Thesis Kecman.

Transcript

thesis Series T 2014 /5 The potential growth in transport demand in the next decade and beyond requires a change from reactive to proactive traffic control to maintain and improve the reliability of railway traffic. In order to enable an anticipative approach to traffic management, it is necessary to develop the tools for monitoring, prediction and optimisation of the traffic operations. This thesis presents the models that can be used as components for a decision support system for predictive traffic management. About the Author Pavle Kecman received his M.Sc. degree from the University of Belgrade in 2008. In June 2010 he joined the Department of Transport and Planning, Delft University of Technology, as a Ph.D. candidate. He currently works as a postdoctoral researcher at the Department of Science and Technology, Linköping University in Sweden. Pavle Kecman Pavle Kecman Models for Predictive Railway Traffic Management Summary Thesis Series TRAIL Research School ISBN 978-90-5584-175-2 Models for Predictive Railway Traffic Management Propositions Pertaining to the dissertation Models for Predictive Railway Traffic Management Pavle Kecman 20 October 2014 1. Extracting and processing information from a real-time data stream requires all typical steps for offline data analysis to be performed simultaneously. (Chapter 3) 2. Having in mind the observed variability of running and dwell times, accurate modelling of the latter requires more attention in order to create valid railway planning and control models. (Chapter 4) 3. A data-driven approach outperforms microscopic simulation tools for real-time prediction in terms of prediction quality, requirements for implementation and computation speed. (Chapter 5) 4. A macroscopic rescheduling model that takes into account minimum headway times in stations and overtaking constraints on open track provides fast solutions of good quality. Thus it is applicable to serve as a decision support system for traffic controllers. (Chapter 6) 5. In order to ensure sustainable mobility, the transport market should be strictly regulated based on the proven (dis)advantages of certain modes for certain trips. 6. The number of citations does not say much about the paper quality just like the number of sold copies or tickets is not a quality indicator of a music record or a movie. 7. George Orwell’s dystopian principle: “Who controls the past, controls the future” is turning out to be correct with the increasing impact of historical data on decision making processes in economy, finance, trade and retail. 8. Rational people push the world forward but it’s the irrational people that make it worth living in. 9. Copyright infringement has a better effect on arts and popular culture than the restrictive intellectual property laws. 10. Everything looks bad if you think about it long enough. These propositions are considered opposable and defendable and have been approved as such by the promotor Prof. Dr.- Ing. I.A. Hansen. Stellingen Behorend bij het proefschrift Models for Predictive Railway Traffic Management Pavle Kecman 20 october 2014 1. Het extraheren en verwerken van informatie vanuit een real-time datastroom vereist dat alle kenmerkende stappen van offline data analyse simultaan worden uitgevoerd. (Hoofdstuk 3) 2. Met de geobserveerde variabiliteit van rijd- en wachttijden in ogenschouw genomen, vereist het nauwkeurig modelleren van wachttijden meer aandacht, om zodoende goede rail planning en regeling modellen te cre¨eren. (Hoofdstuk 4) 3. Een data gestuurde aanpak presteert beter dan microscopische simulatie voor real-time voorspellingen met betrekking tot de kwaliteit van de voorspelling, benodigdheden voor implementatie, en rekensnelheid. (Hoofdstuk 5) 4. Een macroscopisch model dat minimale volgafstand op stations en inhaalmogelijkheden op de open baan meeneemt voor de herplanning van de dienstregeling levert snelle en goede oplossingen. Daarom is het bruikbaar als beslissingsondersteunend systeem voor treindienstleiders. (Hoofdstuk 6) 5. Om duurzame mobiliteit te bewerkstelligen moet de transportmarkt strikt gereguleerd worden op basis van bewezen voor- en nadelen van bepaalde vervoerswijzen voor bepaalde reizen. 6. Het aantal citaties van een artikel zegt niet veel over de kwaliteit, net zoals het aantal verkochte kopie¨en of toegangsbewijzen geen kwaliteitsindicator is voor een muziekstuk of film. 7. George Orwells dystopisch principe: “Wie het verleden beheerst, beheerst de toekomst” blijkt correct met de toenemende invloed van historische gegevens op besluitvormingsprocessen in de economie, financi¨en, en detailhandel. 8. Rationele mensen brengen de wereld voorwaarts, maar het zijn de irrationele mensen die het waard maken erop te leven. 9. Schending van auteursrecht heeft een beter effect op kunst en populaire cultuur dan de beperkende eigendomsrecht wetten. 10. Alles lijkt slecht als je er maar lang genoeg over nadenkt. Deze stellingen worden opponeerbaar en verdedigbaar geacht en zijn als goedgekeurd door de promotor Prof. Dr.- Ing. I.A. Hansen. Models for Predictive Railway Traffic Management Pavle Kecman Delft University of Technology, 2014 This research is supported by the Dutch Technology Foundation STW, which is part of the Netherlands Organisation for Scientific Research (NWO), and which is partly funded by the Ministry of Economic Affairs. Cover illustration: Aleksandar Marti´c Models for Predictive Railway Traffic Management Proefschrift ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus prof. ir. K.Ch.A.M. Luyben, voorzitter van het College voor Promoties, in het openbaar te verdedigen op maandag 20 oktober 2014 om 10:00 uur door Pavle Kecman Master of Science in Traffic & Transport Engineering University of Belgrade geboren te Belgrado, Servi¨e Dit proefschrift is goedgekeurd door de promotor: Prof. Dr.- Ing. I.A. Hansen Toegevoegd promotor: Dr. R.M.P. Goverde Samenstelling promotiecommissie: Rector Magnificus Prof. Dr.- Ing. I.A. Hansen Dr. R.M.P. Goverde Prof. dr. ir. R.P.B.J. Dollevoet Prof. dr. L.G. Kroon Prof. Dr.-Ing. J. Pachl Prof. dr. C. Roberts Prof. dr. D. Mandi´c Prof. dr. ir. S.P. Hoogendoorn voorzitter Technische Universiteit Delft, promotor Technische Universiteit Delft, toegevoegd promotor Technische Universiteit Delft Erasmus Universiteit Rotterdam Technische Universit¨at Braunschweig University of Birmingham University of Belgrade Technische Universiteit Delft, reserve This thesis is the result of a Ph.D. study carried out from 2010 to 2014 at Delft University of Technology, Faculty of Civil Engineering and Geosciences, Department of Transport and Planning. TRAIL Thesis Series no. T2014/5, the Netherlands TRAIL Research School TRAIL P.O. Box 5017 2600 GA Delft The Netherlands Phone: Fax: E-mail: +31 (0) 15 278 6046 +31 (0) 15 278 4333 [email protected] ISBN 978-90-5584-175-2 Copyright cbe 2014 by Pavle Kecman. This work is licensed under the Creative Commons Attribution-NonCommercial 3.0 Unported License. It may be freely shared, copied and redistributed in any medium or format. Transformation and building upon the material is permitted for non-commercial purposes under the condition that the work is properly cited. Printed in the Netherlands ”Nigdar ni tak bilo da ni nekak bilo, pak ni vezda ne bu da nam nekak ne bu.” - Miroslav Krleˇza (Khevenhiller, Balade Petrice Kerempuha) Preface All successful Ph.D. projects are the same, every problematic Ph.D. project is problematic in its own way. The so-called Anna Karenina Principle, named after the first sentence of the great book by Leo Tolstoy, applied in the context of a Ph.D. research implies that little can be said about a project that went according to plan during its whole course. An interesting and well defined topic, good supervision and my passion for railways and research made the previous four years an enjoyable and fruitful period. Or is just my memory playing tricks because the work is finally completed? The work presented in this thesis was carried out as a part of the joint project “Modelpredictive railway traffic management” between the Department of Transport and Planning (T&P) and Delft Centre for Systems and Control (DCSC) of the Delft University of Technology (DUT). The project was funded by the Dutch Technology Foundation STW. The goal was to develop new models and a new model-predictive controller for anticipative management of railway networks. The work described in this thesis represents the first step towards reaching this objective. It is planned to have the presented models integrated in a closed-loop control approach that will be presented in a separate Ph.D. thesis completed at DCSC. There are many people who have directly and indirectly helped me to produce this dissertation. My supervisors and colleagues deserve special gratitude for their help and dedication. My direct supervisor Rob Goverde has been closely involved in this research from the very beginning to the final approval of the thesis. He was always available to help and I am very grateful for his contribution and guidance at the difficult points in my research. It was a great pleasure and an honour to work with him a learn from him. I would also like to thank my advisor Professor Ingo Hansen for his critical point of view that was always followed by useful advice to help me improve my work. On a more general note, I am also grateful to him for the fact that his enthusiasm helped to establish railway operations research as an independent scientific discipline represented with the high quality journals and conferences. I would furthermore like to thank the other people involved in the research project: Bart Kersbergen, Nicol´as Weiss, Ton van den Boom and Bart De Schutter on behalf of DCSC. Keeping up to schedule was to a great extent helped by the fact that we presented our main findings and research plans on a regular basis to the user committee consisting of experts from academia and industry. The committee members: Bob vii viii Models for Predictive Railway Traffic Management Jansen, Leo Kroon, Edo Nugteren, Alfons Schaafsma, Ello Weits, Jianxin Yuan helped a lot with their comments, questions and advice. Direct support for this research was provided by ProRail, Dick Middelkoop in particular, who helped by providing the data sets needed to build and test our models. Furthermore, I would like to thank Francesco Corman and Andrea D’Ariano for their help and contribution to the part of this thesis related to real-time rescheduling. Working with them at the early stage of my research was truly a lesson in efficiency, precision and quality that helped me adopt such attitude in my work. I owe a lot of gratitude to my dear colleagues from the rail group at T&P and the University of Belgrade. Not a single part of this work remained undiscussed between us. From the problem definition, via methodology and programming, to the clarity of the figures, they provided a useful feedback for each aspect at any time and place. I am very grateful to Daniel Sparing, Francesco Corman, Lingyun Meng, Egidio Quaglietta, Nikola Beˇsinovi´c, Nadjla Ghaemi and Peca Jovanovi´c for their time and patience in many casual brainstorming sessions. It was surely the most fun part of doing research. Finally, I would like to thank all technical and administrative staff of T&P and TRAIL Research School for taking care of many practical issues, which allowed me to fully focus on the project. During the last four years spent in the Netherlands I was lucky to be surrounded by many wonderful people to rely on and have fun with at work and outside. My dear friends in Delft, Rotterdam, The Hague, Amsterdam, Almelo and Groningen were there for me to offer a good laugh and their advice and solution to all problems from doing laundry to weltschmertz and existential crises. Having them in my life is definitely my most important achievement from this period. Having a delayed train is not that bad if you’re in a good company. And of course, I am always most grateful to my family for their unreserved support and love that makes the physical distance between us seem so unimportant. Pavle Kecman Belgrade, August 2014 Contents Preface 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Railway traffic control in the Netherlands . . . . . . . . . . . . . . . 2 1.3 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.1 Short-term traffic prediction . . . . . . . . . . . . . . . . . . 5 1.3.2 Network-wide traffic management . . . . . . . . . . . . . . . 6 1.3.3 Model-predictive control . . . . . . . . . . . . . . . . . . . . 7 Thesis objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4.1 Research objective 1 – Monitoring and traffic state prediction 8 1.4.2 Research objective 2 – Rescheduling models for network-wide traffic control . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Thesis contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.5.1 Monitoring and real-time traffic state prediction . . . . . . . . 12 1.5.2 Macroscopic models for network-wide traffic rescheduling . . 15 Thesis outline and scope . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 1.5 1.6 2 vii An overview of railway operation planning and control 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Terminology and basic concepts of railway traffic . . . . . . . . . . . 20 2.2.1 Railway timetable . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Signalling and interlocking . . . . . . . . . . . . . . . . . . . 21 2.2.3 Blocking time theory . . . . . . . . . . . . . . . . . . . . . . 22 ix x Models for Predictive Railway Traffic Management 2.2.4 Train position detection . . . . . . . . . . . . . . . . . . . . 24 2.2.5 Classification of train delays . . . . . . . . . . . . . . . . . . 26 2.2.6 Operational control of railway traffic and transport . . . . . . 26 2.3 Review of approaches for data mining of traffic realisation data . . . . 30 2.4 Review of approaches for process time estimation . . . . . . . . . . . 32 2.4.1 Running time estimation . . . . . . . . . . . . . . . . . . . . 32 2.4.2 Dwell time estimation . . . . . . . . . . . . . . . . . . . . . 33 2.4.3 Headway times . . . . . . . . . . . . . . . . . . . . . . . . . 34 Review of delay propagation analysis and prediction models . . . . . 35 2.5.1 Delay propagation analysis . . . . . . . . . . . . . . . . . . . 35 2.5.2 Identifying structural timetable errors and systematic delays . 37 2.5.3 Delay propagation models . . . . . . . . . . . . . . . . . . . 38 2.5.4 Models for delay prediction in real-time . . . . . . . . . . . . 41 2.6 Review of rescheduling models . . . . . . . . . . . . . . . . . . . . . 43 2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.5 3 Process mining of train describer event data 51 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Methodological framework of the process mining tool . . . . . . . . . 53 3.2.1 Process mining . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.2.2 Process model . . . . . . . . . . . . . . . . . . . . . . . . . 53 The Dutch train describer system . . . . . . . . . . . . . . . . . . . . 55 3.3.1 System architecture . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.2 Data structure and information contained in log archives . . . 56 3.3.3 Shortcomings in TROTS log files . . . . . . . . . . . . . . . 57 Traffic monitoring on open track and in stations . . . . . . . . . . . . 58 3.4.1 Associating signal messages to train number steps . . . . . . 58 3.4.2 Logging of automatic block signal passing events . . . . . . . 59 3.4.3 Logging of station events . . . . . . . . . . . . . . . . . . . . 60 Train route recovery and route conflict identification . . . . . . . . . . 60 3.3 3.4 3.5 CONTENTS 3.6 3.7 4 xi 3.5.1 Process mining train describer data . . . . . . . . . . . . . . 60 3.5.2 Main algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.5.3 Process discovery . . . . . . . . . . . . . . . . . . . . . . . . 64 3.5.4 Automatic identification of route conflicts . . . . . . . . . . . 65 3.5.5 Identification of hindering trains . . . . . . . . . . . . . . . . 65 3.5.6 Estimation of departure and arrival times . . . . . . . . . . . 65 Process mining tool . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.6.1 Case study . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.6.2 Graphical user interface . . . . . . . . . . . . . . . . . . . . 67 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Data analysis and estimation of process times 73 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2 Methodological framework for statistical analysis . . . . . . . . . . . 75 4.2.1 Description of the data set . . . . . . . . . . . . . . . . . . . 75 4.2.2 Global model . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2.3 Local model . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Statistical learning methods . . . . . . . . . . . . . . . . . . . . . . . 77 4.3.1 Multiple linear regression . . . . . . . . . . . . . . . . . . . 77 4.3.2 Tree-based non-linear methods . . . . . . . . . . . . . . . . . 79 Process time estimates – global model . . . . . . . . . . . . . . . . . 81 4.4.1 Running time estimates derived from the global model . . . . 81 4.4.2 Dwell time estimates derived from the global model . . . . . 85 Process time estimates - local model . . . . . . . . . . . . . . . . . . 90 4.5.1 Estimation of running times over a particular block . . . . . . 90 4.5.2 Estimation of dwell times for a particular station . . . . . . . 93 Comparison of statistical models . . . . . . . . . . . . . . . . . . . . 95 4.6.1 Comparison of running time estimation models . . . . . . . . 95 4.6.2 Comparison of dwell time estimation models . . . . . . . . . 96 4.6.3 Comparison of prediction accuracy for scheduled processes . 96 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.3 4.4 4.5 4.6 4.7 xii 5 Models for Predictive Railway Traffic Management Real-time prediction of train event times 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2 Framework of the real-time prediction tool . . . . . . . . . . . . . . . 102 5.3 Microscopic graph based model . . . . . . . . . . . . . . . . . . . . 104 5.4 5.5 5.6 5.7 6 101 5.3.1 The graph model . . . . . . . . . . . . . . . . . . . . . . . . 104 5.3.2 Graph construction . . . . . . . . . . . . . . . . . . . . . . . 105 Computation of arc weights . . . . . . . . . . . . . . . . . . . . . . . 107 5.4.1 Running and dwell arc weights . . . . . . . . . . . . . . . . . 109 5.4.2 Headway and connection arc weights . . . . . . . . . . . . . 109 5.4.3 Online process time estimation . . . . . . . . . . . . . . . . . 110 5.4.4 Time loss due to route conflicts . . . . . . . . . . . . . . . . 110 Online prediction of event times . . . . . . . . . . . . . . . . . . . . 113 5.5.1 Prediction algorithm . . . . . . . . . . . . . . . . . . . . . . 113 5.5.2 Adjusting the running time estimates due to route conflicts . . 115 5.5.3 Adaptive adjustments of running time predictions . . . . . . . 116 Application on a case study . . . . . . . . . . . . . . . . . . . . . . . 118 5.6.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . 118 5.6.2 Description of the case study . . . . . . . . . . . . . . . . . . 118 5.6.3 Comprehensive analysis . . . . . . . . . . . . . . . . . . . . 119 5.6.4 Example of algorithm execution . . . . . . . . . . . . . . . . 122 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . 125 Rescheduling models for real-time traffic management in large networks 127 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6.2 Macroscopic modelling of railway operations . . . . . . . . . . . . . 128 6.2.1 Timed event graphs . . . . . . . . . . . . . . . . . . . . . . . 128 6.2.2 Alternative graphs . . . . . . . . . . . . . . . . . . . . . . . 129 6.2.3 Conversion of timed event graphs to alternative graphs . . . . 132 6.2.4 Resources as building blocks of alternative graphs . . . . . . 133 6.2.5 Sequence-dependent setup times . . . . . . . . . . . . . . . . 136 CONTENTS 6.3 6.4 6.5 6.6 7 xiii Models examined . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.3.1 Macroscopic models . . . . . . . . . . . . . . . . . . . . . . 137 6.3.2 Mesoscopic model . . . . . . . . . . . . . . . . . . . . . . . 141 6.3.3 Overview of the five models . . . . . . . . . . . . . . . . . . 141 Test case A: corridor Utrecht - Den Bosch . . . . . . . . . . . . . . . 141 6.4.1 Test case settings . . . . . . . . . . . . . . . . . . . . . . . . 142 6.4.2 Comprehensive evaluation . . . . . . . . . . . . . . . . . . . 144 Test case B: Dutch national railway network . . . . . . . . . . . . . . 146 6.5.1 Description of the tested instances . . . . . . . . . . . . . . . 146 6.5.2 Comprehensive evaluation . . . . . . . . . . . . . . . . . . . 147 6.5.3 Network-wide effects of reducing delay propagation . . . . . 149 Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . 149 Conclusions 7.1 7.2 153 Summary of the main findings and contributions . . . . . . . . . . . . 153 7.1.1 Monitoring and traffic state prediction . . . . . . . . . . . . . 154 7.1.2 Network-wide traffic rescheduling . . . . . . . . . . . . . . . 157 Recommendations for future work . . . . . . . . . . . . . . . . . . . 158 Bibliography 161 List of acronyms 178 Summary 179 Samenvatting 183 About the author 187 TRAIL Thesis Series 189 xiv Models for Predictive Railway Traffic Management List of Figures 1.1 Hierarchical structure of traffic control . . . . . . . . . . . . . . . . . 2 1.2 Railway map of the Netherlands . . . . . . . . . . . . . . . . . . . . 3 1.3 Workplace of a local traffic controller in Amsterdam . . . . . . . . . 4 1.4 Cascade MPC framework for traffic control . . . . . . . . . . . . . . 8 1.5 Research objectives integrated in a closed loop . . . . . . . . . . . . 9 1.6 Integration of requirements for real-time prediction tool . . . . . . . . 11 1.7 Flowchart of the thesis structure . . . . . . . . . . . . . . . . . . . . 16 2.1 Blocking time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.2 Route conflict . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.3 Blocking time stairways . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Infrastructure based train detection . . . . . . . . . . . . . . . . . . . 25 2.5 Regular time interval train detection . . . . . . . . . . . . . . . . . . 25 2.6 Structure and information flow within operational planning level . . . 27 2.7 Illustrative example of the parallel-shift prediction method . . . . . . 29 3.1 Process mining framework . . . . . . . . . . . . . . . . . . . . . . . 54 3.2 Events and processes in micro and mesoscopic models . . . . . . . . 54 3.3 Three-layer process model . . . . . . . . . . . . . . . . . . . . . . . 55 3.4 Screen shot of a TROTS log flle . . . . . . . . . . . . . . . . . . . . 59 3.5 Process mining TROTS data . . . . . . . . . . . . . . . . . . . . . . 61 3.6 Flowchart of the process mining algorithm . . . . . . . . . . . . . . . 63 3.7 Example network . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.8 Observed area for the case study . . . . . . . . . . . . . . . . . . . . 67 xv xvi Models for Predictive Railway Traffic Management 3.9 Graphical user interface . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.10 Train selection panel . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.11 Infrastructure selection panel . . . . . . . . . . . . . . . . . . . . . . 69 3.12 Time distance diagram . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.13 Blocking time diagram . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1 Regression tree for running time estimation . . . . . . . . . . . . . . 83 4.2 Relative running time prediction error depending on the tree size . . . 84 4.3 R2 of the running time model depending on the tree size . . . . . . . . 84 4.4 MSE of running time model depending on the number of trees . . . . 85 4.5 R2 of running time model depending on the number of trees . . . . . . 85 4.6 Regression tree for dwell time estimation . . . . . . . . . . . . . . . 88 4.7 Relative dwell time prediction error depending on the tree size . . . . 89 4.8 R2 of the dwell time model depending on the tree size . . . . . . . . . 89 4.9 MSE of the dwell time model depending on the number of trees . . . 89 4.10 R2 of the dwell time model depending on the number of trees . . . . . 90 4.11 Dependence of running time on delay (left) and box-plots of running times for punctual and delayed trains (right) . . . . . . . . . . . . . . 91 4.12 R2 for prediction of running time on The Hague HS – Rotterdam corridor 92 4.13 Delay over corridor Leiden - Dordrecht for train line 2200 . . . . . . 92 4.14 R2 for prediction of dwell times on Leiden – Dordrecht corridor . . . 93 4.15 Dependence of dwell time on delay (left) and box-plots of dwell time (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.16 Dependence of dwell time on scheduled departure time . . . . . . . . 94 4.17 Prediction error for dwell times of delayed trains . . . . . . . . . . . 95 4.18 Prediction error of running time estimation models . . . . . . . . . . 96 4.19 Prediction error of dwell time estimation models . . . . . . . . . . . . 97 4.20 Precision of dwell time and running time estimates . . . . . . . . . . 97 4.21 Precision of dwell time and running time estimates relative to scheduled process time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.1 Monitoring and prediction components in the traffic control loop . . . 103 LIST OF FIGURES xvii 5.2 Space-based train separation . . . . . . . . . . . . . . . . . . . . . . 106 5.3 Time-based train separation . . . . . . . . . . . . . . . . . . . . . . . 106 5.4 An example of a mesoscopic DAG . . . . . . . . . . . . . . . . . . . 108 5.5 Time loss dependence on conflict duration: quadratic fit for short (up) and linear fit for long conflicts (down) . . . . . . . . . . . . . . . . . 112 5.6 An example of execution of Algorithm 2 . . . . . . . . . . . . . . . . 115 5.7 An example of route conflict prediction . . . . . . . . . . . . . . . . 117 5.8 A schematic example of adaptive prediction . . . . . . . . . . . . . . 117 5.9 Network and train lines for the case study . . . . . . . . . . . . . . . 119 5.10 Box plots of prediction error distributions for different prediction horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 5.11 Mean absolute prediction error depending on prediction horizon . . . 121 5.12 MAE comparison for adaptive and nonadaptive prediction . . . . . . 121 5.13 MAE of scheduled event times for a parallel shift strategy and the realtime prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 5.14 Time-distance diagram of predicted (at 7:13) and realised train paths . 123 5.15 Blocking time diagram predicted at 7:13 (up), realized blocking time diagram (down) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 5.16 Effects of adaptive prediction . . . . . . . . . . . . . . . . . . . . . . 125 6.1 Graph representation of resources with infinite capacity . . . . . . . . 133 6.2 Graph representation of resources with infinite capacity and headway constraint (left) and a possible selection (right) . . . . . . . . . . . . 134 6.3 Graph representation of resources with infinite capacity and FIFO constraint (left) and a possible selection (right) . . . . . . . . . . . . . . 135 6.4 Graph representation of resources with finite capacity (left) and a possible selection (right) . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.5 Example of sequence-dependent setup times . . . . . . . . . . . . . . 136 6.6 Layout of the illustrative example . . . . . . . . . . . . . . . . . . . 137 6.7 Illustrative example - Model 1 . . . . . . . . . . . . . . . . . . . . . 138 6.8 Illustrative example - Model 2 . . . . . . . . . . . . . . . . . . . . . 139 6.9 Incompatibility graph of illustrative example . . . . . . . . . . . . . . 140 6.10 Illustrative example - Model 3 . . . . . . . . . . . . . . . . . . . . . 140 xviii Models for Predictive Railway Traffic Management 6.11 Illustrative example - Model 4 . . . . . . . . . . . . . . . . . . . . . 141 6.12 Layout of infrastructure and main stations . . . . . . . . . . . . . . . 142 6.13 Timetable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.14 Dutch railway network considered (in black), with main stations . . . 147 6.15 Maximum secondary delays without (left) and with (right) rescheduling 150 List of Tables 2.1 Summary of presented approaches for real-time rescheduling . . . . . 48 3.1 Train number messages generated by TROTS . . . . . . . . . . . . . 56 4.1 Summary of the training set for running time estimation . . . . . . . . 81 4.2 Summary of the LTS model for running time prediction . . . . . . . . 82 4.3 Summary of the training set for dwell time estimation . . . . . . . . . 86 4.4 Summary of the LTS model for dwell time prediction . . . . . . . . . 87 5.1 Model size for different prediction horizons . . . . . . . . . . . . . . 119 6.1 Operational constraints in models . . . . . . . . . . . . . . . . . . . 142 6.2 Quantitative assessment of the 5 models . . . . . . . . . . . . . . . . 144 6.3 Difference in orders between the mesoscopic and each macroscopic model. Direction Ht → Ut . . . . . . . . . . . . . . . . . . . . . . . 146 6.4 Characteristics of the network-wide test case . . . . . . . . . . . . . . 148 6.5 Quantitative assessment of the macroscopic models on test case B . . 148 xix xx Models for Predictive Railway Traffic Management Chapter 1 Introduction 1.1 Background A railway system integrates infrastructure, rolling-stock, staff and a set of strict operational rules into a functional system for transporting passengers and goods. Each of the afore mentioned subsystems represents a very complex structure of interconnected entities. Following the directive of the European Commission (2001), a number of railway systems in Europe are horizontally separated into infrastructure managers (IMs) and train train operating companys (TOCs). This reform was introduced in order to improve the competitiveness and modal share of railways on the transport market (C. Nash, 2010). The IM is responsible for management and maintenance of railway infrastructure, allocating railway capacity (train paths) to TOCs and organizing and controlling traffic along the network. On the other hand, the main task of a TOC is planning, operation and control of passenger and freight transport. An IM has the task to coordinate train path requests of TOCs, allocate infrastructure capacity through a timetable in the tactical planning stage, and control traffic in real time at the operational planning level. A timetable in railway traffic is the master schedule that reflects the relationship between the supply and demand in the railway sector. It contains the scheduled process time of each operation, i.e., a train dwelling in a station, running between two scheduled stops, passenger transfers, rolling-stock or crew connections, etc. However, daily variations and unforeseeable disruptive events may render the planned timetable infeasible. Such events are inevitable in modern railway systems with many interacting processes that depend on human behaviour, technical devices, and the environment. In busy and heavily utilized networks, a deviation from the planned path of a single train can easily propagate as a secondary delay to other trains that run over the same infrastructure or have a planned passenger, rolling-stock or crew connection. Prevention and minimisation of delay propagation, and maintaining timetable feasibility are the main responsibilities of operational traffic control (Pachl, 2009). Railway traffic control is typically hierarchically structured into a local and a global 1 2 Models for Predictive Railway Traffic Management (network) level (Figure 1.1). Local traffic control has the task to perform all safety related actions, set routes for trains, predict and solve conflicts, and control processes that take place on the designated part of infrastructure (Kecman, Goverde, & Van den Boom, 2011). A train typically crosses multiple traffic control areas controlled by different local controllers (signallers and/or dispatchers). The global level (regional or network controllers) comprises the supervision of the state of traffic on the network level, detection of deviations from the timetable, resolution of conflicts affecting the overall network performance, handling failures and events that may have big impact on performance indicators, etc. Network controller Local controller 1 Local controller 2 Local controller n Figure 1.1: Hierarchical structure of traffic control 1.2 Railway traffic control in the Netherlands The railway system in the Netherlands is a typical example of a highly interconnected transport and traffic network. It is one of the most densely utilized networks in the world (Hansen, Wiggenraad, & Wolff, 2013). More than 3000 km of railway tracks that connect 404 stations in virtually all cities in the country (Figure 1.2), are managed by the infrastructure manager ProRail (ProRail, 2013). With regard to the traffic volumes, i.e., the number of trains, train kilometres and the amount of passengers and goods transported per line kilometre, the Dutch railway network performs almost as well as Switzerland and Japan (Wolff, 2011). At any moment in time during peak hours, there are approximately 400 running trains, 93% of which are passenger trains, mainly operated by the national operator Netherlands Railways (NS) (NS Group, 2013). In the heavily utilized Dutch network, trains are scheduled with short headway times and small time supplements using the advanced timetabling tool Design of Network Schedules (DONS) (Hooghiemstra, Kroon, Odijk, Salomon, & Zwaneveld, 1999). Due to dense traffic and interconnected services, delays and incidents that occur in one part of the network may easily propagate through the whole network (Goverde, 2007). Therefore, operational traffic control has an important task to maintain the planned schedule and recover from disruptions as quickly as possible in order to minimise delays and increase traffic reliability. Railway traffic control in the Netherlands is divided in two levels with 13 local traffic control centres (Figure 1.3) and a network control centre Operational Control Centre Rail (OCCR). Timetable updates are negotiated and determined in the OCCR between Chapter 1. Introduction 3 In use for passenger and cargo trains Cargo trains only Figure 1.2: Railway map of the Netherlands the network traffic controllers of ProRail and transport controllers of NS. The network traffic controllers receive information about the current delays and traffic condition from the local level. Their task, in cooperation with the TOC, is to coordinate the dispatching measures that would decrease deviations from the planned timetable on the network level. The controllers on behalf of NS verify that the proposed updates are feasible with respect to rolling-stock and crew circulation plans. Finally, timetable updates are transmitted to the local level for implementation. Local traffic control centres are staffed with signallers, who are in charge of controlling the signals and setting routes in stations, and dispatchers, who observe and supervise traffic conditions and resolve conflicting train routes. In order to manage the complex tasks of controlling dense traffic, controllers and signallers are supported by computer tools for traffic monitoring, remote control of signals and switches, and automatic route setting (Renkema & Van Visser, 1996). Process plans containing the planned routes and schedules for each train are transmitted to the traffic control centres one day in advance. Train positions are monitored and reported by the train describer system Train Tracking and Observation System (TROTS). TROTS messages are received by the traffic control system Verkeersleiding (VKL) that compares the train arrival and departure times with the daily process plan. Train delays are derived by the VKL system at home and exit signals, adjusted with a fixed correction term and rounded to 4 Models for Predictive Railway Traffic Management full minutes. Figure 1.3: Workplace of a local traffic controller in Amsterdam (Source: ProRail, photo by Jos van Zetten) The dispatching support system Procesleiding (PRL) is provided with the process plan and actual train delays from VKL. It comprises the command and monitoring interface with the signalling and interlocking system. Traffic is visualised in the form of a dynamic planned time-distance diagram. PRL is furthermore equipped with an automatic route setting system Automatische Rijweginstelling (ARI), which sets the routes for trains according to the actual process plan and train delays (Berends & Ouburg, 2005). Route conflicts are resolved according to the first-come-first-served principle based on the current train positions or according to the relative train order defined in the timetable. Signallers may also use PRL to set the routes manually in order to manage disruptions and resolve conflicting routes in the process plan. However, they have to rely on their own experience and a set of predetermined what-if scenarios. More details about the Dutch railway system and practice in traffic control are given by Goverde (2005). 1.3 Motivation While a timetable is carefully planned a year in advance using the sophisticated mathematical models, the daily operational control of disruptions and delays still relies predominantly on the predetermined rules and experience and skills of personnel. Traffic controllers do not commonly have any intelligent support tools such as short-term traffic prognosis, conflict detection and prediction or optimal dispatching. Moreover, working in a preventive manner is poorly supported and train traffic controllers are usually restricted to just solving problems as they occur (Kauppi, Wikstr¨om, Sandblad, & Andersson, 2005). Chapter 1. Introduction 1.3.1 5 Short-term traffic prediction Signallers typically do not have any intelligent decision support system to estimate the expected running times. In the current Dutch practice, their situational awareness about current train delays is further limited due to the imprecision in the measurements of actual delays. Local traffic control usually takes the expected arrival delay equal to the current upstream delay as they have no information about the possible recovery times (except from experience) (D’Ariano, 2008). This method neglects the fact that some trains make up for their delays by running in the maximum performance regime and exploiting the running time supplements incorporated in the timetable. On the other hand, other trains may get even more delayed due to a possible time loss in route conflicts. Delay propagation could be prevented or reduced if the traffic was managed proactively, i.e., if controllers had a reliable prediction of a route and connection conflict with a possibility to prevent it. The main advantage of predictive traffic management is that the traffic controllers can anticipate the occurrence of conflicts so that they have enough time to prevent them using a conflict resolution method. Potential conflicting train paths in the current process plan need to be predicted in advance based on the accurate monitoring of train positions, speed and infrastructure conditions such as temporary track or speed restrictions that can affect the process times. As a result, unscheduled stops before red signals could be avoided by resolving such conflicts in advance using e.g. rerouteing, changing the train order at the conflict point, retiming or giving speed advice to the drivers. Several approaches to traffic state prediction can be found in the current practice or academic literature. Macroscopic models (Berger, Gebhardt, M¨uller-Hannemann, & Ostrowski, 2011; Hansen, Goverde, & Van der Meer, 2010) focus on predicting only the event times in stations (departures, arrivals and through rides). That way, the train separation principles cannot be accurately modelled and the route conflicts on open track sections (between two stations) cannot be predicted. On the other hand, the mesoscopic prediction models that are integrated in the traffic control systems such as Rail Control System (RCS) in Switzerland (Dolder, Krista, & Voelcker, 2009), Styrning av T˚ag genom Elektronisk Graf (STEG) in Sweden (Isaksson-Lutteman, 2012) or the short-term prediction module of the rescheduling system Railway traffic Optimization by Means of Alternative Graphs (ROMA) (D’Ariano, 2008) may support the traffic controllers in conflict detection. Every signal passing event is explicitly included in those models. However, the estimates of running and dwell times are computed independently from the actual traffic state. Different performance regimes between delayed and punctual trains, the impact of peak hours or behavioural factors are not considered in the predictions. We use the terminology and definition of macroscopic and microscopic level of modelling described by Radtke (2008) and Schlechte, Bornd¨orfer, Erol, Graffagnino, and Swarat (2011). Macroscopic models consider only station events such as departures, arrivals and through rides. On the other hand, a train run is modelled microscopically on a detailed level of track-clear detection sections. In this aspect 6 Models for Predictive Railway Traffic Management we also define a class of mesoscopic models that model traffic on the level of block sections and station routes. The different levels of modelling are explicitly defined in Section 3.5. 1.3.2 Network-wide traffic management The current practice in operational control of disruptions and delays still relies predominantly on the predetermined rules and experience and skills of personnel. Neither local nor network traffic controllers have a reliable supporting tool to make despatching decisions, predict their effect and evaluate them. For local traffic controllers, that often leads to creating new conflicts in the adjacent areas and suboptimal effects on the network wide level. Even the advanced, recently developed tools that can produce optimal solutions for traffic disruptions for a single traffic control area (Caimi, Fuchsberger, Laumanns, & L¨uthi, 2012; D’Ariano, Pranzo, & Hansen, 2007) or multiple areas (Corman, D’Ariano, Pacciarelli, & Pranzo, 2012b) are not able to tackle large, network-wide instances, due to the demanding computational requirements of inter-area coordination (Corman, D’Ariano, Pacciarelli, & Pranzo, 2014). A decision support system for network-wide traffic management is required to continuously supervise traffic on the network and create network-optimal updates to the timetable, that can be used as a reference master schedule by the local control level. Due to a high combinatorial complexity of the train rescheduling problem (T¨ornquist, 2006) the tools developed so far are not directly applicable for large and busy networks. An important requirement for real-time railway applications is the knowledge of the actual train positions, speed and the time needed for computing the solution, perception, decision and subsequent execution of the dispatching measure (e.g. lock of signal or set-up of a new route). A feasible or (near) optimal solution has to be produced and implemented before it is outdated. In other words, the computation time must not exceed the validity of prediction of the traffic state that is given as an input to the rescheduling problem (L¨uthi, 2009). Simplifications to the existing microscopic and mesoscopic tools that reduce the problem complexity were therefore required for applications on the level of national networks. A series of macroscopic models was developed that do not regard all capacity constraints on open track sections and in station areas (T¨ornquist, 2007; Van den Boom & De Schutter, 2007). Another stream of research was directed to so called delay management with the purpose to optimise passenger delays by controlling the planned passenger transfer connections (Dollevoet, Huisman, Schmidt, & Sch¨obel, 2012; Schachtebeck & Sch¨obel, 2010; Sch¨obel, 2007). However, these macroscopic models for rescheduling were tested mostly on subnetworks of a national network or large urban networks. Therefore, the problem of controlling railway traffic on the level of national network still remains unsolved. An important aspect with high impact on applicability of the existing rescheduling tools in practice is the way they handle uncertainty. Running and dwell times are in Chapter 1. Introduction 7 practice characterised by variability. Moreover, the interdependence of train runs may increase the uncertainty of delays in future. Apart from the recent contributions directed at non-anticipative delay management (Bauer & Sch¨obel, 2014; Gatto, 2007), other rescheduling models assume full knowledge of current delays and their propagation (with or without any rescheduling actions) which a limitation for practical application. For that reason, the accuracy of delay predictions is very important for validity of offline rescheduling models. 1.3.3 Model-predictive control A possible way to model and optimize railway traffic control and overcome the problem of uncertainty is through a closed-loop control paradigm, called model-predictive control (MPC) (Maciejowski, 2002). The essential characteristic of the proposed framework is that it suggests proactive and anticipative (in contrast to reactive) traffic management. Real-time information can be used to predict the occurrence of potential conflicts. Moreover, delay propagation, resulting from route conflicts and planned connections, is prevented by computing optimal control actions. The theoretical framework of the closed-loop railway traffic control is presented in Figure 1.4. A cascade control system is used to model the hierarchical relationship between the global and a local control level L¨uthi (2009). Trains are operated according to a timetable and a daily process plan. Due to inevitable disturbances and deviations from the planned schedule, train runs need to be continuously monitored. By monitoring we assume keeping track of all performance indicators such as the actual train positions, delays, realised running and dwell times of all trains, etc. Monitoring therefore provides the actual traffic state that can be used to predict the future evolution of traffic on the network. A predictive traffic model continuously provides local control level with the information about the expected traffic conditions. It can further be used to evaluate the impact of traffic control actions. In case of larger disruptions that may affect the traffic in a wider area, network traffic controllers can use the prediction model to optimise the traffic on the network, compute the network-optimal timetable updates and transmit them as a reference to the local level. That way all traffic control actions on the local level will lead to the network-optimal traffic state. MPC has been implemented on the station level by Caimi et al. (2012) and on the level of a corridor by Quaglietta, Corman, and Goverde (2013). Whereas the rescheduling models embedded in these approaches can efficiently control traffic in (multiple) control areas, the prediction component relies on the theoretical estimation of running times and minimum dwell times. Variability of the process times is therefore not incorporated in predictions and prediction accuracy has not been tested against the realised train event times. Moreover, due to high computational requirements, these approaches are not directly applicable for controlling traffic on the level of national network. 8 Models for Predictive Railway Traffic Management Railway operations Monitoring Local controller Prediction Reference Timetable Network controller Reference Figure 1.4: Cascade MPC framework for traffic control 1.4 Thesis objectives The main objective of the research presented in this thesis is to develop the systems for monitoring, traffic state prediction and network-wide rescheduling that can be embedded in the cascade MPC framework presented in the previous section. The main objective is divided into two research objectives: • Research objective 1 (RO1) – Develop a tool for monitoring and traffic state prediction • Research objective 2 (RO2) – Develop a macroscopic model for network-wide traffic control Research objectives integrated in a feedback loop are illustrated in Figure 1.5. Train positions are reported by a train describer system. The system for monitoring and traffic state prediction (RO1) uses the live stream to determine the actual and future traffic conditions. In case of deviations with impact on a large part of the network, a rescheduling tool (RO2) can produce a network-optimal timetable update as the new reference for railway operation. 1.4.1 Research objective 1 – Monitoring and traffic state prediction The first research objective in this thesis is to develop a system for monitoring and traffic state prediction. A way to overcome the drawbacks of the current practice and the existing tools for monitoring and short-term traffic prediction (§1.3.1) emerged with the availability of historical traffic realisation data. A real-time stream of raw train describer data can be processed in a way that extracts the actual traffic conditions in Chapter 1. Introduction 9 the network: train positions, accurate estimates of current delays and realised running and dwell times. Moreover, the archives of event logs can be used to learn how trains behave depending on the traffic conditions. The variability of process times can thus be explained by isolating the factors with a high impact on the corresponding process time. Estimates of future process times depend on the current or predicted values of explanatory variables. Therefore, the predictions will incorporate the empirically determined variation of process times due to e.g. driving style, passenger behaviour or peak hours. In order to develop a system for for monitoring and traffic state prediction, a number of requirements needs to be fulfilled. The first requirement is to develop a data processing tool consisting of a detailed process model of railway traffic and an environment comprising the objects that represent the infrastructure elements and trains. The archives of event logs of the Dutch train describer systems have already been used for reconstructing the realised train paths (Goverde & Hansen, 2000) and identifying route conflicts (Daamen, Goverde, & Hansen, 2008). However, the changes in the data structure and system architecture require a fundamentally different approach. Such environment should be compatible with an online stream of train describer messages. All objects need to be updated in real time, thus providing the current state of traffic in order to raise the situational awareness of controllers. Moreover, the data processing tool needs to be applicable for ex post processing of traffic realisation data. The second requirement is to derive robust estimates of process times. The application of the data processing tool results in the clean, structured data that are prepared for analysis. The earlier efforts in punctuality analysis (Goverde, 2005; Yuan, 2006) focused on computing the descriptive statistical parameters and deriving probability distributions. The resulting distributions can be used for an ex ante timetable analysis and development of stochastic models (B¨uker & Seybold, 2012; Medeossi, Longo, & Monitoringyandy prediction (RO1) Actualyandyfuture ytrafficystate Liveydatay stream Railway operations Networkywide rescheduling (RO2) Figure 1.5: Research objectives integrated in a closed loop 10 Models for Predictive Railway Traffic Management de Fabris, 2011). However, in order to develop the predictive models, it is necessary to determine a set of explanatory variables and quantify their impact on process times. The aim is therefore to apply statistical predictive modelling on a training set of historical traffic realisation data. The predictive models can be used to compute the estimates of process times with respect to the actual values of explanatory variables that reflect the state of the traffic on the network. An important requirement in predictive modelling of process times is that the resulting process time estimates need to be robust to noise, missing data and the outliers in the real-time data stream. The robustness of the models is considered as one of the key criteria for selection of the appropriate statistical learning technique (§4.3). Robust estimates and coefficients that reflect the dependence of process times on explanatory variables are stored in a database of historical data. The third requirement is to build a real-time prediction tool. A live stream of train describer data can be processed with the processing tool and give the actual state of the network. Given the information on the current train positions, actual delays, and the realised running and dwell times, the robust estimates of process times can be computed in real time using the predefined predictive models. However, accurate modelling of dependencies between processes of a single train as well as among multiple trains is required in order to predict traffic in large networks over long prediction horizons. The third requirement is therefore to create a fast prediction algorithm that calibrates the realistic traffic model in real time based on the current (future) traffic condition, and estimates the realisation times of all events within a prediction horizon. The integration of the three requirements defined to reach the first research objective in this thesis is illustrated in Figure 1.6. The approach consists of two parts. The offline part (dash-dot box) comprises data mining of a training set of raw train describer data and creating a database of historical traffic realisation data. The online part (dashed box) processes the live data stream, determines the traffic conditions in the network and predicts the future traffic state using the historical database. Note that the processing tool can be used in both parts of the tool. 1.4.2 Research objective 2 – Rescheduling models for networkwide traffic control The second research objective in this thesis is to develop and validate a macroscopic rescheduling model that can be applied for optimal control of traffic in large and heavily utilised networks. An approach of representing the railway rescheduling problem as a job-shop scheduling problem modelled with alternative graphs has been developed by Mascis and Pacciarelli (2002) and Mascis, Pacciarelli, and Pranzo (2002). In a series of improvements of the solution procedure, the model has been successfully applied for optimal control of traffic in station areas (Mazzarello & Ottaviani, 2007), on a corridor (D’Ariano, Pranzo, & Hansen, 2007) or in multiple traffic control areas (Corman et al., 2012b). However, the problem of controlling country-wide traffic is Chapter 1. Introduction 11 Online Live data stream Offline Data processing Actual traffic state Prediction Training data set Process time estimates Database Future traffic state Figure 1.6: Integration of requirements for real-time prediction tool still open since the coordination of local areas is hard to tackle within a short time and there are multiple interdependencies between trains across the whole network. Therefore, the granularity of the alternative graph model needs to be modified in order to become applicable to problems of rescheduling trains on a large scale network-wide level. An important objective of this work is to search for a compromise between the precise modelling of railway capacity constraints and a reasonable time to compute the alternative solutions for the large scale railway traffic management instances. A suitable choice of granularity of the macroscopic model is crucial in order to find the balance between limiting the problem complexity and maintaining the feasibility of produced solutions. 