Data-Driven Approaches for Traffic State and Emission Estimation

Data-Driven Approaches for Traffic State and Emission Estimation

Nikolaos Tsanakas

Linköping Studies in Science and Technology Dissertation No. 2144

Nikolaos TsanakasData-Driven Approaches for Traffic State and Emission Estimation2021

Link¨oping Studies in Science and Technology.

Dissertation No. 2144

Data-Driven Approaches for Traffic State and Emission

Estimation

Nikolaos Tsanakas

Department of Science and Technology Link¨oping University, SE-601 74 Norrk¨oping, Sweden

Data-Driven Approaches for Traffic State and Emission Estimation Nikolaos Tsanakas

Cover photo taken by the author.

Link¨oping Studies in Science and Technology. Dissertation No. 2144 Copyright © 2021 Nikolaos Tsanakas, unless otherwise noted isbn978-91-7929-645-2

issn0345–7524 Link¨oping University

Department of Science and Technology SE-601 74 Norrk¨oping

This work is licensed under a Creative Commons Attribution- NonCommercial 4.0 International License.

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Abstract

Traffic congestion is one of the most severe problems in mod- ern urban areas. Besides the amplified travel times, traffic conges- tion intensifies the amount of emitted pollutants impacting human health and the environment. By making the appropriate interven- tions in traffic, transportation planners can mitigate congestion and enhance the performance of a traffic system. One crucial step in traffic planning and management is the estimation of the current or historical traffic state of a network. The estimation of the traffic state variables (traffic flow, density and speed) reveals the problematic parts of a network, namely, the parts associated with severe conges- tion and high emission rates. Traffic-related observations and traffic models constitute two core elements of a traffic state estimation ap- proach. While the available observation data explicitly or implicitly provide partial information on the traffic state, traffic models de- fine the traffic behaviour and contribute to estimating the variables when they are not directly observable. The estimated traffic state variables form the input to the so-called emission models, which estimate the mass of the emitted pollutants.

The type and availability level of the observation data play a key role in traffic state and emission estimation. Traditionally, the primary source of traffic-related field data are stationary detectors (loop detectors, radar sensors or cameras). Today, following the late advances in communication systems, a vast amount of traffic-related data from mobile sources (GPS or cellular networks) is available.

Such high data availability may give transportation planners new insights into understanding traffic behaviour. Appropriate exploita- tion of data coming from mobile sources can improve the existing approaches for estimating the traffic state and emissions.

The broad aim of this thesis is to enhance the quality of traf- fic state and emission estimation. A special focus is given to the development of methods for exploiting the growing availability of traffic-related field data. By combining traffic data and models, the thesis proposes data-driven approaches for traffic state and emis- sion estimation.

The first part of the thesis (Paper I and Paper II) focuses on improving the current approaches for network-wide emission esti- mation. Traditionally, network-wide emission estimations rely on a static traffic-modelling framework. In Paper I, we suggest an al-

ternative emission estimation approach, which is based on a quasi- dynamic traffic model. To evaluate our approach, we perform field experiments on a 19 km long highway stretch in Stockholm. The results show that our method can improve the spatiotemporal dis- tribution of the estimated emissions. In Paper II, the approach sug- gested in Paper I is applied to a more extensive network covering the city of Norrk¨oping. The results indicate that our approach yields a realistic spatial layout of emissions.

The second part of the thesis (Paper III and Paper IV) suggests novel data-driven approaches for estimating network-wide traffic flows and demand. More specifically, in Paper III, we develop a data-driven traffic-flow propagation approach by utilising travel- time observations. Our method is based on a piecewise linear ap- proximation of the travel time function, which allows the use of an efficient event-based structure for propagating the traffic flow. We evaluate our approach through simulation-based experiments, and the results provide proof of the concept. In Paper IV, we exploit the approach suggested in Paper III to develop an efficient data- driven scheme for estimating the traffic demand. The results of the simulation-based experiments indicate that our approach might lead to more accurate estimations compared to other data-driven estimation approaches suggested in the literature.

Finally, the last part of the thesis (Paper V) focuses on the es- timation of fuel consumption and emissions at a vehicle level. In paper V, we propose a novel method for generating virtual vehi- cle trajectories by fusing data from different sources. Our approach provides a detailed description of vehicle kinematics, and thus, it permits the use of the underlying virtual vehicle trajectories to ve- hicle dynamics-sensitive applications, such as emission modelling.

The results of our experiments show that the advanced modelling of vehicle kinematics can enhance the accuracy of the estimated emis- sions.

Popul¨arvetenskaplig sammanfattning

Tr¨angsel i v¨agtrafiken har en negativ p˚averkan p˚a m¨anniskors liv genom ¨okade restider och utsl¨appsniv˚aer. Ett av de fr¨amsta m˚alen f¨or planering och styrning av trafik ¨ar d¨arf¨or att minska tr¨angseln genom att vidta l¨ampliga ˚atg¨arder.

Innan beslut om ˚atg¨arder i ett trafiksystem (exempelvis r¨orande utbyggnad, v¨agavgifter eller trafikstyrning) ¨ar det viktigt att ha bra information om nuvarande trafikf¨orh˚allanden. Trafikdata i form av fl¨odes- och hastighetsm¨atningar ¨ar dock vanligen begr¨ansade till vissa v¨agavsnitt. D¨ar s˚adana observationer saknas beh¨over trafiktillst˚andet ist¨allet uppskattas. Med trafikestimering avses att skatta trafiktillst˚andet utifr˚an just s˚adan partiell information, och grundstenarna i metoder f¨or trafikestimering ¨ar trafikmodeller och trafikm¨atningar. Trafikestimering ¨ar viktigt vid planering och styrning av trafiksystem eftersom det g¨or det m¨ojligt att se var i trafikn¨atet som problemen uppst˚ar.

Traditionellt har de m¨atningar som anv¨ands vid trafikestimer- ing i huvudsak kommit fr˚an station¨ara punktm¨atningar (slingde- tektorer, radarsensorer och kameror). Idag erbjuder ny teknik ett

¨overfl¨od av trafikrelaterade data fr˚an mobila k¨allor (GPS eller mo- biln¨at). Dessa nya data kan hj¨alpa trafikplanerare att b¨attre f¨orst˚a resen¨arers val och trafikbeteende.

Denna avhandling unders¨oker hur den ¨okande tillg˚angen p˚a trafikdata kan utnyttjas f¨or att f¨orb¨attra estimering av trafiktillst˚and och emissioner fr˚an v¨agtrafiken. De artiklar som ing˚ar i avhandlin- gen f¨oresl˚ar och utv¨arderar datadrivna trafikestimeringsmetoder, som utv¨arderas genom empiriska eller simuleringsbaserade exper- iment. Resultaten visar att de metoder som f¨oresl˚as i avhandlin-

gen kan f¨orb¨attra noggrannheten vid skattning av trafiktillst˚and och emissioner.

