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KATHOLIEKE UNIVERSITEIT LEUVEN FACULTEIT TOEGEPASTE WETENSCHAPPEN DEPARTEMENT BURGERLIJKE BOUWKUNDE AFDELING VERKEER EN INFRASTRUCTUUR Kasteelpark Arenberg 40, B - 3001 Heverlee (Belgium)

Dynamic modeling of heterogeneous vehicular traffic

Promotor: Proefschrift voorgedragen tot prof. ir. L.H. Immers het behalen van het doctoraat

in de toegepaste wetenschappen

door

ir. Steven Logghe

June 2003

KATHOLIEKE UNIVERSITEIT LEUVEN FACULTEIT TOEGEPASTE WETENSCHAPPEN DEPARTEMENT BURGERLIJKE BOUWKUNDE AFDELING VERKEER EN INFRASTRUCTUUR Kasteelpark Arenberg 40, B - 3001 Heverlee (Belgium)

Dynamic modeling of heterogeneous vehicular traffic

Jury Members:

prof. dr. ir. J. Vandewalle, Chairman Dissertation submitted to the prof. ir. L.H. Immers, Promotor Faculty of Applied Science prof. dr. ir. B. van Arem, University of Twente for the Degree of Doctor in prof. dr. ir. E.C. van Berkum, University of Twente Civil Engineering prof. dr. ir. P.H.L. Bovy, University of Delft prof. dr. ir. B. De Moor prof. dr. ir. J. Monbaliu by prof. ir. J. Vanderheyden

ir. Steven Logghe U.D.C. 656.13

June 2003

Onderzoeksinstelling Katholieke Universiteit Leuven Departement Burgerlijke Bouwkunde Afdeling Verkeer en Infrastructuur Kasteelpark Arenberg 40 B - 3001 Heverlee - België Research Institute Katholieke Universiteit Leuven Department of Civil Engineering Division Transportation Planning and Highway Engineering Kasteelpark Arenberg 40 B - 3001 Heverlee - Belgium ©2003 Faculteit Toegepaste Wetenschappen, Katholieke Universiteit Leuven Alle rechten voorbehouden. Niets uit deze uitgave mag worden vermenigvuldigd en/of openbaar gemaakt worden door middel van druk, fotokopie, microfilm, elektronisch of op welke andere wijze ook zonder voorafgaande schriftelijke toestemming van de uitgever, Departement Burgerlijke Bouwkunde, Katholieke Universiteit Leuven, Kasteelpark Arenberg 40, 3001 Heverlee, België. All rights reserved. No part of this book may be reproduced, stored in a database or retrieval system or published in any form or in any way - electronically, mechanically, by print, photoprint, microfilm or by any others means - without the prior written permission of the publisher, Department of Civil Engineering, Katholieke Universiteit Leuven, Kasteelpark Arenberg 40, 3001 Heverlee, Belgium. D/2003/7515/29x ISBN 90-5682-418-X

i

VOORWOORD

Met dit proefschrift wordt een vijfjarige periode afgesloten waarin ik de luxe had om bezoldigd verder te studeren. Daarenboven had ik de mogelijkheid om over een boeiend thema te kunnen ‘file’-soferen. Dit voorwoord vind ik dan ook de geschikte plaats om op deze periode even terug te blikken en enkele mensen te bedanken.

Eerst en vooral wil ik m’n promotor professor Ben Immers bedanken. Met zijn aanstekelijk enthousiasme overtuigde hij me om me verder in de verkeerskunde te verdiepen en een doctoraatsonderzoek op te starten. Tijdens mijn onderzoek kon ik steeds rekenen op zijn verfrissende ideeën en energieke aanmoedigingen.

Professor Eric van Berkum van de Universiteit Twente zorgde de voorbije jaren voor heel wat opbouwende kritiek. Ook professor Bart De Moor en professor Johan Vanderheyden gaven als assessor de nodige bijsturingen en reflecties. Hartelijk dank aan deze leden van mijn begeleidingscommissie.

Professor Piet Bovy van de Technische Universiteit Delft wil ik bedanken voor zijn opmerkingen die tot enkele verduidelijkingen in dit proefschrift leidden en voor het vele multiclass onderzoek in Delft dat onrechtstreeks dit proefschrift beïnvloedde. Professor Bart van Arem van de Universiteit Twente, professor Jaak Monbaliu en voorzitter Professor Joos Vandewalle dank ik om in mijn jury te willen zetelen.

Heel belangrijk voor dit engelstalig proefschrift was ook het werk van Leni Hurley. Zij heeft met veel enthousiasme het basiswerk vertaald wat gezien de vele vaktermen geen gemakkelijke opdracht was. Hartelijk dank, Leni, voor het vele werk dat je in de laatste maanden tegen een hoog tempo hebt verzet.

Verder wil ik mijn collega Jim bedanken voor de diepgaande discussies over verkeerskunde. Hij gaf mij de basiskennis van de verkeersstroomtheorie en leerde mij inzicht te krijgen in de transporteconomie en vertrektijd modellering. Jim heeft ook

Voorwoord ii

met engelengeduld de volledige tekst nagelezen en mij telkens op kleine onvolkomenheden gewezen.

Tijdens de voorbije jaren heb ik nooit het gevoel gehad alleen te werken aan dit proefschrift. Alle collega’s op het gebouw van burgerlijke bouwkunde hebben op één of andere manier tot dit werk bijgedragen : een verschillende zienswijze of een diepgaande discussie gaven mij soms een andere kijk op de zaak. Speciaal wil ik hiervoor Griet, Jasper, Michaël, Kristof, Christophe, Filip, Leen, Bart, Isaak en ook Chris, Tom en Sven bedanken.

Verder wil ik alle mensen bedanken met wie ik de voorbije vijf jaar in projecten samenwerkte. Zowel collega onderzoekers als mensen uit de praktijk gaven me vaak een duwtje in de juiste richting.

De wegenadministraties uit Vlaanderen en Nederland bezorgden me de nodige telgegevens die voor het ontwikkelen van een model noodzakelijk zijn. DWTC steunde me financieel in dit onderzoek.

Tenslotte bedank ik alle vrienden en familieleden die regelmatig interesse toonden in mijn onderzoek. Het fileprobleem is duidelijk iets wat heel veel mensen bezighoudt. Een speciaal woordje van dank gaat naar mijn ouders en schoonouders voor alle praktische en morele steun tijdens de voorbije vijf jaar. Ook mijn vrouw Hilde en de kindjes Janne en Lukas wil ik graag bedanken. Zij waren op de lastigere momenten steeds een stimulans om verder te zetten.

Steven Logghe, Mei 2003.

iii

NEDERLANDSTALIGE ABSTRACT

Het oorspronkelijke macroscopische verkeersstroommodel idealiseert verkeer op een snelweg als een homogeen fluidum. De voertuigen met bestuurders worden in dit model als gelijke deeltjes in een vloeistofbuis beschreven.

In dit proefschrift wordt met de heterogene eigenschappen van het verkeer rekening gehouden. Hiervoor wordt de verkeersstroom opgesplitst in homogene klassen. Elke klasse bestaat uit voertuigen en bestuurders met gelijke kenmerken. Modelleren van heterogeen verkeer omvat dan de beschrijving van homogene klassen en de interacties tussen de verschillende klasses.

Per wegsectie wordt een klasse gekenmerkt door de maximale snelheid, de voertuiglengte en de capaciteit. De capaciteit van een klasse is de maximale intensiteit wanneer alleen voertuigen uit die specifieke klasse op de weg rijden.

De interacties tussen de verschillende klassen is gebaseerd op het useroptimum: elke bestuurder wordt verondersteld zijn eigen snelheid te maximaliseren. Verder wordt aangenomen dat snelle voertuigen de snelheid van trage voertuigen niet kunnen beïnvloeden. Op die manier gedragen de tragere voertuigen zich als bewegende bottlenecks.

Het opgestelde heterogene model bestaat uit een wiskundige formulering die analytisch en grafisch opgelost kan worden. Daarnaast is een numeriek schema opgezet. Hierdoor kan het model met een computer geïmplementeerd worden en kunnen benaderende oplossingen snel berekend worden.

Het ontwikkelde model is verder uitgebreid voor gebruik op complete verkeersnetwerken. Een toepassing van het model in een case-study illustreert de praktische inzetbaarheid. Afsluitend volgt een kritische discussie van aannames en eigenschappen van het model en worden mogelijke uitbreidingen aangestipt.

v

ABSTRACT

In the original macroscopic traffic flow model the traffic on a motorway is idealized to a homogeneous fluidum. In this model the vehicles and their drivers are represented by identical fluid particles in a tube.

This dissertation takes into account the heterogeneous properties of traffic. To this end, the traffic flow is subdivided into homogeneous classes. Each class consists of vehicles and drivers that share the same characteristics. Modeling heterogeneous traffic, in this case, comprises the description of homogeneous classes and the interactions between the different classes.

For each road section, a class is characterised by its maximum speed, its vehicle length and its capacity. The capacity pertaining to a class signifies the maximum traffic intensity that prevails when vehicles from that specific class have exclusive use of the road.

The interactions between the different classes are based on the user-optimum: it is assumed that each driver maximises his own speed and that fast vehicles are unable to affect the speed of slow vehicles. Slower vehicles, in this view, behave like moving bottlenecks.

The heterogeneous model presented in this dissertation comprises a mathematical formulation that can be solved analytically and graphically. In addition, a numerical scheme has been constructed. This enables a computer implementation of the model allowing for a rapid computation of approximating solutions.

The developed model has been extended for use on complete transport networks. A case-study application of the model illustrates its practical usefulness. The dissertation concludes with a critical review of the assumptions and properties pertaining to the model, and a brief mention of possible model extensions.

vii

CONTENTS

Voorwoord – Forword i

Nederlandstalige abstract – Dutch abstract iii

Abstract v

Contents vii

1 Introduction 1 1.1 Background 1 1.2 Motivation 2 1.3 Objective 3 1.4 Approach 4

2 Traffic dynamics 7 2.1 Traffic variables 8 2.1.1 Microscopic 8 2.1.2 Macroscopic 9 2.1.3 Mesoscopic 12 2.2 Traffic observations 12 2.2.1 Fundamentel diagram 13 2.2.2 Bottlenecks 15 2.3 Traffic operations models 16 2.3.1 Microscopic 16 2.3.2 Macroscopic 17 2.3.3 Mesoscopic 18 2.4 Conclusions 19

Contents viii

3 Review of the LWR model 21 3.1Traffic conservation law 21 3.2 Fundamental diagram 23 3.3 Numerical scheme 26 3.4 Discussion 29 3.5 Extensions 30 3.5.1 Non-stationary 30 3.5.2 Non-homogeneous 30 3.5.3 Network 31 3.6 Conclusions 32

4 Classes in the LWR model 33 4.1 Fundamentals of multi-class LWR models 34 4.1.1 Philosophy of multi-class traffic 34 4.1.2 Extended LWR formulation 35 4.2 Class interactions 37 4.2.1 User-classes 39 4.2.2 Special lanes 40 4.2.3 Equal space 41 4.2.4 Equal distance gap 42 4.2.5 Distance gap proportional to vehicle length 43 4.2.6 Equal speed 44 4.2.7 User-optimum 46 4.2.8 System optimum 47 4.2.9 Multi-lane models 48 4.3 Homogeneous fundamental diagrams 49 4.4 Choice of approach 50

5 Transformation of the fundamental diagram 53 5.1 Analytic framework 53 5.2 Numerical scheme 56 5.3 Analytical and numerical case study 60 5.4 Discussion 64

6 Heterogeneous free-flow traffic 67 6.1 Stationary traffic states 68 6.2 Transititions 73 6.3 Numerical scheme 84 6.4 Analytical and numerical case study 92 6.5 Conclusions 95

7 Heterogeneous congested traffic 97 7.1 Stationary traffic states 98 7.2 Transititions 103 7.3 Numerical scheme 105 7.4 Analytical and numerical case study 107 7.5 Conclusions 110

Contents

ix

8 The heterogeneous LWR model 111 8.1 Stationary traffic states 112 8.2 Transitions 117 8.2.1 Transitions where class 2 maintains the free-flow speed 117 8.2.2 Transitions within the congestion regime 119 8.2.3 Transitions from the free-flow or semi-congestion regime to the congestion regime

119

8.2.4 Transitions from congestion to free-flow or semi- congestion regimes

121

8.3 Numerical scheme 125 8.3.1 Up- and downstream, class 2 travels at free-flow speed 126 8.3.2 Up- and downstream congestion 127 8.3.3 Upstream free-flow or semi-congestion; downstream congestion

128

8.3.4 Upstream congestion, donwstream free-flow or semi- congestion

129

8.4 Conclusions 130

9 Network traffic 133 9.1 Origin node 135 9.2 Destination node 136 9.3 Inhomogeneous node 136 9.4 Diverge node 139 9.4.1 Diverge nodes in het LWR model 140 9.4.2 A heterogeneous diverge node 141 9.4.3 A simplified diverge node 143 9.5 Merge node 145 9.5.1 Merge nodes in het LWR model 145 9.5.2 A heterogeneous merge node 146 9.6 Conclusions 147

10 Case study 149 10.1 The study area 150 10.2 Analysing the observations 150 10.2.1 The spatio-temporal traffic filter 150 10.2.2 Oblique N-plots 152 10.3 Model construction 155 10.4 Calibration 158 10.5 Validation 162 10.6 Conclusions 165

11 Discussion 167 11.1 General properties 167 11.2 The weighted congestion branch 168 11.3 Traffic lanes 170 11.4 Homogeneity of classes 173 11.4.1 A curved fundamental diagram 173

Contents x

11.4.2 Several classes 175 11.5 Non-optimal road-use 177 11.6 Non-concave fundamental diagram 178 11.7 Relation with other dynamic models 182 11.7.1 Macroscopic 182 11.7.2 Mesoscopic 185 11.7.3 Microscopic 186 11.8 Conclusions 186

12 Conclusions 187 12.1 Brief summary 187 12.2 Findings 188 12.3 Further research 189

Bibliography 191

Appendix A : The spatio-temporal traffic filter 199

Appendix B : Oblique N-plots 203

Samenvatting – Dutch summary : Dynamisch modelleren van heterogene verkeersstromen

207

About the author 221

1

1 INTRODUCTION

Although the perspective of mobility is much broader, traffic jams belong to the most important and intriguing traffic phenomena. Tailbacks are rows of waiting vehicles that mark the temporary and local overloading of the road network. Discovering the mechanisms inherent in the interacting vehicles and drivers in the tailback remains a driving force in the development of traffic flow theory.

In traffic flow theory, mathematical models are constructed to accurately describe the traffic flow process. The increased number of observations and expanding computational possibilities appear to render these models open for continuing improvement. The first dynamic traffic model, developed by Lighthill and Whitham (1955) and Richards (1956), known as the LWR model, describes traffic as a flow of identical vehicles and drivers. This dissertation extends this original model in order to account for a variety of vehicle- and driver characteristics, enabling a more accurate description and explanation of the tailback phenomenon.

This chapter will begin by outlining the general background to the mathematical models used to describe dynamic traffic flows. The following sections set out the motivation and the objective of this study. The final section briefly outlines the structure of the dissertation.

1.1 Background The interactions between different vehicles, their drivers and the infrastructure give rise to many complex phenomena on our roads. Traffic flow theory describes these traffic phenomena and reproduces them through mathematical models.

Chapter 1 2

These dynamic traffic models are of practical significance in a variety of ways. First and foremost, they help explain traffic phenomena. They both steer and stimulate the search into the true mechanisms of the traffic process.

The models are increasingly used in the design of traffic infrastructure. Interaction with a model renders the probabilistic design process more manageable which, in turn, improves the ultimate design.

Dynamic traffic models play an important role in the operation of the traffic process. Predicting the effects of various scenarios, computing traffic control strategies and making real-time forecasts of traffic situations are some of the practical application areas of these models.

The significance of models in policy and planning is also increasing. The development of long-term objectives needs to consider the actual options and restrictions of traffic operations.

The formulation of several approaches and assumptions has led to a range of dynamic traffic models. Each time, the question centres on the accurate reproduction of the essentially stochastic traffic process.

Detailed models accurately replicate the behaviour of individual vehicles. Since it is practically impossible to reproduce the exact decision-making mechanism of individual drivers, these models resort to a probabilistic approach. Assumptions regarding driver behaviour result in a set of mainly stochastic parameters.

Robust models compute the traffic flow more rapidly. These models describe the traffic properties with less sensitive parameters, resulting in a less detailed description of the traffic process.

This dissertation concentrates on macroscopic models, where traffic is described as a continuous fluïdum. The first dynamic traffic model of this type was formulated by Lighthill and Whitham (1955) and Richards (1956). The macroscopic LWR model assumes traffic to be stationary, homogeneous and deterministic. This allows us to classify it as fairly robust.

1.2 Motivation In the first order macroscopic model, better known as the LWR model, the traffic flow is assumed to be homogeneous. All vehicles behave in an identical and deterministic way.

Real traffic, however, is not homogeneous. Empirical studies (e.g. Tilch and Helbing 1999, Dijker et al. 1997) show that differences between vehicles and drivers have important effects on traffic flow.

The differences in vehicle properties are obvious. There is a large variation in vehicle lengths between lorries and private cars. Legal maximum speed limits and acceleration and deceleration characteristics also may differ significantly. The driver

Introduction

3

population is not homogeneous either (Kockelman 1998). Differences in the style of driving clearly affect the traffic flow (Kijk in de Vegte 2002).

Taking account of these heterogeneous vehicle and driver properties will lead to a more detailed and more accurate model improving the replication of the actual traffic flow process.

The evaluation of measures aimed at target groups, moreover, requires a heterogeneous model. These measures affect a section of the vehicle population. Calculating the effects, for example, of a regulation prohibiting lorries from overtaking or of the introduction of a pay-lane for High Occupancy Vehicles (HOV) requires a heterogeneous model.

1.3 Objective This dissertation presents an extension of the LWR model that takes the heterogeneous characteristics of the traffic flow into account. The study has a two-fold objective.

First, the construction of an analytical model of a heterogeneous traffic flow can lead to new insights into the interpretation of traffic observations. New explanations for traffic phenomena become possible. As in the original LWR model, the preferred option is a model that can be solved both analytically and graphically. The advantage of adhering to a stationary LWR type of model is that explanations can be formulated exclusively based on the heterogeneity aspect. It also prevents needless complexity of the model.

Second, we strive for an heterogeneous LWR model that has potential for practical application. This objective is achieved with the development of a numerical scheme alongside the construction of an analytical model. An extension to a comprehensive network model is also desirable.

An important requirement in model development is to avoid a situation where the gap between model developers and model users becomes too large. This requires a gradual stepwise development of the model. As in the original LWR model, the option to construct a graphic solution must also be retained.

In order to refine the homogeneous LWR model so that it can handle heterogeneous traffic flows, the vehicles are subdivided into classes. Describing the interactions between vehicles of different classes now becomes part of the model, though the basic characteristics of the LWR model are retained. The new model comprises analytical, numerical and graphical solution methods. In addition, an extension to the description of traffic networks is proposed.

Chapter 1 4

1.4 Approach The dissertation is divided into twelve chapters outlining and clarifying the construction of a heterogeneous LWR model.

The introductory chapter is followed by a summary of traffic flow theory in chapter 2. The summary defines a consistent mathematical framework to describe traffic flow and documents a number of empirical phenomena.

Chapter 3 takes a closer look at the LWR model. It explains the analytical background and the required input, and it elucidates the numerical scheme allowing for the calculation of approximate solutions. The chapter ends with a critical discussion and possible extensions to this model.

Chapter 4 lays the foundation for the heterogeneous LWR model. The heterogeneous traffic flow is subdivided into homogeneous class. The general approach to the classes is explained. Subsequently, we discuss all heterogeneous stationary LWR models developed to date, but place them within a new frame of reference. It will appear that the most important difference between all of these models is the way in which classes interact. A new interaction model, based on the user-optimum, is formulated.

Figure 1.1 : Step by step development of the heterogeneous LWR model

The step by step development of the heterogeneous LWR model according to the user-optimum principle, is outlined in figure 1.1. The first four steps each present an analytical and a numerical computation of the model. This is followed by case studies to illustrate the first three steps.

step 1

Transformation of the fundamental diagram

Chapter 5

step 2

Heterogeneous free-flow traffic

Chapter 6

step 3

Heterogeneous congested traffic

Chapter 7

step 4

Heterogeneous LWR model

Chapter 8

step 5

Network traffic

Chapter 9

Introduction

5

Chapter 5 elucidates the transformation of geometrically similar fundamental diagrams. This leads to a simple heterogeneous model based on passenger-car equivalents.

Chapter 6 formulates a heterogeneous free-flow model. In this step, the driving behaviour during congestion is assumed to be identical for all classes.

In chapter 7, we focus on heterogeneous congested traffic. In this step, driving behaviour during free-flow is assumed to be identical for all classes.

The previous three chapters lead to a comprehensive heterogeneous LWR model, to be presented in chapter 8. This comprehensive model enables the description of the traffic process for a flow of interacting classes on a homogeneous link.

Chapter 9 formulates a network extension that renders the model suitable for practical purposes. The implicit assumption here is that links depart and arrive in a node, where definition of different node types results in building blocks for the construction of virtually all motorway network figurations.

Chapter 10 applies the heterogeneous LWR model and the network extension to a motorway in The Netherlands. A thorough study of the observations and a careful model construction is followed by an extensive calibration and validation. This provides a key to the practical prospects of the model.

Chapter 11 presents a critical discussion of the assumptions, the properties and further extensions to the heterogeneous model. The relation to other dynamic models is also discussed.

Chapter 12 concludes this dissertation with a number of its most important findings and suggests recommendations for further research.

Chapter 1 6

7

2 TRAFFIC DYNAMICS

Since the 1950's, scientists and engineers have developed their own views on the dynamics of traffic. In traffic flow theory mathematical instruments are used to describe traffic, especially on motorways. Fifty years later, the search for the mechanisms behind the complex interactions between drivers, their vehicles and the road infrastructure continues. Before we contribute to this fascinating theory in the next chapters, we begin with an outline of current insights.

While the description of traffic phenomena in traffic flow theory is interesting in its own right, it is also of much practical use. It has led to considerable improvements in the design of traffic infrastructure (e.g. Highway Capacity Manual 2000). Evaluating new designs by performing dynamic traffic model calculations enables prior estimation of their possible effects (e.g. Yperman and Immers 2003).

The use of traffic flow theory also makes the process of traffic manageable. New measures that intervene dynamically in the traffic flow became possible alongside the familiar control by traffic lights (e.g. Bellemans 2003, Hegyi et al. 2003). Active steering of the traffic-process increases the efficiency and liveability of the transport system (e.g. Int Panis et al. 2003).

A new industrial sector is developing that uses new technologies to apply the insights gained from traffic flow theory. The impact of Intelligent Transport Systems (ITS) on the present transport system is hard to gauge. Further elaboration of ITS insights and their effects will lead to a new chapter in traffic flow theory (e.g. Tampère et al. 2003).

This chapter gives an overview of the current knowledge. To begin with, we formulate a framework which is used to describe vehicle traffic mathematically. This

Chapter 2 8

abstraction from reality leads to a set of consistent definitions. Next, we discuss the traffic patterns observed. Interpretation of these observations leads to a variety of models. In these dynamic traffic models, assumptions regarding the mechanisms of traffic flow are translated into a set of mathematical equations. Model implementation enables the description of traffic flow and the computation of the repercussions of scenarios. The properties of drivers, vehicles and infrastructure may vary in these scenarios.

2.1 Traffic variables This section defines a number of mathematical variables that can be used to describe traffic. These definitions will be used consistently throughout this dissertation when describing observations and the design of dynamic models. Since we pay particular attention to motorways in this thesis, non-interrupted traffic streams will be examined more closely.

2.1.1 Microscopic When describing a stream of traffic microscopically, we examine each vehicle separately. The position taken by vehicle α on a road can be indicated by xα. The car ahead of this vehicle will be called α-1. Since both vehicles travel along the road, their positions are time dependent. Figure 2.1 shows both vehicles in a t-x co-ordinate system.

Figure 2.1 : Two vehicles in a t-x co-ordinate system.

A trajectory represents the position of a vehicle through time. If we consider the vehicle to be a point with no dimensions, then xα(t) is a pure mathematical function.

x

t

xα(t)

xα-1(t)

hα

gα oα

sα(t)

dα(t)

Lα

t0

Traffic dynamics

9

Normally, we use the back part of a vehicle, the rear bumper, as reference point for the trajectory of this vehicle.

Two trajectories can not intersect one another when both vehicles travel along the same traffic lane. The speed of vehicles is given by the derivative of a trajectory, acceleration by the second derivative.

A vehicle occupies a certain space on the road. This space sα consists of the physical length Lα of the vehicle and the distance gap dα that a driver maintains between himself and the vehicle ahead of him. Analogous to space there is a vehicle's time headway hα. This time headway can be subdivided into a time gap gα and an occupancy oα.

The difference in speed ∆vα is given by :

dttdstvtvtv )()()()( 1

αααα =−=∆ −

(2.1)

The trajectories of vehicles enable a perfect description of dynamic traffic. To this day, the problems associated with direct observation have inhibited the use of trajectories to describe traffic flow. This will hopefully change with the advance of aerial observations (Hoogendoorn et al. 2003) and with the introduction of communication instruments in the vehicles themselves (Westerman 1995).

2.1.2 Macroscopic At the macroscopic scale, vehicles are not treated as separate entities. Here, the discrete nature of traffic is aggregated into continuous variables. This is, therefore, also called the continuum approach to traffic description. Three main variables are defined: density k, intensity q and average speed u. The definitions for a random measurement interval S are always formulated in the t-x space. Figure 2.2 illustrates such a measurement interval. Two specific measurement intervals are given : a space interval ∆X and a time interval ∆T. In practice the last interval is the one most frequently used.

• Density k is a variable typically used in physics that has been adopted by traffic science. Density k indicates the number of vehicles for each kilometre of road. Measuring k across a road section with ∆X length at a certain point in time is simple and straightforward

Xnk

∆=

(2.2)

Here, n represents the number of vehicles present in the road interval ∆X at that point in time. This definition can be extended to a random measurement interval S, an area in the t-x space. In that case we define k as:

k(t,x,S)= Area(S)

Total time of the vehicles in S (2.3)

Chapter 2 10

• Flow q can be compared to the flux of a stream. The flow reflects the number of vehicles passing a point during a certain time interval. For a time interval ∆T, the flow at a particular location can be calculated as follows :

Tmq∆

= (2.4)

Here m represents the number of vehicles that passed the location under consideration during ∆T. Again, this definition can be extended to a random measurement interval S :

• We define the average speed u as the quotient of flow and density. Average speed, therefore, also depends on location, the point in time and the measurement interval.

For a space interval ∆X, this definition of u leads to the arithmetic average of the individual speeds. One also speaks of the space mean speed. In a time interval ∆T, this definition corresponds to the harmonic average of the individual speeds. The arithmetic average of the speeds in a time interval is called the time mean speed. The definition of the (space) mean speed can be rewritten to the relation q = k.u. The three aggregate variables are, therefore, always connected. Knowing two variables automatically leads to the third.

Figure 2.2 : A general measurement interval S, a location interval S1 and a time interval S2

u(t,x,S) = Total time of the vehicles in S

Total distance covered by the vehicles in S (2.6)

S

t

x

S1

S2

∆T

∆X

dx

dt

q(t,x,S) = Area(S)

Total distance covered by the vehicles in S (2.5)

Traffic dynamics

11

The formulated definitions always apply to a measurement interval S. In practical observations determining the size of this interval often leads to a dilemma. When small measurement intervals are observed, the discrete nature of the traffic often inhibits a continuum treatment. In larger measurement intervals certain dynamic properties of the traffic are overlooked. In a theoretical model environment the measurement interval can be transformed to an abstraction. The macroscopic traffic variables are then continuous variables in the t-x space.

The cumulative number of vehicles is often used to complement these aggregate variables. In such cases, each vehicle is assigned a serial number with respect to a reference vehicle. Figure 2.3 outlines a set of trajectories with serial numbers. Overtaking vehicles exchange their serial numbers. The cumulative function N(t,x) then gives the serial number of the last passing vehicle.

Figure 2.3 : Numbered vehicle trajectories

This function only changes on trajectories and remains constant between trajectories. To make this function differentiable, we define ),(~ xtN as a smooth approximation to N(t,x). We can now combine flow and density as follows :

txtNxtq

∂∂= ),(~

),( (2.7)

xxtNxtk

∂∂−= ),(~

),( (2.8)

Due to the use of the approximating function for N(t,x), flow and density are now defined independent of a measurement interval.

The power of the cumulative function will become apparent in chapter 10 when traffic observations need to be interpreted. For each detector location, the traffic counts are then represented by a cumulative function, as explained in appendix B.

1 2 3 4 5 6 7 8

9 10 11 12 13

t

6

6 7 8 9 10 11 12 13

7

1 2 3 4 5

x

Chapter 2 12

2.1.3 Mesoscopic The concept of density is extended at the mesoscopic level. Here we assume that the density can be divided by speed. This leads to the phase-density f(t,x,v). This phase-density represents the number of vehicles of speed v, at location x, at time t. Then, a link can be made with the macroscopic defined density as follows :

∫∞=

=

=v

v

dvvxtfxtk0

).,,(),( (2.9)

The macroscopic flow q is also related to the phase-density :

∫∞=

=

=v

v

dvvvxtfxtq0

.).,,(),( (2.10)

The concept of phase-density requires a more specific approximation of traffic. In actual traffic, a natural number is used to indicate density on a finite road section. The concept of phase-density (Prigogine and Herman 1971) already assumes a range of vehicles with different speeds on an infinitesimal small road section [x,x+dx] during an infinitesimal time [t,t+dt].

This can lead to meaningful concepts in molecular fluids, where the physical size of particles is small compared to the tube in which they flow. In traffic terms, it is difficult to conceive of a phase-density on a road. A mesoscopic approach implies a probabilistic description of traffic flow. In that case the phase-density acts as a probability distribution which reflects the probability that at time t, a vehicle of speed v is at location x.

This phase-concept can be further extended by dividing density in more ways than by vehicle speed alone. Additional divisions, meanwhile, have been proposed according to the maximum speed of vehicles (Pavari-Fontana 1975), and by vehicle class and lane (Hoogendoorn 1999).

Phase-density f(t,x,v) intrinsically reflects a speed distribution that is dependent on time and location. Independent of time and location, the distribution concept is also applied to describe microscopic variables. Ease of pedestrian crossing, for example, can be calculated using a time headway distribution (May 1990).

2.2 Traffic observations The pattern of traffic on a road section is always the result of the interaction between traffic demand and traffic supply. Traffic demand is a term used to indicate the number of vehicles that arrives upstream of the section under observation. Traffic supply is the number of vehicles that can depart from the section under consideration.

Observations regarding these traffic patterns on different motorways have led to some established facts. This section outlines empirical properties based on observed macroscopic variables. Chapter 10 will illustrate a number of these characteristics in a set of observations on a motorway.

Traffic dynamics

13

2.2.1 Fundamental diagram Based on the macroscopic definitions of flow q, density k and (space) mean speed u, we know that we have two independent macroscopic variables. Based on observations, Greenshield (1934) was the first to attempt to establish a relation between the remaining independent variables. It is certain that when traffic is stationary and homogeneous and when all vehicles behave deterministically, this relation can be described by a mathematical function (Cassidy 1998).

Since this equilibrium relation is usually represented in a graphic diagram, traffic engineers speak of the 'fundamental diagram'.

Figure 2.4 outlines a possible fundamental k-q diagram. Usually, the equilibrium flow Qe(k) equals zero at zero density and at jam density kJ . The function achieves a maximum flow qM, also called the capacity, somewhere in between. As indicated in the figure, the speed of the traffic flow in this diagram is represented by a slope. The rising part of the curve corresponds to the free-flow regime, the descending part to congestion. The exact shape of the fundamental diagram is dealt with in chapter 3.2.

Figure 2.4 : A fundamental k-q diagram.

Based on observations at a fixed location, deviations from the unambiguous equilibrium function are discussed. These deviations can be explained by the fact that actual traffic is not necessarily stationary, homogeneous and deterministic.

When real observations are represented in a k-q diagram, the observation points deviate from an unambiguous relation and show a great deal of scatter. The deviations around the virtually straight free-flow branch are usually smaller than the cloud-like set of observation points during congestion (Logghe and Immers 2000). It has also been established that the maximum flow achieved during the free-flow regime differs from the observed capacity during congestion (Gartner et al. 1997). Congested observations around the maximum are less numerous and display a lower flow. This finding gives rise to the two-capacity phenomenon (Hall and Agyemang-Duah 1991). This states that capacity depends on the regime from which it is approached.

These findings are substantiated by linking the various observation points in chronological order. Figure 2.5 shows a set of chronologically linked observation points on the E17 motorway Gent-Antwerp in a k-q diagram. This shows that the

q

k

u

qM

kJ

congestion

free flow

uf

Qe(k)

Chapter 2 14

transition from the free-flow branch to the congestion branch runs along a path that lies above the path taken by the reverse movement. This phenomenon is called the hysteresis effect.

These observations also show that traffic displays oscillations during congestion. Congested traffic often leads to this type of unstable behaviour when vehicles successively accelerate and decelerate. Drivers experience this as start- and-stop waves.

Figure 2.5 : Chronologically linked observation points on the E17 in a k-q diagram

The fundamental diagram can also be calculated for each separate lane. These diagrams show that maximum flow during free-flow is considerably higher on the inside (left) lane than on the outer (right) lane. To a lesser extent, these findings are also found during congestion, as we see in figure 2.6. This lower flow on the outer lane could possibly indicate a less than optimal use of this lane.

Figure 2.6 : 1- minute observations for each lane in a k-q diagram (a) inside (left) lane (b) centre lane and (c) outer (right) lane.

The speed differences between the different lanes are largest during free-flow. During this regime each vehicle chooses its own speed. The speed differences are smaller during congestion.

0

1000

2000

3000

4000

5000

6000

7000

8000

0 20 40 60 80 100 120 140 160 180 200

Density k

Flow q [veh/h]

Traffic dynamics

15

2.2.2 Bottlenecks The previous section examined traffic in a k-q diagram. Transitions between several regimes were observed. These transitions are mainly caused by the presence of bottlenecks. A bottleneck indicates a non-homogeneity in the traffic supply.

Changing road characteristics on through roads are categorised as infrastructural bottlenecks. Examples include a decrease in the number of traffic lanes or an alteration in the grade of the road.

In network bottlenecks, traffic is exchanged between several roads. On motorways this means merging traffic and branching traffic.

When traffic merges, as on a feeder road, it appears self-evident that traffic from the arriving links could become congested. This happens when the capacity of the road downstream from the merge is less than the sum of the capacities of the merging roads. When the traffic demand on the merging links grows too large a bottleneck comes into force and causes congestion.

However, observations confirm this analysis only partly. The critical point in merging traffic only manifests itself at some distance beyond the merging itself (Cassidy and Bertini 1999a,b). Congestion develops approximately 1 kilometre downstream from the merging traffic. This phenomenon is called the capacity funnel (Immers 1980). In the case study in chapter 10 the bottleneck also lies downstream from the feeder road location.

Before the bottleneck begins to form, we often see a period of high flows. These flows, sometimes lasting over half an hour, considerably exceed the subsequent flows during the actual bottleneck. This phenomenon gives rise to the hysteresis effect in the fundamental diagram.

Bottlenecks also play a role in fork junctions, such as exit lanes. (Muñoz et al. 2002b) A downstream congested road can also cause congestion on the road upstream from the road junction.

When the congestion rebounds beyond nodes upstream, we speak about a blocking back. In such cases, vehicles that need not pass through the actual bottleneck are also impeded and congestion disruption levels rise rapidly.

Temporary obstructions also cause bottlenecks. Accidents, for example, cause temporary disruption and act as bottlenecks. Traffic lights also interrupt an even flow of traffic. When a temporary obstruction is discontinued, the first fast vehicles are observed some distance downstream from the location of the bottleneck. Only when this has happened are slow vehicles observed in the traffic stream. This phenomenon is called the dispersion-effect (Lebacque and Lesort 1999).

In addition, slow moving vehicles can also be catalogued as moving bottlenecks in the stream of traffic. These slow vehicles can lower capacity locally and they can also cause congestion.

All kinds of bottlenecks can be pinpointed as the cause of congestion. New theories nevertheless emerge that do not necessarily link the origin of congestion to

Chapter 2 16

bottlenecks. (e.g. Kerner 1999) These new theories regard small perturbations in the homogeneous traffic flow as triggers for congestion. But because it is difficult to observe these small fluctuations empirically, there is no consensus regarding this matter, and bottlenecks remain the only established cause for congestion. In principle, however, these perturbations in the homogeneous traffic flow consist of deviations between different vehicles and drivers. A heterogeneous model, therefore, describes these perturbations more accurately.

2.3 Traffic operations models A dynamic traffic model describes the traffic operations on a road on the basis of a set of mathematical formulas. It is the result of a theory regarding the mechanisms that take place between vehicles, drivers and infrastructure. Some contain possible explanations of the mechanisms. Usually, however, they attempt purely to give an accurate description of the processes in a traffic stream.

The classes of traffic models to be discussed in this section focus on traffic flow operations. They basically compliment the prediction models that are used in transportation planning (De Schutter et al. 1999). The traditional four step prediction model (Ortúzar and Willumsen 1995) examines the successive decision processes of traffic participants in four separate submodels. The generation submodel examines where and when trips will be made. The distribution model looks at driver destinations. In the modal split model, the drivers choose the transport mode. The traffic assignment model discusses the route choice of individual traffic participants. This last step takes the traffic flow operations into account by describing the travel time on a road as a function of flow. The dynamic traffic demand models can, in fact, be added as a fifth step. These traffic demand models give an accurate description of traffic operations on a link and they refine the travel time function of a road during route choice. However, the contribution of dynamic models to longer term forecasting, for example for traffic flow planning, is minimal. Too many uncertainties render accurate modeling of traffic operations virtually impossible.

The traffic operation models can be categorised in several ways. This section catalogues the models according to the traffic variables they use. We will first look at the microscopic models, where we treat each vehicle separately. We then examine the macroscopic models, to be followed by the mesoscopic models.

2.3.1 Microscopic In a microscopic traffic model the individual drivers and vehicles are dealt with separately. Since the behaviour of each driver is not accurately predictable, these models are usually stochastic ones. They are implemented with a computer as simulation models (Aimsun – Barceló and Ferrer (1997), Paramics – Quadstone (2002), Vissim – PTV(2002)). The driver- and vehicle characteristics at time t + ∆t are calculated according to their characteristics at time t. In this way the positions and speeds (among others) of all vehicles are computed. In contrast to macroscopic

Traffic dynamics

17

dynamic models, different kinds of vehicles and drivers are easier specified. Most micro-simulation models fit into one of the following classes:

• The car-following model

In this model, the acceleration of a vehicle is determined by the behaviour of the vehicle ahead.

• The lane-change-model

This model examines the way in which a vehicle changes driving lanes based on the movements of the immediately surrounding vehicles.

• Route choice model

As in the prediction model, vehicles must find the shortest route through the infrastructure network. Input of traffic demand data happens on the basis of a number of Origin-Destination matrices. The route choice is made on the basis of a generalised travel resistance. Travel time, as part of travel resistance, can be computed simultaneously.

• Additional modules

Since the position, speed and acceleration for each vehicle for each simulation interval are known, derived effects such as chemical pollution, noise pollution, time loss and economic costs are easily calculated by these modules.

Besides the vehicles, dynamic characteristics pertaining to the infrastructure system, such as traffic lights, weather conditions and accidents, can also be modeled. The user-friendly nature and the numerous parameters of these microscopic models allow for a realistic representation of the traffic system (Logghe and Immers 1999).

The multitude of parameters in these models can render them somewhat obscure and require a lot of computing power. This makes them extremely suitable for off-line scenario computations.

Microscopic simulation models describe traffic operations during discrete time intervals over continuous time. For each time interval the locations of vehicles are computed in the t-x diagram. In a numerically faster microscopic model the location can also be split into discrete elements. To that end, in cellular automata, the road is divided into cells that can contain only one vehicle. A probability parameter is traditionally added and the car-following model is implemented using simple rules (Nagel and Schreckenberg 1992, Nagel 1996). Since both time and place are discretised, speed also is a natural number. This allows rapid computation of extensive traffic networks (Esser et al. 1999).

2.3.2 Macroscopic In the macroscopic model an analytical relation between flow q and density k is constructed. Conservation of vehicles is applied to a road without feeder- and exit lanes. This can, just as in models for fluids, be expressed by a partial differential equation :

Chapter 2 18

0),(),( =∂

∂+∂

∂t

xtkx

xtq (2.11)

This conservation equation can be complemented by the speed definition (2.6) :

),().,(),( xtuxtkxtq = (2.12)

A third equation is required to achieve an unambiguous description of the traffic flow. Lighthill and Whitham (1955) and Richards (1956) (LWR) append the following fundamental diagram :

( )),(),( xtkQxtq e= (2.13)

This LWR model assumes that the stationary, homogeneous and deterministic relation of the fundamental diagram applies to non-stationary states as well. This critical assumption has led to a number of extensions.

Higher order models emphasise the non-stationary character of traffic flow. To this end, the fundamental diagram (2.13) is replaced by a second partial differential equation that describes the speed more accurately.

Other extensions focus on non-homogeneous characteristics or on the non-deterministic character of traffic. The next chapter discusses the LWR model in depth, and it examines the various extensions. The remaining dissertation then formulates its own proposal for the inclusion of the non-homogeneous character of vehicles and drivers in the LWR model.

2.3.3 Mesoscopic Using phase-density f(t,x,v) a traffic model is constructed analogous to gas kinetic models. As in the macroscopic models, a conservation of vehicles applies here. When vehicles do not change speed, we get :

0),,(),,( =∂

∂+∂

∂x

vxtfvt

vxtf (2.14)

This equation is only valid under light traffic conditions. In all other circumstances, slow vehicles will obstruct faster ones, so that the conservation of speed no longer applies. In response, an interaction term can be appended to (2.14). This conveys the extent of phase density change when faster vehicles decrease speed due to interactions with slow vehicles that can not be overtaken.

Vehicles that have slowed down will, meanwhile, attempt to regain their maximum speed. Increasing speed can be done by acceleration and is represented by a relaxation term. The basic model of Prigogine and Herman (1971) is thus completed as in (2.15):

relaxationactioninter tf

tf

xfv

tf

∂∂+

∂∂=

∂∂+

∂∂

(2.15)

Starting from detailed phase densities, for example on the basis of classes or lanes, additional assumptions can be made regarding the alteration of the phase density (Hoogendoorn 1999).

Traffic dynamics

19

These kinetic models can be used to distil macroscopic models (Hoogendoorn and Bovy 1999). This is how heterogeneous and non-stationary macroscopic models are formulated in which the mesoscopic character, however, remains clearly visible.

2.4 Conclusions The development of vehicular traffic has led to a number of theoretical formulations that examine vehicular traffic mathematically. This enables the interpretation of empirical phenomena using an extensive set of analytical instruments. Describing these complex mechanisms between vehicles has led to a range of dynamic traffic models.

These models are used in the planning and the design of the road infrastructure. Regulating and controlling traffic flows can, in addition, be achieved and predicted.

The end of the development of these dynamic traffic models is not yet in sight. Increased understanding of traffic flow can be achieved through ever more accurate models describing the interaction between the various vehicles, their drivers and the infrastructure. This thesis proposes an extension to the macroscopic model. In the proposed methodology, the various interactions between different drivers and vehicles will be taken into account.