1.5 Thesis contributions This section presents the main theoretical, methodological and practical contributions of the research project presented in this thesis. As outlined in the previous section, the research focused on studying, extending knowledge and improving the two main aspects of operational traffic control. 12 Models for Predictive Railway Traffic Management 1.5.1 Monitoring and real-time traffic state prediction Data processing tool The presented work on fulfilling this requirement provides an insight into data structure and system architecture, and into the advantages and drawbacks of using the new Dutch train describer system TROTS for traffic performance analysis. The previously developed algorithms for processing train describer event data (Goverde & Hansen, 2000) and automatic conflict identification (Daamen et al., 2008) have been taken as a starting point in developing the new data processing tool. However, the new data structure implemented in TROTS required a fundamental modification of the existing algorithms. The tool is developed in an object-oriented environment which makes it suitable for real-time application in monitoring train movements over the network. In other words the tool is able to process large data sets in short time, thus it is applicable for processing live data streams as well as large archives of traffic realisation data. An important contribution is a data mining algorithm that can learn the mutual dependence between track sections and signals implemented in signalling and interlocking systems from the TROTS data. This allows straightforward coupling of signal aspect changes to the train numbers that have caused them, since TROTS data structure does not reveal a connection between messages coming from signals and train number messages. Moreover, the automatic block signals on open tracks between stations are not logged. This problem has been overcome by incorporating the signalling and interlocking logic in the data mining algorithm, thus enabling accurate monitoring or reconstruction of the realised train paths even in these ‘dark territories’. The methodology of process mining (Van der Aalst, 2011) has been applied for the first time for mining the train describer event data. This required a development of a 3-level process model that reflects the majority of microscopic operational constraints of railway traffic. The work resulted in a software tool for automatic recovery of train paths and route conflict identification. Application of the developed tool on a set of train describer data significantly increases the precision of delay estimates and enables distinction between primary and secondary delays. The tool is equipped with a graphical user interface that simplifies analysis of the realised or actual traffic conditions. Traffic data for a particular corridor or station can be selected and visualised to enable the analyst to focus on a specific instance. Moreover, the tabular output of occupation and blocking times of infrastructure elements can be exported and used for analysis of process times and realised capacity utilisation. Realised train paths and route conflicts are visualised using time-distance and blocking time diagrams. The main contributions are summarised in the following list: • process mining algorithm for TROTS data archives and live data stream Chapter 1. Introduction 13 • recovery of process times on the level of signals and block sections in station areas and open tracks • automatic identification of route conflicts in station areas and open tracks • computation of delays for all scheduled arrivals and departures Robust estimates of process times Advanced predictive modelling and statistical learning techniques are used to develop process time prediction models with strong predictive power. Two different approaches are presented. A single general predictive model is developed that, given the current traffic condition, accurately predicts all running and dwell time estimates. The results of this generic approach can be generalised to the parts of the network and train lines that are not included in the training set. Strong and quantified predictive power of the presented models indicate the applicability of presented approach for deriving accurate process time estimates. Moreover, the data structures obtained using the data processing tool motivated the development of separate statistical models for each block section (station route, platform) and each train line. The variability of running and dwell times was explained with greater precision by significantly reducing the number of predictors. Both approaches are validated on an independent test set. We show that the application of the local statistical models produces more accurate predictions. Earlier approaches in this direction (Van der Meer, Goverde, & Hansen, 2010) are improved to include running times on the level of block sections, headway times and time loss due to route conflicts. Robust regression (Rousseeuw & Driessen, 2006), regression trees (Breiman, Friedman, Ohlsen, & Stone, 1984) and random forests (Breiman, 2001) were used for computing the process time estimates. The resulting estimates are insensitive to outliers and data errors which is crucial for real-time applications. Therefore, stability of process time predictions is ensured, which is of utmost importance for reliability of estimates and controlling the error propagation to other dependent processes. Moreover, all captured dependencies and results are interpreted and validated using domain knowledge. The main contributions are summarised in the following list: • a set of predictors for estimation of dwell times and running times on the level of block sections • global predictive models for process time estimation based on robust linear regression, regression trees and random forests • robust regression models for process time estimation for each combination of block, station and train lines 14 Models for Predictive Railway Traffic Management Real-time traffic state prediction A real-time prediction of train event times is the main contribution of this thesis. Very little work on real-time prediction exists in the current literature and the existing approaches rely on the static predictions that are independent of the actual traffic conditions. Therefore, this work required a development of a new methodology to predict the traffic state. The data-driven approach monitors the current traffic conditions on the network and performs the prediction of the future events. A mesoscopic traffic model is developed that reflects the microscopic traffic constraints on open track sections and in station areas. The graph model can be continuously updated with new information about the train positions or traffic control actions. Furthermore, a fast prediction algorithm has been implemented that in a single execution calibrates the model and computes the predicted realisation times of all events within a prediction horizon. The model is calibrated depending on the actual traffic conditions on the observed parts of the network. The mesoscopic character of the tool allows the accurate prediction of route and connection conflicts. For every predicted route conflict, the time loss of the hindered train due to braking, re-acceleration, running with lower speed and unscheduled stops is modelled realistically. The dependence of time loss on the conflict duration is determined from the historical traffic realisation data and quantified. The train dynamics can thus be accurately modelled and the computationally demanding iterative approach to deriving the running times of hindered trains (D’Ariano, Pranzo, & Hansen, 2007) can be avoided. In order to further increase the accuracy of predictions in real-time, an adaptive online error-smoothing component has been implemented. The prediction errors for running trains are monitored and an adaptive filter computes the adjustment to the downstream process time estimates. The trains that significantly deviate from the estimated trajectories are therefore identified in real time and the prediction error for future processes is decreased. A comprehensive analysis of algorithm performance has been carried out on a real-life case study. The computation speed and accuracy of predictions prove the applicability of the concept of data-driven predictions. The obtained results indicate a significant improvement of precision compared to the approaches used in the current practice and implemented in the relevant academic tools. The stability of predictions over different horizons is examined and the optimal prediction horizon is determined with respect to the accuracy of predicted arrival and departure times and accurate prediction of route conflicts. The main contributions are summarised in the following list: • a mesoscopic traffic model that reflects microscopic constraints on open track sections and in stations Chapter 1. Introduction 15 • a prediction algorithm that can quickly predict the traffic evolution in large and busy networks over a long prediction horizon • adjustment of running times of the trains hindered in route conflicts • adaptive adjustment of process time estimates in real-time 1.5.2 Macroscopic models for network-wide traffic rescheduling The main methodological contribution of the work on this objective is a realistic macroscopic model for real-time rescheduling that can solve the network-wide problem instances in short time. The appropriate macroscopic rescheduling model is created as a result of investigating the trade-off between the quality of solutions and the computation time. The effect of increasing the number of considered macroscopic constraints on solution quality, feasibility and the corresponding computation time is presented. The macroscopic models are validated by comparing their performance with the results obtained using a detailed mesoscopic model model. Aggregation of mesoscopic constraints to the macroscopic level was performed in a realistic manner that ensures the feasibility of the solutions produced by the macroscopic model. This includes computation of minimum headway times with respect to blocking time theory instead of using the predetermined norms which is a common approach in current practice and academic research. The modification of the existing alternative graph models is therefore presented that enables computation of minimum headway times with respect to train orders. The feasibility of the approach is demonstrated by a real-world case study for the Dutch national railway network. It is based on the DONS database represented in the form of a timed event graph (TEG) (Goverde, 2007). A data mining algorithm was developed that sweeps through a TEG, builds the macroscopic resources, i.e., stations, open track sections, and converts the TEG into an alternative graph based on running, dwell, headway and connection constraints. The model is applied to a substantial number of realistic disruption scenarios in a large instance that includes a peak hour of traffic in the complete Dutch railway network. The main contributions are summarised in the following list: • a mesoscopic alternative graph model (D’Ariano, 2008) modified in order to incorporate macroscopic operational constraints • an approach to convert a timed event graph into an alternative graph • four macroscopic rescheduling models created to investigate the trade-off between solution quality and computation time • the most complex model built with respect to macroscopic operational constraints produces feasible solutions in short computation time 16 1.6 Models for Predictive Railway Traffic Management Thesis outline and scope This thesis consists of seven chapters (including this one). Based on the content, the first two chapters can be grouped into an introductory part. Similarly the three chapters focusing on monitoring and traffic state prediction can intuitively be grouped into a coherent content. The structure of the parts and their relationship is illustrated in a flowchart in Figure 1.7. Related chapters are grouped and arrows indicate the order in which the chapters could be read. Chapter 1 Introduction Chapter 2 An overview of railway operation planning and control Chapter 4 Process mining of train describer event data Chapter 7 Rescheduling models for real-time traffic management in large networks Chapter 5 Data analysis and estimation of process times Chapter 8 Conclusions Chapter 6 Real-time prediction of train event times Figure 1.7: Flowchart of the thesis structure Part I contains the first two chapters of this thesis. In Chapter 2 the main concepts of railway systems, definitions and terminology needed for understanding the remainder of the thesis are introduced. Moreover, a review of the most important contributions in the scientific literature related to the problem of railway operation and traffic control is presented. Part II of the thesis consists of the chapters related to data-driven decision support systems for monitoring and real-time traffic state prediction. Chapter 3 presents a process model that is used for mining the traffic realization data. The developed process mining method is applied for real-time monitoring of railway traffic and ex post analysis, i.e. Chapter 1. Introduction 17 recovery of realized train paths and identification of route conflicts. The chapter first describes the data structure of TROTS files, preprocessing steps and input preparation. Moreover, the underlying algorithms and procedures are described with a great level of detail. Finally, the graphical user interface and visualisation component is presented that can be used to raise the situational awareness of traffic controllers or simplify performance analysis, depending on the application of the tool. Chapter 4 focuses on statistical analysis of traffic realisation data and computing robust estimates of process times. The used statistical learning tools are described, followed by the the descriptive and inferential statistical analyses of process times and route conflicts. Furthermore, we use the real-life data set to test and validate the common assumptions used to describe the variability of process times. We test the impact of delays and peak-hours on process times. Finally, the results of model performance in an application to a test set of historical data are presented. Chapter 5 gives a description of a real-time prediction tool as well as its position in the railway traffic control loop. The main prediction algorithm is presented followed by a description of the adaptive components that modify the estimates of process times with respect to the current (unexpected) traffic conditions. A real life case study is further described that is used to test the performance of the complete tool for monitoring and traffic state prediction. The integration of the components for data processing, analysis and prediction is described and model accuracy is extensively discussed. Chapter 6 presents different rescheduling models for dynamic management of largescale networks. The principles of deriving alternative graph models from macroscopic data are described, as well as the procedure to convert a timed event graph into a rescheduling model. Furthermore, this part focuses on the procedure to find an appropriate level of granularity for modelling railway traffic on a macroscopic scale with respect to the basic requirements for rescheduling problems such as the solution quality and computation time. A comprehensive analysis of the model performance with respect to the validated mesoscopic model is given, followed by the results of the application to a real-life case study of the Dutch national network. Finally, Chapter 7 summarises the main findings and contributions of the thesis. Limitations of the performed research are also discussed and clear directions for further research are given. As illustrated in Figure 1.7 there are multiple ways to read this thesis depending on the prior knowledge and interest of the reader. Readers with good knowledge of railway terminology and system properties may proceed directly to Chapter 3. Similarly, readers with particular interest in rescheduling aspect of dynamic traffic control can, after the introductory part, proceed directly to Chapter 6. 18 Models for Predictive Railway Traffic Management Chapter 2 An overview of railway operation planning and control 2.1 Introduction The research motivation and main objectives addressed in this thesis were described in the previous chapter. Before presenting the main contributions of this research in the following chapters, it is important to define the problems of traffic control in more detail and review the existing contributions from the scientific community. This chapter first presents the adopted terminology and the basic concepts of railway traffic. The operational rules, implemented in the signalling system and timetable, are of crucial importance for developing new and analysing the existing mathematical models of railway traffic. Moreover, the current practice of traffic control is presented and the main problems are identified. The problems related to railway traffic control and performance analysis have been addressed by numerous contributions. We give a critical review of the existing approaches and emphasise the gaps in the state-of-the-art models that are filled by the tools presented in this thesis. In the first part of the chapter, the basic definitions related to railway timetables, signalling and safety systems, train delays and traffic control are given (§2.2). This is followed by a separate literature review for each research objective and the corresponding requirements. Section 2.3 gives the literature review of processing and mining the train describer and traffic realisation data. The description of running, dwell and headway times, and approaches to their computation and estimation is given in Section 2.4. Section 2.5 presents the recent works on delay analysis, propagation modelling and prediction. The recent real-time rescheduling models are presented in Section 2.6. Finally, we discuss the existing practice and models and analyse their applicability for monitoring, traffic state prediction and network-wide rescheduling (§2.7). 19 20 2.2 2.2.1 Models for Predictive Railway Traffic Management Terminology and basic concepts of railway traffic Railway timetable Railway traffic usually operates according to a timetable. The railway timetable in the Netherlands is periodic, meaning that the pattern of arrivals and departures of all trains is repeated in regular intervals. The timetable construction for the Dutch network is supported by a sophisticated mathematical optimisation model based on the periodic event scheduling problem (PESP) (Serafini & Ukovich, 1989), that was applied to the train timetabling problem by Schrijver and Steenbeek (1993). DONS database contains the running and dwell times for each train, as well as the headway and connection constrains that need to be respected in order to design a feasible timetable for dense traffic of interconnected train lines (Hooghiemstra, 1996). The running time comprises the period between the train departure and the complete halt at the arrival station. It contains the outbound running time from the platform track to the departure signal, the running time from the departure signal to the home signal at arrival station and the inbound running time between the home signal until standstill at the platform track. We distinguish between minimum running times and scheduled running times. Minimum running times are computed with respect to the defined maximum speed on the train route and dynamic properties of rolling-stock and infrastructure which reflect the acceleration and breaking characteristics (Br¨unger & Dahlhaus, 2008). In terms of dynamic properties, a train run in full performance regime between two scheduled stops can be distinguished into acceleration, cruising at the maximum speed and braking continuously at the standard braking rate until stop at the platform (Albrecht, Goverde, Weeda, & van Luipen, 2006). The scheduled running times are given in the timetable. In order to increase the reliability and robustness of the timetable to varying running times and decrease energy consumption, the scheduled running times contain a certain amount of running time supplements (Goverde, 2005). The value of the running time supplement, currently in use in the Netherlands, is 5% of the minimum running time. The running time supplements can also be used for energy efficient driving. The typical strategies are cruising at a lower speed than maximal and/or by coasting before braking to standstill (Albrecht et al., 2006). The dwell time is the time between arrival of a train to standstill at the platform track and subsequent departure after the scheduled stop. Weidman (1995) determined the factors with high impact on the duration of dwell times. They include among other: the number, structure and distribution of passengers on the platform as well as the vehicle and platform design. Dwell times are modelled to a great level of detail by distinguishing them into several sub-processes: door unblocking, door opening, passenger boarding and alighting, door closing and train dispatching (Buchmueller, Weidmann, & Nash, 2008). Chapter 2. An overview of railway operation planning and control 21 In the Dutch practice of timetable design, minimum dwell times are determined based on the train type, station type and estimated passenger demand. The dwell time buffer is introduced to absorb the seasonal and daily variation in boarding and alighting time which is the longest sub process of train dwell time. Moreover, similarly to the running time supplements, dwell buffer times can be used to partially or completely absorb arrival delays. The scheduled dwell time may also be extended with the synchronisation times for passenger transfers and rolling-stock or crew connections. Finally, all events in a published timetable (arrivals and departures) are usually rounded up to full minutes for passenger convenience. The rounding procedure may affect the values of running and dwell time reserves and their distribution over the train route between terminal stations. A timetable is a result of a careful planning process that may take several months to complete and is usually valid for one year (with possible minor modifications). However, the final process plan, that may consider short-term allocated freight train paths and track possessions due to maintenance works, is developed one day in advance. It contains a detailed plan of traffic execution and represents the reference for operational traffic and transport control. A detailed description of the operational control layer is given in Section 2.2.6 after explaining the basic constraints of railway traffic incorporated in the signalling and safety system. 2.2.2 Signalling and interlocking Safety and signalling systems are an essential part of modern railways. Their main purpose is to ensure safe train runs by preventing derailments and collisions between trains that share the same infrastructure elements, and accidents between trains and other vehicles and objects. This overview is focused on the fixed-block signalling system that gives the movement authority for all trains on open tracks and in interlocking areas. A comprehensive description of components and functions of safety and signalling systems is given by Theeg and Vlasenko (2009). Bailey (1995) gives a comparative overview of the signalling systems in different infrastructure companies in Europe. The main signals in the Netherlands can be partitioned into automatic block signals on open tracks and controlled signals in station areas. Automatic block signals operate based on the information from train detection devices and interdependence with the neighbouring signals. Controlled signals are operated manually or activated by the automatic route setting system ARI (Berends & Ouburg, 2005). They are dependent on the logic of interlocking systems with the purpose to prevent head-on, rear-end and flank collisions. A home signal is a signal that protects the station area and prevents the incoming trains from entering if their route is not set up. The departure signal gives a movement authority after the corresponding outbound route has been locked. A basic element of safety and signalling systems is the track-clear detection system. Railway tracks are divided into track sections. The task of the track-clear detection 22 Models for Predictive Railway Traffic Management systems is to detect the presence of a train on a track section. They provide the binary information about the state of a track section which can be: (i) occupied if at least one axle is within track section borders or (ii) free if no axles are present at the section. We define the moment when the first axle of a train occupies the section as occupation time and the moment when the last axle leaves the section as release time. The presence of a train is continuously monitored either by means of track circuits or axle counters in each track section (Pachl, 2009). Track circuits rely on the conductive properties of axles and tracks. The presence of a train is detected when the electric circuit is closed between two rails and a vehicle axle. On the other hand, axle counters detect the presence of a train on a track section by comparing the number of axles counted on each end of the section. The interlocking is a safety system that integrates (interlocked) signals and switches to prevent conflicting or improperly set routes. Switches are movable track elements that enable trains to move from one track to another. In order to be used safely, a switch needs to be set in the appropriate position and locked. Track-clear detection of switches is performed in the same way as for track sections. A comprehensive overview of interlocking systems and principles is given by Theeg and Vlasenko (2009). The Dutch system is described by Goverde (2005). The safety principles required for route setting need to hold as long as the route is being used by a train or until it is cancelled by the controller. A route is released only after the train has cleared it. In order to increase the capacity of interlocking areas, especially in complex and busy stations, the modern interlocking systems employ the sectional-release route setting principle. Each section of the route becomes available for another route as soon as it is released by the last axle on the rear of the train. Route holding ensures that the occupied and non traversed sections still stay locked in the route. 2.2.3 Blocking time theory Fixed-block signalling is efficiently implemented in the railway traffic models using blocking time theory (Hansen & Pachl, 2008). The blocking time can be defined as the time during which a block between two signals is reserved exclusively to one train and therefore blocked for all other trains. It consists of the sight and reaction time of the train driver, approaching time, which is equivalent to the running time over the preceding block, the running time, clearing time needed for the full train length to leave the block, and setup and release time of the signalling system (2.1). Note that a block is physically occupied only during the running and clearing time (represented by shaded box). Using the blocking time theory, a route conflict between two trains corresponds to an overlap of their blocking times. Figure 2.2 depicts an overlap of blocking times of two successive trains. The second train is within the sight distance of approaching signal and the first train has still not left the block. Consequently, the aspect of the Running time Sight distance Distance 'g' Approach time Setup time Sight and reaction time 'y' Clearing time Release time Occupation time 'r' 23 Train lenght Chapter 2. An overview of railway operation planning and control Time Figure 2.1: Blocking time approaching signal is ‘yellow’, causing the second train to brake. The conflict duration is the width of the overlap indicated in red. Running time Approach time tr ed er Hi nd Hi nd er ing tr ain 'r' TIme loss ai n Setup time Sight and reaction time Duration Clearing time Release time 'g' Distance 'y' Time Figure 2.2: Route conflict A train run over the signalled open track section and interlocking areas can be represented as a sequence of blocking times. That sequence models the time slots reserved for train operation. It is represented by the blocking time diagram. Figure 2.3 shows 24 Models for Predictive Railway Traffic Management the blocking time diagrams of two successive trains. The minimum headway time between departures from the same station is determined by compressing the blocking time stairways as much as possible without causing an overlap. The block where the blocking times would first overlap is called the critical block. The resulting minimum line headway can be used for time-based train separation. Distance Critical block headway Minimum line headway Time Figure 2.3: Blocking time stairways 2.2.4 Train position detection Keeping track of train positions is a basic requirement for the monitoring of railway traffic. Traditionally, train positions were monitored only at staffed stations and other timetable points. However, the recent developments in sensor and communications technologies enable a more detailed observation of running trains. Train positioning can be track-based or train-based (L¨uthi, 2009). Figure 2.4 illustrates the track-based approach to train positioning. Train describers are the most commonly employed system for track-based train positioning (Exer, 1995; Pachl, 2009). A train describer system keeps track of train positions in discrete steps over the route based on the messages received from the track-clear detection devices. Moreover, an important function of train describers is logging of incoming infrastructure element messages and the generation of train number messages. The train describer system TROTS is used in the Netherlands since 2007 (ProRail, 2008). Train number steps are followed on the level of track section, i.e. a message reporting the new train position is recorded with every section occupation and release. Moreover, the system also logs binary messages reporting aspect changes of controlled signals (‘stop’ or ‘go’), as well as a position change of switches (‘left’ or ‘right’). Chapter 2. An overview of railway operation planning and control 25 B time A Figure 2.4: Infrastructure based train detection Alternatively, the actual train position may be determined with a certain frequency by means of the Global Positioning System (GPS), possibly in combination with an infrastructure based detector to reduce the measurement errors (Figure 2.5). The exact position of each train is communicated to the traffic control centre in regular intervals e.g. by means of the Global System for Mobile Communications-Railway (GSM-R) (Winter, 2009). Note that GPS signals may not be continuously available and sufficiently accurate to distinguish between parallel tracks. Therefore, GPS cannot ensure safety of train operation in densely occupied railway networks. B A time Dt Dt Dt Dt Dt Dt Dt Dt Figure 2.5: Regular time interval train detection For the purpose of this study, log archives of the Dutch train describer system TROTS have been made available by the infrastructure manager ProRail. Section 2.3 presents earlier contributions related to data mining and extraction of information from historical traffic realisation data. 26 2.2.5 Models for Predictive Railway Traffic Management Classification of train delays Delays in railway traffic occur due to variability of process times, capacity and synchronisation processes, and dependence on availability of infrastructure, rolling-stock and crew. Small deviations from the scheduled process times as a consequence of variability result in disturbances. Goverde (2005) described disturbances as structural deviations that reflect stochasticity of process times due to internal and external factors. The issue of minimising their impact on timetable reliability is addressed both in the tactical and operational control and planning levels. Time supplements and buffer times are added in the process of timetable construction as discussed in Section 2.2.1. Moreover, operational traffic control aims to minimise deviations from the timetable during real-time operations (§2.2.6). On the other hand, disruptions are caused by major deviations of timetable and logistic schedules due to failures of infrastructure, rolling-stock, line blockages, extreme, weather conditions, etc. (Nielsen, 2011). Major disruptions in general do not happen frequently and they are resolved by special disruption and incident management strategies (Jespersen-Groth et al., 2009). Primary delay is an extension of the scheduled process time caused by a disruption within the process (Goverde, 2005). Primary delays may result in secondary (consecutive, knock-on) delays. Occurrence of secondary delays is called delay propagation. Secondary delays occur as a result of interdependences between trains, i.e. due to route conflicts or waiting for scheduled connections. They may be a consequence of primary and secondary delays but also due to early trains and timetable errors. Capacity constraints are a common reason for secondary delays. Extended running time of a train may cause knock-on delays to successive trains on the saturated line. Similarly, extended dwell time in a station often results in consecutive delays of other trains in busy stations due to occupied platform track or station routes. Whereas primary delays are independent from traffic control, preventing delay propagation through the network is one of its most complex tasks that will be addressed in detail in Chapter 6. If the time allowances, i.e. running time supplements and dwell buffer times are not sufficient to absorb the primary delay, the same train suffers follow-up (unavoidable) delays in subsequent stations. For example, extended dwell time in a station may cause a delay of the same train in other stations along its route until the running time supplements and dwell time buffers have absorbed the delay. Note that the follow-up delays of an operating train cannot be reduced or avoided by any traffic control action. 2.2.6 Operational control of railway traffic and transport Operational planning is performed by traffic control centres. Their task is to create updates to the process plans determined on the tactical planning level. In case of disruptions and disturbances, timetable, rolling-stock and crew circulations may become infeasible. Controllers on behalf of an IM (traffic controllers) and the TOCs (transport controllers) need to perform rescheduling actions in real time. Chapter 2. An overview of railway operation planning and control 27 Jespersen-Groth et al. (2009) presented the structure of an integrated operational control level (Figure 2.6). The information flow between different levels of control, and IM and TOCs explains the process of disturbance and disruption management. Local traffic controllers observe traffic in their area and implement the process plans derived on the network level. Disturbances and disruptions with the effect that exceeds their area are reported to the network traffic control. The timetable updates, derived at the network control level, are transmitted to local controllers who need to implement it. Computation of the working network timetable is a cooperative process between the traffic and transport process control. The network traffic control derives the timetable updates, whereas the network controllers on behalf of TOCs, create updates to resource circulation schedules. In an iterative procedure, IM and TOCs derive a feasible working timetable that is given as a master plan for local control. On the local level, traffic and transport controllers cooperate in order to perform all necessary shunting operations. IM TOC Traffic controller Local Signaller and dispatcher Proposed timetable Transport controller Rolling -stock and crew rescheduling Actual timetable Delays and traffic state Network Shunting requests Shunting time slots Local operations control Figure 2.6: Structure and information flow within operational planning level (Jespersen-Groth et al., 2009) The problems of traffic and transport process control are highly interconnected both on the network-wide and on the local level. However, a high complexity of each problem individually prevents an integrated approach to computing the timetable and logistic plans adjustments simultaneously. In the current literature, the problems of real-time transport and traffic control are addressed separately. It is assumed that a feasible solution to both problems is reached in a negotiation process by iteratively solving both problems in a closed loop until a feasible solution is found. A recent review on models for crew rescheduling is given by Potthoff (2010) and Potthoff, Huisman, and Desaulniers (2010). Relevant contributions from the field of rolling-stock rescheduling are reviewed by Nielsen (2011) and Nielsen, Kroon, and Mar´oti (2012). Rolling-stock 28 Models for Predictive Railway Traffic Management and crew scheduling are beyond the scope of this thesis. However, the delay management, with the purpose to decide which passenger connections to keep in case of delays, is closely related to network traffic control. TOC controllers perform delay management to minimise passenger delays. The main tasks of traffic controllers on both levels are monitoring, prediction and rescheduling (L¨uthi, 2009). The traffic state described by actual train positions and speeds must be continuously observed in order to detect deviations from scheduled train paths. The consequences of such deviations need to be accurately predicted as the rescheduling process is performed based on these predictions. Moreover, before implementing a rescheduling decision, the effects on the corresponding area need to be predicted. Monitoring and traffic state prediction Accurate information on train positions can be used to derive performance indicators and parameters needed for estimation of train running and dwell times such as: approximation of train speed, actual train delays, registered values of train running and dwell times, etc. These parameters give an indication about the current traffic state on the network that can further be used to predict the traffic in the period defined by a prediction horizon. In the Netherlands, traffic controllers monitor train positions using infrastructure-based position detectors (§2.2.4). As explained in Section 1.2, the actual train delays are measured at home and departure signals, corrected with a fixed correction term and rounded to full minutes. The precision of such measurements is insufficient for a reliable estimation of the actual traffic state. Traffic controllers use the actual traffic state to predict the future train positions and the evolution of traffic in their area of observation (D’Ariano, 2008). The accuracy of these predictions has a big impact on quality of traffic control decisions and rescheduling actions. Current practice in traffic state prediction has a macroscopic character and relies on the so-called parallel shift method. The actual delay of a train, observed in a timetable point, is extrapolated to subsequent timetable points as illustrated in Figure 5.13. This method neglects the fact that some trains may reduce their delays by running with maximum performance and using the running time supplements. Moreover, some trains may get (more) delays due to route conflicts. The improvement of the described drawbacks of the current practice in monitoring and traffic state prediction represents the first research objective of this thesis (§1.4). Sections 2.3–2.5 review the existing approaches relevant for developing the data-driven monitoring and prediction tool. Rescheduling After predicting the expected conflicts and delays that make the planned timetable infeasible, traffic control needs to find a new feasible schedule for train operations. That procedure is called real-time rescheduling. It is performed both on the level of Chapter 2. An overview of railway operation planning and control Scheduled arrival B 29 Predicted arrival Departure delay A Scheduled departure Real departure Figure 2.7: Illustrative example of the parallel-shift prediction method local and network traffic control. Operational requirements of rescheduling tasks of traffic control are summarised, among others, by Cacchiani et al. (2014); D’Ariano (2008); Harrod (2012); L¨uthi (2009). Network traffic controllers deal with disruptions and disturbances with effects that can propagate and affect the global network performance. They need to take into account macroscopic constraints of railway traffic, such as running times of trains between timetable points, dwell times, minimum headway times between successive dependent events in timetable points, and synchronisation constraints. The objectives of rescheduling on this level depend on the traffic situation and the magnitude of disruption. They vary from minimising the deviations from the published timetable in case of disturbances, to maintaining passenger flows and maximising throughput in case of line blockages and major incidents. An important task of network traffic controllers is to coordinate the controllers on the local level whose situational awareness is limited to their own area, and try to minimise delay propagation multiple areas. Apart form changing the scheduled times and relative train orders defined in the timetable, network traffic controllers may reroute trains over different lines, cancel or add trains, implement short turns, skip-stop operation, etc. Local traffic controllers manage route conflicts, delays and disturbances within their control area. Microscopic train routes, signalling and interlocking principles need to be considered by traffic controllers on this level. The dispatchers and signallers in a local traffic control area implement rescheduling decisions. That includes changing the relative order of trains that simultaneously claim the same block (platform track or station route), changing a train route in a station area or modifying departure times. The objective is to minimise the deviation from a target trajectory set by the hierarchically higher network control level. In the current Dutch practice however, traffic controllers do not have any decision support to pursue the described objectives. In case of a major disruption, on the network level the tendency is to isolate the incident and prevent delay propagation to nonaffected parts of the network. On the local level, traffic controllers rely on the ARI 30 Models for Predictive Railway Traffic Management system that automatically sets routes for the approaching and departing trains. Conflicting routes are resolved using the predetermined rules depending on the magnitude of delays. Specifically, the priority is given on the basis of the first come first served (FCFS) rule for routes crossing each other while for merging or identical routes the orders specified in the timetable are followed. For larger delays, dispatchers take train ordering decisions with the support of a list of what-if scenarios (Corman, D’Ariano, Pranzo, & Hansen, 2011). The current practice in real-time rescheduling may produce suboptimal effects on punctuality and reliability of traffic. The second research objective in this thesis aims to develop a decision support rescheduling system for network traffic controllers (§1.4). Section 2.6 gives a review of the existing scientific approaches to real-time rescheduling. 2.3 Review of approaches for data mining of traffic realisation data The first step in developing a data-driven tool for monitoring and traffic state prediction is to create a data mining procedure for processing the incoming messages about train positions, as well as the archives of historical track occupation data. This section covers the relevant existing approaches for retrieval of traffic related information from the automatically logged event messages. The historical traffic realisation data sources used in the current scientific literature range from the arrival and departure times recorded manually at specific stations to detailed traffic realization data extracted from train describer log files. Longo, Medeossi, and Nash (2012) classify the automatically collected railway operation data sources to sensors embedded in the infrastructure, sensors in rolling-stock and mobile GPS devices. We emphasise that the granularity of data from each data source varies depending on the system employed by a particular infrastructure manager or train operating company. The most frequently used data source for recovery of realised train paths are train describer event log files. The level of detail in the data, as well as the structure of the resulting log files vary in different train describer systems. For that reason, a separate approach is required, in order to develop a data processing algorithm, suitable for a specific system. The purpose of such processing tools is to clean the raw data and produce data structures that are convenient for performance analysis and development of data-driven models. Train describer data were originally archived for maintenance and accident investigation purposes. Their importance in direction of performance analysis has been recognised relatively recently. Goverde and Hansen (2000) presented a tool TNV-Prepare that couples infrastructure messages to train number steps. The track occupation and release messages, as well as signal aspect change messages are in the earlier Dutch Chapter 2. An overview of railway operation planning and control 31 train describer system Treinnummer Volgsysteem (TNV) logged separately from train number messages. The developed algorithms are able to assign infrastructure messages reporting state changes of signals, sections and switches to the train numbers that caused them. The application of this tool results in an ordered list of section and block occupation and release times for each train run. More insight in train interactions and route conflicts from historical track occupation data was gained by expanding the data processing tool with signalling logic (Goverde, Daamen, & Hansen, 2008). Daamen et al. (2008) formalised the signalling logic model through a coloured Petri net and implemented it in the TNV-Conflict tool. The main contribution of this work is an automatic identification of all route conflicts that a train suffered along its path. Hindering trains are also identified by finding the train number that occupied the block section or route protected by the signal of conflict. The identification of route conflicts is a fundamental requirement for distinction between primary and secondary delays and for developing accurate data-driven models that need to rely on conflict-free running times. Results from processing train describer files in Switzerland are presented by Labermeier (2013). The work exploits data from the traffic control system RCS that has been in use as a dispatching system in Switzerland since 2010. Using the actual timetable, realized train event times and connection plans, the author is able to distinguish between primary and secondary delays. Furthermore, a comparative analysis of primary and secondary delays was performed. The results show that secondary delays that occur due to waiting for late feeder trains are the main contributors to low punctuality levels in Switzerland. Train describer data often do not provide information about train runs on open track sections because automatic block signals are not logged. Therefore, an alternative data source for recovery of complete train paths with a great level of detail are train event recorders. The main advantage of using on-board train event recorders is the high frequency of incoming messages about train position and speed (Allotta, Toni, Malvezzi, Presciani, & Colla, 2001). Moreover, the actual stopping and departure times as well as the door opening and closing times are also recorded, thus enabling more detailed modelling of train running and dwell times. De Fabris, Longo, and Medeossi (2008) presented a method that enables the analysis of train event recorder data. The detailed recovery of train movements results in a continuous estimation of speed profiles, acceleration rates and breaking curves. Dwell times are derived accurately from train door sensors. The tool is further applied for calibrating the parameters that can improve the quality of microscopic simulation tools. The described works on data mining train describer data archives in the enable a train path recovery, route conflict identification and separation between primary and secondary delays. However, the implementation of the new train describer system in the Netherlands and corresponding changes in data structure and information contained in the event logs, requires the construction of new algorithms for track occupation data 32 Models for Predictive Railway Traffic Management processing, discovery of processes, and route conflict identification. An important drawback of the Dutch train describer system TROTS is the fact that train events on open tracks are not logged with the precision and frequency required to recover train paths and identify route conflicts. The existing approaches do not provide a solution to this problem. 