Acknowledgements

First and foremost, I would like to thank my supervisor Jan Lund- gren and my co-supervisors, Joakim Ekstr¨om, Johan Olstam and David Gundleg˚ard for their continuous scientific guidance, gener- ous support and enthusiastic encouragement during my studies. It is thanks to their extensive knowledge and experience that this the- sis was made possible. Having the opportunity to collaborate with them has been a great honour.

I gratefully acknowledge the funding sources of my research work. The research in this thesis was part of the research projects BE3AT (Better Estimations of Energy use and Emissions when Analysing Traffic), TENS (Network-wide monitoring of road traffic induced energy consumption based on alternative traffic sensor data), EVY (Energy and emission estimation from radar trajectory data) and POST2 (Prediction and scenario-based traffic manage- ment), which were financed by the Swedish Energy Agency and the Swedish Transport Administration. I would like to acknowledge all the BE3AT, TENS, EVY and POST2 project members and especially Ian Marsh for his valuable insights and feedback.

Special thanks should go to all my colleagues at the Commu- nications and Transport Systems division, especially to the traffic research group, for creating a pleasant and constructive working environment. I am particularly grateful to Clas Rydergren for his valuable advice and suggestions. I would also like to extend my sin- cere thanks to Ida Kristoffersson from the Swedish National Road and Transport Research Institute. Her insightful comments and sug- gestions have greatly improved the quality of this dissertation.

I am grateful to all my friends here in Norrk¨oping for being like a second family to me. Alan, Cristian, Eleni, Malvina, Mano, Maria, Mario, Michela, Niko, Tania, Taso, Tobias, Wenjian, Xeno, Yanni A., Yanni P. and Zheng, thank you for all the beautiful moments we spent together throughout these years. Also, I should not fail to express my gratitude to my office mates, Antzela and Ahmet, and to all my dear friends in Greece, whom I have immensely missed all these years. Thanks should also go to my sweet Elena for always being supportive and caring.

Above all, I would like to thank my parents, Θαν´αση and Θεoδ ´ωρα, my brother Aντ ´ωνη and my sister ⁰Eλενα whose un- conditional love and support are with me in whatever I pursue.

Finally, I would like to dedicate this thesis to my grandfathers, N´ικo and Aντ ´ωνη since I did not have the opportunity to say a proper goodbye to them.

Norrk¨oping, May 2021 Nikolaos Tsanakas

Abbreviations

AVI Automated Vehicle Identification CFM Car Following Model

COPERT COmputer Programme to calculate Emissions from Road Transport

CSD Cross-Sectional Data

DDNA Data-Driven Network Assignment DDNL Data-Driven Network Loading DNL Dynamic Network Loading DTA Dynamic Traffic Assignment DUE Dynamic User Equilibrium FCD Floating-Car Data

GPS Global Positioning Systems

HBEFA HandBook on Emissions FActors for road transport ITS Intelligent Transportation Systems

LWR Lighthill, Whitham and Richards OD Origin–Destination

ODMEP OD Matrix Estimation Problem

PHEM Passenger Car and Heavy Duty Emission Model STA Static Traffic Assignment

TAP Traffic Assignment Problem TSE Traffic State Estimation UE User Equilibrium

VDF Volume Delay Function VVT Virtual Vehicle Trajectories

Contents

Abstract iii

Popul¨arvetenskaplig sammanfattning v

Acknowledgements vii

Abbreviations ix

I Introduction and Overview 1

1 Introduction 3

2 Traffic Data and State Estimation 7

2.1 Cross-Sectional Data 7

2.2 Floating-Car Data 9

2.3 Traffic State Estimation 10

3 Traffic Assignment Modelling 13

3.1 The Traffic Assignment Problem 15

3.1.1 Formulation and Notations 16

3.1.2 Route Choice 17

3.1.3 Network Loading 18

3.1.4 User-Equilibrium Assignment 19

3.2 The Dynamic Case 20

3.2.1 Formulation and Solution Approaches 20

3.2.2 Dynamic Network Loading 23

3.3 The OD Matrix Estimation Problem 27

3.3.1 Formulation 28

3.3.2 Solution Approaches 29

Contents

4 Traffic-Related Emission Modelling 33

4.1 Emissions from Road Traffic 33

4.2 Emission Modelling 34

4.2.1 Microscopic Emission Models 35

4.2.2 Macroscopic Emission Models 35

5 The Present Thesis 37

5.1 Motivation and Objectives 37

5.2 Research Methods 41

5.3 Delimitations 42

5.4 Contributions 43

5.5 Summary of the Papers in the Thesis 44

5.6 Future Research 49

Bibliography 51

II Papers 61

Paper I 65

Paper II 109

Paper III 125

Paper IV 181

Paper V 251

Part I

Introduction and Overview

Chapter 1

Introduction

The development of our modern society is strongly interrelated with urbanisation and a vigorous need for mobility. The current infrastructure supply is often unable to serve such raised demand for travelling, leading to congestion in several big cities worldwide.

Besides the increased travel times, which is an effect directly expe- rienced by road users, congestion aggravates the external impacts of traffic such as vehicular emissions. Investing in supply reinforce- ment can alleviate the traffic conditions and potentially compensate for the social welfare losses created due to congestion. Building new infrastructure may be an option, although it entails the risk of increasing the demand and potentially leading to the same conges- tion levels (Pfaffenbichler, 2011). Increasing the current infrastruc- ture capacity through traffic planning and management has been proven a more cost-effective alternative. Traffic models are useful tools commonly employed for management and planning purposes to appraise congestion effects and also assess congestion mitigation strategies. Alongside traffic models, emission models are used to estimate the emission rates so that their impact on the environment and human health can be evaluated. Emission models estimate the mass of pollutant emitted having as input the traffic state, typically expressed in terms of average speed, and the corresponding traffic activity, i.e., vehicle flow.

Traffic assignment models, which form a sub-family of traffic models commonly used for planning purposes, consider an implicit relationship between travelling demand and infrastructure supply to determine traffic patterns. Urban space is divided into zones

Introduction

by balancing certain surface and socio-economic criteria. Traffic de- mand is typically represented by a matrix called Origin–Destination (OD) matrix, which denotes the number of trips from an origin to a destination zone. Then, a traffic assignment process distributes the demanded trips among the available routes that connect an OD pair. Such assignment of demand to the network attempts to satisfy the underlying modelling hypothesis of travellers’ behaviour. Typi- cally, the behavioural rules that govern the travellers’ choices, such as the user equilibrium (Wardrop, 1952), are expressed in terms of in- dividual costs and benefits. The process that defines the number of trips at each OD route (as a portion of the OD demand), given the generalised routes’ cost, is called route choice. Accordingly, a process called network loading determines the cost of each route by propa- gating the traffic through the route links. A typical network loading process incorporates mechanisms for capturing congestion effects, and, therefore, the route costs depend on the underlying traffic pat- tern obtained by the route choice. This interdependency problem is referred to as the traffic assignment problem (Sheffi, 1985; Patriksson, 1994), and the solution approaches typically iterate among the route choice and loading processes.