Chapter 2 20

21

3 REVIEW OF THE LWR MODEL

When the retention of vehicles is combined with the empirical relationship between flow and density, better-known as the fundamental diagram, we get a model that can be used to describe a range of dynamic traffic characteristics. In this chapter, this LWR model, also called kinematic wave model, will be studied in depth. Besides the model itself, we will evaluate its input requirements. We will also look at the numerical method that can be used to calculate adequate approximating solutions for the model. This will be followed by a critical examination of the model which will provide for a discussion of possible improvements and extensions.

3.1 Traffic conservation law In the nineteen fifties Lighthill and Whitham (1955) and Richards (1956) independently developed the first dynamic traffic flow model. The LWR model describes the traffic on a link using a conservation law. It is, in addition, assumed that flow q is related to density k. This equilibrium relation Qe(k) is better known as the fundamental diagram. The formulation below gives a partial differential equation for a freeway without access or exit ramps:

0),(.)(),( =∂

∂+∂

∂x

xtkdk

kdQt

xtk e (3.1)

The fundamental diagram Qe(k) was initially assumed to be smooth and concave, with a maximum flow qM for density kM. The equilibrium flow equals zero at 'jam' density kJ and at zero density. In the latter situation free-flow speed uf equals Qe’(0).

Chapter 3 22

There are a number of solutions to the partial differential equation (3.1). A trick is used to find a unique and, in terms of traffic science, meaningful solution. To this end we look at a similar equation having a viscosity term as in (3.2).

2

2 ),(),(.)(),(x

xtkDx

xtkdk

kdQt

xtk e

∂∂=

∂∂+

∂∂ ε

(3.2)

This equation does have a unique solution. When epsilon approaches zero, we get a meaningful and unambiguous solution for the traffic conservation law (3.1). This vanishing viscosity solution is stable. When a discontinuous boundary condition in an infinitesimal interval is averaged to a continuous condition, we rapidly approach the original solution. This is why this solution is also said to satisfy the entropy condition.

In the case of piece-wise initial and boundary conditions a meaningful solution to equation (3.1) can be derived graphically using characteristics, also called kinematic waves. Here characteristics are represented by straight lines with a slope of Q’e(k0). These are drawn from a point with density k0 in the t-x plane. The traffic state at these solution lines equals the state at the boundary condition. Characteristics that travel against the traffic flow, in this case Q’e(k) < 0 and k > kM, correspond to congestion. Free-flowing traffic corresponds to characteristics with a positive slope so that Q’e(k) > 0 and k < kM..

Figure 3.1: (a) a fundamental diagram; (b) a shock wave and (c) a fan.

When, as in figure 3.1b, traffic density increases in the x-direction for t = 0, a shock wave develops which shows a slope of ∆q/∆k. The traffic state on this wave changes discontinuously. Decreasing density in the x-direction for t = 0, as in figure 3.1c, gives rise to a fan of characteristics in which all intermediate densities occur. This shows that the flow from congestion to a downstream free-flowing situation always runs via the congestion regime. The traffic flow leaving congestion is, therefore, optimal in the LWR model.

Some initial- and boundary conditions do not lead to solutions. This kind of ill-posed problem occurs, for example, when a congested characteristic continues on into the upstream boundary condition. When this happens, the tailback reverberates into the upstream observation post.

Shock waves and characteristics can both be represented in the fundamental k-q diagram and in the t-x diagram. By scaling these diagrams in such a way that parallels

t

x

x0

trajectory t

x

x0

k1, q1

k2, q2 k1, q1

k2, q2

trajectory

c1

q

Qe(k)

c2

k k2

u2

q2 q1

k1

u1

U12

kJ

qM

kM

uf

q1 qM

Review of the LWR model

23

in both diagrams correspond to equal speeds, the analytical solution can be constructed graphically. Starting from the initial- and boundary conditions, characteristics, shock waves and fans can by drawn, thus enabling the calculation of density, speed and flow in the study area during the study period. The use of this graphical method is limited to small networks with piece-wise constant boundary conditions.

Newell (1993a,b,c) shows that the LWR model also can be studied with the help of the cumulative number of vehicles N(t,x). This function gives the amount of vehicles that pass some location x by time t starting from the passage of some reference vehicle. The partial derivatives of this function are related to the density and the flow as:

),(),( xtqdt

xtdN = (3.3)

),(),( xtkdx

xtdN −= (3.4)

The function N(t,x) agrees to Greens theorem and can be calculated as :

∫∫ −+=x

x

t

t

kdxqdtxtNxtN00

),(),( 00 (3.5)

The integrals may be taken along any curve connecting (t0,x0) and (t,x). By definition, the cumulative number does not change along trajectories and N(t,x) is continuous over shock waves. When different values of N(t,x) can be calculated from several boundary conditions, the smallest value is the only valid one. The latter defines the location of shock waves that agree with the entropy condition.

3.2 Fundamental diagram The fundamental diagram Qe(k) is the prime parameter of the LWR model. In the model it is assumed that the fundamental diagram contains all the properties of the vehicles and the road. Together with the initial- and boundary conditions it determines the location of the characteristics, shock waves and fans.

The fundamental diagram is an equilibrium diagram. It shows the relation between flow q and density k when traffic is stationary and homogeneous. This is why the relation is nearly always an approximation of reality. Traffic flow is, after all, never completely homogeneous and consists of several types of drivers and vehicles, all with their own characteristics. Equally, the stationary character of traffic flow is, at certain densities, barely observable empirically. Density varies considerably as vehicles accelerate or decelerate at a given speed. In addition, this relation applies to a specific link and it is subject to external factors such as weather, visibility and local situations along the road.

The designation 'fundamental diagram' refers to the graphic representation of this relation as shown in figure 3.1a. This relation can, however, also be shown between

Chapter 3 24

two other traffic variables. Because density k, flow q and average speed u are linked, we know that:

kkQ

kUu ee

)()( ==

(3.6)

A relation can then also be derived between speed and flow:

)(qUu e= (3.7)

A microscopic interpretation of this relation is equally possible. We define the space s of a vehicle as the distance between the rear bumper of the vehicle in front and the rear bumper of the vehicle under consideration. Then, in a homogeneous and stationary traffic flow, space is the reverse of density. An equilibrium relation between speed and space is then given by:

)/1(.)( sQssUu ee == (3.8)

This function indicates the speed of a vehicle in terms of the space it occupies on the road. Note that the speed is zero for spaces smaller than 1/kJ and that the free-flow speed applies to infinitely large spaces.

Over time, numerous expressions have been proposed to describe traffic using an unambiguous equilibrium relation. A first series is based on empirical observations (e.g. Greenshields 1935). By observing two traffic variables a function is then fitted. This method is not obvious. Traffic flow must be stationary and homogeneous across the measurement interval, which requires a sufficiently small interval. On the other hand the variables show a discrete character in a measurement interval that is too small.

The observations can be given in a k-q, a k-u or a q-u diagram. The choice of diagram in which the equilibrium relation is fitted is equally important. (Duncan 1979).

The observations must be carried out under equal circumstances and independent of the numerous factors that influence outcome. It is, in addition, questionable, if all densities can be measured on a road section. It is impossible, for example, to observe capacity downstream of a bottleneck. (May 1990).

Dynamic models, such as queuing systems (e.g. Van Woensel 2003, Vandaele et al. 2000), car-following models (May 1990), cellular automata (i.e. Nagel and Schreckenberg 1992) or statistical processes (Haight 1963) have led to various proposals of a number of fundamental diagrams. The fundamental diagram then presents a special case in which traffic is homogeneous and stationary. In the development of these models, the fundamental diagram is often used as a reference to illustrate that stationary traffic can be adequately modeled.

The shape of the fundamental diagram can, to a degree, take the non-homogeneous and non-stationary character of traffic flow into account. The free-flow branch of the diagram, the rising part of the curve, describes the flow of traffic when a variety of vehicles travels freely along the road. In a heterogeneous vehicle flow, each vehicle will try to maintain its own 'desired speed'. Fast vehicles, however, will be obstructed


25

by slower ones and be forced to temporarily reduce speed until overtaking becomes possible. In actual situations average speed appears to remain constant for small densities and so the first part of the free-flow branch is a straight line. At greater densities fast vehicles will be obstructed and average speed will, presumably, decrease. Thus, the diagram shows a curve at higher free-flow densities (Newell 1993b and Highway Capacity Manual 2000).

When density exceeds capacity density kM ,traffic is congested. Opinions differ strongly regarding the shape of the congested diagram. What is certain is that the function continuously decreases between capacity density and jam density. Due to frequent instabilities in congested traffic, stationary congested traffic states are seldom observed. As a result an average speed is often combined with an average density. This relation, however, depends strongly on the measurement interval and the average values have large variances. The non-homogeneous character of traffic flow also becomes more important in this part of the curve. The difference in vehicle length, a container lorry is up to four times the length of a private car, becomes more significant because, at high density levels the road space occupied by vehicles becomes large in comparison to the headways between vehicles. This is why traffic density is highly dependent on vehicle composition.

As a result of non-homogeneous and non-stationary effects, observations of congested traffic often show large variations. Graphically this leads to a cloud of observation points in the k-q diagram which enables different forms of curves to be fitted.

Some researchers assume a non-concave fundamental diagram (Helbing 1997). In mathematical terms this means that the diagram does not fully comply with

0)(

2

2

≤dk

kQd e (3.9)

In this case fans can occur during increasing traffic density in the positive x-direction. Conversely, decompression shock waves appear in the solutions.

Fundamental diagrams are not exclusively used as input parameter for the LWR model. They are also used in road design, evaluations of management measures and qualitative studies of traffic performance. Since they are often used, there is a large number of expressions for the diagrams and the effects of external influences can be easily calculated. However, the LWR model makes large demands on the diagram. It is assumed, for example, that the equilibrium relation also applies to non-homogeneous and non-stationary circumstances. The shape of the diagram and the assumed capacity value strongly influence the location, length and duration of modeled tailbacks.

The triangular k-q diagram, as drawn in figure 3.2, is a simple and much-used version. In it, the free flow speed uf applies to all traffic states during free-flow traffic. This is why the characteristics run parallel to the trajectories at densities smaller than kM. Shock waves between two free-flow traffic states also occur at this speed and are called slips or contact discontinuities. The congestion branch of the fundamental diagram has a constant speed w(<0) so that, consequently, all characteristics inside the congestion regime also have the same speed. The derivative of Qe(k) is

Chapter 3 26

discontinuous in the capacity point, but in a continuous expansion it is assumed that all values between uf and w feature in kM .

Figure 3.2: The triangular k-q fundamental diagram

3.3 Numerical scheme The analytical computation with characteristics, shock waves and fans allows us to gain a qualitative insight into the LWR model. This method rapidly becomes too complicated and time-consuming for practical problems. This is why a number of numerical methods have been developed which can be used to fairly rapidly calculate an approximate solution for the LWR model. Here the numerical scheme must approximate the analytical solution closely, and it should not be used to eliminate possible short-comings of the model itself.

In one numerical approach car-following models are developed on the basis of the LWR model. These are called particle based numerical methods, used for example in the micro-simulation package Integration (Van Aerde 1994). We refrain from a further discussion of these methods.

In a cell method the road under consideration is divided into cells with a length of ∆x where the traffic situation is calculated per time interval ∆t. The aim is to calculate the average cell density K(t,x) based on the traffic situation during earlier time intervals. This density is an approximation of the real density and is considered to be constant in the whole cell. Therefore :

),(),( xtKrtK = ∀

∆+≤<

∆−

22xxrxx

(3.10)

With the aid of the fundamental diagram, knowledge of this traffic density also gives the cell flow Q(t,x) and cell speed U(t,x).

Based on the conservation of vehicles per cell, density is calculated as :

[ ])21,()2

1,(),(),( xxtGxxtGxtxtKxttK ∆+−∆−

∆∆+=∆+

(3.11)

kJ

uf

w

q

k

kM

qM


27

Here the function )21,( xxtG ∆− shows the flow between cell x-∆x and cell x during

[t, t+∆t[.

Flow G across the cell border depends on the flow that can be passed on from the upstream cell and from the flow that the downstream cell can receive. We define these respectively as the sending and the receiving flows.

The sending flow S(k), sometimes called the local traffic demand, indicates how much a cell can emit when there is no downstream obstruction. This function depends on the density in the cell and the equilibrium-relation as shown in figure 3.3. During free-flow the sending flow runs parallel to the flow from the fundamental diagram. At densities above the critical value the sending flow equals the capacity flow.

Analogously a receiving flow, sometimes called the local traffic supply, can be defined. The function R(k) represents the maximum flow that a cell can receive when the upstream flow is unlimited. For densities above the critical value the receiving flow is equal to the flow in the fundamental diagram and equal to the capacity flow at free-flow densities as shown in figure 3.3.

Figure 3.3 : De sending en receiving flows

The actual flow across the cell boundary then becomes the minimum of the sending flow in the upstream cell and the receiving flow of the downstream cell :

( ) ( )[ ])),(,),()21,( xtKRxxtKSMinxxtG ∆−=∆− (3.12)

The combination of the formulas (3.11) en (3.12) leads to a numerical scheme that is known as the Godunov method. It was first applied to the LWR model by Daganzo (1995b) and Lebacque (1996). Compared to other cell based numerical methods this Godunov method takes full account of the direction in which the characteristics propagate through the traffic flow. To avoid stability problems, the maximum speed of the characteristics, traditionally equal to uf , must be smaller than cell length ∆x divided by time interval ∆t.

This method can be interpreted as each time again solving a Riemann problem at the cell boundaries. In a Riemann problem an infinite road is considered where a constant density applies upstream from a particular point and a different constant density downward from that point. Both figures 3.1b and 3.1c solved a Riemann problem. The transition flow G corresponds to the flow at the level of the discontinuity in a

q

Qe(k)

k

S(k)

qM R(k)

Chapter 3 28

Riemann problem with the up- and downstream cell densities. The transition flow for figure 3.1b becomes q1. This is the capacity flow qM in figure 3.1c.

This method also applies to non-homogeneous links. The sending and receiving flows are then determined starting from the cell and are thus determined by the road properties.

Application of the numerical method introduces dispersion into the solution. Shock waves are averaged and attain finite length. This effect renders the numerical solution somewhat more realistic. This is so because, in actual traffic patterns, shock waves show a finite length also. On the other hand this dispersion is introduced by the numerical scheme and not by the model itself. The numerical method, therefore, alters the model.

The impact of the numerical method is illustrated in an example in figure 3.4. We consider a homogeneous road with a constant traffic state. We assume the speed of the traffic flow to be equal to half that of uf = ∆X/∆T. This means that a vehicle in the numerical scheme must change cells every two time intervals.

Upstream from a particular point x0 we assume white coloured vehicles, downwards black vehicles. The driving behaviour of all vehicles is identical.

In an analytical computation of the solution the white and black coloured vehicles will remain completely separated as shown in figure 3.4a. Implementing this example with the numerical scheme causes a mixing of the vehicles. Applying the numerical method to this example always leads to a mixing of vehicles.

Figure 3.4b gives an average colour per cell. The zone with vehicles of both colours enlarges over time. This is caused by the fact that the contents of a cell are assumed to be homogeneous in each time interval. Incoming white cars are mixed with the cell contents and already during the next time interval some of these newly arrived vehicles are transferred to the next cell. This gives the impression that there are white vehicles driving with a speed of uf and black vehicles that are practically stationary. A mixing of black and white cars is more realistic in multi-lane traffic, but it is a characteristic introduced by the numerical scheme. However, a numerical scheme should provide an optimum approach to the analytical model but should not serve to improve lesser properties of the analytical model.

Figure 3.4 : Homogeneous traffic with various vehicle-colours (a) analytical (b) numerical and (c) with an improved numerical method.

x

t (a) (c) (b)


29

Daganzo (1995a) introduced an additional condition for this situation. Per cell the time of entry for the vehicles that enter the cell is recorded. In the eventual transfer flow the first vehicles to have entered the cell are sent on first. This reduces the dispersion considerably, as can be seen in figure 3.4c.

In another extension of the numerical scheme (Daganzo 1999) for traffic flows where a triangular fundamental diagram applies, attention is not restricted to the density in the immediate up- and downstream cells. This enables a closer approach to the analytical solution.

3.4 Discussion In this section we will take a short look at the properties of the LWR model. Starting from the deficiencies we introduce an overview in the following section of the most important improvements and extensions to the model.

The strongest point of the LWR model is, undoubtedly, its capacity for analytical solutions. With discrete constant initial- and boundary conditions a solution can even be constructed graphically. This enables the properties of the model to be studied independently of the numerical scheme.

Moreover, the amount of input parameters is confined to the much-used fundamental diagram. A rapid and adequate numerical scheme also allows for the calculation of an approximate solution for practical problems.

The properties of the LWR model are largely determined by the assumptions. Thus a fundamental diagram is used in which traffic is assumed to be stationary, homogeneous and deterministic. It is assumed that this relation continues to apply in all circumstances.

The shock waves that emanate from the model are a direct consequence of this. In these waves, traffic changes discontinuously between two stationary traffic states and, in principle, an infinite acceleration applies. Consequently, these waves can not occur in actual traffic states. A shock wave could, possibly, be seen as an abstract representation of a transition zone between two stationary traffic states. A more accurate description of these transition zones requires the modeling of accelerations of non-stationary traffic.

The impossibility of the LWR model to model start- and stop waves and other instabilities is also a result of the assumption of continuous stationary traffic. Because traffic is assumed to be homogeneous, the LWR model does not allow for the modeling of differences in driver behaviour (e.g. desired speed) and vehicle characteristics (e.g.. vehicle length). The LWR model, therefore, behaves like a first-in-first-out system (FIFO) and multilane effects such as overtaking can not be described. The heterogeneous desired speeds during free-flow traffic do not feature either. Therefore, the so-called dispersion effect, whereby downstream from a temporary delay fast vehicles pass first, can not be modeled.

Chapter 3 30

3.5 Extensions Over the years, the LWR model was extended and refined. A number of improvements were advanced to generalise the assumptions of the LWR model and to describe the non-stationary and heterogeneous characteristics of traffic flow. In addition, attempts have been made to expand the LWR model from one single link to a comprehensive network model. Each time an extension of the LWR model requires new assumptions regarding the traffic mechanism while new parameters increase the input for the model.

3.5.1 Non-stationary The original publication of Lighthill and Whitham (1955) already carries suggestions for the non-stationary character of traffic flow to be modeled. However, the first to expound a comprehensive higher order model is Payne (1971). A higher order model no longer assumes that all traffic states are described by the fundamental diagram. A momentum equation with additional parameters, such as relaxation time and an anticipation coefficient, is introduced. Besides improved modeling of the non-stationary characteristics of traffic, new deficiencies are ascertained in these models (Daganzo 1995c). In reality drivers are primarily influenced by the traffic situation ahead of them. This anisotropic property is not observed in certain higher order models because the characteristic speed can be larger than the free flow speed. Chapter 4 will look further at the anisotropy of traffic. The analytical tractability disappears, furthermore, in the higher order models. The discussion that has started about the usefulness and errors of these models (see also Papageorgiou 1998, Lebacque and Lesort 1999, Heidemann 1999, Zhang 2003b) has led to greater attention in recently developed higher order models being directed at the typical characteristics of traffic flow (Zhang 1998, 1999; Aw and Rascle 2000). Hence, the interpretation of numerical methods gain in importance.

3.5.2 Non-homogeneous A second type of improvements tries to describe the heterogeneous character of traffic more correctly. Here we can distinguish between two approaches

Munjal and Pipes (1971), Munjal et. al (1971), Holland and Woods (1997) and Greenberg et. al. (2003) divide the homogeneous road into traffic lanes. This leads to a parallel coupling of different LWR models by applying exchange terms. These non-homogeneous road models assume that a slightly different fundamental diagram applies to each traffic lane. On the basis of different densities between the lanes, lane-change flows are then taken into consideration.

A second approach divides the homogeneous vehicle population into different classes. By a class we mean a homogeneous set of vehicle driver entities. The different classes show different driving characteristics. The interaction between these different classes then provides for a better description of heterogeneous traffic flow. This approach forms the basis of this dissertation and is further clarified in chapter 4.


31

In combination with heterogeneous vehicle characteristics Daganzo (2002a,b) recently introduced the concept of motivation. This indicates that passing drivers will, temporarily, accept smaller headways. Elaboration of this concept lead to an analytical extension of the LWR model that is used to describe a range of effects.

3.5.3 Network Until now the LWR model on a link was looked at under unchanging road properties. The extension to a comprehensive network model was described by Daganzo (1995a) and Lebacque (1996). These network models are closely inter-linked with the numerical Godunov method. The network approach is extensively elaborated in chapter 9.

In a network, nodes form the connection between a number of homogeneous links. These nodes occupy no physical space, but take care of the exchange of traffic between the different links.

Modeling traffic behaviour at nodes can not be derived from a fixed mathematical equation. Vehicle behaviour at a node is the result of interactions with other vehicles and traffic rules that are in force. Modeling these nodes is, therefore, based on behavioural rules. We can consider origin, destination, diverge and merge nodes in a network model. From origines nodes new vehicles depart for trips across the network. The input to the origin nodes is the traffic demand. Vehicles leave the network at destination nodes. These are modeled as infinite "drains" because it is assumed that the outflow is not restrained. The road divides in a diverge node. These nodes are usually modeled using a fixed splitting ratio where the upstream traffic behaves like a FIFO system. Two links join in a merge node. The behaviour in this node is modeled using priority proportions for each joining link. The proportions reflect the composition of the downstream traffic flow at the saturation point of the downstream cell and sufficient upstream inflow.

A further refinement of the network models divides traffic flow into classes. The various vehicles are then subdivided according to destination. The classes show no differences in driving behaviour and traffic flow on a link is independent of composition. At the network level these classes do have an impact because they behave differently on the nodes.

When modeling a diverge node the classes replace the use of splitting factors. Starting from the composition of the traffic flow upstream from the diverge node, the traffic splits in the node. Because of the classes, the splitting proportions become time-independent, as it were.

Modeling filter lanes at a diverge can be done using such classes. The next chapter will take a closer look at the splitting of a road in lanes for each exit. This was elaborated by Daganzo (1997b) and Daganzo et al. (1997).

In a next step towards a comprehensive dynamic network model, route choice also plays a role. Lo (1999) computed a dynamic assignment for a LWR model. Here, the dynamic traffic demand is assigned by an equilibrium assignment model.

Chapter 3 32

3.6 Conclusions The formulation of the LWR model, at the time, gave an important impulse to traffic flow theory. In spite of its deficiencies the model continues to be applicable in a variety of instances to this day. Its graphic solution method renders a good insight into model properties and due to the formulation of an effective numerical method it is experiencing a true revival.

Rigorous extensions to the model are still ongoing. Recently insight into the effect of non-stationary extensions increased considerably and the model can also be used for networks.

The remainder of this dissertation discusses the homogeneous character of the LWR model. One condition remains: it should be possible to resolve the model analytically as well as numerically. This includes the extension to a network model.

33

4 CLASSES IN THE LWR MODEL

Some characteristics of observed traffic patterns can not be described by the LWR model. These deficiencies in the LWR model are a consequence of assumptions that can not always be proven. This chapter will discuss the non-homogeneous character of traffic and the possible heterogeneous extensions of the LWR model.

In a homogeneous traffic flow all vehicle-driver entities react identically under identical circumstances. The LWR model is homogeneous because it uses a fundamental diagram that applies to the entire traffic flow. The traffic density is, therefore, unambiguously linked to a flow at a homogeneous speed. Extending the LWR model to model heterogeneous properties can be done in two ways.

A first approach is to make a distinction between lanes. For each traffic lane, a separate LWR model is formulated with separate fundamental diagrams for each lane. Exchange terms are provided between these parallel models, based on the driving conditions on the lanes. This approach was proposed by Munjal and Pipes (1971) and Munjal et al. (1971) and further extended by Holland and Woods (1997). Greenberg et al. (2003) proceed along the same line.

A second approach subdivides the flow of vehicles into classes. A class encompasses a group of driver-vehicle entities that share the same properties. Thus the heterogeneous characteristics of the modeled traffic flow are the result of interactions between homogeneous subgroups within the traffic flow.

In this chapter, we further elaborate on the multi-class approach. This subdivision into classes has the advantage that the heterogeneous properties are allocated to the vehicles, not to the infrastructure. Differences in vehicle- and driver characteristics are, as a result, fully accounted for.

Chapter 4 34

Following on the explanation of a number of basic principles, a number of assumptions are discussed with regard to class interactions. This is followed by the fundamental diagram that must now be formulated for each class. Finally, a set of fundamental diagrams and a way of interaction between the different classes is chosen. The assumptions will be elaborated in the following chapters and will lead to a comprehensive heterogeneous LWR model.

4.1 Fundamentals of multi-class LWR models Before we add classes to the LWR model, we will give an exact description of the concept of a class, for there are different ways in which traffic flow can be sub-divided into classes.

In a following section the LWR model will be formulated for several classes. These expressions are generally applicable in extensions of the LWR model to classes. The remaining assumptions regarding driver behaviour and the interactions between classes is dealt with in subsequent sections.

4.1.1 Philosophy of multi-class traffic The traffic flow in the LWR model is assumed to be homogeneous. All vehicles and drivers are considered to be identical. In multi-class traffic the traffic flow is divided into subgroups. All vehicles belonging to a class share the same properties. A variety of classes can be distinguished according to the characteristics around which they are grouped. Classes can be created pertaining to route choice, vehicle properties and driver characteristics.

We shall first discuss so-called 'user-classes'. The flow of traffic is divided into user-classes according to properties that do not affect driver behaviour on a link.

The use of user-classes does have its repercussions on the network level because it influences driver behaviour on intersections and nodes. Trip purpose, origin, destination, or the value of time of drivers can lead to user-classes. They are used in network extensions to the LWR model, to properly model route choice behaviour. They offer no added value on a link. Note that the term 'user-classes' is applied here to classed where driver behaviour is not essentially different. This term is less narrowly defined in urban transport models (e.g. Bliemer 2001).

In a second type of classes, on the other hand, driver behaviour on a link is fundamentally influenced. Vehicles react and interact according to the class to which they belong. When modeling heterogeneous traffic flow, the focus is on this type of multi-behaviour classes. When subdividing the traffic flow in multi-behaviour classes, no qualifications are made regarding the causes of the differences in driving behaviour. Both vehicle- and driver characteristics may be responsible.

In the LWR model, traffic flow is assumed to be homogeneous. All vehicles react identically to the same situations. Strictly taken, this assumption can not fully apply. In fact, the LWR model implicitly assumes that average vehicle behaviour can be

Classes in the LWR model

35

extrapolated to the entire vehicle population. Averaging traffic flow is an inherent characteristic of homogeneous models.

In the subdivision of vehicle flow into classes, joint characteristics are taken as the basis. It is also assumed that all vehicles belonging to the same class react identically and that they show the same driving behaviour. Also in a homogeneous subgroup we assume average vehicle behaviour. By working with classes the variation around this mean vehicle behaviour is, however, smaller; classes are chosen on this basis.

Crucial in a subdivision of traffic flow into classes is the required number of classes. The larger the number of classes, the more homogeneous driver behaviour per class will be. On the other hand, the amount of input required increases considerably when we deal with a growing number of classes and calculation time of the model increases. A deliberation between the uniformity of the classes, in particular regarding their driving behaviour, and the quantity of classes must be made each time.

Empirical studies clearly show the effect of vehicle- and driver characteristics on the traffic process (e.g. Kockelman 1998, HCM 2000). When classes are introduced these heterogeneous characteristics are converted into an interaction of homogeneous classes. Subdividing the traffic flow is also vindicated by observations.

When detectors are used, the most obvious option is to apply a subdivision based on vehicle-length. This class-division shows a clear difference in traffic flow (e.g. Hoogendoorn 1999). Not only is there a difference in maximum speed but behaviour during congestion also differs (Dijker et al. 1997). The effect of vehicle-length plays a role at low speeds, but at higher congested speeds the following behaviour is also asymmetrical: lorries maintain a larger distance to the vehicle ahead.

Class-division bases on driver characteristics can also be justified empirically (e.g. Kockelman 2001, Kijk in de Vegte 2002). They clarify the differences between commuters versus non-commuters, aggressive- versus non-aggressive drivers and even the effect of age and sex.

4.1.2 Extended LWR formulation In the following derivations we study a heterogeneous traffic flow on a link with unchanging road characteristics. Here, traffic flow is composed of vehicles that belong to n different classes. For each class, density ki, flow qi and average speed ui can be defined. For the road overall, indicated with the index tot, the following relations apply:

∑= itot kk (4.1)

∑= itot qq (4.2)

The average speed of the total heterogeneous traffic flow is defined as follows :

∑== itot

i

tot

tottot u

kk

kqu .

(4.3)

Chapter 4 36

A class is assumed to be homogeneous when all vehicles are identical and all share the same speed ui. In the total heterogeneous traffic flow there is a certain variation across the different vehicles. Here the heterogeneous traffic variables indicate the average values of the traffic flow only. By defining the average speed of the heterogeneous flow as in (4.3), the relation q = k.u continues to apply both for each separate class and for the total flow.

In a k-q diagram the total heterogeneous traffic flow can, therefore, be seen as a vector-sum of the joint homogeneous classes. The traffic conservation law, that forms the basis of the LWR model, applies both to each class i apart and to the overall traffic flow :

0=∂∂+

∂∂

xq

tk ii

(4.4)

As in the homogeneous LWR model, a relation is used that expresses the flow of a class in function of the density. However, in a heterogeneous model this equilibrium flow is a function of all class densities.

),...,( 1 neii kkQq = (4.5)

Introducing these equilibrium relations into the traffic conservation law leads to a system of n partial differential equations:

0' =∂∂+

∂∂

xt eKQK

(4.6)

The following matrix notation is used :

=

nk

kM1

K en

∂∂

∂∂

∂∂

∂∂

=

n

enen

n

ee

e

kQ

kQ

kQ

kQ

L

MOM

L

1

1

1

1

'Q

(4.7)

This system of partial differential equations is convex, if the matrix 'eQ can be resolved into eigenvalues.

When only two vehicle classes are considered, we get a hyperbolic system of partial differential equations if 'eQ has real and different eigenvalues. For this, it is sufficient that the non-diagonal elements of 'eQ have the same sign. In other words :

∂

∂=

∂∂

1

2

2

1

kQ

signkQ

sign ee (4.8)

This condition for the hyperbolicity shows that the system of partial differential equations is well-posed. (Leveque 1992).

When computing the model shock waves will occur. These are lines in the t-x solution zone where class densities change discontinuously. The jump condition shows how


37

the changing class densities and flows across the shock wave are related to the speed z of the wave:

i

i

kq

dtdxz

∆∆==

(4.9)

The speed of the shock wave is, therefore, related to changes in density and flow. Note that this shock wave condition applies both to the separate class variables and to the total traffic flow.

It is assumed that traffic flow is anisotropic (Daganzo 1995c). This means that a driver reacts only to stimuli ahead of him and that the traffic situation behind the vehicle does not influence driver behaviour. Exceptions, such as bumper tailing or an advancing police car are, therefore, not taken into consideration.

In a homogeneous model this anisotropic condition has been met when the speed of the characteristics is smaller than or equal to the speed of the vehicles (Zhang 2003a). As a result, the speed of the shock waves is also smaller than the vehicle speed.

In a multi-class model this condition remains valid, though only per class. Thus, the speed of a shock wave needs only be smaller than the vehicle speeds of those classes that are influenced by the particular shock wave. When a slow class does not change across a shock wave, the speed of this wave may exceed the speed of this vehicle class.

4.2 Class interactions As in the original LWR model, the characteristics of a multi-class LWR model strongly depend on the equilibrium relation that is used (4.5). Besides the typical characteristics of the class, this equation also incorporates the interactions between the different classes. In this section we will see how vehicles from the different classes interact on a homogeneous link. Several assumptions will be reviewed.

For each class, a fundamental k-q diagram can be drawn up for a link. This diagram then applies to a flow of traffic that is homogeneously composed of vehicles of that well-defined class. We regard these homogeneous fundamental diagrams as given and they are specified as :

)( iheii kQq = (4.10)

The characteristics of a class can be completely contained by this fundamental diagram. A link carrying vehicles from one class only can be described by the original LWR model with this fundamental diagram.

To describe a mixed flow of traffic, class-flow must be expressed as (4.5) :

),...,( 1 neii kkQq = (4.11)

Starting from the assumption relating to this equilibrium flow, several heterogeneous models can be constructed. Just as we do with the vehicle population, we divide the

Chapter 4 38

road space into different fractions. Each class can then make exclusive use of a fraction of road space assigned to it.

When we assume that a class behaves itself on an assigned fraction as being subject to its fundamental diagram, then :

=

i

iheiii

kQqα

α . (4.12)

Here αi is the fraction of the total road that is assigned to class i. In this case we can consider the road as divided into parallel spaces used by each class separately.

In all the stationary multi-class LWR models developed so far, the equilibrium flow from (4.11) can be rewritten using road fractions according to (4.12). Formulating a heterogeneous fundamental diagram then comes down to determining the way in which the road fractions are assigned to the different classes. The assignment of fractions of road space to the various classes derives from the interaction between the vehicles. The equilibrium flows for each class are, therefore, the result of the homogeneous fundamental diagrams )( i

hei kQ and of the assumed interactions between

the classes.

The road fractions defined above are always positive :

iα≤0 (4.13)

In addition, the sum of all fractions can not exceed 1 :

1≤∑ iα (4.14)

Here the equality symbol applies when the road is completely used.

A microscopic interpretation can also be given to the concept of road fractions. We define a vehicle's space s as the space between the rear bumper of the vehicle ahead and the rear bumper of the vehicle under consideration. Chapter 2 demonstrated that space s equals the distance gap d for the vehicle plus the length of the vehicle L. In the case of a homogeneous flow of vehicles, the space equals the inverse of density :

k = 1/s (4.15)

We can also rewrite this relation in a different way to clarify that the number of vehicles multiplied by the space per vehicle equals the total road space, or :

1= s.k (4.16)

This relation can be formulated for each class separately if the vehicles of class i proceed on a fraction αi of the total road space. The space si of the class i vehicles and the density ki of class i vehicles are then related as follows:

iii ks .=α (4.17)

Therefore, we can rewrite class flow (4.12) with (4.17) to :


39

=

i

heiiii s

Qksq 1.. (4.18)

In this case the speed of class i is a function of the space :

=

i

heiii s

Qsu 1. (4.19)

Instead of assigning a fraction of the road per class, we can also interpret this choice as a choice for the different spaces si per class.

In the framework of formula (4.12) all first order heterogeneous LWR models developed so far can be described. In the paragraphs that follow, the existing models will be reformulated and discussed using this formula. In addition, a number of new heterogeneous models will be developed that are based on different assumptions regarding the choice of road fractions αi for the various classes.

Characteristic for all models that can be developed within this framework is that the interactions of classes do not profoundly influence the homogeneous fundamental diagram that is used, and thus driver behaviour. A vehicle behaves in the same way as if it was in a homogeneous flow of traffic. Driving mechanisms that deviate from this pattern, as, for example, with the motivated drivers of Daganzo (2002a,b), can thus not be described using this fraction model.

A first paragraph deals with the user-class subdivision of the LWR model. Next, Daganzo's special lane model (1997b), which models filter lanes, will be examined within this framework. Wong et al.'s heterogeneous model (2002) and the model of Zhu et al. (2003) appear to be based on an equal space assumption. Practically analogously, the equal distance gap model of Benzoni-Gavage and Colombo (2002) can be dealt with. The free-flow model of Chanut and Buisson (2003) is based on the proportionality between distance gaps and vehicle-length. The assumption of a homogeneous vehicle speed is implicated in the congested part of the heterogeneous models of Zhang and Lin (2002) and Chanut and Buisson (2003).

New assumptions for heterogeneous models are subsequently developed and analogies with the route choice algorithms (Wardrop 1952) from the predicition models are sought.

In the user-optimum model presented, all vehicles try to maximise speed, while in the system-optimum total flow is maximised. Lastly we look at an adaptation that takes account of the number of lanes.

4.2.1 User-classes When a flow of traffic is subdivided into classes where driver behaviour does not differ between vehicles of the various classes, we speak of user-classes. Thus, the traffic flow is subdivided into user-classes for assignment problems. A class is then assumed for each OD-relation. The example with various coloured vehicles to show

Chapter 4 40

dispersion occurring in the numerical scheme in chapter 3.3 is also based on user-classes.

Characteristic for user-classes is that the total flow is not dependent on the composition of traffic flow. Consequently, the homogeneous fundamental diagrams for a class are identical to the fundamental diagram Qe(k) for the total traffic flow.

By equating the road fractions αi with the relative class density, the equilibrium flow of a class also becomes a fraction of the total flow. Here, the fractions are defined as :

tot

ii k

k=α (4.20)

Therefore, the flow per class can be described as a fraction of total flow :

totitotetot

ii qkQ

kkq .)(. α==

(4.21)

By choosing these fractions the speed of the traffic flow for all vehicles from all classes is identical :

uk

kQkqu

tot

tote

i

ii === )(

(4.22)

The traffic flow has, therefore, a homogeneous speed u and class densities and flows can be expressed fractions of total traffic flow. Writing this system of conservation laws per class gives :

0.. =∂

∂+∂

∂xq

tk totitoti αα

(4.23)

Further computation leads to :

0=∂∂+

∂∂+

∂∂+

∂∂

xq

xq

tk

tk i

tottot

ii

tottot

iαααα

(4.24)

Applying the law of conservation of the total traffic flow (4.4) we get :

0=∂∂+

∂∂

xu

tii αα

(4.25)

This formula shows that vehicle composition along a trajectory is constant. Therefore, traffic flow composition is unable to change faster than vehicle speeds. This amounts to a first in first out (FIFO) rule.

4.2.2 Special lanes Daganzo (1997b) extended the LWR model for two classes both with the same fundamental diagram Qe(k). But a set of special lanes was taken that are accessible to one of the two classes only. This model is used to model diverges as outlined in figure 4.1. Exiting traffic is then confined to a few traffic lanes, while the remaining traffic can use all traffic lanes. Non-exiting traffic, therefore, has exclusive access to 'special' lanes and spreads itself in such a way as to allow for the largest possible flow.


41

Figure 4.1 : The type of freeway diverge that can be modeled using the special lanes method.

This model can also be written using the general fraction equation (4.12) :

=

i

iheiii

kQqα

α . (4.26)

The exiting class then has a fraction that is limited due to the road section that it can use. Suppose that the class 1 vehicles are confined to the md lanes of the total m lanes of the road. We can then write the following fraction equations :

),min( 11 m

mkk d

tot

=α (4.27)

),max(1 212 m

mkk d

tot

=−= αα (4.28)

Besides an analytical computation of this model for a triangular fundamental diagram, the numerical framework was also adapted and interpreted (Daganzo et al. 1997c).

Recent research into observations (Muñoz et al. 2002b) have shown that this model suffices to model the effect of congested exits on motorways. The number of special lanes md assigned to exiting traffic must, however, increase in relation to the distance to the exit lane.

4.2.3 Equal space Wong et al. (2002) recently formulated a heterogeneous model with various vehicle classes. A homogeneous fundamental diagram )(kQh

ei applies per vehicle class and each class is supposed to proceed on a fraction to the total road. These fractions are equated to relative class density :

tot

ii k

k=α (4.29)

Flow per class then equals :

tot

tothei

ii kkQkq )(=

(4.30)

In this approach flow is calculated per vehicle type using the homogeneous fundamental diagram, at total density. In this way the intrinsic properties of a class remain valid while total traffic flow is taken into account.

Chapter 4 42

The model of Zhu et al. (2003) also operates according to this principle. Here, the homogeneous fundamental diagrams, however have a more specific formulation.

In a microscopic interpretation of (4.28) according to (4.17), the space si of class i vehicles appears to be computed as follows :

toti k

s 1= (4.31)

This shows that the space si of all vehicles is identical. This model assumes, therefore, that each vehicle, independent of class, maintains a speed that leads to a fixed user space. The choice of an equal user space for all vehicles leads to different speeds per class. This is why passing becomes an option and why this model sees a road implicitly as multilane where the various classes proceed independently. These parallel traffic lanes lead to class segregation whereby each class has its own speed.

When we look at an actual road with m traffic lanes, then the assigned fraction, at a complete class segregation of the road, must equal a multiple of 1/m. This assumption applies better, therefore, to road with a large number of traffic lanes. It is questionable whether vehicles from the various classes will, in the case of class segregation, search for an identical space.

The approximate description of microscopic lane change behaviour can, however suffice for an extended LWR model.

The assumption to take relative density as class fraction limits the workable homogeneous fundamental diagrams )(kQh

ei . Since the space is identical for all classes, minimum space for al vehicles must also be identical. This is why this model requires identical jam densities for all classes.

The original publication of Wong et al. (2002) demonstrates the properties of the model somewhat awkwardly, using numerical simulations. In addition, the choice for non-concave fundamental diagrams, where the critical densities of all classes are identical, renders the conclusions less general. An analytical computation whereby the intrinsic properties of the model are further examined, seems desirable.

An analytical computation will immediately give rise to the question whether the anisotropic condition can be met. After all, slow vehicles are also influenced by the behaviour of faster vehicles. This implies that slow vehicles are possibly influenced by the properties of vehicles coming from behind. This is why this type of heterogeneous model is open to question on exactly the same point where higher order models fail (Daganzo 1995c).

4.2.4 Equal distance gap Analogous to the homogeneous space model, Benzoni-Gavage and Colombo (2002) formulated a homogeneous gap model. Here it is assumed that the distance gap for all vehicles in a heterogeneous traffic flow is identical. From the combination with the microscopic interpretation of the fractions (4.17) with the distance gap definition we know that :


43

iiii kLkd .. +=α (4.32)

Here Li is the length of the vehicles from class i and d is the gap that is equal for all vehicles. If we take account of a fully utilised road, then the gap becomes equal to :

tot

ii

kkL

d ∑−=

.1

(4.33)

This implicates that the equilibrium flow for a class is given by :

+−+−=

=

∑∑itotii

totheiitotii

tot

i

i

iheiii LkLk

kQLkLkkkkQq

..1)...1.(.

αα

(4.34)

This model can be further simplified for two classes. For the flow from class 1 this gives :

−+

+−++

=2212

2112212

21

11 ..1

)...1.(LkLk

kkQLkLkkk

kq he

(4.35)

The properties of this model agree fairly well with those of the homogeneous space model. When the length of all vehicle classes is equal, both models are identical.

Using the gap rather than the space compensates for the difference in vehicle length. This allows classes to be modeled with various jam densities. The restraint imposed on the homogeneous fundamental diagrams is, therefore, no longer valid.

The anisotropy is also a weak point in this model. Faster vehicles help determine the distance gap of slower vehicles and thus lead to an influence of upstream traffic.

Also, the question as to whether the fractions should be connected to real lanes occurs here. In addition, it remains doubtful whether striving towards equal space gaps is the motivation of the various vehicles.

4.2.5 Distance gap proportional to vehicle length The free-flow model of Chanut and Buisson (2003) can also be worked out within the framework of (4.12). This model distinguishes two classes with different free-flow speed ufi. If the speed is below a fixed capacity speed uM, the free-flow regime applies. The homogeneous fundamental diagrams for both classes are then given by:

−−=

Ji

iMfifiii

hei k

kuuukkQ

.).(

)(β

(4.36)

Here the expression N/Li from the original publication, with N for the number of traffic lanes of the road, and Li for the vehicle-length of vehicles in class i was equated to the jam density kJi of class i. In the model of Chanut and Buisson (2003) the flow of a class in a heterogeneous flow is described as :

Chapter 4 44

+

−−=

2

2

1

121

)(),(

JJ

Mfifiiei k

kkkuu

ukkkQβ

(4.37)

Applying (4.12) to (4.36) and equating to (4.37) gives an expression for the fraction of the road space that is assigned to class i:

+

=

2

2

1

1

1

1

JJ

Ji

kk

kk

kk

α

(4.38)

The microscopic interpretation of this expression, using (4.17) leads to :

( )2211

2211

....1.