2.4 Review of approaches for process time estimation The second step in reaching the first research objective of this thesis (§1.4) is estimation of process times. This section covers the most relevant approaches for running, dwell and headway time estimation. 2.4.1 Running time estimation The minimum running times are usually computed by means of train motion equations (Wende, 2003). This approach considers the dynamic properties of rolling-stock and infrastructure that are represented in equation parameters. The empirical parameters for minimum running time computation are typically given for a particular line or rolling-stock type and train composition. The parameters are usually determined by experts and tuned in practical operation in a particular railway company. This method is often used for minimum running time computation in the process of timetable construction (Hooghiemstra, 1996) and microscopic traffic simulation (A. Nash & Huerlimann, 2004). However, greater precision in calibrating the parameters of the train motion equations for different traffic conditions can be achieved using the actually realised running time data derived from track occupation or train event recorders data. Longo et al. (2012) define a single parameter for each dynamic motion phase. The variability in running times can be modelled by fine tuning the corresponding parameter. The parameters are calibrated against the train event recorders data and corresponding probability distributions are derived. This approach is a convenient way of estimating the robustness of running times in the timetabling stage. Beˇsinovi´c, Quaglietta, and Goverde (2013) extend this approach by calibrating each tractive effort and resistance parameter separately. These parameters are optimised by a procedure that minimises the gap between the simulated and actual train positions and speed profiles. A probability distribution is computed for each considered parameter. Moreover, the parameters with the greatest impact on variability of speed profiles are identified, which can be useful for calibrating prediction models. The accuracy of the model was tested in a real-life case study in the Netherlands. Another stream of research in the modelling and analysis of realized train running and dwell times is related to defining explanatory variables and quantifying their impact on process times. The departure delay has been recognized as a potential predictor for running times of trains of a particular train line on an open track section. Similarly, Chapter 2. An overview of railway operation planning and control 33 arrival delays and peak-hours are used to derive estimates of dwell times in a station. Van der Meer et al. (2010) presented an approach based on robust regression analysis to investigate the correlation between process times and delays. The results show a strong correlation between arrival delays and dwell times. The correlation between running times and departure delay is much weaker. Similar results are obtained from the set of track occupation data from Switzerland by L¨uthi (2009). Both approaches for modelling running times rely on macroscopic data, aggregated over open track sections. The computational requirements for solving the train motion equation prevent straightforward application of this method to simultaneous running time estimation of a large number of trains in busy networks. Moreover, such approach is static and offline in the sense that it does not consider the current traffic state and potential impact on running times. Even the approaches based on data-driven calibration of train motion equations for the purpose of running time estimation require computationally demanding multiple simulations which makes them unsuitable for real-time applications. These drawbacks are overcome by the data-driven approaches based on creating robust estimates of running times dependent on current values of traffic state indicators. Even though this method gives precise free running time estimates on the macroscopic level, it lacks the exact modelling of train interactions on the line. Therefore, blocking times, minimum headway times and the time losses due to route conflicts cannot be captured. 2.4.2 Dwell time estimation Current approaches for the estimation of dwell times used in timetable construction and rescheduling rely to a great extent on the measurements of realised dwell times. Wiggenraad (2001) performed a detailed analysis of dwell times, and passenger boarding and alighting processes using a set of manually collected data from seven busy stations in the Netherlands. The impact of platform and vehicle characteristics, delays, station types and peak-hours was analysed with the purpose of detailed analysis of dwell times. The analysis determined the average boarding and alighting time per individual passenger as well as per passenger within a cluster. An interesting insight is that peak-hours do not have a significant impact on the duration of dwell times. Lee, Daamen, and Wiggenraad (2007) performed a similar study using manually collected data from two busy stations in the Netherlands. They focused on the factors that determine passenger behaviour and its influence on dwell times. Platform and vehicle design, passenger mobility characteristics (age, disabilities, luggage) and crowding effects were analysed in order to enable realistic modelling of dwell times. A non-linear relationship between boarding time and the number of passengers was determined empirically. Recent advancements in sensor technologies and availability of data from on-board event recorders inspired a stream of research on dwell time modelling. More precise and larger data sources are analysed with the purpose of deriving general conclusions 34 Models for Predictive Railway Traffic Management about dwelling processes of trains in stations. Buchmueller et al. (2008) collected the data from door sensors, passenger counters and train event recorders. They analyse the duration of each sub process separately with respect to vehicle and platform design, and passenger demand. The case study and data set for model calibration comprises a large amount of data collected from different train lines in Switzerland. Longo and Medeossi (2013) present a complex model for dwell time estimation that separates dwell time into a set of deterministic and a set of stochastic sub-processes. They focus on the detailed modelling of stochastic processes such as boarding and alighting time, waiting time and departure imprecision time. Boarding and alighting time are considered to be dependent on the train set property and the number of passengers. By treating the number of passengers as a random variable for different train sets, estimates of stochastic sub processes of dwell times can be derived. Detailed dwell time analysis of traffic realisation data in the Netherlands was performed by Stam-Van den Berg and Weeda (2007). The model relies on determining the exact location of access points to the platform and estimation of the actual stopping point of the train. The authors approximate the running time of trains from the moment of occupation of the platform track section to the stopping point of the head of the train by assuming constant deceleration. Similarly, they estimate the exact departure time by assuming constant acceleration between the stopping point of the train and the first track section after the platform track. Even though this approach reduces the estimation error based solely on track occupation data it relies on the knowledge of platform design and assumption about the stopping point of the train. The limited scope of the studies that are based on manual data collection creates difficulties for deriving general conclusions. On the other hand, application of the detailed data-based approach that relies on sensor and train event recorders data or platform layout strongly depends on data availability. Finally, the approaches that incorporate uncertainty by computing probability distributions, possibly dependent on peak hours and arrival delays, are not directly applicable for real-time application due to the large number of stochastic simulations required to derive robust estimates of dwell time duration. 2.4.3 Headway times The estimation of minimum headway times depends on the level of modelling. On the micro or mesoscopic level, most models use a space-based separation of trains, which is analogous to the actual operation controlled by the signalling and interlocking systems. On the other hand, macroscopic models employ a time-based separation between trains, enforced between events at the relevant timetable points (arrivals, departures and through events) (Ciuffini, Longo, Medeossi, & Vaghi, 2013; Harrod & Schlechte, 2013). The space-based separation fully corresponds to the requirements of signalling systems (D’Ariano, Pranzo, & Hansen, 2007) and enables an accurate modelling of route Chapter 2. An overview of railway operation planning and control 35 conflicts and speed profiles of hindered trains and conflict-free train runs. In microand mesoscopic models, that typically employ a space-based principle of train separation, a train cannot pass a main signal protecting a physically occupied or reserved block and station route. Thus the collision prevention is accurately modelled. The separation with at least two free blocks between successive trains is used to model the conflict-free train runs (green wave policy). Corman, D’Ariano, Pacciarelli, and Pranzo (2009) investigated the application of the green wave policy in real-time traffic management on the mesoscopic level by incorporating a two-block separation between successive trains, including a time and distance, respectively for driver’s reaction time and sighting. The current practice and the majority of macroscopic models rely on time-based headway computation based on the empirically determined norms. In the Netherlands different headway norms are applied for each type of conflicting train movements (ProRail, 2013). Time-based headways aim at separating events in stations to achieve conflict-free traffic on open tracks with all signals in the train route showing ‘green’ aspects. In timetabling models, minimum headway time norms are increased with buffer times (one or two minutes in the Netherlands) with the intention to absorb the variation in running times and prevent route conflicts in case of small deviations of train runs from their scheduled slots. Buffer times lead to a decrease of capacity. The problem of their optimal distribution reflects the compromise between schedule reliability and consumed capacity (Yuan & Hansen, 2008). Schlechte et al. (2011) presented a procedure for aggregating capacity constraints without compromising the feasibility of macroscopic models derived from the microscopic level. An earlier approach for the integration of the two levels of modelling was described by Kettner, Sewcyk, and Eickmann (2003). The minimum headway times are estimated on the basis of blocking times derived from microsimulation and may be used for optimisation of timetables and rescheduling. 2.5 2.5.1 Review of delay propagation analysis and prediction models Delay propagation analysis Traffic realization data can be used to analyse delays, punctuality and timetable robustness and stability. Approaches based on historical data give insight into probability distributions of delays. Moreover, the mutual dependency between processes and delays can be used for modelling delay propagation and traffic state prediction. Goverde (2005) performed a statistical analysis of train delays in a complex and busy station of Eindhoven in the Netherlands. The purpose of punctuality analysis was to discover and explain the systematic delay propagation resulting from minor disturbances. Descriptive statistical analysis of arrival and departure delays, and dwell times 36 Models for Predictive Railway Traffic Management was presented. Moreover, corresponding probability distributions were computed to fit the empirically observed values. The author selected the train lines with planned passenger connections and applied robust linear regression to investigate the correlation between arrival delays of feeder trains and departure delays of connecting trains. Yuan (2006) contributed to detailed modelling of train delays in stations by incorporating the microscopic layout of the station and route setting and release principles in analysis. The main goal was to estimate the probability distributions of secondary delays due to route conflicts in complex and busy station areas. Statistical parameters of arrival and departure delays, and dwell times were computed. An important objective in this context was to investigate the impact of peak-hours on train delay propagation. A separate probability distribution for morning and evening peak, and off-peak period was computed. Delay propagation due to route conflict was analysed by identifying the critical points (switches or platform sections) that the two conflicting routes have in common. The time lag between release and subsequent occupation of critical points was determined and used to estimate the probability of route conflicts. A punctuality analysis on a saturated corridor was performed by Richter (2013). Probability distributions of running times were derived from the historical traffic realisation data. Evolution of delays over a busy corridor in Denmark for individual train lines was analysed and correlation between delays of different trains was established. The analysis was separated into several levels. First, the train lines with systematic departure and arrival delays were identified. The routes of selected train lines were analysed and potential conflicting train lines were identified. Finally, linear regression analysis of delays of conflicting train lines was performed to prove the correlation and systematic dependency. A generalised approach for timetable robustness and stability evaluation, based on train describers data in Switzerland was presented by Ullius (2004). Delays in stations and running times of trains on the corridor were analysed. Moreover, global punctuality indicators on the network level were computed. The methodology has been implemented in the OpenTimetable software (A. Nash & Ullius, 2004). Input files contain planned and realized arrival and departure time for each train number. Users can query particular corridors, time-slots and train lines. The output contains the realized time-distance diagrams, delay distributions and capacity utilization. A comprehensive evaluation of the tool and application on large data sets of realised traffic in Switzerland was presented by Graffagnino (2012). Goverde and Meng (2011) developed a data mining tool TNV-Statistics and applied it on a set of track occupation data in order to isolate secondary delays that occurred due to route conflicts. The tool is equipped with a module that computes time loss, delay jumps and the number of conflicts per signal. An important feature of the tool is the identification of conflict chains, i.e., linked lists of trains in successive route conflicts. The tool provides a convenient way to identify systematic delay dependencies due to capacity constraints. Moreover, by analysing the number of route conflicts occurrence per individual signal, the capacity bottlenecks can be identified. Chapter 2. An overview of railway operation planning and control 37 The presented punctuality analysis methodologies can be used to identify typical delay predictors and causes. Moreover, they are useful for developing stochastic and simulation models of railway traffic that can be used to compute robustness measures of timetables. However, their application for predicting delay propagation in real time requires a (stochastic) model of railway traffic that would capture the complex interdependencies between train delays in busy networks. Thus they are not directly applicable for real-time traffic state prediction. 2.5.2 Identifying structural timetable errors and systematic delays The identification of causes and prediction of delays that repeatedly occur on the network-wide level is a complex task that involves applying advanced data mining techniques for analysing historical traffic realisation data. Delay dependencies and identified structural errors in a timetable, that result in systematic delays, can be effectively used not only for timetable improvement but also for real-time predictions. Conte (2007) incorporated the dependencies of train events (due to headway or connection constraints) by modelling them with a stochastic Tri-graph approach. Individual dependencies are then incorporated in a large graph that models the traffic in the observed part of the network. The tri-graph method was chosen to circumvent the drawbacks that occur due to the complexity of the conventional conflict graph models while exploiting their individual advantages. The method corresponds to a combined use of a full conditional independence graph and covariance graph for modelling delay dependencies. Dependencies due to secondary delays are taken into account. This approach was applied on a real-life case study of a large sub network in Germany. The problem size reduction as a result of using the tri-graph justifies the approach despite the increase of prediction error. Flier, Gelashvili, Graffagnino, and Nunkesser (2009) developed a data mining tool for identifying delay dependencies in large networks. In this approach they distinguish between secondary delay due to capacity constraints and due to synchronisation constraints. A separate model has been developed for each type of dependencies. These models are further used to sweep through the aggregated traffic realisation data in order to identify dependencies between delays. Further extensions include identifying multiple (capacity and connection) delay dependencies and improving robustness of the approach to measurement errors and outliers. The model was applied to a set of large-scale traffic realisation data. Important dependencies due to capacity and synchronisation constraints that were difficult to identify using correlation were discovered. Cule, Goethals, Tassenoy, and Verboven (2011) applied pattern recognition algorithms to isolate train delays that frequently occur in the network within a certain time interval. The large set of resulting patterns is further reduced by the closed episode mining algorithm. However, since this method does not incorporate the operational constraints of railway traffic, it is possible to discover mutually independent events within the same 38 Models for Predictive Railway Traffic Management pattern. The method is therefore only applicable for corridors or bottlenecks in which all train events are dependent. The method was applied to a set of traffic realisation data from Belgium. The validity of the discovered patterns of dependent delays was ensured by network decomposition and separate analysis of each interconnected sub network. The described data mining approaches may be used for real-time predictions of train delays in stations. However, the aggregated macroscopic models prevent prediction of route conflicts and train interactions on the level of block sections. For that reason, they are not applicable in the context of real-time traffic control. 2.5.3 Delay propagation models Delay propagation models predict delay values for all trains within the observed part of the network and prediction horizon, based on the current traffic state and actual delays. The basic requirement of delay propagation models is an accurate traffic model. Mutual dependence of the process times of a particular train, as well as time reserves incorporated in the schedule need to be adequately considered. Moreover, the model must take into account all possible train interdependencies that can cause delay propagations over the network due to capacity or synchronisation constraints. A possible way to compute delay propagation with an accurate model is by using microscopic simulation tools (Janecek & Weymann, 2010; Middelkoop & Loeve, 2006; A. Nash & Huerlimann, 2004; Quaglietta, 2013; Siefer & Radtke, 2006). These microscopic simulation models typically rely on detailed modelling of a train run with respect to rolling-stock and infrastructure dynamic properties. On top of that, train interactions are modelled on a detailed level by incorporating the signalling and interlocking principles. Microscopic simulation models can realistically represent train traffic. However, computational requirements prevent the application of such models to dense traffic on large networks with strongly interconnected lines or long prediction horizons. In this review we cover the two fundamentally different approaches to the delay propagation problem. First, we focus on deterministic models with fixed relative train orders and process times. In the second part, an overview of stochastic models is presented where process times are modelled as random variables. Deterministic models Landex (2008) gave an overview of analytical approaches for computing delay propagation on a single line depending on traffic heterogeneity and type of the line (single track or double track). Given the initial delay, minimum headway times and buffer times between trains, the sum of secondary delays is computed. Moreover, the delay propagation between every pair of successive trains is given. Whereas this approach is useful for capacity analysis of a single line, real-time prediction of delays requires generalisation on the network level. Chapter 2. An overview of railway operation planning and control 39 Braker (1993) modelled the Dutch railway timetable as a discrete event dynamic system in max-plus algebra. Running, dwelling and connection precedence constrains are included on a macroscopic level. Periodicity of the system is exploited to model its dynamics using recursive max-plus relations. Timetable stability is evaluated by computing the maximum eigenvalue which is equivalent to minimum cycle time. Goverde (2007) also exploited the convenience of using max-plus algebra for modelling periodic discrete event systems. A periodic timetable is modelled with a timedevent graph that includes running, dwelling, connection and headway constraints while maintaining the macroscopic level of the model. System dynamics is represented with a general higher-order max-plus linear system. Moreover, max-plus spectral analysis is used to evaluate stability and robustness of a timetable. Finally, a bucket-based delay propagation algorithm (Goverde, 2010) predicts the evolution of current traffic condition over multiple periods in a time efficient manner. A mesoscopic approach that integrates a microscopic simulation model (Siefer & Radtke, 2006) with a macroscopic simulation tool for large networks (Kettner, Prinz, & Sewcyk, 2001) is presented by Kettner et al. (2003). The accuracy of the macroscopic model that models railway traffic with a great level of abstraction is increased by computing all running and headway times with a detailed microscopic simulation tool. The two-level approach enables modelling and simulating traffic in large networks with realistic process times. The required computational efforts for simulating process times in large areas are distributed by using geographic decomposition of the network. Stochastic models Carey and Kwieci´nski (1995) presented a stochastic model of a complex scheduled transport system. The event times of all trains in the model are computed in a recursive manner, based on the realised event times of preceding events, timetable, and process times modelled as random variables. The generic model is independent from the choice of suitable probability distributions of running and dwell times. An important aspect of the model is probabilistic modelling of train orders which reflects the potential impact of traffic control decisions. The model was further used to develop and evaluate a series of reliability measures for scheduled services (Carey, 1999). Higgins and Kozan (1998) presented a stochastic delay propagation model and applied it on a case study of a busy urban train network. The underlying traffic model includes the basic dispatching actions. The delay of each train is estimated by summing up solutions to a set of equations that model the probability of primary delay, probability of secondary delay due to capacity constraints and probability of secondary delay due to connection constraints. The solution is obtained using a numerical iterative refinement algorithm. Middelkoop and Bouwman (2001) presented a simulation tool Simone that simulates train running and dwell times over the complete Dutch network. The models are generated automatically from the Dutch timetable database that is used by DONS 40 Models for Predictive Railway Traffic Management (Hooghiemstra, 1996). Train interactions are modelled in stations only. Primary delays, defined by the user, are propagated through the rest of the network. Yuan and Hansen (2007) described a detailed analytical stochastic delay propagation model for complex station areas. First, analytical formulas are given for deriving secondary delays of departing and arriving trains separately. Conditional distributions of arrival and departure times are computed using convolutions for computing the distribution of the sum of random variables. The model is validated on a case study from a complex railway station The Hague HS in the Netherlands. Meester and Muns (2007) formalised the model presented by Carey and Kwieci´nski (1995) as a stochastic event graph. The distributions of free running times are given and the process dependencies are computed as a mixture of the corresponding distributions. A set of performance measures is derived as a linear combination of delay distributions. The method for computing the values of proposed measures relies on approximating delay distributions with phase-type distributions. The upper bound of approximation error is presented and the approach is applied on a small part of the Dutch national network. A stochastic delay propagation approach based on processed train event recorders data was presented by Medeossi et al. (2011). Delay propagation is computed by means of stochastic blocking times. The method relies on asynchronous simulation of individual train runs based on probability distributions of train motion parameters (Longo et al., 2012). Each simulation run generates a blocking time ‘stairway’. Superimposition of blocking time stairways for each train run results in a blocking probability, as well as route conflict probability when other train runs are considered. B¨uker and Seybold (2012) modelled delays as random variables, described with suitable distribution functions, and applied analytical methods to compute delay propagation in a mesoscopic graph-based model. Running, dwelling, connection and headway precedence relations are included in the graph that comprises all scheduled events. Primary and secondary delay distributions are obtained by computing the conditional convolutions of corresponding extended exponential distributions. The model was applied for timetable evaluation and prediction of event times on a real-life case study in Switzerland. The presented deterministic and stochastic delay propagation models rely on the scheduled timetable, train orders, routes and connection plans. Therefore, the actual state of traffic at the moment of prediction cannot be exploited to derive more accurate predictions. Moreover, the large-scale character of the models does not allow precise modelling of train operation, capacity constraints and the resulting impact on running times. They are thus suitable for estimating timetable robustness and stability but applications in real-time traffic control require more detailed modelling and continuous updates of train positions and traffic control actions. Chapter 2. An overview of railway operation planning and control 2.5.4 41 Models for delay prediction in real-time Real-time prediction models can be mesoscopic or macroscopic depending on the traffic control level. Mesoscopic prediction includes predicting the train paths on a detailed level, i.e. each signal passage is predicted. Prediction of train operation with such a level of granularity can be used to detect and resolve route conflicts (Albrecht, 2009) and conflicts due to synchronisation constraints. In order to do so, it is necessary to take into account signalling and interlocking logic that controls train traffic in real time. On the other hand, macroscopic predictions only estimate the realisation time of station events (departures and arrivals) and aim at predicting delay propagation over large networks. Real-time models for traffic state prediction may rely on the actual train positions, speeds, relative train orders and routes. The prediction methods vary from simple extrapolation of current delays to application of methods of statistical learning from historical traffic realisation data. Real-time prediction models presented in this review can be classified according to their granularity to mesoscopic and macroscopic. Berger, Gebhardt, et al. (2011) created a stochastic graph-based macroscopic model for delay prediction. The approach is suitable for online applications where updates about train positions are frequent. Running times are modelled as random variables with probability distributions conditional on departure time and train type. The approach allows testing the model with different types of discrete distributions for running times. By using a set of waiting policies for passenger connections, the future delay propagation of the current delays that are continuously updated can be predicted in the observed part of the network. The model has been applied on a large scale case study in Germany for predicting delays over the entire network over an hours long prediction horizon. Another traffic state prediction approach that relies on the actual state of traffic and exploit it to derive estimates for train running and dwell times is presented by Hansen et al. (2010). A macroscopic model for prediction of train running times is calibrated from historical track occupation data. The robust estimates of minimum running times on the level of open track sections are computed and used to predict subsequent arrival times. The prediction algorithm estimates the realisation time of an event by computing the critical path through the macroscopic graph starting from the last realised event. The main contribution of this data-driven approach is that the estimates of process times reflect phenomena of railway traffic such as the dependence on delay, peakhours, weather, and rolling stock. More detailed traffic prediction models have been in the focus of relevant literature due to their applicability for real-time rescheduling models. L¨uthi and Laube (2007) studied the role of real-time prediction systems in the traffic control environment. In that context their purpose is twofold. The first requirement is prediction of train trajectories until the next controllable point on the network. In other words, train running times to the next station or point in the network that can accommodate reordering or re-routing 42 Models for Predictive Railway Traffic Management need to be computed. That way the accuracy of the rescheduling models, that assume full knowledge of future traffic state, can be improved. The second role is connected to the procedure of finding a new feasible schedule as a result of the rescheduling process. During the optimisation stage, rescheduling systems evaluate different solutions in the search for the optimum. Traffic evolution according to each observed potential solution is predicted and summarised in the corresponding value of the objective function. A mesoscopic graph based rescheduling model (D’Ariano, Pranzo, & Hansen, 2007) that was further extended to distributed control over multiple traffic control area (Corman et al., 2012b), considers the majority of operational constraints of railway traffic. D’Ariano (2008) used the temporal decomposition to apply the model for predictions over a time horizon of several hours. The model predicts the future traffic evolution for each considered rescheduling action. Static arc weights in the graph require an iterative approach to recompute a feasible speed profile of a train based on the train dynamics and detailed infrastructure data. Moreover, running times are estimated based on theoretical values and dwell times based on minimum dwell times, which does not reflect the impact of delays, peak hours and passenger volumes on process times. Fukami and Yamamoto (2001) presented a real-time prediction tool for a high-speed line in Japan. The system follows positions of all trains in the network using train describers messages and estimates the speed of a running train by assuming a constant velocity over track-clear detection sections. The train trajectories until the arrival at the next station are then simulated with respect to all microscopic operational constraints. However, predicted arrival delays at the next station are extrapolated to succeeding stations by a simple parallel shift method. That drastically reduces the complexity of the computationally demanding simulations but also makes the system less reliable for long corridors or complex networks. An online prediction tool has been implemented in the Swiss traffic control system RCS (Dolder et al., 2009). The prediction model is based on a directed acyclic graph, with nodes in the graph corresponding to arrival and departure events at timetable points and signals, and arcs representing precedence relations between nodes corresponding to running, dwell, headway and connection arcs. Arc weights are computed offline by solving the train motion equations using a detailed description of infrastructure and train characteristics. After each train position update, the running times until the next station are computed and a critical path algorithm derives predictions of all event times on the graph. Prediction errors smaller than 1 minute are reported for events within a 10 min prediction horizon. The granularity of the presented macroscopic real-time prediction models is insufficient for applications in traffic control because the interdependencies of train runs on open track sections and in stations cannot be accurately modelled. On the other hand, more detailed models are able to capture route conflicts. However, accurate modelling of train dynamics in route conflicts relies on the computationally demanding procedure for solving train motion equation. Moreover, the predictions are performed based on the pre-computed running and dwell times. Therefore, this method does not consider Chapter 2. An overview of railway operation planning and control 43 the impact that the current traffic conditions, such as delays, peak hours and passenger volumes may have on process times. Finally, the real-time information about running trains is not used to adapt the process time estimates. Such models can therefore not be adjusted online to capture a particular driving style or malfunctioning trains. 2.6 Review of rescheduling models The second research objective of this thesis is concerned with the development of a rescheduling model for network-wide traffic management (§1.4). In Section 2.2.5, capacity restrictions and synchronisation times are presented as the major reasons for delay propagation in railway networks. In the current literature, the problem of rescheduling has been addressed mainly separately for each type of delay propagation. The delay management problem has been defined as deciding whether the planned connections should be kept or cancelled in order to minimise passenger delays and inconvenience in case of disruptions and delays (Sch¨obel, 2007). The more advanced versions of the delay management problem model consider macroscopic capacity constraints in a realistic manner (Dollevoet, Huisman, Kroon, Schmidt, & Sch¨obel, 2014; Schachtebeck & Sch¨obel, 2010). A recent overview of other relevant contributions to delay management is given by Dollevoet (2013). The problem of optimising train traffic with respect to both connection and capacity constraints is too complex to tackle in an integrated formulation. Recently, Corman, D’Ariano, Pacciarelli, and Pranzo (2012a) described a bi-objective optimisation approach where a Pareto front is obtained that minimises secondary delays and the number of broken connections. An iterative approach that integrates a mesoscopic rescheduling tool with a macroscopic delay management model was presented by Dollevoet, Corman, D’Ariano, and Huisman (2013). However, these methods are only able to solve problems not larger than a single traffic control area due to high complexity of the problem and requirements for short computation times. Therefore, for network-wide traffic control, the traffic and delay management problem are still addressed separately. In this thesis and literature review, we focus on real-time traffic rescheduling as a problem of solving a possible schedule infeasibility caused by delays. Various formulations of the rescheduling problem exist and most of them rely on mixed integer linear programming which belongs to a class of difficult NP-hard problems (Schrijver, 1986). The models most commonly contain a binary control variable that models the relative train order on an infrastructure resource or a broken or kept passenger connection. In an extensive review of applicable models and algorithms developed until 2006, T¨ornquist (2006) argues that problem complexity depends not only on the generic formulation but also to a great extent on the actual instance that is modelled. The author states that an increase in the number of binary variables does not automatically imply the increase in complexity due to interdependencies between binary variables and implications of fixing a value for a binary variable. That is important because it shows that the domain 44 Models for Predictive Railway Traffic Management knowledge of railway operations and process dependencies can be employed to reduce complexity of generic optimisation problems. An important property of the real-time rescheduling models is the way they handle uncertainty. The majority of the existing models assume full knowledge of delays and traffic evolution. In contrast to that, Gatto (2007) defined the online version of delay management. In their approach, uncertainty of delay values, that are given as an input to the optimisation procedure, is recognised as a major factor for model accuracy and applicability. For that reason they developed a family of competitive online algorithms that do not anticipate any values of future train delays. Due to high complexity, this approach could only be applied for delay management on a single corridor (Gatto, Jacob, Peeters, & Widmayer, 2007) or a simplified suburban network (Berger, Hoffmann, Lorenz, & Stiller, 2011). A recent contribution to online non-anticipative delay management problem focuses on defining online strategies and waiting policies (Bauer & Sch¨obel, 2014). The main contribution of this work is the design and implementation of a learning strategy for online delay management. Corman and Meng (2014) presented a comprehensive review and classification of rescheduling tools and applications. The relevant models developed in the period 20072013 are classified by the problem scope, model and solution. A special focus in this review was on availability of information on current train positions and traffic state prediction. Models were classified into those where a full deterministic knowledge of future traffic condition is assumed, models with full but stochastic knowledge on future and finally, models with continuous updates of the current and future traffic condition. The first two types are described as static (open-loop) models, whereas the third type as dynamic (closed-loop) models. Note that the rescheduling module of the most closedloop models still assume full knowledge of the present and future traffic state. The rescheduling problem is solved after each update of traffic state estimate (prediction). The performance of the closed-loop rescheduling systems depends to a great extent on the reliability of traffic state predictions. In this literature review we discuss the existing contributions that are applicable for traffic control of large networks or traffic control areas. The approaches for optimal control of train traffic over junctions or single-track lines will not be covered. More details on approaches for controlling railway junctions can be found in Rodriguez (2007) and Milinkovi´c, Markovi´c, Veskovi´c, Ivi´c, and Pavlovi´c (2013). Relevant contributions from the field of optimal rescheduling on single-track lines include S¸ahin (1999), Zhou and Zhong (2007) and Meng and Zhou (2011). Since the purpose of this thesis is development of real-time models applicable for implementation in a model-predictive control loop, a special focus in this review will be on closed-loop models. Pellegrini, Marli`ere, and Rodriguez (2014) presented an approach that models train runs on the level of track sections. The model is able to optimise train routes and relative orders in a complex station area using a mixed integer linear programming (MILP) formulation. The authors consider two objective functions: minimisation of the total secondary delay and minimisation of the maximum secondary delay for any Chapter 2. An overview of railway operation planning and control 45 considered train. The model is implemented in a rolling horizon framework. A twenty minute horizon is considered with a ten minutes update frequency. Finally, the model performance is assessed based on three types of disruption scenarios that differ by magnitude and severity. The high granularity of the model prevents applications to larger instances. A different approach to microscopic rescheduling was presented by Caimi, Chudak, Fuchsberger, Laumanns, and Zenklusen (2010) who focused on rescheduling traffic in and around major stations (condensation zones). The approach represents a modification of the train routing problem that was previously presented by Zwaneveld, Kroon, and Hoesel (2001). In the latter approach, train routes are represented by vertices and conflicting routes are connected by arcs. The problem of finding conflict-free routing is then formulated as an NP-hard problem of finding an independent set. Caimi et al. (2010) improved this approach for use in real-time. First they apply time indexing and predefine a number of possible routes (in space and time) for each train. A conflict graph is then created for each used resource (block section). The problem is further formulated as finding an independent set in each resource graph of conflicting routes and solved using integer linear programming (ILP). The same model was recently included in a model-predictive control framework for closed-loop rescheduling (Caimi et al., 2012). An influential MILP formulation of a train rescheduling problem on a macroscopic level was given by T¨ornquist and Persson (2007). Furthermore, the authors defined three heuristics to reduce the search space and decrease the computation time. Each heuristic limits the number of reordering and rerouting actions. All three methods were applied to a case study of a subnetwork in Sweden and performance in terms of optimality margin and computation time was used to evaluate the strategies. High quality solutions in short time were obtained by predefining a number of permitted reorderings relative to the train that suffered primary delay. The most promising strategy was further extended in T¨ornquist (2007) by introducing a parameter that defines the number of permitted reorderings relative to trains suffering secondary delay. Moreover, a comprehensive analysis of different objective functions and different lengths of prediction horizon was presented. T¨ornquist Krasemann (2011) recently introduced a greedy algorithm to tackle the complex cases of the MILP formulation. The idea was to obtain reasonably good feasible solutions in a very short time and use the rest of the predefined computation time to try to improve it by backtracking and reversing decisions made in the first stage. Acuna-Agost and Michelon (2011) presented an extension to the MILP formulation of T¨ornquist and Persson (2007). The model is improved by increasing the granularity to the level of block sections. Furthermore, running time adjustments were implemented in case of route conflicts due to braking and reacceleration. Presented solution approaches include ‘right shift’ retiming, local search and iterative local search optimisation. A local search approach looks for the solutions close to the planned schedule in the search space. An iterative local search improves the solution iteratively until a 46 Models for Predictive Railway Traffic Management time limit or other end criterion is reached. The model is applied on a corridor case study since the increase in model complexity came with a price in computation time. Further improvement of this approach is presented by Acuna-Agost, Michelon, Feillet, and Gueye (2011). By analysing delay propagation in case of primary delays, the authors are able to assign a probability of being affected to each event included in the model. The probabilities are computed with logistic regression considering the prediction variables such as scheduled headway times around the considered event and time window between the primary delayed event and the considered event. The search space can thus be reduced by focusing on events with high probability of suffering secondary delays. Another extension of the T¨ornquist and Persson (2007) problem formulation and a new solution approach was presented by Min, Park, Hong, and Hong (2011). The original model is extended by introducing a constraint that models conflicts in stations between trains that depart to or arrive from different open track sections. On the other hand, the new model assumes infinite capacity of all stations and unidirectional traffic on all lines. By exploiting these assumptions, a decomposition of the problem to a number of separate subproblems is enabled. The authors prove that solving small separate problems in topological order yields near-optimal global solutions. The proposed solution method is column generation that was applied to a case study comprising a large urban network of a metropolitan area. Van den Boom and De Schutter (2006, 2007) presented a way to overcome the limitation of the conventional max-plus models that rely on a fixed structure, i.e., fixed train orders, sequences, and routes. They proposed an approach called switching maxplus linear systems that can be used to incorporate discrete dispatching actions, such as changing the order of trains, cancelling a train or a connection, into the max-plus framework. In their approach, the structure of the timed event graph can be changed. Every change corresponds to a dispatching decision and results in a new structure (mode) which represents a railway traffic model with the specified order of events and synchronization constraints. The system is managed by switching between different modes, thus allowing the inclusion of discrete decisions into the model. They recast the optimal switching problem as an MILP problem and propose a commercial software or metaheuristic algorithms to obtain solutions. The model performance is tested on the intercity network in the Netherlands. Recently, different formulations of the model and the resulting effect on the computation time were considered (Kersbergen, , Van den Boom, & De Schutter, 2013). The explicit formulation of the model where the state vector does not depend on the values in the previous period caused a considerate increase of computation time. A separate stream of research on real-time traffic rescheduling is characterised by the job-shop scheduling formulation of the problem. The general problem can be defined as the problem of assigning a set of machines to a set of competing jobs where a machine can handle only one job at a time. Mascis and Pacciarelli (2002) analysed the complexity of the job-shop scheduling problem with blocking and no-wait constraints. Chapter 2. An overview of railway operation planning and control 47 Blocking constraints imply that a job keeps blocking a machine until the next machine in the sequence becomes available. A no-wait constraint implies that two consecutive operations of a job must be completed without any waiting time in-between. alternative graph (AG) were introduced as a convenient way of modelling job-shop scheduling problems with specific signalling and safety constraints. The resulting formulation was exploited for applications in railway traffic rescheduling by Mazzarello and Ottaviani (2007). The AG representation of the job-shop scheduling problem was used to develop a mesoscopic model for optimal rescheduling of railway traffic. Apart from the standard constraints for three aspect fixed signalling, a way to model green wave policy and moving block signalling system was also presented. The model was calibrated using a speed profile generation module and a heuristic approach for route conflict resolution was presented. The approach was assessed on a case study of a bottleneck part of the corridor in the Netherlands. D’Ariano, Pacciarelli, and Pranzo (2007) presented and efficient branch and bound algorithm to minimise secondary delays in an alternative graph based rescheduling problem implemented in the real-time traffic management decision support system ROMA. The problem is first reduced by exploiting the fact that a relative train order cannot change on open track lines. The heuristic defined by Mazzarello and Ottaviani (2007) is used to compute the initial solution for the branch and bound algorithm. This model is further extended with a speed coordination component (D’Ariano, Pranzo, & Hansen, 2007). The speed profiles of hindered trains are adjusted to model braking and reacceleration. The conflict resolution and speed coordination components are integrated into a closed-loop framework where the feasibility of the solution computed by the conflict resolution part is verified after computing the adjusted speed profiles. The model was applied on a realistic case study of a busy traffic control area. Optimal results for different kinds of disruption scenarios were obtained in a short time. The problem of coordinating two dispatching areas was tackled by Corman, D’Ariano, Pacciarelli, and Pranzo (2010). A coordination level was added to the rescheduling problem that was formulated with a separate alternative graph for each local dispatching area. The coordination level consists of a border graph that is used to verify the global feasibility of the outputs from each separate area. Global infeasibility is solved by adjusting the solutions of the subproblems. This approach was further extended to coordinate multiple traffic control areas (Corman et al., 2012b). Moreover, the coordinator graph formulation was also improved with optimality conditions. An iterative approach is adopted as a way of communication between each subproblem and the coordination level. A branch and bound algorithm was presented that solves the coordination problem. This approach has been tested on a large and complex subnetwork in the Netherlands. The impact of number of dispatching areas and their sizes on the quality of solutions and computation time was analysed. Other approaches with an alternative graph representation of the job-shop scheduling problem include Mannino and Mascis (2009) who created a decision support system for train control in metro stations that is applied in practice. Moreover, Liu and Kozan 48 Models for Predictive Railway Traffic Management (2011) included train priories in their macroscopic model for train scheduling on a single track corridor. Finally, a recent contribution focused on integrating an alternative graph model into a closed-loop control framework (Quaglietta et al., 2013). Reordering actions computed by ROMA tool (D’Ariano, 2008) are implemented in a microscopic simulator (Quaglietta, 2011). The conflict detection component of ROMA predicts the train traffic during a rolling prediction horizon, based on the current train positions obtained from the simulator. Predictions are performed in a time-driven manner after each predefined rescheduling interval. If route conflicts are predicted, a new schedule is computed by ROMA. Table 2.1: Summary of presented approaches for real-time rescheduling Scope Dispatching area DM Gatto et al. (2007) Berger, Hoffmann, et al. (2011) TM D’Ariano (2008) Caimi et al. (2010) Mannino and Mascis (2009) Pellegrini et al. (2014) Subnetwork Sch¨obel (2007) Schachtebeck and Sch¨obel (2010) Dollevoet et al. (2014) Bauer and Sch¨obel (2014) T¨ornquist and Persson (2007) Min et al. (2011) Acuna-Agost and Michelon (2011) Corman et al. (2012b) Table 2.1 summarises the delay management (DM) and real-time traffic management (TM) models presented in Section 2.2.6. The papers are classified by scope of application. Some papers focused on detailed modelling of infrastructure, train dynamics or uncertainty but were able to cope with a problem size limited to a single dispatching area. On the other hand, larger instances are tackled by macroscopic models with a great deal of abstraction. An exception is the work of Corman et al. (2012b) who coordinated multiple detailed models to tackle the traffic rescheduling problem over several dispatching areas. However, the problem of controlling country-wide traffic is still unsolved since the coordination of local areas modelled micro- meso-scopically is computationally demanding, while the presented macroscopic models are applied to subnetworks of a national network or metropolitan area networks. 