, 2005). However, global fit often results in highly non-linear, non-smooth objective functions. In this situation, the four deterministic methods described above are nolonger viable because of the difficulties of derivatives calculations (Li et al., 2016). Therefore, stochastic optimization methods need to be used to escape such minimum. Stochastic optimization methods are optimization methods that generate and use randomvariables. Introducing randomness into an optimization method, can, on the one hand
speed up progress, and on the other hand, allow the method to escape from the local
optimum and eventually approach the global (approximate) optimum. In addition, algorithms using stochastic optimization methods can often achieve almost uniformly good
performance on many types of problems. In practice, a variety of algorithms based onstochastic optimization have been widely used in calibration problems for global fitting.
Genetic algorithms (GA) and particle swarm algorithms (PSO) are popular metaheuristics that have been successfully applied in recent years to calibrate car-following models.
 Genetic algorithm (GA)Genetic Algorithms (GA) is a metaheuristic inspired by Darwinian evolutionary theory, inwhich the survival of fitter creatures and their genes were simulated. In the 1960s and1970s, John Holland and his students invented and extended genetic algorithms(Holland, 1992). Most of the models of GA that exist today are based on Holland's theory.

Literature Review

12GA is an algorithm based on population. In GA, each possible solution corresponds to achromosome and each parameter to be optimized represents a gene on the chromosome. A fitness function is used to assess the fitness of each individual in the population.
GA simulates natural selection mechanisms, with a greater probability of selecting individuals with higher fitness values. After selection, chromosomes crossover to improve
solutions. To prevent the solution from being trapped in a local optimum, in each iteration
the operator randomly changes the genes of a portion of the individuals in the population.
This part is called variation, which increases the exploration behaviour of the GA.
Through selection, crossover, and mutation, the more suitable solution is maximally retained and perpetuated, while the diversity of the population is maintained.
GA is widely used in multi-objective optimization problems due to its global, parallel, androbust characteristics. Moreover, GA is simple in terms of search improvement and does
not suffer from the restrictive assumption of search space. GA has been used to solve
problems in a variety of fields, including engineering, medicine, and finance. In the transportation field, GA has now been successfully applied to many aspects, such as dynamic
traffic signal control (Sun et al., 2006), air traffic control (Hu & Di Paolo, 2009), and trafficmodel calibration (Yu & Fan, 2017).
 Particle Swarm Optimization (PSO)
The Particle Swarm Optimization (PSO) algorithm of continuous nonlinear functions was
proposed by R. Eberhart and J. Kennedy in 1995 (Eberhart & Kennedy, 1995). The concept was inspired on the one hand by the behaviour of social animals, especially theforaging behaviour of flocks of birds and fish (Marini & Walczak, 2015). On the other
hand, the PSO algorithm is also related to genetic algorithms and evolutionary strategies.
PSO can be described imaginatively as a flock of birds searching for food in a region.
The flock's task is to find the largest food source (the global optimal solution). Throughout
the search process, the flock lets other birds know the location of the food source bypassing information about their respective positions to each other. Eventually, the whole
flock will gather around the food source, i.e., what we call the optimal solution is foundand the problem converges.
An advantage of PSO is that it does not require the optimization function to be differentiable, divisible, and continuous. In addition, PSO has a fast convergence rate and asimple algorithm structure that is easy to program and implement. However, a disadvantage of the PSO algorithm is that it tends to cause premature convergence due to therapid disappearance of particle diversity. For a search space with multiple local optima,
the PSO algorithm can easily fall into a local optimum and fail to obtain a global optimum.

Literature Review

13

As a result, the PSO algorithm is generally suitable for optimization problems with a highdimensional search space, where have multiple local optimum points, and for problem
which is not necessary to obtain a very exact solution. Several researchers have used
PSO to solve the problems in traffic field, such as calibrating the car-following model incongested traffic conditions (Aghabayk et al., 2016), the simulation of large-scale traffic
flow (Liu & Yang, 2022), and developing a lane change model for autonomous based on
PSO (Sheppard & Haberman, 2021).
2.3.3. The current state of calibration of Wiedemann models
In terms of car-following model calibration, in terms of three dimensions, including thetype of car-following model, the calibration methods, and the road scenarios, numerous
researchers have published a large number of studies in more than half a century since
the introduction of car-following models.
On the calibration of the Wiedemann model, Chaudhari et al. calibrated the W-99 modelby using trajectory data based on Root-mean-square deviation (Chaudhari et al., 2022).
Motion cameras are used to calibrate driving behaviour parameters of different types ofvehicle-following pairs using parameter estimation methods on a main section of a motorway in Los Angeles, California, U.S.A. (Durrani et al., 2016). The GA is also used in
the calibration of the Wiedemann model, such as the calibration using real-world driving
data in Sweden (Pourabdollah et al., 2017), the real-world data collected in the Shanghai
Naturalistic Driving Study (SH-NDS) of urban expressway in China (Zhu et al., 2018),
and the trajectory data at the High accident-prone roads in USA (Hamdar et al., 2015).
In 2015, PSO is used to calibration the parameters in VISSIM (Rrecaj & MBombol, 2015).
Aghabayk et al. used PSO to calibrate the Wiedemann model for a mixed traffic flow
including heavy vehicles (Aghabayk et al., 2016).
Focusing on different traffic scenarios, the GA is used to calibrate the Wiedemann model
for an actuated signalized intersection (Park & Qi, 2005), in mixed traffic (Manjunatha etal., 2013), and the traffic environment with connected and autonomous vehicles (Liu &Fan, 2020). Leyn and Vortisch performed manual simulations of theGerman motorway
following models using the Vissim tool (Leyn & Vortisch, 2015).

2.4. Literature Gap
As mentioned in the previous sections, a great deal of research has been published on
the car-following model in the nearly 70 years since its introduction, but at the same time
there are still a large number of gaps to be explored.

Literature Review

14

1)

In the current mainstream research on the calibration of car-following models using real-world data, the data is mainly collected through sensors and cameras
mounted on the vehicle. This approach has the following drawbacks: (a) only information on the trajectory of a particular vehicle can be collected, and the collection of information grows exponentially with the number of vehicles. As a result,traffic trajectory data collected based on sensor information is typically on the order of 10 squared; (b) individual samples are susceptible to traffic conditions(e.g., congestion, accidents, etc.) which can lead to biased estimates of the entirety, and the resulting outcome does not accurately represent the full picture ofthis traffic scenario in general.

2)

Car-following model calibration for German highway ramp areas using real-world
data has not yet been reported. German motorways are unique in that they (a)have many sections with no speed limits and (b) German drivers have driving
habits that place a high premium on the experience of driving at high speeds.
Therefore, car-following models calibrated by previous researchers using traffic
data from other countries and other road types cannot be directly applied to thetraffic scenario of German motorway ramps and need to be calibrated using datafrom the same traffic scenario.

Based on the two gaps mentioned above, the contributions of this thesis are: (a) the useof overall traffic trajectory data taken by drones for calibration, which has a large data
sample size and can reduce the impact of bias caused by individual differences on theoverall estimation; (b) the sampling location is the area near the entrance/exit of a German highway, which is calibrated separately by distinction the type of vehicle in following
pairs (truck or car) to obtain specific car-following model parameters for the following
behaviour of different vehicle types in this specific traffic scenario.

Literature Review

153. Methodology
In this chapter, the methodology used in this thesis will be introduced. The introductionof the methodology consists of four parts. Section 3.1 introduces briefly the software andtools used in this study. Section 3.3 defines the Wiedemann 99 model. In section 3.4 theprocesses of the GA and PSO methods are explained separately.

3.1. Software and Tools
The programming language used in this thesis is Python 3.8. Python is a widely used
programming language in scientific research because of its large number of libraries ondata processing, optimization algorithms, and visualization. The list of python libraries
used in this thesis is as follows: Data processing and computation: Pandas, NumPy
 Visualization and plotting: Matplotlib
 GA and PSO: Scikit-opt
 Other operations: Black, Sys, Pathlib
The hardware used in this thesis is a HP laptop with i7-1065G7 CPU and 16 GB RAM.

3.2. Operating procedures

Figure 3.1 Thesis methodology

Methodology

16

3.3. Car-following model
As we use data from the Highway, the Wiedemann 99 model (W-99) is better suited tothis road scenario. Figure 3.2 describes the thresholds and the driving regimes. The
complete flow chart for calculating the acceleration𝑎𝑛 (𝑡 + 1) by using W-99 model isshown in Figure 3.3 below (Zhu et al., 2018).