LkLkLkLkLLs iii +

+−+= (4.39)

This leads to the following expression for the distance gap of a class i vehicle :

∑∑−

=ii

iiii Lk

LkLd

..1

. (4.40)

Since 2211 .. LkLk + equals the physical space of all vehicles on the road under consideration, (4.40) demonstrates that the distance gap of the vehicles is proportional to the vehicle-length.

The fraction assigned to a class is, therefore, proportional to the physical space occupied by the vehicles of a class on the road.

In this assumption, headway is not determined by speed, but by vehicle length. The anisotropic condition is, therefore, not met. Because each class may show different speeds, the classes operate in parallel.

During the capacity regime, all classes in the model share the same speed uM. Therefore the boundary with the congestion regime, that is modeled according to a different assumption, is unequivocally fixed. In general, however, the assumption of proportionality of distance gap to vehicle length imposes no restrictions on the shape of the fundamental diagram.

4.2.6 Equal speed Another way to model heterogeneous traffic is to assume that the speed of all vehicles is equal. Again a class is described by a homogeneous fundamental diagram )(kQh

ei . This implies a speedfunction inherent to each class. In heterogeneous traffic, the speed of class i vehicles then becomes equal to:

=

i

ihei

i

ii

kQk

uα

α . (4.41)


45

By equating the speed of all classes, the fractions αi are determined. We then know what space the various classes occupy on the road. If we look at a flow of traffic that consists of two classes only and that occupies the road completely, then equating the speeds leads to:

−

−=

1

22

2

1

1

11

1

1

1.1.

αα

αα kQ

kkQ

khe

he

(4.42)

Solving this equation gives fraction αi in function of class densities k1 and k2.

Speed and flow can then be deduced using formulas (4.41) and (4.12).

Both Zhang and Jin (2002) and Chanut and Buisson (2003) use this equal speed assumption during the congestion regime.

In the model of Zhang et al. (2003) we can derive the homogeneous fundamental diagrams as follows:

i

ii

ii

hei

LkkQττ.1)( −=

(4.43)

Using the equal speed assumption (4.42) we find the following fraction for class 1:

2211

122121111 ..

)...(..ττ

ττταkk

LLkkk+

−+= (4.44)

This fraction results in the speed that, for all classes, equals :

2211

2211

2

22

1

112,1 ..

..1)()(ττ kk

LkLkk

kQk

kQu ee

+−−===

(4.45)

A homogeneous fundamental diagram can also be derived in the model of Chanut en Buisson (2003). To this end, we use the same notation as we used in paragraph 4.2.5:

−

−=

Ji

iJiMi

hei k

kkukQ 11

..)(β

β (4.46)

Applying equality (4.42) yields the following fraction for class 1:

2

2

1

1

1

1

1

JJ

J

kk

kk

kk

+=α

(4.47)

This means that the speed for both classes equals the postulated formulation:

∑

∑−

−===

Ji

i

Ji

i

Mee

kkkk

uk

kQk

kQu1

.1

.)()(

2

22

1

112,1 β

β

(4.48)

During the free-flow regime, the equal speed assumptions restrict the homogeneous fundamental diagrams. Because the speed of all vehicles is the same, the maximum

Chapter 4 46

speed, the free-flow speed, of the various classes must also be the same. This equal speed, moreover, makes it impossible to model overtaking. This particularly limits the usefulness of the model in the free-flow regime.

An equal speed for all vehicles corresponds with a perfect compliance with the FIFO rule. All vehicles follow one another at the same speed and to that end use, depending on the class to which they belong, the necessary space. The assumption of equal speeds is, therefore, best achieved with a one-lane road. For traffic flow on a multi-lane road, the assumption of equal speeds only appears achievable during congestion. The term 'synchronised traffic' that is used by Kerner (1999), points at this congested traffic with equal speeds on the various lanes.

4.2.7 User-optimum In this section we develop our own heterogeneous model that is based on the user-optimum. Here, as in the user optimum of Wardrop (1952) all vehicles try to minimise their travel time, which amounts to maximising their speed. During a user-optimum the classes divide themselves across the total road space in such a way that a vehicle is unable to increase its speed without decreasing the speed of slower vehicles.

This definition explicitly takes account of anisotropy. The speed of a vehicle depends solely on vehicles with an equal or lower speed. Faster vehicles can never decrease the speed of slower vehicles which means that vehicles approaching from behind can have no influence.

In a further specification of this user-optimum we assume that slow vehicles do not occupy more space than is strictly necessary. This efficient road use also appears in European rules on overtaking on multi-lane roads. Slow vehicles cut in early in order that they do not take more space than is strictly necessary.

Assuming a fully occupied road of two classes, the speed of the slowest class is as high as possible :

−

−

=

)1(.)1(,.minmax

1

22

2

1

1

11

1

1

αα

αα kQ

kkQ

ku h

ehe

slowesti

(4.49)

The slowest class has maximised its speed when it is unable to increase its speed by taking extra road space.

Here the following scenarios are possible :

• The speed of both classes is equal. Increasing the speed of one class causes a reduction in speed for the other class.

• Increasing the road fraction of the slowest class does not affect speed in any way. This means that the derivative of the homogeneous fundamental equilibrium relation equals zero :


47

0)(

=i

i

ihei

d

kdQ

αα

(4.50)

A difference in speed between the two classes only applies in the second scenario. This shows that this is only possible during free-flow and where there is a (local) straight free-flow branch in the homogeneous fundamental diagram of the slowest class. At that moment the extra assumption that stated that the slowest class does not occupy more space than is strictly necessary becomes useful. For further considerations on the shape of homogeneous fundamental diagrams for classes we refer to paragraph 4.3.

During congested traffic the user-optimum always gives a homogeneous speed of the traffic flow and the homogeneous speed assumption becomes valid.

During the free-flow regime, classes may have different speeds. Thus the model assumes that passing is possible. Since the fraction values do not take the number of traffic lanes into account, an infinite number of traffic lanes is, implicitly, assumed. Section 4.2.9 presents an adjustment that can take this into account.

4.2.8 System optimum To this point, a number of assumptions regarding the interaction of classes was discussed. Road space was sectioned and divided amongst the classes. We determined the fractions using the assumptions regarding driving behaviour.

This section develops a model in which the fractions are imposed on the classes. In allocating the fractions, we strife for a system optimum. During a system optimum total flow can not be increased through a redistribution of the fractions.

The total flow on a fully utilised road of two classes equals :

−

−+

=+=

)1().1(.

1

221

1

11121 α

αα

α kQkQqqq he

hetot

(4.51)

This total flow is maximal when the derivative with regard to fraction α1 equals zero :

01

=αd

dqtot (4.52)

or

−−

−

−

=

−

1

22

1

2

1

22

1

11

1

1

1

11 1

'11

'αααααα

kQkkQkQkkQ he

he

he

he

(4.53)

Solving this equation yields a value for α1. Thus the fraction for class 2 and the flows of the various classes are also known by (4.12).

This system optimum differs from the user-optimum. Slow classes are forced to yield space to faster classes that generate a larger flow on a similar road fraction. This can reduce the speed of slow vehicles in comparison to the user-optimum. Since faster

Chapter 4 48

vehicles can get more space allocated to them, it is also possible that their speed increases.

A system optimum usually aims to minimise total generalised travel time. When the total flow on each location is maximised, travel time at the link level is minimal. Route effects combined with specific priority regulations on intersections could alter the system optimum at the network level.

Since not all vehicle classes share the same value-of-time, a weighted total flow can be maximised. Economically important traffic then has a larger weight and will, therefore, benefit. Thus it is very well possible that slow classes, such as trucks, have greater economic importance and thus receive more space, at the expense of the less important faster vehicles.

A system optimum is aimed for by a network manager. The probability that it occurs spontaneously is small, and active intervention seems unavoidable. The philosophy behind the European overtaking regulations corresponds with the aspiration towards a system optimum. Slow vehicles must cut back in as soon as possible in order to free space for faster vehicles. In that way, flow is maximised while a user-optimum still applies.

Full implementation of this proposed system optimum requires an automatic vehicle guidance system.

Again the number of traffic lanes limits the calculated fractions. Since passing happens explicitly, single lane roads can not be considered.

4.2.9 Multi-lane models In all the interaction models described above that work with different speeds, only fractions in multiples of lane numbers can, strictly taken, occur. This section looks at the way in which models can be adjusted so that the finite number of traffic lanes on a road can be taken into account. The remainder of the discussion is illustrated with the user-optimum model from section 4.2.6 as an example.

We assume a road with m traffic lanes that are numbered increasingly from the slowest to the fastest traffic lane. Two classes use this road of which class 2 is the slowest. Assuming a user-optimum during free-flow, the computed fraction for class 1 not necessarily is a multiple of 1/m. We assume that the computed speeds are respectively u1 and u2, where fraction α2 complies to :

mj

mj 1

2+≤< α

(4.54)

It then follows that class 1 is limited by :

mjm

mjm )1(

1+−≥>− α

(4.55)


49

Because the user-optimum applies, the fraction of class 2 can not be decreased. Only the fraction of class 1 could possibly be adjusted to take account of the number of lanes.

• In a first option, class 1 only utilises the m-(j+1) lanes. There are no class 2 vehicles on these lanes and class 1 can proceed on a parallel basis, at a different speed. This will only happen when the speed of class 1 on this limited number of lanes remains larger than the class 2 speed or :

−−

−−=≤1

..

1 11

112 jm

kmQkmjmuu h

e (4.56)

• If the computed speed of class 1 at a fraction α1 = (m-j-1)/m were to be smaller than u2, then the traffic flow acquires a homogeneous speed u2 and traffic becomes mixed.

In the interaction model according to the user-optimum the model can, consequently, be easily adjusted to take the number of lanes into account. In this adjustment, slow vehicles are still not influenced by fast ones. The anisotropic condition remains valid.

When these lane corrections are added, the dynamics of traffic flow are expected to increase. The question then arises as to how rigorous the homogeneous fundamental diagrams are then still adhered to. The assumption concerned becomes more critical.

4.3 Homogeneous fundamental diagrams In the previous section we looked at various assumptions in regard to the way in which the various classes in a heterogeneous LWR model interact. Per class a homogeneous fundamental diagram applied to traffic flow that contained only vehicles from that class. Almost all interaction models assumed that this homogeneous fundamental diagram for a class remains valid in a mixed traffic flow.

Because a separate homogeneous fundamental diagram is necessary for each class, heterogeneous LWR models require more input. Some things, however, can already be stated about these diagrams.

A free-flow branch of the fundamental diagram usually curves with increasing density. This decrease in average speed is caused by an increased interaction between faster and slower vehicles. When homogeneous fundamental diagrams are drafted, this interaction does not take place. Vehicles in a class do not interact with one another and the way in which vehicles behave in regard to other types is described in the interaction model. This argument fully applies to the slowest class. Due to the anisotropy during free-flow, these vehicles are never impeded by other vehicles. A straight free-flow branch for this class is, therefore, a sound assumption. A straight free-flow branch can also be assumed for faster classes. Interactions with other classes should not be included in this fundamental diagram. These assumptions are even more valid when the vehicles belonging to a class are of a homogeneous nature.

Chapter 4 50

Distinguishing several classes leads to a better approach through this straight free-flow branch, but it requires more input.

The assumptions about the fundamental diagram are more difficult during the congestion regime. What is certain is that the jam densities between the classes can differ. The difference in length between freight- and private cars alone is a good argument in its favour. Opinions differ regarding the shape of the stationary congestion branch. A straight branch will be assumed in the following model development. This approach does not complicate the model unnecessarily and puts no large constraints on the applicability of the model.

In view of the limited amount of input that is required, the use of triangular diagrams with which to describe classes, is justified. The required input per class is confined to three parameters : speed ufi during the free-flow regime, the slope wi of the congestion branch and the jam density kJi.

4.4 Choice of approach The remaining development of a heterogeneous LWR model confines itself to two classes. This renders the model more readable and verification of the assumptions in the model development stage is simplified. When the outcomes of this model have been extensively verified by detailed observations, it is possible to work out an extension to more classes.

The homogeneous fundamental diagrams are assumed to be triangular. These triangular fundamental diagrams are also used as a basis in empirical studies (Kockelman 2000). This can be justified particularly well during the free-flow regime. More complex assumptions regarding the congestion branch seem premature at this stage of model development. The use of these diagrams, moreover, simplifies the analytical development of the model and considerably decreases the input required.

The user-optimum model is assumed to describe the way in which vehicles interact on a link. This seems to be a promising assumption for the description of light free-flow traffic. It also leads to a realistic FIFO approach during congestion at equal speeds. This approach satisfies the anisotropic condition and vehicle following behaviour develops according to its own homogeneous fundamental diagram. Thus empirical data regarding asymmetric following behaviour is taken into account (Dijker et al. 1997).

Extensions to a model that explicitly take account of the number of lanes is possible in a later stage.

In the following development, therefore, a heterogeneous LWR model is formulated in which two classes, each of which complies with a arbitrary triangular k-q fundamental diagram, interact according to the user-optimum.


51

Figure 4.2 : Construction heterogeneous LWR model

The construction of this model is carried out in 5 steps as shown in figure 4.2.

1. In a first step a transformation for homogeneous fundamental diagrams is developed. This model starts from two geometrically similar fundamental diagrams and an equal speed assumption. The use of a scale factor turns this heterogeneous model into a user-class model. The scale factors are, moreover, related to the passenger-car equivalents (pce).

2. A heterogeneous free-flow model is drawn up in a second step. Here the assumption is that only the free-flow branches in the homogeneous diagrams of the classes differ. Behaviour during congestion is equal for both classes. This user-optimum during free-flow is worked out in chapter 6.

3. The third step, presented in chapter 7, focuses on heterogeneous congested traffic only. Then, the free-flow branches of the homogeneous fundamental diagrams coincide, and the equal speed assumptions during congestion are dealt with.

4. In chapter 8, the previous three steps are combined to form a comprehensive heterogeneous LWR model. Assuming two different triangular fundamental diagrams, the heterogeneous traffic flow is described when the classes comply to the user-optimum principle.

5. In a fifth step, discussed in chapter 9, the heterogeneous model is further extended to a network application. Additional assumptions now become necessary to enable an accurate description of behaviour at nodes.

These five steps are discussed in the next five chapters of this thesis. The first four steps each consist of an analytical development, the formulation of a matching practical numerical scheme and an illustration of the model using a small case study.

step 1


Chapter 5

step 2 Heterogeneous free-flow

LWR model

Chapter 6

step 3

Heterogeneous congested LWR model

Chapter 7

step 4 The heterogeneous LWR

model

Chapter 8

step 5

Network traffic

Chapter 9

Chapter 4 52

53

5 TRANSFORMATION OF THE

FUNDAMENTAL DIAGRAM

In this chapter the LWR model is extended for two classes that interact according to the equal speed assumption. Central to this assumption is that both classes can be described by similar homogeneous fundamental diagrams. This enables a transformation of a class in which the scale factor is related to the passenger-car equivalents (pce). In order to show the generality of this method we use a general fundamental diagram.

In the following section we examine the analytical framework in which the heterogeneous flow will be modeled. We then develop a numerical scheme that is based on the existing LWR scheme. Next, we examine an analytical and numerical case study. We conclude with a number of observations that illustrate the relation with the much used passenger-car equivalents.

5.1 Analytic framework In this section we extend the LWR model for a link with constant road characteristics to several classes of vehicles and/or drivers. The remainder of the discussion is confined itself to two classes only, without loss of generality. Its homogeneous fundamental diagram )( i

heii kQq = describes the characteristic property of each class.

Superscript h in the formula indicates the homogeneous flow of vehicles: only vehicles from the class under consideration travel this link.

Chapter 5 54

For the two classes of vehicles we assume fundamental diagrams of similar shape, though they need not share the same capacities and maximal densities. Figure 5.1 shows two such diagrams for the two vehicle classes. Since the shape of both diagrams is identical, the fundamental diagram for class 2 vehicles can be seen as a scaled version of the class 1 diagram:

)(.)( 21 rkQrkQ h

ehe = (5.1)

Figure 5.1 : Two similarly shaped fundamental diagrams for two vehicle classes

The scaling factor r denotes the proportionality between the two diagrams. This also determines the relation between a number of special points:

21

21

21

21

...

ff

JJ

MM

MM

uukrkqrqkrk

====

(5.2)

Knowing the scaling factor r, we can describe the homogenous class 2 flow using the class 1 diagram as follows:

rkrQkQq

heh

e).()( 21

222 == or ).(. 212 krQqr he=

(5.3)

This last formula leads us to the introduction of a transformation. Consider a new vehicle class 2* that relates to class 2 as follows:

2*2 .qrq = (5.4)

2*2 .krk = (5.5)

k

q

kM1 kJ2 kJ1

qM1

qM2

kM2

uf

)(1 kQ he

)(2 kQ he


55

Speeds remain insensitive to this transformation:

22

2*2

*2*

2 .. ukrqr

kqu ===

(5.6)

The same fundamental diagram of class 1 applies to this new class:

)( *21

*2 kQq h

e= (5.7)

The scale-factor r can also be interpreted as a passenger-car equivalent (pce). As is the case with pce factors, a vehicle class is transformed to the properties of another representative class. The value of r and the pce always indicate the relation between the capacity of one class in regard to the reference-class.

As we proceed with the derivation of the model, the heterogeneous model is converted to a model for a traffic flow with a number of classes, each of which corresponds to the same fundamental diagram. When we accept that the mixed traffic flow also adheres to this fundamental diagram, the heterogeneous traffic flow can be described as follows:

0)( *

*

*1

*

=∂

∂+∂

∂x

kdk

kdQt

k tot

tot

tothetot

(5.8)

Here the total transformed density of the mixed traffic flow equals:

21*21

* .krkkkktot +=+= (5.9)

Analogously the intensity and average speed become:

21*21

* .qrqqqqtot +=+= (5.10)

21

21*

**

.

.krkqrq

kqu

tot

tottot +

+== (5.11)

We also assume that the speed of both classes is identical in all cases. An equal speed per class, therefore, also applies to a mixed traffic flow. Fraction αi of the total road space that is allocated to each class in this case is achieved using expression (4.42). Using the scaled variables, this gives the following expression for the fraction of class 1 :

−

−=

1

*2

1*2

1

1

11

1

1

1.1.

αα

αα kQ

kkQ

khe

he

(5.12)

The fractions are, therefore, arrived at by :

*1

*1

1tottot q

qkk ==α and *

*2

*

*2

2tottot q

qkk ==α

(5.13)

Class 1 and the transformed class 2* consequently behave like two classes in a user-class problem. The composition of the traffic flow, therefore, is unable to change faster than the speed of the vehicles. This amounts to a first in first out (FIFO) rule.

Chapter 5 56

Thus, traffic composition, in transformed variables, does not affect the course of the entire traffic flow.

5.2 Numerical scheme In a numeric scheme, the given partial differential equations find a practical application. Starting from initial- and boundary conditions we compute an approximate solution to the heterogeneous LWR model using similarly shaped fundamental diagrams. The scheme is based on the Godunov scheme as reinterpreted for the LWR model by Daganzo (1995b) and Lebacque (1996). In doing so, the freeway to be modeled is divided into cells with a length of x∆ in which the values of the traffic variables are assumed to be homogenous. The change of these variables is calculated per time interval t∆ . Cell length and time unit are adjusted to each other, in such a way that:

futx =

∆∆

(5.14)

The approximating density K of class i in the cell bordered by 2xx ∆− and 2xx ∆+ is calculated at time tt ∆+ as:

[ ])21,()2

1,(),(),( xxtGxxtGxtxtKxttK iiii ∆+−∆−

∆∆+=∆+

(5.15)

Gi represents the intensities at the cell-transitions. From time t to t+∆t this equals )21,( xxtGi ∆− along the upstream cell boundary. Taking the given transformation

into account, these transition intensities per vehicle class are further deduced.

In computing these transitional flows, we look at two cells using a simplified notation as shown in figure 5.2. The upstream cell is indicated by superscript U and the downstream one by D. In the upstream cell the fundamental diagram )(1 kQ hU

e applies to the first class while a scaled version of this diagram with scaling factor rU applies to the second class. The diagram )(1 kQ hD

e and the scaling factor rD apply downstream from the cell transition. This numeric scheme explicitly addresses changing freeway characteristics. Both the shape of the fundamental diagram and the class 2 scale factor in the up- and downstream cells can vary.

Figure 5.2 : Upstream and downstream cell in the numeric scheme

U UiK

Ur )(kQhU

ei US

DDiK Dr

)(kQ hDei

DR

iG


57

The Godunov scheme calculates the transitional flow as a minimum between the sending flow from the upstream cell and the receiving flow in the downstream cell. This sending flow S reflects the volume of traffic that can leave a cell. This equals the flow in the cell under consideration during free-flow, the capacity value applies during congestion. The receiving flow reflects the flow that can enter a cell. During free-flow the capacity applies here, during congestion it equals the flow in the cell:

M

M

e

M

M

M

M

e

kkkk

kQq

kR

kkkk

qkQ

kS

≥∀<∀

=

≥∀<∀

=

)()(

)()(

(5.16)

From this point in the text onward SU denotes the sending flow of the fundamental diagram of the upstream cell. RD denotes the receiving flow of the downstream cell.

Keeping the transformation of class 2 in mind, the total scaled sending flow of the upstream cell must equal ).( 21

UUUUtot KrKS + . The downstream cell's total scaled

receiving flow takes account of the downstream scale factor and equals ).( 21

DDDDtot KrKR + . Since the up- and downstream cells incorporate other scale

factors, one can not simply compute total transition flow as a minimum of UtotS and

DtotR . If we want to compare these flows, the sending flow must be transformed from

the upstream to the downstream scale factor. To do this, the sending flow is first subdivided by vehicle class:

UtotUUU

UU S

KrKKS

21

11 .+

= (5.17)

UtotUUU

UUU S

KrKKrS

21

22 .

.+

= (5.18)

The values of these sending flows from the upstream cell, that have however been scaled according to the downstream transformation, are denoted by superscript D and they are equal to:

UtotUUU

UUD S

KrKKSS

21

111 .+

== (5.19)

UtotUUU

UDU

U

DD S

KrKKrS

rrS

21

222 .

.+

== (5.20)

The total downstream scaled sending flow can be compared to the receiving flow DtotR

and equals:

UtotUUU

UDUDDD

tot SKrKKrKSSS .

21

2121 +

+=+= (5.21)

Chapter 5 58

Since the sending and receiving flows are now known in the same classes, the total transition flow G in the downward scaled variables can be calculated as:

[ ]

++== D

totUtotUUU

UDUDtot

Dtot

Dtot RS

KrKKrKRSG ,.min,min

21

21 (5.22)

This total transition flow can be separated by class according to the contribution of each class in the sending flow. For the moment we stay with the downstream scaling so that they equal:

DtotD

tot

DiD

i GSSG =

(5.23)

Re-scaling to the original classes happens when:

DtotD

tot

DiD G

SSGG == 11

(5.24)

DtotD

totD

Di

D

D

GSr

SrGG

.2

2 == (5.25)

Further computation, by substituting (5.19), (5.21) and (5.22) in (5.24) and (5.20), (5.21) and (5.22) in (5.25) gives the general formulation for transition flows per class:

+

++

+= ).(

).(),.(

).(min 21

2121

21

DDDDUDU

UiUUUU

UUU

Ui

i KrKRKrK

KKrKSKrK

KG (5.26)

In combination with basic equation (5.15) a complete numeric scheme has been formulated that can be used to compute a link's dynamic traffic evolution.

When a vehicle stream is described by several scaled classes, the transition flow per class equals:

= ∑∑

∑∑ =

=

=

=

).(.).(

),.(..

min1

1

1

1

n

j

Dj

Dj

Dn

j

Uj

Dj

Ui

n

j

Uj

Uj

Un

j

Uj

Uj

Ui

i KrRKr

KKrSKr

KG

(5.27)

in which the scaling factors for the first reference class equal 111 == DU rr .

When the freeway characteristics along a freeway remain unchanged, the numeric scheme for two vehicle classes can be simplified. In this case the up- and downstream scaling factors are equal so that:

[ ] *

212121

21

.).(

).(),.(min.).( totUU

UiDDDUUU

UU

Ui

i GKrK

KKrKRKrKSKrK

KG+

=+++

= (5.28)

The total scaled transition flow is defined by:

[ ]).(),.(min)( 2121* DDDUUUtot KrKRKrKStG ++= (5.29)


59

Dividing this total transition flow over the classes is only carried out in a subsequent step. This enables the inclusion of vehicle delay-periods in the calculation of transition flow per class. Daganzo (1995a) formulated a rule of delay that takes account of the length of delays in the composition of the transition flow. This rule of delay decreases the dispersion of the numeric scheme when the composition of the traffic flow alters.

To this end, new densities and transition flows are defined as a function of the delay-period.

The amount of traffic per class i in the upstream cell with a delay τ at time t is written as: )(. tKx U

iτ∆ . This gives the number of class i vehicles that entered this cell immediately following on time interval (t-τ).

Analogously, the required transition flows )(tGi can be divided according to the delay-periods τ of which the traffic is composed )(tGiτ .

The following updating rules apply to this extended notation:

∑∆=∆=∆+∆ ∆ )(.)(.)(. tGttGtttKx iiD

ti τ (5.30)

Traffic that enters a cell receives delay-period ∆t at the next time-unit.

)(.)()()( tGxttKttK i

Ui

Uti τττ ∆

∆−=∆+∆+ (5.31)

Traffic remaining in the cell increases its delay-period with the time interval ∆t.

In order to respect the FIFO rule, delays are recorded. The computed total transition flow )(* tGtot is then divided according to class and composed by traffic with the longest delays. Formulating transition flows per class is done as follows:

)(.)( tKtxftG U

ii τττ ∆∆=

(5.32)

Calculating factor τf starts at the longest delay τmax in the upstream cell. Using diminishing delay-periods all factors per cell are subsequently calculated.

When the total transition flow has not yet been achieved, that section of traffic that has experienced the longest delay is sent on to the next cell. Mathematically this means that:

1=τf as long as *21

max

2

max

1 )(.)()(.)( totUU

tpp

tpp GtKrtK

txtGrtG ≤+

∆∆++ ∑∑

∆+=∆+=ττ

τ

τ

τ

τ

(5.33)

If this equation no longer applies it means that not all of the traffic experiencing this length of delay can be sent on to the next cell. τf is then calculated as follows:

Chapter 5 60

( ))(.)(

)(.)(

21

max

2

max

1*

tKrtK

tGrtGG

xtf UU

tpp

tpptot

ττ

τ

τ

τ

ττ +

+−

∆∆=

∑∑∆+=∆+=

(5.34)

0=τf applies to all traffic experiencing a shorter delay.

In conclusion, this gives:

∑∑ ∆∆== )(.)()( tK

txftGtG U

iii τττ (5.35)

where the factor τf for each diminishing delay τ is calculated as follows:

( )

+

+−

∆∆=

∑∑∆+=∆+=

)(.)(

)(.)(

,0max,0min21

max

2

max

1*

tKrtK

tGrtGG

xtf UU

tpp

tpptot

ττ

τ

τ

τ

ττ

(5.36)

The transition flows by class are composed, therefore, of the longest waiting traffic from the upstream cell and meet the following criterion:

)()(.)( *21 tGtGrtG tot=+ (5.37)

5.3 Analytical and numerical case study In this section we develop an analytical solution to a freeway traffic problem and also present a numerical analysis.

Figure 5.3 : Schematic representation of the case-study.

The case study considers a freeway without access and exit roads as represented in figure 5.3 . The freeway is divided into three sections. In the first section, where x<x1, the fundamental diagram )(1 kQh

e for class 1 vehicles applies. Class 2 vehicles behave

x1

x2


61

according to a scaled version of this k-q fundamental diagram. The scaling factor in this first section is rA = 2.

In a second section, where x1<x<x2, the same fundamental diagram )(1 kQhe applies to

the class 1 vehicles. The scaling factor for the scaled diagram that describes class 2 vehicles, is rB = 4.

The third section, where x > x2, shares the properties of the first section. The fundamental diagram )(1 kQ h

e applies to class 1 and the scaling factor for class 2 is rA = 2.

This theoretical case study can be seen as a freeway carrying passenger and goods traffic, and where the freeway inclines sharply at the level of the second section.

An empty freeway is assumed as the initial condition:

0),( 0 =xtk 0xx >∀

The upstream traffic demand starts at t0 with an equal inflow of class 1 and class 2 vehicles:

10 41),( Mi qxtq = 10 ttt <<∀

Figure 5.4 : The fundamental diagram used in the analytical and numerical case study.

At time t1 traffic composition changes to a homogenous class 1 traffic flow.

11 21),( Mqxtq = ; 0),(2 =xtq 1tt ≥∀

q qM1

1*

43

Mtot qq =

1*

53

Mtot qq =

11 21

Mqq =

kJ1 kM1 k

)(1 kQhe

)(.)( 12 Ahe

AhAe r

kQrkQ =

)(2 kQhBe

Chapter 5 62

Total traffic demand, therefore, does not change at t1 , only the composition of the traffic changes.

Figure 5.4 depicts the fundamental diagram, figure 5.5 gives the analytic computation of this problem. The characteristics, the constant flow solution lines, are indicated by a thin line, shock waves by a bold line.

Figure 5.5 : Analytical solution of the case study in the t-x diagram.

The construction of the solution starts from the origin with a fan between the empty traffic state and the state where 121

*

43. M

Atot qqrqq =+= . This fan is followed by

kinematic waves that arise from the boundary condition and travel to t1. A segregation trajectory beginning at t1 indicates the end of the class 2 vehicles. A shock wave starting at the same point reflects the break with the new traffic demand:

11*

21

Mtot qqq == . Although total traffic demand remains the same, a shock wave

does occur. This happens, of course, because the total scaled traffic demand actually does differ.

Traffic demand in the first period 0<t<t1, is too large to be processed by the second section. This is due to the fact that the transformed total flow with scaling factor rB is

121*

45. M

Btot qqrqq =+= . This causes a regressive shock wave during the opening

fan in the first section. Those characteristics that formerly flowed into the second section, show smaller slopes in this second section.

t

x

x1

x2

x0 t0 t1

Legend Shock wave Characteristic Trajectory


63

The congestion zone in the first section grows up to the shock wave of the changing traffic demand. At this point a forward shock wave diminishes the length of the congestion.

When the segregating trajectory reaches the second section, the upstream bottleneck for section 1 disappears. This gives rise to a fan that dissolves the congestion in the first section thus increasing the speed of the closing shock wave.

The trajectory that marks the changing composition of traffic demand causes a new fan when it arrives in the third section. Before this trajectory reaches the third section, the speed of the characteristics increases at the border of section 2 and 3. Because the class 2 vehicles in the traffic flow suddenly assume a smaller scale factor, it is as if the capacity of the total traffic flow increases at that point. Subsequent to this segregating trajectory q2 equals zero. This is why the same fundamental diagram applies to section 2 and 3 from that point.

Figure 5.6 : Numerical solution of the case study in the t-x diagram.

Figure 5.6 shows the t-x diagram of the numeric simulation. The figure gives the total flow qtot (= q1 + q2) using a colour scale. Black equals zero intensity, white indicates capacity flow qM1.

In this numeric method the t-x solution space is subdivided into cells. For each time interval ∆t, density according to (5.15) was computed for each cell with a length of ∆x. When computing transition flows, the delay rule was applied for constant road properties. This dispersion correction was not used for the section transitions at x1 and x2 .

Legend

Total flow

t

x

x2

x1

x0 t1 t0

21 qqqtot +=

1Mq

0

Chapter 5 64

The similarities between the numeric calculation in figure 5.6 and the analytic solution in figure 5.5 are clearly visible.

5.4 Discussion This section takes a closer look at the significance of the scale factor in the presented heterogeneous model. The flow of vehicles was divided into classes. All vehicles inside a class are assumed to share the same properties. Scale factor r denotes how vehicles from various classes differ. It corresponds to the relation between the 'jam' densities and the capacities. The space occupied by a vehicle of a particular class at speed u is indicated by si(u). Since spacing is the inverse of density, the proportionality factor between the average spaces is also related to the scale factor.

)()(

1

2

2

1

2

1

usus

kk

kkr

J

J

M

M === (5.38)

Using scale factors to divide traffic flow into various classes enables us to highlight the following heterogeneous characteristics:

• Differences in vehicle characteristics, e.g. the physical length of vehicles

• Differences in driver characteristics, e.g. calm and aggressive drivers

• Differences in road characteristics that affect the capacity for particular types of vehicles, e.g. a tunnel.

Conversely, this extended LWR model cannot describe the following characteristics:

• Different maximum speed limits. This is relevant both in terms of vehicle characteristics (e.g. heavy vehicles versus private cars), road characteristics (e.g. grades) and driver characteristics (e.g. fast drivers)

• Differences in speed. A homogeneous traffic speed is assumed. This is why overtaking and multilane highways are insufficiently modeled.

• Differences in acceleration and reaction time: The LWR model is an equilibrium model where acceleration is assumed to be infinite and reaction time is zero.

If we take passenger-cars as a class of reference, then the scaling factor equals the generally used passenger-car equivalents (pce). As in the Highway Capacity Manual 2000, pce includes differences in vehicle characteristics (e.g. freight), driver characteristics (e.g. commuters vs. recreational drivers) and road characteristics (e.g. grade, multilane, ..). To do this, an equivalent is determined on an hourly basis and this also takes account of differences in (maximum) speed, acceleration and reaction time. Averaging the heterogeneous effects on an hourly basis, enables the inclusion of differences that could never otherwise be described dynamically by that kind of factor. Using the known pce values in the heterogeneous LWR model gives us a readily applicable heterogeneous LWR model that can also indirectly address speed differences.


65

This heterogeneous model can be simplified to the classic LWR model when the vehicle composition on a link remains unaltered during the modeling period. We then use a weighted fundamental diagram. Assume that the vehicle composition is characterised by the relative class 1 density :

21

1

21

1

qqq

kkk

+=

+=β

(5.39)

Since vehicle composition does not change, the fraction taken by each class on the road is also fixed. This is for class 1 :

)1.(1 βββα

−+=

r

(5.40)

The class 1 flow can, therefore, be expressed in function of the total density, the composition, the scale factor using the homogeneous fundamental diagram of class 1 :

( )tothe krQ

rq )).1.((

)1.( 11 ββββ

β −+−+

= (5.41)

Therefore, we can express the weighted fundamental diagram that must be used with a constant vehicle composition β , as :

( ))1.()).1.(()( 1

ββββ

β −+−+==

rkrQkQq tot

he

totetot (5.42)

When the scale factor r remains the same across the entire network, the homogeneous LWR model can also be applied. Traffic demand can then be scaled to the reference class, after which the LWR model with the homogeneous class 1 fundamental diagram can be applied.

The surplus value of this scale model, therefore, is only revealed in a network with changing vehicle composition and with pce factors that change across the network.

Besides the applicable model, this chapter offers the idea of a scaling of fundamental diagrams. Scaling of fundamental diagrams will also be applied in the final heterogeneous LWR model that is formulated in chapter 8.

Chapter 5 66

67

6 HETEROGENEOUS

FREE-FLOW TRAFFIC

This chapter presents a second step in the extension of the LWR model to a comprehensive heterogeneous traffic flow model. Having derived the scaling of fundamental diagrams in the previous chapter, we now look at the various forms of driver behaviour in the free-flow regime.

The traffic flow is divided into two classes yet again. A triangular fundamental diagram applies for each class. It is assumed that each vehicle, also those in a heterogeneous traffic flow, continue to behave according to this homogeneous fundamental diagram. In this chapter we focus on the difference in driving behaviour of vehicles during the free-flow regime. To that end, we assume identical driving behaviour for both classes during the congestion regime. This implies coinciding congestion branches and jam densities in the fundamental diagrams for both classes.

When modeling the interactions between the vehicles belonging to different classes, a user-optimum is assumed. This means that the vehicles of the classes spread themselves across the total road space in such a way that a vehicle increasing its speed necessarily reduces the speed of slower vehicles. We also assume that vehicles limit their use of the road space to that which is strictly required.

These assumptions run parallel to the non-co-operative Nash equilibrium (Nash 1950) known from game theory. Here, each road-user optimises his own speed and does not take other drivers into account. These assumptions appear to be realistic for a heterogeneous model. This aspect will be examined more closely in chapter 11.

Chapter 6 68

In this chapter we only examine traffic on a link with homogeneous road characteristics. Inhomogeneous links and nodes are examined in chapter 9, subsequent to the presentation of the general heterogeneous LWR model.

Based on the assumptions, we begin by clarifying the various possible stationary traffic states. We then compute the transitions between all possible stationary states. To gain a comprehensive picture of the model these transitions will be qualitatively described.

This analytical description will be followed by the formulation of a numerical scheme. This scheme enables a rapid approximate computer computation of the analytical solution.

Newell's moving bottleneck problem (1998) is included in a case study that will be computed both analytically and numerically. Here the original problem definition will be translated to an interaction between two vehicle classes with different speeds.

6.1 Stationary traffic states In this chapter we look at two vehicle classes which differ only in their respective maximum speed. Each class is characterised by a homogeneous triangular fundamental diagram, as illustrated in figure 6.1

Both classes are assumed to share the same jam density kJ and the same angle of the congestion branch w. Only maximum speed, and hence the free flow branch of the fundamental diagrams differ. If we indicate the fastest class by index 1, the capacity point of the 'slow' vehicle class 2 (kM2,qM2) lies on the congestion branch of class 1.

We can express the diagrams using the following function :

MiJ

Mifihei kkkkw

kkukkQ

>−≤

=).(

.)(

(6.1)

Figure 6.1 : Two homogeneous fundamental diagrams that apply to two vehicle classes.

As pointed out in chapter 4, a heterogeneously composed traffic flow can be described using the general equilibrium relation (4.12) :

)(2 kQ he

k

q

kM2 kJ kM1

qM2

qM1

uf2 uf1

w

M1

M2

)(1 kQhe


69

=

i

iheiii

kQq

αα .

(6.2)

Fraction αi, here, is that part of total road space used by class i. When the road is entirely used, the following relation applies between the fractions :

12 1 αα −= (6.3)

This equilibrium relation (6.2) can only be used if it is assumed that the vehicles, also in a mixed traffic flow, stick to the homogeneous fundamental diagrams.

Assigning the fractions to the different classes illustrates the assumed manner of interaction between the classes. We assume a user-optimum for the remaining computation. This implies that all vehicles increase their speed to the point where an increase in speed of one single vehicle leads to a decrease in the speed of slower driving vehicles. It is, in addition, assumed that none of the classes occupies more space then is strictly necessary to maintain its speed.

We examine the significance of these assumptions for the specific fundamental diagrams using a k1-k2 phase-diagram as put forward in figure 6.2. This phase diagram shows the possible combinations of class densities. The following conditions apply :

0≥ik (6.4)

Jtot kk ≤ (6.5)

This shows up as a triangular region of possible densities in the phase-diagram. We can indicate regimes or phases in this phase-diagram. We define a regime as a collection of traffic states with common characteristics. We distinguish three regimes: free-flow (A), semi-congestion (B) and congestion (C). For each regime, the properties and boudaries are expounded in the phase-diagram.

Figure 6.2 : The phase-diagram of the heterogeneous vehicle flow.

k1

k2

kM2 kJ

kJ

kM2

kM1

C

A B

Chapter 6 70

Regime A : Free-flow Traffic states where both vehicle classes can maintain their maximum speed are classified under the free-flow regime. From (6.2) we know the speed of a class i:

=

i

ihei

i

ii

kQ

ku

αα

. (6.6)

This speed equals the free-flow speed ufi subject to :

Mii

i kk

≤α

(6.7)

Allowing for (6.3) this leads to the following condition for the free-flow regime :

12

2

1

1 ≤+MM kk

kk

(6.8)

In figure (6.2) this regime is shown graphically on the phase-diagram. Flow per vehicle class is then given by :

fiii ukq .= (6.9)

The vehicles of the different classes behave completely independent of each other and can overtake without problem. Both classes are operated on a parallel basis. Because all the vehicles maintain their maximum speed, the user-optimum has been met.

Figure (6.3) illustrates a traffic state from the free-flow regime. For each class, the fundamental diagrams are reduced by the fractions. For each class, the state points lie on the free-flow branches of the diagrams. The total traffic state is the vector addition of both classes.

In the k-q diagram all the possible total traffic states that belong to the free-flow regime lie in the grey zone.

Figure 6.3 : The free-flow regime in the k-q diagram

222. αα kQh

e

k

q M1

α1.qM1

qtot

q2 = α2..qM2

q1

M2

111. αα kQh

e


71

Regime B : Semi-congestion In the semi-congestion regime, class 2 vehicles behave as they do in the free-flow regime. The speed of these vehicles equals uf2. In this regime, class 1 vehicles are congested : their speed is less than uf, but exceeds or equals the speed uf2 of class 2. The term semi-congestion was taken from Daganzo (2002a). It indicates that intrinsically slow vehicles can function in a free-flow regime, although faster vehicles are already experiencing congestion.

Road space in this regime is also optimally divided over both classes. The class 2 speed is at a maximum when fraction α2 is restricted by

2

22

Mkk

≥α (6.10)

When the equal sign applies, we get the smallest possible fraction for class 2 for which the speed is maximal. In that case class 2 lies in the capacity point of the homogeneous fundamental diagram. This minimal fraction for class 2, gives the largest possible fraction for class 1 :

2

221

M

M

kkk −

=α (6.11)

The speed of class 1 inside the semi-congestion regime lies between uf1 and uf2. Substituting the class 1 fraction in (6.2) gives the following expression for the speed of vehicle class 1 :

−−=

−=

=

2

22

11

1

1

1

1

11

1

11 .1....

M

MJJ

he k

kkkk

wkk

wk

kQ

ku

αα

αα

(6.12)

This expression is only valid if :

11

1Mk

k>

α

(6.13)

Substitution of (6.3) in this condition leads to an expression that compliments the boundary condition of the free-flow regime :

12

2

1

1 >+MM kk

kk

(6.14)

The class 1 speed was assumed to exceed uf2. Assuming (6.12), this gives :

22

22

11 .1. f

M

MJ uk

kkkk

wu ≥

−−=

(6.15)

Further computation leads to a second condition that defines the semi-congestion regime :

221 Mkkk ≤+ (6.16)

Chapter 6 72

In figure 6.2 this regime is indicated with the index B in the phases-diagram. In figure 6.4 the semi-congestion regime is shown in a k-q diagram. The class 2 vehicles are located on the free-flow branch. Class 1 is described by a fraction α1 of the original diagram. The traffic state lies on the congestion branch of this reduced fundamental diagram.

Once again, the total traffic state is the vector addition of both classes. All possible traffic states in the semi-congestion regime lie on the congestion branch between the two capacity points M1 and M2.

Figure 6.4 : The semi-congestion regime in the k-q diagram

The semi-congestion regime complies with the user-optimum. The slowest class 2 vehicles are able to maintain their maximum speed, while the class 1 vehicles can maximise their speed without influencing the class 2 vehicles.

Regime C : Congestion In the third regime, the speed of both vehicle classes is less than the free-flow speed of the slowest class. In that case the speed of both classes is identical and the congestion branch of the fundamental diagram applies. Therefore, both classes comply with the same fundamental diagram at homogeneous speeds. The heterogeneous vehicle classes in this regime are converted into user-classes. Class fractions are, therefore, proportional to relative density :

tot

ii k

k=α

(6.17)

The speed of the traffic flow will be homogeneous and total flow complies with the fundamental diagram :

).()( Jtottotheitot kkwkQq −== (6.18)

The congestion regime in the phase-diagram of figure 6.2 is indicated with C.

When the traffic state becomes congested, the user-equilibrium always leads to homogeneous speeds. When the congestion branches coincide, the classes behave as user-classes. At that stage, traffic composition is of no importance anymore.