Static traffic assignment models are the simplest type of assign- ment models, which assume that both demand and network load- ing are static. Even though these models have been proposed for over 50 years ago, they are still widely used due to their robust- ness, tractability, reduced computational cost and low requirements of input data (Brederode et al., 2019). Along with the strategic plan- ning usage, static traffic assignment usually constitutes the basis for environmental analyses because they can provide the required traffic state and activity in a straightforward manner (Wang et al., 2018). On the other hand, dynamic traffic assignment models con- sider both dynamic demand and network loading, and thus, they are acknowledged as more capable of modelling traffic congestion.

The underlying network model commonly incorporates advanced mechanisms for capturing the variations of congestion, which, how- ever, extend the required computational resources.

Traditional traffic assignment approaches typically employ en- dogenous variables for capturing congestion. Namely, the conges- tion level explicitly or implicitly depends on traffic state variables such as speed, density and flow, which at the same time are the variables under estimation. This modelling setting causes the inter-

Introduction

dependency problem between the route choice and network loading mechanisms and establishes an inherent complexity to the traffic as- signment problem. Despite this fact, the conventional endogenous- congestion approaches are an essential and widely used tool for modelling long-term variations on demand or supply. Traffic as- signment models are also commonly used for short-term planning or monitoring purposes, such as the estimation of current or his- torical flows and emissions. In such applications, the endogenous modelling of congestion is not strictly required since exogenous variables, such as traffic-related field data, can be deployed for deter- mining the congestion level. Thereby, we can bypass the computa- tionally costly route choice–network loading iterative process.

The latest advances in information and communication systems have provided a vast amount of traffic-related data, which can give new insights into traffic modelling. Taking advantage of the avail- able field data in traffic modelling is not something new since the traditional cross-sectional data from loop detectors or radar sensors have been widely used for several applications, such as traffic con- trol or OD demand matrix estimation. Today, besides the cross- sectional data, cost-effective floating-car data become more and more accessible from vehicle identification devices or cellular networks.

By fusing the data from various sources, we can directly estimate the traffic variables, i.e., speed, flow or travel time, which accord- ingly can be used for applications such as emission estimation. Fur- thermore, we can assimilate the data to existing modelling frame- works. Assuming that congestion is implicitly incorporated in the field observations, we can utilise the corresponding travel time esti- mations to determine the routes’ cost or flow propagation. Conse- quently, the available field observations allow us to use alternative data-driven formulations of traffic assignment.

The broad objective of this thesis is to provide new insights on the estimation of traffic flows, travel times and emissions. A special focus is given to the development of data-driven estimation meth- ods. The thesis examines how the growing availability of traffic- related field data can be exploited to enhance the traffic state and emission estimation. The five papers included in the thesis con- tribute to the fields of traffic assignment, traffic flow theory, traffic state estimation and OD demand estimation.

The thesis is divided into two parts: Part I provides an introduc- tion and an overview of the theoretical background of the thesis.

Introduction

Part II includes the five research papers that complete this disserta- tion. The remainder of Part I is outlined as follows: Chapter 2 de- scribes the main types of traffic-related data and gives an overview of the traffic estimation approaches. Chapter 3 discusses traffic as- signment modelling and presents its variants. The main approaches for estimating vehicular emissions are presented in Chapter 4. Fi- nally, Chapter 5 specifies the motivation, objectives, delimitations and contributions of the thesis, while it also provides a summary of the included papers.

Chapter 2

Traffic Data and State Estimation

The collection of traffic-related data is an essential tool for under- standing traffic behaviour, and it has led to significant contributions in traffic planning and management. Greenshields et al. (1935) were among the first who collected (see Figure 2.1) and exploited field data to develop a fundamental relationship between traffic flow and density. Since the seminal work of Greenshields et al., the data types and collection techniques have evolved to their contemporary form, which is associated with a vast amount of data coming from nu- merous sources. The various types of field data that are available today can be classified into two major categories: cross-sectional and floating-car data (Treiber and Kesting, 2013).

2.1 Cross-Sectional Data

Most of the existing devices for measuring traffic are stationary de- tectors, which collect Cross-Sectional Data (CSD). Stationary detec- tors are installed at specific cross-sections along a road and detect the vehicles that traverse their location. They can explicitly provide microscopic information (i.e., for individual vehicles), including the detection time, speed or length of a vehicle (Treiber and Kesting, 2013). However, most of the sensors typically aggregate in time such microscopic data to provide macroscopic quantities, such as

Traffic Data and State Estimation

Figure 2.1: Greenshields measuring speed and density by the help of a simple camera and a photographic timer. Image source: Kuhne (2008).

flow, mean speed or occupancy. This thesis mainly focuses on the macroscopic quantities collected by stationary sensors, and there- fore, the term CSD will only refer to macroscopic quantities.

The most commonly used devices for collecting cross-sectional data are inductive loop detectors and radar sensors, which are ex- tensively installed along the major streets of many European cities.

Loop detectors are installed on the roadway surface and sense the presence of a conductive metal object (Neudorff et al., 2003;

Auberlet et al., 2014). Radar sensors, which are usually installed on bridges above the traffic, transmit energy, and a portion of this energy is scattered by the vehicle back towards the sensor, where it is detected and converted into traffic information (Neudorff et al., 2003). Other devices commonly used for collecting CSD are pressure, acoustic and magnetic detectors (Leduc, 2008; Auberlet et al., 2014).

CSD provide information on traffic variables (flow, speed and occupancy) at fixed locations (cross-sections) expressed as a tempo- ral mean over each counting period. Figure 2.2 illustrates the flow (left-hand side) and speed (right-hand side) counts for a typical day of 2016 obtained by 92 radar sensors installed along a motorway stretch in Stockholm, Sweden. The average distance between two consecutive sensors is 240 meters, and the counting period is one minute.

Traffic Data and State Estimation

2016-04-20

00:00 06:00 12:00 18:00 00:00

Time 5

10 15 20

Location (km)

0 2000 4000 6000 8000

Flow (veh/h) 2016-04-20

00:00 06:00 12:00 18:00 00:00

Time 5

10 15 20

Location (km)

0 20 40 60 80 100 120 Speed (km/h)

Figure 2.2: CSD from stationary radar sensors installed along a 21 km long highway segment in Stockholm, Sweden. Source:

Tsanakas et al. (2017).