Figure 3.2 Thresholds and driving regimes in W-99 model (Zhu et al., 2018)

The corresponding thresholds in the flow chart are represented as:∆𝑋(𝑡) = 𝑋𝑛−1 (𝑡) − 𝑋𝑛 (𝑡) − 𝐿𝑛−1
∆𝑉(𝑡) = 𝑉𝑛−1 (𝑡) − 𝑉𝑛 (𝑡)𝑆𝐷𝑋𝑐 = 𝐶𝐶0 + 𝐶𝐶1 ∙ 𝑉𝑠𝑙𝑜𝑤𝑒𝑟
𝑉𝑛 (𝑡),
𝑉𝑠𝑙𝑜𝑤𝑒𝑟 = {𝑉𝑛 (𝑡) − ∆𝑉(𝑡) ∙ 𝑅𝑁𝐷,

𝑖𝑓 ∆𝑉(𝑡) > 0 𝑜𝑟 𝐿𝑉𝑎𝑐𝑐 (𝑡) < −1 𝑚/𝑠 2
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

𝑅𝑁𝐷 = 𝑈𝑛𝑖𝑓𝑜𝑟𝑚 𝑅𝑎𝑛𝑑𝑜𝑚 [−0.5, 0.5]
𝑆𝐷𝑉 = 𝐶𝐶6 ∙ (∆𝑋(𝑡) − 𝐿𝑛−1 )2
𝑆𝐷𝑋0 = 𝑆𝐷𝑋𝑐 + 𝐶𝐶2
𝑆𝐷𝑋𝑣 = 𝑆𝐷𝑋0 + 𝐶𝐶3 (∆𝑉(𝑡) − 𝐶𝐶4 )Methodology

17

−𝑆𝐷𝑉 + 𝐶𝐶4 ,𝐶𝐿𝐷𝑉 = {0,

𝑖𝑓 𝑉𝑛−1 (𝑡) > 0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

𝑆𝐷𝑉 + 𝐶𝐶5 ,𝑆𝐷𝑉,

𝑖𝑓 𝑉𝑛 (𝑡) > 𝐶𝐶5
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

𝑂𝑃𝐷𝑉 = {where:𝑋𝑛 , 𝑋𝑛−1 = position of the following and leading vehicle respectively,∆𝑋 = following gap (m),𝑉𝑛 , 𝑉𝑛−1 = velocity of the following and leading vehicle respectively (m/s),
𝐿𝑛−1 = length of the leading vehicle (m),∆𝑉 = speed difference between the following and leading vehicle (m/s),
𝑆𝐷𝑋𝑐 = minimum safe following distance (m),𝐿𝑉𝑎𝑐𝑐 = acceleration of the lead vehicle (m/s2),
𝑆𝐷𝑉 = perception threshold of speed difference (m/s),
𝑆𝐷𝑋0 = maximum following distance (m),𝑆𝐷𝑋𝑣 = distance threshold of following vehicle perceiving its approach to a slower leader(m),
𝐶𝐿𝐷𝑉 = perception threshold of speed difference at short decreasing distances (m/s),
𝑂𝑃𝐷𝑉 = perception threshold of speed difference at short but increasing distances (m/s),
𝑉𝐷𝐸𝑆 = desired speed of following vehicle (km/h),
𝑎𝑚𝑎𝑥 = maximum acceleration of vehicle (m/s2), and𝐶𝐶0 to 𝐶𝐶9 = model parameters.
Table 3.3 shows the constraints applied to the Wiedemann 99 parameters according toZhu et al. in 2018. The following data and boundaries are used as a reference for the
initial value setting when initialising the following model.

Methodology

18

Table 3.3 Wiedemann parameters value (Zhu et al., 2018)

Parameter

description

BoundsMean

Std. dev. 5%95%

𝐶𝐶0 [m]Standstill gap[0, 20]1.01920.96960.10562.5673𝐶𝐶1 [m]Headway time

[0, 5]

1.42420.43260.82002.1980𝐶𝐶2 [m]“Following” variation[0, 10]5.87153.95000.52089.9986𝐶𝐶3 [s]Threshold for entering “following”[-20, 0]-17.1579

4.7313-19.9700 -4.5030

𝐶𝐶4 [m/s]

Negative “following” threshold[-5, 0]-0.23120.2604-0.9616-0.1006

𝐶𝐶5 [m/s]

Positive “following” threshold[0.1, 5]1.79461.57060.1391

4.7846𝐶𝐶6 [10-4 rad/s] Speed dependency of oscillation[0.1, 20]

3.55193.8974

0.266411.0473

𝐶𝐶7 [m/s2]

Oscillation acceleration[−1, 1]0.53500.33430.0974

0.9729𝐶𝐶8 [m/s2]Standstill acceleration[0, 8]

4.0101

3.1826

0.10877.9984𝐶𝐶9 [m/s2]

Acceleration at 80 km/h

[0, 8]

2.65493.06120.10957.8852𝑉𝐷𝐸𝑆 [km/h]

Desired speed of following vehicle [1, 150]

86.0492

17.8998

67.2759 124.1050

Methodology

19Figure 3.2 Calculation process in Wiedemann 99 model for acceleraton of following vehicle (Zhu et al., 2018)

Methodology

203.4. Calibration Methods
Calibration refers to the adjustment of model parameters to improve the ability of themodel to replicate driving behaviour and traffic characteristics. Figure 3.4 shows a general framework of calibration. According to this framework, calibration is an iterative process by which the error between the simulated and real values is reduced until a stopping
criterion is achieved.

Figure 3.3 General framework of calibration

As mentioned before, in this study, the calibration methods are GA and PSO. There areseveral reasons for choosing these two optimisation methods. Firstly, as the Wiedemannmodel has multiple parameters and thresholds and there is no direct correlation between
the thresholds, it is difficult to calibrate it directly using numerical methods. Using metaheuristic algorithms such as GA and PSO, by introducing randomness, largely reduces
the amount of computation involved in the search process and can avoid the problem ofdifficult solutions using numerical methods. Secondly, the GA and PSO process is simple, the results are intuitive and the meaning of the parameters in the algorithm is easy
to understand and adjust. The algorithms make few or no assumptions about the problem
being optimised and, because of the stochastic nature, can search a very large space ofcandidate solutions. And by adjusting the parameters and repeating the experiments,
convergence to the optimal solution can be achieved.
Methodology

21

3.4.1. GA
The genetic algorithm consists of three main steps in each iteration: selection, crossover,and mutation. Figure 3.4 illustrates the basic processes of the GA.

Figure 3.4 Flow chart of GA

1)

Selection

According to Darwin’s theory of evolution, in the natural world, the most adapted individuals in nature have a greater chance of surviving and carrying on their genes to the next
generation through mating. This causes their gene frequencies to increase in the next
generation. To simulate natural selection, the GA used in this thesis selects individuals
within a population by fitness proportionate selection. This selection method uses thefitness level to associate a probability of selection. For individual 𝑖 in the population with
𝑁 individuals, 𝑓𝑖 is its fitness, and its probability of being selected is expressed as:
𝑝𝑖 =

𝑓𝑖
𝑁
∑𝑗=1 𝑓𝑗This can be imagined as similar to a roulette wheel in a casino. Figure 3.4 shows anexample of the roulette wheel. A proportion of the wheel is allocated to each of the possible selections based on their fitness value. It can be observed that the best individual
Methodology

22

(with the highest Fitness value), i.e., individual 4, had the largest proportion of the roulette
wheel, while the worst individual (individual 1) had the lowest proportion. In each selection phase of GA, two individual is selected by the roulette wheel.

Figure 3.5 An example of the mechanism of fitness proportionate selection (roulette wheel selection)

2)

Crossover

The two solutions selected by the selection process (the parent solutions) will be used
to create the next generation. The two parent solutions will be crossed over to create two
new child solutions. GA in this thesis uses the double-point crossover method. Figure
3.5 shows the concept of double-point crossover. Two crossover points are picked randomly from the parent chromosomes and the bits in between these two points areswapped between the parent chromosomes.

Figure 3.6 Concept of double-point crossover3)

Mutation

After the child solutions have been created, one or more of their genes will be randomly
mutated. Mutations introduce another layer of randomness to maintain the diversity ofthe population. This operator reduces the probability that the GA will fall into a local optimum solution. In this thesis bit string mutation is the strategy of mutation, which geneat random locations was flipped.

Methodology

23

3.4.2. PSO
The PSO algorithm for tracking the global optimum 𝐺𝐵𝐸𝑆𝑇 published by R. Eberhart and
J. Kennedy in 1995 has a simple form.

Figure 3.5 Flow chart: Steps of PSO algorithm.

The position of the 𝑖 𝑡ℎ particle in D-dimensional hyperspace is described by a vector 𝒙𝑖 :𝒙𝑖 = [𝑥𝑖1 , 𝑥𝑖2 , 𝑥𝑖3 , . . . , 𝑥𝑖𝐷 ]And the particle swarm is formed by the population of N candidate solutions:
𝑿 = {𝒙𝟏 , 𝒙𝟐 , 𝒙𝟑 , . . . , 𝒙𝑵 }In each time iteration, the position of particle changes by:
Methodology24𝒙𝒊 (𝑡 + 1) = 𝒙𝒊 (𝑡) + 𝒗𝒊 (𝑡 + 1)
The mathematical formula of changing the velocity is:𝒗𝒊 (𝑡 + 1) = 𝑤 ∙ 𝒗𝒊 (𝑡) + 𝑐1 ∙ 𝑹1 (𝑡) ∙ (𝑃𝐵𝐸𝑆𝑇𝑖 − 𝒙𝒊 (𝑡)) + 𝑐2 ∙ 𝑹2 (𝑡) ∙ (𝐺𝐵𝐸𝑆𝑇 − 𝒙𝒊 (𝑡)),where 𝑤 is inertia factor of particle swarm, 𝑐1 and 𝑐2 are two acceleration constants representing “cognitive coefficient” and “social coefficient” respectively. In general, 𝑐1 and𝑐2 take values in the interval [0, 4]. They weight the two control the behaviour of particles.
𝑹1 and 𝑹𝟐 are two matrices of random numbers generated from a uniform distribution in[0, 1). For the tendency of the particles to move, they give a random effect. 𝑃𝐵𝐸𝑆𝑇𝑖 is the
personal optimum recorded of the 𝑖 𝑡ℎ particle.
The stopping criterion can usually be one of the following three conditions: Achieve a specified number of iterations, 𝐺𝐵𝐸𝑆𝑇 remains unchanged after a specified number of iterations since the last update, or attain the prespecified target of fitness value (Marini & Walczak, 2015).