222. αα kQh

e

k

q M1

α1.qM1

qtot

q2 = α2..qM2

q1

M2

111. αα kQh

e


73

Summary The boundaries of the three regimes are summarised in table 6.1. Based on the two class densities, the regime of the stationary traffic state can be determined.

Table 6.1 : Summary stationary traffic regimes Regimes Regime boundaries Speed Free-flow 0≥ik

12

2

1

1 ≤+MM kk

kk

fii uu =

Semi-congestion 0≥ik

12

2

1

1 >+MM kk

kk

221 Mkkk ≤+

−−=

2

22

11 .1.

M

MJ

kkk

kk

wu

22 fuu =

Congestion 0≥ik

221 Mkkk >+

Jtot kk ≤ tot

Jtoti k

kkwu ).( −=

6.2 Transitions Among the strong points in the LWR model is the ability to compute the solution entirely analytically, using characteristics, also called kinematic waves. This option is retained when extending the LWR model to two vehicle classes. This enables detachment of the numerical scheme, that is used for the implementation of the model, from the intrinsic analytical properties of the model.

This section gives a qualitative summary of all the possible Riemann problems. In a Riemann problem we consider an infinite road, with, as initial condition, two traffic states. Upstream, one traffic state applies for x < x0. At x0 this changes discontinuously to the second downstream traffic state. By discussing all possible combinations of traffic states, we describe the various transitions. This gives a qualitative summary of this new model.

A phase-diagram, a fundamental diagram and a t-x diagram is depicted for each Riemann problem. Here, the t-x diagram is scaled so that straight lines with the same angle in the k-q fundamental diagram and in the t-x diagram correspond with equal speed. This allows for further application of the graphic power of the LWR model.

The various traffic states are indicated with a letter that corresponds to the regime to which the traffic state belongs. The index A indicates traffic states in the free-flow regime, B indicates the semi-congestion regime and C represents congestion. Superscript U represents the upstream state and D the downstream state. Intermediate states appearing in the solution are allocated indices G or H. Those traffic states that occur exclusively in the phase-diagram and that do not appear in the t-x diagram are indicated with I.

Chapter 6 74

In the t-x diagram, shock waves are represented by a double line. Slips, shock waves with the speed of the vehicles, are indicated with a bold line. Trajectories are shown as arrows. In the t-x diagram as well as the k-q diagram, the properties of the slow and fast classes are shown in respectively light and dark grey.

The various transitions will show that the basic principles of the LWR model remain valid. There is a conservation of vehicles, also for each class, and no vehicle can be influenced by upstream vehicles. The speed of class 2 vehicles, furthermore, is always less than or equal to that of the class 1 vehicles.

All transitions are separated from one another by shock waves, slips and fans. Each class complies, hereby, with the stationary fixed traffic states. Higher order models based on this heterogeneous model, whereby deviation from the stationary traffic state is temporarily possible, can be developed from this stationary model. They are, however, not discussed here.

We will first look at transitions within one regime, then at inter-regime transitions.

Intra-regime transitions Intra-regime transitions encompass the Riemann problems with up- and downstream a traffic state belonging to the same regime. It nevertheless appears that some of these Riemann problems can lead to intermediate traffic states from other regimes.

• Transition from free-flow to free-flow (A⇒⇒⇒⇒A)

In this first transition we look at two free-flow traffic states, upstream AU and downstream AD as in figure 6.5. The transition between both traffic states occurs in two stages. First, a slip with speed uf2 separates the upstream and downstream initial conditions of the class 2 vehicles. A second slip with speed uf1 then separates the up- and downstream characteristics of the class 1 vehicles.

Due to the separate shock waves for each vehicle class a zone appears in the t-x diagram in which the upstream characteristics of the class 1 vehicles apply alongside the downstream class 2 characteristics. This new traffic state AG can also be found in the phase-diagram. The density of the class 1 vehicles remains constant during the transitions from AU to AG. This leads to a vertical in the phase-diagram. This is followed by a horizontal class 1 change in the phase-diagram. This transition is valid for all Riemann problems between two free-flow regimes, as long as the new state AG is also maintained in this free-flow regime. This condition can be expressed mathematically using the free-flow boundary condition (6.8) :

12

2

1

1 ≤+M

D

M

U

kk

kk

(6.19)

Graphically, this condition requires that the point of intersection of a vertical line across the upstream initial state and the horizontal line across the downstream state lies in the free-flow regime. If this condition is not met, the semi-congested


75

regime appears. This kind of transition is further elaborated after the treatment of the inter-regime transitions between free-flow and semi-congestion regimes.

Figure 6.5 : A free-flow to free-flow transition (A⇒A) sub-case 1 (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

• Transition from semi-congestion to semi-congestion (B⇒⇒⇒⇒B)

In a second type of intra-regime transitions, we study a Riemann problem between the semi-congestion states BU and BD, as shown in figure 6.6. During semi-congestion regimes, class 2 vehicles behave as they do in a free-flow regime. The up- and downstream properties of the class 2 vehicles are constantly separated by a uf2-slip. Across such a slip the change in class 2 vehicles is compensated by a change in class 1 vehicles. The total density of the traffic flow does not change across this shock wave. In the phase-diagram, this slip is indicated by a transition line of -45°. Traffic flow composition alone changes across this uf2 shock wave, while total traffic flow remains unchanged.

This uf2-slip is preceded by a wave with congestion branch speed w. This initial shock wave influences the class 1 vehicles only, and can be indicated by a horizontal transition in the phase-diagram. The intermediate state BG consequently shows the upstream class 2 density and the downstream total density.

This intra-regime transition also has conditions attached to it. The transition from upstream to downstream is only possible when the graphic construction, a horizontal line through the upstream and a sloping -45° line through the downstream state, gives a point of intersection inside the semi-congestion regime. This leads to the following condition :

12

12122 .

MM

MDD

MU

kkkkk

kk−

−+<

(6.20)

If this condition is not met we get an intermediate free-flow state. For example, if the up- and downstream traffic state in figure 6.6 is switched we get an impossible intra-regime transition. We will elaborate on these kind of transitions subsequent to the discussion on inter-regime transitions between free-flow and semi-congestion regimes

t

x

x0

k

q

kM2 kJ kM1

k1

k2

kM2 kJ

kJ

kM2

kM1

AD

AU

AG

AG

AD

AU

AG AD

AU

Chapter 6 76

Figure 6.6 : A semi-congestion to semi-congestion transition (B⇒B) sub-case 1 (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

• Transition from congestion to congestion (C⇒⇒⇒⇒C)

A final sort of intra-regime transitions happens in the case of a congested traffic state up- and downstream. This transition, as shown in figure 6.7, is characterised by an initial shock wave with speed w. Total traffic flow alters across this wave but vehicle composition remains unaltered. In the phase-diagram this wave is shown as a line through the origin. The initial wave is followed by a slip with the downstream vehicle speed. Across this slip, vehicle composition alone alters. This part of the transition is visible in the phase-diagram along a line of constant total density, i.e. a line at an angle of -45°.

This type of Riemann problems with up- and downstream congested traffic states always remains inside the congestion regime.

Figure 6.7 : A congestion to congestion transition (C⇒C) shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

t

x

x0

k

q

kM2 kJ kM1

k1

k2

kM2 kJ

kJ

kM2

kM1

BD

BU

BG BD=BG

BU

BG

BD

BU

t

x

x0

k

q

kM2 kJ kM1

k1

k2

kM2 kJ

kJ

kM2

kM1

CD

CU CG

CD=CG

CU

CG

CU

CD


77

Inter-regime transitions This type of transitions involves a number of regimes. We begin with a summary of the transitions based on all possible regime combinations. These are applied afterwards to further develop those intra-regime transitions that collide with the regime boundaries.

• Transition from free-flow to semi-congestion (A⇒⇒⇒⇒B)

Both during free-flow and during semi-congestion, class 2 vehicles retain free-flow speed uf2. This is why a uf2-slip always separates the class 2 up- and downstream properties. Subject to the up- and downstream traffic states, we can now distinguish two sub-cases.

A first kind, shown in figure 6.8, comprises an extension of the higher intra-regime semi-congestion transition. During an initial shock-wave, the upstream traffic regime AU changes to a semi-congestion state BG. Total density in this state equals that of the downstream state BD. Traffic flow composition subsequently changes across a uf2 -slip from state BG to BD.

A condition to this transition is that the shock wave between AU and BG happens first and, consequently, has a speed less than slip-speed uf2. This means in the fundamental diagram that the upstream state AU lies above the dotted line with speed uf2 . A state-point overhead will evolve to BG across a shock wave of a speed less than uf2. This condition agrees to an upstream class 1 density Uk1 that exceeds Hk1 . Free-flow traffic states with density less than Hk1 lie below the dotted line in the fundamental diagram, a state with higher class 1 density lies above the dotted line in the fundamental diagram. Density Hk1 is the lowest class 1 density that can lead to a semi-congestion state with total density as in the downstream state. This enables a graphic calculation Hk1 in the phase-diagram. Free-flow traffic states of a class 1 density that exceeds Hk1 lead to transitions to the semi-congestion regime of this type.

This density can be expressed as follows :

21

1112211

...

MM

MD

MD

MMH

kkkkkkkk

k+

+−=

(6.21)

Upstream class 1 densities less than Hk1 comprise a second sub-case as shown in figure 6.9. Here, the transition across the uf2-slip happens first. This leads to a free traffic state AG of the upstream class 1 density and the downstream class 2 density. This transition is followed by a wave that concerns type 1 vehicles only. The speed of this second shock wave, from AG to BD, depends on the up- and downstream state, and definitely exceeds uf2.

In conclusion, we can say that the transition from free-flowing traffic to semi-congestion happens via an intermediate state. Subject to the up- and downstream states, this intermediate state belongs to the free-flow or the semi-congestion regime.

Chapter 6 78

Figure 6.8 : A free-flow to semi-congestion transition (A⇒B) sub-case 1 shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

Figure 6.9 : A free-flow to semi-congestion transition (A⇒B) sub-case 2 shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

• Transition from semi-congestion to free-flow (B⇒⇒⇒⇒A)

Again, the transition from semi-congestion to free flow, consists of two sub-cases. With this transition the class 2 vehicles again retain their free-flow states. Changes in their properties necessarily occur along a slip with speed uf2. The structure of the transition will, each time, consist of a class 1 change along a wave with speed w, followed by the class 2 slip, which is, once again, followed by a shock wave with exclusively class 1 changes.

In an initial kind of transitions, the class 1 vehicles switch in the initial shock wave with speed w to the A-B regime boundary as in figure 6.10. The class 1 vehicles increase speed until they drive at maximum speed in state BG. The free-flow-free-flow transition from figure 6.5 applies from this state to the downstream free-flow state AD. Two slips with speeds uf2 and uf1 allow the traffic state to evolve from BG via AH to AD. A condition is that the AH class 2 density is less than the BG class 2 density. Dependent on the up- and downstream starting states this leads to the following condition :

t

x

x0

k

q

kM2 kJ kM1

k1

k2

kM2 kJ

kJ

kM2

kM1

BD

AU

BG

BD=BG

AU

BG BD

AU k1

H k1H

t

x

x0

k

q

kM2 kJ kM1

k1

k2

kM2 kJ

kJ

kM2

kM1

BD

AU

AG

BD

AU AG BD

AU k1

Hk1H

AG


79

UD kk 22 ≤ (6.22)

If condition (6.22) is not met, the second sub-case applies, as in figure 6.11. Again, class 1 vehicles change their speed across an initial shock wave of speed w. However, intermediate state BG now lies fully in the semi-congestion regime. Vehicle composition alters across the subsequent uf2-slip, as it did in transition 2. Total vehicle flow remains the same, and only vehicle composition alters across this slip. Since the type 2 vehicles in state BH must equal the type 2 density of the downstream state AD, BG, which is where the first shock wave led, is now also determined. The uf1-slip from BH to AD completes the transition.

Figure 6.10 : A semi-congestion to free-flow transition (B⇒A)sub-case 1 shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

Figure 6.11 : A semi-congestion to free-flow transition (B⇒A)sub-case 2 shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

Intra-regime transitions via other transitions The inter-regime transitions between free-flow and the semi-congestion regime enable completion of the intra-regime transformations in the free-flow and in the semi-congestion regime.

t

x

x0

k

q

kM2 kJ kM1

k1

k2

kM2 kJ

kJ

kM

kM1

AD

BU

AH

AH BU BG

BG AD

BU AD

BG

AH

t

x

x0

k

q

kM2 kJ kM1

k1

k2

kM2 kJ

kJ

kM2

kM1

AD

BU

BH

BU BG=BH

BG AD BU

AD

BG

BH

Chapter 6 80

• Transition from free-flow to free-flow (A⇒⇒⇒⇒A sub-case 2)

The above showed that these types of transitions consisted of two slips as in figure 6.5. This led to an intermediate state, characterised by the upstream type 1 and the downstream type 2 states. When this intermediate state is absent in the free-flow regime, this second sub-case occurs.

We can see this type of transition as a composite from a free-flow to semi-congestion transition (sub-case 1, as in figure 6.8) and a semi-congestion to free-flow transition (sub-case 2, as in figure 6.11). Figure 6.12 clarifies the transition. The A⇒B transition belongs to the first sub-case because the class 1 density in BH in figure 6.12 exceeds the upstream class 1 density. If this were not the case, the transition would happen directly in the free-flow regime as it does in the first sub-case of this intra-regime transition. The transition from the semi-congestion to the free-flow regime happens along the second sub-case as in figure 6.11. This is so because the downstream class 2 density always exceeds the upstream density. If this were not the case, the first sub-case of the intra free-flow transition would pose no problem here either.

Thus, the shock wave pattern in the t-x diagram consists of a w shock wave for type 1 vehicles to the semi-congestion regime, a uf2-slip that alters vehicle composition alone, and a uf1-slip in the direction of the downstream state.

Figure 6.12 : A free-flow to free-flow transition (A⇒A) sub-case 2 shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

• Transition from semi-congestion to semi-congestion (B⇒⇒⇒⇒B sub-case 2)

Sometimes, transitions inside the semi-congestion regime collide on the boundary with the free-flow regime. When condition (6.20) can not be met, we get a composite from semi-congestion to a free-flow transition (sub-case 2, as in figure 6.11) and a free-flow to semi-congestion transition (sub-case 1, as in figure 6.8). Figure 6.13 clarifies this composite transition. Increased speed in the class 1 vehicles across a shock wave with speed w to BG is followed by a change in class 2 density across a uf2-slip to AH. From this free-flow state a shock wave leads to BD.

t

x

x0

k

q

kM2 kJ kM1

k1

k2

kM2 kJ

kJ

kM2

kM1

AD

AU

BH

AU BG=BH

BG

BH AD

BG

AU AD


81

Figure 6.13 : A semi-congestion to semi-congestion transition (B⇒B) sub-case 2 shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

Inter-regime transitions with the congestion regime

• Transition from semi-congestion to congestion (B⇒⇒⇒⇒C)

Figure 6.14 examines these types of transitions. Along an initial wave, with speed w, vehicle-flow changes to the downstream speed. This wave is followed by a slip with downstream speed uD, along which only the flow composition changes. The compression of the traffic flow across the initial wave actually occurs in two stages. At first, the speed of the class 1 vehicles decreases to uf2. From that point the flow is congested and vehicle composition remains the same. These two aspects are clearly visible in the phase-diagram. Slowing the class 1 vehicles down happens along a horizontal line, the remaining transitions happen along a straight line through the origin. We indicate the intersection of this path with the B-C regime boundary by index I. This state does not appear in the t-x diagram, but is visible in the k1-k2 phase-diagram.

Figure 6.14 : A semi-congestion to congestion transition (B⇒C) shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

t

x

x0

k

q

kM2 kJ kM1

k1

k2

kM2 kJ

kJ

kM2

kM1

BD

BU

AH

BU BG

BG AH

BD BG

BU

BD

AH

t

x

x0

k

q

kM2 kJ kM1

k1

k2

kM2 kJ

kJ

kM2

kM1

CD

BU

BU

CD=CG CG

BU

CG

I

CD

Chapter 6 82

• Transition from free-flow to congestion (A⇒⇒⇒⇒C)

This type of transition, shown in figure 6.15, occurs via a direct shock wave between the up- and downstream traffic states. The speed of this wave will exceed w and will be less than the downstream state speed. This direct shock wave is followed by a slip with the downstream speed uD along which the flow-composition alters. The direct wave in the phase-diagram can be divided somewhat artificially in a purely type 1 vehicle change to an intermediate state I, followed by a path that leaves vehicle composition unchanged. The state CG density equals that of the downstream state CD.

Figure 6.15 :A free-flow to congestion (A⇒C) transition shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

• Transition from congestion to semi-congestion (C⇒⇒⇒⇒B)

Transitions from congestion to semi-congestion can best be seen as a composite transition. To this end, we examine a virtual intermediate state I at the boudary between the congestion and semi-congestion regimes. According to our perspective, we consider state I as a congested or a semi-congested state. The composition then consists of :

o a transition from the congested state CU to a congested state I.

o a transition from the semi-congested state I to the semi-congested state BD.

Since there are two sub-cases for the last stage, this type of transitions subdivides itself.

In an initial sub-case, put forward in figure 6.16, the state between CU and BG changes via the shock wave with speed w. Here, the intermediate state I only occurs in the phase-diagram, and is not visible in the t-x diagram. Since the total densities of BG and BD are equal, a slip with speed uf2 only adapts the composition of the vehicles.

The free-flow regime appears in the second sub-case. Figure 6.17 shows how the initial shock wave transforms the upstream state CU in BG on the boundary between the semi-congestion and the free-flow regimes. This wave, with speed w, has a broken path in the phase-diagram via intermediate state I. From BG a uf2-slip

t

x

x0

k

q

kM2 kJ kM1

k1

k2

kM2 kJ

kJ

kM2

kM1

CD

AU AU CD=CG

CG

AU

CG

CD


83

re-routes the traffic flow to AH. A final wave, at a speed between uf1 and uf2, introduces the downstream semi-congestion regime BD.

Figure 6.16 : Congestion to semi-congestion transition (C⇒B) sub-case 1 shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

Figure 6.17 : Congestion to semi-congestion transition (C⇒B) sub-case 2 shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

• Transition from congestion to free-flow (C⇒⇒⇒⇒A)

The argumentation for the last type of transitions in our discussion, runs along the same lines as the congestion to semi-congestion transition. We treat this transition as a composite of earlier discussed cases that can be subdivided into two sub-cases. Figures 6.18 and 6.19 clarify the transitions.

t

x

x0

k

q

kM2 kJ kM1

k1

k2

kM2 kJ

kJ

kM2

kM1

BD

CU

BD=BG

CU I

BD

BG

CU BG

t

x

x0

k

q

kM2 kJ kM1

k1

k2

kM2 kJ

kJ

kM2

kM1

BD

CU

AH

CU

BG

BG AH

BD BG

I

BD

AHCU

Chapter 6 84

Figure 6.18 : Congestion to free-flow transition (C⇒A) sub-case 1 shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

Figure 6.19 : Congestion to free-flow transition (C⇒A) sub-case 2 shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

In conclusion All possible Riemann problems share an unambiguous solution. The basic assumptions remain valid and anisotropy is assured. Based on these Riemann problems, each problem can be solved analytically using peacewise constant initial- and boundary conditions The solution is then achieved using the 15 Riemann-transitions above.

6.3 Numerical scheme In this section we formulate a numerical scheme for the two-classes model with different maximum speeds. The scheme applies to a homogeneous road section, the fundamental diagrams remain the same for the entire link. The construction process is analogous to the Godunov scheme of the LWR model.

t

x

x0

k

q

kJ

k1

k2 kJ

kM2

kM1

AD

CU

BH

CU

BG=BH

BG AD

AD

BG

I CD

BH

kM2kM1 kM2 kJ

kM2 kM1 kM2 kJ t

x

x0

k

q

kJ

k1

k2 kJ

kM2

kM1

AD

CU

AH

CU

BG

AD AH

AD

BG

I CD

BG

AH


85

The road to be modeled is subdivided into cells with length x∆ of constant traffic variables. Per time interval t∆ the following conservation equation applies for each vehicle class :

[ ])21,()2


∆∆+=∆+

(6.22)

In the Godunov scheme, the transition flows Gi, that are applied during time interval t∆ across the cell borders, correspond with the flow at the cell boundary of the

corresponding Riemann problem. This means that for the transition flow at xx ∆− 21

an infinite road is considered with for all xxx ∆−< 21 the upstream cell densities

),( xxtKi ∆− and downstream cell densities ),( xtKi . The flows on this infinite road

at the discontinuity xx ∆− 21 are the transition flows )2

1,( xxtG ∆− looked for.

Based on the analytically computed transitions it is possible to calculate the Godunov transition flows and to apply the numerical scheme. In this section, however, we look for an expression for the transition flows. In the remaining computation we indicate the up- and downstream cells with superscripts U and D. For both classes, the same set of triangular fundamental diagrams )(1 kQh

e and )(2 kQUe apply to the two cells. In

the computation, a set of sending and receiving flows is defined. The sending flows indicate the maximum outflow in the upstream cell. The receiving flow is the maximum inflow to the downstream cell. These flows are defined for both classes separately so that the transition flows can be determined as :

[ ]iii RSMinG ,= (6.23)

The sending en receiving flows are computed based on the densities in the up- and downstream cells. Based on the class densities, fractions and regimes in the cells can be determined. When the free-flow or semi-congestion regime applies in the upstream cell, the class 2 fraction equals :

2

22

M

UU

kK

=α (6.24)

During the congestion regime, the following applies :

)( 21

22 UU

UU

KKK+

=α (6.25)

The class 1 fraction then becomes : UU21 1 αα −= (6.26)

Analogously, the fractions in the downstream cell can be determined.

The sending flows depend on the regime in the upstream cell :

• Free-flow regime

The sending flow for both classes equals class flow :

Chapter 6 86

UQS 11 = (6.27)

UQS 22 = (6.28)

• Semi-congestion regime

Class 1 is congested and class 2 behaves as in the free-flow regime :

111 . MU qS α= (6.29)

UQS 22 = (6.30)

• Congestion regime

Both classes are congested and the sending flow equals the following fractions of the maximum flow :

111 . MU qS α= (6.31)

222 . MU qS α= (6.32)

Computing the receiving flows is somewhat more complicated because they depend both on the up- and the downstream cell densities.

• Free-flow and semi-congestion regimes

During these regimes, the class 2 receiving flow depends on upstream composition and downstream capacity :

222 . MU qR α= (6.33)

The computation of the class 1 receiving flow depends on the up- and downstream class 1 fraction. This is calculated from a projection, according to the class 2 free-flow speed, from downstream state points on the upstream fundamental diagram. The computation depends on the class 1 fractions in the up- and downstream cells.

a) DU11 αα >

When class 1 has more space up- than downstream, the fundamental diagrams can be outlined as in figure 6.20. In this case, class 2 acts as a moving bottleneck for class 1. When the free-flow regime applies downstream, the receiving flow becomes identical to the capacity point in the downstream diagram. However, one must bear the upstream diagram in mind. That is why this capacity point is projected on the upstream diagram following a straight line with the class 2 free-flow speed. Mathematically, the receiving flow then equals :

211111 ).(. MDU

MD qqR ααα −+= (6.34)

This receiving flow is valid when the following condition is met :

111 . MDD kK α< (6.35)


87

When this condition can not be fulfilled, the downstream state point is projected on the upstream branch according to an uf2 slope. This results in a receiving flow for class 1 of :

21111 ).( MDUD qQR αα −+= (6.36)

Figure 6.20 : Calculating the class 1 receiving flow when the upstream class 1 fraction exceeds the downstream fraction.

b) DU11 αα ≤

In this case, the upstream class 1 fraction is less then the downstream fraction. The associate projections from the downstream to the upstream diagram are illustrated in figure 6.21. During an upstream free-flow regime and even partly during the upstream semi-congestion regime, the receiving flow equals the upstream fraction of the class 1 capacity. The receiving flow then is :

111 . MU qR α= (6.37)

and it remains thus as long as class 1 density is less than :

211111 ).(. MUD

MUD kkK ααα −+≤ (6.38)

Figure 6.21 : Calculating the class 1 receiving flow when the upstream class 1 fraction exceeds the downstream fraction.

)(1 kQ he

U1α .qM1

k

q

kM2 kJ kM1

qM2

qM1

uf2 uf1

w D

1α .qM1

)(1 kQ he

D1α .qM1

k

q

kM2 kJ kM1

qM2

qM1

uf2 uf1

wU1α .qM1

R1

Chapter 6 88

When this is not the case, a projection is introduced which results in :

21111 ).( MDUD qQR αα −+= (6.39)

When calculating the class 1 receiving flows, therefore, the class 2 free-flow speeds are taken into account. Since triangular class 1 diagrams are used, the capacity point coincides with the point where the derivative of the class 1 fundamental diagram equals uf2. This makes a calculation of the receiving flows easier than it was in the original moving bottleneck problem as described by Newell (1998). In chapter 11.4.1, the analogy to Newell is extrapolated by assuming a curved fundamental diagram for class 1.

• Congestion regime

During congestion total receiving flow equals total flow in the downstream cell : Dtottot QR = (6.40)

In this case, the receiving flows for each class are also dependent on the upstream regime. For an upstream semi-congestion or congestion regime, the following applies ;

Dtot

Utot

U QRR .. 111 αα == (6.41)

Dtot

Utot

U QRR .. 222 αα == (6.42)

When the free-flow regime applies upstream, the analytical solution consists of a direct wave, as illustrated in figure 6.22. In that case, the receiving flow for each class depends on wave-speed wUD :

Dtot

Utot

Dtot

UtotUD

KKQQ

w−−

= (6.43)

Figure 6.22 : Transition from free-flow to congestion

The receiving flows per class can then be calculated as a fraction of the total receiving flow, as follows :

QtotU

k

q

ktotD kJ K1

U

R2

M1

uD

w Rtot

M2

wUD

R1

K2U


89

D

UD

UDUU

uw

wKQR

−

−=

1

.111

(6.44)

D

UD

UDUU

uw

wKQR

−

−=

1

.222

(6.45)

The model can be applied numerically on the basis of the updating rule (6.22) with the definitions of the transition flows (6.23), and on this definition of the sending and receiving flows.

To indicate its correctness, all possible transitions from the previous section 6.2 are computed numerically in figures 6.23 through 6.25. Cell lengths ∆x and ∆t were chosen in order that the free-flow speed of the fastest class complies with :

txu f ∆

∆=1 (6.46)

In addition, the scale equals those in figures 6.3 through 6.17. This enables a visual comparison of the numerical method with the analytical solution. For each transition, density for class 1, class 2 and for the entire traffic flow can be shown.

Figure 6.23: Transitions computed by a numerical scheme.

A⇒A sub1

A⇒A sub2

B⇒B sub1

k1

x

t k2 ktot

Chapter 6 90


B⇒B sub2

C⇒C

A⇒B sub1

A⇒B sub2

B⇒A sub1

B⇒A sub2

k1

x

t k2 ktot


91


B⇒C

A⇒C

C⇒B sub1

C⇒B sub2

C⇒A sub1

C⇒A sub2

k1

x

t k2 ktot

Chapter 6 92

6.4 Analytical and numerical case study Newell (1998) studied the way in which a moving bottleneck influenced the traffic system in the LWR model. The traffic flow on a homogeneous road is studied where a fundamental diagram with lower capacity applies at the local level. Because this inhomogeneity moves at constant speed, we speak of a moving bottleneck.

This original problem definition can also studied using a heterogeneous class LWR model. The local presence of vehicles of the slowest class then acts as the moving bottleneck for the fastest class. Here, Newell's case study is converted to a problem of two interacting classes. The use of triangular homogeneous fundamental diagrams simplifies the case study.

We consider a homogeneous road with exclusively class 1 vehicles in the free-flow regime. As illustrated in figure 6.26, there is a feeder road at location x1, where a column of class 2 vehicles enters the road from t1 to t2. It is assumed that the vehicles that filter in at this point have absolute priority with regard to the advancing class 1 vehicles on the primary road itself. When the flow of the class 2 vehicles entering the road is too large, this convoy behaves like a static bottleneck for the upstream class 1 vehicles.

Figure 6.26 : Analytical solution of the moving bottleneck in the t-x diagram.

All class 2 vehicles drive at free-flow speed uf2 until they leave the primary road at location x2. The effect of this column is shown using characteristics, shock waves, slips and fans in the t-x diagram of figure 6.26. In figure 6.27 the class 1 homogeneous initial state is indicated with A on the fundamental diagram. We assume

C

x

A

D

t

B

A

B

A

M1 B

x1

x2

t1 t2


93

that the column occupies half of the available road, so that the class 2 flow equals half the class 2 capacity qM2.

Figure 6.27 : Analytical solution to the moving bottleneck in the k-q diagram.

When the convoy enters the road, only half the road space is available to class 1 vehicles. Upstream from the feeder, therefore, state C applies, where the flow equals half of the class 1 capacity qM1. The same class 1 flow will be processed near the column, but at the free-flow speed uf1 . Since the speed of the class 1 vehicles exceeds that of the incoming column, namely the class 2 vehicles, this stream passes the head of the column. At that moment, more space is available to the class 1 vehicles, although the same state B continues to apply. This state B is separated from the initial state A by a uf1 slip.

When the incoming column has fully entered the road, at t2, a semi-congestion state D applies upstream from the column's tail. This state is separated from state C by a wave with speed w. In the example, the flow of state D exceeds the flow of the initial state A, hence the shock wave continues on from A to D. When all class 2 vehicles have left the primary road, we get a fan in which the class 1 capacity qM1 is attained.

Here we must note that Newell's original paper (1998) uses a curved fundamental diagram. In that case the fan of characteristics near the column certainly does influence traffic flow. This will only lead to the stationary 'moving bottleneck flow' at the head of the column. However, the original paper does not allow for this, which, in the view of this author, caused the 'eight evenly split referee reports' received by Newell at publication (see Daganzo (1997a) p 159 note 43). When the fan of characteristics is born in mind, regime D also becomes fan-shaped while the traffic regime upstream of the moving bottleneck changes also. This leads to a curved wave between initial state A and D, invalidating some conclusions in the remainder of Newell's paper (1998).

A homogeneous class 1 vehicle flow is simulated in the numerical computation for this case. The transition flows are adapted near the feeder road at location x1, between t1 and t2. G2 = qM2/2 is used for the class 2 transition flow for the cell downstream

qM1/2 C

D

B

M1

A M2

convoy qM2/2

qM1/2+qM2/2

qA

k

q

Chapter 6 94

from the feeder road. This limits the transition flow for class 1 to a maximum of qM1/2.

The class 2 transition flow downstream from the exit road at location x2, is set at zero. This prohibits class 2 traffic downstream from the exit road.

Class 1 and class 2 densities are illustrated in figures 6.28 and 6.29, using the same scale as in the analytical solution shown in figure 6.26. In each case, white signifies higher density levels and black lower density.

Figure 6.28 : Numerical calculation of the class 1 flow in the t-x diagram.

In the numerical computation, the cell-interval and the time-interval were synchronised in order to comply with (6.46). Since this discretisation is optimal only for class 1, there is an increased dispersion for the class 2 vehicles. Figure 6.29 clearly shows how the rigorously discontinuous properties of the column have dispersed near x2 . Use of the delay rule, as was done in chapters 3.3 and 5.3, only applies to a flow of traffic with homogeneous speeds, and provides no improvement here. The dispersion of the numerical Godunov scheme limits its application somewhat. To anticipate this problem, modeling a rigorous discontinuous class pattern requires a much smaller cell-interval as compared to the physical length of the column. When this is not the case, the bottleneck is flattened which enables the processing of a higher class 1 flow than would seem possible from the analytical solution. Lebacque's method (2002) could possibly be used with localised moving bottlenecks of smaller physical length. Locally moving cell boundaries are then inserted to incorporate the impact of a bus.

t

x


95

Figure 6.29 : Numerical calculation of the class 2 flow in the t-x diagram

6.5 Conclusions In this chapter we presented a heterogeneous free-flow LWR model for two classes. To this end we chose a 'user-optimum', in which slower vehicles are not affected by faster ones, while all vehicles maximise their speed.

Each class was described by a triangular fundamental diagram, in which the congestion branches of both classes coincided. During congestion, therefore, the classes behave as user-classes of homogeneous speed where the first-in-first-out rule applies.

During free-flow the road is used optimally and we implicitly accept completely parallel traffic operations for both classes. This approach does not allow for physical separation by means of traffic lanes. The actual flow is, therefore, slightly overestimated.

The user-optimum in this heterogeneous free-flow model describes slow vehicles according to the LWR model, while modeling the interaction between the fast and the slow classes as if the slow vehicles were moving bottlenecks. This model, therefore, perfectly describes Newell's (1998) moving bottleneck, also numerically. The numerical solution of the first order Godunov scheme, however, results in greater dispersion, thereby spreading out the sharply outlined bottleneck.

When heterogeneously congested traffic has been dealt with in the next chapter, this model will be integrated in a heterogeneous description of a traffic flow consisting of

t

x

Chapter 6 96

two classes, each of which will be described with an arbitrary triangular homogeneous fundamental diagram.

97

7 HETEROGENEOUS

CONGESTED TRAFFIC

To this point, we have discussed two steps that are necessary to extend the LWR model to a heterogeneous model. We dealt with the scaling of fundamental diagrams in chapter 5 and examined heterogeneous free-flow traffic in chapter 6. The emphasis in this chapter will be on differences in driving behaviour during congestion.

Again we examine two vehicles classes, each of which is described by a triangular fundamental diagram. In this chapter we assume that only the congestion branch of these diagrams differs for the two vehicle classes. The two classes show the same behaviour during free flow. In addition, we assume that the mixed vehicle flow always shows a homogeneous speed during congestion.

First we deal with the analytical derivation of this extended LWR model. In addition to the computation of the stationary regimes, we will look at all possible transitions between the regimes. We then extend the numerical method of the LWR model. In this way, we can implement the extensions using a practical computation scheme.

We then compute the analytical and numeric solution for a case study with traffic signals.

As in the two previous chapters, the model will be developed for a homogeneous section of the road. Together with the extensions formulated earlier, this will lead to a global heterogeneous LWR model, to be discussed in the next chapter. Then we will be able to describe a stream of traffic consisting of two classes with arbitrary

Chapter 7 98

triangular fundamental diagrams on a homogeneous link. We subsequently extend this to a dynamic network model in chapter 9.

At the same time, but independent of the research presented here, Zhang et al. (2002) en Chanut and Buisson (2003) presented a similar model. The assumptions in the empirical investigations of Koppelman (1999) regarding vehicle- and driver characteristics (1999) also run along the same lines.

7.1 Stationary traffic states In this chapter, the LWR model is extended to two classes where the fundamental diagrams differ during congestion only. The homogeneous fundamental diagrams are shown in figure 7.1. Again we work with two triangular k-q diagrams. The free flow branches of both fundamental diagrams coincide. Free flow speed uf, and capacity point M(kM,qM) are the same for both classes. The classes differ from one another through the jam densities kJ1 and kJ2 and the congestion branch speeds w1 and w2. In mathematical terms these equilibrium relations can be expressed using the following function :

MJiJiM

fM

Mfhei kkkk

kkuk

kkukkQ

>−−

≤=

).()(

..

)( (7.1)

Figure 7.1 : Two homogeneous fundamental k-q diagrams with differing congestion branches

As with the previous extensions of the LWR model, we assume that vehicle behaviour in a mixed stream of traffic is closely related to the homogeneous fundamental diagram of the constituent classes. Each class gets a part of the total road space assigned to it. This fraction αi allows the equilibrium flow per class to be defined by :

=

i

iheiii

kQq

αα .

(7.2)

We assume that the road is fully utilised :

121 =+αα (7.3)

M

k kJ1kJ2 kM

qM q

uf

w1 w2

)( 11 kQ he

)( 22 kQhe


99

Central to this chapter is the fact that the speed of the heterogeneous traffic flow is homogeneous. Because both class share the same maximum speed, this corresponds with the user optimum: no vehicle is able to increase its speed without decreasing the speed of slower vehicles. In concrete terms, this means that we assume that vehicles from different classes adapt their used space to achieve equal speed. The fraction of total road space that is assigned to each class, can then be derived by equalising the speed per class :

=

2

22

2

2

1

11

1

1 ..α

αα

α kQ

kk

Qk

he

he

(7.4)

Starting from the class densities, the fraction and hence the speed and flow per class can then be computed. All possible combinations of class densities can be reflected in a k1-k2 phase-diagram, as is shown in figure 7.2. Based on the characteristics of the traffic flow, we can distinguish and indicate two regimes or phases in this phase diagram. Traffic in the free flow regime will have the free-flow speed, while speed in the congestion regime will be less.

Figure 7.2 : The phase-diagram for a mixed flow of traffic.

Regime A : Free flow As long as total density remains smaller than capacity density kM, speed remains maximal and equal to uf. Both classes in this regime are described by an identical fundamental diagram. The classes, therefore, function as user-classes. Traffic flow composition does not influence the actual traffic service at link level. Flow per class is given by :

fiiei ukkQ .)( = (7.5)

In a phase-diagram, as in figure 7.2., the free-flow regime can be indicated as zone A. The following set of inequalities define this zone :

• Class density is positive

0≥ik (7.6)

• Total density is smaller than capacity density

kJ1

kJ2

kM

kM

k2

A

B

Chapter 7 100

Mkkk ≤+ 21 (7.7)

Traffic flow speed is homogeneous and equation (7.4) always applies.

Regime B : Congestion In the congestion regime, total density exceeds capacity density kM and traffic flow speed lies below free-flow speed uf. The central assumption is that vehicles from both classes share the same speed. Per class, space use per vehicle does differ due to the different congestion branches. We calculate fraction αi that a class occupies by equalising the speed per class as in (7.4). If we use the congestion branch of the fundamental diagram (7.1) and if we take account of total road utilisation (7.3) we get:

−

−−−

=

−

− 21

2

22

11

1

1

11

1

1.

)(.

.1

.)(

.. J

JM

fMJ

JM

fM kk

kkuk

kk

kkk

ukk α

αα

α

(7.8)

Additional computation leads to an expression for the fraction used by class 1 in function of class densities and a number of parameters of the homogeneous fundamental diagrams :

2112112212

211211212211 ........

........

JJJMJMJJ

JJJMJJ

kkkkkkkkkkkkkkkkkkkkkkkk

+−−+−−

=α (7.9)

Based on this fraction, the speed of the traffic flow can be computed by :

==

1

11

1

1 .α

α kQ

kuu h

ei (7.10)

Computing (7.10) by substitution of (7.9) gives the following expression for the speed during congestion :

−−+

−−

=

2

2

1

1.21

2

2

1

11..

JJM

JJfM

kk

kk

kkk

kk

kk

uku

(7.11)

A mixed stream of traffic in the congestion regime is characterised by a homogeneous speed u, the class densities and the flows. In the k-q diagram the state point of the total traffic flow will lie between the congestion branches of the constituent classes. Figure 7.3 sketches a congested traffic situation


101

Figure 7.3 : A congested traffic situation with an indication of a virtual congestion branch and accompanying jam density kJtot.

A congested state point of total traffic stream enables the assumption of a virtual congestion branch. This straight line then crosses through the capacity point as shown in figure 7.3. The slope wtot of this congestion branch is expressed as :

totM

totfMtot kk

ukukw

−−

=..

(7.12)

Further computation by substitution of the speed according to formula (7.11), gives the following expression :

totJJ

M

JJfM

tot

kkk

kk

k

kk

kk

ukw

−

+

+

=

2

2

1

1

2

2

1

1

.

..

(7.13)

Starting from this virtual congestion branch we once again define a total jam density of kJtot. We determine this density as the point of intersection of the congestion axis with the weighted congestion branch:

JtotM

fMtot kk

ukw

−=

.

(7.14)

Reworking this equation leads to the following expression :

2

2

1

1

JJ

totJtot

kk

kk

kk

+=

(7.15)

This shows that total jam density is the harmonic weighted average of the two jam densities for each class. Inputting this new variable considerably simplifies the expression for the speed (7.11) :

)().(.

JtotMtot

JtottotfM

kkkkkuk

u−

−=

(7.16)

Based on the speed, the flow for each class can be calculated as :

u kJ2 kM

qM

k

q

kJ1kJtot ktot k1 k2

q1 q2 wtot

qtot w1

w2

Chapter 7 102

)().(.

...JtotM

JtottotfM

tot

itot

tot

iii kk

kkukkk

qkk

ukq−

−===

(7.17)

The usefulness of the congestion branch with speed wtot and the weighted jam density kJtot that we introduced before, is not confined to the simplification of expressions. This weighted congestion branch can also be interpreted as the set of congested states with the same traffic composition. We define the relative density βi of a class as:

tot

ii k

k=β

(7.18)

Using this relative class density, the congestion branch speed wtot becomes equal to :

1.

..

2

2

1

1

2

2

1

1

−

+

+

=

JJM

JJfM

tot

kkk

kkuk

wββ

ββ

(7.19)

This speed does, therefore, not depend on total density, but merely on relative density. This weighted congestion branch applies to a traffic flow of a given composition. This means, off course, that the weighted jam density is also only dependent on traffic composition.

2

2

1

1

1

JJ

Jtot

kk

kββ

+=

(7.20)

This weighted jam density then indicates the highest density that can be achieved by a stream of traffic of a given composition. When we generalise this condition, we can state that total density must always be less than weighted maximum density. Elaboration of this statement leads to a boundary condition of the congestion regime.

The possible traffic states of the congestion regime are confined by :

• Again, class densities are positive

0≥ik (7.21)

• The boundary with the free-flow regime

Mtot kk > (7.22)

• Density is restricted by the weighted jam density :

12

2

1

1 ≤+JJ k

kkk

(7.23)

In the phase diagram of figure 7.2 the congestion regime is indicated by B.


103

7.2 Transitions The transition between two stationary states occurs in shock waves, slips and fans. In this section all possible combinations of stationary regimes are studied using Riemann problems. Here we examine an infinite road with a discontinuity at x0. Upstream, at x < x0, a constant traffic state is considered that is indicated by superscript U.

Downstream from this discontinuity a different homogeneous traffic state applies that is indicated by superscript D. All possible combinations of up- and downstream traffic regimes are examined qualitatively. In each case, the transition is shown in a fundamental k-q diagram, a phase diagram and a t-x diagram.

• Transition from free-flow to free-flow (A⇒⇒⇒⇒A)

This kind of transitions develops quite simply via a slip with a speed of uf. The discontinuity between the up- and downstream states propagates with the speed of the traffic flow in the t-x diagram. A different vehicle composition, flow and density applies up- and downstream from the shock wave. Figure 7.4 gives an example.

Figure 7.4: A free-flow to free-flow transition (A⇒A) in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

• Transition from congestion to congestion (B⇒⇒⇒⇒B)

This transition is made up of a shock wave and a slip as shown in figure 7.5. The speed of the traffic flow changes in an initial shock wave. Here, the composition of the traffic flow remains identical to that of the upstream traffic state. The speed of the shock wave equals wtot, as defined in (7.13). The vehicle composition then changes along a slip by speed uD. Total density and flow also change across this slip because vehicle composition also determines the location of the total congestion branch.

k

q

kJ2 kJ1

x

t

k1

k2

kM kJ1

kJ2

kM

x0

AD

AU AD AU

AU AD

Chapter 7 104

Figure 7.5: A congestion to congestion transition (B⇒B) in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram

• Transition from free-flow to congestion (A⇒⇒⇒⇒B)

In an initial direct wave the upstream free-flow state is transformed into a state of congestion at the downstream speed. The vehicle composition of the upstream traffic state, however, determines the correct total flow and density. Depending on the upstream total flow, the upstream total density, the upstream vehicle composition and the downstream speed, this wave can propagate itself both with and against the traffic flow. A second shock wave, a slip at speed uD, separates the up- and downstream vehicle composition. An example is shown in figure 7.6.