2.7 Discussion This chapter presented the terminology and basic concepts of railway traffic. The essential problems in the current traffic control practice were identified. Moreover, we analysed the relevant scientific contributions directed at improvement of the current practice. Finally, the gaps in the current research were identified and translated into research objectives of this thesis. The drawbacks of the current traffic control practice in the Netherlands include the lack of intelligent computer-based support to traffic controllers to monitor the traffic Chapter 2. An overview of railway operation planning and control 49 conditions on the network, predict the future train movements, reduce delays and resolve route conflicts in a network-optimal way. Traffic monitoring consists of a delay registration system with insufficient precision. Moreover, the controllers have no support to predict the future evolution of traffic or the consequences of their dispatching decisions. Finally, dispatching actions are made based on the predetermined scenarios and rules-of-thumb which may lead to suboptimal effects on traffic state. For each objective of this thesis a review of the existing literature was performed. Applicability of the existing data mining methods for processing and extracting information from the train describer logs depends strongly on the data structure and information logged by the system. Therefore, a new data mining approach for extracting train event times and route conflicts from the Dutch system TROTS is presented in this thesis. The tool also overcomes the limitations of earlier approaches that were limited to station areas and controlled signals. In the approach presented in this thesis, train paths and route conflicts can be recovered on open track sections. The full list of contributions is presented in Section 1.5.1 and a detailed description is given in Chapter 3. Static estimation of process times irrespective of the current traffic conditions of the network was recognised as the main drawback of the existing approaches. The second objective of this thesis aims at bridging this gap in an accurate and computationally efficient manner that relies on historical traffic data. Running and dwell times are estimated from the TROTS log files without relying on the manually collected data, on-board units or station design data. The detailed description of this approach is given in Chapter 4 and the main contributions are summarised in Section 1.5.1. The analysis of the relevant literature on delay prediction identified the low granularity in modelling, fixed structure of the models, and high computational requirements as the main drawbacks for straightforward application in real-time traffic control. The research in this direction presented in Chapter 5 resulted in a model that can quickly and accurately predict train event times over large areas and long prediction horizons. The considered level of detail is suitable for prediction of route conflicts which is an important requirement to support traffic controllers. The model can be continuously updated with incoming information on train positions or traffic control actions. Finally, the online character of the model allows adjustments of the process time estimates in real-time thus overcoming the limitation of static models with pre-computed process time estimates. Finally, the relevant contributions from the field of real-time rescheduling were discussed from the perspective of their applicability for rescheduling traffic over largescale networks. None of the reviewed approaches is suitable for applications on the network control level. In Chapter 6 we present a macroscopic model with an appropriate level of model granularity that allows application of the previously developed procedure (D’Ariano, Pacciarelli, & Pranzo, 2007) for solving large instances comprising the Dutch national network. 50 Models for Predictive Railway Traffic Management Chapter 3 Process mining of train describer event data This chapter is an edited version of the article Kecman, P. and Goverde, R. M. P. (2012). Process mining train describer event data and automatic conflict identification. In C. A. Brebbia, N. Tomii, & J. M. Mera (Eds.), Computers in Railways XIII, WIT Transactions on The Built Environment (Vol. 127, pp. 227–238). Southampton: WIT Press. 3.1 Introduction Monitoring of current and prediction of the future train positions and delays are important tasks of traffic control, as discussed in the previous chapter. Train describers are a typical way of centralised monitoring of train positions in discrete points in the network. A message is received in the traffic control centre after every available update of train positions. In complex and busy railway networks, multiple train describer messages can be received within a second. Consequently, message archives contain separate large text files for a particular area for each day. Train describer messages and archives were recently recognised as an important source of information about train traffic (Goverde et al., 2008). Real-time stream of messages can be used for monitoring traffic conditions in the network. At the same time, archives of train event messages provide the necessary information for an ex post analysis of the realised traffic. Efficient data processing tools are thus necessary that are able to extract the information from a train describer message in real time and update the train positions, actual delays and register route conflicts. Likewise, the tool has to be able to quickly process the large archives and retrieve all relevant information about train traffic in a certain area and period. The necessary information include: realised 51 52 Models for Predictive Railway Traffic Management running and dwell times, headway times between trains in bottlenecks, actual arrival and departure times, and delays (primary and secondary). Automatic identification of route conflicts is an important requirement. A case study on a busy corridor in the Netherlands showed that 55% of arrival delays exceeding 3 minutes are caused by route conflicts (Daamen, Houben, Goverde, Hansen, & Weeda, 2006). Registered delays at stations cannot be with certainty attributed to route conflicts, therefore, it is difficult to identify and analyse them. Typically, train delays at stations are monitored and registered online using train detection at main signals and timetable databases, but the accuracy is insufficient for process improvements. Railway operations thus require a feedback of operations data to improve planning and control. Accurate data on the level of track sections and signal blocks can be used to gain a better understanding of the realized train paths and conflicts between them. In an earlier work, Daamen et al. (2008) developed algorithms for automatic route conflict identification based on data records of the Dutch train describer system TNV, which were implemented in the tool TNV-conflict. The TNV system has recently been replaced by the new train describer system TROTS which contains an essential new approach to train number steps. This came with a new format for the log files. In particular, train number steps are no longer given with respect to a route block to a next signal, but at section level. This means that a train number step does not predict to which signal the train is heading, as was customary with TNV. Thus, we cannot look ahead at the aspect of the signal at the end of a block to identify a conflict. The algorithms described in Daamen et al. (2008) had to be modified in a way described in this chapter. In this thesis, a process mining (Van der Aalst, 2011) approach is applied on the log files of the Dutch train describer system TROTS. The resulting tool recovers and visualizes the realized train paths, blocking times, and route conflicts, and thus provides essential information for analysing railway operations that can be used for fine-tuning the railway timetable and operational processes or development of data-driven models. The tool supports a tabular output for statistical analysis, as discussed in Goverde and Meng (2011), and visualizations of the realized time-distance and blocking time diagrams with highlighted route conflicts. A separate procedure is presented for identification of route conflicts suffered by departing trains after a scheduled stop because extended dwell time of a train cannot directly be attributed to a route conflict. Moreover, several improvements to earlier approaches have been implemented in terms of retrieving traffic information from open track segments where traffic is controlled by non-logged automatic block signals. Blocking times over open track sections are determined so that route conflict identification is applicable over entire corridors, including ‘dark territories’ with aggregated track sections. The remainder of the chapter is structured as follows. Section 3.2 describes the methodological framework of the process mining tool and formalizes the blocking time theory as the employed process model. Furthermore, Section 3.3 explains the system architecture, data structure and drawbacks of the Dutch train describer system TROTS. Chapter 3. Process mining of train describer event data 53 Section 3.5 explains the process mining algorithm and subroutines. A case study with a description of the graphical user interface is given in Section 3.6. Finally, Section 3.7 gives a brief summary and presents further application of processed train describer data in the data-driven prediction tool. 3.2 3.2.1 Methodological framework of the process mining tool Process mining Process mining is a method for discovering processes and extracting information about them from event data using a process model (Van der Aalst, 2011). It combines data mining with domain knowledge about the specific processes that are analysed. The principle idea of the concept is to extract the necessary information from large data sets and obtain an output containing clean and structured data ready for analysis. Figure 3.1 presents the background of process mining. Recent advancements in sensor technology and telecommunications enabled continuous monitoring of processes in complex systems. The corresponding software systems often store the event messages and measurements of processes in event logs. Using a process model, which is built based on the domain knowledge of the real-world system, event logs can be searched and relevant processes can be discovered and retrieved. This recently developed data mining paradigm has been applied successfully in analysing business processes (Van der Aalst et al., 2007) and activities in social networks (Medeiros, Weijters, & Van der Aalst, 2007). In this thesis, we apply this method to mining historical train describer event data, recovery of train paths and identification of route conflicts. The following section gives more details about the process model, whereas the event log files of the Dutch train describer system TROTS are described in Section 3.3. 3.2.2 Process model The process model is built according to the principles of signalling systems (§2.2.2) and timetables (§2.2.1). A three-layer model of railway traffic is used to represent railway operations on multiple levels. A microscopic traffic model represents a train run on the level of track-clear detection sections. Each track section occupation and release represents an event. A section is occupied between the occupation and release events. By keeping track of section events, the actual position of the head of the train is determined at every occupation event and the position of the rear of the train at every release event. On the mesoscopic level, a train run is modelled as a sequence of signal passages. Signal passages are events that initiate processes such as blocking a part of the infrastructure and running over a block. The topology of the signalling and interlocking system 54 Models for Predictive Railway Traffic Management Real world process Software system knowledge principles operational rules event messages measurements Process model Event logs Process mining Figure 3.1: Process mining framework provides a way to directly map the microscopic layer of the model to mesoscopic level. A block is occupied when a train occupies the first section in the block. Similarly, a block is cleared by a train when the last axle of the train clears the last section in the block. Therefore, detailed knowledge of the infrastructure layout provides a way to build the mesoscopic model from the microscopic model in a straightforward manner. Blocking time theory provides the logic implemented in this layer of the process model in order to identify route conflicts. Figure 3.2 emphasises the events and processes in micro and mesoscopic models. A train run over a track or block section can be represented by occupation (o) and release (r) events that are connected by running and clearing processes. time o running o r clearing Figure 3.2: Events and processes in micro and mesoscopic models A macroscopic traffic model is the top level of the multilayer model. It represents a train run over the network as a sequence of departure and arrival events separated by Chapter 3. Process mining of train describer event data 55 running and dwelling processes. A similar bottom-up approach can be used to create this layer from the low-level microscopic model. A way to do that is by incorporating the station layout and topology of track circuits. Departure and arrival events can than be determined by occupation or release events of the relevant track sections. More details about the exact method for determining arrival and departure times employed in this thesis is given in Section 3.5.6. Processes in this level are the train runs between two scheduled stops and dwellings in stations. The superimposed three-layer model of a train run between two stations is depicted in Figure 3.3. Using the topology of the signalling system, the occupation and release times of block sections can be directly derived from occupation and release times of individual sections. Moreover, the station layout allows attributing platform section events to be arrival or departure events which allows direct computation of delays. Note that all boxes in the figure represent only physical occupation times of the corresponding infrastructure elements. In order to enable identification of route conflicts using blocking time theory, the mesoscopic model needs to be extended with additional times as described in Section 2.2.3. Station B Station A time A Micro D Meso D A Macro Figure 3.3: Three-layer process model 3.3 3.3.1 The Dutch train describer system System architecture The train describer system in the Netherlands TROTS keeps track of train movements at discrete points in the network and monitors the state of infrastructure elements such as track sections, signals and switches (ProRail, 2008). The system contains two components. The first component keeps track of the status of infrastructure elements. Every change caused by a train (section occupation or release), signalling system (signal 56 Models for Predictive Railway Traffic Management change to ‘stop’ or ‘proceed’) or traffic controller (switch position or signal change) is registered and logged as an infrastructure message. The second component keeps track of train number steps. A train is identified by its number which is inserted manually by the traffic controller before (or just after) the first departure. TROTS assigns each track section occupation or release to the train number that caused it. These messages are logged as train number steps. The Dutch railway network is divided into 13 TROTS areas. Each area comprises one or more major stations with complex topologies and 30–40 km of surrounding railway infrastructure. A communication protocol between the systems enables coordination between adjacent areas so that a train number inserted in one area is transmitted to other areas that the train crosses on its route. In order to reconstruct the train traffic over multiple TROTS areas, it is necessary to merge the corresponding log files. TROTS log files are archived per day and area in large files of TROTS format of approximately 75 MB. An important improvement compared to the earlier TNV system (Goverde, 2005; Goverde & Hansen, 2000) is that the train number step messages are coupled to track section messages. Another modification is that the current train route from signal to signal is no longer available in the system. Therefore a train step message does not predict the destination signal any more. At any moment, only the past train step and infrastructure messages are known which motivated a modification of the existing algorithms for recovery of train paths and route conflict identification. 3.3.2 Data structure and information contained in log archives TROTS generates train number messages and infrastructure messages and logs them into a comma separated values (CSV) file. The content of a train number message is given in Table 3.1. Each successive train number step message contains either a new occupied track section at the front or a released track section at the rear. Table 3.1: Train number messages generated by TROTS Filed 1 2 3 4 5 6 7 Message content Time stamp Message code Type: insert, remove, train step Running direction km position of last section km position of next section All occupied sections Infrastructure messages contain binary values that indicate the state change caused by a train, signalling system or a signaller. The following information is contained: time stamp, event code, element type (section, signal, switch), element name, and new state Chapter 3. Process mining of train describer event data 57 (‘occupied’/‘released’, ‘stop’/‘go’, ‘left’/‘right’). The event code of a train number step corresponds to a section message with the same event code. This coding is used to match a message about a section occupation or release with a message of a train number step. A train run can therefore be tracked along the route on the level of track sections. Infrastructure element names are given in form of a string that contains the station area code and a numerical element identifier (e.g. section RTD$282T). 3.3.3 Shortcomings in TROTS log files There are several issues in the TROTS log files that represent a potential source of inaccuracy and complicate the discovery of processes defined in Section 3.5 and subsequent performance analysis. • Time-lag between infrastructure and train messages. The system architecture (§3.3.1) reveals that infrastructure messages and train number step messages are generated by different components of the system, which sometimes results in a significant difference between the time stamps of the corresponding messages. Analyses show random delays of up to 7 seconds of the train number step messages. In order to avoid the possible inconsistencies, the developed tool does not use the time stamps of the train number step messages but only the ones of the corresponding infrastructure element messages. This imperfection of the train describer system is in the remainder of this thesis considered as a source of unavoidable errors with limited effects on the overall results. • Signal messages cannot be coupled to train step messages. Infrastructure messages of a signal aspect change to ‘stop’ cannot be coupled directly to any train number step or section occupation message. Therefore, the current data structure of the log files does not allow identification of the signals passed along a train route. • Automatic block signals are not logged. Without intermediate logged signals, an open track segment between two stations can be regarded as one block from the exit signal at the station of departure to the home signal at the station of arrival. Thus, route conflicts between successive trains on an open track section cannot be identified. • Track sections on open track are aggregated. Moreover, multiple track sections can be aggregated within the TROTS system. They are reported to be occupied and released at the same time as a group. If a signal is located in the middle of such group, the system misses the occupation and release times of associated track sections. • Scheduled stops cannot be identified. In the infrastructure messages in TROTS log files, no distinction can be made between platform tracks and other track sections. Therefore, scheduled stops can not be detected in a straightforward way. 58 3.4 Models for Predictive Railway Traffic Management Traffic monitoring on open track and in stations The previous section presented the data structure and information contained in the TROTS log files. The train positions are reported with each occupation and release of a track section. However, the traffic performance indicators such as, actual delays, route conflicts, realised running and dwell times and headways require monitoring of the signal passing and station events (departures and arrivals) of each train. The essential requirement for developing a tool for monitoring the traffic conditions on the network is to overcome the listed limitations of TROTS related to signal and section messages on open track and in stations. 3.4.1 Associating signal messages to train number steps In order to determine train blocking times and identify route conflicts, signal passing times for each train need to be known. Level of information in TROTS log files does not allow straightforward identification of the signals passed on a train route. Signal messages only indicate if the signal aspect is ‘stop’ or not. Signal change cannot be associated to a passing train or a traffic control action. This can be overcome by creating an additional input file that lists each signal together with the first section it protects and the section that releases it. This enables a bottom-up derivation of the mesoscopic model from the microscopic level. The corresponding input file can be created automatically by data mining the TROTS log archives in a preprocessing step. A pattern discovery algorithm has been developed that finds events that frequently occur together within a predefined time window. We look for section occupation messages from the corresponding station area, that are registered shortly before or after a message reporting signal aspect change. A 10 seconds wide window was used that is moved and positioned around each message that reports a signal aspect change to ‘stop’. All relevant track section occupation messages within the window are noted. After parsing a significant number of messages for each signal, the section that most often gets occupied within the moving time window is registered as the section protected by the signal. We use this input in the main algorithm to identify the signal passing time of a train number via the corresponding section that got occupied. Figure 3.4 presents a screen shot of a TROTS file that illustrates the procedure of coupling signals with the sections they protect. A time window around the selected message is presented (highlighted). The message reports a change of signal DT$72 to ‘stop’. A message reporting occupation of a section within the same area (DT$71BT) is discovered (boxed) and can be coupled to train number 2122 using the identical message tag (BM3359596). Chapter 3. Process mining of train describer event data 59 Figure 3.4: Screen shot of a TROTS log flle 3.4.2 Logging of automatic block signal passing events The major limitation of using TROTS event logs for traffic monitoring is that events at automatic block signals on are not logged. This means that the train blocking times and route conflicts on open track section cannot be determined form the TROTS data. Moreover, aggregation of track sections on open track additionally complicates the train path recovery if a signal is located in the middle of the group. An additional input, containing a list of automatic block signals, interdependent signals and sections and a list of aggregated sections is therefore required for keeping track of train runs on open track. Using the additional infrastructure data, passing times and aspects of automatic block signals can be determined based on the section occupation and release messages. In order to simulate the three-aspect fixed-block signalling based on track section messages, each signal in the list needs to be connected with dependent track sections and signals. The list therefore contains each block on the open track section defined with delimiting signals and comprised track sections. Occupation time of the first section in a block can be interpreted as the switching time of the signal that protects the block to ‘stop’. Similarly the release time of the last section in the block corresponds to signal change to ‘proceed’. For distinction between signal aspects required to identify route conflicts, interdependence with the neighbouring signals is used. 60 Models for Predictive Railway Traffic Management Passing times of signals that are located within the aggregated section groups cannot be determined explicitly by relying on the TROTS section messages. In order to overcome this, a linear interpolation procedure can be used. The running time over each section is computed as a fragment of total running time over the aggregated group proportional to the section length. This approximation assumes constant speed over the aggregated section. After the running time over each section in the group is approximated, the three-aspect fixed-block signalling logic can be implemented in the way described above. 3.4.3 Logging of station events An important aspect of traffic monitoring is keeping track of actual delays, running and dwell times. Since TROTS log files make no distinction between platform tracks and other track sections, an additional data source is required to recognise station events from the event logs. In order to determine the exact departure and arrival times for a scheduled stop of a train, a list of platform sections in each considered station is necessary. Moreover, a timetable, which indicates the trains that are scheduled to stop in each station, is required. A registered occupation or release of a section in a station is a candidate event that represents an arrival or departure of the train. The exact procedure to estimate the times of station events from the list of candidate events is described in Section 3.5.6. A timetable with scheduled arrival and departure times for each train is also needed to compute delays. By comparing the realised with the scheduled event times, the actual delays can be computed and updated. 3.5 3.5.1 Train route recovery and route conflict identification Process mining train describer data The concept of the process mining tool is presented in Figure 3.5. An important property is that this method can be applied both for processing a live stream of incoming train describer messages for the purpose of monitoring traffic, and processing archives of log files to extract the structured historical traffic realisation data. The core of the tool is an environment containing section, signal, block and train objects. All objects are created and updated on-the-fly while parsing a TROTS log file using the described infrastructure and timetable files. Static attributes in each object are fixed when objects are created, using additional infrastructure and timetable input files described in the previous section. Section objects are attributed by name, platform flag that describes platform sections, open track (OT) flag that indicates aggregated track sections on an open track, and signal that protects Chapter 3. Process mining of train describer event data 61 the section (only for the first section in a block). Signal objects are described by name, protected section (first section of the protected block) and previous section (last section of the previous block). The static attributes of block objects include the delimiting signals of the block and comprised track sections. Finally, each train is described by the corresponding object using the attributes number and timetable that contains the list of scheduled arrival and departure times in stations. TS1 TS2 S1 TS3 S2 TROTS Block Start signal End signal Sections Train list Train Number Timetable Section list Signal list Section Name Platform Signal OT Train list Signal Name Protected section Train list Stop/go list Process model Output Realised delays of scheduled events Realised train paths (time-distance) and blocking times Realised rоute conflicts Figure 3.5: Process mining TROTS data Each infrastructure object keeps track of occupation, release and passing times of all trains that are reported by the train describer system. This data is stored in the train list of the corresponding object. The dynamic list ‘Stop/go’ in a signal object is updated with every signal message. Each row in the list contains the time of an aspect change to ‘stop’ and subsequent change to ‘go’. A Train object is attributed with the lists of traversed sections and signals, that are updated with every message from the log file related to the train. Information passing between different object classes and methods within the same class reflect the operational constraints of railway traffic such as route setting and release principles and train separation on open track according to blocking time theory. The output of process mining includes, realised running and dwell times, route conflicts, realised departure and arrival times, as well as realised train paths (blocking time or time distance diagram). 62 3.5.2 Models for Predictive Railway Traffic Management Main algorithm For the purpose of this study, we simulate the real-time environment by parsing the chronologically ordered messages line-by-line. Since the time lag between two successive messages is often less than one second, an efficient algorithm is needed to extract the relevant information from each message, update the corresponding objects and identify a route conflict or determine the actual event time. The relevant data from the log files are saved in the infrastructure and train number objects which enables the algorithm to revisit them, and use and update the information therein (Daamen et al., 2008). At any moment in time the values of object attributes provide the current traffic state with all train positions, actual delays and infrastructure availability. The major advantage of this approach is the data flow between objects. The subroutines depicted in the main loop of the algorithm (Figure 3.6) are implemented as methods for the correspodning object classes. They are able to compare the values of the corresponding attributes for the relevant objects and identify route conflicts, process times, and the actual arrival and departure times. The algorithm first reads each line of the log file and updates the corresponding object. TROTS logs each section infrastructure message followed (one or more messages later) by the corresponding train step message. Thus after a train step message is received, the necessary information about the section event is complete, i.e. time stamp, section name, train number and event type are known. Signal passages can be registered using the predefined interdependence between signals and protected sections (§3.4). The first step of information processing is to update the dynamic attributes of the corresponding objects in the subroutine ‘updateObj’.The remainder of the algorithm contains subroutines that reflect the fixed-block signalling principles in order to detect route conflicts and identify hindering trains. Sections that are relevant for identification of route conflicts are the sections interdependent with main signals. Occupation of the first section after each signal and release of the last section in a block initiate the aspect changes of the corresponding signals. These events are used for identification of route conflicts. For occupations of other sections, the procedure ‘identifyHindering’ is activated to identify the hindering train if a route conflict has been identified at the previous signal passage. The branch for an open-track section is applied for the non-logged signals, that are processed by the subroutine ‘logSignal’. The signal information is further treated in the same manner as for logged signals. Other (non open track) relevant sections are processed if an occupation message is received. The subroutine ‘routeConflicts’ checks if the train is hindered by an earlier train. For departing trains a similar subroutine ‘departureConflicts’ is activated after the exact arrival and departure times were determined by the ‘getEventTimes’ subroutine. For all identified conflicts a subroutine ‘identifyHindering’ is activated. All subroutines are explained in detail in the following subsections using the toy network depicted in Figure 3.7. The network consists of signals S1–S4, open-track sec- Chapter 3. Process mining of train describer event data 63 read line identifyHindering N updateObj occupied? Y N relevant? Y Y N open track? N occupied? released? N Y Y N departure? Y getEventTimes logSignal routeConflict departureConflict Figure 3.6: Flowchart of the process mining algorithm 64 Models for Predictive Railway Traffic Management tions TS1–TS4, platform track sections TS5–TS7 and section TS8 in the interlocking area. Sections TS1 and TS2 are aggregated into TS1/TS2 and aspect changes of signals S1 and S2 are not logged. TS1/TS2 S1 TS3 S2 TS4 TS5 S3 TS6 Platform TS7 TS8 S4 Figure 3.7: Example network 3.5.3 Process discovery The discovery of processes on a micro and mesoscopic level is performed by subroutines ‘updateObj’ and ‘logSignal’. For each infrastructure and train message, ‘updateObj’ creates a new object or updates the dynamic attributes in the corresponding object. Section and train step messages can be directly connected using the unique message code and together they carry the complete information about a section event. Signal messages are only used to update the ‘Stop/go’ attribute of Signal objects since a signal aspect of a controlled signal can be changed by traffic control and not only by a passing train. The problems of non-logged signals and aggregated sections on open tracks are resolved with the ‘logSignal’ subroutine. For each message reporting a release of the first or last section in a block on an open track, this subroutine updates the corresponding Signal and Train objects. The time of the aspect change to stop is equal to the occupation time of the first section (e.g. in Figure 3.7 S2 changes aspect to ‘stop’ when TS3 is occupied). Similarly, the corresponding object of a non-logged automatic block signal is updated with an aspect change to ‘go’ at the time of release of the last section in a block (S2 changes to ‘go’ when TS4 is released). Aggregated sections are occupied and released at the same time (e.g. for TS1/TS2, TS2 is occupied when a train occupies TS1, and TS1 is released when a train releases TS2). In order to estimate the time of the aspect change of S1 to ‘stop’ (or previous signal to ‘go’) we need to estimate the actual occupation time of TS2 (release time of TS1). The procedure is performed at the moment of release of the aggregated segment so the total running time is known. We exploit the assumption that trains move with constant speed over the aggregated sections on an open track. The running time over TS1 is computed as a part of the total running time proportional to the length of TS1. The presented subroutines perform full route recovery of a train run on a micro and mesoscopic level. Moreover, by keeping track of section occupation and release times, as well as signal passing times, the realised headway times between successive departures, occupations of critical track sections and signals passages are implicitly determined. Chapter 3. Process mining of train describer event data 3.5.4 65 Automatic identification of route conflicts This subroutine is activated for every signal passing event. Train separation principles described in Section 2.2.3 are incorporated in this subroutine. If a train did not have a scheduled stop in the previous block, the algorithm checks if a route conflict exists, i.e. if the current signal (e.g. S3 in Figure 3.7) displayed ‘stop’ at the moment when the train passed the previous signal (S2). The time stamp of the approach signal message is modified with a constant value of 12 seconds, representing the sight and reaction time. If a route conflict is identified, the time of conflict is the passing time of the approach signal (S2) by the hindered train. Identification of route conflicts suffered by a departing train requires a different procedure. We assume that the departing train was hindered if the exit signal was showing ‘stop’ at the earliest possible departure time. It is considered that the earliest departure time equals the scheduled departure time if a train had the arrival delay that is smaller than the dwell time buffer. Otherwise, the earliest departure time is after the minimum dwell time has passed since the arrival time. The subroutine ‘departureConflict’ lists potential outbound route conflicts. However, extended dwell times in stations cannot directly be explained by route conflicts. In order to exclude the trains that waited for a feeder train to realize a connection, or the ones that had an extended dwell time for some other reason, additional information from signallers and dispatchers is necessary. If a route conflict is identified, the time of conflict is the earliest possible departure time of the hindered train. 3.5.5 Identification of hindering trains After a route conflict has been identified, the subroutine ‘identifyHindering’ identifies the hindering train. As the hindered train progresses along the block protected by the signal of conflict (e.g. S2 in Figure 3.7), the algorithm compares the previous release times of each section (TS3 and TS4) with the time of conflict (as defined in the previous section). The train that released the section after the time of conflict is the hindering train. Identification of the hindering train completes the necessary attributes for a route conflict object (Figure 3.1). Recall that the conflict duration is defined as an overlap of blocking times of two trains (§2.2.3). For departure conflicts, the conflict duration is a period between the earliest possible departure time and passing time of the exit signal. 3.5.6 Estimation of departure and arrival times This subroutine derives the realized arrival and departure times from TROTS log files. This corresponds to discovery of macroscopic processes, running and dwell times. When a train passes the exit signal (S4 in Figure 3.7) after a scheduled stop (i.e., a message reporting the occupation of the first section after the exit signal is received) a list of times of all section occupations and releases in the platform block (including 66 Models for Predictive Railway Traffic Management the passing time of the exit signal) is created (sections TS5–TS8). Note that not all release times of platform sections are recorded by the time the train passes the exit signal, however that does not affect the method we propose. The period of standstill is determined as the longest time gap between two successive events. The time of the last section message before the standstill is set as the arrival time and the time of the first section message after the standstill is the departure time. This method is an improvement of the current practice in the Netherlands which relies on the measurements of signal passing times adjusted with a fixed correction term (§1.2). Moreover, the approaches presented by Longo et al. (2012), Stam-Van den Berg and Weeda (2007) and Richter (2013) are based on occupation times of predetermined sections. The approach described above is a generic procedure that relies not only on section occupation times but also on section releases times which increases the precision of estimates and requires only train describer data as input without exact knowledge of station topology. The error of arrival (departure) time estimates depends on the number of platform track sections and the distance between the stop location of the rear (front) of the train and the used section border. 3.6 Process mining tool The algorithms discussed in the previous section are implemented in a software tool for processing TROTS log files developed in MATLAB. The tool is able to process large sets of historical data and extract the relevant processes, route conflicts and delays. Moreover, for analysing traffic in particular instances, i.e. station areas or corridors during a specific time interval, a graphical user interface (GUI) has been developed that simplifies selection of a particular instance and provides the graphical and tabular output. 3.6.1 Case study This section illustrates the application of the presented algorithm in an ex post traffic analysis for the TROTS areas The Hague and Rotterdam in the Netherlands. The area comprises the busy corridor Leiden–The Hague HS–Rotterdam–Dordrecht and surrounding tracks. Figure 3.8 shows the macroscopic layout of the observed area with indicated large stations. Since the messages from each TROTS area are logged into separate files, analysis of traffic over multiple areas requires merging the files while maintaining the chronological order of the messages. The performance of the tool is demonstrated by processing a data set for one day of traffic. The algorithm parses the merged files and reconstructs the realized train paths of 2048 trains on micro, meso ad macroscopic level. Moreover, all occupation times of 1396 track sections and all blocking times of 733 blocks are determined, as well as the aspect changes of 624 signals and the arrival and departure time estimates of all trains at 21 stations. Finally, 1011 route conflicts are identified. The time required to Chapter 3. Process mining of train describer event data 67 Leiden The Hague CS The Hague NOI The Hague HS Delft H. v. Holland Schiedam Rotterdam Dordrecht Figure 3.8: Observed area for the case study process all 600 000 messages, describing the traffic over one complete day, was around ten minutes. 3.6.2 Graphical user interface In order to simplify the analysis of the output, a GUI has been created (Figure 3.9). The left part of the GUI contains tabbed panels for data loading (top left panel), visualization control (top right) and displaying results in tables (lower panel). The right part of the GUI is reserved for the visualization of traffic in either time-distance or blocking time diagrams. The tab panel for loading data enables the user to either load the raw data and start the algorithm or load already processed data and display the results. In the lower tab panel, the user can choose which results to display. In the tab Trains (Figure 3.10), a train line can be selected from the pop-up menu which enables selecting a train number from the chosen line. We can then select the whole train path or a part of it by selecting a start and end station. The results are then displayed in the tables on the left and the visualization panel on the right. The selected part of the train route is visualized together with all other trains that operated on the selected corridor 15 minutes before and after the selected train. The tables represent the list of conflicts in which the selected train participated, the running times on all sections, the blocking times, and actual arrival and departure times and delays at all stations. The panel Infrastructure (Figure 3.11) enables the user to choose the corridor and the 68 Models for Predictive Railway Traffic Management Figure 3.9: Graphical user interface Figure 3.10: Train selection panel Chapter 3. Process mining of train describer event data 69 time interval and get the corresponding list of conflicts, list of sections, signals, blocks, and stations that were utilized by trains on the corridor within the selected time interval. Selection of the infrastructure element from the corresponding pop-up menu displays all the state changes of that element with the associated train number and time instants (in seconds from midnight). Figure 3.11: Infrastructure selection panel The visualization control panel (upper right panel Figure 3.9) enables the user to switch between the blocking time diagram and time-distance diagram of traffic on the selected corridor and time interval. Also it is possible to turn on/off the zoom and pan tools and rotate the axis of the diagrams. Finally the selection of the check-box Scheduled also visualizes the scheduled train paths. Figure 3.12 shows the time-distance diagram on the busy corridor between The Hague HS and Rotterdam in the Netherlands between 9:00 and 9:40. The number of tracks between the stations is indicated (the number of lines between station name abbreviations on the left side of the figure indicates the number of tracks) as well as the conflicts (red squares on the hindered train path at the location of the signal of conflict). Intercity trains are presented in blue colour and local trains in magenta. Many minor disturbances are captured in the figure. Moreover, a major disruption, possibly due to a broken train just after departure from station Rotterdam (RTD) is visible. Finally, a non-scheduled overtaking of train 5133 by train 9220 occurred in station delft (DT). Closer analysis of the route conflicts requires a representation of the traffic situation using blocking time theory. Figure 3.13 displays the corresponding blocking time diagram for one direction that appears after selecting the appropriate radio button on the visualization control panel. 70 Models for Predictive Railway Traffic Management Figure 3.12: Time distance diagram Overlaps in blocking times indicating conflicts are denoted in red colour. Note that trains on parallel tracks of four-track lines may overtake each other. Blocking times that appear to be overlapping but are not shown in red are non-conflicting parallel processes. The figure shows a departure conflict of train 5133 in station Delft. The train was hindered due to the unplanned overtaking. Moreover, a sequence of route conflicts is captured in station Schiedam (SDM) where trains had to wait for an available platform track. 3.7 Conclusions This chapter presented a tool for recovery of train paths and automatic conflict identification based on process mining of train describer data. Historical archives from the Dutch train describer system TROTS were used for developing the algorithms. The drawbacks of TROTS data for performance analysis have been overcome by including additional input containing the necessary infrastructure and timetable data. The algorithms have been implemented in a software tool for data processing and performance analysis. The tool provides flexibility in analysing particular train paths and traffic on the corridor. Visual and tabular output simplify analysis and highlight severe disruptions as well as minor disturbances as a result of variability of process times. The process mining method allows application of domain knowledge to data understanding and extracting the relevant information which are the first essential steps for any data-driven application (Fayyad, Piatetsky-Shapiro, Smyth, & Uthurusamy, 1996). Moreover, model-based processing allows anticipation of a future event and thus sim- Chapter 3. Process mining of train describer event data 71 Figure 3.13: Blocking time diagram plifies the detection of errors in the data by detecting an unexpected event. This is of great importance for real-time applications that do not rely on clean and structured data, but noisy live data streams that need to be quickly processed. Straightforward applicability for other train describer systems strongly depends on their data structure. However, using the principles of blocking time theory as a process model in mining the event log files is a generic method for analysis of running times and dwell times, and identification of route conflicts for fixed block signalling systems. Potential developments are mainly directed towards automatic analysis by providing useful statistical indicators for structural flaws in the timetable, as well as detecting severe disruptions and identifying primary delays, see also Goverde and Meng (2011). Finally, the presented approach is important for development of the data-driven models discussed in the following chapters. The tool is used to process a large set of historical data comprising three months of traffic in Rotterdam and The Hague TROTS areas. Chapter 4 presents statistical models for analysing the processed data and deriving robust estimates of process times. Moreover, the tool has been applied in the simulated real-time environment for monitoring train positions, infrastructure availability and actual delays (Chapter 5). 