Methodology

25

4. Data Description
This chapter describes the processing and filtering of data. Section 4.1 describes how
and where the data were collected and section 4.2 explains pre-processing operations
on the data, including outlier removal and aggregation. Section 4.3 performs exploratory
data analysis on the data. Section 4.4 describes the fundamental diagram of the roadsegment. In section 4.5, the following pairs are filtered out. The process is illustrated in
Figure 4.1.

Figure 4.1 Flow chart of data processing and analysis

4.1. Data Collection
The following dataset used in this thesis is The Exits and Entries Drone Dataset (exiD)from fka GmbH (Moers et al., 2022). The exiD dataset is a new dataset that records the
trajectories of natural road users at the exits and entrances of German highways. Using
drones, the dataset has 92 individual records at seven different measurement locations
at highway entrances and exits between Aachen and Cologne, Germany, where the trajectories of each road user and their type are extracted. In total, the traffic trajectories of
69,172 vehicles were extracted. The dataset uses state-of-the-art computer vision algorithms, the positional error is typically less than 10 cm.

Data Description

26

The data containing information on traffic trajectories are recorded by camera-equipped
drones. All videos were recorded in fine weather conditions with high visibility and little
wind. At the flight altitude of the drone, approximately 420 meters of the highway arecaptured, which can cover a large part of the entrance and exit ramps. The video resolution is 4K and 25 frames per second. A deep neural network is used to detect road
users in records and the detections from each frame are assigned to each other using a
tracking-by-detection approach based on location and size to generate trajectories. The
dataset was smoothed for the position, direction, velocity, and acceleration by using a
Rauch-Tung-Striebel (RTS) smoother, which is a linear-Gaussian smoothing algorithm.
The dataset provides the position, heading, lateral and longitudinal speed, and acceleration of the recorded vehicles at each time step. In addition, the type of vehicle and the
ID of the surrounding vehicles are also provided.
An overview of one recording from the exiD dataset is shown in Figure 4.1. This image
visualizes the traffic trajectories of an entrance and exit on a German highway.

Figure 4.1 Overview image of one recording from the exiD dataset (Moers et al., 2022)

4.2. Data pre-processing
Data pre-processing refers to the manipulation or removal of data before it is used toensure or improve the quality of the data. Data pre-processing is crucial before the startof an experiment as the quality of the data is directly related to the quality of the final
results. Real data may inevitably contain a large number of missing values, and noise at
the time of collection, or there may be outliers present as a result of manual entry errors.
Therefore, pre-processing the data can reduce the impact of "dirty" data on the results.
The exiD dataset was post-processed after data collection. ExiD using an RTS smootherto obtain the smoothed position, direction, velocity, and acceleration of the traffic trajectory at each timestamp. Most missing values and data errors were automatically detected
and resolved in this step. For errors that could not be resolved automatically, the developers of this dataset performed manual checks and processing. As a result, in the
Data Description

27

datasets that we obtained, the missing values and most of the outliers during the datacollection process have been processed and replaced. For the calibration of the carfollowing model, the data needs to be further filtered and aggregated based on conventional data processing.
4.2.1. Detecting and treating outlier
The main basis for further filtering is the reasonableness of the range of values of theparameters. Considering the reality of German highway traffic, in this study, the longitudinal velocity range of vehicles was set to be greater than 0 and less than 150 km/h.
Observations outside this range are considered outliers. Manual inspection of the outliers
revealed that the outliers did not appear in discrete timeframes, but always in the formof a continuous sequence in the data of a certain vehicle. One explanation for this phenomenon might be that the driving style of some vehicles differs too much from the totality, which could be caused by special circumstances or the personal style of the driver.
As individual data deviate from the overall distribution, calibration using a dataset containing these data can also lead to a decrease in the accuracy of the results. Therefore,
vehicle information containing outliers was removed from the training dataset.
4.2.2. Data aggregation
In the exiD dataset, the frame rate of the recordings is 25 Hz, which means that the timeinterval between two adjacent frames is 0.04 seconds. The calibrated car-following
model aims to simulate as accurately as possible the behaviour of each vehicle in arealistic traffic scenario. The average human driver's reaction time is around one second,so a data accuracy of 0.04 seconds is clearly too high for this study. Aggregation of thetime series data is required to match the realistic situation. In this study, data points within
one second are aggregated. This means that every twenty-five consecutive data points
in the time series will be aggregated into one data point. After aggregation, we obtaintime series data with a time interval of one second.

4.3. Exploratory data analysis
Exploratory data analysis of the acquired data helps to gain the understanding of thedata and discovering the patterns.
In this thesis, we conducted an exploratory analysis of the car and truck data separately
to observe the distribution of the variables to be used in the calibration part of the experiment. The four variables observed were longitudinal velocity, longitudinal acceleration,
time headway and distance headway. In discussing time headway and distance headway, we focused on vehicles with possible following behaviour by limiting the range of

Data Description

28

distance headway to those with a distance headway of less than 120 meters from thevehicle in front.
As the lateral movement of the vehicle is not considered in the car-following behaviour,
all references to velocity and acceleration in this thesis are in the longitudinal direction
unless otherwise stated.
4.3.1. Variables distributions of car

Figure 4.2 Distribution and kernel density estimation of velocity and acceleration of car: (a) acceleration distribution, (b)kernel density estimation for acceleration, (c) velocity distribution, and (d) kernel density estimation for velocity.

Figure 4.2 shows the velocity distribution and the acceleration distribution of car. Kernel
density estimation helps us to observe the probability density distribution of the data in amore intuitive way. As illustrated, the acceleration follows an almost symmetrical unimodal distribution with a very sharp peak. The peak value is approximately 0 m/s2. Most
acceleration values are in the range of -2 m/s2 and 2 m/s2.

Data Description

29

The situation differs in terms of the distribution of velocity. There are obviously bimodalin the probability density curve about the velocity, with a first small peak occurring at
about 4 m/s and a second large peak at about 24 m/s. A possible explanation for this isthat there was unusual congestion on certain sections of the road during certain sampling
periods, resulting in a lower speed distribution for this part of vehicles. This type of congestion is not common and therefore the volume of data is small in the overall picture.
The majority of vehicles had speeds in the range of 10 to 40 m/s.

Figure 4.3 Distribution and kernel density estimation of distance headway and time headway of the following car: (a)distance headway distribution, (b) kernel density estimation for distance headway, (c) time headway distribution, and (d)kernel density estimation for time headway.

As mentioned above, in the analysis of the distance headway and time headway, theobservations are restricted to vehicles with possible following behaviour, i.e., the distance headway falls within the range (0, 120) meters. The distributions and the kernel
density estimations are shown in Figure 4.3. It is obviously observed that both the time
headway and distance headway show a unimodal and right-skewed distribution, whichapproximates the Rayleigh distribution in terms of shape. The kernel density estimationof time headway is relatively smooth. The distance headway peaks about 20 m and thetime headway peaks at about 0.8 m/s.
Data Description

30

The feature values for the probability density distributions are listed in Table 4.3 below.
For the distribution of velocity and acceleration, a total of 755,793 observations wereselected. For the time headway and distance headway, a total of 450,780 observations
were selected. As the table shown, the mean value of car velocity is 26.5 m/s, the mean
acceleration is 0.07 m/s2, the mean time headway is 1.68 s, and the mean distance
headway is 42.68 m.
Table 4.2 Description of the features of cars𝒗 [m/s]𝒂 [m/s2]

𝒉 [s]

𝒔 [m]

count

755793755793450780450780mean

26.50140.06991.679542.6810std

6.7774

0.49120.985227.2075min

0.0139-6.5093

0.01

0.23

25%

22.0163-0.1543

0.92

21.48

50%

26.5283

0.039

1.42

35.004975%

31.6847

0.3026

2.24

58.6125max

41.6498

4.47654.99

119.99Where 𝒗 is the velocity, 𝒂 is the acceleration, 𝒉 is the time headway to lead vehicle, and𝒔 is the distance headway to lead vehicle. 25%, 50%, and 75% represents the first, seconds and the third quartiles respectively.
4.3.2. Variables distributions of truck
Figure 4.4 illustrates the velocity and acceleration distribution of the truck. It is noticeable
that the velocity of the truck distributes more concentrated than the velocity of the car,which with a steep and slightly left-skewed unimodal distribution. The kernel density estimation curve of the truck's velocity is unsmooth. Most of the trucks have a concentrated
distribution of velocities between 20 and 26 m/s.
The acceleration distribution pattern for trucks is similar to that of cars, also with a sharp,
approximately symmetrical, and unimodal distribution, which the concentration of values
around 0 m/s2. The distribution of trucks is more concentrated than that of cars, mostacceleration values of trucks are in the range of -1 m/s2 and 1 m/s2.