2.2 Floating-Car Data

During the last decades, the so-called Floating-Car Data (FCD) data have also become highly accessible due to extensive research on communication systems. FCD are typically obtained by probe vehi- cles, which, once projected on a time-space diagram, seem to ”float”

in the traffic flow (Treiber and Kesting, 2013). The time-space plot of Figure 2.3 shows the different spatiotemporal structure of FCD and CSD. FCD have been proven to be effective tools in traffic man- agement and an essential component in developing new Intelligent Transportation Systems (ITS) (Leduc, 2008).

The primary sources of FCD are vehicles equipped with Global Positioning Systems (GPS) devices or cellular phones (they are also called mobile sources). GPS devices collect the geo-referenced coor- dinates of probe vehicles at regular time intervals, which are then map-matched to the traffic network links. The major disadvantage of GPS-based FCD is the limited number and the commercial na- ture of vehicles equipped with such devices (Treiber and Kesting, 2013; Leduc, 2008). Due to that limited number of probe vehi- cles, the estimation of macroscopic variables, such as flow or speed, can be performed only under extensive assumptions. In contrast to the GPS-based approaches, the cellular-based methods of col- lecting FCD commonly employs a larger number of probe vehicles.

However, the map-matching of raw cellular data typically demands more complex algorithms, and precision is generally lower than the corresponding GPS location precision (Auberlet et al., 2014).

An alternative way to collect FCD is through Automated Vehi-

Traffic Data and State Estimation

FCD CSD

Figure 2.3: Time-space plot of virtual cross-sectional and floating-car data.

cle Identification (AVI) devices installed at the road infrastructure, such as license plate recognition or Bluetooth. AVI devices typi- cally require higher installation and maintenance cost compared to mobile sources. A more extended overview of traffic sensors and floating-data collection techniques is given by Allstr¨om et al. (2017).

A special type of FCD are the so-called trajectory data, which con- tain information on how every vehicle’s position within a given road section evolves over time. Such information can be obtained from video footages or series of photographs taken by cameras on the top of roadside buildings. Trajectory data can be considered as one of the most comprehensive types of traffic data available (Treiber and Kesting, 2013). Microscopic variables, such as position, speed and acceleration, can be explicitly or implicitly calculated given in- dividual trajectories, while macroscopic variables can be estimated from a data set of trajectories. Figure 2.4 illustrates the time-space plot of a trajectory data set collected at a stretch of Highway 101, Los Angeles, California, in 2005. Eight synchronised video cameras mounted from the top of an adjacent building were utilised to track the position of each vehicle passing through that highway stretch.

2.3 Traffic State Estimation

In practice, due to extensive installation and maintenance costs, it is not economically feasible to equip every part of a road network with stationary detectors or AVI devices. Furthermore, the available mobile sources very seldom provide data for the entire vehicle fleet (also because of privacy concerns, among other reasons). Therefore, the various sets of traffic data available for a road network typi-

Traffic Data and State Estimation

Figure 2.4: Time-space plot of trajectory data. Data source:

https://ops.fhwa.dot.gov/trafficanalysistools/ngsim.htm.

cally contain partial and noisy information on the traffic variables.

Figure 2.3 provides an illustrative example of such partial informa- tion. Speed is observed for a limited number of vehicles and only at specific points in time and space. However, based on the available observations, one can estimate the traffic variables for every vehicle and at every point in time and space. The term Traffic State Estima- tion (TSE)¹refers to the process of inferring the traffic state variables (e.g., flow, density or speed) on road segments relying on partially observed traffic data (Seo et al., 2017). Traffic state estimators con- stitute essential mechanisms in traffic modelling, and their latest advances have provided substantial contributions to traffic manage- ment and control. The main tools employed by the TSE techniques include filtering and extrapolation of the available data, fusion of data coming from heterogeneous sources and assimilation of the data to traffic models.

The various traffic state estimators suggested in the literature consider different levels of detail (ranging from a network to a ve- hicle level) and spatiotemporal granularity, depending on the po- tential application. Irrespectively of the required level of detail, the development of a typical TSE approach encompasses two elements:

the available partial data and a priori information on how traffic be- haves (such information could be independent of the partial data).

Depending on the prior information’s source, TSE techniques pro- posed in the literature can be categorised into three major method- ological groups (Seo et al., 2017): i) Model-driven or parametric es-

1In the literature, the term TSE is also referred to as highway traffic state esti- mation since it typically regards the estimation of traffic variables at a highway level. Accordingly, the term TSE introduced in this chapter exclusively refers to a highway-level estimation.

Traffic Data and State Estimation

timation approaches, where simple linear modelling structures (Ni and Wang, 2008) or more complex traffic models (Coifman, 2002;

Wang and Papageorgiou, 2005; Work et al., 2010; Mehran et al., 2012) are employed to reflect the traffic behaviour, ii) data-driven or non-parametric approaches, where the modelling structure is not a priori specified but instead adapts to the available data (e.g., kernel estimations) or to historical data, e.g., estimations based on regres- sion or neural networks (Sharma et al., 1999; Zhong et al., 2004;

Gastaldi et al., 2013), iii) a combination of the two previous groups (Treiber and Helbing, 2002; Van Lint and Hoogendoorn, 2010; All- str¨om et al., 2016).

Chapter 3

Traffic Assignment Modelling

The flow patterns throughout a traffic network are typically esti- mated by a modelling framework consisting of two stages: trip or traffic generation and trip or traffic assignment. At the first stage, the so-called activity system defines the total number of pro- duced or attracted trips for each zone by comprising land-use, socio- economic and demographic factors. This number of trips (or system users) reflects the system’s travel demand and forms the second stage’s input. At the second stage, the result of two competitive mechanisms, namely, system users (demand) and transportation sys- tem (supply), determine the final flow patterns (Sheffi, 1985). This second stage can be subdivided into three steps, forming an alter- native four-step formulation of the two-stage modelling framework (Mcnally, 2007): i) trip generation, ii) trip distribution, iii) modal split and iv) route assignment (see Figure 3.1). The trip distribu- tion process distributes the total trips produced at an origin to each destination, attempting to match the corresponding production and attraction patterns. Such trip demand is then split among the avail- able transportation modes, relying on the generalised cost or the disutility of each alternative. This way, the modal split process de- termines the trip demand per transportation mode, which is typi- cally expressed in a matrix form (the rows represent the origins and the columns the destinations). Finally, the route assignment process aims to distribute the demand among the routes that connect an OD

Traffic Assignment Modelling

Trip

distribution Modal split

Route assignment Trip

generation Production and attraction

patterns

Trip demand

per OD pair OD matrix per transportation

mode

Trip costs, flow patterns Activity

system

Transportation system

Figure 3.1: The two stages of the four-step modelling frame- work.

pair based again on each alternative route’s generalised cost.

Therefore, in a typical traffic assignment process, we distribute the demand among the different alternatives according to their cost.