72 Models for Predictive Railway Traffic Management Chapter 4 Data analysis and estimation of process times This chapter is an edited version of the article Kecman, P. and Goverde, R. M. P. (2013). Calibration of a data-driven railway traffic prediction model. In T. Albrecht, B. Jaekel, & M. Lehnert (Eds.), Proceedings of the 3rd International Conference on Models and Technologies for Intelligent Transport Systems 2013 (pp. 459–469). Dresden: DUTpress. Submitted to Public Transport 4.1 Introduction Processed historical traffic realisation data can be used to derive robust estimates of process times. The tool for processing raw train describer data, presented in the previous chapter, discovers train running times over track sections, blocks, and between scheduled stops. Dwell times at scheduled stops are also discovered, as well as all route conflicts and their duration. An important property of the resulting data structures is that they contain indicators of the traffic state for every discovered process. This enables analysis of processes depending on a particular train line, time of day and actual train delays, which is described in this chapter. The work presented here is carried out as a contribution to the second requirement in development of the tool for monitoring and traffic state prediction (§1.4.1). Accurate estimation of running, dwell and headway times is important for all planning and control levels of railway traffic. The validity of capacity analysis and planning at strategic and tactical level depends to a great extent on the accuracy of process time estimation (Abril et al., 2008; UIC, 2013). Similarly, timetabling models assume 73 74 Models for Predictive Railway Traffic Management full knowledge of process times. Moreover, stochastic models developed in order to improve timetable robustness (B¨uker & Seybold, 2012; Medeossi et al., 2011) rely on probability distributions of process times derived from historical traffic realisation data. Finally, on operational control level, process times are estimated for real-time traffic prediction and conflict detection (D’Ariano, Pranzo, & Hansen, 2007; Dolder et al., 2009), as well as to provide reliable passenger information (Berger, Gebhardt, et al., 2011). The relevant processes in railway traffic and earlier attempts in estimating their duration are described in Section 2.4. The essential drawback of the existing approaches for estimation of running times (Br¨unger & Dahlhaus, 2008; Dolder et al., 2009) and dwell times (Buchmueller et al., 2008; Stam-Van den Berg & Weeda, 2007) is that they do not consider the actual traffic conditions on the network at the moment of estimation. In other words, process time estimates do not differ depending on the time of the day, train positions or delays. An initial work in overcoming this problem was presented by Van der Meer et al. (2010). However, the macroscopic character of the model prevents estimation of train runs on the level of block sections, which is essential for prediction of route conflicts. In this chapter we analyse the processed historical data and build statistical models for estimation of process times. Two approaches for deriving process time estimates are examined. The global approach consists of a generic statistical model applied on the aggregated set of historical data. The data about all running times on the level of block sections and all dwell times are aggregated and used to train and validate the statistical models. A set of predictor variables is identified for the purpose of building the global model. A series of advanced supervised learning methods is used for computing accurate process time estimates. The accuracy of the robust linear regression model is improved using state-of-the-art tree-based regression methods (Hastie, Tibshirani, & Friedman, 2009). On the the other hand, multiple local models are developed that estimate process times for particular blocks, stations and train lines. Each local model represents an independent statistical model. Robust linear regression is used to neutralise the impact of outliers which is important for real-time applications that need to process noisy data, as well as missing values (Rousseeuw, 2005). The train describer log files are the single information source for developing the models. The two approaches are compared by their accuracy and applicability for calibrating the real-time traffic prediction model described in Chapter 5. The methodology used to build the statistical learning models is described in the following section. Section 4.3 describes the advanced supervised learning methods for estimation of conflict-free running and dwell times followed by their implementation in the global (§4.4) and local model (§4.5). Model validation is presented in Section 4.6. Finally, the main conclusions and recommendations for further research are given in Section 4.7. Chapter 4. Data analysis and estimation of process times 4.2 4.2.1 75 Methodological framework for statistical analysis Description of the data set For this study, a set of track occupation data comprising TROTS archives for three months (March – May, 2010) from the areas Rotterdam and The Hague (Figure 3.8) was made available by ProRail. The raw data archives are processed with the process mining tool (Chapter 3) resulting in running, dwell, blocking and headway times of all trains. The data archives from 82 days was used to train and calibrate the statistical models (training set). The remaining 10 days of data was used to test and compare the prediction accuracy of the models (test set). In the processed files, running times are given on the level of block sections. An important aspect is the distinction between conflict-free runs and hindered train runs. Hindered train runs are filtered out and only conflict-free running times are included in the data set. Dwell times at each station are provided using the method described in Section 3.5.6. 4.2.2 Global model The global model for process time estimation aggregates the recovered process times of all trains into a set of running times and a set of dwell times. A separate model is created for each process type. Global model for running time estimation Predictor variables used to estimate the train running time over a block are determined. An obvious indicator of train running times is the block length, which can be derived from train number step messages (§3.3.2). Moreover, we include the block distance from the last scheduled stop, as well as the distance to the next scheduled stop in order to include the effects of extended running times due to braking and acceleration. The distances are computed between the middle of the platform and the middle of the block. Running times over blocks in a cruising stage depend on the maximum speed limit. On the main lines in the Netherlands, excluding high-speed and freight lines, the maximum speed limit on is either 130 or 140 km/h (ProRail, 2013). Maximum speed limit in the part of the network used for this case study is 140 km/h. Furthermore, we consider the impact of peak-hours on train running times. A binary variable is created that indicates whether the observed process takes place during a peak-hour. The difficulty in separating the data set to peak and off-peak events stems from the fact that the limits of peak hours can be fuzzy, as well as train line and station dependent. The exact limits of peak periods are difficult to obtain without the additional data sets that reflect passenger demand such as passenger counts, ticket sales information or smart card data. Therefore, in the global model, we use the definition of peak-hours from the Dutch national train operator NS. Morning peak is the period 76 Models for Predictive Railway Traffic Management between 6.30 – 9.00 and the afternoon peak is between 16.00 – 18.30. Peak-hours are considered only on working days, therefore, weekends and holidays have no peakhours. Note that the drawback in accuracy of using the predefined limits for peak periods has been overcome in the local model (4.5.2). The categorical variable that indicates the train type is also considered as a predictor. In order to create a generic model, this variable has only two levels: intercity trains with scheduled stops in large stations and local trains that stop in every station along their route. Freight trains are not included in the data set due to a small corresponding sample in the considered area. Moreover, even though hindered train runs are excluded from the data set, the headway time between successive trains is included as a predictor that may explain the impact of the preceding train on train running time. Headway time is in this context defined as the time since the previous occupation of the same block. Finally, we test the validity of the assumption that the running time of a train depends on the value of delay at the previous departure. It is assumed that delayed trains may run with full performance in order to use the running time supplements to reduce delay. On the other hand, trains running on time or ahead of their schedule run in a lower performance regime, thus avoiding early arrivals and achieving energy efficient driving. This assumption was not validated in earlier approaches (L¨uthi, 2009; Van der Meer et al., 2010) on the macroscopic level. In this thesis, the impact of departure delays on train running times over block sections is examined with respect to acceleration, cruising, coasting and braking. Global model for dwell time estimation Dwell time predictors, obtainable from train describer data are presented in this section. Scheduled dwell time for each scheduled stop is an obvious choice for a predictor variable. Furthermore, the fact that trains do not depart before their scheduled departure time indicates that arrival delay may have a major impact on train dwell time. Early trains have longer dwell times than scheduled in order to avoid early departures. On the other hand, trains with a positive arrival delay that is larger than dwell time buffer spend a minimum dwell time in order to minimize the departure delay. The impact of train and station type is examined by including corresponding two-level categorical variables. Stations are separated into small and large stations and trains into intercity and local trains. A station where only local trains stop (it is skipped by intercity trains) is considered as small. On the other hand, a station where both local and intercity trains stop is considered as large. Finally, the impact of peak-hours on dwell times due to the increased number of alighting and boarding passengers is included in the same manner as in the described model for running time estimation. Note that all considered predictors are obtainable from track occupation data and the timetable. Dwell times are predicted as an integrated process since no rolling-stock or passenger data, needed to analyse sub-processes of train dwellings, were available for this study. Moreover, the imprecision in estimating the exact arrival and departure times , described in Section 3.5.6, may influence the accuracy of dwell time estimates. Chapter 4. Data analysis and estimation of process times 77 The significance and predictive power of each presented variable is tested using the statistical learning methods described in Section 4.3. 4.2.3 Local model This chapter also focuses on the local model for running and dwell time prediction. The processed train describer data enable creating a separate process time estimation model for each block, station and train line. The goal of the local model is to explain the variation of running times of trains of the same line over a particular block. For that reason, many of the predictors used for running time estimation in a global model, such as block length, distance to and from the last scheduled stop, train type, headway become redundant. In order to estimate process times for a particular instance (train line, block section or station), we investigate the impact of departure delay and attempt to verify the assumption that delayed trains run faster to reduce their delay. Similarly, the local model for dwell time estimation that is created for each station and train line considers only the impact of arrival delay and peak-hours. The applicability of such models is limited by data availability. For example a local model for estimating a process time of trains of a certain train line over a particular block cannot be generalised to other block sections or train lines. Therefore, a sufficient amount of data is required to build each local model. It is important to have this in mind because train lines operate with different frequencies and some parts of the network may be utilised less than the busy main lines or station routes. 4.3 Statistical learning methods This section describes the statistical learning methods used in this thesis for developing the predictive models for process time estimation. The criteria used to select these methods are: prediction accuracy, the simplicity of implementation, computational requirements and the interpretability of results. Moreover, an important aspect of supervised learning techniques is the trade off between bias and variance, i.e, between underfitting and overfitting the models (Hastie et al., 2009). Due to the envisaged realtime application of the obtained process time estimates, it is essential that they are robust against the outliers in the data. Thus the methods and models with high bias are favoured compared to the models with high variance that overfit the data. We first test the applicability of the linear models due to their simplicity and high bias. The accuracy of predictions is further tested with the regression tree based method, which can capture the nonlinear relations between the predictors and the response. Finally, the application of random forests, a method that overcomes the high variance property of regression trees, is tested. 4.3.1 Multiple linear regression Running and dwell times in the global model can be predicted as outcomes of a linear model. The approach relies on the multiple regression model due to its simplicity. 78 Models for Predictive Railway Traffic Management The individual impact of each predictor is computed as well as the goodness of fit of the global model. The generic model is given in equation (4.1) which estimates the value of the response variable Y with respect to p predictors. A regression coefficient β j estimates the expected change in Y per unit change in X j , where j ∈ {1, . . . , p} assuming that all other predictors held fixed. Coefficient β0 is called the intercept and ε is the error term. Y = β0 + β1 X1 + β2 X2 + · · · + β p X p + ε. (4.1) The goal of multiple linear regression is to compute estimates βˆ 0 , βˆ 1 , . . . , βˆ p based on n observations, so that the residual sum of squares (RSS) is minimal. RSS is computed as RSS = ∑ni=1 (yi − yˆi )2 , where yˆ = βˆ 0 + βˆ 1 x1 + βˆ 2 x2 + · · · + βˆ p x p . Discussion of a multiple linear regression model requires analysis of overall model accuracy, as well as the analysis of impact of each of p predictors included in the model. The model accuracy is reflected through the residual standard error (RSE) s RSE = 1 RSS n− p−1 (4.2) and the fraction of explained variance of Y R2 = 1 − RSS . TSS (4.3) Total sum of squares (TSS) is computed by TSS = ∑ni=1 (yi − y) ¯ 2 , where y¯ is the mean value of all observations of y, represents the sum of squares without any model. Furthermore, the F statistic shows the predictive power of the model F= (TSS − RSS)/p ∼ Fp,n−p−1 . RSS/(n − p − 1) (4.4) The impact of each predictor X j is estimated by computing the p-value which indicates the probability of accepting the hypothesis that no relationship between X j and Y exists. More details on linear models and their interpretation can be found in Hastie et al. (2009). Robust linear regression Robust linear regression (Rousseeuw, 2005) represents a modification to the described least-squares method with the intention to identify outliers in the data set and exclude them from the computations. Estimates that are resistant to outliers both in x and y direction can be obtained by fitting the regression curve to the majority of data and subsequently identifying outliers as data points with large residuals from the robust solution. Chapter 4. Data analysis and estimation of process times 79 Rousseeuw and Driessen (2006) presented an efficient algorithm for computing robust linear regression coefficients using the least trimmed squares (LTS) method. The objective is to find a h-subset of the data set n and minimise h ∑ (yi − yˆi)2i:n (4.5) i=1 where (y1 − yˆ1 )21:n ≤ · · · ≤ (yn − yˆn )2n:n are ordered squared residuals and h is a point that reflects the percentage α of resisted outliers h = dn(1 − α)e. The simplicity of the linear model comes with a price of inaccuracy and inability to model interactions between predictors and their non-linear impact on the output variable. An example of the non-linear relationship between a predictor and the output variable are discrete categorical variables. Interactions between predictors indicate that they are correlated and the impact of one predictor is dependent on the value of another. It is therefore difficult to distinguish the impact of correlated predictors separately. An example of interacting predictors may be that delayed trains run faster in the acceleration and braking phase. Thus the impact of block position with respect to previous and next scheduled stop could indicate how important a departure delay is on running time estimation. The correlation between distance from the previous and distance to the next scheduled stop is clear. 4.3.2 Tree-based non-linear methods Regression trees A way to overcome the drawbacks of linear methods emerged with the development of the tree-based methods (Breiman et al., 1984). The basic concept of these methods and their application in regression is to segment the predictor space into simple regions. The output variable is predicted in each region separately. The predictor space represents the set of values for X1 , X2 , . . . , X p which is divided into J distinct and non-overlapping regions R1 , R2 , . . . , RJ . For every observation of p predictors an appropriate region R j (terminal node in the tree) exists. The terminal node for each observation is reached by applying splitting rules, i.e., binary decisions at internal nodes that direct the observation towards its corresponding region. The estimated value of the output variable is computed as the mean of the response values for the training observations in R j . Similarly to linear models, regression trees are created as a result of the optimisation procedure that minimises RSS = ∑Jj=1 ∑i∈R j (yi − yˆR j )2 , where yˆR j is the mean of all output values from training observations in R j . Due to the high computational complexity of this problem, an algorithm has been developed that recursively partitions the predictor space in a greedy manner (Therneau, Atkinson, & Ripley, 2014). The tree is built by the following procedure: first a single variable X j and point s is found 80 Models for Predictive Railway Traffic Management that splits the data into two groups {X|X j < s} and {X|X j ≥ s}. The procedure is recursively repeated in each partition until a threshold is reached in terms of number of training observations in the region. The described algorithm always selects the splitting variable that contributes to minimisation of RSS the most. In principle that may cause that certain variables with low contribution to the main objective may be completely left out from the model and not chosen as splitting variables. An indicator of importance is obtained for each variable used for growing the tree. It is computed as the sum of improvements of the objective function for each split for which the variable was used as the splitting variable. Furthermore, in order to increase the interpretability of a tree and avoid overfitting the model to the training set, the tree can be pruned. The resulting tree has fewer regions and performs better on the test set. More details on pruning can be found in Breiman et al. (1984). The major advantage of regression trees is that they are transparent and easy to interpret and validate by experts. They are able to handle non-linear dependencies, interaction between predictors, and categorical variables. However, the prediction accuracy is often unsatisfactory when applied on a test set. Even after pruning the trees, the prediction in terminal nodes may be significantly affected by outliers. Random forests The drawbacks of regression trees can be overcome by generating a large number of trees on the training set and using average values of all responses to estimate an instance from the test set. The fundamental concept is called bagging and relies on repeated sampling of the training set and obtaining B different training sets (Breiman, 1996). Each sample Sb of size 2B/3 where b ∈ {1, . . . , B} is used to build a regression tree. The predicted response of the model to a test observation is computed as the average over all trees 1 B yˆ = ∑ yˆSb (4.6) B b=1 In each sample the data that is left out is used to estimate the so called out of bag (OOB) error. The response for the bth observation is predicted using each regression tree for which this observation was left out from the training set (B/3 observations on average). All errors are averaged to obtain the OOB error as a cross-validated indicator of model accuracy. Random forests have been introduced by Breiman (2001) in order to further improve the accuracy of bagging models. They rely on randomisation of the previously described recursive algorithm for construction of each tree. The major modification is that not all predictors are considered for choosing the best split of the predictor space but only randomly chosen m variables. The prediction is again performed by averaging the response of each of the B trees thus further reducing the response error of regression trees. Chapter 4. Data analysis and estimation of process times 4.4 81 Process time estimates – global model Predictor variables used to derive process time estimates form the global model were described in Section 4.2.2. This section presents the results of applying the statistical learning methods described in the previous section on the available data set (§4.2.1). The algorithms for creating the statistical models have been implemented in a programming language for statistical computing R (R Core Team, 2013). The packages for robust linear regression (Rousseeuw et al., 2014), regression trees (Therneau et al., 2014) and random forests (Liaw & Wiener, 2002) were used to build the respective models. 4.4.1 Running time estimates derived from the global model Table 4.1 summarises the training set used to derive running time estimates. Response variable ‘running time’ represents the running time of a train over a block. It is predicted with respect to departure delay (‘departure delay’), block length (‘block length’), distance from the last (‘distance from’) and to the next (‘distance to’) scheduled stop, and time headway from the preceding train (‘headway’). Two categorical binary variables: train type (68% of data points are related to intercity trains), and peak-hours (27% of data relate to peak-hours) are also included in the model. Training data set comprises 101481 data points describing the running times of nine train lines over 143 blocks in 82 days. Table 4.1: Summary of the training set for running time estimation running time (sec) departure delay (sec) block length (m) distance from (m) distance to (m) headway (sec) Mean Median St. Dev. Min Max 43.16 100.29 1137.82 5300.93 6986.98 691.58 41.57 53.97 1185.00 3685.00 5140.00 610.58 19.10 148.90 384.70 5291.24 5500.12 556.48 10.01 −147.73 255.00 131.00 1190.00 93.66 179.35 1199.13 1915.00 24440.00 2376.00 21349.03 Robust linear model for running time estimation Results of applying LTS robust multiple linear regression (Rousseeuw et al., 2014) to fit the data are given in Table 4.2. The coefficient is given for each variable. Moreover, we give an indicator of importance of a particular variable for the overall model, which is a representation of the corresponding p-value. Note that the categorical variables are presented with only one level since the variable value for the second level is equal to zero. The results indicate that all considered variables have a significant impact on running times. The running times during peak-hours are slightly shorter than off peak. A negative correlation with departure delay is determined which indicates that in general, 82 Models for Predictive Railway Traffic Management Table 4.2: Summary of the LTS model for running time prediction Dependent variable: Coefficient peak hour = 1 departure delay headway distance to distance from block length train type = ‘local’ Intercept −0.1608 −0.0019 −0.0013 −0.0002 −0.0002 0.0239 0.6803 13.0600 R2 Residual Std. Error F Statistic 0.6514 6.5810 19320.0800 Note: running time p-value ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ ∗∗∗ p<0.01 delayed trains run slightly faster to recover the delay. Furthermore, a negative correlation between running times and headways can indicate that in case of short headways after the preceding trains, trains tend to run slower to reduce the possibility of running into a route conflict. The position of the block with respect to the station of the previous and next scheduled stop may reflect the phase of running time as defined in Section 2.2.1. Negative coefficients of the corresponding variables indicate shorter running times with increased distance from/to the scheduled stop. Furthermore, block length has an expected positive impact on train running times. Finally, local trains are estimated to have slightly longer running times than intercity trains. The lower part of Table 4.2 presents the predictive quality of the model. The R2 value indicates that 65% of variation of running times can be explained by the presented model. Having in mind that the variation within the training set is relatively low (Table 4.1), this implies that the presented model is useful for estimating running times. This is also demonstrated by the low RSE of less than 7 seconds. Large F statistic and low p-value indicate strong correlation between response and explanatory variables. Regression tree model for running time estimation The non-linear relationship between predictors and response, as well as interactions among predictors can be resolved using regression trees. Figure 4.1 presents the tree obtained after applying the recursive partitioning algorithm (Therneau et al., 2014) on the training set. The large training set enabled the construction of a complex regression tree with 16 internal nodes (splits), indicated by oval nodes, and 17 terminal nodes (rectangular nodes). Each node contains the mean value of response (running time) and the number of data points n in the corresponding region. The tree indicates the relative impor- distance_to>=3692 50.199 n=7055 50.843 n=1466 84.987 n=567 length>=1212 length=1655 41.619 n=27332 60.366 n=2033 43.379 n=34387 headway=698.5 47.494 n=84135 52.267 n=9483 distance_to=1173 length=195 distance_from=2306 distance_to=1499 53.726 33.904 n=18357 n=1083 52.621 n=19440 distance_from>=7324 distance_from=1540 distance_from=3380 89.428 n=1975 58.672 n=11458 distance_to>=1970 distance_to=1189 length>=1128 45.106 n=62718 length=214.4 11.86 n=2917 length=408 22.138 n=17346 length=84.16 266.78 n=6026 delay< 84.16 234.28 n=8755 scheduled< 150 scheduled>=150 delay>=−65.34 delay< −65.34 delay>=−25.23 186.72 n=21370 scheduled>=45 scheduled>=90 144.51 n=18412 scheduled< 45 111.18 n=106248 scheduled< 90 140.37 | n=141355 228.7 n=35107 294 n=13737 242.16 n=7082 327.46 n=1909 delay>=−127.6 delay< −127.6 260.27 n=8991 357.91 n=4746 scheduled< 210 scheduled>=210 delay< −25.23 88 Models for Predictive Railway Traffic Management 0.4 0.6 0.8 1.0 89 0.2 X Relative Error Chapter 4. Data analysis and estimation of process times 0 2 4 6 8 Number of Splits 0.4 0.0 R−square 0.8 Figure 4.7: Relative dwell time prediction error depending on the tree size 0 2 4 6 8 Number of Splits Figure 4.8: R2 of the dwell time model depending on the tree size Random forest model for dwell time estimation We attempt to improve robustness against outliers and prediction accuracy of the global model by applying the random forest method on the training set (Liaw & Wiener, 2002). The resulting random forest contains 300 trees. Each split in each tree is created by choosing the best out of three randomly selected predictors. 1700 1500 MSE 1900 The indicators used to examine the quality of the model are MSE and R2 . Figure 4.9 shows the reduction of MSE with increasing number of trees in the forest. No significant decrease of error is achieved for forests larger than 100 trees. 0 50 100 150 200 250 300 Number of trees Figure 4.9: MSE of the dwell time model depending on the number of trees The coefficient of determination R2 is also significantly improved for forest size of up to 100 trees (Figure 4.10). The final value shows that by using the random forest 90 Models for Predictive Railway Traffic Management 0.74 0.72 0.70 R−squared 0.76 algorithm, 76% of dwell time variability can be explained and predicted. Thus, the predictive power has been improved compared to the regression tree model. This comes at the cost of computation time which is much longer than for the tree model or LTS robust regression. 0 50 100 150 200 250 300 Number of trees Figure 4.10: R2 of the dwell time model depending on the number of trees 4.5 Process time estimates - local model The data structure resulting from the process mining algorithm (Chapter 3) can be exploited to derive a separate statistical model for each process. A running process is defined by the train line and block section, and a dwell process by the train line and station of the scheduled stop. The LTS robust linear regression model is used to predict running and dwell times depending on departure and arrival delay, respectively. The assumption about the different behaviour of delayed (delay larger than 60 seconds) and punctual or early trains is tested by separating the set of observed running and dwell times into corresponding sets of delayed and punctual trains and applying the Wilcoxon rank-sum test at 5% significance level. The null hypothesis is that samples have continuous distributions with equal medians. 4.5.1 Estimation of running times over a particular block The difference in running times is expected to be the largest in the last part of the open track section before the scheduled stop where punctual or early trains have longer running times due to coasting or cruising with lower speed. In Figure 4.11, the realised running times are presented relative to the departure delay. A weak correlation between running times and departure delays was found on the level of block sections. This is illustrated in Figure 4.11 (left) which shows the dependence of running time over the last block before the scheduled stop in Delft station of train line 2200. The solid red line in the left part of the figure represents the robust fit. The black dashed line represents the 10th percentile of running times. The small percentile is selected and used as a robust estimator of minimum running times Van der Meer et al. (2010). It used in real-time running time predictions as the lower bound in order to avoid unrealistically low values for large delays. Chapter 4. Data analysis and estimation of process times 91 The Wilcoxon rank-sum test rejected the null hypothesis with p ≈ 0 thus indicating the different distributions of running times of delayed trains. Box-plots in Figure 4.11 (right) show small differences in distributions of six data samples specified based on the value of departure delay. The box-plots used in this thesis indicate the median (line in the middle of the box), the 1st and the 3rd quartiles (upper and lower bound of the box) and data maximum and minimum (ends of the upper and lower whisker). Note that the outliers are excluded from the plots for the sake of clarity of the figures. Outliers are detected in a conventional procedure by adding (subtracting) the interquartile difference multiplied by 1.5 to (from) the upper (lower) quartile. All values outside of the obtained range are considered as outliers. Running1time1over1the1approaching1block1to1Delft1[s] Running1time1over1the1approaching1block1to1Delft1[s] 80 75 70 65 60 55 50 45 40 35 30 0 100 200 300 400 500 600 700 800 Departure1delay1from1The1Hague1HS1of1line122001[s] 900 1000 54 52 50 48 46 44 42 40 38 36 34 1 2 3 4 Departure1delay1[min] 5 Figure 4.11: Dependence of running time on delay (left) and box-plots of running times for punctual and delayed trains (right) Figure 4.12 shows weak correlation between running times and departure delays for all train lines on the corridor The Hague HS – Rotterdam. Each circle corresponds to a train line and block pair. All blocks on the corridor, represented by the start and end signal code, are given on the horizontal axis. The colour of each circle represents the value of R2 for the particular local model, according to the colour map given on the right side of the figure. Since no or only weak correlation between running times and actual delays was discovered, it is important to determine how the running time supplements are actually used. In order to do so, delay accumulation over all scheduled stops for each line was analysed. Figure 4.13 shows how the delay of line 2200 trains changes over the route along the corridor Leiden – Dordrecht. The solid red line indicates the mean of delay change over distance. No distinction can be made between early, punctual and delayed trains. It is visible that time reserves are spent on extended dwell times. Trains generally run with full performance thus compensating for departure delay (delayed trains) or having more slack during dwell times (punctual trains). Delay [s] -200 V6 0 V6 7_G D 15 V6 TA _D 1 62 T 5 A D 3_D 62 TA 3 46 TA4 D 6_D 66 T8 0 T6 D 3_D 25 T8 D 09 T80 T _ 9 SD 815 DT M _S 815 82 D SD 1_ M8 M SD 21 82 M SD 7_ 82 M SD 7 3 SD 8_ M3 S 8 SD M7 DM M 0_ 7 94 SD 0 _ M G DH 94 V3 S 0 A1 G 4_ 16 V3 G 44 V3 4 G V3 _G 4 4 V G 2_ 607 V6 G V G 05 60 V6 _G 5 D 13 V6 TA _D 1 62 TA 3 1 6 D DT _D 21 H A4 TA SA 6 4 6 1 4 D 16 _DT 4 H SA _DH 62 15 SA 5 6_ 1 5 D RTD 6 T6 25 170 D _D T T G 14 14 V3 _D 02 T _ 34 D GV T3 3 4 D 4_D 2 T7 2_ T72 D T8 03 G G Train - line 92 Models for Predictive Railway Traffic Management S1900 -0 -0.1 S2100 -0.2 S2200 -0.3 -0.4 S5000 -0.5 S5100 -0.6 -0.7 S9200 -0.8 -0.9 -1 Block Figure 4.12: R2 for prediction of running time on The Hague HS – Rotterdam corridor 300 250 200 150 100 50 -50 0 -100 -150 LEDN LAA GV DT SDM RTD RTB RLB Figure 4.13: Delay over corridor Leiden - Dordrecht for train line 2200 DDR Chapter 4. Data analysis and estimation of process times 4.5.2 93 Estimation of dwell times for a particular station Train1line Availability of data from door sensors and on board equipment has inspired recent research in detailed modelling of train dwell times (Medeossi et al., 2011). In this thesis we rely solely on train describer data, thus a detailed analysis of different phases of dwelling in scheduled stops was not possible. The dependence of dwell times on arrival delays was examined. Figure 4.14 shows the correlation between arrival delay and dwell time for each observed train line and station pair. The correlation is particularly strong for the large stations Leiden (LEDN), The Hague HS (GV), Delft (DT), Schiedam (SDM), Rotterdam (RTD) and Dordrecht (DDR). In smaller stations where only local trains are scheduled to stop, no significant correlation between dwell times and arrival delays was established. That can be explained by the fact that these stops are scheduled as short stops as described in Section 4.4.2. The trains only stop for boarding and alighting and depart as soon as possible. S1900 -0.1 S2100 -0.2 S2200 -0.3 S5000 -0.4 S5100 -0.5 S6300 -0.6 S3400 -0.7 S9200 -0.8 S2600 D R D D ZW D BR LB R R TZ R TB TD R M SD T TZ D D W R SW V G VM G T A LA K VS VN D LE D N -0.9 Station R2 Figure 4.14: R2 for prediction of dwell times on Leiden – Dordrecht corridor Figure 4.15 (left) shows the dependence of dwell times on arrival delays for the train line 2200 in station Delft. The horizontal black dashed line represents the 10th percentile of all dwell times, whereas the red line represents the robust linear fit for punctual trains. The scheduled dwell time is 60 seconds. Strong correlation (R2 = 0.8704) was captured for early and punctual trains. The Wilcoxon rank-sum test rejected the null hypothesis (p ≈ 0) and different distributions of dwell times for punctual and late trains are clear from the box-plots in Figure 4.15 (right). However, the variation of dwell times for delayed trains needs to be explained by other factors and therefore, the data set is divided into a set of punctual and delayed trains at the threshold of 60 sec. Models for Predictive Railway Traffic Management 400 220 350 200 Dwell time of 2200 trains in Delft [s] Dwell time of 2200 trains in Delft [s] 94 300 250 200 150 100 50 180 160 140 120 100 80 60 0 -200 -100 0 100 200 300 400 500 600 Arrival delay of 2200 train line in Delft [s] 700 800 0 1 2 3 Arrival delay [min] 4 Figure 4.15: Dependence of dwell time on delay (left) and box-plots of dwell time (right) The variability of dwell times of delayed trains is explained by modelling dwell time as a time series to determine the dependence on the peak-hours. Dwell times of delayed trains normally equal the minimum dwell time required for passenger operations and route setting if the delay exceeds the dwell buffer time. We assumed that passenger volumes and consequently the time needed for alighting and boarding increases during peak-hours. Figure 4.16 shows dwell times (weekends and holidays were not considered) relative to scheduled arrival times of the train line 2200 in Delft. The increase in dwell times during peak-hours is clearly visible. The red line indicates the median dwell time. Recorded9dwell9rimes9of922009trains9[s] 290 250 200 150 100 50 6:00 8:00 10:00 12:00 14:00 16:00 18:00 Scheduled9arrival9time9to9Delft 20:00 22:00 24:00 Figure 4.16: Dependence of dwell time on scheduled departure time This clear distinction between causes of variability of dwell time for punctual and delayed trains requires a separate approach to prediction of dwell times. Therefore, for punctual and early trains, dwell time can be predicted based on the correlation with arrival delay. On the other hand, dwell time for a delayed train is estimated from historical data based on dwell times of the same train number and adjacent train Chapter 4. Data analysis and estimation of process times 95 numbers of the same series (e.g. if train 2245 arrived with a delay, the dwell time will be predicted as the average dwell time of trains 2243, 2245 and 2247 obtained from the data set of delayed trains). The reason for including the data from the adjacent train numbers is to ensure the sufficient sample size and robustness of the moving average estimate. Note that the described moving average smoothing method incorporates the effects of the peak hour dependence of dwell times without assuming a clear limit for peak periods. That way the limitation of including the peak-hour dependence in the global model as a categorical variable has been overcome (§4.2.2). 40 0 −80 −40 Prediction error (sec) 80 Figure 4.17 shows the effects of using the described moving average approach for predicting the dwell times of delayed trains on a test set. The prediction accuracy is compared to the approach based on LTS robust regression. The prediction error is computed by subtracting the estimate from the realised dwell time. Positive bias of the LTS estimate error indicates that dwell times of delayed trains are underestimated. This approach assumes the minimum dwell time for delayed trains thus disregarding the effects of peak-hours. The moving average approach significantly improves the prediction accuracy. LTS regression Moving average Figure 4.17: Prediction error for dwell times of delayed trains 4.6 Comparison of statistical models All presented global and local models for estimating running and dwell times are validated on a test set consisting of processed data for 10 days of traffic in TROTS areas Rotterdam and The Hague. The test set for running time estimation contains 18684 data points. The size of the test set for dwell times is 12225. The results of the individual local models are combined in one data set in order to be comparable with the global model. 4.6.1 Comparison of running time estimation models Figure 4.18 shows the distribution of prediction error of each model for running time prediction. Random forests clearly give the most accurate estimates of running times with respect to other global models. The performance of random forests is comparable to the performance of local models that give the most accurate predictions of running 96 Models for Predictive Railway Traffic Management 20 10 0 −10 Prediction error (sec) 30 times. The most significant predictors used in the global model are block length and position with respect to the previous and the following scheduled stop. These predictors do not need to be considered in the local models that are created for a particular block and train line pair. The correlation with other explanatory variables such as departure delay in both model types is weak and therefore the two models give similar output. Note the very small prediction errors within ±10 seconds for the local LTS and random forest estimates. Global LTS Regression tree Random forest Local LTS Figure 4.18: Prediction error of running time estimation models 4.6.2 Comparison of dwell time estimation models Similar results are obtained for dwell time estimation (Figure 4.19). Random forests are the best performing global model. However, local models, consisting of a LTS robust linear regression model for punctual trains and a time series (TS) model for delayed trains, give more accurate estimates of dwell times. A high standard deviation of prediction error even for the most precise model indicates that relying on train describer data as the sole source for developing prediction models may not be enough for accurate estimation of dwell times. An important aspect for comparing the global and local models for dwell time estimation is how sensitive they are to imprecision in estimating actual arrival and departure times. As explained in Section 3.5.6 the measurement error depends on the topology of track circuits, stopping position of the train and train length. It is expected that for the trains of a single train line in the same station these measurement errors are identical. Therefore, the accuracy of local models is not significantly affected. However, the global model, that aggregates the dwell time data from all stations and train lines, may have lower prediction accuracy due to measurement errors of actual arrival and departure times. 4.6.3 Comparison of prediction accuracy for scheduled processes Finally, we compare the accuracy of dwell time and running time estimates. The quality of the presented models for estimating the duration of scheduled processes can thus be analysed. Recall that the running times analysis presented in this chapter relates 50 100 97 0 −100 −50 Prediction error (sec) Chapter 4. Data analysis and estimation of process times Global LTS Regression tree Random forest Local LTS+TS Figure 4.19: Prediction error of dwell time estimation models to running time over blocks sections. This is important for calibrating the mesoscopic traffic prediction model presented in the next chapter. However, in order to offer a fair comparison of the presented models, the accuracy of scheduled process time estimates needs to be considered. 40 20 0 −40 −20 Predictionn error (sec) The running time between two scheduled stops can be computed as the sum of the running times over blocks in the train route including the outbound route from the station of departure and the inbound route at arrival station. In order to exclude the impact that other trains may have had on the running times of trains in the test set, all hindered train runs are excluded from analysis. Figure 4.20 shows a comparison of prediction error for dwell time and running time estimates. The approach based on the local LTS models is selected for the analysis. Running times between two scheduled stops are clearly predicted more precisely than dwell times. This is also demonstrated in Figure 4.21 which compares the relative errors for dwell time and running time estimates. The relative errors are obtained with respect to the scheduled time of the corresponding process. The errors of running time estimates are within 10% of the corresponding scheduled running times. The variability of the relative error of dwell time estimates is much larger. The errors of dwell time estimates may be even larger than the corresponding scheduled dwell times. Dwell time Running time Figure 4.20: Precision of dwell time and running time estimates −1.0 −0.5 0.0 0.5 1.0 Models for Predictive Railway Traffic Management Relative prediction error 98 Dwell time Running time Figure 4.21: Precision of dwell time and running time estimates relative to scheduled process time 4.7 Conclusions This chapter presented two data driven approaches for estimation of conflict-free running times and dwell times. Global models are developed by collecting all running time and dwell time data from the training set and creating a separate predictive model for estimation of each type of process times. Advanced supervised learning methods were tested and compared by predictive power, interpretability of results, and accuracy. On the other hand, the data structure and large size of the test set were exploited to develop local running time and dwell time models for a particular block and station. Both approaches are validated on the test set. Estimates of the local models provided on average more accurate predictions of process times. The running times showed small variation which was to a great extent explained by predictors in both models. Weak dependence on actual delays has been established for running times. The analysis on one corridor showed that the majority of trains run in full performance regime regardless of departure delays. In the last few years, NS, the main train operator in the Netherlands, is promoting the concepts of energy efficient driving which may have an effect on the running times of punctual or early trains (Scheepmaker, 2013). Furthermore, running times seem to be weakly affected by peak-hours and do not have a significant daily variation. An interesting observation is that even for conflict-free train runs short minimum headway after the preceding train may cause extended running time in order to prevent a route conflict. Dwell times of punctual trains show strong correlation with arrival delays, in particular in large stations. On the other hand, the dwell times of delayed trains are more sensitive to impact of passenger volume variability in peak and off-peak periods. Despite the strong predictive power of the presented models, the validation on an independent test set showed that variability of dwell times cannot be fully explained by selected predictor variables. Dwell times need to be modelled with a higher precision since the variation of prediction error is significantly larger than for running times. One way to Chapter 4. Data analysis and estimation of process times 99 do it is to include other data sources on platform design and rolling-stock to estimate. That would enable computation of more precise estimates of arrival and departure events. Moreover, the data sources related to behavioural properties of passengers and train drivers can be used to derive more accurate estimates of dwell times. Finally, we discuss the advantages of the two presented approaches based on the local and global models. The major advantage of the global model is that the results can be generalised and applied to other parts of the network and different train lines that were possibly not included in the training data set. However, the accuracy of process time estimation is the most important criterion for selecting the appropriate model. Since the running time and dwell time estimates are used for real-time calibration of the traffic prediction model presented in the following chapter, the overall accuracy of the model can be severely affected by propagation of process time estimation error over the prediction horizon. Moreover, calibration of the global model as well as the application in real-time is computationally more demanding than creating the multiple local models and using them for prediction. The approach based on the multiple local models is therefore used for calibrating the traffic prediction model presented in the next chapter. 100 Models for Predictive Railway Traffic Management Chapter 5 Real-time prediction of train event times This chapter is an edited version of the article Kecman, P. and Goverde, R. M. P. (2014). Online data-driven adaptive prediction of train event times. IEEE Transactions on Intelligent Transportation Systems. (in press) 5.1 Introduction Real-time prediction of train positions in time and space is a basic requirement for effective route setting, traffic control, rescheduling, and passenger information. The previous chapter described how conflict-free running times and dwell times can be predicted from historical traffic realisation data. However, accurate prediction of train event times, requires detailed modelling of other operational constraints of railway traffic such as interdependencies between trains that share the same infrastructure or have a planned synchronisation constraint. Railway traffic controllers in the Netherlands currently have no support to predict train traffic or the effects of their control actions. Train positions are monitored using the train describer system and actual delays are measured with low accuracy and rounded to full minutes (§2.2.4). Existing tools developed in Switzerland (Dolder et al., 2009), Sweden (Isaksson-Lutteman, 2012) and Japan (Fukami & Yamamoto, 2001) provide controllers with real-time traffic prediction and conflict detection. Moreover, conflict detection modules of the state-of-the-art rescheduling models (Caimi et al., 2012; D’Ariano, 2008) predict train event times and route conflicts in order to derive rescheduling actions. However, these approaches do not exploit the dependence of process times on the actual traffic state or the interdependence with other trains. Process times are predicted based on theoretically obtained values independent of the actual 101 102 Models for Predictive Railway Traffic Management condition of traffic on the network. Running times are typically estimated using predetermined empirical values or microscopic simulation. Similarly, fixed scheduled or minimum dwell times are used to estimate the duration of a scheduled stop for each train. In this chapter, an online approach to traffic state prediction is presented. The main idea is that the prediction is performed by propagating the actual traffic information (current train positions, process times, delays) through a realistic, mesoscopic model of railway traffic. The level of detail included in the model is an essential difference from the earlier approaches (Van der Meer et al., 2010). We use the actual route plans, timetable and current positions of all trains to build a graph model. Microscopic operational constraints are reflected in the graph topology that captures all scheduled events and signal passages. Modelled processes represent the precedence relations between events such as train runs and stops, connections, and minimum headways. The graph is calibrated dynamically, in real-time, using historical track occupation data with predetermined and quantified dependency of running and dwell times on departure and arrival delays, respectively. When an update of train positions becomes available, a depth-first search based algorithm sweeps through the graph. Robust estimates of arc weights are computed in real-time using the estimates derived with the LTS method as described in Chapter 4 and an efficient algorithm computes the predicted realization time for all events within the prediction horizon. This approach is extended by precise modelling of route conflicts and incorporating time losses, due to braking and re-accelerating of hindered trains, in the predictions. Moreover, we present an adaptive component that exploits feedback information about the actually realized blocking times of running trains. The realized process times are monitored and trains with process times that continuously deviate from computed estimates in a certain pattern are detected. The estimates of downstream process times can subsequently be adjusted to minimize the expected prediction error. The described prediction tool is tested on a busy corridor Leiden – Dordrecht in the Netherlands. The accuracy of predictions, size of the model and computation speed are presented and analysed depending on the length of the prediction horizon. The next section (§5.2) describes the framework of the system design. Sections 5.3 and 5.4 give a detailed description of the model and data-driven calibration, respectively. The online prediction algorithm is presented in Section 5.5 and its performance in reallife case study in Section 5.6. Finally, Section 5.7 summarises the presented model and gives guidelines for further research and improvements. 