Data Description

31

Figure 4.4 Distribution and kernel density estimation of velocity and acceleration of truck: (a) velocity distribution, (b) kernel
density estimation for velocity, (c) acceleration distribution, and (d) kernel density estimation for accelration.

The distribution of time headway and distance headway between a truck, as a vehicle
with following behaviour, and the vehicle in front of it is shown in Figure 4.5. The distribution of the distance headway of the truck in the interval (0, 120) meter can be approximated as a right-skewed single-peaked distribution. The kernel density estimation plot
presents a more intuitive visualisation of this. With the help of the quantile values listed
in Table 4.3, it can be observed that most of the time headway observations are between
1.4 s and 3.15 s, and most of the distance headway values are between 32.08 m and73.74 m.
In graphical terms, the distance headway and the time headway have coherence. Thisis obviously visible in the kernel density estimation graph. The absence of smooth edges
on the right-hand side of the graphics indicates that the distance headway and time
headway of some of the trucks between the front vehicle are too large and beyond therange of values taken in this thesis. This situation is also consistent with the observationin real world that trucks are usually further away from the vehicle in front than cars.

Data Description

32

Figure 4.5 Distribution and kernel density estimation of distance headway and time headway of the following truck: (a)distance headway distribution, (b) kernel density estimation for distance headway, (c) time headway distribution, and (d)kernel density estimation for time headway.

Table 4.3 shows the distribution features of trucks. In total, there were 214,165 observations included in the velocity and acceleration distributions. For the time headway anddistance headway, 105,940 observations were extracted. The mean value of truck velocity is 23.26 m/s, the mean acceleration is 0.02 m/s2, the mean time headway is 2.33
s, and the mean distance headway is 54.43 m.
Table 4.3 Discription of the features of trucks𝒗 [m/s]𝒂 [m/s2]

𝒉 [s]

𝒔 [m]

count

214165214165105940105940mean

23.2561

0.02322.331954.4347std

2.99940.23951.150727.6877min

0.7971-3.4437

0.02

0.86

Data Description

33

𝒗 [m/s]𝒂 [m/s2]

𝒉 [s]

𝒔 [m]

25%

22.6953-0.0474

1.4

32.08

50%

23.6591

0.0066

2.19

50.73

75%

24.69410.0761

3.15

73.74

max

40.70152.68314.99

119.99Where 𝒗 is the velocity, 𝒂 is the acceleration, 𝒉 is the time headway to lead vehicle, and𝒔 is the distance headway to lead vehicle. 25%, 50%, and 75% represents the first, seconds and the third quartiles respectively.
A comparison of the distribution of the variables for trucks and cars follows. It can benoticed that cars have a wider distribution interval than trucks for both the velocity andacceleration variables. The distribution for trucks is more concentrated. This demonstrates that truck drivers are less likely to exhibit acceleration/deceleration behaviour oflarge magnitude compared to car drivers. In addition, driving styles and habits of individual truck drivers differ less from each other than from the group of car drivers.
The situation becomes different in the variables time headway and distance headway.
Trucks show a wider distribution, while cars have a more concentrated peak. The explanation for this phenomenon is that car drivers tend to keep a smaller following distancefrom the vehicle in front, whereas truck drivers usually keep a larger following distancebecause of the longer braking distance trucks required compared to cars.

4.4. Fundamental Diagram of Traffic Flow
The fundamental diagram gives the relationship between traffic flow speed, density, andvolume in a section of road. It is generally used to represent the traffic capability of theroad segment and to simulate changes in traffic flow under regulatory measures such as
speed limits.
Figure 4.6 shows the fundamental diagram for the data collection road section numbered
01 in exiD dataset. In this figure 𝑞 repesents traffic volume, 𝑣 is the speed of the traffic
flow, and 𝑘 is the traffic density in the road segment. As the seven road sections
contained in this dataset have similar fundamental diagrams, this subsection uses theroad segment number 01 as an example to analyze the traffic flow characteristics in thisdataset.

Data Description

34

Figure 4.6 Fundamental diagram of Data collection site 01 in exiD dataset

It can be clearly observed from the 𝑞 − 𝑘 diagram that traffic density and traffic flux show
a significant positive correlation. In a more typical k-q diagram, the traffic density reaches
a maximum when the traffic flux increases to a certain value and then gradually decreases as the traffic flux increases. However, in Figure 4.6, this change is not observed.
A possible explanation for this situation is the low density of traffic flow on the observed
sections of this dataset. As the traffic flux is below the capacity of this road segment, themaximum traffic ability of the road is not reached and therefore the traffic density does
not reach the maximum value of the capacity of the road section.
Negative correlations between variables can be observed in both the 𝑞 − 𝑣 and 𝑘 − 𝑣plots. Compared to the 𝑞 − 𝑘 plots, the 𝑞 − 𝑣 and 𝑘 − 𝑣 plots present a less pronounced
correlation. The explanation for this phenomenon is also because of the low density of
traffic. In this case, the speed of the traffic flow depends mainly on the desired speed ofeach individual vehicle, which in turn is related to the driving habits of the driver and thecharacteristics of the vehicle type. Traffic flow and density have some influence on speed
but are not the main factors.

4.5. Selection of Following Pairs
To begin with, we need to define the constraints of car-following behaviour. The constraints generally consist of three components, namely the lateral distance between thetwo vehicles (to determine that the lead vehicle is in the same lane as the following vehicle), the longitudinal distance (when the front vehicle is beyond a certain distance fromthe vehicle behind, the behind vehicle is in free driving and the behaviour of the frontvehicle has no effect on the behind vehicle), and the length of time that the following
state is maintained (to determine that the car-following behaviour between two vehicles
does exist and do not briefly exist in a front-to-back relationship). Table 4.4 summarize
the parameter ranges of the car-following periods in several studies and the final criteria
for car-following extraction used in this thesis based on previous studies.
Data Description

35

Table 4.4 Summary of extraction criteria for car-following behaviour

Lateral distance [m]Longitudinal distance [m](LeBlanc et al., 2013)(Chong et al., 2013)

> 15
< 1.9

(Fernandez, 2011)(Zhu et al., 2018)

This study

Duration [s]

< 120

> 30

< 100
< 2.5
At the same lane
(Lane ID known)

< 120

> 15

< 120

> 15

According to most of the literature listed, the longitudinal distance between the lead vehicle and the following vehicle was set to less than 120 meters and the following time
was longer than 15 seconds in this study. By using these conditions, the following pairs
between different types of vehicles in the dataset are filtered out. The flow chart of thefollowing-pair selection, in which the following vehicle is a truck and the leading vehicle
is a car, is illustrated in Figure 4.7.

Figure 4.7 Flowchart of following-pair selection: by truck follows car

Table 4.5 shows the number of car-following pairs. As can be seen in Table 4.5, in thedataset used in this thesis, the largest quantity of following pairs was of type car-car, witha total of 4,466 pairs, and the smallest quantity of following pairs was of type car-truck,

Data Description

36

with only 224 pairs in 92 datasets across seven observed locations. Trucks followed cars
the least, which might be explained by the more frequent lane changing behaviour ofcars or by the driving habits of truck drivers who prefer to follow trucks.
Table 4.5 Amount of different car-following type in dataset

Type of following vehicle

Type of leading
vehicleData Description

Car

Truck

Car

4466

224

Truck

715

625

37

5. Results and Discussion
This chapter introduces the experimental procedures and results. Section 5.1 explains
the experimental design. In sections 5.2 and 5.3, the convergence and the performanceof the optimization algorithms are evaluated. In section 5.4, the velocities estimated by
the calibrated model and the observed values are compared.