Figure 7.6: A free flow to congestion transition (A⇒B) in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram

• Transition from congestion to free flow (B⇒⇒⇒⇒A)

This transition involves a negative shock wave at a speed of wtot and a slip at a speed of uf. The speed of the first shock wave is determined on the basis of the upstream vehicle composition. This wave changes the state of congestion to the capacity-state. The slip then separates the capacity-state with the upstream vehicle composition from the downstream state. Note that a fan of characteristics in fact becomes visible between these two shock waves. A graphical solution is drawn in figure 7.7.

x

t

k

q

kJ2 kJ1

k1

k2

kM kJ1

kJ2

kM

x0

BU

BU

BD

BG

BU

BD

wtot

kJtot

BG

BD

x

t

k

q

kJ2 kJ1

k1

k2

kM kJ1

kJ2

kM

x0BD

AU BG

BG BD

BG

AU

BD


105

Figure 7.7: A congestion to free flow transition (B⇒A) in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram

Based on this qualitative outline, every transition between two piece-wise constant initial conditions can be solved. The analytic and graphic computations render the solution obvious and enable one to check that individual vehicles are never influenced by upcoming traffic.

7.3 Numerical scheme In this section, the formulated analytical model is converted to a numeric scheme. Using this scheme, an approximate solution can be calculated. As in the traditional Godunov scheme for the LWR model, the link to be modeled is divided into cells with length ∆x. Per cell, an approximate homogeneous density Ki(t,x) is assumed for each class. Per time interval ∆t, a new density is computed according to the rule :

[ ])21,()2


∆∆+=∆+

(7.24)

The transition flows )21,( xxtGi ∆± give the flows per vehicle class across the cell

transitions during time interval ∆t. When we formulate these transition flows, we introduce a simplified notation, as we did in the two previous chapters. The cell upstream from the cell transition under examination will, from now on, be designated by U, the downstream cell gets a superscript D.

In this Godunov scheme, the transition flow is calculated as the flow at the cell transition when the Riemann problem with the up- and downstream cell densities is examined. In elaborating this numeric scheme for the LWR model, we propose a simplified expression for this transition flow: the minimum between the sending flow from the downstream cell and the receiving flow into the downstream cell. This sending flow S shows how much traffic can leave a cell and the receiving flow reflects the flow that can enter a cell. In this extended scheme for two classes, the transition flow per class will also be computed as the minimum of the upstream sending and the downstream receiving flow :

[ ]iii RSMinG ,= (7.25)

k

q

kJ2 kJ1

k1

k2

kM kJ1

kJ2

kM BU

AD BU

AD

BG

BG

x

t

BU

AD

BGx0

Chapter 7 106

Depending on the regime of the upstream cell, the sending flows of both classes are computed as follows :

• Free-flow regime :

The sending flow equals the flow in the upstream cell Uii QS = (7.26)

• Congestion regime :

Total sending flow equals capacity. Starting from the traffic stream composition, the sending flows are calculated as :

MUtot

Ui

i qKK

S .= (7.27)

The receiving flows per class are dependent on the upstream traffic composition and the downstream regime :

• Free-flow regime :

These receiving flows are maximally :

MUtot

Ui

i qKK

R .= (7.28)

• Congestion regime :

The receiving flow is de flow that applies to the downstream speed at the upstream composition of the stream of traffic. Speed uD in the downstream cell is known. The weighted congestion branch that applies to the upstream composition is characterised by the jam density KJtot and the congestion branch speed Wtot. Substituting the upstream traffic composition in formulas (7.15) and (7.14) gives :

2

2

1

1

21

J

U

J

U

UU

Jtot

kK

kK

KKK

+

+=

(7.29)

JtotM

fMtot Kk

ukW

−=

.

(7.30)

The total receiving flow corresponds with the flow at the downstream speed uD near to this upstream weighted congestion branch. This leads to :

Dtot

DtotJtot

tot uWuWK

R−

=..

(7.31)

Again, a subdivision per class takes account of the composition upstream :

totUtot

Ui

i RKKR =

(7.32)


107

For each class, the transition flow can then be calculated as the minimum of these sending and receiving flows. Applying these transition flows in (7.24) gives a numeric scheme for a homogeneous link. The four different transitions, as solved analytically in figures 7.4 - 7.7, are appoximated quite well by this numeric scheme. In figure 7.8, the four transitions are solved with the numeric scheme. In each figure, density is shown by a colour scale. The highest density is shown in white, the lowest in black. The time interval and the cell length are tuned to one another so that the product of the free flow speed and the time interval equal the cell length.

Figure 7.8 : The four types of transition computed by a numerical scheme.

7.4 Analytical and numerical case study In this section a case study is worked out analytically and numerically. To this end, we look at a homogeneous road where circulation at location x1 is temporary

k1

x

t k2 ktot

A⇒B

B⇒A

B⇒B

A⇒A

Chapter 7 108

completely blocked. The traffic flow on this road consists of two classes. Each class on this link complies with the homogeneous fundamental diagrams of figure 7.9. Traffic flows q1

A and q2A are taken as initial boundary condition. The stream of traffic

consists mainly of type 1 vehicles. At t = 0 the composition changes in figure 7.10 to q1

B and q2B. The total traffic flow remains the same but the type 2 vehicles gain the

upper hand.

Figure 7.9 : Analytical computation of the case study in (a) the phase diagram and (b) the fundamental diagram

On t = tb the obstruction starts at location xstop. At that moment, the upstream state changes to a completely saturated state. Downstream of the obstruction, a slip separates the initial state (A) from the empty road (O). When the obstruction disappears at te, the capacity flow is achieved in a fan.

Figure 7.10 : Analytical solution of a temporary obstruction during an altered composition of the traffic flow in a t-x diagram.

MU

AU

JUAD

JD

MD O

t

x

xstop

tb te x0

t0 ts

k

q

kJ2 kJ1

MU=MD

AU= AD

JD JU

MD k1

k2

kM kJ1

kJ2

kM

AU AD

MU

O

JD

JU


109

The changed composition of the traffic flow reaches the queue that has developed upstream from the obstruction at ts. In spite of an unaltered total flow, the shock wave changes between the free and the saturated state. This is because the different traffic composition changes the weighted jam density. A larger share of type 2 vehicles leads to a smaller jam density. On average, a stationary vehicle now occupies more space, thus increasing the expansion of the queue.

Since capacity is independent of traffic composition, the less dense queue also dissipates faster. Therefore, travel time is not influenced by the traffic composition.

Figure 7.11 : Numerical solution of the case study in an x-t diagram (a) class 1 density and (b) class 2 density.

Figure 7.12 : Analytical and numerical solution (total density) to a temporary obstruction during an altered traffic stream composition in a t-x diagram.

x

t

k1 k2

t

x

Chapter 7 110

The analytical case study can also be solved by the numerical scheme. To do this, the densities in the cells are calculated using (7.24). During the temporary obstruction, the transition flow becomes temporary zero at the stop location. Figure 7.11 shows the calculated densities in the t-x diagram. Here, black corresponds to a low density, and white to a high density. Figure 7.12 projects the analytical solution on the numerical solution.

7.5 Conclusions This chapter presented a heterogeneous LWR model for congested traffic. For a stream of traffic consisting of two classes, it was assumed that each class could be described by a triangular fundamental diagram where the capacity and free-flow speed for both classes is the same.

It was also assumed that the heterogeneously composed traffic stream has a homogeneous speed. This is why the various vehicles are unable to overtake one another, and why traffic on a link is a FIFO system. During congestion, this assumption corresponds with the user-optimum : no vehicle can increase its speed without decreasing that of another vehicle.

The assumptions resulted in a LWR model that works with a weighted fundamental diagram. Based on the assumed vehicle composition, a new congestion branch with accompanying weighted jam density applies. During changes in the composition of vehicles, slips divide the various traffic states in this FIFO system.

111

8 THE HETEROGENEOUS

LWR MODEL

This chapter presents a general heterogeneous LWR model that is based on the combined LWR extensions developed so far. The model describes a traffic flow in which the vehicles are divided into two classes.

A limiting condition is assumed, namely that each class is described by a homogeneous triangular k-q diagram. Vehicle behaviour is deterministic and the vehicles conform to the homogeneous fundamental diagram, even when the composition of the traffic flow is heterogeneous. A user-optimum is assumed for the interactions between the two classes.

This states that no vehicle can increase its speed any further without influencing the speed of slower vehicles. This hypothesis leads to equal speeds during congestion, as derived in the heterogeneous congestion model in chapter 7. Different speeds are possible during free-flow traffic and the model develops analogously to that in chapter 6.

We begin this chapter with an analytical formulation of the model. Based on the various assumptions, all possible stationary traffic states are first examined. The transitions between the different stationary states are, subsequently, developed. We also assume, as we did in the LWR model, that stationary states are the only state-types possible during the transitions, and that these are separated by shock waves, fans or slips.

Chapter 8 112

A numerical scheme is drafted representing a powerful calculation method to arrive at a rapid approximating solution for the analytical model. This scheme is based on the Godunov scheme that calculates cell-density using discrete time intervals.

8.1 Stationary traffic states This initial section examines all possible stationary traffic states. Based on the class-densities of the two vehicle classes, the properties of the vehicle flow are fully explained.

Each of the two vehicle classes on the link under consideration is described on the basis of a triangular fundamental diagram. When a flow of traffic is composed of vehicles belonging exclusively to one class, the LWR model with this homogeneous fundamental diagram applies. In the remaining description of the interactions between the two classes, the triangular diagrams are defined as :

MiJiJiMi

fiMi

Mifihei kkkk

kkuk

kkukkQ >−

−

≤= ).(

)(.

.)(

(8.1)

Besides the free flow speed ufi, the critical density kMi and the jam density kJi, the congestion-wave speed wi and the capacity flow qMi can also be defined as independent parameters :

fiMiMi ukq .= (8.2)

JiMi

Mii kk

qw

−=

(8.3)

Figure 8.1 : The two triangular homogeneous fundamental diagrams.

In the remaining computation the fastest class is indicated by the index 1. Figure 8.1 sketches two homogeneous fundamental k-q diagrams. Without loss of generality, a

k

kJ2 kJ1r.kM2

q

kM2 kM1

w1 w2

qM1

r.qM2

qM2

uf2 uf1

The heterogeneous LWR model

113

smaller jam density, a smaller capacity and a more negative congestion-wave speed for class 2 were assumed.

As in chapter 5, we can define a scale factor for class 2. This scale factor links both diagrams. We define the scale factor r here as indicated in figure 8.1. The capacity point for class 2 lies on the class-1 congestion branch when class 2 is scaled by a factor r. The value of r then becomes :

).(.

212

11

fM

J

uwkkw

r−

= (8.4)

The transformation from class 2 to a scaled class is not carried through because the two classes lack a similarly shaped fundamental diagram. However, this scale factor can establish a relationships between both classes. This is due to the fact that the flow for a class 1 density of r.kM2 equals r.qM2 while the class 2 free-flow speed applies.

A user-optimum is assumed in the interactions between both classes. In this situation, both classes spread themselves across the entire road space in such a way that no vehicle can increase its speed without reducing the speed of slower vehicles. All vehicles, therefore, aim to maximise their speed which amounts to minimising travel time. Since slower vehicles can not be influenced by faster vehicles, the anisotropic condition is met. This means that drivers are only influenced by traffic circumstances ahead of them.

Figure 8.2 : The k1-k2 phase diagram with the three traffic regimes.

It is also assumed that a vehicle class never occupies more space than is strictly necessary. This principle causes optimal road-use, which applies in the LWR model also.

k1 r.kM2 kM1 kJ1

kM2

k2 kJ2

B A

C

kMtot

β2/β1

kJtot

Chapter 8 114

Applying these principles leads to a unambiguous description of a heterogeneous traffic flow. The stationary properties of the traffic flow are examined for all admissible combinations of class-densities. Three regimes or phases can be distinguished according to the class densities: the free-flow regime, the semi-congestion regime and the congestion regime. Figure 8.2 gives the three regimes in a k1-k2 diagram.

In this phase-diagram, the traffic regime is represented in function of the class-densities. We will now look at the properties and the boundaries of the three regimes.

Regime A : Free flow With the free-flow regime we mean all traffic states in which the speed of the classes equals the respective free-flow speeds. The speed of a class depends both on the homogeneous fundamental diagram and the road-fraction occupied by the class, following from (4.12):

=

i

ihei

i

ii

kQk

uα

α . (8.5)

From (8.1) it follows that this speed equals the free-flow speed ufi when the following condition is met :

Mii

i kk ≤α

(8.6)

Since the sum of the fractions equals 1, the following boundary in the phase-diagram is derived :

12

2

1

1 ≤+MM kk

kk

(8.7)

Due to the fact that both classes operate in parallel alongside each other during the free-flow regime, the user-optimum applies. The classes do not influence each other and the user-optimum is met. The class flow then becomes :

fiii ukq .= (8.8)

Regime B : Semi-congestion At a greater density, the class 1 vehicles will be the first to decrease speed. Those traffic states in which the speed of the class 1 vehicles is less than uf1 but greater than - or equal to - the class 2 free-flow speed, belong to the semi-congestion regime.

The speed of class 2 remains equal to the maximum speed uf2, respecting the user-optimum. The minimum fraction for class 2 whereby this speed is maintained is :

2

22

Mkk

=α (8.9)


115

This gives class 1 the largest possible fraction, which in turn, maximises the speed of class 1 vehicles.

)....)....(.

.121121

12122111

1

11

1

11

JMMM

JJMMfMhe kkkkkk

kkkkkkukkQk

u−

+−=

=

αα

(8.10)

This regime remains valid as long as the speed of class 1 exceeds or equals the class 2 free-flow speed, or :

1. 2

2

2

1 ≤+MM kk

krk

(8.11)

In the phase-diagram, the boundaries of the semi-congestion regime are, therefore, given as follows :

• The class-densities are positive :

0≥ik (8.12)

• The boundary with the free-flow regime :

12

2

1

1 >+MM kk

kk

(8.13)

• The speed does not go below that of the class 2 free-flow speed :

1. 2

2

2

1 ≤+MM kk

krk

(8.14)

Regime C : Congestion When the traffic-flow speed is less than the class 2 free-flow speed, the speed of both classes must be equal. If this were no so, then the slowest class could increase its speed by taking road space from the fastest class. During congestion, the fractions are, consequently, divided to achieve a homogeneous speed that is less than the smallest free-flow speed.

The congestion regime in the phases diagram in figure 8.2 is restricted by :


0≥ik (8.15)

• The boundary with the semi-congestion regime :

1. 2

2

2

1 >+MM kk

krk

(8.16)

• The maximum density restricts the phase-diagram :

12

2

1

1 ≤+JJ k

kkk

(8.17)

Chapter 8 116

This last condition was already derived in chapter 7 (7.23)

The fractions can be extracted by equating the class speeds :

−

−−−

=

−

− 21

2

22

22

2

11

1

1

11

11

1

1

1.

)(.

.1

.)(

.. J

JM

fMJ

JM

fM kkkk

ukk

kkkk

ukk α

αα

α

(8.18)

As in chapter 7, a weighted congestion branch can be introduced that applies to the given composition. Again, we define the relative density βi of a class during congestion as follows :

tot

ii k

k=β (8.19)

Using this traffic variable we can define a weighted capacity density kMtot, a weighted capacity flow kMtot, and a weighted jam density kJtot. Determining these three variables for a given traffic composition can be done with the aid of a graphic construction in the phase diagram as is shown in figure 8.2. The weighted congestion branch corresponds to a straight line through the origin with slope β2 /β1. The points of intersection with the boundaries of the congestion regime then reflect the weighted capacity and the jam traffic state. Further computation leads to the expression from chapter 7 (7.20) for the weighted jam density :

2

2

1

1

1

JJ

Jtot

kk

k ββ +=

(8.20)

Analogously, expressions for the weighted capacity point, at speed uf2, are found:

21

2

..

ββ rkrk M

Mtot +=

(8.21)

The slope of the weighted congestion branch wtot can then be further derived to :

JtotMtot

Mtottot kk

qw

−=

(8.22)

A simple formulation of the speed for the given class densities using this weighted fundamental diagram is as follows :

)1(21 kk

kwu Jtot

tot +−=

(8.23)

Since the speed is homogeneous, a FIFO regime applies during congestion. The composition can only change along a trajectory and the class flows are proportional to the class densities :

ukq ii .= (8.24)


117

8.2 Transitions A model must not only describe the stationary traffic states but also the transitions between such states. All possible Riemann problems are examined in this section. In a Riemann problem we consider an infinite road where a stationary traffic state applies at t0, upstream from a point x0. In this point, the traffic state changes discontinuously to another stationary state that applies downstream. The evolution of this discontinuous change through time is studied in this process.

As in the LWR model, transitions consist of clear changes between stationary traffic states that are separated by shock waves, fans and slips. The traffic flow is, therefore, still considered to be stationary.

By studying the Riemann problems pertaining to all possible combinations of stationary traffic states, we obtain a qualitative overview of the transitions in this heterogeneous model. In this overview, the assumptions can be inspected: All vehicles attempt to maximise their speed, but not at the expense of slower vehicles. Meantime, the road is used optimally. The dynamic properties of the model are also clarified.

In the overview of all Riemann problems, some transitions are almost identical to transitions from previous chapters. All transitions between stationary traffic states in the congestion and semi-congestion regimes closely resemble the transitions in chapter 6. The transition in the congestion regime itself is related to that of chapter 7. In conclusion, the transitions between the congestion regime on the one hand and the free-flow or the semi-congestion regimes on the other hand, are dealt with in two separate categories.

8.2.1 Transitions where class 2 maintains the free-flow speed

We can refer the reader to chapter 6 for transitions in this category. However, the expressions become somewhat more complex through the introduction of the scale factor. The transitions that, in chapter 6, were proposed within the semi-congestion regime in the phase diagram with a slope of –45° now have a slope that equals –arctg(1/r). The following eight kinds of transitions can be distinguished.

• Free-flow to free-flow (A⇒⇒⇒⇒A) transition subcase 1

Within this kind of transition, the free-flow regime continues to apply. A required condition here is, analogously to condition (6.19) :

12

2

1

1 ≤+M

D

M

U

kk

kk

(8.25)

This transition corresponds with that in figure 6.5.

Chapter 8 118

• Free-flow to free-flow (A⇒⇒⇒⇒A) transition subcase 2

When condition (8.25) can not be met, the semi-congestion regime appears in the solution. Except for the slope in the semi-congestion regime, this transition develops analogously to that in figure 6.12.

• Free-flow to semi-congestion (A⇒⇒⇒⇒B) transition subcase 1

Dependent on the upstream class 1 density, the uf2-slip follows first or last. When condition (8.26) is met, the class 1 vehicles first decelerate prior to a change in the composition of the class 2 vehicles.

21

1112211 .

.....

MM

MD

MD

MMH

krkkkkkrkkrk

++−=

(8.26)

Figure 6.8 gives a good insight into this transition.

• Free-flow to semi-congestion (A⇒⇒⇒⇒B) transition subcase 2

When condition (8.26) is not met, the class 2 vehicles change first across the uf2-slip, as in figure 6.9.

• Semi-congestion to semi-congestion (B⇒⇒⇒⇒B) transition subcase 1

In an initial kind, the transition remains entirely within the semi-congestion regime. To achieve this, the class 2 flow must comply with :

12

12122 .

MM

MDD

MU

kkkkkkk

−−+<

(8.27)

This transition consists of a shock wave with speed w1 across which the class 1 speed changes and is followed by a uf2-slip, as in figure 6.6.

• Semi-congestion to semi-congestion (B⇒⇒⇒⇒B) transition subcase 2

When condition (8.27) is not met, the free-flow regime appears in the solution analogous to the transition in figure 6.13.

• Semi-congestion to free-flow (B⇒⇒⇒⇒A) transition subcase 1

Again, this transition separates into two kinds. A first type applies when the following condition is met :

UD kk 22 ≤ (8.28)

As in figure 6.10, this transition is made up of a w1 shock wave, followed by a uf2 and a uf1-slip.

• Semi-congestion to free-flow (B⇒⇒⇒⇒A) transition subcase 2

A second kind occurs when (8.28) is not met and it develops analogous to the transition in figure 6.11.

These eight categories give a qualitative insight into the range of Riemann problems that can occur when both the up- and downstream free-flow speed applies for class 2.


119

8.2.2 Transitions within the congestion regime This type of transition develops parallel to the congestion-congestion transition from chapter 7. The speed changes across an initial wave. This shock wave has the speed of the weighted congestion branch wtot when the last is calculated with the upstream traffic composition according to (8.22).

This shock wave is followed by a slip with the speed of the downstream traffic flow. The composition of the traffic flow changes across this slip. Since the total flow at this speed is subject to the traffic composition, the total flow also changes across this slip. Figure 8.3 shows this transition in the phase diagram and in a t-x diagram.

Figure 8.3 : Congestion to congestion transition (C ⇒ C) shown in (a) the phase- diagram; (b) the fundamental diagram and (c) a t-x diagram.

8.2.3 Transitions from the free-flow or semi-congestion regime to the congestion regime

The transitions from semi-congestion or free-flow to a congestion regime cannot be derived from the earlier formulated heterogeneous models, and can be broken down into two kinds.

In a first type we distinguish two shock waves and a slip, as in figure 8.4. In the first wave, only the properties of the class 1 vehicles change. The speed of these vehicles decreases to uf2 . This results in a homogeneous traffic flow in this intermediate state G. The class 2 density is not affected. Therefore, the class densities are given by :

UG kk 22 = (8.29)

rkkk UM

G ).( 221 −= (8.30)

This traffic state G corresponds to the weighted capacity state Mtot. The second shock wave develops along the weighted congestion branch with speed wtot. Here, the speed of the traffic flow decreases to the downstream speed uD. The vehicle composition of traffic state G continues to apply, notably :

GG

GiG

i kkk

21 +=β

(8.31)

t

x

x0

CD

CU CG

k1

k2

r.kM2 kJ1

kJ2

kM2

kM1

CG

CD CU

q

k

CU CD

CG kJ2 kJtot kJ1

wtot

Chapter 8 120

The weighted congestion speed wtot and the weighted jam density kJtot can be calculated at this vehicle composition according to (8.22) and (8.20). The class densities of the resulting traffic state H at the downstream speed uD can be calculated with (8.23) as follows :

tot

DJtot

GiH

i

wukk

−=

1

.β (8.32)

Further computation of (8.32) leads to :

DJJ

DfJMJJ

Df

UJJf

GiH

i ukkuukkrkrkuukkkuk

k..).(..).)((

...

212222122

212

+−+−−=

(8.33)

A slip with the speed of the downstream traffic flow concludes this process. This slip separates the up- and downstream vehicle composition.

Figure 8.4 :Semi-congestion to congestion (B ⇒ C) subcase 1, transition via two shock waves shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

This first type of transitions is only possible when the slope of the class 1 shock wave is less than that of the subsequent wave along the weighted congestion branch, or :

2

2

1

1

2

11

121

11

..

J

G

J

G

ftotGU

Gf

UU

kk

kk

uw

kkkuku

+−

=<−−

(8.34)

Where this is not the case, a direct wave applies between the upstream state and a state at the downstream speed. These kinds of transitions are also rounded off by a slip, as sketched in figure 8.5.

The speed of the direct shock wave uUG and the class densities of the intermediate state G require some calculations.

As an initial condition, the changes in the class variables across the direct shock wave must, for both classes, lead to the same shock wave speed :

k1

k2

r.kM2 kJ1

kJ2

kM2

kM1

CG

CD BU BG

t

x

x0

CD

BU CH

BG

q

k

BU

CD CH

BG

kJ2 kJ1


121

2

2

1

1

kq

kqU UG

∆∆

=∆∆

= (8.35)

Further derivation of this condition results in a relation between the unknown class densities :

).().(

.).(

11

221

1

1222 UDU

fDU

GUD

Uf

UG

uukuuk

kuu

uukk

−−

+−

−=

(8.36)

We also know that the speed of the state G equals uD. This, too, gives rise to a relation between the two class densities of state G. Rewriting (8.23) leads to :

rkkukkrukukrkkukkrukukrkkkukr

kJJ

DJM

DJfM

JJD

JMD

JfMG

JJfMG

.........)........(....

2112122

2122222121222 +−

+−−=

(8.37)

Combining (8.36) and (8.37) gives an analytical expression for the class densities in state G and enables the computation of the shock wave speed UUG with (8.35).

Figure 8.5: Free-flow to congestion (A ⇒ C) subcase 2, transition via a direct shock wave shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

We see here that the transition from the free-flow or semi-congestion regimes to a congested state develops via one or two shock waves, followed by a slip. Note that the double shock wave is only possible when the congestion branch speed of class 2 exceeds that of class 1. In such double shock waves it is as if the heterogeneous traffic flow complies with a non-concave fundamental diagram.

8.2.4 Transitions from congestion to free-flow or semi-congestion regimes

Again, the transition from the congestion regime to the free-flow or semi-congestion regime can be broken down into two groups of transitions.

We can distinguish two stages in the first kind of transition. In an initial stage, the speed of the congested upstream state increases across a shock wave until the weighted congestion state with speed uf2 is achieved. This shock wave, therefore, has the same speed as the weighted congestion branch wtot ,which can be calculated at the

k1

k2

r.kM2 kJ1

kJ2

kM2

kM1

CG

CD AU

t

x

x0

CD

AU

CG

q

k

AU

CD CG

kJ2 kJ1

Chapter 8 122

upstream traffic composition. This state qMtot lies at the border of the congestion and the semi-congestion regimes. This state of 'semi-congestion' subsequently leads to a transition to the downstream free-flow or semi-congestion state. There are four possibilities depending on the downstream conditions as shown in table 8.1 :

Table 8.1 :Possible transitions subsequent to a wtot-wave from congestion

In each of the four possible transitions in table 8.1, the initial wave with speed wtot is followed by a wave of speed w1. This only applies, however, when wtot is less than w1, or :

12 ww < (8.36)

When this is not the case, the second transition type applies. In that case, the heterogeneous model behaves again like a LWR model with a non-concave fundamental diagram. In this case, a direct shock wave occurs characterised by decreasing density and increasing speed.

For this direct wave, as was the case in the previous paragraph, the changes in the class variables across the shock waves are related to its speed.

2

2

1

1

kq

kqU UG

∆∆=

∆∆=

(8.37)

Further derivation of this condition results in a relation between the unknown class densities in the intermediate free-flow or semi-congestion state G :

)()()(.

12121

1212 GGGGUU

GUUGG

uukuukuukkk

−+−−=

(8.38)

We know that, in a direct wave from the congestion regime to the semi-congestion or free-flow regimes, the class 2 speed will equal the free-flow speed during the intermediate state G:

22 fG uu = (8.39)

The value of the class 1 speed Gu1 depends on the downstream state. Figure 8.6 sketches the various possible G states.

When a free-flow state (AD) applies downstream, a free-flow regime can only occur in state G if the class 2 density is less than the downstream class 2 density. This clearly transpires in the semi-congestion to free-flow transitions subcase 1. Assuming the indicated downstream free-flow state, the possible states G then lie on the straight line indicated with (*).

Transitions : B=>A subcase 1 B=>A subcase 2 B=>B subcase 1 B=>B subcase 2


123

Subcase 2 of the semi-congestion to free-flow transition shows that, when a smaller class 2 density applies, state G is located in the semi-congestion area. The location of this G state is indicated by the sloping straight line (°) in figure (8.6).

Figure 8.6 : The phases diagram with an indication of the possible traffic states subsequent to a direct shock wave from congestion.

When the congestion regime applies downstream, state G can still belong to the free-flow regime. It follows from the B⇒B transition subcase 2 that this situation again requires a high class 2 density for state G. In figure (8.6), the straight line (*) again applies as the set of possible G states, at an upstream semi-congestion state (BU). The remaining possible states G then lie on the second straight line (°).

On the basis of this discussion and from figure 8.6, it appears possible to use the upstream free-flow or semi-congestion states, to define two straight lines on which state G should lie. From now on, we will indicate these straight lines in the further computations, with density L

ik

Here, point L is the intersection point of the straight lines. It is related to the upstream state as follows:

At upstream free-flow : DL kk 22 =

At upstream semi-congestion : 12

21211 .

...MM

DDM

ML

kkrkrkkrkk

−−−=

(8.40)

(8.41)

The two straight lines (*) and (°) in figure (8.6) lead to a second condition alongside (8.38). Based on these two conditions it is possible to formulate a mathematical expression for the class densities of state G.

When state G is a free-flow state, the straight line (*) applies. In that case:

11 fG uu =

1

2122 .

M

MGM

G

kkkkk −=

(8.42)

(8.43)

Combining (8.38) and (8.39) with (8.42) and (8.43) results in a quadratic equation in the unknown class 1 density :

° °° ° °

° ° ° °

* * *

* * * * *

* * * *

k1

k2

r.kM21

kM2

Lk2 BD

AD

kM1 Lk1

Chapter 8 124

( ) ( )

0).(..

).(.).(.).(..).(.

2121

121212122112122

1

=−+

−−−−−+−

4444 34444 21

444444444444 3444444444444 2144 344 21

C

fUU

MM

B

fUU

MfUU

MffMMG

A

ffMG

uukkk

uukkuukkuukkkuukk

(8.44)

Two solutions are possible:

AACBBk G

242

1−±−=

(8.45)

Here, the limitations indicated by the straight line (*) in the phases diagram must still be taken into consideration.

LG kk 110 ≤≤ (8.46)

If (8.45) does not result in a class 1 density that complies with (8.46), then state G lies on the second straight line (°) in figure 8.6. In that case the following equation applies between the class densities :

rkk

rkk

GL

LG 1

21

2 −+= (8.47)

In addition, the class 1 speed in this semi-congestion state is given by :

G

L

fffG

kkuuuu

1

12121 ).( −+=

(8.48)

Substituting (8.47), (8.48) and (8.39) in (8.38) consequently gives an expression for the class 1 density of state G :

)().)(()..)(().)((

121212

21212121211

ffLUU

fU

LLUff

LLLf

UUG

uukkrkuukrkkruukkrkuuk

k−++−

−−−−+−=

(8.49)

This is how the downstream semi-congestion state G can be computed.

The direct wave from congestion to the semi-congestion or free-flow regimes are followed by a uf2 wave. Dependent on the downstream regime and the regime to which the direct wave transformed - free-flow (*) or semi-congestion (°) - the class 1 properties change across a shock wave, a uf1 slip, or they no longer change.

To illustrate this situation, two analytical computations are sketched. In figure (8.7) state G lies on the straight line (*). The direct wave is, subsequently, followed by a uf2 and uf1 slip.


125

Figure 8.7 Congestion to free-flow transition via a direct wave shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

Figure 8.8 sketches a transition from congestion to semi-congestion. In this transition, state G lies on the second straight line (°) from figure (8.6). Subsequent to the direct shock wave, the uf2 wave completes the solution. Figure 8.8 Congestion to semi-congestion transition via a direct wave shown in (a) the phase-diagram; (b) the fundamental diagram and (c) a t-x diagram.

We can state, in conclusion, that the transition from the congestion regime to a free-flow or semi-congestion regime depends on the congestion branch speeds wi of both classes. When the class 2 congestion branch is less steep than w1, the heterogeneous model behaves like a LWR model with an non-concave fundamental diagram. Decompression shock waves then appear in the solution. In the alternative case, the weighted congestion state appears in the shape of an intermediate state.

8.3 Numerical scheme The numerical scheme retains the same properties seen in the previous sub-steps of this heterogeneous LWR model. A link is subdivided into equal cells with length ∆X in which an approximate value for the class densities is calculated per time interval ∆T. Using the Godunov rule, density is calculated as follows :

[ ])21,()2


∆∆+=∆+

(8.50)

k1

k2

r.kM2 kJ1

kJ2

kM2

kM1

CG

AD AH

BG L

t

x

x0

AD

CU

AH

BG q

k

BG AD

AH

kJ1 kJ2

t

x

x0

BD

CU

BG

k1

k2

r.kM2

kJ2

kM2

CD BU

BG Lk1

Lk2 L

q

k

BU BG

CU

kJ2 kJ1

Chapter 8 126

Here, the transition flow Gi is equated to the intensity at the cell boundaries when a Riemann problem between both cells is solved.

When looking for an expression for the transition flow, the traditionally used sending and receiving flows are not always utilised. The direct computation of the transition flow, based on the associate Riemann problem, often simplifies things considerably.

In parallel to the previous section concerning transitions, the transition flows are developed in four steps. Based on the regimes in the up- or downstream cells, one of the following methods becomes of relevance in calculating the transition flow. In combination with (8.50), the numerical scheme then can be used for a homogeneous link.

8.3.1 Up- and downstream, class 2 travels at free-flow speed

Calculating the transition flows when there is no congestion regime develops analogously to the heterogeneous free-flow numerical scheme in chapter 6. In this section, however, one does calculate the transition flow as a minimum of the sending flows Si from the upstream cell, and the receiving flows Ri in the downstream cell :

[ ]iii RSMinG ,= (8.51)

Sending flows

The sending flows are exclusively dependent on the traffic state in the upstream cell.

• Free-flow regime

The sending flow for both classes equals the class flow : UQS 11 = (8.52)

UQS 22 = (8.53)

• Semi-congestion regime

Class 1 is congested and class 2 behaves as it does in the free-flow regime :

111 . MU qS α= (8.54)

UQS 22 = (8.55)

Receiving flows

No restrictions are imposed on class 2 from the downstream cell during semi-congestion and free-flow. For this reason, the receiving flow for class 2 is equated with the sending flow :

2222 . SqR MU == α (8.56)

The class 1 receiving flows are treated in the same way as in chapter 6. The introduction of the scale factor leads to slightly changed formulas.


127

• DU11 αα >

211111 .).(. MDU

MD qrqR ααα −+= when 111 . M

DD kK α< (8.57)

21111 .).( MDUD qrQR αα −+= when 111 . M

DD kK α≥ (8.58)

• DU11 αα ≤

111 . MUD qR α= when 211111 .).(. M

UDM

UD krkK ααα −+≤ (8.59)

21111 .).( MDUD qrQR αα −+= when 211111 .).(. M

UDM

UD krkK ααα −+> (8.60)

These expressions for the sending and receiving flows are used to calculate the transition flow according to (8.51). On the basis of these formulas, all eight transition types from section 8.2.1 are dealt with correctly.

8.3.2 Up- and downstream congestion Based on the analytical transition, the transition flow is computed.

Based on the relative class density Uiβ (8.19) in the upstream cell, the weighted

capacity density UMtotK (8.21), the weighted jam density U

JtotK (8.20) and the weighted

congestion branch speed UtotW (8.22) can be computed. Dependent on these upstream

parameters, the transition flows are then computed, using the downstream speed as follows :

tot

D

DJtotU

ii

WuuKG

−=

1

..β

(8.61)

The calculated transition flows correspond to the class flows of the intermediate state G in the transition of section 8.2.2. To illustrate this situation, we show in figure 8.9 the numerical solution to the analytical example from figure 8.3. As in the previous chapters, a colour scale from black (low density) to white (high density) shows the different densities.

Figure 8.9 :Numerical solution of the congestion to congestion transition (C ⇒ C) of figure 8.3: (a) class 1 density; (b) class 2 density and (c) total density.

k1

x

t k2 ktot

Chapter 8 128

8.3.3 Upstream free-flow or semi-congestion; downstream congestion

The calculation of the transition flows is based on the analytical computation in paragraph 8.2.3.

First, we verify that the weighted capacity state appears in the solution. When (8.34) is complied with, the solution contains two shock waves and a slip. On the basis of the up- and downstream cell densities and using (8.29) and (8.30), this condition becomes:

2

2

1

22

2

221

2221

).(11).(

)..(.

J

U

J

UM

fU

MU

UMf

UU

kK

krKk

urKkK

rKkuKU

+−−

<−−

−−

(8.62)

When (8.62) is met, the transition flow equals the class flows of state H in section 8.2.3. Using (8.32) this results in :

DJJ

DfJMJJ

Df

UJJf

DUM

UkkUukkrkrkUuKkkuUrKk

G..).(..).)((

....)(

212222122

212221 +−+−−

−=

(8.63)

DJJ

DfJMJJ

Df

UJJf

DU

UkkUukkrkrkUuKkkuUK

G..).(..).)((

....

212222122

21222 +−+−−

= (8.64)

When condition (8.62) is not met, the transition develops across a direct wave. Since the speed of this wave can be both positive and negative, the transition flow is calculated as the minimum of the up- and downstream flow for this wave.

).,.min( DGi

UUii UKUKG = (8.65)

In this case, the class densities pertaining to state G are computed using (8.36) and (8.37).

Figure 8.10 : numerical solution of the free-flow to congestion transition (A ⇒ C) subcase 1 as derived analytically in figure 8.4: (a) class 1 density; (b) class 2 density and (c) total density

Figures 8.10 and 8.11 numerically compute a Riemann problem between a free-flow and a congestion regime. The wave in the first case develops direct, as was computed

k1

x

t k2 ktot


129

analytically in figure 8.4. The direct shock wave case from figure 8.5 is given in figure 8.11.

Figure 8.11 : numerical solution of the free-flow to congestion transition (A ⇒ C) subcase 2 as analytically derived in figure 8.5: (a) class 1 density; (b) class 2 density and (c) total density.

8.3.4 Upstream congestion, downstream free-flow or semi-congestion

An analytic computation of the solution to the transitions from the congestion regime to the free-flow or semi-congestion regimes was also carried out in paragraph 8.2.4. Based on the slope of the congestion branches of the homogeneous fundamental diagrams, a decompression shock wave or the weighted congestion state Mtot appeared in the solution. We take another look at this condition (8.36) :

12 ww < (8.66)

When (8.66) is met, the weighted congestion state appears in the solution. This initial shock wave is followed by the transition from the weighted congestion state to the downstream state. In this last step, the weighted congestion state functions as the upstream semi-congestion state. It is expedient, therefore, to take the weighted congestion state in the upstream cell into consideration, when calculating the transition flow. Using (8.21), the upstream class densities can then be recalculated as follows :

UU

UiM

iMtot KrKKkrK

21

2

...

+=

(8.67)

By considering the weighted congestion state upstream , the transition flows can be calculated according to section 8.3.1. The free-flow regime or the semi-congestion regimes now apply both in the up- and downstream cells. This is how the appropriate transition flows according to (8.51) are arrived at.

A decompression shock wave appears in the analytical solution, when condition (8.66) is not met. The state changes across this wave from congestion to an intermediate state G in the free-flow or semi-congestion regimes.

k1

x

t k2 ktot

Chapter 8 130

In this case, the transition flows correspond to the class flows pertaining to state G. The transition flows are then formulated on the basis of the calculations described in section 8.2.3.

The analytical solutions from figures 8.7 and 8.8 are recalculated in figure 8.12. In both cases, the numerical method gives a satisfactory approximate solution in which the speed of the traffic flow increases across a decompression shock wave.

Figure 8.12 : numerical solutions to transitions from congestion with a direct shock wave as computed analytically in figures 8.7 and 8.8.

8.4 Conclusions In this chapter, the LWR model was extended for a traffic flow consisting of two classes. It was assumed that each class on the homogeneous link under consideration complies with a triangular fundamental diagram. The LWR model applies when the traffic flow is composed of vehicles belonging to one class only. Using the triangular fundamental diagram, the traffic operations are then accurately described

This heterogeneous extension of the LWR model assumed that vehicles belonging to different classes interact according to the 'user-optimum'. This implies that each vehicle aims to maximise its speed without decreasing the speed of slower vehicles. The anisotropic condition is, therefore, met: vehicles are never influenced from behind.

It was also assumed that each vehicle continues to behave according to its fundamental diagram. In addition, a vehicle never occupies more space than is strictly necessary. Faster traffic is, therefore, not impeded by slower vehicles.

In the analytical computation, we used a scale factor to couple the fundamental diagrams of both classes. All possible stationary traffic states were divided into the following three classes: the free-flow regime, the semi-congestion regime and the

k1

x

t k2 ktot

C⇒B(fig 8.8)

C⇒A(fig 8.7)


131

congestion regime. The transitions between the various regimes always developed via slips, shock waves and fans and were always separated by intermediate stationary traffic states.

As in the traditional LWR model, a numerical method is desirable when computing approximating solutions. On the basis of the Godunov scheme, this type of method was developed.

In the next chapter, this model is further extended to model traffic networks. Chapter 11 gives a critical discussion of this heterogeneous model with possible extensions to and relations with other models.

Chapter 8 132

133

9 NETWORK TRAFFIC

In the previous chapters the LWR model was extended to a heterogeneous model. For a road with unchanging road characteristics, the assumptions regarding class interactions resulted in an unambiguous description of the traffic process. In addition the application of the numerical methods put forward in this thesis enabled a rapid computation of an approximate solution.

However, to make practical model applications possible, the model needs to be extended so that the flow of traffic can also be described on smaller networks. To this end, the numerical version of the heterogeneous LWR model is, in this chapter, extended to a network model.

As in the classical traffic models, it is assumed that a road network consists of nodes and links. In this philosophy a homogenous link always departs from and ends in a node. Nodes have no physical length and act as flow exchange locations. Based on the number of arriving and departing links, we distinguish the following node types in this chapter.

• Origin node

Traffic on the network departs from an origin node. These nodes function as feeders to the transport network. We assume that only one link can depart from these nodes while none can arrive, as in figure 9.1a

• Destination node

Traffic leaves the network in destination nodes. These 'drains' are thought to have an infinite capacity. As shown in figure 9.1b, only one link can arrive at a destination node.

Chapter 9 134

• Inhomogeneous node

A road with changing road properties is divided into homogeneous links that change road properties in inhomogeneous nodes. In such nodes, one link arrives and all traffic is directed to that one departing link as shown in figure 9.1c.

• Diverge node

In a diverge node one link arrives and exactly two links depart as shown in figure 9.1d. Using these nodes road exits can be modeled.

• Merge node

In a merge node two links arrive and one link departs as in figure 9.1e. These points enable the modeling of feeder- and merging roads.

The limited set of nodes above serve as building blocks. They enable the construction of practically all possible motorway networks. The modeling of urban networks requires a further extension of these types. For example, if one wants to model a signal controlled intersection, temporary capacity limitations must be possible in a node where several links arrive and depart.

Figure 9.1. : Node types

For the further elaboration's in this chapter, the classical methods of the LWR theory at the network level are examined per node type. This will show that, in contrast to the entropy condition on a homogeneous link, no singular fixed solution exists in this case. Different assumptions regarding traffic behaviour on nodes co-exist. Taking the assumptions on driver-behaviour, complexity and numerical aspects into account, a method will be worked out for each node type.

Origin node

Destination node

Inhomogeneous node

Diverge node

(a)

(b)

(c)

Merge node (e)

(d)

Network traffic

135

9.1 Origin node In an origin node, traffic is added to the network. We assume that only one link departs from the origin node.

A traffic model always uses a simplified representation of the actual traffic network. Thus, the traffic demand that is fed into the model in an origin node, corresponds with the assumed traffic flow of the real network.

When constructing a traffic model, the optimum model boundaries to be adopted are those that describe the full congestion area in space and time. This prevents a possible rebounding back of congestion into the origin nodes.

In spite of these principles, a buffer is often modeled in an origin node. The traffic demand is then added to this reservoir per time interval. In the classical LWR model, the flow into the network then equals the minimum of the receiving flow of the first true cell of the departing link and the sending flow leaving this buffer.

A similar philosophy is applied to the heterogeneous model. Traffic demand Demandi(t) represents, for each class, the number of vehicles that wants to enter the network per time unit in an origin node. We represent the number of vehicles in the accompanying buffer as ni(t). Thus, per time interval, the number of vehicles in the buffer is accounted for by :

))(.()()( iiii GtDemandttnttn −∆+=∆+ (9.1)

The transition flow Gi here, is the flow from the buffer to the first cell of the link that departs from the origin node. To calculate this transition flow, we turn this buffer into a fully fictitious cell. As indicated in figure 9.2, this buffer then forms an upstream extension of the homogeneous link that departs from the origin node.