The cost associated with each alternative is not fixed but rather de- pends on the assigned demand and the infrastructure supply. Thus, in analogy to microeconomic models, we seek to equilibrate the sys- tem, given a demand to supply relationship. Assuming that the traffic system constitutes a forward-feed system, such equilibration may be successively but separately performed for each of the three steps (trip distribution, modal split and route assignment). How- ever, since the last step might alter the generalised costs considered at the modal split (or even at the trip distribution), one can use the new trip costs as feedback to the previous steps (dashed arrows in Figure 3.1). Accordingly, the updated trip costs modify each trans- portation mode’s demand to be fed into the route assignment. This loop is followed until some convergence criteria are met. Such an iterative process is referred to as traffic assignment with elastic or variable demand.

Although the term traffic assignment, in principle, regards the last three steps of the four-step model, from here onwards in this thesis, it will refer only to the last step, route assignment. The de- scription and the investigation of the methods used in the first three steps lie beyond the scope of this thesis. The modelling approaches developed and presented in the thesis concern the route assignment step and only for private vehicle traffic. It is assumed that a reliable OD matrix (or a prior OD matrix that needs correction) is available, having been defined at the previous three stages. Moreover, it is as- sumed that the estimated trip costs from the route assignment step do not affect the trip distribution nor the modal split, namely, the demand is non-elastic.

Traffic Assignment Modelling

3.1 The Traffic Assignment Problem

At the last step of the four-step model, the demand for travelling between an OD pair (expressed either in terms of trips or system users) is assigned among the routes that connect that OD pair. On the one hand, we assume that system users choose their route fol- lowing some behavioural rules and given the routes’ disutility or generalised cost. On the other hand, the generalised cost of each route depends on the amount of traffic traversing that route (and other routes) due to congestion effects. Therefore, the Traffic Assign- ment Problem (TAP) is the problem of assigning the OD demand to the network so that the route volumes and route costs are equili- brated, reflecting the underlying behavioural rules.

The most commonly assumed behavioural rule adheres to the so-called Wardrop’s first principle (Wardrop, 1952), stating that:

The journey times on all routes actually used are equal and are not greater than those which would be experi- enced by a single vehicle on any unused route.

In other words, Wardrop’s first principle describes an equilibrium condition, called User Equilibrium (UE), where no traveller can improve her/his generalised cost¹ by unilaterally changing routes.

Hence, it is assumed that individual travellers act independently attempting to minimise their cost of travelling. Even though the as- sumption of such an ”egoistic” behaviour may seem reasonable, a UE condition presupposes that travellers know exactly the cost of each route a priori (this implies that travellers are experienced users of the network and/or have information on the costs). This is a pre- sumption that cannot always be assumed to hold true (Sheffi, 1985).

Alternatively, one can assume that travellers choose their route by unilaterally attempting to minimise their perceived cost. This as- sumption leads to a more ”realistic” condition called stochastic user equilibrium. In this case, a random error term representing the per- ception error is added to the generalised costs.

Nevertheless, it is still debatable within the research community to what extent and under which circumstances a traffic equilibrium exists in reality. Despite though the recent criticism on traffic equi- librium (Di and Liu, 2016; Yildirimoglu and Kahraman, 2018), the

1Wardrop refers to journey times, but there might be more factors that affect a route choice.

Traffic Assignment Modelling

equilibrium-based assignment remains the dominant approach for modelling the travellers’ route choices.

3.1.1 Formulation and Notations

Consider a transportation network represented by a directed graph G(N^,A)^{, where}N is the set of nodes and A is the set of links. Let R be the set of nodes that represent origin zones with R ⊆ N^, andS be the set of nodes that correspond to destination zones with S ⊆ N. The number of trips from an origin node, r ∈ R^{, to a} destination one, s∈ S, is denoted by x_rsand represents the demand of OD pair(r, s). Such demand is typically stored in a matrix having

|R| ^{rows and} |S| ^columns2, called OD matrix. Moreover, let Krs

be the set of different alternative link sequences, called routes, that connect origin node r ∈ Rto destination node s ∈ S. The number of travellers departing from origin r and travelling towards their destination s via the k^th route of OD pair (r, s), is denoted by f_rsk. Note that, by definition, the sum of route volumes, f_rsk, equals the total demand of OD pair,(r, s), namely,

k∈K

∑

f_rsk =xrs, ∀^r ∈ R^{, s}∈ S^{. (3.1)}

Each route k ∈ Krs is also associated with a route cost denoted by c_rsk. This cost includes all the aspects affecting how onerous the travel through that route is, such as travel time, monetary costs, etc.

Finally, let the link volumes³ be denoted by ya, a∈ A. We can then derive an explicit relationship between the route and link volumes as

y_a =

∑

r∈R

∑

s∈S

∑

k∈Krs

f_rskδ_rsk^a , ∀^a∈ A^{, (3.2)}

where δ_rsk^a is an incidence indicator, taking the value of 1 if link a is part of route k that connects r to s, and 0 otherwise.

2|C|denotes the cardinality of setC.

3Very often, the term flow is used in the literature instead of volume. While by definition, volume is a spatial quantity, denoting the number of vehicles per link, flow is a temporal quantity, i.e., number of vehicles passing by a cross-section;

this distinction makes sense only in dynamic models. To retain a terminological consistency, the term volume is used throughout this thesis to signify the absolute number of vehicles per link, route or OD pair.

Traffic Assignment Modelling

A traffic assignment model (UE-based or not) aims at determin- ing the optimal route volumes, f_rsk^∗ , whose optimal costs, c^∗_rsk, reflect the underlying assumed behavioural rules. Let us now reshape the variables f_rsk and c_rsk to the column vectors f and c, respectively.

Furthermore, let us define a process R : c → f . Process R, which is commonly called route choice, determines how travellers choose their route according to the assumed behavioural rules and given the route costs. Thus, the optimal route volume vector, f^∗, can be obtained by the route choice process as f^∗ = R(c^∗). However, loading different route volumes, f , to the network alters the sys- tem performance, which in turn affects the route costs, c. Consider now that a process which maps the route volumes to route costs is available, defined as L : f → c. We can employ process L, com- monly referred to as network loading, to calculate the optimal costs as c^∗ =L(f^∗). Therefore, the TAP can be formulated as a fixed-point problem (Bellei et al., 2005; Liu et al., 2009; Szeto et al., 2011; Yildiri- moglu and Kahraman, 2018), since we can derive the optimal route volumes by the joint process of route choice and network loading as

f^∗ = R(L(f^∗)).

3.1.2 Route Choice

In principle, a route choice process defines the portion of demand, x_rs, that chooses each route k∈ Krs for each r ∈ R^{, s} ∈ S^{. Let such} portions be denoted by p_rsk with

f_rsk = p_rskx_rs, ∀^k∈ Krs, r∈ R^{, s}∈ S^{, (3.3)} and

k∈K

∑

p_rsk =1, ∀^r∈ R^{, s}∈ S^{. (3.4)} Depending on how travellers are assumed to perceive route costs, modelling of route choices can either be deterministic or stochastic.