5.1. Experimental Design

Figure 5.1 Flowchart of Wiedemann 99 model calibration by GA and PSO

As shown in Figure 5.1, the experimental procedure consists of two parts, which are thecalibration of the W-99 model and the evaluation of the calibration results. In the calibration section, the filtered car-following data is fed into the GA and PSO calibrators respectively. The calibrator uses the chosen optimization algorithm (GA or PSO) to adjust theparameters of the model with the goal of making the difference between the model simulated output and the real observations as minimal as possible.
5.1.1. Initialization of models
Before calibration, the car-following model and calibrator need to be initialized. The parameters to be calibrated in the car-following model and the parameters in calibrators
are initialized with the values recommended by the existing literature. The setting of theResults and Discussion

38

calibrator parameters has a direct impact on the calibration results, so it is important tochoose the appropriate parameters.
 Wiedemann 99 model
The initial values and bounds for the parameters to be calibrated in the W-99 model are
given in Table 5.1. The bounds for the values are referenced from Vortisch and Zhu's
study (Leyn & Vortisch, 2015; Zhu et al., 2018). Compared to the references, the desired
speed boundaries 𝑉𝐷𝐸𝑆 are more narrowly defined at 72 to 108 km/h, which is set basedon the high-speed characteristics of the traffic flow on the highway.
Table 5.1 The default value and bound of parameters in W-99 modelParameter

description

BoundsDefault

𝐶𝐶0 [m]Standstill gap[0, 20]1.0192𝐶𝐶1 [m]Headway time

[0, 5]

1.4242𝐶𝐶2 [m]“Following” variation[0, 10]5.8715𝐶𝐶3 [s]Threshold for entering “following”[-20, 0]-17.1579

𝐶𝐶4 [m/s]

Negative “following” threshold[-5, 0]-0.2312𝐶𝐶5 [m/s]

Positive “following” threshold[0.1, 5]1.7946𝐶𝐶6 [10-4 rad/s]

Speed dependency of oscillation[0.1, 20]

3.5519𝐶𝐶7 [m/s2]

Oscillation acceleration[−1, 1]0.5350𝐶𝐶8 [m/s2]Standstill acceleration[0, 8]

4.0101𝐶𝐶9 [m/s2]

Acceleration at 80 km/h

[0, 8]

2.6549𝑉𝐷𝐸𝑆 [km/h]

Desired speed of following vehicle[72, 108]86.0492

 GA and PSO Calibrator
Table 5.2 and 5.3 shows the initial parameter values for GA and PSO calibrator respectively. In this thesis, GA and PSO algorithms used in the calibrator are called from an
open-source Python library named scikit-opt, which is a module built specifically for heuristic algorithms (Guo, 2020).
The initial values are set with reference to published literature on the calibration of carfollowing models (Dadashzadeh et al., 2019). As PSO typically has a faster convergence
speed than GA, a lesser maximum iteration is set for PSO in the experiments.

Results and Discussion

39

Table 5.2 Initial parameter values of PSO calibrator

Parameter

description

Default

𝑚𝑎𝑥𝐼𝑡𝑒𝑟Maximum number of iterations

50

𝑛𝑃𝑜𝑝

Population size

10

𝑛𝐷𝑖𝑚

Dimension of objective function

11

𝑝𝑀𝑢𝑡𝑎𝑡𝑖𝑜𝑛

Mutation rate

0.001

𝑝𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛

Precision: int/float or list

1e-4

Table 5.3 Initial parameter values of PSO calibrator

Parameter

description

Default

𝑚𝑎𝑥𝐼𝑡𝑒𝑟Maximum number of iterations

40

𝑛𝑃𝑜𝑝

Swarm size

10

𝑛𝐷𝑖𝑚

Dimension of objective function

11𝑤

Inertia weight

0.5

𝑐1

Personal learning coefficient1

𝑐2

Global learning coefficient1

5.1.2. Goodness-of-Fit Function Definition
Goodness-of-Fit (GoF) function measures the accuracy of the calibrated model, which
serves as the objective function of GA and PSO. The fitness function plays an importantrole in the selection process of an individual solution for the next generation in GA anddetermines the tendency of the particle swarm to move in PSO. In this thesis, the rootmean-square error (RMSE) is used to define the fitness function. The RMSE is definedas:∑𝑁 (𝑥𝑖 − 𝑥̂𝑖 )2
𝑅𝑀𝑆𝐸 = √ 𝑖=1
𝑁
where 𝑥𝑖 is the actual observation from real-world data in frame 𝑖, 𝑥̂𝑖 is the simulated
value, and 𝑁 is the total number of frames input.

Results and Discussion40

Since the output of the Wiedemann model is the predicted acceleration of the vehicle atthe next time stop, the fitness function is defined as the RMSE of acceleration:

𝐹𝑖𝑡𝑛𝑒𝑠𝑠 = 𝑅𝑀𝑆𝐸𝑎𝑐𝑐 = √

∑𝑁
̂𝑖 )2
𝑖=1(𝑎𝑖 − 𝑎𝑁where 𝑎𝑖 is the observed acceleration in frame 𝑖, 𝑥̂𝑖 is the simulated acceleration from
W-99 model, and 𝑁 is the total number of frames input.

5.2. Optimization Convergence Analysis
In an optimization algorithm, convergence represents a steady state at the end of theprocess, when more iterations will not lead to further changes or improvements in the
results. During optimization, premature convergence might be observed, where the algorithm finds a stable point (usually referred to as a sub-optimal solution) too early andthe evaluation results are worse than expected.
5.2.1. Convergence of GA

Figure 5.2 Convergence of GA calibrator: by following-pair type with car-truck, 𝑛𝑃𝑜𝑝 = 10 , 𝑚𝑎𝑥𝐼𝑡𝑒𝑟 = 50 , and𝑝𝑀𝑢𝑡𝑎𝑡𝑖𝑜𝑛 = 0.004

Figure 5.2 illustrates the convergence process for calibration of the W-99 model by using
the GA calibrator. The input following-pairs data is in type car-truck, in which the leadvehicle is a car, and the following vehicle is a truck. The scatter plot above shows thefitness value of each individual in every generation, and the line plot below presents thelowest fitness value achieved in each iteration. In the scatter plot we can observe that,in general, the distribution of fitness values for the population tends to converge as the
Results and Discussion

41

iterations increase. During the iterations, some outliers appear occasionally. This is attributed to the randomness of the mutation in the algorithm. Since the direction of variation is uncertain, from the scatter plot we can observe the individuals with high fitness
values after mutation, which are usually dropped out of the selection and do not survive
into the next generation of inheritance. In contrast, when individuals with lower fitness
values than their parents appear, there is a reduction in the minimum fitness value recorded by the line plot. This mutation is usually passed on to the next generation. As canbe noticed from the line plot, there is a significant decrease in the lowest fitness value ofthe population as the number of iterations increases.

Figure 5.3 The optimization process by GA-calibrator in different following-pair type: (a) car-car, (b) car-truck, (c) truckcar, and (d) truck-truck.

As can be seen in Figure 5.3 that for the four different types of following pairs, the final
fitness value (i.e., the RMSE between predicted acceleration and observed acceleration)that the GA calibrator converges to differs. In comparison, the GA calibrator has a better
convergence for the calibration of the following pair of types of truck-cars. For data oftype car-car and truck-truck, convergence is significantly worse. Possible reasons for thisresult are as follows: Algorithm features. Because of the random characteristics of both mutation and selection in GA, it is probable that during this optimization process, no better mutations

Results and Discussion

42

occurred or that genes with low fitness values were mutated but not selected by roulette wheel selection. Therefore, the good mutations were not passed on to the next
generation.
 Dataset features. As the traffic flow conditions in the entrance/exit area of the highway
are complex, vehicles would accelerate, decelerate, and change lanes more frequently in the entrance/exit area than on a normal highway. Consequently, car-following behaviour might also be impacted and exhibits instability.
As mentioned earlier, mutation has a significant impact on the convergence of the GA.
Therefore, mutation rate is a key parameter in changing the GA convergence. The appropriate mutation rate varies for different optimisation problems. A mutation rate that is
too low makes "profitable" mutations less likely to occur and will reduce the speed ofconvergence, while a mutation rate that is too high may increase the probability of a“good” gene being changed, which might also impact GA convergence. Therefore, it isimportant to find the appropriate mutation rate for a specific optimization problem. Figure
5.4 shows the effect of varying the mutation rate on the convergence of the GA calibrator,
with all other parameters kept constant.

Figure 5.4 The influence of different mutation rates on the convergence of the GA calibrator under identical conditions:following-pair type car-truck, 𝑛𝑃𝑜𝑝 = 10, 𝑚𝑎𝑥𝐼𝑡𝑒𝑟 = 50.

After comparison it is observable that, under the same conditions, the optimization process with a mutation rate of 0.002 converges significantly worse than the other two mutation rates. This is because the lower mutation rate in this scenario makes the probability
Results and Discussion43

of obtaining a good gene lower. In the initial iterations, the optimization with a mutation
rate of 0.008 has a higher adaptation value, but after about 30 iterations, it shows similar
convergence to the optimization with a mutation rate of 0.004.
5.2.2. Convergence of PSO

Figure 5.5 Convergence of PSO calibrator: by following-pair type with truck-car, 𝑛𝑃𝑜𝑝 = 10, 𝑚𝑎𝑥𝐼𝑡𝑒𝑟 = 40, 𝑤 = 0.8, 𝑐1 =1, and 𝑐2 = 1

The calibration convergence process by the PSO calibrator using car-truck data is shownin Figure 5.5. The line graph records the change in the global optimum 𝐺𝐵𝐸𝑆𝑇 as thenumber of iterations increases. From the plot, we can observe that the fitness value of
𝐺𝐵𝐸𝑆𝑇 decreases rapidly and converges in the first few iterations. After about 15 iterations, the particles converge on a single point and do not move to other positions anymore. Fast convergence is a property of the PSO algorithm, which is consistent with ourobservations.
Figure 5.6 below illustrates the convergence of the optimisation using the PSO calibrator
for different types of follow-up data. It can be seen that, in general, the PSO calibrator
converges rapidly. The calibration of type car-car data converged the slowest of the fourtypes, completing convergence at the 28th iteration. For the other three types, convergence was over before the 15th generation.
From the convergence results, despite the slowest convergence rate, the lowest fitness
value was obtained for the calibration of the car-car data. For calibrations of type trucktruck, despite having the fastest convergence, the convergence is the worst in comparison in terms of convergence values. Possible explanations for this situation are as follows: Premature convergence of the PSO. This is the biggest drawback of the original PSO
algorithm. Since the particles converge too quickly and always tend to move towards
Results and Discussion

44

the GBEST obtained in this iteration, PSO could easily fall into one of the local optima
before it has even searched for a global optimum in space. Although PSO can reduce
the probability of falling into a local optimum by assigning random terms to the velocity
and direction of particle motion, the standard PSO is weakly able to escape from local
optima compared to algorithms such as GA.
 Dataset characteristics. Since the GA calibrator also converged the worst for trucktruck data, it is also possible that there is relatively large variability in individual following behaviour in this combination of following-pairs.