5.2 Framework of the real-time prediction tool The main components of the tool in the real-time environment and the flow of data between them are depicted in Figure 5.1. The parts of the tool presented in this chapter are shown with shaded boxes. The traffic model is based on a directed acyclic graph Chapter 5. Real-time prediction of train event times 103 (DAG) with dynamic arc weights. The graph topology is built and updated based on the actual process plan (train orders, route and connection plan) and current positions of trains on the network. We assume that the actual route and connection plans are continuously provided by traffic control for the duration of prediction horizon. The route plan for a train is given as a planned sequence of block sections in the train route. A route plan can be translated to the level of track sections (Chapter 3) and used to determine the necessary headway arcs for routes with common track sections. Each change of the actual plans or information from the real-time operations, i.e., changing the relative order of trains, adding or cancelling trains, modifying train routes, updating connections, and removing passed events, results in an update of the graph topology. Railway operations Ti m eta bl e Actual process plan Arc weights computation Traffic model Route conflict adjustment Predicted process times Realised process times co Rout nfl e ict s Actual traffic state rs ns de es tio Or out nec R on C Monitoring Predicted event times Traffic control Adaptive adjustment Figure 5.1: Monitoring and prediction components in the traffic control loop Arc weights represent the estimated process times which are computed based on the actual (predicted) traffic state and processed historical data. The actual traffic state, comprising the current train positions and delays, is provided continuously by the monitoring component and the future traffic state is obtained from the traffic model. The weight of an arc is time-dependent and assigned in a dynamic way depending on the (estimated) starting time of the modelled process (Nachtigall, 1995). By comparing the actually realised event times with the scheduled times, the actual delays are obtained. Similarly, we obtain predicted delays by subtracting the predicted event times from the scheduled ones. Arc weights are computed from the database of historical data that contains the predetermined dependencies of process times on delays. The database is 104 Models for Predictive Railway Traffic Management obtained using the methodology described in the previous chapter. This way the dependence of running and dwell times on the current (predicted) delays is incorporated in the model. This primary prediction loop is extended with an adaptive component (Section 5.5.3) that compares the actually realised process times of the running trains with the predicted values and adapts the running time until the next scheduled stop to minimise prediction error. The adaptive component of the prediction model enables online detection of the train runs with process times that continuously exceed or fall behind the computed estimates, and adjusts the predictions of future train behaviour accordingly. Furthermore, the accuracy of predictions is increased by adjusting the running times over the approaching block for the hindered trains (Section 5.5.2). Route conflict duration is predicted and the corresponding adjustment factor is retrieved from the predetermined dependence of running time increase on conflict duration. After every graph update, a prediction of event times of all reachable events is performed by applying a depth-first search based algorithm on the graph-based traffic model (Section 5.5). 5.3 5.3.1 Microscopic graph based model The graph model The railway traffic, represented as a discrete-event dynamic system is modelled with a DAG G = (V, E), where V is a set of nodes and E is a set of arcs. Each event is modelled by a node. We distinguish between signal events (passing of a signal by a running train) and station events (arrival and departure to and from a platform track). The microscopic graph model needs to support rescheduling actions such as reordering, rerouting, revising services (cancelling transfers or trains, adding extra trains) and retiming. Therefore, graph G is constructed in a form of an adjacency list which is a suitable data structure that supports operations on dynamic sets (Cormen, Leiserson, Rivest, & Stein, 2009). The existing routines for implementation of the dispatching actions in a graph-based traffic model represented as an adjacency list are thus applicable (Van der Meer, 2008). pred A node i ∈ V is described by (ni , infrai , typei , previ , nexti , ti , tirec ), representing the train number, infrastructure element (signal or platform track), type, direct predecessors, direct successors, predicted realization time and the recorded time (when available), respectively. Nodes that model scheduled events, i.e., arrivals and departures are also attributed with the scheduled event time t sch . By comparing the recorded (predicted) event times with the scheduled event times, the current (predicted) delay is obtained for a specific train and used to estimate the duration of its subsequent processes (dwell and running times). Scheduled departure times are also used to incorporate the timetable constraints (a train cannot depart before the scheduled departure time). Chapter 5. Real-time prediction of train event times 105 An arc models a precedence relation between events. Apart from modelling the running and dwelling processes related to a specific train, directed arcs are also used to model interactions between trains, namely minimum headway and connection constraints. Connection arcs can be used to model synchronisation constrains such as passenger transfers, and rolling-stock and crew circulation constraints. Arc (i, j) ∈ E is described by (i, j, wi, j , typei, j ) representing the tail event, the head event, the arc weight and the arc type (‘dwell’, ‘run’, ‘headway’, ‘connection’). 5.3.2 Graph construction The graph is constructed based on the actual process plan that includes the given train routes, scheduled event times (actual timetable) and connection plan (§2.2.1). The connection plan contains all planned synchronisation constraints. A train route is represented as a sequence of track sections and signals. For events belonging to the same train, running arcs connect all signal passing events, as well as signal passing events with station events. Dwell arcs connect station events, i.e. an arrival event with a subsequent departure event. An inbound running arc connects a home signal event with a subsequent arrival event, whereas an outbound running arc connects a departure event with a subsequent exit signal event. Headway arcs separate the successive occupations of a block between two signals or a station route by different trains. Typically, a signal changes to a permissive aspect as soon as all sections in a block (station route of the approaching train) protected by the signal, have been released. As discussed in Section 2.4.3, minimum headways need to be modelled differently for interactions of trains on open track or station areas with overlapping and merging routes on the one hand, and for interactions of trains with diverging or intersecting routes on the other. Note that the relative train orders on each infrastructure element (switch, track section, block, platform track) are needed to construct the headway arcs. The relative orders can be determined from the actual process plans (train routes and actual timetable). On open tracks and for station routes with the same end signal, the critical section that constrains a signal release is the section before the end signal of the block or route. This situation holds for trains that run over the same block or station route or for trains with merging routes (routes that have different starting signals and the same end signal) in interlocking areas (Figure 5.2). An accurate space-based train separation is ensured by adding a headway arc that constrains the realisation of a signal passing event of an approaching train until the protected block was cleared by the previous train. The head event is the start signal passing event of the approaching train, the tail event is the end signal passing event of the preceding train. However, in station areas, conflicting routes are often diverging (with the same starting signal and different end signals, as shown in Figure 5.3) or intersecting (different starting and different end signals). For such route interactions, the ‘sectional release’ route locking principle applies. Since all events in the mesoscopic model are signal passages 106 Models for Predictive Railway Traffic Management TS4 S1 S3 S2 TS1 S4 TS2 TS5 TS3 S5 S7 Train1-route S1 TS1 TS3 TS5 S7 Train2-route S3 TS2 TS3 TS5 S7 r Train1 S1 S8 S7 r - running time h - clearing time+ signal release time h r Train2 S3 S6 S7 Figure 5.2: Space-based train separation or station events, the event of the critical section release has not been included. We model the train separation in a time-based manner by adding a headway arc between passing events of signals that initiate running processes over the protected switches. The head event and the tail event are the start signal passing events of the approaching and the preceding train, respectively. The procedure to compute the weights for both types of headway arcs is described in Section 5.4.2. TS4 S1 S3 S2 TS1 S4 TS2 Train1-route S1 TS1 TS3 TS5 S7 Train2-route S1 TS1 TS3 TS4 S5 Train1 S1 TS5 TS3 r S5 S7 S6 S8 S7 r - running time h - 10th percentile from the data h Train2 S1 r S5 Figure 5.3: Time-based train separation Finally, a connection arc models the commercial constraints (passenger transfers), or logistic constraints (rolling-stock and crew connections). The arrival event of a feeder train is the tail event and the departure event of a connecting train is the head event. The graph can be constructed based on the train list (each train is described by the Chapter 5. Real-time prediction of train event times 107 route and timetable), train orders, and the list of planned synchronisation constraints. Given G = (V, E), the dynamics of railway traffic can be simulated with the following constraints: pred pred (5.1) t j ≥ ti + wi, j , ∀i, j ∈ V, (i, j) ∈ E pred ti ≥ tisch , i ∈ {V | typei = ‘departure’}. (5.2) Constraint (5.1) defines the precedence relation between the tail and the head event of an arc. Inequality (5.2) represents the timetable constraint for all departure events. Figure 5.4 shows an illustrative example of a directed acyclic graph for two trains. The planned route for each train can be described by the sequence of signals: S1, S2, S3, S4, S6 for train T 1 and S1, S2, S3, S5, S6 for train T 2. Every signal passage is modelled as a node. Both trains have a scheduled stop at the station which is modelled with arrival and departure nodes. Each node is represented as an object with the corresponding attributes and their values. Time related attributes were left out from the figure for the sake of clarity. Nodes belonging to one train run are connected by running and dwell arcs. Since the trains run over the same infrastructure, the necessary minimum headway times are ensured with headway arcs. The route between signals S1 and S3 is the same for both trains, thus requiring at least one block separation between trains, which is modelled with headway arcs (T 1, S2) → (T 2, S1) and (T 1, S3) → (T 2, S2). Recall that for conflict-free traffic, the two-blocks separation is required (§2.2.3). The sectional release principle between diverging inbound routes of two trains is enabled with the headway arc (T 1, S3) → (T 2, S3). Finally, train T 1 can leave the station when the block between S5 and S6 has been released by train T 2, which is modelled by the headway arc (T 2, S6) → (T 1, S4). A planned connection is secured with the arc between the arrival event of T 1 and the departure event of T 2. Note that the direction of the headway arcs indicate the order of trains. In Figure 5.4 train T 2 overtakes train T 1 in the station. The graph topology is continuously updated according to the rolling prediction horizon and traffic control decisions. Possible new trains, planned to operate within the actual horizon, are added to the graph with their planned route on the level of block sections. The necessary headway arcs are built per block between consecutive trains that use at least one shared track section covered by the block. With each update of train positions (signal passage, departure or arrival of a train), the nodes describing events from the past and their incoming and outgoing arcs are removed from the graph (and stored with the realized event times). The size of the graph is thus stable within a certain time interval. 5.4 Computation of arc weights Track occupation data, obtained by processing the train describer log files of the Dutch train describer system TROTS (Chapter 3), are used to calibrate the graph model with name = (T2,D) n = T2 infra = platform2 type = departure prev.= {(T2,A),(T1,A)} next = {(T2,D)} name = (T2,S5) n = T2 infra = S5 type = pass prev.= {(T2,D)} next = {(T2,S6)} name = (T2,S3) n = T2 infra = S3 type = pass prev.= {(T1,S3),(T2,S2)} next = {(T2,A)} name = (T2,S2) n = T2 infra = S2 type = pass prev.= {(T1,S3),(T2,S1)} next = {(T2,S3)} name = (T2,S1) n = T2 infra = S1 type = pass prev.= {(T1,S2)} next = {(T2,S2)} name = (T2,A) n = T2 infra = platform2 type = arrival prev.= {(T2,S3)} next = {(T2,D)} name = (T1,S4) n = T1 infra = S4 type = pass prev.= {(T1,D),(T2,S6)} next = {(T1,S6)} name = (T1,D) n = T1 infra = platform1 type = departure prev.= {(T1,A)} next = {(T1,S4)} name = (T1,S3) n = T1 infra = S3 type = pass prev.= {(T1,S2)} next = {(T1,A),(T2,S2)} name = (T1,S2) n = T1 infra = S2 type = pass prev.= {(T1,S1)} next = {(T1,S3),(T2,S1)} name = (T1,S1) n = T1 infra = S1 type = pass prev.= [ ] next = {(T1,S2)} name = (T1,A) n = T1 infra = platform1 type = arrival prev.= {(T1,S3)} next = {(T1,D),(T2,D)} S5 S3 S2 S1 PLATFORM S4 name = (T2,S6) n = T2 infra = S5 type = pass prev.= {(T2,S5)} next = {(T1,S4)} name = (T1,S6) n = T1 infra = S6 type = pass prev.= {(T1,S4)} next = [ ] S6 108 Models for Predictive Railway Traffic Management Figure 5.4: An example of a mesoscopic DAG Chapter 5. Real-time prediction of train event times 109 actually realized rather than theoretical process times. Three months of data (March – May 2010) from the busy corridor between Leiden and Dordrecht in the Netherlands were used to develop statistical models for estimation of process times depending on the actual traffic conditions (Chapter 4). Recall that the route conflicts were identified and only conflict-free process times are used in the analysis of process times and model calibration. In order to include the dependence of process times on actual traffic conditions, a dynamic, time-dependent computation of arc weights (Nachtigall, 1995) is implemented in the prediction algorithm. The main idea behind this approach is that the running and dwell time of a train may vary depending on the current delay of the train, time of the day, realised process times and route conflicts. 5.4.1 Running and dwell arc weights Track occupation data were analysed separately for each train line, thus ensuring that the stopping pattern and routes of all observed trains are the same. Correlations between running and dwell times with actual delays are determined using LTS robust linear regression resisting 25% of outliers (Rousseeuw & Driessen, 2006). For the purpose of obtaining the weight of a running arc dynamically, each block section and station route in the infrastructure database has been attributed with regression coefficients a0 and a1 that are computed for each train line. For a known delay value zn of a train n, the expected running time over a block can than be computed as a0 + a1 zn . We also include the 10th percentile of a process time and use it as the absolute minimum process time to avoid infeasible predictions in case of large delays. Recall that the method for dwell time estimation depends on the actual delay value of a train (§4.5.2). Dwell time of early or punctual trains (delayed less than 60 seconds) is computed using the robust linear regression coefficients as b0 + b1 zn . The regression coefficients b0 and b1 are computed for each station and train line. On the other hand, dwell time of a delayed train is computed using the moving average over a corresponding time series as a function of the train number n. 5.4.2 Headway and connection arc weights Headway arc weights The weight of a headway arc that models space-based train separation represents the minimum time from the moment when the head of the first train leaves a block section to the moment when the next train can occupy the same block. The arc weight equals the sum of block clearing time by the first train, and setup and release time of the signalling system (§2.2.2). In this thesis a constant value of 2 seconds is used for the setup and release time on open track and 12 seconds for route setting time in stations. Clearing time is estimated from the data as the 10th percentile of the clearing times of a block by a specific train line. Note that this approach only models the constraint of the 110 Models for Predictive Railway Traffic Management signalling system that does not allow multiple trains in the same block. The identification of route conflicts is not possible without including the remaining components of the blocking time. In Section 5.5.2 the approaching time, as well as sight and reaction time of the train driver are included in the module for route conflict identification. In order to model the principle of sectional release using time-based train separation, the minimum headway time between two trains with diverging or intersecting routes is estimated from the data as the 10th percentile of the time headways between train runs of the corresponding train lines from the historical track occupation data. By choosing a small percentile of the realised time headways, the impact of buffer times on minimum headway times estimates is excluded. Connection arc weights The weight of a connection arc is equal to the minimum transfer time for passenger connections or the time needed to perform activities that enable planned rolling-stock and crew circulations, for logistic connections. Minimum connection times do not depend on the current delay of trains and the possible effect of delays on headway times was not considered in this work. Therefore, these values are computed offline and the corresponding arc weights are fixed. 5.4.3 Online process time estimation The regression coefficients used to predict running and dwell times based on actual delays, time series models. as well as the 10th percentiles of process times and headway and connection arc weights are precomputed using the method described in Chapter 4 and stored in the data structure of processed historical data W . The database contains a separate object for each block section, inbound and outbound route, and station. Each object is attributed with a process time estimate and coefficients for each train line. We define a separate mapping for retrieving the necessary regression coefficients and 10th percentiles for each process type of arc (i, j). Running time estimates are obtained by fr (W, (i, j)); fd 0 (W, (i, j)) is used for dwell time estimates for punctual and early trains, and fd 00 (W, ni ) for dwell time estimates for delayed trains. Finally, f f ix (W, (i, j)) is used for fixed values of headway and connection time estimate from W . The procedure to estimate process time for arc (i, j) from the historical database W based on the actual (predicted) delay zni is given in Algorithm 1. Running time estimation is presented in lines 2–4, dwell time in lines 5–10 and fixed estimates for headway and connection times in line 12. Note the different method for estimating the dwell times for delayed trains (line 10) that is based on the train number as explained in Section 4.5.2. 5.4.4 Time loss due to route conflicts The running time estimates are computed based on the free running times. Therefore, if a route conflict is detected, the arc weights that model the running processes of Chapter 5. Real-time prediction of train event times Algorithm 1 E STIMATE P ROCESS T IMES 1: Input: W , zni , (i, j) 2: if typei, j =‘run’ then 3: (a0 , a1 , p10 ) ← fr (W, (i, j)) 4: wi, j ← max(p10 , a0 + a1 zni ) 5: else if typei, j =‘dwell’ then 6: if zni ≤ 60 then 7: (b0 , b1 , p1 0) ← fd 0 (W, (i, j)) 8: wi, j ← max(p10 , b0 + b1 zni ) 9: else 10: wi, j ← fd 00 (W, ni ) 11: else 12: wi, j ← f f ix (W, (i, j)) 13: return wi, j 111 {retrieve the coefficients} {compute the estimate} {retrieve the coefficients} {compute the estimate} {retrieve the estimate for delayed trains} {retrieve the fixed estimate} the hindered train over the affected blocks need to be adjusted. The running time adjustments need to incorporate time loss in running time estimates of the hindered train. The time loss consists of braking, possible waiting time in front of the signal, running at a lower speed and re-acceleration. The impact of a route conflict on the running time of the hindered train over the subsequent block depends on the conflict duration and the route and running time of the hindering train. The typical situation that occurs in practice when the two conflicting trains follow the same route is the ‘conflict wave’, where the hindered train keeps passing signals that show yellow aspect and is thus unable to re-accelerate to full speed (Goverde & Meng, 2011). We therefore consider the time loss due to re-acceleration only after the hindered train has passed a green aspect signal. In order to estimate the effects of route conflicts on train running times, all route conflicts within 82 days of traffic on the busy corridor Leiden–Dordrecht in the Netherlands were filtered out (Chapter 3). A quadratic robust fit was used to determine the correlation between conflict duration and the resulting time loss. Time loss is obtained as the difference between the realized running time over a block and the predicted conflict-free running time derived depending on the current train delay. Figure 5.5 shows the regression analysis that was performed based on 20130 data points split into conflicts shorter (left) and longer than 150 seconds (right). A robust quadratic fit resisting 25% of the outliers showed the best performance in terms of coefficient of determination R2 = 0.79 for conflicts shorter than 150 seconds. Even though the data points are scarce for conflict durations longer than 150 seconds, the linear regression line (R2 = 0.92) can be interpreted as the waiting time in rear of the signal, which is the greater part of time loss in long conflicts. This approach enables the adjustment of running time estimates for hindered trains after a route conflict and its duration have been predicted. 112 Models for Predictive Railway Traffic Management 350 300 Time loss [s] 250 200 150 100 50 0 0 50 Conflict duration [s] 100 150 1000 Time loss [s] 800 600 400 200 0 200 300 400 500 600 700 Conflict duration [s] 800 900 1000 Figure 5.5: Time loss dependence on conflict duration: quadratic fit for short (up) and linear fit for long conflicts (down) Chapter 5. Real-time prediction of train event times 5.5 113 Online prediction of event times This section describes the online prediction based algorithm for prediction of event times over graphs with dynamic arc weights. When an event happens, the corresponding node is selected as a root node. The algorithm then visits all reachable nodes in the graph with dynamic arc weights and predicts all event times within the prediction horizon. If route conflicts are predicted, the running times of affected trains are adjusted. Finally, the actual information from the running trains is used for smoothing the prediction error for future process times. 5.5.1 Prediction algorithm The two typical methods of traversing the graph are breadth-first and depth-first search (Cormen et al., 2009). A recursive depth-first search (DFS) is chosen as the method of traversing the graph, due to its low memory requirements, which is an important constraint for large graphs. This version of the DFS algorithm does not rely on queues or stacks to keep track of the already visited nodes. Moreover, the prediction algorithm needs to traverse the graph in the topological order and DFS is typically used to determine the topological structure of a graph. After each event realisation, the reachable set of nodes is traversed, where the root node is the node that models the realised event. The prediction algorithm then updates the predicted event times of all events in the reachable set. Note that if a node is not reachable, the corresponding event time can in no way be affected by the new information. Therefore, it is not necessary to visit that node in the prediction process. The weights of running and dwell arcs are determined online with every graph traversal using the functional dependence of process time on the current train delay (§5.4.3). During the algorithm execution, the predicted event times of a scheduled event will provide predicted delays of trains. Therefore, subsequent process time estimates are computed with respect to zˆ which is a vector that contains the predicted delays for each train. For implementation purposes, an attribute colour is added to each arc. Its values:‘white’ and ‘black’ indicate that the arc has not been discovered or has been discovered (appropriate weight has been assigned), respectively. Finally, every first node in the planned route of a train, modelling the entrance time of the train (the first departure or the first event within the observed network), is connected to a dummy node 0 by an arc with weight that is equal to the expected entrance time. After processing each event realization, the graph is updated by removing the realized event together with all incoming and outgoing arcs. An arc between node 0 and the next node in the event sequence of a train is added. The weight of the added arc is equal to the predicted realization time of the next event of the train. Moreover, every traffic control action also results in a graph update. The process times of each added train are initially calibrated with respect to the actual delays and expected entrance times. 114 Models for Predictive Railway Traffic Management When an update about the occurrence of event i ∈ V arrives, a set of reachable nodes Vˆ is computed that comprises all nodes reachable from and including i. The recorded pred ← tirec and colour attribute of each arc in the subgraph is set to event time is set ti ‘white’. If typei ∈ {‘departure’,‘arrival’}, the current delay value of the corresponding train is updated zni ← tirec − tisch . The information is propagated through the graph and predicted event times of all reachable events are computed according to Algorithm 2. Algorithm 2 P REDICT E VENT T IMES 1: Input: G, Vˆ ,W, z, i, Thor 2: zˆ ← z 3: for all j ∈ nexti do 4: wi, j ← EstimateProcessTimes(W, zˆni , (i, j)) 5: colouri, j =‘black’ 6: if colourk, j =‘black’, k ∈ prev j ∩ Vˆ then pred pred 7: t j ← max (tk + wk, j ) k∈prev j 8: 9: 10: if type j ∈ {arrival,departure} then if type j = departure then pred pred t j ← max(t j ,t sch j ) 11: 12: 13: 14: if pred zˆn j ← t j − t sch j pred t j ≤ Thor then PredictEventTimes(G, Vˆ ,W, zˆ, j, Thor ) ˆ zˆ return G, {arc weight} {predicted event time} {timetable constraint} {predicted delay} {recursive call} The main loop of the prediction algorithm is initiated in line 3. In line 4 the actual weight of an outgoing arc is computed using the procedure described in Algorithm 1. If all constraints on the event realization time are known, i.e., all direct predecessors within the subgraph were visited and all incoming arcs traversed (line 6), the predicted event time is computed in line 7. Otherwise, a new iteration of the main loop is initiated. The timetable constraint for departure events is included in line 10. For all scheduled events, the predicted delay vector is updated in line 11. Finally, if the predicted event time is within the prediction horizon Thor , a recursive call of the algorithm is performed in line 13. Note that the predicted event time may also depend on events that are not reachable from the realized event and thus do not belong to the subgraph. For that reason it is required to explicitly define the set of reachable nodes Vˆ . In such cases arc weight wk, j for k ∈ prev j \ Vˆ in line 7 can be computed using the same procedure wk, j ← EstimateProcessTimes(W, zˆnk , (k, j)). The predicted event time of k is is retrieved from G as a prediction during an earlier algorithm call. Since the Algorithm 2 represents a modified version of a DFS algorithm, its complexity can be determined in a similar way Cormen et al. (2009). The modification restricts the generic DFS algorithm to follow the topological order of the graph. The prediction algorithm sweeps through the subgraph of reachable nodes Vˆ and it is called exactly Chapter 5. Real-time prediction of train event times 115 once for each node (line 6). For an algorithm call for node i ∈ Vˆ , the main loop (lines ˆ where Eˆ = {( j, k) | { j, k} ∈ Vˆ }, the 3–13) is called |nexti | times. Since ∑i∈Vˆ nexti = E, ˆ running time of Algorithm 2 is O(|E|). Figure 5.6 shows an example of algorithm performance when event (q, S1) is realized. Solid arcs illustrate arcs with computed arc weights. The weights of dashed arcs are still unknown. The events with predicted event times are shown in grey colour. Note that an event time can be predicted only after all incoming arcs in the corresponding node are solid. This is visible in the figure when the algorithm backtracks after step 4 to determine the weight of (q, S2) → (r, S2) in order to compute the predicted event time of (r, S2). S2 S3 (q,S1) (q,S2) (q,S3) (r,S1) (r,S2) (r,S4) S1 (1) (3) (5) S4 (q,S1) (q,S1) (q,S2) (q,S3) (r,S2) (r,S4) (q,S2) (q,S3) (r,S2) (r,S4) (2) (4) (6) (q,S1) (q,S1) (q,S1) (q,S2) (q,S3) (r,S2) (r,S4) (q,S2) (q,S3) (r,S2) (r,S4) (q,S2) (q,S3) (r,S2) (r,S4) Figure 5.6: An example of execution of Algorithm 2 5.5.2 Adjusting the running time estimates due to route conflicts The prediction of route conflicts can be performed by extending the microscopic model with the principles of blocking time theory. Section 5.4 describes how running and clearing times, and setup and release times are determined for each train run over a block. After including sight and reaction, and approaching time, the blocking times can be determined. Route conflicts are identified by the overlapping blocking times (Figure 2.2). Since the running time estimates are computed based on the free running times, arc weights that model the running processes over affected blocks need to be adjusted to take into account braking (and possible waiting time in front of the signal), running at a lower speed, and re-acceleration for every predicted route conflict. Algorithm 2 is 116 Models for Predictive Railway Traffic Management therefore extended with a running time adjustment procedure. For each predicted route conflict, the increase of running time of the hindered train is computed depending on the conflict duration. In this thesis we consider a conventional three-aspect signalling system. The weight wi, j of running arc (i, j) needs to be adjusted if it is estimated that Signali will be showing a ‘yellow’ aspect at the moment when the train arrives at the sighting distance of the signal. This moment is obtained by modifying the predicted signal passing time pred ti with a fixed value of 12 seconds for the sight and reaction time of the train driver. The signal aspect can be determined by comparison with the release time (switch to a permissive aspect) of the following signal, Signal j , rel tSignal = j max pred (tk k∈{prev j \i} + wk, j ). (5.3) The procedure for adjustment of the running time of hindered trains is shown in Algorithm 3. Algorithm 3 ROUTE C ONFLICT P REDICTION pred 1: Input: ti 2: if ti pred rel ,tSignal , wi, j j rel − 12 < tSignal then j pred rel d ← tSignal − (ti j 4: ∆ ← f (d) 5: wi, j ← wi, j + ∆ 6: return wi, j 3: − 12) {compute duration} {compute time loss} {update arc weight} If a route conflict is predicted (line 2) the duration d of the conflict is computed in line 3 and the predicted time loss ∆ as a function of d determined from historical data as explained in Section 5.4.4, in line 4. The running time estimate is updated in line 5. An example of a route conflict prediction is given in Figure 5.7. The graph shows three trains q, r, s (the trains are planned to pass signal S2 in that order) with their planned routes over the given subnetwork. Using the introduced notation, we denote pred by t8 the time when train s passes signal S2. A route conflict can be identified pred by comparing the passing time of train s at signal S1, t7 with the earliest possible release time of signal S2 due to minimum headway times after passing of trains q and pred pred r, max(t3 + w3,8 ,t5 + w5,8 ). If train s passes signal S1 before the release time of S2 the conflict is identified and the running time estimate of train s between signals S1 and S2 can be adjusted according to lines 3–5 in Algorithm 3. 5.5.3 Adaptive adjustments of running time predictions In the presented prediction model, the estimated running times over block sections depend on departure delay from the last scheduled stop. In order to exploit the realtime information received since the last departure, an adaptive component has been Chapter 5. Real-time prediction of train event times S1 117 S3 S2 v1=(q,S1) S4 v3=(q,S3) v2=(q,S2) v4=(r,S1) v6=(r,S4) v5=(r,S2) v7=(s,S1) v8=(s,S2) v9=(s,S3) Figure 5.7: An example of route conflict prediction developed that keeps track of the actually realized running times of a running train and adjusts the running times estimates until the next scheduled stop. A moving average smoothing method is used to incorporate the prediction error observed during the train run into future predictions until the next stop. A schematic example of adaptive prediction is given in Figure 5.8. The running train departed from station A and, in the situation from the figure, has just cleared the jth out of m blocks to station B where it is scheduled to stop. The grey solid line starting at station A represents the predicted running time of the train based on the actually registered departure delay. For the sake of clarity, for subsequent realized signal passages only the predicted running time over the following block is shown. A B bj b1 bj+1 bi bm δ1 δj-1 δj j+1 i-1 i m-1 m Time Figure 5.8: A schematic example of adaptive prediction The prediction error of the running time over block bk is denoted by δk and computed after each observed signal passage. For subsequent blocks until some block m0 ∈ { j + 1, ..., m} we derive the estimated prediction error δˆ and adjust the estimates of running 118 Models for Predictive Railway Traffic Management times over the remaining blocks by j ˆδi = 1 δk , ∀i = j + 1, ..., m0 l k=∑ j−l (5.4) where l is a parameter l ∈ {1, ..., j − 1} that specifies the length of the moving average. A separate value of parameters l and m0 is selected for each train type. The red dotted line in Figure 5.8 denotes the adjusted prediction of running times to station B. By applying this adaptive prediction strategy, the continuous delay sources of the conflict-free run of a single train (e.g. due to particular driving style or defective rolling-stock) as well as temporary speed restrictions (due to infrastructure malfunctions or maintenance), will be possible to identify and include in the predictions. 5.6 5.6.1 Application on a case study Experimental setup In order to test the performance of the described algorithms, an experimental environment was set up that includes a static and a dynamic component. The static component consists of the database of historical track occupation data used for dynamic arc weight assignment and running time adjustments (Section 5.4). The dynamic component of the experimental environment consist of the actual process plans for all trains within the prediction horizon and actual train event times. The actual route for each train is given on the level of track sections, which is crucial for accurate modelling of route conflicts and building the mesoscopic graph model. As the prediction horizon moves, new trains are added to the model and passed events are removed. As explained in Section 3.3.1, the train describer log files contain chronologically ordered infrastructure and train step messages. We created a real-time environment for model validation by sweeping through the train describer file for one day of traffic. Every train step message (signal passage) represents the new information that is propagated through the graph using Algorithm 2. The actually realized train event times are used to test the accuracy of predictions. 5.6.2 Description of the case study The experimental setup was built for the busy corridor Leiden–The Hague–Rotterdam– Dordrecht in the Netherlands. The 60 km long corridor is (partially) traversed daily by approximately 300 trains per direction. Figure 5.9 shows the schematic representation of the observed network along with the train lines and the corresponding stopping pattern for the 2010 timetable, which was available for this study. The thin line illustrates a train line that runs once per hour, whereas the other lines operate twice per hour. Chapter 5. Real-time prediction of train event times 119 Leiden DevVink Dordrecht ht D d Zwijndrecht Barendrecht R’damvLombardijen R’damvZuid R’damvBlaak SchiedamvCentrum Delft Zuid Delft DenvHaag Moerwijk Rijswijk DenvHaagvHS DenvHaagvCentraal DenvHaag Mariahoeve DenvHaag LaanvvanvN.O.I RotterdamvCentraal Voorschoten Figure 5.9: Network and train lines for the case study The selected corridor and train routes enable testing the model with all possible train interactions – merging, diverging and intersecting routes. The part of the corridor between Delft and Rotterdam Centraal as well as the branches towards Den Haag Centraal is a double track line. The remaining part is a four-track line, where two tracks are dedicated for each direction. 5.6.3 Comprehensive analysis This section presents a comprehensive analysis of the prediction tool performance based on the application to one day of traffic on the corridor. The prediction algorithm is initiated and the rolling horizon moves after receiving each of the 9776 messages that report the realization of signal and station events that occurred on the observed day. Table 5.1 shows the average number of events that are predicted in each algorithm execution, and the average number of arcs for prediction horizons of 2 hours, 1 hour, 30 minutes, 20 minutes and 10 minutes. The average number of nodes and arcs are monotonically decreasing as shorter prediction horizons are considered. Table 5.1: Model size for different prediction horizons Average no. events Average no. arcs Prediction horizon [min] 120 60 30 20 10 1040 532 269 180 90 2288 1117 590 389 202 Figure 5.10 shows the box-plots of errors of event time predictions for each considered prediction horizon. The prediction error is computed as the difference between the actually realised event time and the predicted event time. The standard deviation of prediction error reduces with the decrease of horizon length. The median of the prediction error also follows a monotonically decreasing trend as a smaller prediction 120 Models for Predictive Railway Traffic Management horizon is considered. Medians in each box plot show a slight positive bias. In order to explain this, we define a negative error that occurs if the predicted event occurred earlier then predicted and the positive error if it occurred later then predicted. Thus the negative error of a process time estimate is bounded by the physical and operational constraints for the duration of the corresponding process. On the other hand, no such bound exists for the positive error, which explains the positive bias of prediction errors. 200 Prediction error [s] 150 100 50 0 -50 -100 -150 120 60 30 20 Prediction horizon [min] 10 Figure 5.10: Box plots of prediction error distributions for different prediction horizons The accuracy of predictions is indicated by the mean absolute error (MAE). The prediction horizon of 120 minutes is divided into 10 second wide intervals. The absolute prediction error is computed as the absolute value of the difference between the actually realised event time and the predicted event time. MAE is obtained in each interval by computing the mean value of all corresponding absolute prediction errors. The dependence of the MAE on the length of the prediction horizon is shown in Figure 5.11. The MAE is within 45 seconds even for the longest prediction horizon. The accuracy of predictions that are within a 30 minutes prediction horizon is significantly increased since more accurate information is available on events that have a direct impact on the realization time of an event. For the prediction horizons shorter than 30 minutes, the MAE is monotonically decreasing with horizon length. We obtain an average error shorter than 1 minute for all prediction horizons. The 10 minutes horizon shows that in terms of average prediction error, our model outperforms the approach of predicting event times using train motion equations (Dolder et al., 2009). The benefit of adaptive prediction when applied to all observed train runs in one day of traffic is shown in Figure 5.12. Improvements are noticeable for a prediction horizon of up to 10 minutes due to parameter l that defines the width of the moving average and m0 that defines the smoothing horizon (Section 5.5.3). Different combination of parameter Chapter 5. Real-time prediction of train event times 121 45 Mean absolute error [s] 40 35 30 25 20 15 10 5 0 0 20 40 60 80 100 120 Prediction horizon [min] Figure 5.11: Mean absolute prediction error depending on prediction horizon values were tested. Values l = 3 and m0 = 3 for intercity trains and l = j −1 and m0 = m for local trains showed the best performance in terms of MAE. Therefore, the running time estimates are adapted for the next three blocks for intercities and until the next scheduled stop for local trains. Similarly, the moving average is computed over the last three block sections for intercities and over all blocks since the last departure for local trains. 25 Meanlabsolutellerrorl[s] 20 Nonadaptivelprediction Adaptivelprediction 15 10 5 0 -5 0 100 200 300 400 500 600 700 Predictionlhorizonl[s] Figure 5.12: MAE comparison for adaptive and nonadaptive prediction The accuracy of route conflict predictions for different prediction horizons is strongly correlated with the mean absolute error. For prediction horizons longer than 30 minutes, approximately 80% of route conflicts longer than 30 seconds are accurately predicted. As shorter prediction horizons are considered, the accuracy of route conflicts prediction increases. For a 10 minutes horizon, 95% of route conflicts are accurately predicted. The running time adjustment does not show a noticeable global effect when the MAE 122 Models for Predictive Railway Traffic Management of all predictions of all events is considered. However, an accuracy analysis performed on the isolated set of running times of hindered trains, shows an increase in prediction accuracy of 8 seconds on average for a 30 minutes prediction horizon and 13 seconds for 10 minutes prediction horizon. Since the prediction algorithm is linear, the computational complexity, which depends on the size of the input graph, is not considered as a criterion for choosing the most appropriate prediction horizon. Even for the longest prediction horizon, the algorithm execution takes less than one second. Finally, Figure 5.13 shows the comparison of prediction accuracy of the model presented in this chapter with the conventional parallel shift strategy, which is typically applied for estimating train arrival and departure times. The analysis is performed for scheduled event times only. The benefits of real-time prediction are noticeable for every prediction horizon. 120 Parallel-shift Real-time-prediction-tool Mean-absolute--error-[s] 100 80 60 40 20 0 0 1000 2000 3000 4000 5000 6000 7000 8000 Prediction-horizon-[s] Figure 5.13: MAE of scheduled event times for a parallel shift strategy and the realtime prediction 5.6.4 Example of algorithm execution An example of predictions is shown in Figure 5.14. The presented time-distance diagram shows the predicted train paths (solid lines). The realized train paths in space and time are presented with dashed lines. The prediction is performed at the departure of train ST5025 from Den Haag HS (GV). The complete routes of the seven trains that enter the network within the 30 minutes prediction horizon are included in the predictions. The mean absolute prediction error for 161 predicted events (including signal passages) is 19.33 s, while the maximum prediction absolute error is 68.71 s. ST502 7 ST50 2 ST5 127 IC21 27 7 27 ST5 0 S22 IC9 27 216 27 IC212 25 ST51 IC21 27 ST 512 7 S22 IC19 S2 22 IC9 7 216 19 25 6 IC212 7 RSW IC921 IC19 25 DT IC DTZ S222 7 ST 50 25 ST5 025 SDM 27 IC92 16 ST 50 25 IC1 925 RTD 123 7 Chapter 5. Real-time prediction of train event times GVMW GV 07:13 07:21 07:30 07:38 07:46 07:55 08:03 08:11 08:20 Figure 5.14: Time-distance diagram of predicted (at 7:13) and realised train paths The major advantage of the presented model for traffic controllers is the ability to predict all route conflicts within the prediction horizon. We use the principle of overlapping blocking times (Hansen & Pachl, 2008) to predict and visualize route conflicts. Figure 5.15 shows the predicted (up) and realized (down) blocking time diagram. Local trains are presented in magenta and intercity trains in blue. Overlaps in blocking times that indicate route conflicts are given in red. The three out of the four major route conflicts that occurred, one in Schiedam (SDM) and two in Rotterdam (RTD), were predicted by the model. The fourth conflict that was not captured occurred more than 30 minutes after the moment of prediction. The very short running time of IC2127 between stations Delft South (DTZ) and Schiedam (that is visible in Figure 5.14) caused a route conflict with the preceding ST5127 at SDM, which was not captured by the model. An example of adaptive prediction that minimizes the prediction error for running trains is shown in Figure 5.16. The example considers intercity train IC1919 that does not have scheduled stops between The Hague HS (GV) and Delft (DT). The first prediction, derived at the moment of departure from The Hague HS, is represented by the blue line. After three corrections of running time estimates resulting in predictions, the prediction error of less than 1 second was achieved (the final running time estimate practically overlaps the realized running time given with dashed line). The predicted arrival time error is monotonically decreasing as the train progresses towards station Delft. Therefore the propagation of prediction error to connected events of other trains is reduced thus affecting the overall performance of the model. 07:13 RSW 07:21 07:30 07:38 25 19 07:46 07:55 027 SDM ST502 7 27 IC21 ST51 27 5 27 07:55 ST50 ST5 127 IC2 127 S22 27 IC9 216 5 92 502 IC1 07:46 ST5 27 IC21 7 16 IC92 S222 925 IC1 07:38 7 IC S2 22 IC 7 92 16 ST 51 27 RTD 7 IC212 6 IC921 27 ST5 027 IC21 27 ST 51 S2 22 IC9 7 216 25 IC 19 25 27 ST502 7 IC212 7 ST51 S22 27 IC92 16 IC19 25 25 ST5 0 ST50 2 7 ST5 127 IC21 27 S22 I C9 27 216 ST 50 25 IC1 925 RTD IC212 DT 07:30 16 ST 07:21 S222 7 50 ST DT IC92 25 ST5 0 RSW IC19 25 DTZ S222 25 50 DTZ ST SDM 7 925 07:13 IC1 124 Models for Predictive Railway Traffic Management GVMW GV 08:03 08:03 08:11 08:11 08:20 GVMW GV 08:20 Figure 5.15: Blocking time diagram predicted at 7:13 (up), realized blocking time diagram (down) Chapter 5. Real-time prediction of train event times 125 19 19 DT IC RSW GVMW GV 05:56:40 05:58:20 06:00:00 06:01:40 06:03:20 Figure 5.16: Effects of adaptive prediction 5.7 Conclusions and outlook This chapter presented a tool for accurate prediction of event times based on a directed acyclic graph with dynamic arc weights. The process times in the model are obtained dynamically using processed historical train describer data, thus reflecting all phenomena of railway traffic captured by the train describer systems and preprocessing tools. The graph structure of the model allows applying fast algorithms to compute prediction of event times even for large and busy networks. The main contribution of this approach is the dynamic estimation of process times for each train by using the predetermined functional dependence of process times on actual delays. Train interactions are modelled with high accuracy by including the main operational constraints and relying on the actually realised corresponding minimum headway times (obtained from the historical data) rather than on theoretical values. The recursive depth-first search algorithm with dynamic arc weights gives predictions for all event times within the horizon. The predictive traffic model supports route setting and traffic control decisions and could be interactively used by signallers and traffic controllers. First, the model predicts route conflicts for a given actual route plan and train positions. This could be used by the signaller to pro-actively resolve the conflict by e.g. changing routes or the order of trains. The impact of any control decision can be checked by an update of the predictive model leading to new conflict and arrival time predictions. If a control 126 Models for Predictive Railway Traffic Management decision leads to satisfying results it can be implemented in the actual process plan. If on the other hand a route conflict cannot be avoided, the signaller could give speed advice (or new target passage times) to the relevant train drivers so that the impact of the route conflict is minimal and energy can be saved by preventing unnecessary braking and re-acceleration. Second, the arrival time predictions could be used to check connections in the case of arrival delays. When a connection conflict is detected, the signaller may decide to secure or cancel a connection in advance. This way up-todate passenger information can be provided, both at stations and in the delayed trains. Similarly, endangered logistic connections (crew or rolling-stock) can be predicted in advance. The model has been applied in a case study on a busy corridor in the Netherlands in a real-time environment using train describer log files, and produced accurate estimates for train traffic and route conflicts within 30 minutes. Application of the model to a wider area is possible either by enlarging the observed area or by coordinating multiple areas. Finally, the model structure enables straightforward application of the networkwide delay propagation algorithm (Goverde, 2010) to estimate the further effect of current traffic conditions (or examined traffic control actions). For larger examples that model dense traffic, it is expectable that more events can occur almost simultaneously, i.e., more than one update can arrive within one second. The presented event-driven prediction algorithm can be modified to a time-driven version where the prediction process is performed in regular time intervals based on the information that arrived within the interval. Finally, performance analysis has been conducted using raw data from train describer log files. Even though data are very detailed and of high quality, various errors in logging of event times still occur which may affect the accuracy of subsequent predictions. In future research robustness of the model to noise and errors in the data need to be increased in order to provide more stable and accurate predictions. Chapter 6 Rescheduling models for real-time traffic management in large networks This chapter is an edited version of the article Kecman, P., Corman, F., D’Ariano, A., and Goverde, R. M. P. (2013). Rescheduling models for railway traffic management in large-scale networks. Public Transport, 5(12), 95–123. 