On the one hand, deterministic route choices imply that network users have a ”perfect” knowledge of route costs, and they opt for the route that actually has the minimum cost. On the other hand, the stochastic variant assumes that system users have different knowl- edge and perception of the route costs (Sheffi, 1985). Typically, a random variable, ξ_rsk, is distributed across the network users to re- flect the perception differences. Then, the demand portion on route

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k can be expressed in terms of probabilities,

p_rsk =prob(c_rsk+ξ_rsk ≤^crsl+ξ_rsl), ∀^l∈ Krs. (3.5) The form of this random utility model is defined by the assumed distribution of the perception differences. Irrespectively of the pre- sumptions on travellers’ perception, the route choice is not com- monly performed a priori because, in congested networks, we can- not know the route costs before the network loading.

3.1.3 Network Loading

A network loading model aims at determining the route costs given the route volumes. Each network link, a ∈ A, is associated with a generalised cost, ˆca. Typically, it is assumed that the cost of trav- elling through link a gets higher as the link volume, y_a, increases (due to congestion effects). Hence, the link costs can be expressed as a function of the link volumes, ˆca(ya), known as link-cost function.

Consequently, the route costs also implicitly or explicitly depend on the route volumes and are contingent upon the underlying mapping relationship of link to route volumes.

In the simple static case, where propagation effects are ne- glected, network loading is a rather straightforward process. We can obtain an explicit analytical relationship between the route costs and the link volumes as

c_rsk =

∑

a ˆca(ya)δ_rsk^a , ∀^k∈ Krs, r ∈ R^{, s}∈ S^{, (3.6)} where y_a is given by (3.2).

Assuming that travel time is the only factor affecting a link’s disutility, the link costs are given by the so-called Volume Delay Function (VDF). VDFs provide mathematical relationships between traversal link time, ta, and its volume, ya. One commonly used VDF was suggested by the Bureau of Public Roads of U.S. in 1964, as

ˆca(y_a) =t_a(y_a) =t_0a 1+γ ya

C_a

β!

, ∀^a ∈ A^{, (3.7)}

where C_a is the capacity of link a and t_0a the free-flow travel time of link a, and γ, β are parameters. Numerous types of VDFs have

Traffic Assignment Modelling

been proposed and used in practice, including Davidson’s (David- son, 1966) and the asymmetric Akcelik’s function (Akcelik, 1991), or even more complex functions incorporating delays and queue lengths (Skabardonis and Dowling, 1997).

3.1.4 User-Equilibrium Assignment

According to Wardrop’s first principle, UE is a stable condition where all the used routes between origin r and destination s have the same cost, and no traveller can improve her/his individual cost by unilaterally changing routes. It can be shown (Sheffi, 1985; Flo- rian and Hearn, 1995; Patriksson, 1994) that the UE conditions are satisfied if

(c_rsk−^θrs)f_rsk =0, ∀^k ∈ Krs, r∈ R^{, s} ∈ S^{, (3.8)} c_rsk−^θrs≥^0, ∀^k ∈ Krs, r∈ R^{, s} ∈ S^{, (3.9)} c_rsk, θ_rs, f_rsk ≥^0, ∀^k ∈ Krs, r∈ R^{, s} ∈ S^{, (3.10)} where θ_rsis the cost of the shortest route from r to s. Equation (3.8) essentially states that either the cost of a route equals to the shortest one or that this route is unused. Equation (3.8) is coupled with inequality (3.9), which constrains the costs of the unused routes to be greater than the shortest cost.

The UE problem was formulated as an optimisation problem with linear constraints by Beckmann et al. (1956), as

miny z(y) =

∑

a∈A Z _ya

0 ˆca(ω)d(ω), (3.11a)

s.t.

∑

k∈Krs

f_rsk = x_rs, (3.11b)

ya =

∑

r∈R

∑

s∈S

∑

k∈K^rs

f_rskδ^a_rsk, (3.11c) f_rsk ≥^0, ∀^k∈ Krs, r∈ R^{, s} ∈ S^{, (3.11d)} where the cost function, ˆc(y), is assumed to be a strictly increas- ing and positive function of y. It has been proven that Beckmann’s formulation satisfies the UE conditions and ensures the solution’s uniqueness (Sheffi, 1985; Patriksson, 1994). Several approaches for solving the UE problem have been suggested, including heuristic equilibration techniques, such as the incremental assignment (Sheffi,

Traffic Assignment Modelling

1985), or standard techniques for solving constrained convex prob- lems, such as the convex combination algorithm (Frank and Wolfe, 1956).

3.2 The Dynamic Case

Section 3.1 describes the static variant of TAP, which is usually re- ferred to as Static Traffic Assignment (STA). Such static formulation implies that both the demand and the network loading are static;

two assumptions that in most cases may be implausible and re- strict the use of STA to strategic planning (e.g., long-term planning for large scale networks). In practice, transportation systems are fast-paced alternating systems, and the negligence of any dynamics might deteriorate the modelling quality. The STA formulation con- siders the demand as constant over the demand period, commonly spanning over a relatively long period of a day (e.g., the morning peak period). Furthermore, the network loading is instantaneous, implying that all the assigned vehicles at a route are simultaneously present on each link of that route as propagation effects are not ex- plicitly considered. In contrast to STA, in Dynamic Traffic Assign- ment (DTA) i) the demand is not static but varies over time since the OD matrix commonly is time-dependent, and ii) the arrival time at a link is different from the departure time due to the Dynamic Network Loading (DNL). Thus, DTA incorporates mechanisms for capturing the temporal variations of traffic and provides a solid ground for operational and short-term planning usage.

3.2.1 Formulation and Solution Approaches

The rationale behind the DTA models is similar to the one consid- ered in STA; namely, each traveller is assumed to choose a certain route according to specific behavioural rules. Similar to STA, a ma- jor class of DTA problems is based on the UE, which can now be extended to the corresponding Dynamic User Equilibrium (DUE).