Figure 5.6 The optimization process by PSO-calibrator in different following-pair type: (a) car-car, (b) car-truck, (c) truckcar, and (d) truck-truck.

For PSO, three parameters have a relatively large impact on the convergence of thealgorithm, namely the inertia weight parameter 𝑤, the cognitive coefficient 𝑐1 , and thesocial coefficient 𝑐2 . These three parameters affect the inertia and tendency of the particles when they move. Figure 5.7 and Figure 5.8 show the effect of the parameter 𝑤 andparameters 𝑐 on the convergence process of the PSO calibrator respectively, under theidentical conditions.

Results and Discussion

45

Figure 5.7 The influence of different inertia weight parameter on the convergence of the PSO calibrator under identicalconditions: following-pair type car-truck, 𝑛𝑃𝑜𝑝 = 10, 𝑚𝑎𝑥𝐼𝑡𝑒𝑟 = 40, 𝑐1 = 𝑐2 = 1.49

As shown in the figure, the PSO calibrator converges to a minimum at 𝑤 = 0.8 for theinertia weight parameter. For values of 0.3 and 0.5, there is no significant difference inthis optimisation problem.

Figure 5.8 The influence of different cognitive and social coefficient on the convergence of the PSO calibrator underidentical conditions: following-pair type car-truck, 𝑛𝑃𝑜𝑝 = 10, 𝑚𝑎𝑥𝐼𝑡𝑒𝑟 = 40, 𝑤 = 0.3

The cognition term represents tendency for individuals to replicate past behaviours thathave proved successful, while the social term represents the tendency to follow the successful experiences of others. In mainstream research, 𝑐1 and 𝑐2 are frequently set to
Results and Discussion

46

2.0, as this allows the search to cover areas which are centred on PBEST and GBEST.
Another commonly used value is 1.49445, which ensures convergence of the PSO algorithm. Carlisle has experimented extensively and has proposed a better set of parametersettings, setting 𝑐1 = 2.8 and 𝑐2 = 1.3 (Carlisle & Dozier, 2001). In the present optimisation problem, as shown in Figure 5.8, the set of parameters from Carlisle has a lower
convergence value. For the other two options, there is no significant difference in convergence.
5.2.3. Comparison of GA and PSO
Comparing the convergence of the GA and PSO calibrators, it can be seen that PSO
significantly outperforms GA in terms of the convergence speed of the algorithm. PSO
requires fewer iterations to converge rapidly, while the undirected nature of mutation
sacrifices some of the efficiency of the GA algorithm as the convergence of GA relies
mainly on 'effective' mutation (i.e., individual in child generation obtains lower fitness
values by mutation). The undirected nature of mutation sacrifices some of the efficiencyof the GA algorithm. Therefore, in general, GA requires more iterations to converge thanPSO.
In terms of convergence performance, the PSO algorithm usually achieves a lower fitness value than the GA as well. This situation is probably because the GA is still notcompletely converged after the maximum number of iterations. Therefore, in future improvements, a better convergence could be considered by increasing the number of iterations of the GA.

5.3. Optimization Performance Analysis
In order to understand the simulation ability of the calibrated model to car-following behaviour, the performance of the calibrated model needs to be assessed by comparing
the simulated results of the model with the real observations.
5.3.1. Calibrated parameter values of W-99 model
Table 5.1 and Table 5.2 present the parameters of the W-99 model after calibration using
GA and PSO respectively. Different parameter values are obtained by using data of different following-pair types.
Table 5.1 Calibrated parameters of W-99 model by GA calibrator

Parameter

description

Car-Car

Car-Truck Truck-Car

Truck-Truck

𝐶𝐶0 [m]Standstill gap

4.87265.81398.0051Results and Discussion

3.106447

Parameter

description

Car-Car

Car-Truck Truck-Car

Truck-Truck

𝐶𝐶1 [m]Headway time

4.58124.9331

4.76984.7660𝐶𝐶2 [m]“Following” variation

7.01579.96599.3601

7.2280𝐶𝐶3 [s]Threshold for entering “following” -16.2409-9.2694-12.8016-12.6292

𝐶𝐶4 [m/s]

Negative “following” threshold-0.9385-2.8122-0.8511-2.0539

𝐶𝐶5 [m/s]

Positive “following” threshold

3.64550.92402.70244.8331𝐶𝐶6 [10-4 rad/s] Speed dependency of oscillation 17.5722

8.735413.6691

17.9360

𝐶𝐶7 [m/s2]

Oscillation acceleration-0.39540.66990.2151-0.7996

𝐶𝐶8 [m/s2]Standstill acceleration

4.3382

0.23714.42607.8375𝐶𝐶9 [m/s2]

Acceleration at 80 km/h

2.23065.69097.65236.9053

𝑉𝐷𝐸𝑆 [km/h]

Desired speed of following vehi84.4924
cle

72.23472.0601

73.3952

From Table 5.1 we could observe that the calibration results for the same parameter
differed using different types of car-following data. Some of these results can be explained with the help of the actual meaning of the parameters. For example, the parameter 𝐶𝐶0, which represents the standstill gap between the two vehicles, and looking at
the value of 𝐶𝐶0, we can see that there is a maximum standstill gap between the truck
and the truck, a result that corresponds to the real situation.
Table 5.2 Calibrated parameters of W-99 model by PSO calibrator

Parameter

description

Car-Car

Car-Truck Truck-Car Truck-Truck

𝐶𝐶0 [m]Standstill gap

3.5862001.2947

𝐶𝐶1 [m]Headway time

5

5

5

5

𝐶𝐶2 [m]“Following” variation

1007.860610

𝐶𝐶3 [s]Threshold for entering “following” -20-7.5925-17.2691-18.8389

𝐶𝐶4 [m/s]

Negative “following” threshold-3.6509-3.4204-50

𝐶𝐶5 [m/s]

Positive “following” threshold

0.1

0.45260.1

3.8594𝐶𝐶6 [10-4 rad/s] Speed dependency of oscillation 0.1

9.192220

0.1

𝐶𝐶7 [m/s2]

0.9752-0.76540.9885

Oscillation acceleration

Results and Discussion

0.319848

Parameter

description

Car-Car

Car-Truck Truck-Car Truck-Truck

𝐶𝐶8 [m/s2]Standstill acceleration

8

0

0

0

𝐶𝐶9 [m/s2]

Acceleration at 80 km/h

5.93780

0

0

𝑉𝐷𝐸𝑆 [km/h]

Desired speed of following vehi72
cle

94.3696

94.0597

72

Comparing the two tables above, we can see that there are significant differences between the calibrated parameters of GA and PSO calibrators. When calibrating with the
PSO calibrator, it is observed that the particles often converge to the parameter boundaries of the W-99 model. This situation did not occur in the GA calibrator. For parameter
𝐶𝐶1, the calibration converged to the upper boundary for all four following-pair types.
Whereas for parameter 𝐶𝐶8, the results for car-car converge to the upper boundary, while
the other three types converge to the lower boundary. It is difficult to conclude the patternin terms of the parameter values. One conjecture for this phenomenon is that the settingof the boundaries may not exactly match the traffic scenario being calibrated. Anotherconjecture is that the parameters that converge to the boundaries are not strongly correlated with the adaptation values. In such cases, due to the algorithmic nature of PSO,
it is likely to be observed that the particles converge to the boundaries.
5.3.2. Estimation performance evaluation
The distribution of relative errors could be used to assess the ability of the model tosimulate realistic situations. Because of the frequent and sudden instantaneous changes
in acceleration, the relative error in velocity is used for error analysis in this thesis in thesection discussing calibration errors. Figures 5.9 and 5.10 illustrate the relative error invelocity for calibration of the W-99 model using the GA and PSO calibrators respectively.
The histograms of the error distribution are differentiated by the type of car-following
pairs.
Figure 5.9 shows the relative error distribution between the predicted velocity and theactual observed values for the W-99 model after calibration by using the GA calibrator.
It is observed that the relative error is generally distributed between the range from -0.1
to 0.1 for the four types of following pairs and the distribution peaks are around zero.
This means that the relative error is less than 10%. The W-99 model calibrated using thedata with the following-pair type of truck-truck has a more concentrated relative error
distribution, indicating that it is better calibrated. One reason for this result is the velocityof trucks cannot change significantly suddenly.