To calculate the class density in the fictitious cell, we divide the number of vehicles of that class in the buffer by the cell length of the homogeneous link :

xtntK iOrigin

i ∆= )()(

(9.2)

We calculate the transition flow from this fictitious cell to the first real cell using the classical numeric method from chapter 8.3. The number of vehicles in the buffer can then be updated for each time interval, according to (9.1).

Figure 9.2 : An origin node as a buffer cell.

When congestion rebounds back to the origin cell, or when the traffic demand exceeds the network capacity, the densities calculated according to (9.2) can become larger

buffer 1 2 3

Origin node

Chapter 9 136

than the jam density. At that moment, the numerical methods of chapter 8 continue to apply.

The only disadvantage is that an immediate mixing of classes in the buffer occurs. The composition of the recently added traffic demand then disturbs the historic class composition in the buffer. This phenomenon, however, only occurs when congestion rebounds to the source or when the traffic demand is too large for the network. Neither of these phenomena should occur in a careful modeling approach where an origin node using a buffer is adequately modeled.

9.2 Destination node In a destination mode traffic disappears from the modeled network. We assume that these drains do not restrict the modeled network. This is why the capacity of these drains is considered to be infinite. But, because the link that arrives in this node has a finite capacity, the outflow from a network does remain restricted.

In practice, the arriving link in a destination node is extended by an empty cell as in figure 9.3. The transition flow between the last real cell and the fictitious empty destination cell is, subsequently, calculated using the numerical scheme for a homogeneous link from chapter 8.3.

Figure 9.3 : The destination node as a fictitious, always empty cell.

In the classical network model for the LWR model, it suffices to assume that this transition flow equals the sending flow from the last real cell. The sending flow in the heterogeneous model, however, is not unambiguously determined. Therefore, the transition flow to a fictitious empty cell must be explicitly verified.

9.3 Inhomogeneous node In the network philosophy applied here there are no links over which road properties can alter. A modeled network consists exclusively of homogeneous links that are connected to each other in the nodes. To model road property changes, inhomogeneous nodes are defined. In such a node, one link only arrives and one departs. Using these nodes, a changing road section can be divided into homogeneous parts that are connected to each other in these inhomogeneous nodes.

An inhomogeneous node does not cause problems in the network versions of the LWR model. As in a homogeneous link, the sending flow is calculated in the last cell of the arriving link and in the receiving flow of the first cell of the departing link.

Destination node

empty n-2 n-1 n

Network traffic

137

The transition flows between the two links then is the minimum of these sending and receiving flows (Lebacque 1996). In the heterogeneous LWR model, however, the receiving flows are dependent on the upstream cell. Consequently, this method does not apply. Therefore, an alternative method has to be drawn up.

The transition flows Gi in an inhomogeneous node can, in principle, be computed analytically using the corresponding Riemann problem. This section, however, formulates a numerical approach. The transition flows from the arriving to the departing link in an inhomogeneous node are calculated in two steps. Each step gives a possible transition flow for each class. The transition flow that will be applied in the end, is then calculated as the minimum of these two values. Figure 9.4 illustrates this approach.

Figure 9.4 : Calculating the transition flow at an inhomogeneous node.

A first step looks exclusively at the upstream link. The eventual transition flow can never exceed the maximum outflow from the arriving link. We compute this maximal sending flow in the same way as we did at the destination node. The transition flow from the last cell of the arriving link to a fictitious empty cell gives us the maximal outflow max

iS leaving that cell.

A second step computes a transition flow between a fictitious cell and the first cell of the departing link. Density in this fictitious cell will be related to the density in the last cell of the arriving link. Since the properties of the departing link apply, the homogeneous-link-method from chapter 8.3 can again be used in this calculation. Figure 9.4 gives an outline of the addition of this fictitious cell Z.

We assume that the speed in the fictitious cell Z equals that of the free-flow speed of the slowest class. In that case the weighted capacity state applies and the class densities in function of a given traffic composition βi (8.21) or a given fraction αi can be computed :

DM

DZZDZ

DM

DZZ kr

rkrk 21

21

211 ..

... α

βββ =

+=

(9.3)

DM

ZZDZ

DM

DZZ k

rkrk 22

21

222 .

... α

βββ =

+=

(9.4)

To compute the class densities in cell Z, we establish a connection between traffic composition and fractions and the traffic state in the last cell U of the upstream link.

Inhomogeneous node

empty n-2 n-1 n

Z 1 2 3

maxiS

ZiG

),( max Ziii GSMinG =

Chapter 9 138

When the congestion regime applies in cell U, the relative density of this upstream cell is used to calculate the state in the fictitious cell Z. In this way, traffic composition does not alter in an upstream congestion state.

Ui

Zi ββ = (9.5)

When the free-flow or the semi-congestion regime applies in the upstream cell U, we assume that both classes in cell Z proportionally occupy the same space. To this end, we define the actual space that a class occupies in an upstream cell as :

UMi

iUi q

S max

=α (9.6)

We can then compute the density in the fictitious cell using (9.3) and (9.4) when the fractions are computed with :

)( 21UU

UiZ

i αααα+

= (9.7)

These arithmetical rules respect the heterogeneous characteristics of traffic in an inhomogeneous node. Upstream congestion maintains traffic composition. Likewise, the effective class fractions in an upward parallel traffic regime remain proportional. When the speed of the class 2 congestion branch exceeds that of the class 1 congestion branch, application of (9.6) and (9.7) gives rise to the same class densities during congestion as given by (9.5). Only when the class 2 congestion branch is steeper than class 1 does (9.5) give different values. When, in following sections, the inhomogeneous node returns in the computations of diverge and merge nodes, the approximate densities in fictitious cells will always be computed using the effective spaces given by (9.6). The approximation pertaining to a steeper class 2 congestion branch, however, is, in that case, small compared to the gain in calculation time.

Thus, the traffic state in the fictitious cell Z is fixed and transition flow ZiG can,

therefore, be computed from the fictitious cell Z to the first cell of the departing link.

The transition flow to be applied is set equal to the minimum of the values of the two steps :

),( max Ziii GSMinG = (9.8)

The consequences of the assumed class densities in the fictitious cell Z require further clarification.

When the congestion regime applies upstream, the following can happen downstream:

• The congestion regime applies downstream also. In that case, the methodology developed causes traffic composition up- and downstream to be preserved. In addition the first step ensures a restricted outflow from the upstream cell. In fact, when the transition flow exceeds the weighted capacity in the upstream cell U

MtotQ , the semi-congestion or the free-flow regime appears in the solution.

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139

• The free-flow or semi-congestion regime applies downstream. Here, the weighted congestion branch wtot upstream from the inhomogeneous node appears in the solution. This solution is not accurate when the congestion branch speed of the slowest class w2 is less than w1. As described in chapter 8.2, a direct wave appears in that case from the congestion to the free-flow or semi-congestion regime. When applying the numerical method around an inhomogeneous node, the approximation with a weighted congestion branch is seen as sufficiently accurate.

The congestion regime can continue to apply between the wave with speed wtot

and the inhomogeneous node. In that case the downstream link shows a smaller capacity than the arriving link. As is to be expected, traffic composition of the inhomogeneous node is retained.

When the free-flow or semi-congestion regime applies upstream:

• The congestion regime applies downstream.

In the case of a homogeneous link a dual wave appears in this situation against the direction of traffic or a straight wave as described in chapter 8.2.3.

The method suggested can show anomalies, especially in the case of a straight wave. On the other hand the extent to which this situation occurs at the level of an inhomogeneous node is uncertain. This state can only be retained when the volume of upstream arriving traffic demand remains identical to the capacity downstream from the inhomogeneous node. This situation, therefore, will only sporadically continue for any length of time at the level of an inhomogeneous node, which renders possible anomalies acceptable.

• The semi-congestion or free-flow regime also applies downstream.

In this case, the proposed method does not seem to lead to significant anomalies. Temporary anomalies can occur only when downstream capacity is significantly less than upstream capacity. However, this situation will rapidly evolve into a congestion regime upstream from the inhomogeneous node.

The computation method proposed for inhomogeneous nodes is far from unique. The assumption regarding the conservation of vehicle composition during a congestion regime upstream, in particular, has been a deciding factor in its formulation. Introducing more exact assumptions, other methods can probably be developed although they would be more computing-intensive. However, the formulated method applies sufficiently for the intended network applications.

9.4 Diverge node In a diverge node, traffic arrives at one link and there are two departing links. A diverge node gives the opportunity to model fork junctions and exit ramps. An outline of the existing diverge node approximations in the LWR model will be followed by the presentation of a generally applicable methodology. We will round off this section

Chapter 9 140

with the formulation of a simplified diverge node approximation that will be applied in the case study of the next chapter.

9.4.1 Diverge nodes in the LWR model For the network version of the LWR model, the following discussion outlines the various diverge models applied. In the remaining discussion, the arriving link will be indicated by U, and the two downstream links by A and B or by the index X.

• FIFO model with split factors (Daganzo 1995a)

A first-in-first-out model usually uses split factors. It is assumed that traffic upstream from the diverge node divides itself in the proportions pA en pB across the departing link. It is also assumed that an obstruction on either one of the two departing links rebounds on the entire arriving link. Consequently, the non-obstructed exit becomes obstructed also. The transition flow from the upstream link to the downstream link X then becomes :

= X

XXX

pRMinSMinpG ,.

(9.9)

The FIFO-assumption, where an obstructed downstream link is able to obstruct the entire upstream traffic, applies fully only to a one-lane road upstream from the diverge node. Figure 9.5a shows how an upstream obstruction propagates itself on the entire link upstream from a FIFO diverge.

• FIFO method with user-classes

Instead of working with split factors per node, the homogeneous traffic flow can be divided into user-classes. The entire sending flow on a node is then, by definition, divided into SA and SB. The FIFO method can then be applied whereby the split factors are calculated for each separate time interval as :

tot

XX

SSp =

(9.10)

• Multi-lane method with split factors (Lebacque 1996)

This method too works with a split factor of pX for each downstream link. In this case it is assumed that the total sending flow is divided so that the transition flow towards the downstream link X equals :

[ ]XXX RpSMinG ,.= (9.11)

Here a downstream obstructed link does not interfere with the flow to the other link. This is illustrated in figure 9.5b. This is how the multi-lane character of the upstream road is modeled. A disadvantage of this method is that the split factors are not used as fixed boundary condition. The relation between the actual transition flows does not automatically agree with the split factors. Especially when one of the two downstream links is structurally obstructed, traffic demand on the diverge node is inadequately divided.

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141

• Multi-lane method with user-classes

In the multi-lane method, user-classes can also be used. In that case, the transition flow is computed as follows :

[ ]XXX RSMinG ,= (9.12)

In regard to the split factors this approach has the advantage that route choice is guaranteed. The portion that wants to enter an obstructed link rises over time, which makes behaviour more realistic. On the other hand, the obstruction is only able to fully continue downstream when the non-obstructed link also becomes obstructed.

• Special lanes (Daganzo 1997b)

When applying user-classes in diverges, the method above continually shows deficiencies. The dynamics of multi-lane roads do not appear in the FIFO method. The multi-lane method only addresses upstream obstruction in the last cell before the diverge.

Lebacque (1996) was the first to try and model the effect of separate auxiliary lanes upstream from a diverge node. Daganzo (1997b) developed a comprehensive computation for an obstructed motorway exit. In this 'special lanes method' the entire arriving link is split up. A fraction of the total road αA is reserved for one of the downstream links, and this fraction can not be used by traffic for the other downstream link. All traffic can use the other fraction 1-αA. Enforcing these requirements to the upstream link and applying the multi-lane method with user-classes, enables adequate modeling of an obstructed motorway exit. Figure 9.5c illustrates this configuration.

Figure 9.5 : (a) FIFO diverge ; (b) a multi-lane diverge and (c) a special lanes diverge.

The assumptions about the traffic mechanisms upstream of a diverge node have led to a range of diverge models. A simple application to the heterogeneous LWR model, however, is not obvious.

9.4.2 A heterogeneous diverge node In this section we formulate a heterogeneous diverge model according to the multi-lane method with split factors. Applying user-classes would increase the complexity of the model considerably. In addition, these user-classes offer no added value at this stage of smaller networks model development.

(a) (b) (c)

Chapter 9 142

The choice for the multi-lane approach is closest to the philosophy of the heterogeneous LWR model. The first-in-first-out behaviour is explicitly avoided during the free-flow and semi-congestion regimes. In the computation of the multi-lane method for the heterogeneous LWR model, a solution will be put forward for the unobserved split factors.

Unlike the classical LWR model, it is not possible to calculate the downstream receiving flow without knowledge of the traffic composition in the upstream cell. That is why we adapt the methodology for the inhomogeneous node in order to model diverge nodes. Starting from the split factors for each link and for each class X

ip , we develop a diverge model as shown schematically in figure 9.6. Note that we assume the split proportions to comply with :

1=+ Bi

Ai pp (9.13)

Figure 9.6 : Calculating transition flows in a diverge node

The inhomogeneous node model can then be applied, whereby the maximal outflow from the upstream cell to link X is restricted by the split factor :

maxmax . iXi

Xi SpS = (9.14)

For each downstream link a fictitious cell can be introduced in the second step. Class densities in these fictitious cells are calculated using effective space (9.6), thus :

)..(.

2211UXUX

Ui

XiXZ

i ppp

αααα

+=

(9.15)

Using (9.3) and (9.4) class densities in the fictitious cells are calculated next. This second step then gives the transition flows XZ

iG per link.

The transition flow for the downstream link that is to be applied then becomes the minimum of the two values arrived at :

empty n-2 n-1 n

BZ 1 2 3

maxiS

BZiG

[ ]BZii

BBi GSpMinG ,. max=

Diverge node

AZ 1 2 3

AZiG

[ ]AZii

AAi GSpMinG ,. max=

Network traffic

143

[ ]XZii

XXi GSpMinG ,. max= (9.16)

The sum of all these transition flows then leaves the last cell of the upstream link :

∑= Xii GG (9.17)

In this multi-lane methodology, the split ratios are not strictly adhered to when a downstream link is obstructed. To maintain the correct split factors on a long term basis, we develop a method to regulate the split factors over time. We begin by defining the cumulative transition flow in the diverge node as follows :

∑=t

t

Xi

Xi tGtH

0

)()( (9.18)

Using this cumulative transition flow we keep track of the actual split ratio. To calculate the split ratio in the next time interval, the following would, ideally, apply :

Xi

iBi

Ai

iXi

Xi p

ttStHtHttSttptH =

∆+++∆+∆++)()()(

)().()(max

max

(9.19)

Starting from this equation, the split ratio to be applied can be calculated in the following time interval )( ttp X

i ∆+ on the basis of the historic transition flow :

[ ])(

)()()()(.)( max

max

ttStHttStHtHpttp

i

Xii

Bi

Ai

XiX

i ∆+−∆+++=∆+

(9.20)

This formulation results in a stable method which enables stricter compliance with the split factors. The resulting split factor is, however, limited by :

1)(0 ≤∆+≤ ttp Xi (9.21)

The application of this multi-lane method with split factors leads to effective modeling of a diverge node, even when a downstream link is obstructed. Refinements are possible at a later stage. To fully express the heterogeneous character, split factors are desirable for each class. An extension to user-classes or even to auxiliary lanes will lead to a more complete diverge model.

9.4.3 A simplified diverge node Specifically for the case-study in the next chapter, a simplified diverge method will be formulated. Here, a diverge node serves to model an exit lane on a primary road. In the case-study, detailed traffic counts are available. In this simplified method, observed exit lane flows are used as fixed boundary conditions. This negates the need for behaviour assumptions in the diverge model. Due to the strong similarity between observed and modeled exit lane flows, the emphasis of the case-study will be on the primary road and not on the nodes.

We assume a diverge node as proposed in figure 9.7. Downstream of the diverge node, we indicate the primary road by link A and the exit lane by B. In an off-line

Chapter 9 144

simulation, the observed exit lane flows are known per time interval. These flows, which are to be strictly adhered to, are indicated for each class by )(tN B

i .

Figure 9.7:A simplified diverge node

We can use these observed flows to dynamically verify the split factor. We aim, after all, to equalise the sum of the cumulative transition flow in the direction of the exit lane and the cumulative counts. This enables us to rewrite (9.19) to the following condition for the split factor in the next time interval :

∑∆+

=∆+∆++tt

t

Bii

Bi

Bi tNttSttptH

0

)()().()( max (9.22)

Using (9.22) to find the unknown split factor gives a way to introduce observed counts as fixed boundary conditions.

This method is only possible in off-line studies when detailed counts are available. When we introduce exit lane flows as fixed boundary conditions into the model, travel behaviour in a diverge is not actually modeled. The counts imposed enable one to focus on through-going traffic.

The impact of an obstructed exit lane is partly included. On the one hand, the obstruction becomes apparent in the lowered observed transition flows to the exit lane that are strictly adhered to. On the other hand, the model does not allow for the build-up of exiting traffic on the primary road. In this simplified model an obstructed exit lane implies that more traffic stays on the primary road. This behaviour is often realistic because an obstructed link affects route choice. At that moment, more vehicles will choose another branch of the junction and use another route to reach their destination.

This diverge model will be applied in the next chapter in the simulation of the traffic operations on a motorway where the exit lanes are not obstructed. Because of this, an exit lane is modeled as a drain with a varying capacity.

empty n-2 n-1 n

1 2 3

maxiS

[ ]Bii

Bi NSMinG ,max=

AZ 1 2 3

AZiG

[ ]AZi

Bii

Ai GGSMinG ,max −=

exit

primary road

Diverge node primary road

exit

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145

9.5 Merge node In a merge node two links arrive and one only link departs. Joining roads and feeder roads can be modeled using merge nodes. Existing merge models from the LWR model will be examined first. Next, a general methodology for the heterogeneous LWR model will be developed.

9.5.1 Merge nodes in the LWR model Several methods exist in the network version of the classical LWR model to model a merge node. In the remaining discussion, the upstream links will be indicated by A, B or index X. The downstream link will be indicated by D.

• Optimal merge model (Daganzo 1995a)

This method assumes that the departing link is used optimally and that the imposed priority proportions are strictly adhered to. When the sum of the sending flows from both upstream links is less than the receiving flow of the downstream link, the transition flow becomes :

XX SG = when RSS BA ≤+ (9.23)

With larger sending flows the total transition flow to the departing link will equal that of the receiving flow. Priority proportions pX will be used when dividing this total transition flow over the upstream links. One assumes here that the proportions between transition flows GX are equal to these priority proportions when the sending flows are large. This leads to the following expression :

[ ]RpSRSMedianG ABAX .,, −= when RSS BA >+ (9.24)

• Fractioned-off merge model (Lebacque 1996)

In this method, the road space of the downstream link is split up. For each upstream link it is assumed that fraction αX of the downstream link is assigned exclusively to the upstream link X. In this case, the transition flow from link X is calculated as follows :

[ ]RSMinG XXX .,α= (9.25)

In this method the outflow from an upstream link can never exceed the assigned fraction of the capacity of the downstream link. Hence the downstream link is not necessarily used optimally, even when the feeding link can send on more traffic.

• Fairness model (Jin and Zhang 2003)

This simplified merge model does not impose fixed priority proportions. The optimal merge model is applied whereby the priority proportions are equal to the proportions of the upstream sending flows. The model thus achieved has

Chapter 9 146

less parameters and is numerically less burdensome. The transition flows are given by :

+

= BA

XXX

SSSRSMinG .,

(9.26)

We can also interpret this model as the fractioned-off model whereby the fractions are determined as being dynamic and proportional to the sending flows.

Again it goes without saying that an extension of these merge models to the heterogeneous LWR model is not self-evident. Since the receiving flow can never be calculated independent of the upstream vehicle composition, simplifications will become necessary.

9.5.2 A heterogeneous merge node In this section, the Fairness model is extended and adapted to the heterogeneous LWR model. Central to this model is that the priority proportions are formulated proportionally to the sending flows. Figure 9.8 gives the method schematically

Figure 9.8 : Calculating the transition flows in a merge node

Starting from the inhomogeneous node method from section 9.3, the final transition flow is calculated in two steps, as presented schematically in figure 9.8. In a first step, the maximum sending flow for each upstream link is computed. This is done as explained above, by computing the transition flow from the last cell to a fictitious empty cell. For each upstream link, this first step gives a maximal transition flow

maxXiS .

Merge node

empty n-2 n-1 n

maxAiS

Z 1 2 3

ZiG

+

= ZiB

iA

i

AiA

iAi G

SSSSMinG maxmax

maxmax ,

empty n-2 n-1 n

maxBiS

+

= ZiB

iAi

BiB

iBi G

SSSSMinG maxmax

maxmax ,

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147

In a second step, a fictitious cell Z is added upstream from the departing link. The traffic state in this cell is connected with the sending flows from the upstream cell. As in the inhomogeneous node method, the weighted congestion state of the free-flow speed of class 2 is assumed. Starting from the sending flows from the two upstream links, the effective space as defined in (9.6) is now computed as follows:

BMi

AMi

Bi

AiU

i qqSS

++=

maxmax

α (9.27)

The fractions of each class are then fixed using (9.7) :

)( 21UU

UiZ

i αααα+

= (9.28)

The proposed approach to compute class densities in the fictitious cell Z relies on the fairness method. When determining the fractions, the relative sending flow from both upstream cells is taken into consideration. In addition, (9.28) takes the class composition of the upstream links into consideration. In this way, an optimal reflection of the upstream traffic flow in the fictitious cell Z is aimed at.

Based on the maximum sending flows and the transition flows from cell Z, the final transitions flows are then computed as :

+

= ZiB

iA

i

XiX

iX

i GSS

SSMinG .)(

, maxmax

maxmax

(9.29)

The method arrived at gives an approximate computation of the transition flows and leads to optimum utilisation of the downstream link. The fairness method is included in this, because the transition flows are proportional to the sending flows

9.7 Conclusions In this chapter, the heterogeneous LWR model was extended to a network model. To this end, nodes were defined in which homogeneous links begin and end, and where an exchange of traffic takes place.

Five types of nodes were defined. Origin nodes function as points of input in the network while destination nodes form drains. Road characteristics on an on-going link can alter at an inhomogeneous node. On- en off-ramps can be modeled, respectively with a merge and a diverge

The accepted node configurations limit the application to a network of motorways. User-classes were not part of this initial, approximate numerical computation for a heterogeneous LWR network. Route choice behaviour, therefore, has to be imposed explicitly. Consequently, it can not be computed dynamically. This, on the other hand, limits the need for calibration data somewhat.

In the next chapter, a case-study will be examined, based on the formulated network version of the heterogeneous LWR model. The heterogeneous model will be applied and tested on a motorway with feeder- and exit roads.

Chapter 9 148

149

10 CASE STUDY

This chapter presents an application of the developed heterogeneous LWR model. Using the network extensions from the previous chapter we model a stretch of 7.5 km of motorway during the morning rush hour. This stretch of motorway features three on-ramps and two off-ramps. We compare the observed and modeled traffic patterns in order to show the capabilities of the model.

This study is part of the first phase of model development. Its emphasis lies on the through traffic on the main road. To this end, the modeling of on- and off-ramps is simplified and route choice modeling is absent. Practical aspects regarding user-friendliness and model versatility are not considered.

Nor does this chapter test the validity of the assumptions in the heterogeneous model against actual data. This is done in Chapter 11, which contains an in-depth theoretical discussion on the subject. Comparing the model assumptions to actual data requires more accurate research in which tools for class-dependent data-analysis and microscopic data-interpretation must also be developed.

The study area and the available traffic observations are first highlighted. The traffic data is analysed in two ways. This outlines the structural congestion in the study area and it illustrates a number of recent insights pertaining to congestion mechanisms. The conversion from the actual network to a set of links and nodes is discussed during the model's construction. The input of traffic counts and the construction of the fundamental diagrams is illustrated.

During the calibration phase the model parameters are adjusted to achieve an optimum representation of the existing traffic pattern. Measurement data from a different day is used to validate the model.

Chapter 10 150

10.1 The study area The study area comprises the Dutch A9 motorway near Badhoevedorp. Figure 10.1 gives a schematic representation of the motorway with three on-ramps and two off-ramps. Gradients or bends that could influence the traffic flow on this two-lane motorway are absent. The traffic lanes are 3.5 meters wide and the permitted maximum speed is 100 km/hour.

Figure 10.1 :Schematic representation of the study area.

The motorway is equipped with 10 detectors that measure individual vehicle data. These dual loop detectors accurately register the detection time of each vehicle to a tenth of a second. Individual speeds, vehicle length and traffic lanes are also recorded.

The measurements were carried out in the autumn of 1994. Due to their accuracy and high quality this data has already been used in a number of studies. Dijker et al.(1997) used them to calibrate the microsimulation model FOSIM. Hoogendoorn (1999) and Hoogendoorn and Bovy (1999) used them in a multi-class and multilane kinetic model. Helbing (1997), Tilch B. and Helbing (1999) and Smulders et al. (2000) used these data to investigate empirical congestion phenomena.

10.2 Analysing the observations In this analysis two methods of analysis are employed to study the observations of 20 October 1994. By filtering these observations according to Treiber and Helbing (2002) we get a global view of the traffic pattern. Applying the N-curves according to Cassidy and Windover (1995) gives a more thorough picture of the congestion.

10.2.1 The spatio-temporal traffic filter The average speed and the flow in a t-x diagram are filtered using the filters from Treiber en Helbing (2002). Appendix A presents a short explanation of the method. The values estimated for the entire t-x space are based on the observed measurement values at the detector locations. The basic idea is that the properties of the traffic flow move with the flow of traffic during free-flow. During congestion they travel against the flow. Since the filter method works with mean measurement values, we determine (space) mean speed and flow per minute at each detector location.

The method is applied to each separate homogeneous road section. The value of the flow and the average speed between kilometres 43.31 and 42.25 is, therefore, estimated based on data from detectors D3 and D4 only. All homogeneous sections in

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the study area are filtered analogously. Figure 10.2 shows the filtered average speed and the flow for the entire section in a t-x diagram.

The value in the figures is shown by a colour scale. At 7.15am the mean speed and flow diagrams show a number of disruptions that move with the flow. This leads us to conclude that the free-flow regime previously applied to the entire section. The fluctuations in flow and speed originate upstream from the study area. Temporary variations in the traffic demand propagate over time and along the section. A number of waves are reinforced by the on-ramps. At 6.40am, a weak wave in the flow diagram downstream from detector D4 increases in size. A temporary increase in traffic demand on the second on-ramp even leads to a wave in the flow diagram at 7.00am. An off-ramp can reduce these fluctuations in the flow somewhat. This phenomenon is clearly visible near the second off-ramp at detector D10.

Figure 10.2 (a) The average speed and (b) the flow in a t-x diagram for 20 October 1994.

During free-flow, the fluctuations in the flow exceed the fluctuations in speed. Because the waves are also present in the speed diagram this analysis shows that the free-flow branch of the global traffic flow in a k-q diagram cannot be a perfectly straight line.

At approximately 7.25am, congestion arises between detectors D8 and D9. Here, the increased flow that was generated via the second on-ramp earlier on, collapses. This location lies more than 1.3 kilometres downstream from the second off-ramp and it confirms that the bottleneck is localised downstream from the actual merge. The concept of funnel capacity is used to describe this phenomenon. From that moment congestion waves start to emerge propagating against the traffic flow. Due to the presence of the different on-ramps, the average value of the speed in these start- and

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stop waves decreases as a function of distance to the bottleneck location. The speed between detectors D3 and D4 hardly ever exceeds 30km/h.

Downstream from the bottleneck location minor waves arise that move with the flow. Thus, the outflow from congestion fluctuates lightly.

Between 9am and 9.15am the location of the bottleneck moved upstream, beyond detector D8. The congestion regime disappears after 9.30 am.

The outflow from the tailback has a lower flow value than the maximum observed flow for 7.25am. These high flows are only observed again beyond the next on-ramp, downstream from D11. The free-flow speed downstream from the bottleneck is also considerably lower than the speed in the free-flow regime observed earlier on.

The filter method gives a global insight in traffic patterns. These figures are somewhat distorted because the filter method uses implicit properties of the LWR model with a triangular fundamental diagram. A more thorough analysis, based on non-averaged and unfiltered measurement values, will be applied in the next section.

10.2.2 Oblique N-plots By counting the number of vehicles that pass a detector, we get the cumulative number of vehicles in function of the time N(t). When these cumulative curves for the different detectors are represented in the same t-N diagram, we can determine the number of vehicles and the travel time between two detectors. The significant rises of the N-curves over time, however, reduces the readability of these diagrams. To overcome this problem, Cassidy and Windover (1995) have developed a method in which the diagrams are scaled in relation to a basic flow. A graphic sketch of N(t) – t.q0 gives an oblique diagram that is easier to analyse. Muñoz and Daganzo (2002a) perfected these diagrams with a basic flow, rendering a fully consistent method. This methodology is elucidated in appendix B.

For the study area, the oblique N-plots are sketched for the four detectors D7, D8, D9 and D10. Conservation of vehicles applies to this section and the congestion regime arises.

At a basic flow of 2160 veh/hour, figure 10.3 gives the oblique N-curves for the free-flow regime between 6 and 7am. The local gradient of the curves indicates the flow, whereby a horizontal line corresponds to the basic flow.

In figure 10.3, the two waves become clearly visible during the free-flow regime. At 6.27am, the flow near detector D7 decreases to 548 veh/hour. After a short interval, this low flow appears on the following detectors as well. This low flow is followed by a period of larger flow. From 6.30 am, the flow at detector D7 increases to no less than 4458 veh/hour. This phenomenon repeats itself and we even see a flow of 4927 veh/hour.

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Figure 10.3 : Oblique N-plots with a basic flow of 2160 veh/h

Both waves are the result of a moving bottleneck. A slow vehicle on the right-hand lane causes a local bottleneck. The fact that trucks overtake this slow vehicle as well, explains the low flow of 548 veh/hour beyond this moving bottleneck. Upstream of this slow vehicle, there is the formation of a tailback of fairly homogeneous speed, in which few vehicle interactions occur. This explains why high flow values are achieved in this stream.

In 1989, Belgium introduced a regulation-strategy of moving bottlenecks, called 'block-driving'. Police vehicles move within the traffic flow at a constant speed of 70 km/hour and prevent other vehicles from overtaking them. These purposely created moving bottlenecks cause a period of high flow upstream of these police vehicles in which speeds are fairly homogeneous. The flow downstream of these vehicles is, necessarily, low. This measure aims to increase safety levels in heavy traffic by harmonising speeds.

It is a misconception to think that this intervention also increases capacity. Since the period of low flow in front of the regulation vehicle must also be included, the average flow on such road sections is lower than capacity and travel time increases for all road users. The potential for capacity gain over a longer term arises only when this measure leads to a significant decrease in the probability of accidents occurring.

Figure 10.4 shows the oblique N-plots between 7am and 10.30am. The flow at detector D7 rises to 4760 veh/hour at 7.13am. As indicated in the figure, the same flow is observed moments later at the downstream detectors. All small perturbations propagate in the travel direction. This high flow, therefore, pertains to the free-flow regime. At 7.25am the flow near detector D9 decreases to 3600 veh/hour. After a short while this decrease is also observed near D10. The free-flow regime continues to

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apply, therefore, downstream from D9. This decreased flow at D9 is not due to a change in traffic demand. The flow in D8 only decreases after 7.25am, followed by a decrease in D7. This upstream continuation of the decreasing flow suggests that we are dealing with the congestion regime. Accordingly, the bottleneck, the location where the congestion regime arises, lies some way downstream from the on-ramp between detectors D8 and D9. In contrast to the predictions of the LWR model, the flow downstream of the bottleneck is lower than the flow that existed before the bottleneck became active.

From 7.25am, the flow at D9 is about 3600 veh/hour. A temporary decrease at about 7.35am is followed by an increase to 3770 veh/hour. Bertini and Cassidy (2002) also note this recuperation of bottleneck flows.

Figure 10.4 : Oblique N-plots with a basic flow of 3600 veh/hour

The flow upstream from the bottleneck becomes more erratic as the distance to the bottleneck increases. This is because the fluctuation in the flow at D7 show a greater amplitude than at D8. The propagation of these perturbations against the traffic flow are sometimes seen in the figure. It is impossible to formulate an unambiguous systematic match.

The free-flow regime downstream from the bottleneck also shows small fluctuations in flow values. These waves in the outflow from the bottleneck clearly propagate with the traffic flow and can be observed both at D9 and D10. Some of these fluctuations are already discernible at detector D8. This suggests that these fluctuations are due to the heterogeneous character of the traffic flow.

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The free-flow regime becomes apparent again after 8.55am. For a quarter of an hour, the shape of the cumulative curve in D8, D9 and D10 is almost identical. This is followed by a short period of congestion. At 3350 veh/hour, the flow is considerably less than it was in the first congestion period. Trucks probably contribute to this situation. Figure 10.5 shows proportion of long vehicles, with lengths greater than 6 meters, as a function of time This figure shows that the share of large vehicles is larger at the end of the peak period.

Figure 10.5 : Proportion of long vehicles (L>6 m) in function of time

After 9.25am, the free-flow regime applies again across the four detectors. Once again, the progressive fluctuations in the traffic demand become clearly visible across the successive detectors.

10.3 Model construction This section explains the construction of the network-model for the study area. The network extension in chapter 9 shows that the road network consist of a set of homogeneous links that are connected at nodes. Five node types are sufficient to construct virtually all motorway configurations.

Figure 10.6 shows the actual road network and the model representation. Letters indicate the nodes, the links are numbered.

Figure 10.6 : (a) the road network and (b) its model representation

% long vehicles

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The road is subdivided into seven homogeneous links. Node A is the upstream origin node. The first on-ramp is modeled in merge node B. Link 8 feeds this on-ramp. The origin Node I receives the traffic demand input for the on-ramp.

The first off-ramp is localised in diverge node C. The traffic leaves the modeled network via link 9 and destination node J. The traffic from the second on-ramp in merge node D arrives from origin node K across link 10.

The road section between merge node D and the next off-ramp in the diverge node F are modeled using two links. From our observations, we know that the actual bottleneck is situated between detectors D8 and D9. Immers (1980) suggested introducing an extra node to the network at the location of the actual bottleneck location. If this were done, the capacity funnel could be described with a (heterogeneous) LWR model after all. In its application, the capacity of link 5 would then have to be less than that of link 4. Here, the inhomogeneous node E, halfway between detectors D8 and D9, functions as the bottleneck location. This trick enables one to model the capacity funnel downstream from the second on-ramp. Additional inhomogeneous nodes downstream from the other merge nodes would be required to model this effect there also.

The second off-ramp is modeled in diverge node F. The exiting vehicles leave the model via link 11 and destination node L. A last on-ramp is shown in merge node G. Arriving traffic enter the road via the upstream link 12 and the origin node M. All traffic leaves the network beyond link 7, via destination node H. All links for the on- and off-ramps are 200 meter long.

In the model, the vehicles are divided into two classes. The length of the class 1 vehicles is less than 6 meters. Longer vehicles belong to class 2. It is not difficult to distinguish these classes in the observations, if we assume that the longer vehicles also have the lowest free-flow speed. It is not so easy to link a more consistent classification, with prevailing driver properties and a range of vehicle properties, to the observations. A vehicle classification based on length appears sufficient for this initial application

The observations provide the traffic demand in the four origin nodes. The input of the model consists of the traffic demand for each node per minute and per class. The counted flow on the acceleration lane gives the traffic demand for the on-ramps. In this way, the queue at the entry point is not taken into account. This means that the traffic demand of the on-ramps in the model will have to enter the primary road directly. Congestion at links 8, 10 and 12 is not desirable. This is how the model uses the counted on-ramp flows as a fixed boundary condition. The emphasis, therefore, lies firmly on the through road and the available data is correctly interpreted.

When applying the merge node from chapter 9.5 we want to give the entering traffic the greatest possible advantage. In the fairness method, the traffic composition downstream from the merge is proportional to the sending flows. Increasing the capacity of the on-ramp links also increases the maximum sending flow and the share of the entering traffic. This is why the road properties of links 8, 10 and 12 are not coupled to the actual road characteristics of these links. Each time, the properties of

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the link downstream of the merge are assigned to these links so that the queues on these links can be kept at a minimum. Table 10.1 shows this assignment.

Table 10.1 : On-ramp links properties

On-ramp link Link properties 8 2 10 4 12 7

The diverge nodes are modeled using the method from chapter 9.4.3. In this simplified diverge node the counts of the exiting traffic are immediately used to calculate the amount of traffic that is branching off. In other words, these counts are imposed on the model. Count fluctuations automatically lead to a change in the number of vehicles leaving the modeled primary road. These imposed fluctuations can lead to 'start-and-stop' wave patterns in the model during congestion upstream of diverge nodes.

Since congestion does not occur on the off-ramp links, the link properties of the off-ramp links are equated to those of the links upstream from the diverge node. Table 10.2 shows the link properties of links 9 and 11.

Table 10.2 : Off-ramp link properties

Off-ramp link Link-properties 9 2 11 5

When modeling the merges and diverges, the counts of the arriving and departing traffic are observed. In this way the emphasis in this case study lies on the ongoing primary road while the complex modeling of filtering in and pulling out is avoided as much as possible. The limited number of count-data, in fact, leaves no room for another approach.

The numerical simulation uses a time interval ∆t of three seconds. The cell length ∆x is determined for each link, in such a way that the number of cells is maximal while at the same time condition (10.1) is complied with.

1futx ≥

∆∆

(10.1)

At a free-flow speed of 120 km/hour this gives a minimum cell-length of 100 meters.

Two homogeneous fundamental diagrams apply for each link. These triangular diagrams function as the basic parameters of the developed heterogeneous LWR model. They are calibrated by minimising the difference between the modeled and the observed traffic pattern.

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10.4 Calibration Two triangular fundamental diagrams must be determined for each link. For each class, the relation )(kQ h

ei is described by the three parameters ufi, qMi en kJi. In this calibration process we look for values for these parameters such that the model and observed traffic patterns closely match.

The maximal class flows qM1 and qM2 greatly affect the location and duration of the congestion area, as in the homogeneous model (Hurdle and Son 2000). That is why these sensitive parameters are determined last. First we search for the free-flow speeds ufi and the jam densities kJi.

The free-flow speed of the slowest class is determined for each link. To this end, all individual speeds from the free-flow regime between 6am and 7am and between 10am and 11am are used. Only vehicles detected on the primary road near the link under consideration are used. The first two columns in Table 10.3 give the detectors used in each link. The harmonic mean value obtained for the class 2 free-flow speed is given in column four.

Table 10.3 : Class 2 free-flow speed for each link.

Link Detectors used uf1 [km/h] uf2 [km/h] 1 D3, D4 112.07 85.79 2 D4, D5 110.05 84.74 3 D5 106.34 82.80 4 D7, D8 110.27 86.08 5 D9, D10 109.01 85.00 6 D10, D11 107.57 84.13 7 D11, D12 105.19 82.30

Determining the free-flow speed for class 1 is somewhat more complex. Due to the slower class 2 vehicles, class 1 vehicles often experience semi-congestion. Their speed is, therefore, lower than the free-flow speed we are looking for. To allow for this, only vehicles with a time headway greater than 5 seconds are examined. This guarantees that these vehicles travel in the free-flow regime because they are not obstructed by vehicles ahead. Analogous to the class 2 vehicles, the average speed of these vehicles is taken for each link. The values obtained are given in table 10.3.

A rough value is assumed for the jam densities. Its significance is small but can be important in the accurate calibration of the congestion regime. This parameter influences the angle of the congestion branch and indirectly determines the length of the tailback. These properties, however, also depend on the maximal flows that are still to be determined. It appears, therefore, that the value of the jam densities is not too significant.

When constructing the case study, a maximal class 1 density of 350 veh/km is accepted for all sections. For class 2 we take kJ2 equal to 150 veh/km.

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These start values were originally introduced into the model as an initial estimate. The results of the case study will show that the results obtained with these start values is satisfactory. Thus the input of additional adjustments is not necessary, while the non-supported values are retained. Adjusting these values afterwards would, after all, require a new calibration of the maximal flows.

We know from the observations that the bottleneck is located between detectors D8 and D9. To ensure that the bottleneck in the model is located at the same point, the maximal flows qMi of link 5 must be less than the other capacity flows. To model the correct tailback duration with the heterogeneous model, we only need to find the capacity flows of link 5. In the calibration procedure we assume, firstly, that the capacity flows of the other links exceed the values of link 5 by 15%. This puts the emphasis during this calibration step firmly on the bottleneck link.

Figure 10.7 : Observed and modeled scaled N-plot at detector D9 with a basic flow of 3600 veh/hour

In the calibration procedure we work with the cumulative flow immediately downstream from the bottleneck at detector D9. We compare the observed cumulative flow )(9 tN obs

D by class to the modeled cumulative flow )(mod9 tN el

D at this location. An iterative search process minimises the difference between the two curves. In mathematical terms this means finding qM1 and qM2 of link 5 where (10.2) is minimal :

∫ − dttNtN iel

DiobsD .)()( mod

99 (10.2)

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The heterogeneous LWR model can not be used to describe the high flow prior to the activation of the bottleneck. As in the traditional LWR model, outflow from the bottleneck is maximal. This will lead to deviations between model and observations, particularly in the period just prior to congestion and during the activation of the bottleneck.

Figure 10.7 shows the observed and modeled oblique N-plots which have a base-flow of 3600 veh/hour. With the obtained values of qM1 = 4068 veh/h and qM2 = 1980 veh/h, expression (10.2) is minimal. The average deviation over the 14000 vehicles amounts to a mere 27 vehicles for class 1 and 7 vehicles for class 2. Both N-curves agree very well, for both the free-flow and semi-congestion regimes. The duration of congestion in the model is somewhat longer because the previous period with high flow is also modeled as congestion. This increases the flow in the bottleneck somewhat also.

The number of class 2 vehicles increases towards the end of the congestion period. This also decreases the total flow. Figure 10.7 clearly shows the fluctuations and the decrease in the modeled capacity flow after 8.30am.

Since the traffic demand has been averaged over one-minute periods, very small fluctuations can not be modeled. This phenomenon is clearly visible at the two moving bottleneck at 6.30am and 6.45am. Here the high flow beyond the moving bottleneck exceeds the obtained capacity value.

Figure 10.8 : The modeled scaled N-curves for detectors D7, D8, D9 and D10

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Figure 10.8 gives the modeled N curves for the four detectors D7, D8, D9 and D10. These model results can be compared to the observed values of figure 10.4. This figure clearly shows how the tailback propagates itself against the flow from D9. Immediately after the start of congestion, the flow near detector D8 exceeds the capacity of the bottleneck. The average travel time and the number of vehicles between the detectors increases. At 7.16am the congestion regime begins at detector D8 and at 7.22am this regime is noticeable at detector D7. During the congestion regime the N-curves run almost parallel to each other. Howerver, several fluctuations in the flow that move with the flow during the congestion regime are clearly visible. These waves are caused by changes in the heterogeneous vehicle composition. The obvious start-and-stop waves from the observations at detectors D7 and D8 are not seen in the model results.

The free-flow regime recommences after 9.25am and the number of vehicles decreases between the detectors.

Figure 10.9 gives the modeled flow and the average speed for the entire main road. These figures can be compared with the filtered observations from figure 10.2. The resemblance between both figures is remarkable.

Figure 10.9 (a) The average speed and (b) the flow in a t-x diagram for the model of 20 October 1994.

Figure 10.9 clearly shows how the inhomogeneous node E functions as a bottleneck. During the passage of both bottlenecks, the capacity is already temporarily exceeded. This causes some congestion and the high flow becomes more balanced.