DUE is a stable condition where Wardrop’s first principle applies only to travellers assumed to depart at the same time. Consider a discrete-time setting, where the analysis period, T, is divided into H equal-length periods, called demand or assignment periods. Then, let x^h_rs represent the demand of OD pair,(r, s), for the demand pe-

Traffic Assignment Modelling

riod h ∈ H^{, where} H = {1, . . . , H}. Assuming that route choices are constant during each demand period, the DUE conditions are satisfied if

(c_rsk^h −^θrs^h)f_rsk^h =0, ∀^k∈ Krs, r∈ R^{, s}∈ S^{, h}∈ H^{, (3.12)} c^h_rsk−^θrs^h ≥^0, ∀^k∈ Krs, r∈ R^{, s}∈ S^{, h}∈ H^{, (3.13)} c^h_rsk, θ^h_rs, f_rsk^h ≥^0, ∀^k∈ Krs, r∈ R^{, s}∈ S^{, h}∈ H^{, (3.14)} where f_rsk^h is the number of vehicles which have departed during the h^th demand period and follow the k^th route that connects origin r to destination s. c^h_rsk is their associated travel cost, and θ^h_rs the minimum cost for the vehicles of x^h_rs. Note that, in accordance to the static case,

f_rsk^h = p_rsk^h x^h_rs, ∀^k∈ Krs, r∈ R^{, s} ∈ S^{, h}∈ H^{, (3.15)} and

k∈K

∑

f_rsk^h = x^h_rs, ∀^r∈ R^{, s}∈ S^{, h}∈ H^{. (3.16)}

However, in contrast to the static case, it becomes challenging to explicitly derive an analytical relationship between the route vol- umes and the route costs. The dynamic link volumes, which com- monly determine the link costs, can be expressed as

y_a(t) =

∑

r∈R

∑

s∈S

∑

k∈Krs

h

∑

f_rsk^h ˜δ_rsk^ah(t), ∀^a∈ A^{, t}∈ H^{, (3.17)}

where ˜δ_rsk^ah(t)is a binary indicator taking the value of 1 if route vol- ume f_rsk^h traverses link a at time instant t, and 0 otherwise. While binary indicator, δ_rsk^a , included in the static formulation is purely a graph property, indicator ˜δ_rsk^ah(t) should reflect propagation effects.

Therefore, a flow-propagation mechanism (network loading) is re- quired for solving the DUE and in general, the DTA problem. If such mechanism accounts for congestion, ˜δ_rsk^ah(t) also depends on the traffic conditions on the rest network links, providing an inher- ent complexity to DTA.

Although several analytical formulations of the DTA problem have been suggested, including the seminal studies of Merchant and Nemhauser (1978) and Janson (1991), none of them is able to

Traffic Assignment Modelling

provide a universal solution for general networks. The supplemen- tary constraints that account for flow propagation disturb the con- vexity of the problem (and all its appealing properties). Moreover, the analytical formulations of DTA typically incorporate rather sim- plified flow-propagation mechanisms in exchange for mathematical tractability. Therefore, such approaches lack in ”realism”, being un- able to adequately capture the complex traffic flow dynamics (Peeta and Ziliaskopoulos, 2001).

The modelling incapabilities of the analytical DTA formulations alongside the increasing computational capabilities of modern com- puters paved the way to the simulation-based DTA models (Smith, 1993; Peeta and Mahmassani, 1995; Ben-Akiva et al., 1998; Tong and Wong, 2000; Florian et al., 2008). The simulation-based DTA models do not explicitly include in their formulation the underlying prop- agation mechanism. The DNL is performed at a separate phase of the solution algorithm, deploying simulation to emulate the com- plex traffic interactions.

A typical solution algorithm iterates between the route choice and network loading (see Figure 3.2). Starting with an initial guess of the route costs (e.g., free-flow costs), the demand of each OD pair, x^h_rs, is distributed among the various routes k ∈ Krs, leading to the initial route volumes, f_rsk^h,0. At the DNL phase of iteration i, the traffic simulator propagates the route volumes, f_rsk^h,i, through the network, determining the route costs, c^h,i_rsk. At the next route choice phase, new route volumes, f_rsk^h,i+1, are defined by the considered optimisation routine (based on the costs c_rsk^h,i). This procedure is re- peated until some convergence criteria are satisfied, e.g., the relative cost difference between the used routes of the same OD pair being less than a predefined threshold. The so-called method of successive averages is a commonly used solution algorithm where, at each it- eration, an auxiliary traffic pattern is generated based on the routes with the minimum cost. Such auxiliary traffic patterns contribute to the algorithm’s direction-finding mechanism.

However, due to the simulation-based modelling setting, it is impossible to derive any mathematical properties of the underlying optimisation routine or optimal solution. Hence, the simulation- based approaches cannot guarantee convergence, uniqueness and all the desirable properties provided by the analytical formulation (at least for the static case). Nevertheless, from a practical perspec-

Traffic Assignment Modelling

Dynamic network loading

Route choice

Route costs, 𝑐_𝑟𝑠𝑘^ℎ Route

volumes, f_𝑟𝑠𝑘^ℎ

Analysis period, 𝑇 Demand period, ℎ

Simulation interval

Figure 3.2: The solution algorithms of the simulation-based DTA models iterate between the route choice and the network loading (adapted from Chiu et al., 2011).

tive, notions such as uniqueness and global optimality might not be meaningful in real-world applications (Peeta and Ziliaskopoulos, 2001). Instead, an effective and robust heuristic DTA implementa- tion that leads to close-to-optimal solutions might be more appreci- ated. One more key limitation of the simulation-based approaches is the expanded computational burden. Simulation-based models in- corporate more ”realistic” propagation mechanisms, which enhance the modelling quality but are computationally expensive. One cru- cial factor that heavily influences DTA’s deployment efficiency is the complexity of the corresponding network loading model since it might be called several times until the attainment of a solution.

3.2.2 Dynamic Network Loading

Network loading models, in principle, ”load” and propagate the demand to the network resulting in the required traffic patterns and their associated costs. A DNL model aims at capturing con- gestion, preserving at the same time fundamental principles that reflect the modelling realism. Depending on the granularity level, DNL models can be classified as microscopic or macroscopic. Traffic in microscopic models is represented by individual driver-vehicle units, while in macroscopic models by traffic streams. The most commonly used microscopic variables are position, speed and accel- eration of each vehicle. The variables most widely used by macro- scopic models are flow, density and average speed. A third mod- elling approach, which is outside the scope of this thesis, is called mesoscopic and can be placed between the microscopic and macro- scopic models. The choice of modelling perspective (micro, macro or meso) is typically performed based on the potential application of DTA (e.g., online or offline) and the demanded level of detail or

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accuracy.

Microscopic traffic simulation

Microscopic traffic simulation models attempt to describe individ- ual vehicles’ behaviour, and they are typically used for small-scale applications or situations where modelling heterogeneity of traffic is important (Treiber and Kesting, 2013). The core model of mi- croscopic longitudinal simulation is the Car Following Model (CFM).

CFM describes how one vehicle following another reacts to its sur- rounding traffic environment and the fluctuations of the leading ve- hicle’s traffic conditions. The longitudinal motion of the following vehicle is typically described by an ordinary differential equation, which relates the instantaneous acceleration of the follower to the leading vehicle’s kinematics.