Results and Discussion

49

Figure 5.9 Relative error of velocity by GA calibrator in different following-pair types: (a) car-car, (b) car-truck, (c) truckcar, and (d) truck-truck.

For the W-99 model calibrated using the PSO calibrator, the distribution of relative errors
is quite similar to that using the GA calibrator. Figure 5.10 demonstrates this intuitively.
Similarly, the relative error of calibration using the PSO calibrator is less than 10% andthe relative error distribution is more concentrated for the truck-truck type.
The previous analysis shows that the truck-truck group, which converged less well in thecalibration process, obtained the most concentrated relative error distribution in the relative error analysis. That is, although the optimization process did not converge well, thecalibrated parameters still simulated the actual situation relatively well. This phenomenon is perhaps an implication, pointing to the fact that the fitness function we have chosen for convergence in this study perhaps does not exactly correspond with the optimization objectives.

Results and Discussion

50

Figure 5.10 Relative error of velocity by PSO calibrator in different following-pair types: (a) car-car, (b) car-truck, (c) truckcar, and (d) truck-truck.

5.3.3. Simulation by calibrated model
The next section will focus on individuals in the traffic flow. An individual vehicle is selected and the acceleration at the next time frame is simulated by using the calibrated
W-99 model from the first frame, and then the predicted velocity for the next time frame
is calculated by using the predicted acceleration and the current velocity. the time seriesof the predicted velocity and the observed velocity for that individual are plotted in thesame graph. By this graph, it is possible to obtain a more intuitive picture of the model'sability to simulate the trajectory of an individual vehicle.
After generating the simulation plots from the model calibrated by GA and PSO, it was
found that the GA and PSO calibrated models presented very similar results in the individual traffic behaviour simulation plots, therefore here only the GA-calibrated W-99
model is analysed here as an example for the single vehicle simulation.

Results and Discussion

51

Figure 5.11 Simulation of single vehicle by GA-calibrated W-99 model of different following-pair in accelration and velocity:(a) car-car, (b) car-truck, (c) truck-car, and (d) truck-truck

As can be seen in Figure 5.11(a), the single-vehicle selected in the car-car following pairis better modelled. The time series of acceleration is plotted on the left and the velocity
Results and Discussion52

is plotted on the right. From the acceleration plot, we can observe that the predicted andobserved values have the same shaped time series curve, but there is a deviation of
about a one-time frame delay between the predicted and observed values. When reflected in the velocities, the time series of predicted and observed values approximately
overlap.
In figures 5.11 (b), (c), and (d), the predicted and observed accelerations maintain shapeconsistency in only a portion of the time frames, while large deviations occur in otherparts. This large deviation of acceleration has an impact on the simulation results forvelocity. In figure 5.11 (c), for example, the predicted acceleration remains at zero after
58 second, whereas in the actual observation the acceleration rises to greater than zerofrom about 60 seconds and remains positive. This is reflected in the velocity plot, where
the predicted speed of the vehicle remains constant after 58 second, while the actual
speed of the vehicle stays up after a small drop until it crosses the curve with the predicted speed value. One possible explanation for this phenomenon of successive anomalies in the acceleration predictions at some of the time frames is due to the limitationsof the W-99 model. The fact that the predicted acceleration remains constant from acertain point onward may be because certain thresholds in the model settings have beenreached. This suggests that for a particular traffic situation there may be further possibilities for the model to be refined from a design perspective.
The relationship between the predicted and observed values of velocity can be obtainedby setting them as the x and y axes respectively. This result is illustrated in Figure 5.12.
Again, as the GA and PSO results are very similar, therefore only the GA results is shown
here as an example.
As shown in Figure 5.12, although there are some deviations, in general, the true speed
variation is simulated by the model. the data points for the car-car following pair areclosest to the 45-degree line, which means that the car-following model calibrated byusing data of this type could simulates good the same car-following scenario in reality
better. For the other three types of following pairs, the speed of the rear vehicle was able
to coincide partially with the 45-degree line (as in figures (c) and (d)), or the speed points
are still generally distributed around the 45-degree line despite the error between them(as in figure (b)).

Results and Discussion

53

Figure 5.12 45-degree-line of simulated velocity by GA calibrated W-99 model and the observed velocity of different
following-pair in accelration and velocity: (a) car-car, (b) car-truck, (c) truck-car, and (d) truck-truck

Results and Discussion

54

6. Conclusion and Outlook
In this chapter, a summary and outlook are given for this study. Section 6.1 reviews andsummarises the methodology, experiments, and analysis of this thesis. In section 6.2,
some shortcomings in this study and possible directions for improvement in subsequent
work are discussed.

6.1. Conclusion
The main objective of this thesis is to calibrate the car-following model, and to obtain amodel that is able to simulate the car-following behaviour and characteristics in microscopic traffic flows in the entrance/exit areas of German highways. The thesis uses theexiD dataset to obtain car-following data for this specific traffic scenario, which was collected for seven entrance/exit areas on a section of the German highway by using adrone. The filtered following pairs were divided into four datasets based on the vehicle
types of the preceding and following vehicles, and in a subsequent process, these data
were fed individually into a calibrator to calibrate the target car-following model and ultimately resulting in different sets of parameters.
The Wiedemann 99 model was chosen as the model to be calibrated in this thesis, as itis the most comprehensive model at present for the logical description of driver behaviour
and can describe most of the observed features of everyday driving behaviour well andis used frequently in the simulation of highway microscopic traffic flows. Because of thelarge number of parameters (11 parameters) that need to be calibrated in the
Wiedemann model, calibration by using traditional numerical methods is difficult andtherefore this thesis has chosen to use meta-heuristic algorithms, namely GA and PSO,
for calibration.
After initialising the parameters of the Wiedemann model and the calibrators of GA and
PSO, the following-pair data is fed into the calibrators respectively and the calibrated
results are obtained after optimization of the algorithm. The performance of both GA andPSO algorithms in this optimisation problem is discussed by analysing the convergenceof the algorithms. In order to evaluate the effectiveness of the calibration, the acceleration and velocity of the following vehicle in all following pairs were simulated using thecalibrated car-following model, and the resulting velocity time series were taken as relative errors to the actual observations in the dataset. By looking at the distribution of therelative errors, it was observed that after calibration, the relative errors in the model predictions were less than 10%.

Conclusion and Outlook55

To observe the simulation of the calibrated model for an individual vehicle, a single vehicle was selected, and the velocity and acceleration of that vehicle were predicted byusing the calibrated W99 model and the time series of predicted values were plotted on
the same line graph as the real values. We can observe that for acceleration the prediction error is greater than for velocity. In some of the line plots, the velocities simulated by
the model essentially overlap with the observed values. At some points in time, the model
appears to fail in its predictions, with the predicted acceleration ceasing to change after
a certain point is reached. Presumably this occurs because a certain threshold of themodel has been reached. This may mean that the model is still not a good enough description for the particular traffic scenario it is calibrated for.

6.2. Outlook
The experiment results show that GA and PSO are applicable to calibrate Wiedemannmodels for highway on- and off-ramps in general. However, there are still some issues
that can be improved in future work.
1)

The lack of convergence in the GA algorithm. Although we have observed convergence in the GA, the minimum of fitness value obtained is significantly higher
than the PSO. By increasing the population size and the number of iterations, it ispossible to increase the possibility of good mutant genes appearing and thus GA
could potentially gain further convergence.

2)

The particles of PSO always converge at the bound for some parameters. For
this problem, the bounds can be set manually in subsequent operations to avoid
particles converging at the boundary values. Another approach that could be attempted is referencing Wang's operation when using PSO for multi-parameter
optimization (Wang et al., 2023). Before the optimization starts, a correlation
analysis is performed on the parameters to be calibrated and those with high correlation are selected for calibration. The distribution of the error values in space isobtained by Monte Carlo methods and Latin hypercube sampling, allowing visualobservation of whether the lowest value of the error is indeed located at theboundary.

3)

The calibrator operates inefficiently. This is one of the reasons for the convergence of the algorithm. To solve this problem, multiprocessing can be added to
the GA and PSO calibrators to speed up the objective function in subsequent operations, which have been shown to significantly reduce the operation time.

4)

The simulated situation for a single vehicle is not representative of the global simulation of traffic flow. When performing single-vehicle simulations, we have also
observed that the model gives better results for some vehicles and worse results

Conclusion and Outlook

56

for others. This is a bias due to the individual driving style and behaviour of thevehicle driver. Therefore, in order to evaluate the simulation of the model for the
traffic flow overall, it is necessary to use traffic simulation software with the
Wiedemann model, such as SUMO, to simulate each particle of the traffic flow
separately using the calibrated parameters. The simulation data is then analysed.
An assessment of the consistency of the traffic flow can be achieved by comparing the fundamental diagram of traffic flow between simulated and observed values.