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The morning congestion begins at 7.10am. The speed during congestion on link 4 is homogeneous. The light fluctuations in the flow, meanwhile, move with the traffic. In the filter method the fluctuations were, during congestion, filtered automatically against the flow. The same fluctuations in figure 10.2 move equally against the traffic flow.

Due to the assumptions in the diverge and merge nodes, the variations in the traffic demand on the on-ramps and in the volume of exiting traffic at off-ramps are important factors. These methods help explain the wave character, of both flow as speed during congestion at links 1, 2 and 3. This explanation was recently confirmed by observations from Mauch and Cassidy (2002).

The congestion regime in the t-x diagram closely matches the observed congestion regime. The location, the length, and, to a lesser extent, the duration are sufficiently modeled.

During the free-flow and semi-congestion regimes, light waves are modeled in flow and in speed. As in the observations, these waves move with traffic flow.

Lower speeds are observed downstream from the bottleneck. In the model, the speed then exceeds the free-flow speed of class 2. In section 10.3 it was estimated that these occurred between 6 and 7am and between 10 and 11am. An accurate modeling of the lower speeds downstream from the bottleneck is not possible.

As was done by Hurdle and Son (2000), the calibration technique was directed towards an optimum approximation of the observed traffic patterns by the model. Again, the fact that the capacity value is exclusively observed in the bottleneck, implies that the direct distillation of these values from observations is not self-evident (Wu 2002). In the next section we will apply the model parameters obtained above to the same road-section, but on a different day.

10.5 Validation In this section we validate the model. Based on a different traffic demand, the model will be applied with the parameters from the calibration phase. The traffic demand is derived from the counts during 19 October 1994.

The filtered traffic observations of 19 October 1994 are given in figure 10.10. The methodology of section 10.2.1 was applied.

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Figure 10.10 (a) The average speed and (b) the flow in a t-x diagram for 19 October 1994.

These observations show that, in this instance, congestion begins between detectors D7 and D8. It now appears that the location of the bottleneck does not lie at a fixed distance from the merge. This contradicts the findings by Cassidy and Bertini (1999). Based on these filtered observations, the period of prolonged high flow prior to the activation of the bottleneck does not appear to exist.

These observations show that the capacity funnel and the increased flow prior to the activation of the bottleneck are not fixed properties of a road section. Conclusive explanations for this mysterious phenomenon are absent.

When the model results are examined, we again compare the cumulative flows in detector D9. Figure 10.11 gives the observed and the modeled N-curves.

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Figure 10.11 : Observed and modeled scaled N-plot at detector D9 with a basic flow of 3600 veh/hour for 19 October 1994

The striking match between both curves is again prominent during the free-flow regime. The match between the curves during congestion is even tighter than in the calibration model. Because the high flow prior to the activation of the bottleneck was absent, the model approaches the observations better. During this validation, the average deviation for class 1 amounts to twenty-four vehicles and eight for class 2. It is again interesting to see that the slope of the modeled N-curve decreases after 8.30am. Here too, the number of lorries increases in the late morning rush.

In figure 10.12 the modeled speed and flow are depicted in a t-x diagram. Comparison with the filtered observations from figure 10.10 indicates a different location of the bottleneck. The assumption in the model that the bottleneck lies between detectors D8 and D9 does not stand up on October 19. In spite of this change, the duration of the congestion is reflected correctly.

The lower speed downstream from the bottleneck in the observations, moreover, is striking. This phenomenon can not be described adequately in the model.

Applying the calibrated model to a different day gives a comprehensive idea of the capabilities of the model. The deviations in the cumulative curves downstream from the bottleneck appear to be even less during validation. Interpretation of the observations, however, indicate that the mechanisms underlying the capacity funnel and the increased flow prior to congestion are not self-evident.

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Figure 10.12 (a) The average speed and (b) the flow in a t-x diagram for the model of 19 October 1994.

10.6 Conclusions The heterogeneous LWR model and its extension to networks on a motorway were applied in this case-study. The model was able to reproduce a good qualitative description of the traffic flow during the calibration and in the validation. Identical wave-formation appeared during the free-flow regime and the cumulative curves were closely approached. The emergence, duration and length of the congestion area are also well described by the model.

In addition to the properties of the LWR, the model is also able to describe the fluctuations in speed during the free-flow regime in function of the traffic composition. In addition the outflow from congestion to free-flow becomes dependent on the vehicle composition. This allows the modeling of light forward moving waves in the flow during congestion.

Applying the network extension enables the modeling of the capacity funnel. Adding an inhomogeneous node downstream from the merge enables the reproduction of the delayed onset of congestion. The strict merge and diverge methods make wave formation during congestion upstream of these nodes possible. The variation in the traffic demand on the on-ramp and the changing amount of traffic leaving the road produce a start-and-stop wave pattern.

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The high observed flow prior to the activation of the bottleneck cannot be described by the heterogeneous LWR model. In this case, averaging the flow during the active bottleneck period and the previous period to a generalised capacity offers a solution. The lower speed downstream from the bottleneck can not be reproduced either. The changing location of the capacity funnel can, moreover, not be modeled.

The stationary character of the model prevents the description of start-and-stop waves between the bottleneck location and the first upstream merge. The speed in this congestion regime remains homogeneous.

The following chapter presents an in-depth discussion of the underlying assumptions and the possible extensions of the heterogeneous LWR model. The observed deficiencies will be part of this discussion.

167

11 DISCUSSION

We now come to a critical examination of the presented heterogeneous LWR model. In this examination, the emphasis lies on the basic model for road sections that was discussed in chapter 8. The network version from chapter 9, which looks at the various node types, is not included in this discussion.

The model properties will be verified, and the assumptions used will be investigated. Possible extensions to the heterogeneous LWR model will be proposed. The relation with other dynamic traffic models will be discussed.

The discussion is broken down in seven steps. First we compare the general model properties to the traffic phenomena discussed in chapter 2.2 and to the observations from the case study in chapter 10.2. We then take a closer look at a number of aspects like the weighted congestion branch, adapting the fractions to the number of traffic lanes, the homogeneity of the classes, non-optional road-use, and the significance of the non-concave fundamental diagram. These aspects can lead to new assumptions and model extensions. We finally discuss the link with other dynamic traffic operations models.

11.1 General properties During its step-wise formulation the homogeneous LWR model was extended to two classes. Traffic flow remained stationary and deterministic. The vehicles pertaining to a class are characterised by a triangular fundamental k-q diagram. This relation is maintained under heterogeneous circumstances.

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Each class operates on a fraction of the road space. The fractions are assigned to the different classes in accordance with the user-optimum. This implies that each vehicle maximises its speed without reducing the speed of slower vehicles. In this way, no vehicle can be influenced by vehicles upstream and the anisotropic condition applies.

The assumptions lead to a model that is fully described using the possible stationary states and the transitions between these states.

Three kinds of stationary states appear in the solution. During the free-flow regime, all vehicles drive at their respective maximum speeds. In the semi-congestion regime, however, the slowest vehicles alone drive at their maximum speeds. In this regime, the fastest class is congested. The speed of these vehicles is below their free-flow speed but above the free-flow speed of the slowest class. The classes operate in parallel both in the semi-congestion and the free-flow regimes. In the congestion regime all vehicles travel at equal speed and the traffic behaves itself as in a FIFO system. The vehicles of the different classes travel chain-wise.

The transitions between the stationary traffic states consist of shock waves, slips and fans. Other stationary states can occur at these transitions between the waves, slips and in the fans.

In the fundamental diagram, the points of the total traffic state lie between the two fundamental diagrams of the classes. The scatter usually observed also appears in this stationary model.

The capacity of an active bottleneck becomes dependent on the traffic composition, and in the congestion regime onward waves with the heterogeneous characteristics of the traffic participants appear.

The heterogeneous model describes the traffic process in light traffic more accurately. The dispersion effect is modeled by the heterogeneous speeds. In the parallel regimes, the average speed becomes dependent on the traffic composition.

The model does not explain the origin of oscillations in the traffic flow. This problem is further discussed in section 11.2.

Nor does the heterogeneous model explain the collapse of the high flow when a bottleneck begins to form. In the fundamental diagram this phenomenon is defined as the hysteresis effect. However, examination of the observations in the case study show that this phenomenon does not occur always during the activation of a bottleneck. There appears to be a connection with the reduced speed downstream from the bottleneck. Again, this is not sufficiently described by the heterogeneous model. The observations also show that the location of the capacity funnel downstream from a merge is not fixed. Again, the heterogeneous model offers no explanation.

11.2 The weighted congestion branch This section takes a closer look at traffic behaviour during congestion. During this regime, the assumed user-optimum results in a homogeneous speed for the

Discussion

169

heterogeneously composed traffic flow. The vehicles spread themselves across the total road space till no vehicle can increase its speed. In this way, all vehicles travel at the same speed and overtaking is not possible. Traffic on a congested link behaves like a First-In-First-Out (FIFO) system.

A FIFO system occurs most likely on a one-lane road. Vehicles can not overtake on these roads and the sequence of the various vehicles is automatically respected.

The model leads to a weighted congestion branch with a speed of wtot (8.22). This speed depends on the composition and on the congestion branches of both classes. However, in real traffic the length of vehicles is finite. Therefore, the weighted congestion branch on a one-lane road will always approach the class 1 congestion branch and the class 2 congestion branch alternately. Figure 11.1 shows an example for a one-lane road using trajectories. In this figure, the class 1 and 2 vehicles are indicated in dark and light grey respectively.

Figure 11.1 : Practical shock wave and theoretically weighted shock wave in (a) the fundamental diagram and (b) the t-x diagram.

The shock wave in the example of figure 11.1 has the speed of the weighted congestion branch. Vehicle composition does not change across this shock wave. The fractions, however, do change. In the example, the class 2 vehicles occupy a relatively larger road space subsequent to the shock wave.

The weighted congestion branch is, consequently, an approximation that applies optimally when the vehicles of the different classes are most evenly spread across the

t

x

class 1

class 2

q

kJ2 kJ1 kJtot k

kD1 kD

2

qMtot

U

(a)

(b)

D

qUtot

qU1

qU2

U D

Chapter 11 170

length of the road. The formation of a cluster of vehicles of a particular class causes local deviations to this pattern.

When we examine a road consisting of several traffic lanes, the class division of the different vehicles across the different lanes must also be the same. If this is not so, the wave speeds on the lanes can differ slightly and local disturbances occur between the lanes. These deviations can bring about lane-change incentives. These lane changes can, in turn, lead to new instabilities. This could also explain the emergence of start- and stop waves in congested traffic.

Start-and-stop waves, as well as other unstable phenomena during congested traffic, have always been explained through non-stationary models. The reaction-time of drivers increases acceleration and deceleration and it reinforces these unstabilities against the flow of traffic. The choice behaviour of drivers also plays a role. They do not decide continuously, but react only once a discreet threshold value has been exceeded.

However, the effect of heterogeneous driver- and vehicle properties can help explain the emergence of these intriguing phenomena. Mauch and Cassidy (2002) confirm this assumption by attributing observed oscillating traffic patterns to lane-change manoeuvres.

In conclusion we can say that, during congestion, the user-optimum leads to equal speeds. This assumption is particularly valid during an even spread of the vehicle classes, both lengthwise and across the different lanes. Disturbances caused by the formation of clusters of vehicles of the same class could lead to unstable phenomena. These local perturbations provide an interesting area of research. However, they are not included in the study of the stationary heterogeneous LWR model.

11.3 Traffic lanes Chapter 4.2.8 gave a concise explanation of the way in which the number of lanes can be taken into account. We resume this discussion in this section and we focus on the possible consequences of traffic lanes on the heterogeneous LWR model.

A fraction αi of the total road space is allocated to each class in the heterogeneous LWR model. The class speeds differ during the free-flow and the semi-congestion regimes. This implies that both classes are also physically separated on a road. The classes must, therefore, operate parallel on their road fraction.

Since the fractions can assume the entire range of values between 0 and 1, there is an implicit assumption that the modeled road consists of an infinite number of traffic lanes during these regimes. The reality is different: there is a limited number of lanes and only fractions with discrete values occur. Since the congestion regime-speed is homogeneous, it is the only regime were the entire range of fractions is allowed.

Let us consider a road of m traffic lanes, numbered from the slowest to the fastest lane. Of the two classes using this road, class 2 is the slowest. On the basis of a user-optimum during free-flow, the computed fraction for class 1 is not necessarily a

Discussion

171

multiple of 1/m. We assume that the calculated speeds are u1 and uf2 respectively, whereby the following applies for the fraction α2:

mj

mj 1

2+≤< α

(11.1)

Class 1 is, therefore, limited by :

mjm

mjm )1(

1+−≥>− α

(11.2)

Since the user-optimum applies, the class 2 fraction can not be reduced. The class 1 fraction could possibly be adapted to allow for the number of lanes.

• In a first option class 1 confines itself to the m-(j+1) lanes. Class 2 vehicles do not use these lanes which is why class 1 can proceed in parallel, with a different speed. This will only happen when the class 1 speed on this limited number of lanes maintains a higher speed than the class 2 speed, or :

−−

−−=≤1

..

1 11

112 jm

kmQkmjmuu h

ef (11.3)

Class 2, therefore, occupies a fraction of the road that is a multiple of 1/m. Thus, the fraction assigned to class 1 is smaller. The free-flow and the semi-congestion regime will, therefore, occupy a smaller space in the phase diagram.

• If the calculated speed of class 1, at a fraction mjm /)1(1 −−=α were to be less than uf2, the traffic flow would achieve the speed uf2 and the traffic would become mixed. This would, in principle, bring the traffic flow into the semi-congestion regime. Still, the road remains under-utilised. The traffic can, in fact, function on a smaller road fraction, at speed uf2. This explains the use of the term “non-optimal semi-congestion regime”.

On the basis of these observations, the traditional phase diagram from chapter 8 is adjusted. Figure 11.2 illustrates the phase-diagram for a road with three traffic lanes.

Besides the unchanged congestion regime we now have three regimes.

The properties pertaining to the free-flow and the semi-congestion regimes are identical to those of the heterogeneous LWR model from chapter 8. However, the fractions for both classes during these regimes are no longer continuous. The only valid values for the fractions are 0, 1/3, 2/3 and 1. This has no implications for class 1 when the class 2 density varies between two of these discrete fraction values. The class 1 properties from the original model where k2 = 2/3 kM2, for example, continue to apply to all class 2 densities in the [1/3 kM2, 2/3 kM2] interval. Here, the lanes assigned to class 2 are not necessarily optimally used.

Chapter 11 172

Figure 11.2 : Phase-diagram for two classes with three lanes

During the non-optimal semi-congestion regime, both classes travel at the class 2 free-flow speed. The entire road is not optimally used during this semi-congestion regime. Without lowering speed in the process, the class densities can increase until they comply with (8.14) :

1. 2

2

2

1 =+MM kk

krk

(11.4)

These stationary states are entirely determined using of the phase-diagram from figure 11.2. The transitions can also be adjusted in order to allow for the traffic lanes.

The numerical Godunov scheme can also be extended to allow for the discrete fractions. In this case, however, the numerical dispersion causes problems. A small transgression of a discrete fraction value leads to the assignment of an extra lane to class 1. Hence, small numerical deviations could be magnified. These undesirable effects can undermine the numerical simulation of the multi-lane model.

In these lane-adjustments, it is indirectly assumed that the class densities are examined over a space interval that does not exceed one vehicle. In that case the fractions are indeed a multiple of 1/m.

When density is studied over longer intervals, the fractions can assume several values again and the general model also applies again.

One of the new elements in the multi-lane extension presented is the fact that vehicles do not always make optimal use of the road. The user-optimum adopted here, explicitly assumed that the class 2 vehicles did not occupy more space than was strictly necessary. These assumptions becomes less certain when traffic lanes are introduced. This aspect will return in the coming section, with the discussion of the homogeneity of classes. Non-optimal use is elaborated on in paragraph 11.5.

k1

k2

r.kM2 kJ1

kJ2

kM2

kM1

32 kM2

31 kM2

Legend Free flow regime Semicongestion regime Non-optimal semicongestion Congestion regime

Discussion

173

11.4 Homogeneity of classes When determining the number of classes, a deliberate choice will have to be made between a limited number of classes with a greater spread or several more homogeneous classes. Here, the analytical transparency and the numerical calculation time are considerably better in a more limited number of classes. Some traffic characteristics, however, are possibly better and more accurately described when dealing with a larger number of classes. In the scope of this dilemma this section discusses two possible tracks. An initial extension changes the shape of the fundamental diagram of class 1. We then look at the extension of the heterogeneous LWR model to a number of classes. In each case the aim is to increase the accuracy of the description of the traffic operations.

11.4.1 A curved fundamental diagram The observations in the case study in chapter 10 show that increased traffic density lowers the speed of the fastest class. This decrease in speed can not be explained solely by the increased disruption of the class 2 vehicles and the emergence of the semi-congestion regime alone. Since the vehicles in class 1 do not share exactly the same desired speed the slower class 1 vehicles hinder the faster ones. In this way, interactions between vehicles of the same class cause a decrease in the average speed of a class when traffic density increases. This phenomenon applies particularly to class 1. Fast class 2 vehicles that are locally obstructed by the slower class 2 vehicles can still overtake unimpeded during the free-flow or the semi-congestion regimes. The average speed of class 2, therefore, does not decrease, but it does affect the assumption regarding the optimal use of road space. Section 11.5 looks at this non-optimal use of the slowest class. Now we focus on the fastest class.

Figure 11.3 : Curved fundamental diagram for class 1.

The fundamental diagram can take lowering average speeds caused by vehicle interactions into consideration. This can happen when the fundamental diagram is allowed to curve at higher speeds. Figure 11.3 sketches two homogeneous fundamental diagrams showing a curved diagram for class 1 for speeds exceeding the free-flow speed of class 2. In this diagram, class 1 achieves the maximal flow qM1 at

kM1

uf2 q

qM1 qE1

r.qM2

qM2

r.kM2 kM2 kE1 kJ2 kJ1 k

uM1

Chapter 11 174

speed uM1. This speed uM1 is less than the free-flow speed uf1. This capacity speed does, however, exceed the free-flow speed of class 2. If this was not so, in other words, if uM1 was less than uf2, class 2 vehicles would not obstruct the class 1 vehicles. In this case, a larger portion of class 2 vehicles could lead to a rise in the total flow. Dividing the different vehicles into classes is, in that case, erroneous, because the class with the highest free-flow speed does not drive at the highest speed at the maximal flow.

Introducing a curved fundamental diagram clarifies the difference between waves over which the traffic density increases and between fans in which traffic density increases. Fans are less frequent in triangular fundamental diagrams because the characteristic speed inside a regime remains the same (Velan and Florian 2002). All free-flow states, for example, share a characteristic speed ufi whereby a transition from dense to less dense traffic happens via a slip. In a curved diagram, a range of characteristics appear in the solution.

In the transitions inside the semi-congestion regime, waves occurred with speed uf2 where both the class 1 and the class 2 states changed simultaneously. These waves should also appear when using the curved diagram. This is why point E1 is defined for which the characteristic speed of class 1 equals the class 2 free-flow speed :

21)(

fEe u

dkkdQ =

(11.5)

This point determines the boundary between the free-flow area and the semi-congestion regime, as shown in figure 11.4. In this phase diagram, the maximal flow α1.qM1 belongs to the semi-congestion regime. In a moving co-ordinate system, however, point E1 represents the capacity, as shown by Newell (1998) in the moving bottleneck.

Figure 11.4 : The phase-diagram at a curved homogeneous class 1 fundamental diagram.

kM1 r.kM2 kE1 kJ1

kM2

kJ2

k1

k2

Legend Free-flow regime Semicongestion regime Congestion regime

Discussion

175

An illustration is used to show that the transitions in the phase-diagram develop similar to the transitions from chapters 6 and 8. Figure 11.5 shows the second sub-case of the free-flow to free-flow transition in the phase diagram, the fundamental diagram and the t-x diagram.

In this transition, the semi-congestion regime is arrived at in the solution via the traffic states BG and BH. The upstream traffic state AU changes initially to the semi-congestion state BG across a shock wave. As shown on the k-q diagram, the class 1 traffic becomes denser across this shock wave. This is followed by a wave with speed uf2 across which both class 1 and class 2 change. This wave is a tangent of the downstream class 1 diagram. Thus, state BH corresponds with point E1 at the downstream fraction D

1α . A fan occurs, lastly, between BH en AD in which the class 1 density decreases. The t-x diagram in figure 11.5 sketches the solution for class 1. The characteristics and two trajectories are indicated on the diagram.

Figure 11.5 : The free-flow to free-flow transition (A⇒A) sub-case 2 with a curved class 1

Analogous to this transition, all transitions can be solved with the curved diagram. When using the curved fundamental class 1 diagram, fans appear and curved shock waves can show up in the solution. This extension allows the inclusion of the non-homogeneous character of class 1 without having to increase the number of classes.

11.4.2 Several classes A second way in which to approach the spread around classes is to extend the number of classes. We assume that the traffic flow consists of n homogeneous classes where each class on the road under consideration is described by a homogeneous triangular fundamental diagram. Each class is accordingly characterised by the free-flow speed ufi, the critical density kMi and the jam density kJi. The classes are numbered in decreasing free-flow speed sequence from 1 to n. As in the heterogeneous LWR model with two classes, the scale factors ri are, analogous to (8.4), defined by :

kM1 r.kM2 kE1

kM2

AD

AU BG

BH

k1

k2

AU

AD BG

BH x

t

x0

kM1

uf2 q

qM1

kD1

k

BH

BG

AD

AU

E1

Chapter 11 176

).(.

1

11

fiMi

Ji uwk

kwr−

= (11.6)

These scale factors convey for class 2 to n how the class fundamental diagram must be scaled in such a way that the capacity point ends up on the congestion branch of class 1. In the subsequent formulation of the model r1 is set equal to 1. Figure 11.6 indicates the scale factor for a class j.

Figure 11.6 : The homogeneous fundamental diagrams of several classes.

The increase in the number of classes also leads to an increase in stationary traffic regimes. When all classes travel at the free-flow speed ufi we can speak of the free-flow regime. Condition (8.7) then becomes :

11

≤∑=

n

i Mi

i

kk

(11.7)

The semi-congestion regime is redefined. We define j-congestion when the j fastest classes are congested and the remaining n-j classes travel at their free-flow speed. Here, the speed of the fastest j classes is the same and lies between ufj and ufj+1. During n-congestion all classes travel at a speed of less than ufn. Thus, n-congestion corresponds to the congestion regime of the old terminology. The boundaries of the j-congestion regime are given analogously to the semi-congestion conditions (8.12), (8.13) and (8.14) as:


0≥ik (11.8)

• The boundary with the j-l congestion regime :

1.

1

1

>+∑∑=

−

=

n

ji Mi

ij

i Mii

i

kk

krk

(11.9)

• The boundary with the j+l congestion regime

1. 11

≤+ ∑∑+==

n

ji Mi

ij

i Mii

i

kk

krk

(11.10)

qMj

rn.qMn

qM1

qMn

rj.qMj

kM1 kMj rj.kMj

kMn rn.kMn kJn kJj kJ1 rj.kJj

q

k

w1

Discussion

177

During fully congested traffic, termed n-congestion here, 11.9 leads to the upper limit of this regime :

1.

1

1

>+∑−

= Mn

nn

i Mnn

i

kk

krk

(11.12)

The extreme limit of congestion from (8.17) is, additionally, converted to :

11

≤∑=

n

i Ji

i

kk

(11.13)

Based on the different conditions it is possible to determine the stationary regimes in this multi-class model. When we deal with more than two classes, the analytical computation of the transitions becomes much more complicated. Practical applications, moreover, require the categorisation of observed vehicles. In our case study vehicle length was the criterion used to divide the traffic into two classes. Several classes that are subdivided on the basis of this vehicle characteristic will not necessarily lead to more homogeneous driver classes. Ascertaining driver and vehicle characteristics on the basis of traffic observations is on-going research (e.g. Kockelman 2001, Kijk in de Vegte 2002). This can lead to improving the homogeneity of classes.

11.5 Non-optimal road-use The assumed user-optimum demands that slow vehicles occupy only the minimum required road-space. This gives the fastest class the largest possible road section without infringing on the anisotropic condition. These assumptions give the largest possible flow at which the user-optimum remains valid. Larger flows can possibly be achieved by assuming a system optimum, but this does not give a realistic representation of the behaviour of the individual vehicles. The anisotropic condition can, moreover, not be met.

In two cases studied above, optimal road use in the user-optimum was endangered.

When the basic model takes the number of traffic lanes into consideration, this can not always be complied with. Traffic lanes break the road up in parallel lanes. Since the number of traffic lanes is finite, not all vehicle compositions can comply optimally with the parallel traffic operations. In practice, therefore, one finds that the outer lane is under-utilised because class 1 private cars are unwilling to filter into the lane used by the slower class 2 trucks.

When the class 2 vehicles are not identical, interactions occur within the same class. Slight differences in the desired speed, therefore, lead to the situation whereby fast trucks overtake the slower ones. These non-homogeneous properties lead to a situation in which the slowest class may occupy more space than is strictly necessary.

These examples of non-optimal road-use do not endanger the user-optimum. This is due to the fact that optimal road-use and the user-optimum are unconnected in a heterogeneous model. In a homogeneous model, on the other hand, these two

Chapter 11 178

assumptions are tightly linked. Optimal road-use is an integral part of the fundamental diagram. The flow in the equilibrium relation Qe(k) is related to a particular density. However, at a given density, it is possible that certain road-users voluntarily assume a lower speed than is strictly possible. In this regard the fundamental diagram functions as the boundary state for all possible state points. Optimal road-use in homogeneous traffic corresponds with the principles of the user-optimum. The only time when the road is not used optimally occurs when certain vehicles fail to maximise their speed.

This is why non-optimal road-use is somewhat more complex in a heterogeneous model. One can overcome these problems by extending the number of lanes and by defining a number of classes.

This effect can also be carried over to the basic model. Let us assume that class 2 systematically exceeds the actual minimal space by p per cent. The actual class fraction then becomes :

2

22 ).1(

M

real

kkp+=α

(11.14)

The basic model can overcome this problem when an adjusted homogeneous fundamental diagram for class 2 is used. Instead of the actual real

Mk 2 , the capacity density now becomes :

)1(2

2 pkk

realM

M +=

(11.15)

Using this adjusted fundamental diagram explicitly implies that class 2 systematically uses more space than is strictly necessary. This is implicitly applied when a fundamental diagram is calibrated to the actual traffic pattern. In terms of the fundamental diagrams discussed in chapter 10, we can assume that the capacities obtained for class 2 are, consequently, somewhat smaller than those that are strictly possible.

11.6 Non-concave fundamental diagram In chapter 8 we saw that the heterogeneous model can feature properties pertaining to the LWR model with a non-concave diagram. These properties appear when the congestion branch of class 1 is steeper than the class 2 congestion branch. In this section, we will pay particular attention to non-concave diagrams and their consequences in the heterogeneous model. We begin with a concise explanation of the non-concave diagram in the traditional LWR model, using an illustration. To this end, we consider a road where traffic complies with the fundamental diagram OMBJ from figure 11.7a. A diagram is concave when there are no points beneath the straight line that connects two random points on the diagram. When, in figure 11.7a, a straight line is drawn between points M and J, point B, with a density between kM en kJ, lies beneath this straight line. This diagram is, therefore, non-concave.

Discussion

179

Figure 11.7 : A temporary obstruction in (a) a non-concave k-q diagram and (b) in a t-x diagram

As an initial condition, we assume that the free-flow traffic state A applies to the entire road under consideration. Near x0, the road is temporarily obstructed between t0 and t1. Figure 11.7b sketches the solution to this problem in the t-x diagram. Downstream from the obstruction, a slip at free-flow speed separates state A and the empty road O. Upstream from the obstruction, transitions occur from the free-flow state A, to the congested state B and on to complete congestion in J. As a consequence, state B appears in the solution, with congested but not stationary traffic. The area in the t-x diagram that is indicated with B actually consists of a fan of characteristics. Observations (Daganzo 1999) confirm that these traffic patterns do occasionally happen.

With the discontinuation of the temporary obstruction, the traffic accelerates directly from the congestion state J to the capacity state M. This happens across a direct shock wave where decompression occurs. In fact, this wave runs outside the fundamental diagram.

An observer in x1, upstream from the temporary obstruction, sees the following transitions. The original traffic state changes across a shock wave from A to B. He then sees stationary traffic in state J. This is followed by the capacity state M to be followed, eventually, by state A again. When these transitions are drawn in the fundamental diagram, a path appears. The various transitions comprise the area ABJMA. The transition from free-flow to congestion runs along a path that differs from the reverse transition. Here, we can, in fact, speak of a hysteresis effect. However, this hysteresis is the opposite of the hysteresis effect that can be seen during traffic observations. As explained in chapter 2, the path from free-flow to congestion lies above the reverse path.

However, the non-concave diagram reinforces the reverse hysteresis effect and thus implies a deterioration, when compared to the concave fundamental diagram. In the heterogeneous LWR model, the properties of the non-concave diagram return when the speed of the class 1 congestion branch is less than that of class 2. In that case, shock waves appear with decreasing traffic density, and fans appear in which traffic density increases. Figure 11.8 shows two fundamental diagrams where w1< w2. To

A

M

J

B

q

k

x

t

x0x1

M

A B

O

O

J

t1 t0

Chapter 11 180

keep things simple, the scale factor r equals 1. This means that the capacity point of the class 2 diagram lies on the congestion branch of diagram 1.

The figure shows the transition from the upstream congestion state U to the downstream free-flow state D. This transition develops along a direct wave. State D was chosen in such a way that the traditional uf1 and uf2 are no longer significant. Chapter 8.2.4 already dealt with this in detail. Figure 11.8b shows the transition in the t-x diagram.

Figure 11.8 : Congestion to free-flow transition in (a) the k-q diagram and (b) the t-x diagram.

The direct shock wave between the up- and downstream traffic states lies above the capacity-point M2 of class 2. As a result, the direct wave develops analogous to the direct shock wave JM in the illustration in figure 11.7.

When working out this transition using trajectories, the true non-concave character of this transition becomes clear. Figure 11.9a sketches an aerial view of a road. The class 1 vehicles are indicated in dark-scale, class 2 vehicles in light-grey. The road is congested upstream from x0 where traffic state U applies. All class 2 vehicles are on the outer lane. The remaining road space is occupied by class 1 vehicles. In this congestion state, the class 1 vehicles occupy more space than those of class 2. This is so because the class 1 congestion branch in the k-q diagram lies beneath the congestion branch of class 2. In this state, each set of thirteen vehicles contains six class 2 vehicles. These six vehicles occupy thirty per cent of the road.

The free-flow state D applies downstream from x0. In D, half the road space is taken by class 1, and half by class 2 vehicles. We know from the fundamental diagram that the class 1 density is less than that of class 2. The flow of class 1, however, is higher due to its greater speed. In this free-flow state, the traffic develops in parallel. Each class keeps to a separate traffic-lane.

Figure 11.9b sketches the transition with trajectories in a t-x diagram. The colour of each trajectory refers to the specific class : dark-grey for class 1 and light-grey for

k

q

U

D

t U

D

x

x0(b)

(a)

Qhe1

Qhe2

w2

w1

M2

M1

Discussion

181

class 2 vehicles. When two trajectories coincide, the colour of the vehicle in the left lane is shown.

Figure 11.9 : (a) aerial view of traffic state at t0, (b) trajectories in t-x diagram and (c) aerial view of traffic state at t1

The solution in the heterogeneous model consists of a direct wave, as shown in figure 11.8b. However, the direct wave becomes a transition zone when the wave with speed w2 is taken into consideration. The four class 1 vehicles on the left lane, meanwhile, increase speed. This develops across a shock wave with speed w1. Since w1 is more negative than w2, the first four class 1 vehicles have already reached their free-flow speed when the last of the six class 2 vehicles are still congested. The congested class 1 vehicle on the outer lane now follows the class 2 vehicles at congestion speed, while the class 1 vehicle beside him on the left lane has already accelerated. At that moment, the class 1 vehicle on the outer lane decides to switch lanes and accelerates.

U

D

x0

t

t0

x

t1(a) (b) (c)

x0

Chapter 11 182

In the figure, this lane-changing is indicated by a circle. The two class 1 vehicles react in the same way: the vehicle on the left lane accelerates, upon which the vehicle on the right lane switches lanes and accelerates.

When these thirteen vehicles have passed, the procedure repeats itself. In the t-x diagram, the direct wave becomes a transition zone of finite dimension. Figure 11.9c shows the aerial view at moment t1. At that moment, almost all vehicles travel in the free-flow regime.

This example with trajectories shows that lane-change behaviour is essential in explaining the 'non-concave' properties of the heterogeneous model. This mechanism, however, offers a possible explanation for the general non-concave fundamental diagram. Following this reasoning, the numerous lane-changes in certain traffic densities lead to a non-concave diagram.

In the example of figure 11.9b, the outflow from congestion is optimal. Class 2 achieves maximal flow on the right lane, while the class 1 vehicles on the left lane also travel at their capacity state. These high flows in the model can only be achieved when the lane-changes are strictly adhered to. Slight deviations propagate themselves irreversibly and decrease the outflow. This high flow also depends on the position of the vehicles. When congested class 2 vehicles also drive on the left lane, they reduce the high flow.

These microscopic observations provide a possible explanation for the lowered outflow from congestion to free-flow that is seen in observations. The heterogeneous LWR model does not predict this decreased discharging flow. When we examine the mechanisms underlying the transition from the serial congestion regime to a parallel regime, we find that a reduced outflow is practically unavoidable. This shows, once again, the power of the heterogeneous LWR model as a basis for further extensions.

11.7 Relation with other dynamic models This section gives a brief overview of the relation between the formulated heterogeneous LWR model and other dynamic traffic models. We begin by establishing a link with other macroscopic models. This is followed by the relation with kinetic models and we finish with a look at the relation to microscopic models.

11.7.1 Macroscopic The link with other heterogeneous stationary macroscopic models has been elaborated in chapter 4. In that chapter, the different models were placed in a general framework where the equilibrium relation is given by (4.12) :

=

i

iheiii

kQqα

α . (11.16)

The way in which the road space is assigned to the different classes differs in these models. Here, the heterogeneous character of the traffic flow at different speeds that

Discussion

183

complies with the anisotropic condition, expresses itself exclusively in the user-optimum. That makes this formulated heterogeneous LWR model unique.

The model is partly related to the rabbit-slug model of Daganzo (2002a, 2002b). That model examines two classes on a road with two traffic lanes. The vehicles belonging to the fastest class (class 1) are called 'rabbits', those of the slowest class 'slugs'. In this model, the homogeneous fundamental diagrams apply to each lane, as shown in figure 11.10.

Figure 11.10 : The homogeneous fundamental diagrams for each lane in the rabbit-slug model.

Class 1 has a discontinuous fundamental diagram. This means that the diagram reflects high flows as long as the speed exceeds that of the free-flow speed of class 2. The explanation of these high flows lies in the relative short distances between these vehicles on the left lane during parallel regimes. At this moment the rabbits are called ‘motivated’.

When the speed drops below the class 2 free-flow speed congestion occurs. During congestion, both classes share the same congestion branch for each lane. The congestion branch upper right in the figure then applies to the entire road. Both classes behave as user-classes.

The discontinuity in the diagram of class 1 necessitates additional assumptions for the transitions between the parallel regimes and the congestion regime.

The discharging flow qD is defined for the transition from congestion to the free-flow regime. The class 1 vehicles will always accelerate into this state point. This results in a lower outflow from congestion while the hysteresis effect is explicitly obtained.

The transition from free-flow or semi-congestion to congestion develops in a more complicated way. Without going into detail we see two situations. In one of these situations, state B on the left lane may increase in the solution, while the class 2

kJ 2.kJ 2.kM2 kD kM1 kB

qD

kM2

qM1

qB

qM2

2.qM2

)(1 kQ he

)(2 kQ he

q

k

Chapter 11 184

vehicles on the outer lane maintain their speed. This state appears to be extremely unstable and might explain the emergence of start-and-stop waves.

This model is, strictly speaking, not a heterogeneous LWR model. The reason for this is that the fundamental class 1 diagram is not a uniquely defined function. It becomes clear from the phase-diagram, as sketched in figure 11.11, that the regime is not singularly defined for specific combinations of class densities. The squared area can represent two regimes. The class combinations in the white area cannot occur.

Figure 11.11 : The phase-diagram of the rabbit-slug model

Due to this non-singular definition of the class densities, this model does not fit in the LWR framework, while a numerical Godunov scheme appears to be quasi impossible.

This phase-diagram corresponds closely to the phase diagram from the multilane extension in figure 11.2. Since the formulation of the rabbit-slug model is fairly abstract and phenomenological, it is not complete (Banks and Amin 2003). The phase diagram from figure 11.11 is, though somewhat altered, but a part of the phase diagram from figure 11.2. The heterogeneous LWR model presented in this thesis could, due to its systematic and complete construction, possibly lead to a practically applicable version of the rabbit-slug model.

The heterogeneous LWR model presented here is stationary. A relation to higher order models is, therefore, absent. A further non-stationary extension seems a promising way to achieve a complete macroscopic description of the traffic flow (Zhang 2002). A combination with Lebacque's bounded aceleration model ( Lebacque 1997) seems to be appropriate to this situation. Preliminary research into the combination of higher order models with multi-class (e.g. Colombo 2002) as well as multi-lane models (e.g. Greenberg et al. 2003) is underway. These macroscopic models can also be constructed on the basis of kinetic models. This last type is further discussed in the chapter on mesoscopic models.

Legend Free flow Semicongestion Congestion Both Free flow and congestion Both Semicongestion and congestion No regime

k2 (slugs)

k1

kM2

2.kJ 2.kM2 kD kM1 kB kJ

kJ

kM2

Discussion

185

11.7.2 Mesoscopic The heterogeneous LWR model can be expressed using mesoscopic variables. We define the phase density f(t,x,v) as follows :

{ }21,0),(

),,(uuv

uvxtkvxtf ii

∉∀=∀

= (11.17)

If we follow that route, the heterogeneous LWR model could also be written in the kinetic form as :

relaxationactioninter tf

tf

xfv

tf

∂∂+

∂∂=

∂∂+

∂∂

(11.18)

Here, the interaction and relaxation terms become fairly complex expressions. At the qualitative level, the following distinctions can be made between it and the original model by Prigogine and Herman (1971).

• In contrast to the kinetic model, traffic in the heterogeneous LWR model is not probabilistic in character. The phase distribution must, therefore, be interpreted as representing the actually occurring densities.

According to 11.17, the phase density never shows dispersions and always consists of two Dirac impulses.

• In the original kinetic model, the relaxation term always takes account of the finite acceleration capacity of a vehicle. The heterogeneous LWR model here is stationary and thus does not consider relaxation time.

• When a fast vehicle approaches a slow vehicle ahead of it, it can, according to the original kinetic model, interact in two ways. It can either overtake the slow vehicle, while maintaining its speed, or it can immediately reduce its speed to that of the slow vehicle ahead. The choice between overtaking or deceleration is decided by an arbitrary fixed parameter.

The interaction term is more complex in the heterogeneous LWR model. The transitions between the free-flow and the semi-congestion regimes showed that the fast vehicles are able to achieve a variety of speeds that are depend directly on the up- and downstream class densities. If the principles alone are used from the original kinetic model, we get a situation in which the interactions on a homogeneous road are never able to lead to speeds other than uf1 and uf2.

In the heterogeneous LWR model, interactions between fast and slow vehicles can, in fact, lead to even higher speeds. When the upstream class 2 density exceeds that of the downstream density, a situation can arise in which semi-congested class 1 vehicles accelerate due to a sudden increase in available space. At such moments, this acceleration process occurs abruptly and is not linked to the relaxation time.

• In the heterogeneous LWR model, vehicles occupy a certain place. Thus, the space taken by vehicles during congestion is not necessarily proportional to the

Chapter 11 186

vehicle density. An accurate elaboration of the heterogeneous LWR model according to the notation of the kinetic models requires, therefore, an additional sub-division of the phase distribution - one that takes the classes into account.

This comparison clearly highlights the critical assumptions in the interaction term of the original kinetic model. It appears that the complexity between vehicles travelling at different speeds is over-simplified when an arbitrary probability of passing is applied. This criticism applies partly to the multi-class kinetic models and the derived macroscopic version that was developed by Hoogendoorn and Bovy (1999).

11.7.3 Microscopic The analytical computation of the heterogeneous model can also be carried out using a particle-based numerical scheme. For the traditional LWR model, this approach is applied in the INTEGRATION package (Van Aerde 1994).

In this chapter, the examples with trajectories illustrate this approach (figures 11.2 and 11.9). In the case of the heterogeneous model, the lane-change manoeuvres, in particular, present a complex problem. For that matter, elaboration of a particle-based numerical scheme also requires an additional extension of the LWR model to include the traffic lanes.

11.8 Conclusion s This chapter undertook a critical examination of the formulated heterogeneous LWR model. To this end, the options pertaining to a heterogeneous stationary model were initially discussed in order to explain empirical phenomena. This was followed by a description of possible additional extensions to this heterogeneous LWR model.

A detailed examination of the assumptions and possible extensions shows that a number of phenomena hitherto exclusively explained by non-stationary models, can also be described by a heterogeneous stationary model. The intriguing start-and-stop waves can be caused by local perturbations in the composition of classes. The changing traffic demand at on-ramps and the variations in the extent of exiting traffic at off-ramps could possible explain their emergence.

In the hysteresis effect, the outflow from congestion to free-flow does not develop optimally. Local variation in traffic composition and small differences in lane-change preferences showed that a diminished outflow is almost unavoidable in practical circumstances.

Non-homogeneous classes, the construction of a lane-specific model, and non-optimal traffic operations were also discussed. To end this chapter, we established a link with other dynamic models.

187

12 CONCLUSIONS

This dissertation extends the original homogeneous LWR model to a heterogeneous version. Observations, after all, show that traffic flow is not homogeneous. The extension formulated in this dissertation incorporates the heterogeneous driver and vehicle characteristics. The motivation behind this approach is two-fold.

First, the systematic development of a stationary continuum model leads to new insights, enabling a possible explanation of observed traffic phenomena that originate in the heterogeneous character of the traffic flow.

Second, a heterogeneous LWR model provides a more accurate description of traffic flow than a homogeneous model. To increase its usefulness, an analytical version of the model should preferably be accompanied by a numerical calculation scheme and the option to apply the model to a larger network.

In this final chapter we review the heterogeneous model developed in this dissertation. A short summary will be followed by the most important findings and by suggestions for further research.

12.1 Brief summary The original LWR model assumes traffic to be stationary, homogeneous and deterministic. The model can be extended to address each of these limitations. In this dissertation we systematically examined the effect of a heterogeneous extension.

Dividing the traffic population into classes turns heterogeneous traffic into a game of interacting homogeneous classes. We assumed that vehicles of a particular class could

Chapter 12 188

be described by the original LWR model if the road is free of other vehicles. This allowed us to formulate a fundamental diagram for each class separately.

When vehicles of different classes interact, a user-optimum is assumed. Each vehicle aims to maximise its speed without decreasing the speed of slower vehicles. Due to this interaction, vehicle behaviour can never be influenced by vehicles coming from behind. We also assumed that the traffic behaves optimally: no vehicle occupies more space than is strictly necessary.

This resulted in the formulation of a set of interaction rules for two classes using triangular fundamental diagrams. The analysis proceeded in a step-wise way. Each step comprised an analytical model and an associated numerical scheme. In an initial approximation, the scaling of similar fundamental diagrams led to an applicable heterogeneous model that is related to passenger-car-equivalents. In a free-flow version, the emphasis was on the speed-differences in light traffic conditions. A third step focused on differences between classes during congestion. Applying the user-optimum, the previous three steps were then used to develop a complete heterogeneous model for two random triangular fundamental diagrams.