6.1 Introduction Real-time rescheduling of railway traffic is an important task of traffic controllers with a great impact on punctuality and reliability of train traffic. Reliable predictions and information from the monitoring system, that are described in the previous chapters, are crucial for developing robust rescheduling models. In case of disturbances, accurate predictions of train delays may serve as an input for traffic controllers or rescheduling models that aim to reduce delay propagation and deviation from the scheduled timetable. This chapter focuses on rescheduling models that can be integrated in a closed loop with monitoring and prediction systems as described in Figure 1.4. In current practice, traffic controllers attempt to reduce the effects of disruptions and delays using a set of predetermined rules and their own experience. Traffic in local control areas or on the network level is controlled without a reliable supporting tool to compute optimal control decisions. That often leads to creating suboptimal solutions with new conflicts and effects on the network level. This problem has been tackled by numerous approaches available in the academic literature (§2.6). The scope of the existing models is limited to a single traffic control area (Caimi et al., 2012; D’Ariano, 2008; Pellegrini et al., 2014) or a subnetwork (Corman et al., 2012b; T¨ornquist Krasemann, 2011). The resulting solutions are (near) optimal in the respective areas. However, Goverde (2010) showed that in large and busy networks delays often propagate 127 128 Models for Predictive Railway Traffic Management across multiple traffic control areas and, depending on their magnitude, often have an impact on the network level. We therefore aim at developing a global, network scale optimization tool that optimizes the actual state over the overall network and controls the traffic from a global perspective with adjustments to the timetable. In this chapter, we examine the applicability of macroscopic models for rescheduling railway traffic at a network-wide level. Railway traffic is represented by a timed event graph that allows computing delay propagation in large and strongly interconnected networks in a short time. The timed event graph is then converted to four alternative graph models with different operational constraints. An efficient solution algorithm D’Ariano, Pacciarelli, and Pranzo (2007) is applied on the alternative graph models to optimize the rescheduling actions. All presented models have been tested on a series of delay scenarios, compared to each other, and evaluated by comparison to the mesoscopic model of D’Ariano (2008), that takes into account detailed infrastructure data and train dynamics. All comparisons have been performed in terms of resulting secondary delay and dispatching decisions on a case study of the corridor between Utrecht and Den Bosch in the Netherlands. Moreover, the macroscopic models have been applied on a test case of one peak hour of the Dutch national timetable in order to test their applicability on large and busy networks. An important objective of the work presented in this chapter was to investigate the trade-off between precise modelling of operational constraints of railway traffic and the time needed to compute a (near) optimal solution for a large network. Macroscopic models with different levels of abstraction, result in different quality of solutions and computation time. The aim is to select such level of the granularity of the macroscopic model that enables computing a feasible solution of high quality in short time. The next section describes the general approach to macroscopic modelling of railway operations as well as the procedure used to solve the rescheduling problem and obtain the new schedule. Section 6.3 gives a specific description of presented models. Sections 6.4 and 6.5 report on a comparison of the models on a railway corridor and on the whole Dutch network, respectively. Finally, we discuss the performance of each model and give directions for future research (§6.6). 6.2 6.2.1 Macroscopic modelling of railway operations Timed event graphs Railway operations can be modelled at the macroscopic level by means of timed event graphs (TEG), as formally defined by Goverde (2007). A TEG is a representation of a discrete-event dynamic system which consists of events, connected by processes that are described by the minimum process times. The major difference from the graphbased model presented in the previous chapter is the pure macroscopic character of the model. Chapter 6. Rescheduling models for large networks 129 An individual train run is modelled as a series of events and processes that connect them. Every node is an event, defined by the train number, the timetable point (station, stop on open track, junction), type (departure, arrival or through) and the scheduled event time. A through event is the moment when a train passes the centre axis of a timetable point without stopping. Every running or dwell process is modelled by an arc, defined by the train number, type (run or dwell), start and completion event, and the minimum process time. Interactions between trains are modelled with headway and connection processes. Headway processes separate events of different trains that have identical, opposite, intersecting, merging or merging routes. The minimum headway time between two trains is computed according to the blocking time theory (Hansen & Pachl, 2008). All events in a TEG take place on the platform tracks or centre axis of a timetable point. Since route conflicts may occur at signals that prevent a train from entering the occupied or reserved block, a minimum headway time needs to be computed between events in the station where conflicting outbound routes start and inbound routes end (minimum line headway). The connection processes separate the departure event of a connecting train and arrival events of each feeder train by a minimum connection time. An event in a TEG can occur only after all processes represented by incoming arcs of the corresponding node have been completed. Events in a TEG occur in a fixed sequence determined by the topology of the graph. The fixed structure of a timed event graph is a major obstacle for the straightforward application in the field of realtime rescheduling since many dispatching decisions imply changes of relative order of occurrence among events. In this chapter, we overcome this limitation by converting a TEG to an alternative graph (Mascis & Pacciarelli, 2002). 6.2.2 Alternative graphs An alternative graph (AG) is a representation of a job-shop scheduling model with additional operational constraints. On a mesoscopic level, the train rescheduling problem posed as a job-shop scheduling problem (D’Ariano, 2008) is to schedule a finite set of jobs (trains), defined by fixed sequences of operations (train runs and dwellings) which cannot be interrupted, on a finite set of resources (block sections or platform tracks) that can perform one operation at a time (no-store or blocking constraint). The objective is to schedule all operations on the corresponding resources and to minimize the secondary delay. We extend this model to a macroscopic scale by aggregating multiple block sections into open track segments and platform tracks into timetable points and use them as resources in macroscopic models. Therefore, the number of operations that can simultaneously be handled by one resource, depends on the capacity of that resource. Alternative graphs consist of nodes N, fixed arcs F, and pairs of alternative arcs A. We add the connection arcs C to this generic formulation as in Corman et al. (2012a). We use the following notation: M, O, T are sets of resources (machines), operations and 130 Models for Predictive Railway Traffic Management trains (jobs), respectively; i, j are indices of resources mi , m j ∈ M; r, s, are indices of trains tr ,ts ∈ T . We denote by xir the starting time and by pri the processing time of operation ori ∈ O of train tr on resource mi . The headway time between the starting times of operations of trains tr and ts on resource mi is denoted by hr,s i . Ti is a set of trains that use resource mi . A node in the graph represents a single operation ori ∈ O of job tr ∈ T , that is performed on resource mi ∈ M. Every node is described by the starting time xir of the corresponding operation. Since one job consists of a predetermined sequence of operations, node xir at the same time represents the completion time of the previous operation. An arc (xir , xsj ) ∈ {F ∪A∪C} with weight pri represents the precedence relation between operations ori and osj given by the following equation. xsj ≥ xir + pri , ∀i, j, r, s : mi , m j ∈ M,tr ,ts ∈ T (6.1) Fixed arcs are used to model fixed precedence relations between operations that have to be performed in a fixed relative order. Alternative arcs are decision variables used to determine the relative order of operations scheduled to be performed on the same resource. If operations ori and osi are scheduled to be performed on the same resource mi , then the relative order of operations can be determined by selecting the appropriate alternative arc. The concept of alternative arcs can be modelled with the binary control variable kir,s such that: kir,s = ( 1 if xir < xis , 0 otherwise ∀i : mi ∈ M, ∀r, s : tr ,ts ∈ Ti , (6.2) with the constraint that exactly one arc from each pair has to be selected: kir,s + kis,r = 1, ∀i, r, s : mi ∈ M, ts ,tr ∈ Ti . (6.3) In sections 6.2.4–6.2.4 we define different resource types, each of them with particular properties in terms of capacity. For each resource mi with limited capacity, the starting times of two operations ori and osi depend on the value of the corresponding decision variables kir,s and kis,r that define the relative order of operations. The constraints that formalize this dependence are given separately for each resource type in the corresponding section. A selection of exactly one arc from each pair is called a complete selection. The objective is to select alternative arcs in a way that would minimize the waiting time of all operations. A valid solution determines the precedence relations between each two operations that are scheduled on the same resource. Two basic properties need to be respected: (i) completeness (exactly one arc from each alternative pair is selected), (ii) consistency (it must not contain positive length cycles). A complete and consistent Chapter 6. Rescheduling models for large networks 131 selection yields the full schedule of all jobs. A possible interpretation of a complete selection of an alternative graph is a mode in a switching max-plus linear system (Van den Boom & De Schutter, 2007). Connections can be represented by a constraint between events of different trains. If there is a scheduled connection between a feeder train ts and the connecting train tr then: xir ≥ xsj + cs,r , (6.4) where xir is the starting time of the operation of train tr on resource mi (a departure event when mi models an open track) and xsj is the starting time of the operation of train ts on resource m j (an arrival event when m j models a station). The arc (xsj , xir ) ∈ C is weighted by cs,r , that represents the minimum connection time. Scheduled starting and completion times of operations (timetable constraints) are incorporated in an AG by means of dummy operations (nodes) 0 and n, with starting times x0 and xn . If operation ori is scheduled to start at time dir , the fixed arc (x0 , xir ) with weight dir (release time) is added to ensure that the operation cannot start before its scheduled starting time. In real operation, scheduled completion time αri (due date) of operation ori may become infeasible due to disturbances that can cause extension to the planned processing time pri (primary delays). This delay can propagate over successive operations of the train tr , thus making their due dates also infeasible. A modified due date can therefore be defined by − max (αri , τri ) where τri is the earliest possible completion time of operation ori , considered isolated from interactions with all other operations not belonging to job tr , that cannot be improved by any rescheduling action. A fixed arc with the weight equal to the modified due date is added to the graph from the node that represents the completion time of the operation to node n. We define an unavoidable delay by r − max (αr , τr )) and a total delay as max (0, τri − αri ), a secondary delay by max (0, xi+1 i i the sum of the unavoidable and secondary delays. D’Ariano (2008) showed that minimisation of the critical path between nodes 0 and n is equivalent to minimizing the maximum secondary delay over all operations. Thus the objective function can be formally expressed with: min xn − x0 . (6.5) The solution procedure (D’Ariano, Pacciarelli, & Pranzo, 2007) determines the starting times xir for every operation, and values of binary variables kir,s , that represent the orders within each pair of trains tr ,ts ∈ T scheduled to use the same resource mi . An exact search is performed in the solution space by means of a branch and bound algorithm. A good initial solution is found by a set of heuristics (first come first served, first leave first served, avoid most critical completion time). The solution procedure is truncated 132 Models for Predictive Railway Traffic Management after a time limit (in this work 5 min) is reached. We refer to Corman et al. (2014) and D’Ariano, Pacciarelli, and Pranzo (2007) for additional information on the solution procedure. 6.2.3 Conversion of timed event graphs to alternative graphs The macroscopic modelling of railway traffic by means of alternative graphs is explained with the terminology introduced in Chapter 2. Each timetable point and each open track segment is modelled by a resource. An individual train run is modelled as a sequence of operations. Every operation is attributed by the starting time, duration and the resource traversed by the train. A train run is represented with nodes and fixed arcs. In order to describe how a TEG is converted to an AG, the difference in meanings of nodes and arcs in these two graphs needs to be resolved. We keep the interpretation of nodes and arcs as in TEG and convert it to AG in the following manner. Fixed arcs represent operations (run or dwell). Weight of each fixed arc is equal to the minimum processing time of the corresponding operation. Every node is an event, i.e. arrival or departure (a through event is included by fixing the dwell time to 0), representing the start of the operation denoted by the outgoing fixed arc and completion of the operation denoted by the incoming fixed arc. Every departure after a scheduled stop is connected to node 0 and every arrival to a station with a scheduled stop to node n, as explained in the previous section. Having in mind that each operation in an AG is associated to a particular resource, we dedicate an additional resource attribute to each event in the corresponding TEG model. Arrival and through events are augmented with the timetable point resource. On the other hand, the open track resource is added to each departure event. The advantage of using AG for macroscopic models is in modelling discrete decisions that manage interactions between trains. If two trains have operations that cannot be simultaneously performed on the same resource with constrained capacity, at least one pair of alternative arcs weighted by the minimum headway time between two operations is added in order to specify the precedence relation between operations. The number of alternative pairs and their start and end nodes depend on the resource type. The order of operations is determined by selecting the appropriate arc from the pair. This way, resolution of intra-track conflicts (conflicts between trains using the same resource) can be appropriately modelled. However, inter-track conflicts on a macroscopic level are modelled in a different manner since they represent conflicts between operations taking place on different resources. Modelling of inter-track conflicts will be explained in detail in Section 6.3.1. Connections in the macroscopic AG models are fixed and modelled in the same way as in timed event graphs. A connection arc is added between the node that models an arrival event of the feeder train and the node that models a departure event of the connecting train with the weight equal to the minimum connection time. Chapter 6. Rescheduling models for large networks 6.2.4 133 Resources as building blocks of alternative graphs Infrastructure elements can be modelled by using resources with different properties in terms of capacity. This results in multiple models with different complexity and number of operational constraints. Before presenting a detailed description of the macroscopic models (Section 6.3), we first describe the essential meaning of each type of resource. Infinite capacity resources (ICR) The simplest way of specifying a resource is by considering only the temporal duration of the scheduled operation. This means that no further restriction is posed on train orders and headways between trains. Therefore, these resources do not model interactions between trains and all trains can use them independently from each other. This resource type is used under the assumption that capacity is sufficient to accommodate demand at all times, thus no conflict can occur and the only binding constraint is the processing time. Figure 6.1 shows how two trains tr and ts are modelled on infinite capacity resource (ICR) type resource mi . Each node represents the starting time of an operation on the resource (xir is a starting time of operation ori of train tr on resource mi ). Arcs represent the operation that started at their parent node and their weight is equal to the minimum processing time of operation (pri is the minimum processing time of operation ori ). There are no arcs between operations associated with different trains in Figure 6.1, thus operations of both trains can be performed independently from each other. The only constraint that has to be respected when scheduling operations on this resource type is given by equation (6.1). xri-1 xsi-1 pri-1 psi-1 xri xsi pri psi xri+1 xsi+1 Figure 6.1: Graph representation of resources with infinite capacity Infinite capacity resource with headway (ICR+H) If a resource is modelled as infinite capacity with headway, the number of operations that can simultaneously be processed is not restricted. However, the starting times of two consecutive operations ori and osi on the same resource mi are separated by a time interval defined with headway hr,s i . Trains are thus prevented to occupy the same infrastructure element within a predefined headway time. Introducing a minimum time separation between the starting times of two operations on a resource does not constrain completion times of operations, which are not constrained 134 Models for Predictive Railway Traffic Management by any headway. They can therefore occur simultaneously and not necessarily in the same relative order in which they started. Figure 6.2 (left) depicts the alternative graph that can be used to compute the starting time of both operations ori and osi on resource (machine) mi . Alternative arcs are shown with dashed lines with weights equal to the minimum headway time between two operations of different trains on the same resource. A possible selection is given on the right side of the figure. Note that in order to independently observe the properties of this type of resources, neighboring resources mi−1 and mi+1 are modelled as non-constrained infinite capacity resources. If we define by M2 ⊂ M a set of machines of type infinite capacity resource with headway (ICR+H), the starting time of an operation scheduled on this resource can be fully defined by equations (6.1)-(6.3) and the additional constraint, representing the choice of train orders: r,s xir ≥ xis + hs,r i − L · ki , ∀i, r, s : mi ∈ M2 , tr ,ts ∈ Ti , (6.6) where L is a sufficiently large number (larger than the latest completion time of the latest operation). xri-1 xri pri-1 psi-1 xri-1 xsi xri pri-1 hir,s his,r xsi-1 xri+1 pri xri+1 pri his,r psi xsi+1 xsi-1 psi-1 xsi psi xsi+1 Figure 6.2: Graph representation of resources with infinite capacity and headway constraint (left) and a possible selection (right) Infinite capacity resources with FIFO property (ICR+FIFO) This type of resource is an extension of the type infinite capacity with headway in the sense that an additional headway constraint is imposed on the completion times of operations (Mascis et al., 2002). Note that capacity is limited only by the headway constraints that separate the starting and completion times of two successive operations. The graph depicting two operations on resource mi of type infinite capacity resource with FIFO property (ICR+FIFO), is shown in the left part of Figure 6.3 (adjacent resources mi−1 and mi+1 are modelled as non-constrained infinite capacity resources). In contrast to other resource types, two operations, performed on the same resource, are separated with two pairs of alternative arcs. Namely, alternative arc (xir , xis ) that s , xr ) that gives assigns precedence to start of operation ori is paired with arc (xi+1 i+1 precedence to completion of operation osi . In the same way, alternative arc (xis , xir ) is r , xs ) (arcs belonging to one pair are shown in the same color). paired with arc (xi+1 i+1 That way, both starting times and completion times of two operations are separated. Chapter 6. Rescheduling models for large networks 135 However, the increase in the number of arcs does not directly contribute to the increase in complexity since the selection of an arc from one pair implies the selection of an arc from the second pair, i.e. two pairs of alternative arcs represent only one decision variable (e.g. selection of arc (xis , xir ) from the blue pair implies the selection of arc s , xr ) from the red pair, as shown in the right part of Figure 6.3). Selection of (xi+1 i+1 arcs (one from each pair) that would violate the first in first out (FIFO) constraint would result in a positive length cycle, which is not permitted neither in a TEG (Goverde, 2007) nor in an AG (D’Ariano, 2008). xri-1 xsi-1 xri pri-1 psi-1 xri+1 pri his,r hi+1r,s hir,s hi+1s,r xsi psi xsi+1 xri-1 pri-1 xri xri+1 pri his,r hi+1s,r xsi-1 psi-1 xsi psi xsi+1 Figure 6.3: Graph representation of resources with infinite capacity and FIFO constraint (left) and a possible selection (right) If M3 ⊂ M is a set of machines of type ICR+FIFO, the starting time of an operation scheduled on those machines can fully be defined by constraints (6.1)-(6.3), (6.6) and the additional constraints: r,s r s xi+1 ≥ xi+1 + hs,r i+1 − L · ki+1 , ∀i, r, s : mi ∈ M3 , tr ,ts ∈ Ti (6.7) r,s kir,s = ki+1 , ∀i, r, s : mi ∈ M3 , tr ,ts ∈ Ti . (6.8) Finite capacity resources (B) The most restrictive resource type allows only one operation to be processed at the same time. Figure 6.4 shows that an operation that is processed second, can be initiated only after the preceding operation has been completed and the required headway time has passed. xi-1,r pi-1,r xi,r xi+1,r pi,r xi-1,r pi-1,r xi,r xi+1,r pi,r hir,s his,r xi-1,s pi-1,s xi,s pi,s his,r xi+1,s xi-1,s pi-1,s xi,s pi,s xi+1,s Figure 6.4: Graph representation of resources with finite capacity (left) and a possible selection (right) 136 Models for Predictive Railway Traffic Management If we define by M4 ⊂ M a set of machines of type B, the starting time of an operation scheduled on those resources can fully be defined by constraints (6.1)–(6.3) and the additional constraint: r,s s xir ≥ xi+1 + hs,r i − L · ki , ∀i, r, s : mi ∈ M4 , tr ,ts ∈ Ti . 6.2.5 (6.9) Sequence-dependent setup times Minimum headway times between two trains depend on blocking time diagrams of both trains (Hansen & Pachl, 2008). Figure 6.5 shows the AG of an example with q three trains: tr , ts , tq using resource mi (type IC+H) with starting times xir , xis , xi , respectively. In the standard AG model, all alternative arcs coming out of a node have equal weights, which is suitable for mesoscopic modelling of railway traffic. However, in the macroscopic models, presented in this chapter, this formulation needed to be extended to allow different weights of outgoing alternative arcs from a node. For q,r q q,s example, hi (headway between xi and xir ) does not have to be equal to hi (headway q between xi and xis ). In that manner, we are able to model minimum headways between trains according to the blocking time theory. hiq,r hiq,s xs i his,r xri xqi his,q hir,s hir,q Figure 6.5: Example of sequence-dependent setup times Kecman, Corman, D’Ariano, and Goverde (2012)presented the models without sequencedependent setup times. In that approach, the weights of all outgoing headway arcs from a node are equal. They are computed as the maximum of all minimum headway times between the event (modelled by the node) and the conflicting events. In the examq,r q,s q,r q,s q,r q,s ple from Figure 6.5, hˆ i = hˆ i = max(hi , hi ), where hˆ i , hˆ i are weights of arcs q q (xi , xir ) and (xi , xis ) without sequence-dependent setup times. A precise modelling of train headways with complex sequence-dependent setup times complicates the setting from an algorithmic point of view (Corman, Goverde, & D’Ariano, 2009), as discussed in detail in Section 6.5.2. Chapter 6. Rescheduling models for large networks S1 T1 T2 OT1 St OT2 137 S2 T3 OT3 S3 Figure 6.6: Layout of the illustrative example 6.3 Models examined 6.3.1 Macroscopic models A description of the four rescheduling models for network-wide traffic management will be given in this section. The resources presented in the previous section will be used to model different infrastructure elements. All macroscopic models assume unidirectional traffic on double track lines. Bidirectional open track segments (single track line segments) are modelled with resource type B under the assumption of low traffic volumes over such segments. That approach is conservative because it limits the capacity of the line segment to one train at a time which in reality is not the case for successive trains running in the same direction. Macroscopic models will be described on an illustrative example shown in Figure 6.6. Infrastructure elements in the example are stations S1, S2 and S3, stop St and open track segments OT1, OT2 and OT3. Trains T1 and T2 run from S1 to S2 on open track segments OT1 and OT2. Train T1 has a scheduled stop at St. Train T3 runs from S3 to S1 on open track segment OT3. Routes of the three trains T1, T2 and T3 are presented in Figure 6.6 with arrows of corresponding colours. Since trains T1 and T2 use the same open track segments, all potential conflicts between them can be characterized as intra-track conflicts. However, conflicts between the inbound route of train T3 and the outbound routes of trains T1 and T2 at station S1 are an example of inter-track conflicts. Figures 6.7, 6.8, 6.10 and 6.11 present the alternative graphs for each described model. Every node is an operation of a train (defined by the colour) on the specified resource. Dummy nodes (0 and n) incorporate timetable constraints in the model. Fixed arcs are presented in colours that correspond to train colours from Figure 6.6. They are marked by type of operation: run or dwell. Train departure is modelled as a start of operation on the open track resource and train arrival as a start of operation on a timetable point resource. The outgoing fixed arcs from node 0 are weighted by the scheduled departure times (SDT). The incoming fixed arcs to node n are weighted by modified due dates (MDD) as explained in Section 6.2.2. 138 Models for Predictive Railway Traffic Management SDTStT1 S1 SDTS1T1 dwell OT1 run St dwell OT2 run MDDStT1 S2 MDDS2T1 0 SDTS1T2 SDTS3T3 S1 S1 dwell run OT1 OT3 run dwell St S3 dwell OT2 run S2 MDDS2T2 n MDDS1T3 Figure 6.7: Illustrative example - Model 1 Alternative arcs are shown in dashed lines. For the sake of clarity their weights (minimum headway times) are not shown in the figures. Model 1 This is the simplest macroscopic model considered in this chapter. The AG of the illustrative example modelled by Model 1 is shown in Figure 6.7. All timetable points are modelled as resources with infinite capacity and no constraints (Section 6.2.4). This black-box approach to modelling stations relies on the assumption that the capacity of each station is at all times sufficient to satisfy demand. Open track segments that connect stations are modelled with resource ICR+H (Section 6.2.4). A pair of alternative arcs is added to ensure the time separation between starting times of two successive operations on the same open track resource (departures). However, headways between arrivals are not considered in this model and the order of arrivals is not implied by the order of departures. Moreover, inter-track conflicts are not included as a constraint in this model which has a great level of idealization and its use can only be justified with low complexity and short computation time. Operational constraints considered in this model satisfy the requirements for modelling homogeneous traffic (all trains have equal speeds) on the line. In that case, trains are separated in time at the departure points and the model assumes fixed running times, thus arrival headways become redundant if trains have the same running time. Model 2 We extend the previous model by considering arrival headway time and sequence of arrivals to a timetable point from the same open track segment. That is achieved by modelling open track segments with resource type ICR+FIFO (Figure 6.8). This ability to model intra-track conflicts between trains with different speeds on the line results in the increased size of AG, since the number of alternative arcs used to model train Chapter 6. Rescheduling models for large networks 139 SDTStT1 S1 dwell OT1 run St dwell OT2 run S2 MDDStT1 SDTS1T1 0 SDTS1T2 SDTS3T3 S1 S1 dwell run OT1 OT3 run dwell St S3 dwell OT2 run MDDS2T1 S2 MDDS2T2 n MDDS1T3 Figure 6.8: Illustrative example - Model 2 interactions on open track segments has doubled when compared with Model 1. However, the complexity of this model is not directly influenced by the increase of the size of the graph as shown in Section 6.2.4. Model 3 None of the previously presented macroscopic models is able to capture potential intertrack conflicts. Since the two potentially conflicting operations are performed on different resources (different open track segments), capacity constraints associated with a single resource are not able to model these conflicts. In order to overcome this, we introduce an additional finite capacity resource with processing time 0. This resource does not have any physical interpretation (we therefore refer to it as a virtual resource) and its purpose is to separate in time events leading to inter-track conflicts. If two trains with conflicting routes through a timetable point arrive to (depart from) the timetable point using different open track segments, we add the virtual resource to the path of each train. An inbound route is represented by an arrival event (the resource is added between the open track and the timetable point) and an outbound route with a departure event (the resource is added between the timetable point and the open track resource). Having an additional resource results in the additional operation (therefore also a node in the AG) with processing time 0. A pair of alternative arcs is then added between every two nodes that represent events leading to an inter-track conflict, in order to regulate the precedence relation between the two events. Figure 6.6 shows an example of potential inter-track conflicts between train T3 and trains T1 and T2 at station S1. Figure 6.9 shows the resulting incompatibility graph. Events that can lead to conflict and can thus not occur within a specified headway time are connected by undirected arcs (red for inter-track and black for intra-track conflicts). The alternative graph for this illustrative example is shown in Figure 6.10. Alternative arcs between virtual resources D1, D2 and A3 are added according to the incompatibility graph (Figure 6.9), where red pairs represent inter-track conflicts and black pairs 140 Models for Predictive Railway Traffic Management represent intra-track conflicts. For example, a possible inter-track conflict in station S1 between T2 and T3 is modelled in the following way. If T2 departs first, T3 can arrive (operation A3 can start) only after T2 has departed (operation D2 has been completed, i.e. operation at resource OT2 of train T2 has started) and corresponding headway time has passed. Similarly, if T3 arrives first, T2 can depart (operation D2 can start) only after operation at station S1 of train T3 has started and the minimum headway time has passed. Since the time separation of trains running on the same open track segment is ensured by the selection of the alternative arcs related to the open track resource, there is no need to separate virtual resources representing D1 and D2 by additional alternative arcs. D2 A3 D1 Figure 6.9: Incompatibility graph of illustrative example Model 4 In this model, we partition the set of timetable points to stations, where overtaking is possible and stops on open tracks (or other timetable points), with no additional tracks to accommodate overtaking. The important property of the latter group is that their capacity allows only one operation (dwelling or through ride) at a time per direction. Stations are modelled with resources type ICR like in the previously described models. Stops are modelled with two resources of type B, one per direction. That way, due to the properties of this resource type (Section 6.2.4), timetable points where overtaking is SDTStT1 S1 SDT S1 dwell D1 OT1 0 run St dwell OT2 run MDDStT1 S2 T1 MDDS2T1 0 SDTS1T2 S1 dwell D2 S1 SDTS3T3 OT2 0 0 A3 run run St OT3 dwell dwell OT2 S3 run S2 MDDS1T3 Figure 6.10: Illustrative example - Model 3 MDDS2T2 n Chapter 6. Rescheduling models for large networks 141 SDTStT1 S1 SDT S1 dwell D1 OT1 0 run St dwell OT2 run MDDStT1 S2 T1 MDDS2T1 S1 0 SDTS1T2 dwell D2 OT2 0 S1 SDTS3T3 0 A3 run run St OT3 dwell dwell OT2 S3 run S2 MDDS2T2 n MDDS1T3 Figure 6.11: Illustrative example - Model 4 not possible cannot be occupied by more than one train per direction at the same time. Overtaking is in this model enabled only in stations with sufficient number of tracks and appropriate layout. The alternative graph of the illustrative example is presented in Figure 6.11. 6.3.2 Mesoscopic model The mesoscopic model of D’Ariano (2008) is used to evaluate the performance of each macroscopic model studied here. This model has been validated and tested on numerous case studies. The model incorporates all operational constraints of railway traffic and provides accurate estimations of train movements at the level of block sections and signals. 6.3.3 Overview of the five models Table 6.1 summarizes operational constraints which are taken into account in the presented models. A gradual increase in number of considered operational constraints in the presented sequence from Model 1 to the mesoscopic model (Meso) is visible. Depending on the network and traffic properties such as: capacity of stations, possibilities for occurrence of inter-track conflicts and heterogeneity of traffic, the appropriate modelling approach can be applied. Another important criterion for selecting the most appropriate model is the size of the resulting graph and the computation time needed to obtain a solution of good quality. The performance of each model in terms of this criterion depends mainly on the network size and the number of trains (as shown in the following sections). 6.4 Test case A: corridor Utrecht - Den Bosch Comprehensive evaluation of the macroscopic models relies on comparison with the mesoscopic model, which requires detailed infrastructure data on the level of block sections, signals and valid rolling-stock dynamics. 142 Models for Predictive Railway Traffic Management Table 6.1: Operational constraints in models Model Model1 Model2 Model3 Model4 Meso 6.4.1 Stations capacity X Stops capacity X X Inter track conflicts X X X Intra Departure track conflicts headway X X X X X X X X X Test case settings All models have been applied to one hour of a timetable for the busy double-track line between Utrecht (Ut) and Den Bosch (Ht) in the Netherlands. We also consider a branch that leads to station Den Bosch Oost (Hto) and merges with the main corridor in Diezebrug junction (Htda). Track layout of the corridor is presented in Figure 6.12. Utrecht Den Bosch Geldermalsen Diezebrug junct. Zaltbommel Den Bosch oost Culemborg Houten Lunetten Figure 6.12: Layout of infrastructure and main stations Sporenplan (2014) The macroscopic infrastructure layout with all timetable points (stations, stops and junctions) is presented in the lower part of Figure 6.13. Big circles represent large stations where overtaking is possible (since Ht and Ut are area limits in this study, overtaking can be performed only in Geldermalsen), small circles represent stops on open track and the red circle in Htda specifies that inter-track route conflicts are possible. In the periodic hourly timetable (Figure 6.13) there are four pairs (one per direction) of intercity trains that run between Utrecht and Den Bosch without stopping in intermediate stations. There are also two pairs of regional trains that stop in Zaltbommel (Zbm), Geldermalsen (Gdm), Culemborg (Cl), Houten (Htn), Utrecht Lunetten (Utl) and two pairs between Ut and Gdm (also stop in Cl, Htn, Utl). Trains operating between Den Bosch and Htda (junction with a branch toward Nijmegen–Nm) in Figure 6.13 are two pairs of intercity trains and two pairs of regional trains running on the service between Nijmegen and Den Bosch. Chapter 6. Rescheduling models for large networks 143 [min] 0 0 10 20 30 40 50 60 Ht Htda Mbh Ozbm Zbm Gdm Cl Htn Utl Ln Utva Ut 4000 Hto 5000 Nm Figure 6.13: Timetable The scheduled departure and arrival times are given in the timetable for each station. The minimum dwell time is 120 s in large stations Ut, Gdm and Ht and 60 s in stops. All minimum running times in the macroscopic models are computed by a standard approach of simulating each train run using the mesoscopic model and summing up minimum running times over the corresponding block sections. Kettner et al. (2003) and Schlechte et al. (2011) used a similar concept. The minimum headway times are in the mesoscopic model computed according to so-called ’departure on yellow’ concept of blocking time theory (Hansen & Pachl, 2008), i.e., a train is allowed to depart as soon as the previous train has released the first block section. This reflects the behaviour of local traffic controllers in disturbed conditions. The logic of blocking time theory is implemented in the mesoscopic model. Therefore, interactions between trains along the open track segments are regulated with high precision (i.e., a block section can never be occupied by more than one train). On the other hand, macroscopic rescheduling models need to mimic the behaviour of network traffic controllers and aim to produce a new operational and conflict-free timetable with minimum deviation from the published timetable. We impose a minimal headway for open track segments at departure (in all macroscopic models) and arrival events (Model 2, 3 and 4). We use the standard approach to compute minimum headway times by compressing the blocking time diagrams (Hansen & Pachl, 2008). For computing arrival headways and inter-track headways, a two-block separation principle of blocking time theory for conflict-free running was used. 144 6.4.2 Models for Predictive Railway Traffic Management Comprehensive evaluation The five models were applied to the corridor test case. The solution procedure described in Section 6.2.2 was used to minimize secondary delay in all models. The complete equivalence of all models is achieved in terms of departure and arrival times of trains, when they were applied without delays. In the following subsections the quality of solutions obtained by the macroscopic models will be evaluated by comparisons with the mesoscopic model (reference model). The smaller the differences, in terms of relative orders of trains, between the solutions obtained using the mesoscopic model and those obtained using a macroscopic model, the better is the performance of the macroscopic model under evaluation. Comparisons between the objective values will be performed only among the macroscopic models due to the different way of computing the minimum headway times in the mesoscopic model. A comprehensive evaluation of the models was performed over 200 delay instances. All trains from the timetable shown in Figure 6.13 are delayed in each instance according to the Weibull distributions as in Corman et al. (2012b). The maximum primary delay is 326.80 s and the average primary delay is 30.15 s (both values are average over all instances). All experiments are performed on a computer with Intel Core i5520M/2.4 GHz processor and 4 GB memory. Quantitative analysis In the quantitative part of evaluation, presented in Table 6.2, the size of the resulting AG for each model is given in number of nodes, number of fixed arcs and number of alternative pairs (Columns 2–4). We also present the average computation time (CTF) to obtain the first solution using initial heuristics and average computation time (CTB) to compute the best solution or prove optimality for the initial solution (Columns 5–6). Moreover, average (ASD) and maximum (MSD) values of secondary delay over all instances are presented for each model (Columns 7–8). Table 6.2: Quantitative assessment of the 5 models Model Model1 Model2 Model3 Model4 Meso Nodes Fixed arcs 394 505 394 505 410 521 410 521 1018 1155 Alt. pairs 558 1116 1164 1636 2312 CTF (s) <1 <1 <1 <1 <1 CTB (s) <1 <1 <1 <1 1.20 ASD (s) 4.17 6.14 7.50 11.05 5.88 MSD (s) 84.16 112.00 118.00 182.00 119.56 As expected, the size of the graph increases together with the number of operational constraints considered in each model. There is a large difference in terms of ratio number of nodes/number of alternative pairs, between Model 1 and the mesoscopic Chapter 6. Rescheduling models for large networks 145 model on the one side, and Models 2, 3 and 4 on the other. That can be explained by the fact that Models 2, 3 and 4 employ ICR+FIFO resource type for modelling open track segments. Therefore, those models need twice as many pairs of alternative arcs to model train runs along open tracks compared to Model 1 (see Section 6.2.4). For applications on this relatively small test case all five models show excellent performance in terms of computation time to obtain the first as well as the best solution. For the macroscopic models, the solution is produced almost instantaneously (around 0.1 s for the best solution in Model 4), whereas the size of the mesoscopic model causes a slightly longer computation time to prove the optimal solution. For this set of instances, the optimal solution was always found for all models. The last two columns of Table 6.2 show that the average and maximum secondary delay increase along with the number of operational constraints taken into account in each macroscopic model, meaning that the more realistic models are able to capture more interactions between trains and therefore compute more realistic delay propagation (the mesoscopic model is not considered in this analysis due to different computation of minimum headway times). Comparison of train reordering actions Reordering trains (changing the order of departures) is a common dispatching action for reducing delay propagation. In this section, we will investigate how close are the solutions of macroscopic models to the solution of the reference mesoscopic model in terms of orders of departures. The analysis has been carried out on trains running from Den Bosch towards Utrecht. There are three checkpoints where the relative order of trains is determined: through runs in Htda, departure from Gdm and arrival in Ut. By checking the orders of through runs in Htda we are able to estimate the effect of considering inter-track conflicts that are possible to occur in the junction Htda. According to the published timetable, intercity trains are scheduled to overtake slower regional trains in Gdm. Therefore, checkpoints in Gdm and Ut are used to verify if some macroscopic models provide solutions with a different point of overtaking (which in reality is infeasible). The first three rows of Table 6.3 give the percentage of train sequences (for each macroscopic model) that are different from the corresponding sequences produced by the mesoscopic model, in each check point on the 200 instances. The last row of the table shows the percentage of different sequences aggregated over all three check points. In almost all instances, the solutions of the four macroscopic models suggest identical sequences of departures from Htda as the mesoscopic model. In this checkpoint, differences in operational constraints among models, only to small extent affect the resulting relative order of trains. By comparing the percentage of different sequences of departures from Gdm and arrivals to Ut for each model, it is visible that in a large number of instances, Models 1, 2 and 3 allow overtaking between Gdm and Ut (the number of differences at arrival to Ut is much smaller than the number of differences at departure from Gdm). In Model 146 Models for Predictive Railway Traffic Management Table 6.3: Difference in orders between the mesoscopic and each macroscopic model. Direction Ht → Ut Model 1 Model 2 Model 3 Model 4 Through run Htda (%) 1.0 1.0 0.5 0.0 Departure from Gdm (%) 33.5 20.0 20.0 2.0 Arrival to Ut (%) 4.5 4.0 4.0 2.0 Average (%) 13.0 8.3 8.2 1.3 4, the percentage of different sequences is the same in both check points which implies that the relative order of trains that depart from Gdm is maintained until Ut. Therefore, we can conclude that Model 4 showed the best performance in terms of feasibility of solutions. This comparison of aggregated differences shows that Model 4 gives solutions closest to the accurate mesoscopic model compared to other macroscopic models. Only 1.3% departure sequences are different on the three checkpoints. Other macroscopic models show greater deviation from solutions provided by the mesoscopic model. This deviation percentage is again correlated to the number of operational constraints included in the models. 6.5 Test case B: Dutch national railway network The primary purpose of this section is to test the applicability of the macroscopic models presented in Section 6.3 for the management of large and busy networks. Figure 6.14 shows the test case of the Dutch national network that represents one of the busiest railway networks in the world with more than 700 passenger trains per hour operating during peak hours. 6.5.1 Description of the tested instances Input data for the macroscopic models of traffic on the Dutch national network is obtained from the macroscopic timetabling tool DONS (Hooghiemstra, 1996), that is able to generate a periodic hourly timetable on the national level with all scheduled event times in all timetable points (departures and arrivals) and scheduled process times (running and dwell times, connection times and headways) rounded to full minutes. Slack times and time reserves are not included in the DONS constraints database, which is used to build the timed event graph. Scheduled headway times are normally used for timetabling purposes. However, in real-time traffic management, trains are separated by minimum headway times. Due to unavailability of exact blocking times, the procedure of computing minimum headways, explained in Section 6.4 could not be applied. Instead, the norms given in the Dutch network statement (ProRail, 2013) were used. In order to reduce the size of the problem without losing validity we have computed all strongly connected components in the graph as explained in Goverde (2007). If a Chapter 6. Rescheduling models for large networks 147 Groningen Leeuwarden Alkmaar Hoorn Zwolle Leiden Almelo Amsterdam Haarlem Deventer Hilversum Schiphol A’foort Den Haag Hengelo Enschede Arnhem Zutphen Utrecht Gouda Nijmegen Rotterdam Dordrecht Den Bosch Roosendaal Breda Eindhoven Venlo Heerlen Maastricht Figure 6.14: Dutch railway network considered (in black), with main stations primary delay occurs within a strongly connected component, it cannot propagate to other strongly connected components. Therefore, each strongly connected component of a TEG corresponds to an autonomous model. The strongly connected component considered in this example comprises the largest part of the Dutch national hourly timetable and takes into account all trains operating on the lines depicted by black solid lines in Figure 6.14. Thick solid lines represent double and multiple-track segments, whereas the thin solid lines stand for single-track segments. Table 6.4 reports specific information on the network-wide test case. We take into account all intercity, regional and freight trains (reserved slots). 6.5.2 Comprehensive evaluation The four macroscopic models have been tested on 200 delay instances in which all trains were delayed according to Weibull distribution, similar as in Section 6.4.2. The maximum primary delay is 18.22 min and the average primary delay is 1.41 min (both values are average over all instances). Table 6.5 reports average results for the network-wide instances on each macroscopic 148 Models for Predictive Railway Traffic Management Table 6.4: Characteristics of the network-wide test case Instance property Stations Other timetable points Unidirectional open track segments Bidirectional open track segments Trains Connections Number 298 294 1119 324 679 84 model (Column 1): the number of nodes, fixed arcs and alternative pairs (Columns 2– 4), the average computation time (CTF) to obtain the first solution using initial heuristics (Column 5), average computation time (CTB) to compute the best solution or prove optimality for the initial solution over all instances (Column 6) and the average (ASD) and maximum (MSD) secondary delays (Columns 7–8). All values in Columns 5–8 are average over 200 instances. Table 6.5: Quantitative assessment of the macroscopic models on test case B Mode Model1 Model2 Model3 Model4 Nodes Fixed arcs 17490 20591 17490 20591 18968 22069 18968 22069 Alt. pairs 16494 32380 33956 42750 CTF (s) 9.89 50.79 52.95 89.43 CTB (s) 10.04 50.85 53.22 89.57 ASD (min) 0.21 0.25 0.29 0.34 MSD (min) 4.75 5.17 6.09 6.72 Model 4, the most realistic macroscopic model, captures the largest amount of secondary delays compared to the other macroscopic models. The more precise information comes with a cost in the alternative graph size and in the computation time of solution algorithms. For Models 1–3, the instances were solvable with the standard setting reported in Section 6.2.2. For Model 4, the increased complexity defined a set, comprising 32% of the instances, that are harder to be solved. A solution for those instances could be found only by considering additional initial heuristics, as in Corman et al. (2014). However, 10% of all instances of Model 4 needed more than 5 minutes to be solved. The branch and bound algorithm proves optimality for 90%, 84%, 82% and 80% of instances of Models 1–4, respectively. For the remaining instances the branch and bound algorithm is not able to compute the optimal solution within the given time limit of computation. Finally, the computational cost of implementing sequence-dependent setup times in our models is analysed. There is an increase of 15% (on average over all models) in terms of computation time, compared to the case without sequence-dependent setup Chapter 6. Rescheduling models for large networks 149 times, where conservative, larger minimum headway times were considered (Kecman et al., 2012). Comparing the results and especially the different feasibility rates in the two studies, it appears that the instances tackled in the current work are more challenging. They require longer computation times and the percentage of feasible solutions is smaller. A possible reason for the identified differences is that the headways considered in the case with sequence-dependent setup times are shorter. The two options of alternative ordering of trains are therefore more similar to each other, thus trains are competing more closely for priority, and there is a higher chance to have multiple operations requesting the same resource at the same time. In other words, the resulting schedule is more compact, and computing a good quality solution is more challenging. 6.5.3 Network-wide effects of reducing delay propagation In order to demonstrate the effect of minimization of secondary delays on the national network, we compare the delay propagation that arises if the relative order of events (departures and arrivals of all trains) remains as scheduled in the timetable, with the secondary delays that occur as a result of the solution procedure on Model 4. The maximum primary delay for a typical instance is 16 min and the average primary delay is 1.24 min. The total secondary delay accumulated in all stations is 3093 min when the order of events is fixed and 1611 min if secondary delays are minimized by applying the solution procedure (Section 6.2.2) on Model 4. Figure 6.15 presents the maximum secondary delays in major stations in the Netherlands with fixed order of events (left) and after modifying the order of events as proposed by the optimal solution (right). Without rescheduling actions, the maximum secondary delays are the largest in the busiest part of the network around Amsterdam (Asd) and Utrecht (Ut), as well as in Leiden (Ledn), Apeldoorn (Apd) and Tilburg (Tb). Secondary delays still occur after optimization in the busiest part of the network but the network-wide effect of rescheduling actions is clearly visible compared to the left part of Figure 6.15. 