Conclusion and Outlook

57

References
Aghabayk, K., Sarvi, M., & Young, W. (2016). Including heavy vehicles in a car ‐following model: modelling, calibrating and validating. Journal of Advanced
Transportation, 50(7), 1432-1446.
Ahmed, H. U., Huang, Y., & Lu, P. (2021). A review of car-following models and modelingtools for human and autonomous-ready driving behaviors in micro-simulation.
Smart Cities, 4(1), 314-335.
Bando, M., Hasebe, K., Nakayama, A., Shibata, A., & Sugiyama, Y. (1995). Dynamical
model of traffic congestion and numerical simulation. Physical Review E, 51(2),1035.
Barceló, J., & Casas, J. (2005). Dynamic network simulation with AIMSUN. In Simulation
approaches in transportation analysis (pp. 57-98). Springer.
Brackstone, M., & McDonald, M. (1999). Car-following: a historical review.
Transportation Research Part F: Traffic Psychology and Behaviour, 2(4), 181196.
Carlisle, A., & Dozier, G. (2001). Tracking changing extrema with particle swarm
optimizer. Auburn Univ., Auburn, AL, Tech. Rep. CSSE01-08.
Chaudhari, A. A., Karthik, K., Chilukuri, B. R., Treiber, M., & Okhrin, O. (2022).
Calibrating Wiedemann-99 model parameters to trajectory data of mixed
vehicular traffic. Transport. Res. Rec.
Chong, L., Abbas, M. M., Flintsch, A. M., & Higgs, B. (2013). A rule-based neural network
approach to model driver naturalistic behavior in traffic. Transportation ResearchPart C: Emerging Technologies, 32, 207-223.
Dadashzadeh, N., Ergun, M., Kesten, S., & Žura, M. (2019). An automatic calibration
procedure of driving behaviour parameters in the presence of high bus volume.
Promet-Traffic&Transportation, 31(5), 491-502.
Durrani, U., Lee, C., & Maoh, H. (2016). Calibrating the Wiedemann’s vehicle-following
model using mixed vehicle-pair interactions. Transportation Research Part C:
Emerging Technologies, 67, 227-242.
Eberhart, R., & Kennedy, J. (1995). A new optimizer using particle swarm theory.
MHS'95. Proceedings of the sixth international symposium on micro machine andhuman science,
Fellendorf, M., & Vortisch, P. (2010). Microscopic traffic flow simulator VISSIM. In
Fundamentals of traffic simulation (pp. 63-93). Springer.
Fernandez, A. (2011). Secondary tasks in steady state car following situations
Gazis, D. C., Herman, R., & Rothery, R. W. (1961). Nonlinear follow-the-leader modelsof traffic flow. Operations research, 9(4), 545-567.
Gipps, P. G. (1981). A behavioural car-following model for computer simulation.
Transportation Research Part B: Methodological, 15(2), 105-111.
Guo, F. (2020). scikit-opt https://github.com/guofei9987/scikit-opt/
Hamdar, S. H., Mahmassani, H. S., & Treiber, M. (2015). From behavioral psychology toacceleration modeling: Calibration, validation, and exploration of drivers’
cognitive and safety parameters in a risk-taking environment. Transportation
Research Part B: Methodological, 78, 32-53.
He, Z., Zheng, L., & Guan, W. (2015). A simple nonparametric car-following model driven
by field data. Transportation Research Part B: Methodological, 80, 185-201.
Holland, J. H. (1992). Adaptation in natural and artificial systems: an introductory
analysis with applications to biology, control, and artificial intelligence. MIT press.
Hu, X.-B., & Di Paolo, E. (2009). An efficient genetic algorithm with uniform crossover for
air traffic control. Computers & Operations Research, 36(1), 245-259.
Koutsopoulos, H. N., & Farah, H. (2012). Latent class model for car following behavior.
Transportation Research Part B: Methodological, 46(5), 563-578.
Lazar, H., Rhoulami, K., & Rahmani, D. (2016). A review analysis of optimal velocity
models. Periodica Polytechnica Transportation Engineering, 44(2), 123-131.
Conclusion and Outlook

58

LeBlanc, D. J., Bao, S., Sayer, J. R., & Bogard, S. (2013). Longitudinal driving behavior
with integrated crash-warning system: Evaluation from naturalistic driving data.
Transportation research record, 2365(1), 17-21.
Leyn, U., & Vortisch, P. (2015). Calibrating VISSIM for the German highway capacity
manual. Transportation research record, 2483(1), 74-79.
Li, L., Chen, X. M., & Zhang, L. (2016). A global optimization algorithm for trajectory data
based car-following model calibration. Transportation Research Part C:
Emerging Technologies, 68, 311-332.
Liu, P., & Fan, W. (2020). Exploring the impact of connected and autonomous vehicles
on freeway capacity using a revised Intelligent Driver Model. Transportation
planning and technology, 43(3), 279-292.
Liu, Y., Zou, B., Ni, A., Gao, L., & Zhang, C. (2021). Calibrating microscopic traffic
simulators using machine learning and particle swarm optimization.
Transportation letters, 13(4), 295-307.
Liu, Z., & Yang, G. (2022). Large-scale traffic flow simulation based on intelligent PSO.
MATEC Web of Conferences,
Manjunatha, P., Vortisch, P., & Mathew, T. V. (2013). Methodology for the Calibration of
VISSIM in Mixed Traffic. Transportation research board 92nd annual meeting,Marini, F., & Walczak, B. (2015). Particle swarm optimization (PSO). A tutorial.
Chemometrics and Intelligent Laboratory Systems, 149, 153-165.
Moers, T., Vater, L., Krajewski, R., Bock, J., Zlocki, A., & Eckstein, L. (2022). The exiD
dataset: a real-world trajectory dataset of highly interactive highway scenarios ingermany. 2022 IEEE Intelligent Vehicles Symposium (IV),
Olstam, J. J., & Tapani, A. (2004). Comparison of Car-following models (Vol. 960).
Swedish National Road and Transport Research Institute Linköping, Sweden.
Park, B., & Qi, H. (2005). Development and Evaluation of a Procedure for the Calibrationof Simulation Models. Transportation research record, 1934(1), 208-217.
Pipes, L. A. (1953). An operational analysis of traffic dynamics. Journal of applied
physics, 24(3), 274-281.
Pourabdollah, M., Bjärkvik, E., Fürer, F., Lindenberg, B., & Burgdorf, K. (2017).
Calibration and evaluation of car following models using real-world driving data.
2017 IEEE 20th International conference on intelligent transportation systems(ITSC),
Punzo, V., & Simonelli, F. (2005). Analysis and comparison of microscopic traffic flow
models with real traffic microscopic data. Transportation research record,
1934(1), 53-63.
Rockwell, T. H., Ernst, R. L., & Hanken, A. (1968). A sensitivity analysis of empiricallyderived car-following models. Transportation Research/UK/.
Rrecaj, A. A., & MBombol, K. (2015). Calibration and Validation of the VISSIM
Parameters-State of the Art. TEM journal, 4(3), 255.
Saifuzzaman, M., & Zheng, Z. (2014). Incorporating human-factors in car-following
models: a review of recent developments and research needs. Transportation
Research Part C: Emerging Technologies, 48, 379-403.
Sheppard, J., & Haberman, B. (2021). Using Particle Swarm Optimization to Learn a
Lane Change Model for Autonomous Vehicle Merging. 2021 IEEE Symposium
Series on Computational Intelligence (SSCI),
Sun, D., Benekohal, R. F., & Waller, S. T. (2006). Bi‐level programming formulationand heuristic solution approach for dynamic traffic signal optimization.
Computer‐Aided Civil and Infrastructure Engineering, 21(5), 321-333.
Treiber, M., & Helbing, D. (2003). Memory effects in microscopic traffic models and wide
scattering in flow-density data. Physical Review E, 68(4), 046119.
Treiber, M., Hennecke, A., & Helbing, D. (2000). Congested traffic states in empiricalobservations and microscopic simulations. Physical Review E, 62(2), 1805.
Treiber, M., & Kesting, A. (2013). Traffic flow dynamics. Traffic Flow Dynamics: Data,
Models and Simulation, Springer-Verlag Berlin Heidelberg, 983-1000.

Conclusion and Outlook

59

Wang, H., Sun, C., Haidn, O., Aliya, A., Manfletti, C., & Slavinskaya, N. (2023). A jointhydrogen and syngas chemical kinetic model optimized by particle swarm
optimization. Fuel, 332, 125945.
Wiedemann, R. (1974). Simulation des Strassenverkehrsflusses.
Wiedemann, R., & Reiter, U. (1992). Microscopic Traffic Simulation: The Simulation
System MISSION, Background and Actual State (Project ICARUS (V1052) Final
Report, Issue.
Yu, M., & Fan, W. D. (2017). Calibration of microscopic traffic simulation models using
metaheuristic algorithms. International Journal of Transportation Science andTechnology, 6(1), 63-77.
Zhu, M., Wang, X., & Tarko, A. (2018). Modeling car-following behavior on urban
expressways in Shanghai: A naturalistic driving study. Transportation ResearchPart C: Emerging Technologies, 93, 425-445.

Conclusion and Outlook60

Declaration

I hereby confirm that the presented thesis work has been done independently and using
only the sources and resources as are listed. This thesis has not previously been submitted elsewhere for purposes of assessment.

Place, Date, Signature

Conclusion and Outlook

61