The early formulations of CFM, including the seminal work of Pipes (1953), were based on empirical driving rules. The follower’s speed was assumed to be a linear function of the relative distance to the leader. Since then, numerous models have been suggested in the literature, including the so-called stimulus-response (Chan- dler et al., 1958; Gazis et al., 1961), safe-distance (Gipps, 1981) and psycho-spacing models (Wiedemann, 1973; Fritzsche and Ag, 1994).

Besides the longitudinal movement’s description, microscopic mod- els commonly provide mechanisms for describing the lateral move- ment based on the gap-acceptance theory, such as the approaches suggested by Gipps (1986) and Ahmed (1999).

The basic output of a microscopic simulation model is the tra- jectory of each individual vehicle. Considering each vehicle indi- vidually enhances the modelling quality but extends the simulation requirements for computational power and memory. Hence, the ap- plication of microscopic traffic simulation is limited to only off-line applications of DTA and for small-scale networks.

Macroscopic traffic flow models

On the other hand, macroscopic flow models are commonly used in more extensive networks and longer analysis time periods. Traffic flow is described at an aggregate level, and drivers’ behaviour is as- sumed to be homogeneous. Therefore, macroscopic traffic models describe collective phenomena, such as the evolution of congestion

Traffic Assignment Modelling

and the propagation of traffic waves (Treiber and Kesting, 2013).

They are also called hydrodynamic models since they are typically derived from the analogy between the traffic flow and the flow of continuous media such as fluid or gases. The basic dynamic macro- scopic variables, in correspondence to fluid dynamics, are density, ρ(x, t), flow, Q(x, t)and local speed, V(x, t), with

Q(x, t) =ρ(x, t)V(x, t), (3.18) where the independent variables are space⁴, x, and time, t. Density refers to the number of vehicles per length unit, x+dx, at time instant t, while flow concerns the number of vehicles per time unit, t+dt, at the location x (Hoogendoorn, 2001).

As traffic flow is treated as a one-dimensional compressible fluid, the movement of vehicles is governed by the continuity equation,

∂ρ(x, t)

∂t + ^∂Q(x, t)

∂x =0. (3.19)

In analogy to mass conservation, this equation describes the corre- sponding vehicle conservation. Vehicle conservation is a fundamental notion considered in macroscopic traffic models, ensuring that at a road segment with finite length, no vehicle appears or disappears, other than the vehicles that have entered or exited the road segment, respectively. Lighthill and Whitham (1955) and Richards (1956) con- sidered a supplementary relation between flow and density,

Q=Q_f(ρ), (3.20)

known as the fundamental diagram of traffic flow. The fundamental diagram is an experimental relation that applies to stationary traffic (Gentile et al., 2010). Furthermore, Lighthill and Whitham (1955) and Richards (1956) assumed that the stationary fundamental dia- gram also holds for non-stationary traffic,

Q(x, t) =Q_f(ρ(x, t)), (3.21) implying that flow, Q(x, t), or local speed, V(x, t), instantaneously and locally adjust to density changes. Thus, the continuity equation

4Although throughout Chapter 3 x denotes the OD demand, in this subsection, x denotes the continuous space. The same letter is used twice to retain a notational consistency with the basic references.

Traffic Assignment Modelling

becomes

∂ρ(x, t)

∂t + ^dQf(ρ) dρ

∂ρ(x, t)

∂x =0. (3.22)

This equation is known as the Lighthill, Whitham and Richards (LWR) model and is a non-linear, first-order partial differential equation that describes the propagation of kinematic waves. The solution of this partial differential equation has the following general form (Immers and Logghe, 2002; Yperman, 2007; Treiber and Kesting, 2013)

ρ(x, t) =F₀(x− ˜ct), (3.23) where F0(x) =ρ(x, 0)represents the initial density at x and ˜c is the kinematic wave speed defined as

˜c= ^dQf(ρ)

dρ . (3.24)

Equation (3.23) implies that a certain density ρ₀ = F₀(x₀) propa- gates with speed ˜c, which, by definition, equals the tangent of the fundamental diagram at ρ₀. Therefore, density is constant along a straight line in the t−x plane, called characteristic line. Any dis- continuity in the flow-density conditions that yields sharp changes from a state [ρ₁, Q_f(ρ₁)]to a state [ρ₂, Q_f(ρ₂)]creates a shock wave, which propagates with speed ¯c_1,2,

¯c1,2 = ^Qf(ρ2)−^Qf(ρ1)

ρ2−^ρ1 . (3.25)

Although the LWR first-order traffic flow model was de- veloped under some unrealistic assumptions (e.g., local speed instantaneously follows density changes), it is widely used in DNL. Numerous solution schemes for the LWR model have been suggested in the literature, including the cell transmission model (Daganzo, 1994) and the link transmission model (Yperman, 2007;

Gentile et al., 2010).

The boundary conditions are exogenous to the flow-propagation mechanism, and they are typically determined by the route choice process (Treiber and Kesting, 2013). The route choice defines the flows on the upstream boundary of the first link of each route and the turning fractions at each node. Then, the so-called node mod- els determine the realised transition flows from each incoming to

Traffic Assignment Modelling

each outgoing link of a node, given the turning fractions and some node or link boundary constraints. Macroscopic node models can be formulated as an optimisation problem that seeks to maximise the transition flows subject to demand and supply and vehicle con- servation constraints (Tamp`ere et al., 2011). Even though node mod- els, in general, have attracted less attention in the literature than the flow-propagation models, they constitute an essential component of DNL.

3.3 The OD Matrix Estimation Problem

OD matrices provide essential input to traffic assignment and nu- merous other transportation models. The modelling accuracy is very tightly related to the quality of the corresponding OD ma- trix. However, since OD trips seldom are directly observable and demand patterns are inherently complex, obtaining a reliable OD matrix has been proven to be extremely difficult. The conventional estimation typically relies on the combination of surveys (e.g., home or road-based interviews) and models (e.g., the first two steps of the four-step model); approaches that usually are rather expensive, both in terms of money and time. Furthermore, the fidelity of the esti- mated OD matrices is highly questionable as the surveys might be outdated.

An alternative estimation approach suggests the exploitation of volume counts from the stationary detectors that might be installed along a network’s links. This approach turns out to be more ben- eficial than the conventional ones because the volume counts pro- vide inexpensive and direct information of the underlying demand patterns. Thus, the OD Matrix Estimation Problem (ODMEP) is the problem of finding the OD matrix, which, once assigned to the network, satisfies the observed volume counts. In principle, the ODMEP can be seen as the inverse or reciprocal TAP: while in the latter we seek to find the optimal link volumes given an OD matrix, in the former, we look for an OD matrix given the link volumes (see Figure 3.3). ODMEP has attracted a lot of research interest lately, of- fering great challenges mainly due to its underdetermined nature:

the number of observed link volumes is typically considerably lower than the number of OD pairs.