6.6 Conclusions and outlook The potential further growth of both passenger and freight flows in already busy railway networks in western Europe will mostly have to be accommodated over the existing railway infrastructure. This will lead to an increase of capacity utilization, thus reducing reliability and punctuality of railway services. Improvements in traffic management and control have to be made in order to prevent a decrease of traffic reliability. In that context, this contribution leads to an improvement of global delay propagation in case of disturbances. This chapter presented four models of railway traffic flows at a macroscopic level. The trade-off between the level of detail included in each model and the number of considered operational constraints was examined in terms of minimization of secondary 150 Models for Predictive Railway Traffic Management 21 18 Gn Lw 16 Asn Hr Gn Lw Asn Hr 14 Amr Amr Zl Aml Asd Ledn Shl Asdz Gvc Gv Rta Rtd Asa Amf Apd Hgl Tb Es 11 Ledn Shl 8 6 Gv Rta Rtd Wt Mt Amf Apd Ut Hrl 1 Max6VsecondaryVdel6 [min] Es Em Ehv Wt 2 Hgl Nm Ht Tb Bh Dv Ah Ddr Bd Rm Std Asa Hm Vl 4 Ehv Asdz Gvc Em Hm Bd Aml Asd Nm Ht Ddr Zl 12 Dv Ah Ut Bh Mt Vl Rm Std Hrl Figure 6.15: Maximum secondary delays without (left) and with (right) rescheduling delay and computation time. A comprehensive evaluation was performed on two realworld case studies. We were able to handle very large instances such as the Dutch national network within short time (less than 90 seconds) even with the most complex macroscopic model, which had the best performance with respect to feasibility of solutions. A major contribution of the work presented in this chapter is a modification to the mesoscopic alternative graph railway rescheduling model of D’Ariano (2008). The macroscopic capacity constraints were implemented by choosing the appropriate resource type for each infrastructure element. By maintaining the structure of the alternative graph models, efficient solution procedures developed by D’Ariano, Pacciarelli, and Pranzo (2007) and Corman et al. (2014) became applicable for for solving rescheduling problems on a network-wide level. Moreover, the transformation of the mesoscopic constraints to the macroscopic level was performed with respect to feasibility of the resulting models. This was achieved by applying the concept of sequencedependent setup times for implementing the realistic minimum headway time values for each considered train sequence. The presented models are applicable for a decision support system for network traffic control. The potential for model improvements is in studying other traffic disturbances and dispatching measures, such as global rerouting. This chapter presented an MILP formulation of the macroscopic rescheduling problem. A large number of efficient commercial software packages exist that provide great flexibility for selecting the objective function. This would enable to examine different passenger oriented Chapter 6. Rescheduling models for large networks 151 objectives, possibly dependent on the magnitude of disruption. However, D’Ariano, D’Ariano, Sam´a, and Pacciarelli (2013) presented a study that compared the solution quality and computation time required by a commercial software and the branch & bound algorithm (D’Ariano, Pacciarelli, & Pranzo, 2007). The commercial software underperformed in both aspects for large instances. Therefore, this direction for future research requires a reduction of problem size in a preprocessing step before applying a commercial software. 152 Models for Predictive Railway Traffic Management Chapter 7 Conclusions 7.1 Summary of the main findings and contributions The work presented in this thesis was dedicated to developing the components and building blocks of a model-predictive controller for railway traffic management. Modelpredictive control can be used to plug in the tools for traffic control that assume full knowledge of the future to an online environment. The main components of modelpredictive control: monitoring, short-term prediction and optimisation are translated in the context of real-time management of railway traffic. These challenging problems from the current traffic control practice have been tackled by numerous contributions from the professional, academic and scientific community. A review of the existing approaches revealed two clear gaps that determined the main research objectives set for this research. The first objective was to develop a system that monitors train traffic and predicts its future evolution. The monitoring system needs to keep track of the traffic state. That includes monitoring of train positions, running times, actual delays, headways and route conflicts. Based on the current traffic state in the network, the evolution of the future traffic state within a certain time horizon needs to be predicted. The second research objective focused on creating a real-time rescheduling model that can produce (near) optimal schedules on the network level. The aim was to create a global traffic model that takes into account all interdependencies between trains in the network. Given the predicted traffic state at the end of the rescheduling computation procedure, the model should provide a solution that minimises the deviation from the reference plan within a short computation time. This chapter gives a summary of the main findings, and scientific and practical contributions of this research. Recommendations for the future research are given in Section 7.2. 153 154 7.1.1 Models for Predictive Railway Traffic Management Monitoring and traffic state prediction The purpose of this research objective was to develop a monitoring and short-term prediction system that can be embedded in an MPC loop, as well as used independently to support traffic controllers in supervising and managing traffic in their part of the network. The availability of high-quality traffic realisation data from the Dutch train describer system TROTS motivated a data-driven approach. The general idea was to analyse and quantify the impact that current traffic conditions may have on the process times. The future process times were predicted using the actual traffic state information from the monitoring system. In order to include interdependencies between trains in the predictions, a model was developed that captures the operational constraints of railway traffic and identifies all route and connection conflicts. The model was calibrated in real time using process time estimates derived with respect to the actual traffic conditions. A fast critical path algorithm predicts all event times within the prediction horizon. Process mining train describer data The first step in the development of the system for monitoring and traffic state prediction was to develop a data mining algorithm that can quickly extract occupancy times of infrastructure elements, recover train paths and identify route conflicts from the train describer log file. The work resulted in a process mining tool implemented in an object-oriented environment applicable for quick processing of large data archives and real-time data streams. The analysis of the system architecture and data structure of TROTS archives revealed several drawbacks for applications in process discovery and performance analysis on open track sections. Consequently, several preprocessing steps have been developed that enable traffic monitoring on open tracks and not just in station areas. Moreover, signal messages have been coupled to section and train messages to enable the analysis of train runs on the level of block sections. The algorithm discovers and keeps track of processes such as train runs on the level of block sections, dwell times, and headway times between all trains at every infrastructure element. Moreover, the tool continuously monitors the actual delays, and the realised running and dwell times of all trains. The accuracy of the arrival and departure time measurements is significantly improved compared to the measurements obtained by the method currently in use in the Netherlands. The resulting data structure is convenient for statistical analysis and model calibration in this and other research projects. Hindered train runs are identified and can be filtered out to calibrate the models with conflict-free running times. Finally, the algorithms have been implemented in a tool equipped with a visualisation component that simplifies the analysis of realised or actual traffic conditions. The applicability of the tool is strongly dependent on the data structure and format of train describer log archives. The level of information captured by a train describer system varies depending on the particular infrastructure manager. However, process mining is a generic method that can be applied for discovering processes and extracting Chapter 7. Conclusions 155 information. In the process mining tool presented in this thesis, a three-level model is implemented which enables extracting information about processes on a micro-, mesoand macroscopic level. Predictive modelling of process times The second step in the development of the monitoring and traffic state prediction system is to develop predictive models that can derive robust estimates of process times, depending on the current traffic conditions. The process time estimates can be used for calibrating traffic prediction models. Having in mind the system requirement for a traffic model that needs to identify and model route conflicts, the estimates of running times were derived on the mesoscopic block section level. The application of transparent statistical learning methods, such as robust linear regression and tree-based non-linear methods, resulted in several insights about running times. Domain knowledge and hypotheses related to railway traffic were used to define a set of predictors for process time estimation. Two approaches were presented. First, a single global predictive model was developed that discovers dependency of process times on a set of predictors. The second approach relies on a separate model for each train line, block section and station. A set of predictors for running times was determined based on the three months of train describer event data from two traffic control areas in the Netherlands. Both linear and tree-based methods reveal a weak dependence of running times on departure delays. Furthermore, the small variation of running times is explained to a great extent by the block length and position with respect to the previous and following scheduled stops. Observations for a particular train line and block section confirmed that no clear distinction can be made between the running times of delayed and punctual trains. The headway time passed since the preceding train run turned out to have an impact on train running time even for conflict-free train runs. The predictive modelling of dwell times required close attention due to the high variation of dwell times observed in the training data set. The arrival delay and scheduled dwell time turned out to be the strongest predictors of dwell times especially in large stations. A statistical analysis of dwell times of a particular train line revealed that the dwell times of delayed trains are responsive to peak-hour variations. Moreover, the analysis of coefficients and intercept after applying robust linear regression revealed the magnitude of the inevitable error that occurs when the dwell times are estimated using only train describer data. A high percentage of variance of dwell times can be explained using the developed predictive models. However, the difficulty to predict the dwell times of local trains still represents the major source of inaccuracy for the prediction model. For more accurate estimates, other data sources than train describers (e.g. on-board units) need to be used. A high predictive power of the presented models was established by cross-validating the models and applying them on an independent test set. Robust linear regression gave insight into the predictive quality of each individual explanatory variable but the 156 Models for Predictive Railway Traffic Management accuracy of this model is still insufficient for real-time applications. The tree-based methods managed to capture the non-linear relationship between the response and explanatory variables. The prediction accuracy was significantly increased especially after applying the random forests method. The large set of available data was exploited to derive the local models that on average outperform the global model in terms of accuracy. The limitation of this approach is that a test set needs to be related to the same infrastructure area and train lines as the training set. Real-time prediction of train event times The final step for developing the monitoring and traffic state prediction system was to create a traffic model. The model is built and updated based on the traffic control actions and current train positions reported by the train describer system. The model topology reflects all capacity and synchronisation interdependencies between trains. The calibration is performed in real time with the robust estimators of process times. With each update of train positions, an efficient prediction algorithm visits all arcs in the graph, retrieves their weights depending on the actual traffic condition, and predicts the realisation times of all signal and station events within the prediction horizon. The mesoscopic character of the graph allows identification of all route and connection conflicts. An improvement of the prediction model was achieved by accurate modelling of the train dynamics for the trains hindered by route conflicts. The process mining tool was used to filter out all route conflicts from the training data set. The time loss due to braking, running at lower speed, waiting in front of the signal at danger, and re-acceleration was determined. Moreover, a robust statistical model established and quantified a strong correlation between the time loss and conflict duration. For every predicted route conflict, the corresponding running times of the hindered trains can be adjusted to take into account the expected time loss. The tool has been further extended with an online adaptive component that keeps track of the realised running times of trains in real time. The trains with running times that deviate from their robust estimates in a certain pattern are identified and downstream estimates are adapted to reduce the expected prediction error. This can be used to identify malfunctioning trains, peculiar driving styles or trains that significantly differ from the trains used in the training set, with respect to dynamic properties. The prediction accuracy was validated against data from the test set. Prediction horizons of different lengths were examined and a significant decrease of prediction errors was revealed for horizons shorter than 30 minutes. An average prediction error smaller than one minute was obtained even for a prediction horizon of two hours. This is a significant improvement compared to the current practice or the approaches described in the literature. The applicability and the quality of results of the presented approach depend significantly on the availability and quality of data for model calibration as well as the frequency and spacial resolution of train position updates. For practical applications a Chapter 7. Conclusions 157 high quality data sources that provide frequent and accurate updates on train positions are required. At the moment, the presented results after the application of the method on the Dutch train describer event data indicate a direct applicability of the proposed approach in practice on the Dutch national network. 7.1.2 Network-wide traffic rescheduling The second research objective in this thesis was directed at real-time rescheduling for large networks with dense traffic and many interdependencies between trains. Operational constraints of railway traffic were translated to the macroscopic level where the only events are arrivals and departures in stations. The traffic was modelled by means of alternative graphs which enabled the implementation of dispatching actions in the model. Four macroscopic models were created, each with a different number of operational constraints included. The impact of including an additional constraint on the model complexity and feasibility of produced solutions was analysed. The models were validated using an accurate mesoscopic model on a case study of a single corridor. The feasibility of solutions produced by the macroscopic models turned out to be strongly correlated with the number of included operational constraints. More realistic models produced a larger number of solutions that were equivalent to the reference obtained by using the detailed model. A large case study of one peak-hour of the Dutch national timetable was used to demonstrate the applicability of the models for real-time applications with respect to the computation time required to produce a solution. Tackling a problem of such size would be infeasible with the existing mesoscopic model. The expected positive correlation between the number of considered constraints and the computation time was confirmed. Even the most complex considered macroscopic model was able to produce optimal solutions in less than 90 seconds which shows its suitability for practical applications. In the context of online application and integration with the predictive model, there are several constraints for the presented macroscopic rescheduling models. The arc weights in the models were fixed and the model assumes full knowledge of future train movements during the prediction horizon. Therefore, the process time estimates are independent of the actual traffic condition. However, different solutions may cause different values of delays and headway times which in the current models has no impact on the process times. This limitation is related to the solution procedure used to compute the optimal schedule that does not support dynamic arc weight computation. The solution procedure (D’Ariano, Pacciarelli, & Pranzo, 2007) minimises the maximum consecutive delay. However, the objectives for rescheduling in the context of network-wide traffic control may differ depending on the magnitude of disruption and the condition of traffic on the network. In this thesis an alternative MILP formulation of the macroscopic alternative graph model was given that enables applications of the generic MILP solvers and heuristics that provide greater flexibility in choosing the appropriate objective function. This enables implementation of the passenger oriented 158 Models for Predictive Railway Traffic Management objectives that minimise the passenger delay rather than minimising the maximum secondary train delays. 7.2 Recommendations for future work In this section we present three general directions for future work on models for predictive railway traffic management. The first direction should be focused on a further improvement of the presented models. Secondly, we analyse a possible research direction towards the integration of the models. Finally, the challenges and opportunities for practical implementation are discussed. A possible improvement of the developed system for monitoring and traffic state prediction mostly depends on the availability of data from sources other than train describer systems. The data-driven approach for real-time prediction of train traffic turned out to produce stable and accurate predictions of train event times. A further improvement of the model accuracy is envisaged through a more detailed modelling of dwell times. Additional data on the station design and train length are required in order to determine the exact stopping position of each train. This can result in a significant reduction of the error for registering the exact arrival and departure times. Moreover, detailed train event recorder data could enable accurate modelling of separate subprocesses of dwell times. Finally, the recent trend of migration from passenger tickets to smart cards in the Netherlands Railways opens a possibility for computing estimates of the number of passengers even in real time, provided that almost all passengers use the latter (Van der Huurk, Kroon, Mar´oti, & Vervest, 2012). The use of accurate passenger counts may result in increased accuracy of dwell time estimation. Further work on improving the presented macroscopic models for real-time rescheduling in large networks should be dedicated to speeding-up the solution procedure. A potential method to increase the computation speed is to investigate the performance of advanced metaheuristics that quickly provide feasible solutions of good quality albeit with no guarantee of optimality. Furthermore, the presented MILP formulation enables the application of commercial software packages that offer great flexibility for choosing the objective function. However, in order to exploit the benefits of flexibility of commercial software, the problem size needs to be reduced. A possible algorithmic way to to that is by reducing the problem size in a preprocessing step. A heuristic can be developed that may reduce the graph size by limiting the number of alternative reordering options to the most probable set. A similar approach, based on stochastic modelling, has been implemented by Acuna-Agost et al. (2011) on a mesoscopic level. An important aspect for the future research on this topic is the integration of the presented models. The impact of uncertainty on the feasibility of solutions produced by a rescheduling system could be significantly decreased by means of a more comprehensive prediction system, similar to the one presented in this thesis. One way for integrating the models into an online process would be by plugging them into a model-predictive control loop. In an laboratory environment this requires the use of Chapter 7. Conclusions 159 a microscopic simulation tool that would simulate the real-time operation and enable the implementation of the rescheduling decisions. Another possible direction is to create a single integrated system that supports a dynamic, time-dependent computation of process times in the optimisation phase. Integration of the monitoring and traffic state prediction with an advanced driver advisory system is also a possible direction for future work. Accurate predictions of route conflicts could be used to compute optimal train trajectories that would prevent a conflict or minimise its consequences related to the waiting time and energy consumption. An interesting challenge in this aspect would be to investigate the mutual impact of the two systems. In particular, it is important to analyse how the future predictions are affected by the driver advisory system. The reaction of the driver and compliance to the given advice need to be considered in a closed-loop regime between the two systems. Finally, we focus on the practical implementation of the described models. The monitoring and traffic state prediction systems have been developed based on the data format of the train describer system TROTS. This system is currently in use for tracking train positions across the Netherlands by the infrastructure manager ProRail and provides data of sufficient quality. Thus the presented tools are practically applicable in a straightforward procedure by plugging the system to a live data stream. Moreover, having in mind the constant increase of the quality and availability of the traffic data in many railway companies, there are promising prospects for the general applicability of the data-driven approach for monitoring and traffic state prediction on other networks. The potential practical application of the rescheduling model depends on the infrastructure managers which are still reluctant to applying computer aided rescheduling systems even as a decision support to traffic controllers. The presented models could provide fast and reliable support for network traffic controllers for example at the network control centre such as the OCCR in the Netherlands. It contains a centralised information system that can provide live data feed to the rescheduling system and distribute the solutions to the corresponding local centres that need to implement them. A technical requirement for application of the presented models is the availability of a continuous and accurate estimate of the traffic state over the whole network that could serve as an input for the rescheduling process. That implies the necessity to integrate the monitoring and prediction systems from the multiple local traffic control centres. 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Routing trains through a railway station based on a node packing model. European Journal Of Opera- 176 Models for Predictive Railway Traffic Management tional Research, 128, 14–33. (Cited on page 45.) Acronyms AG alternative graph. ARI Automatische Rijweginstelling. CSV comma separated values. DAG directed acyclic graph. DCSC Delft Centre for Systems and Control. DFS depth-first search. DONS Design of Network Schedules. DUT Delft University of Technology. FCFS first come first served. FIFO first in first out. GPS Global Positioning System. GSM-R Global System for Mobile Communications-Railway. GUI graphical user interface. ICR infinite capacity resource. ICR+FIFO infinite capacity resource with FIFO property. ICR+H infinite capacity resource with headway. ILP integer linear programming. IM infrastructure manager. LTS least trimmed squares. MAE mean absolute error. 177 178 Models for Predictive Railway Traffic Management MILP mixed integer linear programming. MPC model-predictive control. MSE mean squared error. NS Netherlands Railways. OCCR Operational Control Centre Rail. OOB out of bag. PESP periodic event scheduling problem. PRL Procesleiding. RCS Rail Control System. ROMA Railway traffic Optimization by Means of Alternative Graphs. RSE residual standard error. RSS residual sum of squares. STEG Styrning av T˚ag genom Elektronisk Graf. T&P Department of Transport and Planning. TEG timed event graph. TNV Treinnummer Volgsysteem. TOC train operating company. TROTS Train Tracking and Observation System. TSS total sum of squares. VKL Verkeersleiding. Summary This thesis is focused on predictive management of railway traffic in real time. The main research topics include: (1) monitoring and real-time traffic prediction and (2) rescheduling in large and heavily utilised networks. Railway traffic control is typically hierarchically structured into a local and a global (network) level. Local traffic control (signallers and/or dispatchers) has the task to perform all safety related actions, set routes for trains, predict and solve conflicts, and manage processes that take place on the designated part of infrastructure. A train typically crosses multiple traffic control areas. The global level (regional or network controllers) comprises the supervision of the state of traffic on the network level, detection of deviations from the timetable, resolution of conflicts affecting the overall network performance, handling failures and events that may have big impact on performance indicators, etc. Signallers in general do not have any intelligent decision support system to estimate the expected running times. Delay propagation could be prevented or reduced if the traffic was managed proactively, i.e., if controllers had a reliable prediction of a route and connection conflict with a possibility to prevent it. The current practice in operational control of disruptions and delays still relies predominantly on the predetermined rules and experience and skills of personnel. Neither local nor network traffic controllers have an efficient supporting tool to make dispatching decisions, predict their effect and evaluate them. A possible way to model and optimize railway traffic control is through a closed-loop control approach, called model-predictive control (MPC). This thesis presents an MPC framework and railway traffic control models that can be integrated in the closed control loop. Trains are operated according to a timetable and a daily process plan. Due to inevitable disturbances and deviations from the planned schedule, train runs need to be continuously monitored. Monitoring provides the actual traffic state that can be used to predict the future evolution of traffic on the network. A predictive traffic model is thus required to continuously provide the local control level with the information about the expected traffic conditions. It can further be used to evaluate the impact of traffic control actions. In case of longer disruptions that may affect the traffic in a wider area, network traffic controllers can use the prediction model to optimise traffic on the network, compute network-optimal timetable updates and transmit them as a reference to 179 180 Models for Predictive Railway Traffic Management the local level. That way all traffic control actions on the local level will match with the network-optimal traffic state. Monitoring and traffic state prediction A way to overcome the drawbacks of the current practice and the existing tools for monitoring and short-term traffic prediction emerged with the availability of historical traffic realisation data. This thesis shows how a real-time stream of raw train describer data from the Dutch TROTS system can be processed in a way that extracts the actual traffic condition in the network: train positions, accurate estimates of current delays and realised running and dwell times. Moreover, archives of event logs are used to learn how trains behave depending on the traffic conditions. The variability of process times is explained by isolating the factors with high impact on the corresponding process time. Estimates of future process times depend on the current or predicted values of explanatory variables. Therefore, predictions incorporate the empirically determined variation of process times due to e.g. driving style, passenger behaviour or peak hours. The final step for developing the monitoring and traffic state prediction system was to create a traffic model. The model is built and updated based on the traffic control actions and current train positions reported by the train describer system. The model topology reflects all capacity and synchronisation interdependencies between trains. The calibration is performed in real time with the robust estimates of process times. The monitoring tool is based on the process mining algorithm that discovers and keeps track of processes such as train runs on the level of block sections, dwell times, and headway times between all trains at every infrastructure element. Moreover, the tool continuously monitors the actual delays, and the realised running and dwell times of all trains. The accuracy of the arrival and departure time measurements is significantly improved compared to the current practice. The resulting data structure is convenient for statistical analysis, calibration and validation against real-life data in this and other research projects. Hindered train runs are identified and can be filtered out to calibrate the models with conflict-free running times. Finally, the algorithms have been implemented in a tool equipped with a graphical user interface and a visualisation component that simplifies the analysis of the realised or actual traffic conditions. The running times are estimated based on the correlation and dependence on explanatory variables. Both linear and tree-based methods reveal a weak dependence of running times on departure delays. Furthermore, the small variation of running times was explained to a great extent by the block length and position with respect to the previous and following scheduled stops. Observations for a particular train line and block section confirmed that no clear distinction can be made between the running times of delayed and punctual trains. The headway time passed since the preceding train run turned out to have an impact on train running times even for conflict-free train runs. The predictive modelling of dwell times required close attention due to the high variation of dwell times observed in the training data set. The arrival delay and scheduled Summary 181 dwell time turned out to be the strongest predictors of dwell times especially in large stations. The statistical analysis of dwell times of a particular train line revealed that the dwell times of delayed trains are responsive to peak-hour variations. Moreover, the analysis of coefficients and intercept after applying robust linear regression revealed the magnitude of the inevitable error that occurs when the dwell times are estimated using only train describer data. A high percentage of variance of dwell times can be explained using the developed predictive models. However, the difficulty to predict the dwell times of local trains still represents the major source of inaccuracy for the prediction model. For more accurate estimates, other data sources than train describers (e.g. on-board units) need to be used. A graph-based traffic model has been developed that accurately represents operational constraints of railway traffic. With each update of train positions, an efficient prediction algorithm visits all arcs in the graph, retrieves their weights depending on the actual traffic condition, and predicts the realisation times of all signal and station events within the prediction horizon. The high level of detail in the model allows the identification of all route and connection conflicts. The prediction accuracy was validated against the actual realisation data from the test set. The prediction horizons of different lengths were examined and a significant decrease of prediction errors was revealed for horizons shorter than 30 minutes. An average prediction error smaller than one minute was obtained even for the prediction horizon of two hours. That is a significant improvement compared to the current practice or the approaches described in the literature. A further improvement of the prediction accuracy was achieved by accurate modelling of the train dynamics for the trains hindered by route conflicts. If a route conflict is predicted, the corresponding running times of the hindered train can be adjusted to take into account the expected time loss. The tool has been further extended with an online adaptive component that keeps track of the realised running times of trains in real time. The trains with running times that deviate from their robust estimates in a certain pattern are identified and downstream estimates are adapted to reduce the expected prediction error. This can be used to identify malfunctioning trains, peculiar driving styles or trains that significantly differ from the trains used in the training set, with respect to dynamic properties. Macroscopic models for network-wide rescheduling The second research objective in this thesis is to develop a decision support system for network traffic controllers that can be integrated in an MPC loop. The system is based on a macroscopic rescheduling model that can be applied for optimal control of traffic in large and heavily utilised networks. Alternative graphs are used as a modelling tool for traffic rescheduling. This requires a definition of different resource types to model the macroscopic constraints of railway traffic. A series of models, each with a different level of granularity, is presented with the purpose to search for a compromise between precise modelling of railway capacity constraints and a reasonable time to compute the alternative solutions for the large scale railway traffic management instances. A suitable choice of the granularity of the macroscopic model is determined that reflects 182 Models for Predictive Railway Traffic Management the balance between limiting the problem complexity and maintaining the feasibility of produced solutions. The macroscopic models are validated using an accurate detailed model on a case study of a single corridor. Furthermore, a large case study of one peak-hour of the Dutch national timetable is used to demonstrate the applicability of the models for realtime applications with respect to the computation time required to produce a solution. The expected positive correlation between the number of considered constraints and the computation time was confirmed. However, even the most complex considered model was able to produce optimal solutions in less than 90 seconds which shows its suitability for practical applications. Samenvatting Dit proefschrift richt zich op voorspellend railverkeersmanagement. De belangrijkste onderzoeksthema’s zijn: (1) monitoring en real-time voorspelling van het treinverkeer en (2) de herplanning bij vertragingen in grootschalige en zwaar belaste netwerken. Railverkeersleiding is meestal hi¨erarchisch gestructureerd in een lokaal en een globaal (netwerk) niveau. De lokale verkeersleiding (treindienstleiders) heeft de taak om alle veiligheid gerelateerde acties uit te voeren, rijwegen voor treinen in te stellen, conflicten te voorspelen en op te lossen, en processen die plaats vinden op het aangegeven deel van de infrastructuur te beheersen. Een trein overspant meestal meerdere verkeersleidinggebieden. Het globale verkeersleidingniveau (regionale en netwerkverkeerleiders) omvat het toezicht op de toestand van het verkeer op netwerkniveau, de detectie van afwijkingen van de dienstregeling, het oplossen van conflicten, en de afhandeling van storingen en gebeurtenissen die grote invloed op het railvervoer hebben. Treindienstleiders hebben in het algemeen geen intelligent beslissingsondersteunend systeem beschikbaar om de verwachte rijtijden van treinen in te schatten. Vertragingsvoortplanting zou voorkomen of verminderd kunnen worden als het verkeer proactief werd beheerd, d.w.z., als verkeerleiders een betrouwbare voorspelling zouden hebben van conflicterende treinbewegingen en daarbij de mogelijkheid om het conflict te voorkomen. De huidige praktijk in de operationele beheersing van storingen en vertragingen is nog steeds voornamelijk gebaseerd op vooraf bepaalde regels en de ervaring en vaardigheden van het personeel. Noch lokale noch netwerkverkeersleiders hebben een effici¨ent ondersteunend hulpmiddel om de bijstuurmaatregelen uit te voeren, hun effect te voorspellen en deze te evalueren. Een mogelijke manier om de verkeersmanagement te modelleren en optimaliseren is door middel van een gesloten-lus besturingssysteem, de zogenaamde model-gebaseerde voorspellende regelaar (MPC, model-based predictive control). Dit proefschrift presenteert een MPC kader en verkeersmanagementmodellen die in een gesloten lus kunnen worden ge¨ıntegreerd. Treinen worden bediend volgens een tijdschema en een dagelijks proces plan. Door onvermijdelijke storingen en afwijkingen van het geplande tijdschema moeten de treinritten voortdurend worden bewaakt. De monitoring levert de actuele verkeerstoestand die gebruikt kan worden om de toekomstige ontwikkeling van het treinverkeer op het netwerk te voorspellen. Een voorspellend verkeersmodel is dus noodzakelijk om het lokaal besturingsniveau continu de informatie te leveren 183 184 Models for Predictive Railway Traffic Management over de verwachte verkeerssituatie. Het kan verder gebruikt worden om de impact van mogelijke beslissingen van de verkeersleiding te evalueren. Bij langere storingen die het verkeer in een grotere regio be¨ınvloeden, kan de netwerkverkeersleiding het voorspellingsmodel gebruiken om het verkeer op het netwerk te optimaliseren, de netwerk-optimale dienstregeling te actualiseren, en deze als nieuw plan naar het lokale niveau te sturen. Op die manier zullen alle acties van treindienstleiders op lokaal niveau overeenkomen met de netwerk-optimale toestand. Monitoring en voorspelling verkeerstoestand Een manier om de beperkingen van de huidige praktijk en bestaande tools voor monitoring en verkeersvoorspelling te overbruggen wordt mogelijk door de beschikbaarheid van historische verkeersgegevens. Dit proefschrift laat zien hoe een real-time stroom van onbewerkte gegevens uit het Nederlandse treinnummervolgsysteem TROTS verwerkt kan worden om de actuele verkeerstoestand in het netwerk te tonen: actuele treinposities, nauwkeurige schattingen van de actuele vertragingen en de gerealiseerde rij- en halteertijden. Bovendien worden de archieven van logbestanden van gebeurtenissen gebruikt om te leren hoe treinen zich afhankelijk van de verkeerssituatie gedragen. De variabiliteit van procestijden wordt verklaard door factoren met een hoge impact op het desbetreffende procestijd te isoleren. Schattingen van toekomstige procestijden zijn afhankelijk van de huidige of voorspelde waarden van verklarende variabelen. Daarvoor bevatten voorspellingen de empirisch bepaalde variatie van procestijden als gevolg van bijvoorbeeld rijstijl, reizigersgedrag of spitstijden. De laatste stap voor de ontwikkeling van het monitoring en verkeersvoorspellingssysteem is het cre¨eren van een verkeersmodel. Het model wordt gebouwd en bijgewerkt op basis van de bijstuurmaatregelen van de verkeersleiding en de actuele positie van de treinen die door het treinnummervolgsysteem gemeld worden. De modeltopologie weerspiegelt alle onderlinge capaciteit- en synchronisatie afhankelijkheden tussen treinen. De kalibratie wordt uitgevoerd in real-time met robuuste schattingen van procestijden. Het monitoring instrument is gebaseerd op een process mining algoritme dat de procestijden zoals rijtijd op het niveau van bloksecties, halteertijden op stations, en opvolgtijden tussen treinen op alle infrastructuurelementen, opzoekt en bijhoudt. Bovendien controleert het instrument continu de werkelijke vertragingen en de gerealiseerde rij- en halteertijden van alle treinen. De nauwkeurigheid van de metingen van de aankomst- en vertrektijd is aanzienlijk verbeterd ten opzichte van de huidige praktijk. De resulterende datastructuur is handig voor statistische analyse, kalibratie en validatie van real-life data in deze en andere onderzoeksprojecten. Gehinderde treinritten worden ge¨ıdentificeerd en kunnen gefilterd worden om de modellen met conflictvrije rijtijden te kalibreren. Ten slotte zijn de algoritmes ge¨ımplementeerd in een Matlab tool, uitgerust met een grafische gebruikersinterface en een visualisatie component, dat de analyse van de gerealiseerde verkeerssituatie vereenvoudigt. De rijtijden zijn geschat op basis van de correlatie en de afhankelijkheden van de verklarende variabelen voor een specifieke casus. Zowel lineaire als beslisboom-gebaseerde methoden tonen voor rijtijden een zwakke afhankelijkheid van vertrekvertragingen. De Samenvatting 185 kleine variatie in rijtijden wordt grotendeels verklaard door de bloklengten en de positie ten opzichte van de vorige en volgende geplande haltes. Observaties van een specifieke treinserie en bloksectie bevestigen dat er geen duidelijk onderscheid gemaakt kan worden tussen de rijtijden van vertraagde en van stipte treinen. De opvolgtijd vanaf de vorige treinrit blijkt impact op de treinrijtijd te hebben, zelfs voor conflictvrije treinritten. Het voorspellingsmodel voor halteertijden vereist grote aandacht vanwege de hoge variatie van halteertijden waargenomen in de training dataset. De aankomstvertraging en geplande halteertijd bleken de sterkste voorspellers van halteertijden te zijn, vooral in grote stations. De statistische analyse van de halteertijden van een bepaalde trein toonden dat de halteertijden van vertraagde treinen reageren op de spitsurenvariaties. De analyse van de geschatte co¨effici¨enten van robuuste lineaire regressie toonde de omvang van de onvermijdelijke fout die optreedt wanneer de halteertijden alleen met gegevens van het treinnummervolgsysteem geschat worden. Een hoog percentage van halteertijdvariatie kan verklaard worden met de ontwikkelde voorspellende modellen. De moeilijkheid om de halteertijden van lokale treinen te voorspellen vormt nog steeds de belangrijkste bron van onnauwkeurigheid voor het voorspellingsmodel. Voor meer nauwkeurige schattingen zouden andere gegevensbronnen dan het treinnummervolgsysteem (bv. on-board units) moeten worden gebruikt. Een graaf-gebaseerd verkeersmodel is ontwikkeld dat nauwkeurig operationele beperkingen van het spoorverkeer vertegenwoordigt. Met elke update van de positie van de treinen doorzoekt een effici¨ente voorspellingsalgoritme alle takken in de graaf, bepaalt het gewicht afhankelijk van de actuele verkeerstoestand en voorspelt de realisatietijden van alle sein- en spoorsectiegebeurtenissen binnen de voorspelde tijdhorizon. Het hoge detailniveau van het model maakt de identificatie van alle conflicten op rijwegen en aansluitingen mogelijk. De nauwkeurigheid van de voorspellingen werd gevalideerd tegen de daadwerkelijk gerealiseerde gegevens uit de test set. De voorspellingshorizon voor verschillende lengtes werden onderzocht en een significante daling van de voorspellingsfouten werd aangetoond voor ene horizon korter dan 30 minuten. Zelfs voor de voorspellingsperiode van twee uren werd een gemiddelde voorspellingsfout kleiner dan een minuut verkregen. Dit is een aanzienlijke verbetering ten opzichte van de huidige praktijk of de methoden beschreven in wetenschappelijke literatuur. Een verdere verbetering van de nauwkeurigheid werd bereikt door accurate modellering van de dynamiek van de treinen, die gehinderd worden door rijwegconflicten. Als een rijwegconflict wordt voorspeld, kan de rijtijd van de gehinderde trein worden aangepast om rekening te houden met het verwachte tijdverlies. Het model is verder uitgebreid met een online adaptieve component die de gerealiseerde rijtijden van treinen in real-time bijhoudt. De treinen met rijtijden die afwijken van hun robuuste schattingen in een bepaald patroon worden ge¨ıdentificeerd en schattingen stroomafwaarts worden aangepast om de verwachte voorspellingsfout te verminderen. Dit kan gebruikt worden voor identificatie van defecte treinen, eigenaardige rijstijl, of andere treinen die aanzienlijk verschillen van de treinen in de training set met betrekking tot 186 Models for Predictive Railway Traffic Management de dynamische eigenschappen. Macroscopische modellen voor herplanning in grootschalige netwerken Het tweede doel van het onderzoek in dit proefschrift is de ontwikkeling van een beslissingsondersteunend systeem voor de netwerkverkeersleiding dat in een MPC-lus gentegreerd kan worden. Het systeem is gebaseerd op een macroscopisch railverkeersmodel dat toegepast kan worden voor een optimale besturing van het treinverkeer in grote en zwaar belaste netwerken. Alternative graphs worden gebruikt voor de modellering voor de herplanning van het treinverkeer. Verschillende macroscopische modellen, elk met een verschillend niveau van fijnheid, zijn onderzocht met het doel een compromis te zoeken tussen nauwkeurige modellering van infrastructurele capaciteitsbeperkingen enerzijds en een redelijke rekentijd voor het oplossen van grootschalige verkeersmanagement situaties anderzijds. Een geschikte keuze van de fijnheid van het macroscopische model weerspiegelt het evenwicht tussen de probleemcomplexiteit en de haalbaarheid van geproduceerde oplossingen. De macroscopische modellen zijn gevalideerd met behulp van een gedetailleerd model van een casus van een grote corridor. Daarnaast is een grote casus van een spitsuur van de Nederlandse nationale dienstregeling gebruikt om de toepasbaarheid van de modellen voor real-time landelijke toepassingen te demonstreren met betrekking tot de rekentijd nodig om een oplossing te genereren. De verwachte positieve correlatie tussen het aantal beschouwde beperkingen en de rekentijd werd bevestigd. Zelfs het beschouwde meest complexe model was in staat om optimale oplossingen te produceren in minder dan 90 seconden. Dit toont de geschiktheid van het ontwikkelde model voor praktische toepassingen. About the author Pavle Kecman was born in Belgrade, Serbia in 1982. He studied traffic and transport engineering at the University of Belgrade where he obtained his M.Sc. degree in transportation engineering in 2008. After spending two years as a research and teaching assistant at the Faculty of Transport and Traffic Engineering in Belgrade, in June 2010 he joined the Department of Transport and Planning, Delft University of Technology, as a Ph.D. candidate. He was working on a research project: “Model-predictive railway traffic management” sponsored by the Dutch Technology Foundation STW. After completing his Ph.D. thesis in June 2014, he started the postdoctoral research as a part of EU project Capacity4Rail at the Department of Science and Technology, Link¨oping University, Sweden. His research interests include railway operations and application of data mining and operations research to traffic and transport related problems. 187 188 Models for Predictive Railway Traffic Management TRAIL Thesis Series The TRAIL Thesis Series is a series of the Netherlands TRAIL Research School on transport, infrastructure and logistics. The following list contains the most recent dissertations in the TRAIL Thesis Series. For a complete overview of more than 100 titles see the TRAIL website: www.rsTRAIL.nl. Kecman, P., Models for Predictive Railway Traffic Management, T2014/5, October 2014, TRAIL Thesis Series, the Netherlands Davarynejad, M., Deploying Evolutionary Metaheuristics for Global Optimization, T2014/4, June 2014, TRAIL Thesis Series, the Netherlands Li, J., Characteristics of Chinese Driver Behavior, T2014/3, June 2014, TRAIL Thesis Series, the Netherlands Mouter, N., Cost-Benefit Analysis in Practice: A study of the way Cost-Benefit Analysis is perceived by key actors in the Dutch appraisal practice for spatial-infrastructure projects, T2014/2, June 2014, TRAIL Thesis Series, the Netherlands Ohazulike, A., Road Pricing mechanism: A game theoretic and multi-level approach, T2014/1, January 2014, TRAIL Thesis Series, the Netherlands Cranenburgh, S. van, Vacation Travel Behaviour in a Very Different Future, T2013/12, November 2013, TRAIL Thesis Series, the Netherlands Samsura, D.A.A., Games and the City: Applying game-theoretical approaches to land and property development analysis, T2013/11, November 2013, TRAIL Thesis Series, the Netherlands Huijts, N., Sustainable Energy Technology Acceptance: A psychological perspective, T2013/10, September 2013, TRAIL Thesis Series, the Netherlands Zhang, Mo, A Freight Transport Model for Integrated Network, Service, and Policy Design, T2013/9, August 2013, TRAIL Thesis Series, the Netherlands Wijnen, R., Decision Support for Collaborative Airport Planning, T2013/8, April 2013, TRAIL Thesis Series, the Netherlands Wageningen-Kessels, F.L.M. van, Multi-Class Continuum Traffic Flow Models: Analysis and simulation methods, T2013/7, March 2013, TRAIL Thesis Series, the Netherlands 189 190 Models for Predictive Railway Traffic Management Taneja, P., The Flexible Port, T2013/6, March 2013, TRAIL Thesis Series, the Netherlands Yuan, Y., Lagrangian Multi-Class Traffic State Estimation, T2013/5, March 2013, TRAIL Thesis Series, the Netherlands Schreiter, Th., Vehicle-Class Specific Control of Freeway Traffic, T2013/4, March 2013, TRAIL Thesis Series, the Netherlands Zaerpour, N., Efficient Management of Compact Storage Systems, T2013/3, February 2013, TRAIL Thesis Series, the Netherlands Huibregtse, O.L., Robust Model-Based Optimization of Evacuation Guidance, T2013/2, February 2013, TRAIL Thesis Series, the Netherlands Fortuijn, L.G.H., Turborotonde en turboplein: ontwerp, capaciteit en veiligheid, T2013/1, January 2013, TRAIL Thesis Series, the Netherlands Gharehgozli, A.H., Developing New Methods for Efficient Container Stacking Operations, T2012/7, November 2012, TRAIL Thesis Series, the Netherlands Duin, R. van, Logistics Concept Development in Multi-Actor Environments: Aligning stakeholders for successful development of public/private logistics systems by increased awareness of multi-actor objectives and perceptions, T2012/6, October 2012, TRAIL Thesis Series, the Netherlands Dicke-Ogenia, M., Psychological Aspects of Travel Information Presentation: A psychological and ergonomic view on travellers response to travel information, T2012/5, October 2012, TRAIL Thesis Series, the Netherlands Wismans, L.J.J., Towards Sustainable Dynamic Traffic Management, T2012/4, September 2012, TRAIL Thesis Series, the Netherlands Hoogendoorn, R.G., Swiftly before the World Collapses: Empirics and Modeling of Longitudinal Driving Behavior under Adverse Conditions, T2012/3, July 2012, TRAIL Thesis Series, the Netherlands Carmona Benitez, R., The Design of a Large Scale Airline Network, T2012/2, June 2012, TRAIL Thesis Series, the Netherlands Schaap, T.W., Driving Behaviour in Unexpected Situations: A study into the effects of drivers compensation behaviour to safety-critical situations and the effects of mental workload, event urgency and task prioritization, T2012/1, February 2012, TRAIL Thesis Series, the Netherlands Muizelaar, T.J., Non-recurrent Traffic Situations and Traffic Information: Determining preferences and effects on route choice, T2011/16, December 2011, TRAIL Thesis Series, the Netherlands Cantarelli, C.C., Cost Overruns in Large-Scale Transportation Infrastructure Projects: A theoretical and empirical exploration for the Netherlands and Worldwide, T2011/15, November 2011, TRAIL Thesis Series, the Netherlands