This heterogeneous model describes the traffic process on homogeneous road sections. Nodes were then introduced to extend this model to a network version. Links depart from and arrive in a node. Defining different node types results in building blocks for the construction of virtually all motorway network figurations.

A case study was used to illustrate the application of the heterogeneous model to a motorway with on- and off-ramps. An exhaustive study of the observations and a careful model-construction were followed by a comprehensive calibration. The input parameters were determined so as to ensure a close fit between the modeled and observed traffic pattern. Validation of the model proved that this approach resulted in an adequate model that can realistically reproduce the traffic pattern for a different traffic demand.

A critical discussion examined the assumptions, the properties and further extensions to the heterogeneous model. It appeared that a great number of observed traffic phenomena could be explained using heterogeneous stationary models. Finally, the relation to other dynamic traffic models was discussed

12.1 Findings The most important findings are evaluated in relation to the two-fold objective of this dissertation. We first clarify the practical added value of the model. The focus then moves to the capability of the heterogeneous model to explain observed traffic phenomena.

Extending the original LWR model to two classes resulted in an applicable model. This was achieved by complementing the analytical development by a numerical calculation scheme. The various intermediate steps that led to the complete model are also of practical value.

Conclusions

189

Scaling similar fundamental diagrams resulted in a heterogeneous model that uses passenger car equivalents (chapter 5). This stand-alone model enables the approximate modeling of heterogeneous traffic with minimal input.

A second intermediate step (chapter 6) generated a model that describes the traffic flow at different speeds during free-flow. This model can effectively model the impact of moving bottlenecks and dispersion.

The focus in the third intermediate step was on different forms of driver-behaviour during congestion (chapter 7). Here, vehicle composition affects the flow at a given speed.

With the aid of a network extension that was also developed in this dissertation, the final version of the model can be applied to motorways. This macroscopic model facilitates a rapid and accurate description of heterogeneous traffic. The model can be used in the design of road infrastructure, the implementation of traffic management measures and the real-time control of traffic flow. Additional applications in ITS or in forecasting traffic patterns are options in this model.

The systematic computation of heterogeneous stationary traffic based on the LWR model allows for the explanation of a number of observed traffic phenomena.

In the fundamental diagram, the points of the total traffic state lie between the two fundamental diagrams of the classes. The scatter commonly seen in observations also appears in the stationary model.

The capacity of an active bottleneck becomes dependent on the traffic composition and forward moving waves appear in the congestion regime, when the heterogeneous characteristics of the traffic participants are introduced.

The heterogeneous model enables for a more accurate description of the traffic process in light traffic. The dispersion effect is modeled by using heterogeneous speeds. During the parallel regimes the average speed becomes dependent on traffic composition.

The intriguing start-and-stop waves could be caused by local perturbations in the composition of the classes. Their origin could also be explained by changing traffic demand on on-ramps and by the variations in the amount of traffic leaving the motorway at off-ramps.

In the hysteresis effect the outflow from congestion to free-flow does not develop optimally. Local variations in traffic composition and small differences in lane-change preferences show that a decreased outflow is almost unavoidable in practical heterogeneous circumstances.

12.2 Further research For further research, we once again differentiate between the practical and the theoretical options.

Chapter 12 190

The heterogeneous model developed in this dissertation has potential for practical application. Application within a model-based control strategy is an obvious option. This also applies to the computation and evaluation of traffic operations and traffic management measures. The acquired insight can also be applied within ITS and moreover the model can support the design of the road infrastructure. To this end, a further extension of the theoretical LWR model to a comprehensive model is required.

The heterogeneous model can also be useful in planning models that calculate the costs of congestion, emissions and accidents. A gap currently exists between economic transport models and traffic operations models. The cost-functions in non-dynamic economic models are clearly heterogeneous in character. Emission costs, for example, are much larger for lorries than for private vehicles. To date, the more detailed traffic operations models are largely homogeneous. The heterogeneous LWR model narrows the gap somewhat. Dynamic cost functions render the formulated LWR model extremely suitable for more accurate economic analysis.

In addition to the applications that lie in the range of possibilities, there is no reason why new model developments could not occur. This, however, will require a more extensive validation of the model.

Pairing the heterogeneous model to higher order models with the input of driver related motivation aspects appears to be the ultimate way forward for macroscopic model development. For that purpose, a first step would be to examine the proposed extensions to the heterogeneous model from chapter 11. An explicit lane-model with, possibly, a greater number of classes or more complex class descriptions, appears desirable. An explanation of the mechanisms underlying the hysteresis phenomenon and the capacity funnel would come within the range of such a model.

The network version for the heterogeneous LWR model could be extended. This would imply a greater number of nodes based on a proper analytical foundation.

The construction of route-choice algorithms in a dynamic network model, as formulated recently by Lo (1999), can be extended to the heterogeneous LWR model.

The immediate model improvements must be complemented by the adaptation of the observation techniques for heterogeneous traffic. An adaptation of the oblique N-plots method for each class will provide new insight into traffic observations. This brings the estimation of travel times for each class within reach.

191

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199

APPENDIX A THE SPATIO-TEMPORAL

TRAFFIC FILTER

The method of Treiber and Helbing (2002) filters macroscopic traffic data from detectors. The method is illustrated using a general macroscopic traffic variable Z. The general variable Z can refer to any of the variables flow, density, average speed or occupancy. The detectors provide a measurement value for each separate detector location xDi and for each time interval. The t-x diagram can show these measurement values Z(xDi, tj) in a grid, as in figure A.1.

Figure A.1 : Raw detector data for a traffic variable Z in a t-x diagram.

The filter method is used to estimate Z for the entire t-x space. The surrounding measurement values are extrapolated to estimate a value for Z in an arbitrary point (t0 ,x0).

This interpolation explicitly takes account of the traffic regime. During the free-flow regime the traffic properties propagate themselves in the travel direction. During

t

x

Z(t,x)

xD1 xD2

xD3 xD4

t1t2

t3 t4 t5 t6

Appendix A 200

congestion, the characteristics of the traffic are propagated against the flow. This method calculates a separate Z-value for the two regimes. The ultimate value of Z then becomes a linear combination of both values.

),().1(),(.),( 000000 xtZwxtZwxtZ freecong −+= (A.1)

In this formula the parameter w, that is restricted to values between 0 an 1, will depend on the traffic regime. The determination of w will be elucidated further down this appendix.

In the free-flow regime it is assumed that the traffic flow properties are propagated in the direction of travel at the speed of uF (>0). During congestion, the assumed speed is uC (<0).

An exponential filter is used as a basis for the interpolation. All measurement values are weighted in this filter, according to the distance and the time difference to the point under consideration :

∑ ∑

∑ ∑

= =

= =

−−

−−

−−

−−

=max

1

max

1

00

max

1

max

1

00

00

exp

exp).,(

),(D

i

t

j

jDi

D

i

t

j

jDiDij

ttxx

ttxxxtZ

xt

τσ

τσφ

(A.2)

The parameters σ and τ determine the way in which the measurement values are averaged by the filter. These smoothing values reflect the area that is affected by the measurement values.

When this exponential filter is applied, the direction of the interpolation that was postulated for the free-flow and the congestion regimes is now taken into account. To this end, the time is rescaled along an axis with the speed of the particular regime :

),(),( 00

000 xu

xtxtZ

regime

regime −= φ (A.3)

Figure A.2 : Range of influence of the filter during free-flow and congestion

x

xD1

xD2

xD3

xD4

t1 t2 t3 t4 t5 t6

σ

t τ τ

The spatio-temporal traffic filter

201

In this way, the areas affected by the exponential filter are somewhat distorted, as shown in figure A.2.

Applying (A.2) and (A.3) with uF and uC give a value for the traffic variable Z during both regimes. The weighting parameter w from formula (A.1) is determined in a last step. This parameter reflects to what extent both regimes apply in the point under consideration (t0 ,x0).

Defining the regime happens according to the speed. Applying (A.1) and (A.2) using average speed U as the general traffic variable Z gives the average speed Ucong during congestion and Ufree during free-flow. In function of these two speeds, factor w in now calculated as :

∆

−+=

VUUV

wfreecong

c ),min(tanh1

21

(A.4)

Vc and ∆V are parameters in this formula.

The following values for the different parameters were proposed in the original publication.

Table A.1 : Parameters of the adaptive smoothing method

Parameter Proposed value

σ 0.6 km

τ 1.1 min

uF 80 km/h

uC -15 km/h

Vc 60 km/h

∆V 20 km/h

The method formulated here is implicitly based on the LWR model with a triangular fundamental k-q diagram. A fixed speed uF is put forward for the characteristic speed during free-flow. During congestion, uC corresponds to the speed of the congestion branch. Inaccuracies are filtered out by using the exponential function, and abrupt transitions between different regimes are averaged out. A global picture of the traffic data results, that were interpreted indirectly in accordance with the LWR model.

Appendix A 202

203

APPENDIX B OBLIQUE N-PLOTS

This appendix illustrates the principles of scaled cumulative curves on detectors. The method was proposed by Cassidy and Windover (1995) and refined by Muñoz and Daganzo (2002a).

Figure B.1 : Numbered trajectories.

With the aid of trajectories, individual vehicles on a through-road can be depicted in a t-x diagram. During actual traffic observations, legal and practical objections prevent us from gaining full knowledge of the road trajectories. Detectors enable the

1 2 3 4 5 6 7 8

9 10 11 12 13

t

6

6 7 8 9 10 11 12 13

7

1 3 4 5

x

Detector A

Detector B

Detector C

Appendix B 204

observation of traffic at fixed locations. Figure B.1 sketches a t-x diagram with three trajectories and three detectors.

By numbering all vehicles in relation to a reference vehicle, a cumulative function N(t,x) can be defined. This numbering takes account of the place changes of vehicles during overtaking manoeuvres.

The value of the cumulative function can be plotted at a detector location in a t-N diagram. The value of the cumulative function increases with each passing car. In figure B.2, N is shown at all three detector sites. In this process, the stepwise function is averaged out to a strictly increasing function. Downstream localised detectors always lie lower down in this diagram. When compared to the reference-vehicle, these detectors always show less passing vehicles than those upstream.

Figure B.2 : Cumulative curves for three detectors.

In this t-N diagram, the average travel time between two detectors can be read as the horizontal distance between the two associate curves. The number of vehicles between two detectors, the accumulation, can be read as the vertical distance between two curves.

In this diagram, the flow at the detectors is read as the slope of the curves because :

tNq

∆∆=

(B.1)

The N-curves for a detector are increasing functions. This hinders the readability over longer periods with more vehicles. A re-scaling shows the N-curves in a non-orthogonal co-ordinate system more clearly over longer observation periods. As mentioned this methodology was formulated by Cassidy and Windover (1995) and refined by Muñoz and Daganzo (2002a).

A background flow q0 can be freely deducted from the three N-curves, as shown in figure B.3. This reduces the value of the three N-curves for each time unit by q0. It decreases the slopes of the N-curves which makes it easier to show them in a diagram. By drawing lines with a slope of –q0 it becomes clear that the curves are now plotted in a non-orthogonal co-ordinate system. The vertical axis now depicts the flow minus the background flow. The original N-value can be read off an oblique axis, perpendicular to the lines with slopes –q0.

N

t 0

accumulation

travel time Detector A

Detector B

Detector C

Oblique N-plots

205

Figure B.3 :Scaled cumulative curves for the three detectors.

The number of vehicles between two detectors is still given by a vertical line between two curves. The value can be read both on the vertical as on the oblique axis. The travel time between two curves must now be read on a line with slope–q0. The flow at a detector now represents the slope of the curve augmented by the background flow q0.

When the curves are plotted for each separate detector, the traffic pattern can be studied with optimal accuracy on a road without on- and off-ramps. To show the detector output in this diagram, two additional problems must be addressed:

Since not all vehicles are registered, the cumulative flows at a detector contain errors. Magnetic measurement loops, for example, may find vehicle lane-changes near the detection location problematic.

At a certain time, the number of vehicles between two detectors must be known, in order to determine the vertical distance between different curves.

When applying the N-curves in the case-study in chapter 10, these problems were addressed as follows. During free-flow, we assume that all vehicles drive at the same average speed. If this assumption is valid, the shape of the N-curves of two successive detectors is identical. In the application, the accumulation looked for was the one in which the shapes of two successive N-curves differ the least. In mathematical terms this amounts to finding the number of vehicles acc for which expression B.2 is minimal :

( )∫ −−max

min

2detdet ),(),(min

t

tdownup dtacctxNtxN

(B.2)

Finding the accumulation was solved by applying a least squares method. The cumulative error of a detector is now dealt with by finding this accumulation separately for a free-flow period both at the beginning and at the end of the investigation period. By averaging out the difference in the number of vehicles taken over the entire investigation period, the two N-curves are adjusted to each other..

t

N(xdet,t)-q0.t

travel time

N

0

accumulation -q0

0

Detector A Detector B

Detector C

Appendix B 206

207

SAMENVATTING : DYNAMISCH MODELLEREN VAN

HETEROGENE VERKEERSSTROMEN

Het oorspronkelijke macroscopische verkeersstroommodel idealiseert verkeer op een snelweg als een homogeen fluidum. De voertuigen met bestuurders worden in dit model als gelijke deeltjes in een vloeistofbuis beschreven.

In dit proefschrift wordt met de heterogene eigenschappen van het verkeer rekening gehouden. Hiervoor wordt de verkeersstroom opgesplitst in homogene klassen. Elke klasse bestaat uit voertuigen en bestuurders met gelijke kenmerken. Modelleren van heterogeen verkeer omvat dan de beschrijving van homogene klassen en de interacties tussen de verschillende klassen.

Per wegsectie wordt een klasse gekenmerkt door de maximale snelheid, de voertuiglengte en de capaciteit. De capaciteit van een klasse is de maximale intensiteit wanneer alleen voertuigen uit die specifieke klasse op de weg rijden.

De interacties tussen de verschillende klassen is gebaseerd op het gebruikersoptimum: elke bestuurder wordt verondersteld zijn eigen snelheid te maximaliseren. Verder wordt aangenomen dat snelle voertuigen de snelheid van trage voertuigen niet kunnen beïnvloeden. Op die manier gedragen de tragere voertuigen zich als bewegende knelpunten.

Het opgestelde heterogene model bestaat uit een wiskundige formulering die analytisch en grafisch opgelost kan worden. Daarnaast is een numeriek schema opgezet. Hierdoor kan het model met een computer geïmplementeerd worden en kunnen benaderende oplossingen snel berekend worden.

Samenvatting 208

Het ontwikkelde model is verder uitgebreid voor gebruik op complete verkeersnetwerken. Een toepassing van het model in een praktijkgeval illustreert de praktische inzetbaarheid. Afsluitend volgt een kritische discussie van aannames en eigenschappen van het model en worden mogelijke uitbreidingen aangestipt.

Hoofdstuk 1 : Inleiding Hoewel mobiliteit veel meer dan dat omvat, blijven files een van de meest intrigerende verkeersfenomenen. Files zijn wachtrijen van voertuigen die de tijdelijke en lokale overbelasting van het wegennet markeren. Het achterhalen van de mechanismes van interacterende voertuigen en bestuurders in de file blijft een drijfveer bij het uitwerken van de verkeersstroomtheorie.

Binnen de verkeersstroomtheorie worden wiskundige modellen opgezet om de verkeersafwikkeling accuraat te beschrijven en te voorspellen. Door toenemende observaties en de groeiende rekensnelheid van computers lijken deze modellen nog steeds voor verbetering vatbaar.

Het eerste dynamische macroscopisch verkeersmodel van Lighthill and Whitham (1955) en Richards (1956) (LWR) beschreef het verkeer als bestaande uit identieke voertuigen en bestuurders. In dit proefschrift wordt dit oorspronkelijke model uitgebreid met de bedoeling verschillende voertuigen bestuurderskenmerken in rekening te kunnen brengen.

Door een analytisch model van een heterogene verkeersstroom op te zetten, kan het nodige inzicht verworven worden om verkeersobservaties te interpreteren. Hierdoor wordt het mogelijk nieuwe verklaringen voor verkeersfenomenen te zoeken.

Verder wordt gestreefd naar een toepasbaar heterogeen LWR model. Om deze toepasbaarheid te bereiken moet naast de opzet van een analytisch model een numeriek schema uitgewerkt worden. Het heterogeen LWR model moet ook tot een volwaardig netwerkmodel uitgebreid worden.

Een belangrijke vereiste om tot een toepasbaar model te komen is dat de kloof tussen modelontwikkelaars en modelgebruikers niet te groot wordt. Daarom moet de opbouw van het model in logische stappen gebeuren. Ook de mogelijkheid om, net zoals in het LWR model, de oplossing grafisch te construeren, moet behouden blijven.

Hoofdstuk 2 : Dynamisch verkeer In dit hoofdstuk wordt een overzicht van de verkeersstroomtheorie geschetst. In deze theorie wordt met behulp van wiskundige instrumenten de verkeersafwikkeling op snelwegen beschreven. Door gebruik te maken van de verkeersstroomtheorie kunnen controlemaatregelen bedacht en vooraf geëvalueerd worden en kan het ontwerpen van verkeersinfrastructuur verbeterd worden.

Eerst werken we een kader uit waarbinnen het voertuigverkeer wiskundig wordt beschreven. Deze abstractie van de werkelijkheid leidt tot een set consistente

Dynamisch modelleren van heterogene verkeersstromen

209

definities. Voor dit proefschrift zijn de dichtheid k, de intensiteit q en de gemiddelde snelheid u de belangrijkste. Tussen deze drie variabelen geldt volgende relatie :

ukq .= (S.1)

In verkeersobservaties wordt een relatie tussen de intensteit q en de dichtheid k waargenomen. Omdat deze empirische functie Qe(k) meestal grafisch wordt voorgesteld, spreken we van ‘het fundamenteel diagram’. In figuur S.1 wordt een mogelijk fundamenteel k-q diagram geschetst. Algemeen geldt dat de intensiteit Qe(k) nul is voor een nuldichtheid en voor de maximale dichtheid kJ. Tussenin bereikt de functie een maximale intensiteit qM die ook de capaciteit wordt genoemd. De snelheid van de verkeersstroom wordt in dit diagram als een helling afgelezen zoals aangeduid op de figuur. Het stijgend gedeelte van de curve komt overeen met het ‘vrij-verkeer’ regime, het dalend gedeelte met ‘congestie’. Bij een onbelaste weg is de snelheid maximaal en gelijk aan uf.

Figuur S.1: Een fundamenteel k-q diagram

De interpretatie van waarnemingen leidt vervolgens tot verschillende types modellen. In deze dynamische verkeersmodellen worden aannames omtrent de mechanismes van de verkeersstroom uitgewerkt tot een set wiskundige vergelijkingen. In het verdere proefschrift wordt enkel met macroscopische modellen gewerkt.

Hoofdstuk 3 : Overzicht van het LWR model In dit hoofdstuk wordt het oorspronkelijke macroscopische vekeersmodel van Lighthill and Whitham (1955) en Richards (1956) – het LWR model – uitvoerig toegelicht. In dit LWR model wordt het behoud van voertuigen gecombineerd met het fundamenteel diagram, de empirische relatie tussen de intensiteit q en de dichtheid k. Wiskundig leidt dit tot volgende vergelijking :

0),(.)(),( =∂

∂+∂

∂x

xtkdk

kdQt

xtk e (S.2)

Met behulp van karakteristieken kan de oplossing van deze vergelijking analytisch en grafisch bestudeerd worden. Karakteristieken zijn rechten met een helling Q’e(k0) die vanuit een punt met dichtheid k0 in het t-x vlak getekend worden. In alle punten op

q

k

u

qM

kJ

congestie

vrij- verkeer

uf

Qe(k)

Samenvatting 210

deze oplossingslijn blijft de dichtheid k0 gelden. Deze oplossing kan grafisch geconstrueerd worden wanneer het fundamenteel k-q diagram en het t-x diagram geschaald worden. Dezelfde helling in beide diagrammen komt dan met een zelfde snelheid overeen.

In figuur S.2a wordt een fundamenteel diagram met bijhorende hellingen van karakteristieken geschetst. Tijdens vrij-verkeer, zoals voor dichtheid k1 in de figuur, hebben de karakteristieken een positieve helling. Tijdens congestie gaan de karakteristieken tegen de rijrichting in.

Wanneer we een weg waarop de dichtheid toeneemt in de x-richting beschouwen, ontstaat een schokgolf met helling ∆q/∆k. In figuur S.2b wordt weergegeven hoe een toename van de dichtheid in x0 de dichtheid verandert er van k1 naar k2 een golf genereert. De snelheid U12 van deze golf kan in het fundamenteel diagram als de helling van de verbindingslijn van deze twee verkeerstoestanden afgelezen worden. In deze golf verandert de verkeerstoestand discontinu en vermindert de snelheid van u1 naar u2. Dit wordt met een trajectorie, het pad van een voertuig, geïllustreerd.

Een afname van de dichtheid in de x-richting veroorzaakt een waaier van karakteristieken waarbij alle tussenliggende dichtheden aan bod komen. Figuur S.2c illustreert hoe de intensiteit uit congestie naar een afwaartse vrij-verkeer toestand altijd over het congestieregime verloopt. De uitstroom uit de file is bijgevolg optimaal in het LWR model.

Figuur S.2: (a) een fundamenteel diagram; (b)een schokgolf en (c) een waaier.

Naast de analytische en grafische oplossingsmogelijkheid is er een numerieke methode voor het LWR model. Hiervoor wordt de weg in cellen opgedeeld. Vervolgens wordt per tijdstap de dichtheid in een cel benaderend berekend. Door deze numerieke methode kan het analytische model met een computer snel opgelost worden.

De belangrijkste eigenschappen van het LWR model vloeien rechtstreeks uit het fundamenteel diagram voort. Vooral de vorm hiervan is belangrijk. Verder wordt het verkeer in dit fundamenteel diagram homogeen en stationair verondersteld. Deze stationaire eigenschap zorgt ervoor dat de versnelling nul is, behalve in schokgolven waar de snelheid discontinu verandert. De homogeniteit blijkt uit de gelijke snelheid van alle voertuigen. Voor deze tekorten kunnen modeluitbreidingen voorgesteld worden. In dit proefschrift wordt het LWR model uitgebreid met de bedoeling

t

x

x0

trajectoriet

x

x0

k1, q1

k2, q2 k1, q1

k2, q2

trajectorie

c1

q

Qe(k)

c2

k k2

u2

q2q1

k1

u1

U12

kJ

qM

kM

uf


211

heterogene eigenschappen van voertuigen en bestuurders in rekening te kunnen brengen.

Hoofdstuk 4 : Klassen in het LWR model In dit hoofdstuk wordt de basis gelegd voor het heterogeen LWR model. Daarvoor wordt de heterogene verkeersstroom op basis van gemeenschappelijke eigenschappen in homogene klassen opgesplitst. We nemen aan dat alle voertuigen en bestuurders uit een klasse identiek reageren en hetzelfde rijgedrag vertonen.

De dichtheid ki, de intensiteit qi en de gemiddelde snelheid ui kunnen we per klasse definiëren. De wet van verkeersbehoud, die de basis vormt van het LWR model, geldt dan voor elke klasse i afzonderlijk en voor de globale verkeersstroom :

0=∂∂+

∂∂

xq

tk ii

(S.3)

Net zoals in het homogene LWR model wordt gebruik gemaakt van een relatie die de intensiteit van een klasse qi uitdrukt in functie van de dichtheid. Deze intensiteit is in een heterogeen model echter een functie van alle klassedichtheden:

),...,( 1 neii kkQq = (S.4)

Combineren van (S.4) met het behoud van voertuigen (S.3) levert een stelsel van partiële differentiaalvergelijkingen op, die het heterogeen LWR model vormt

Bij de uitwerking van het model zullen schokgolven ontstaan. Ook hier geldt dat de golfsnelheid z met de verandering van de dichtheden en intensiteiten over de schokgolf gerelateerd is :

i

i

kq

dtdxz

∆∆==

(S.5)

Over een schokgolf moet deze voorwaarde zowel voor de afzonderlijke klassevariabelen als voor de totale verkeersstroom gelden.

Aangenomen wordt dat het verkeer anisotroop is. Dit wil zeggen dat een bestuurder enkel reageert op stimuli vóór zich en dat de verkeerssituatie achter het voertuig geen invloed op het rijgedrag heeft. In een homogeen model is aan deze anisotropie voorwaarde voldaan wanneer de snelheid van de karakteristieken kleiner of gelijk is aan de snelheid van de voertuigen (Zhang 2003). In een heterogeen model blijft deze anisotropie voorwaarde geldig, maar nu per klasse. Hierdoor hoeft de snelheid van een schokgolf slechts kleiner te zijn dan de voertuigsnelheden van de klassen die door de betreffende golf beïnvloed worden. Wanneer een trage klasse over een schokgolf niet verandert, mag de snelheid van deze golf dus wel groter zijn dan de snelheid van de voertuigklasse.

Wanneer op een weg slechts voertuigen uit één klasse rijden, geldt het oorspronkelijk LWR model met een fundamenteel diagram voor die klasse. De dan geldende relatie noemen we voortaan het homogeen fundamenteel diagram en duiden we aan als:

Samenvatting 212

)( iheii kQq = (S.6)

Uit een vergelijking van alle tot nu toe ontwikkelde heterogene stationaire LWR modellen blijkt dat de heterogene relaties (S.4) allen geïnterpreteerd kunnen worden met behulp van deze homogene fundamentele diagrammen :

=

i

iheiii

kQqα

α . (S.7)

Hierin is αi de fractie van de totale weg die klasse i toegewezen krijgt. De weg kunnen we in dit geval beschouwen als opgesplitst in parallelle ruimtes waarover elke klasse zich afzonderlijk afwikkelt.

Voor alle tot nu toe opgestelde heterogene stationaire modellen worden de heterogene relatie (S.4) in de vorm van uitdrukking (S.7) omgezet. De modellen verschillen daarbij op de interactie-aanname: hoe worden de wegfracties toegewezen aan de verschillende klassen in functie van de klassedichtheden.

Naast de ontrafeling van de expliciete aannames in de verschillende modellen wordt ook duidelijk dat de anisotropie voorwaarde nagenoeg overal wordt geschonden. Daarom stellen we in dit proefschrift een eigen aanname voort die daar wel rekening mee houdt.

Tijdens het gebruikersoptimum verdelen de klassen zich over de totale wegruimte zodat elk voertuig zijn snelheid niet meer kan verhogen zonder de snelheid van trager rijdende voertuigen te verminderen. Deze definitie houdt expliciet rekening met de anisotropie. De snelheid van een voertuig is enkel afhankelijk van voertuigen met een gelijke of lagere snelheid. Snellere voertuigen kunnen de snelheid van tragere voertuigen nooit verminderen zodat beïnvloeding door achterop rijdende voertuigen uitgesloten is. In een verdere specificatie van dit gebruikersoptimum nemen we aan dat trage voertuigen niet meer ruimte innemen dan strikt noodzakelijk is. Dit efficiënt gebruik van de weg komt ook in Europese inhaalregels op snelwegen naar voor. Trage voertuigen voegen snel in om zo niet meer wegruimte dan strikt noodzakelijk in te nemen.

In het verdere proefschrift volgt een stapsgewijze uitwerking van het heterogeen LWR model volgens dit gebruikersoptimum. Figuur S.3 schetst hoe de volgende vijf hoofdstukken tot een volwaardig heterogeen netwerkmodel leiden. Bij de eerste vier stappen wordt telkens een analytische en een numerieke uitwerking van het model voorgesteld. Aansluitend volgt voor de eerste drie stappen een illustrerende case.


213

Figuur S.3 : Opbouw heterogeen LWR model

Hoofdstuk 5 : Transformatie van het fundamenteel diagram

In dit hoofdstuk wordt een eerste stap gezet naar een volwaardig heterogeen LWR model. Centraal hierbij staat dat alle voertuigen dezelfde snelheid hebben en dat alle klassen door een gelijkvormig homogeen fundamenteel diagram worden beschreven. Hierdoor is het mogelijk een transformatie van een klasse uit te voeren waarbij de schaalfactor gerelateerd is aan de personenauto-equivalenten (pae). Deze schaling laat toe de heterogene stroom als een homogene stroom te modelleren. In deze methode worden geen beperkingen aan de vorm van het fundamenteel diagram opgelegd.

Naast een analytisch kader wordt een numeriek schema opgesteld dat gebaseerd is op het gebruikelijke schema voor het LWR model. Een uitgewerkte case illustreert het analytische model en de effecten van het numerieke schema.

De grote kennis van pae waarden voor verschillende voertuigklassen en infrastructuurtypes maken een snelle toepassing van dit model in de praktijk mogelijk.

Hoofdstuk 6 : Heterogeen vrij-verkeer In dit hoofdstuk wordt de verkeersstroom in twee klassen opgesplitst. Per klasse geldt een driehoekig homogeen fundamenteel diagram. In deze tweede stap focussen we op het verschillend rijgedrag van voertuigen tijdens vrij-verkeer. Daarom geldt voor beide klassen een identiek rijgedrag tijdens congestie. In figuur S.4 worden de twee

stap 1

Transformatie van het fundamenteel diagram

Hoofdstuk 5

step 2

Heterogeen vrij-verkeer

Hoofdstuk 6

step 3

Heterogeen congestie verkeer

Hoofdstuk 7

stap 4

Het heterogeen LWR model

Hoofdstuk 8

stap 5

Netwerk verkeer

Hoofdstuk 9

Samenvatting 214

fundamentele diagrammen geschetst. De congestietakken van beide diagrammen vallen samen. Klasse 1, met de grootste maximale snelheid, heeft ook de grootste capaciteit.

Figuur S.4 : Twee homogene fundamentele diagrammen die gelden voor twee voertuigklassen

De uitwerking van het gebruikersoptimum voor deze twee klassen leidt tot drie regimes.

• Tijdens vrij-verkeer hebben beide klassen hun maximale snelheid. De voertuigen van de verschillende klassen elkaar perfect voorbijsteken. De voertuigstroom is per klasse parallel op de weg opgesplitst.

• Tijdens semi-congestie blijven voertuigen van de traagste klasse aan hun maximale snelheid uf2 rijden. De snelste klasse bevindt zich echter in congestie: de snelheid is lager dan uf1, maar toch groter dan de klasse 2 snelheid.

• In congestie hebben alle voertuigen dezelfde snelheid die lager is dan de maximale klasse 2 snelheid.

Deze drie regimes worden in figuur S.5 in een fasendiagram geschetst. Dit diagram toont de verschillende regimes volgens de dichtheid van beide klassen. Vrij-verkeer wordt met ‘A’ aangeduid, ‘B’ staat voor semi-congestie en ‘C’ voor congestie.

Met het bepalen van deze stationaire verkeerstoestanden worden de klasse intensiteiten qi volgens (S.4) bepaald. In een tweede stap werken we overgangen tussen alle mogelijke stationaire toestanden uit. Dit leidt tot een kwalitatieve beschrijving van 15 types van transities. Deze transities zijn uit schokgolven of waaiers van karakteristieken samengesteld. De beschrijving van de transities geeft een volledig overzicht van het model.

)(2 kQ he

k

q

kM2 kJ kM1

qM2

qM1

uf2 uf1

w

M1

M2

)(1 kQhe


215

Figuur S.5 : Fasendiagram voor een gemengde verkeersstroom

Na deze analytische uitwerking wordt een numeriek schema opgesteld. In een gevalstudie wordt ‘het bewegend knelpunt’ van Newell (1998) zowel analytisch als numeriek uitgewerkt De originele probleemstelling van een bewegende hindernis wordt vertaald naar een interactie tussen twee voertuigklassen met verschillende snelheden. De voertuigen van de traagste klasse fungeren dan als de bewegende knelpunten voor snellere voertuigen.

Hoofdstuk 7 : Heterogeen congestie verkeer In dit hoofdstuk beschouwen we opnieuw twee voertuigklassen waarbij de klemtoon nu op het verschillend rijgedrag tijdens congestie ligt. Per klasse geldt een driehoekig homogeen fundamenteel diagram. Enkel de congestietakken van deze diagrammen verschillen voor beide klassen zoals weergegeven in figuur S.6. Gedurende vrij-verkeer gedragen beide klassen zich op dezelfde manier. We nemen aan dat alle voertuigen gedurende congestie altijd dezelfde snelheid hebben.

Figuur S.6 : Twee homogene fundamentele k-q diagrammen met verschillende congestietakken

De uitwerking van de stationaire verkeerstoestanden leidt tot twee regimes. Er ontstaat een driehoekig fundamenteel diagram waarvan de congestietak afhangt van de verkeerssamenstelling.

k1

k2

kM2 kJ

kJ

kM2

kM1

C

A B

M

k kJ1kJ2 kM

qM q

uf

w1 w2

)( 11 kQ he

)( 22 kQhe

Samenvatting 216

Na deze stationaire verkeerstoestanden komen alle mogelijke transities tussen de regimes aan bod. De vier soorten transities worden allen kwalitatief beschreven.

Aansluitend wordt de numerieke methode voor dit model ontwikkeld. In een gevalstudie wordt een verkeerssituatie met verkeerslichten zowel analytisch als numeriek opgelost.

Hoofdstuk 8 : Het heterogeen LWR model De combinatie van de drie ontwikkelde LWR uitbreidingen leidt in dit hoofdstuk tot een algemeen heterogeen LWR model. Dit model beschrijft een verkeersstroom waarbij de voertuigen in twee klassen zijn opgedeeld.

Elke klasse kenmerkt zich door een homogeen driehoekig k-q fundamenteel diagram zoals weergegeven in figuur S.7 . Het gedrag van de voertuigen is deterministisch en de voertuigen blijven zich naar deze homogene fundamentele diagrammen schikken, ook wanneer de verkeersstroom heterogeen is samengesteld.

Figuur S.7 : Twee driehoekige homogene fundamentele diagrammen.

Voor de interacties tussen de twee klassen wordt het gebruikersoptimum aangenomen. Dit stelt dat een voertuig z’n snelheid niet meer kan verhogen zonder de snelheid van trager rijdende voertuigen te beïnvloeden. Deze aanname leidt tot gelijke snelheden tijdens congestie zoals in het heterogeen congestie model uit hoofdstuk 7. Tijdens vrij-verkeer zijn verschillende snelheden mogelijk en verloopt het model analoog aan de uitbreiding in hoofdstuk 6.

Om het verband tussen de twee diagrammen te leggen wordt een schaalfactor r gebruikt zoals in hoofdstuk 5. De capaciteit van de traagste klasse wordt op de congestietak van de snelste klasse geschaald zoals weergegeven in figuur S.7.

Net zoals in hoofdstuk 6 ontstaan drie regimes : vrij-verkeer, semi-congestie en congestie. De uitwerking van alle mogelijke transities leidt vaak tot complexe uitdrukkingen. Wanneer de congestietak van de traagste klasse minder steil is dan die van de snelste klasse, ontstaan mogelijks schokgolven waarover het verkeer versnelt.

k

kJ2 kJ1 r.kM2

q

kM2kM1

w1 w2

qM1

r.qM2

qM2

uf2 uf1


217

Deze eigenschappen verlopen zoals in het klassiek LWR model met een niet-concaaf fundamenteel diagram.

Het opgestelde numeriek schema berekent op discrete tijdstappen de dichtheid in een cel. De numerieke methode vertoont echter dispersie.

Hoofdstuk 9 : Netwerk verkeer Het heterogeen model uit het vorig hoofdstuk beschrijft de verkeersafwikkeling op een doorgaande weg. Om praktische modeltoepassingen mogelijk te maken, wordt het model uitgebreid voor netwerken.

Zoals in de klassieke verkeersmodellen nemen we aan dat een wegennetwerk bestaat uit knooppunten en schakels. Binnen deze filosofie vertrekt en eindigt een homogene schakel altijd in een knooppunt. Knooppunten hebben geen fysieke lengte en fungeren als uitwisselingslocatie van verkeersintensiteiten. Naargelang het aantal toekomende en vertrekkende schakels onderscheiden we in dit hoofdstuk volgende types knooppunten.

• Herkomstknooppunt

In een herkomstknooppunt wordt verkeer op het netwerk losgelaten. Deze knooppunten fungeren als de voedingspunten van het verkeersnetwerk. We nemen aan dat er vanuit deze bronnen slechts 1 schakel kan vertrekken terwijl er geen kunnen toekomen zoals in figuur S.8a.

• Bestemmingsknooppunt

Verkeer verlaat het netwerk in bestemmingsknooppunten. We veronderstellen dat deze ‘putten’ een oneindige capaciteit hebben. Zoals weergegeven in figuur S.8b. kan slechts 1 schakel toekomen in een bestemmingsknoopunt.

• Inhomogeen knooppunt

Een weg met veranderende wegeigenschappen wordt opgesplitst in homogene schakels die in inhomogene knooppunten van wegeigenschappen veranderen. In dergelijk knooppunt komt 1 schakel toe en wordt al het verkeer doorgesluisd naar die ene vertrekkende schakel zoals weergegeven in figuur S.8c.

• Splitsingsknooppunt

In een splitsingsknooppunt komt 1 schakel toe en vertrekken precies twee schakel zoals weergegeven in figuur S.8d. Met behulp van deze splitsingspunten kunnen afslagen gemodelleerd worden

• Samenvoegingsknooppunt

In een samenvoegingsknooppunt komen twee schakels toe en vertrekt 1 schakel zoals in figuur S.8e. Deze samenvoegpunten maken het mogelijk opritten en samenvloeiingen te modelleren.

Samenvatting 218

Figuur S.8 : Type knooppunten

Deze beperkte set knooppunten fungeren als bouwblokken. Nagenoeg alle mogelijke netwerken van snelwegen kunnen ermeer opgebouwd worden. Het modelleren van stedelijke netwerken vergt een verdere uitbreiding van deze types.

Per type knooppunt worden de klassieke methodes uit de LWR theorie op netwerkniveau doorgelicht. Vervolgens wordt per type knooppunt een aangepaste heterogene methode uitgewerkt. Daarbij houden we rekening met aangenomen rijgedrag, de complexiteit en numerieke aspecten.

Hoofdstuk 10 : Gevalstudie In dit hoofdstuk passen we het ontwikkelde heterogeen LWR model toe. Door het geobserveerde en gemodelleerde verkeerspatroon met elkaar te vergelijken worden de mogelijkheden van het model aangetoond. Met behulp van de netwerkuitbreiding uit het vorig hoofdstuk wordt een snelweg van 7.5 kilometer met drie opritten en twee afritten tijdens de ochtendspits gemodelleerd. In figuur S.9 wordt de werkelijke snelweg en de modelweergave geschetst.

Eerst worden het studiegebied en de beschikbare verkeersobservaties toegelicht. De verkeersdata worden volgens twee methodes, die toegelicht worden in de appendices, geanalyseerd. Dit geeft een overzicht van de structurele congestie in het studiegebied en illustreert enkele recente inzichten in congestiemechanismes. Deze telgegevens bepalen eveneens de verkeersvraag die op het gemodelleerde netwerk wordt losgelaten.

Tijdens de modelopbouw komt de omzetting van het werkelijke netwerk tot een set van schakels en knooppunten aan bod.

Herkomstknooppunt

Bestemmingsknooppunt

Inhomogeen knooppunt

Splitsingsknooppunt

(a)

(b)

(c)

Samenvoegings- knooppunt

(e)

(d)


219

Figuur S.9 : (a) het wegennetwerk en (b) de modelvoorstelling ervan

In de calibratiefase worden de modelparameters bijgestuurd om een zo goed mogelijke weergave van het verkeerspatroon te bekomen. Hiervoor worden per schakel de homogene fundamentele diagrammen aangepast. Aan de hand van een nieuwe verkeersvraag wordt het opgestelde model gevalideerd. Tijdens de calibratie en validatie blijkt dat het model in staat is een goede kwalitatieve beschrijving van de verkeersstroom te reproduceren.

Hoofdstuk 11 : Discussie In dit hoofdstuk wordt het opgezette heterogene LWR model kritisch bekeken. We baseren ons hiervoor op het basismodel voor een schakel uit hoofdstuk 8. De netwerkversie uit hoofdstuk 9, met de behandeling van verschillende types knooppunten, komt niet aan bod.

Eerst toetsen we de algemene eigenschappen van het model aan de besproken verkeersfenomenen uit hoofdstuk 2.2 en aan de waarnemingen tijdens de case uit hoofdstuk 10.2. Vervolgens wordt dieper ingegaan op enkele losse aspecten zoals de gewogen congestietak, het afstemmen van de fracties op het aantal rijstroken, de homogeniteit van klassen, het niet optimaal gebruik van de weg en de betekenis van het niet concaaf fundamenteel diagram. Bij deze aspecten worden de gebruikte aannames in vraag gesteld en worden nieuwe aannames en modeluitbreidingen voorgesteld. Tot slot wordt de relatie met andere dynamische verkeersstroommodellen besproken.

Hoofdstuk 12 : Besluit In dit proefschrift werd het klassieke LWR model uitgebreid waarbij heterogene bestuurder- en voertuigkenmerken in rekening werden gebracht. Uit waarnemingen blijkt immers dat de verkeersstroom niet homogeen is.

De uitbreiding van het klassieke LWR model naar twee klassen heeft geleid tot een toepasbaar model. Hiervoor werd naast de analytische uitwerking ook het numeriek schema aangepast. De verschillende tussenstappen die tot het complete model leiden, hebben eveneens praktisch nut. Het uiteindelijk model is met behulp van de ontwikkelde netwerkuitbreiding inzetbaar voor snelwegen. Een snelle en accurate

D3 D4 D5 D6 D7 D8 D9 D10 D11 D12

rijrichting


Strook 21

A 1 B 2 C 3 D 4 E 5 F 6 G 7 H

8 9 10

I J K L M

11 12

(a) (b)

Samenvatting 220

beschrijving van heterogeen verkeer wordt bijgevolg met een macroscopisch model mogelijk. Dergelijk model kan ingezet worden bij het ontwerpen van weginfrastructuur, het implementeren van verkeersbeheersmaatregelen of het real-time controleren van verkeersstromen. Verdere toepassingen binnen ITS of voorspellen van verkeerspatronen zijn eveneens mogelijk.

De systematische uitwerking van heterogeen stationair verkeer op basis van het LWR model laat toe om enkele geobserveerde verkeersfenomenen te verklaren. Alle mogelijke verkeerstoestanden liggen tussen de twee homogene fundamentele diagrammen. Daardoor wordt de mythe van het vaste fundamenteel diagram doorprikt. De capaciteit van een actieve bottleneck wordt afhankelijk van de verkeerssamenstelling en in het congestieregime verschijnen voorwaartse golven volgens de heterogene kenmerken van de verkeersdeelnemers. In het discussie hoofdstuk werd duidelijk dat lokale variaties in de verkeerssamenstelling nagenoeg onvermijdelijk de uitstroom van congestie naar vrij-verkeer beperken. Dit biedt een nieuwe mogelijke verklaring voor het hysteresis effect.

221

ABOUT THE AUTHOR

Steven Logghe was born in 1975 in Oostend, Belgium. He finished highschool in the St. Jozefsinstituut Torhout in 1993. In 1998 he received a master’s degree in civil engineering from the K.U.Leuven. Together with Joost Lismont he wrote the masters’ thesis ‘Evaluation and application of a method to design a public transportation network on national and regional level’.

Since 1998, Steven is working as a research assistant at the Transportation Planning and Highway Engineering division of the K.U.Leuven under the leadership of Prof. Ben Immers. In this period he did research on the OSTC projects “Traffic congestion problems in Belgium: mathematical models, simulation, control and actions” and “Sustainability effects of traffic management systems”. He specialised in traffic flow theory and the application of microsimulation models. He wrote course texts and lectured on traffic signals and traffic flow theory. Furthermore he was involved in microsimulation modeling for Antwerp and Ghent and time-of-day modeling. His interest in macroscopic models resulted in this phD dissertation about dynamic modeling of heterogeneous vehicular traffic.

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