Optimization Approaches for Design of Congestion Pricing Schemes Joakim Ekstr¨om Norrk¨oping 2012 Optimization Approaches for Design of Congestion Pricing Schemes Joakim Ekstr¨om Cover illustration by Ida Lundqvist Link¨oping studies in science and technology. Dissertations, No. 1443 Copyright c 2012 Joakim Ekstr¨om, unless otherwise noted ISBN 978-91-7519-903-0 ISSN 0345-7524 Printed by LiU-Tryck, Link¨oping 2012 Abstract In recent years, there has been a growing interest in congestion pricing as a tool for solving traffic congestion problems in urban areas. However, the transportation system is complex and to design a congestion pricing scheme, i.e. to decide where and how much to charge the road users, is not trivial. This thesis considers congestion pricing schemes based on road tolls, and the efficiency of a pricing scheme is evaluated by a social welfare measure. To assist in the process of designing congestion pricing schemes, the toll design problem (TDP) is formulated as an optimiza- tion problem with the objective function describing the change in social welfare. In the TDP, the road users are assumed to be distributed in the traffic network according to a Wardrop equilibrium. The TDP is a non- convex optimization problem, and its objective function is non-smooth. Thus, the TDP is considered as a hard optimization problem to solve. This thesis aims to develop methods capable of optimizing both toll locations and their corresponding toll levels for real world traffic net- works; methods which can be used in a decision support framework when designing new congestion pricing schemes or tuning already im- plemented ones. Also, this thesis addresses the global optimality of the TDP. In this thesis, a smoothening technique is applied which approxi- mates the discrete toll location variables by continuous functions (Pa- per I). This allows for simultaneous optimization of both toll locations and their corresponding toll levels, using a sensitivity analysis based as- cent algorithm. The smoothening technique is applied in a Stockholm case study (Paper II), which shows the potential of using optimization when designing congestion pricing schemes. Global optimality of the TDP is addressed by piecewise linear ap- proximations of the non-linear functions in the TDP (Papers III and IV), resulting in a mixed integer linear program (MILP). The MILP can be solved to global optimality by branch and bound/cut methods which are implemented in commercially available software. iii Popul¨arvetenskaplig sammanfattning Om du n˚agon g˚ang har k¨ort bil i en st¨orre stad under rusningstrafik s˚a har du kanske m¨arkt att det tog lite l¨angre tid ¨an om du varit en- sam p˚av¨agen. Vad du d˚a upplevde var en av v¨agtrafikens negativa effekter, mer specifikt tr¨angsel. Tr¨angsel i v¨agtrafiken ¨ar inte bara ett problem f¨or dig som ¨arute p˚av¨agen, utanaven ¨ f¨or samh¨allet i stort och den samh¨allsekonomiska f¨orlusten av tr¨angsel, i form av b˚ade tid och monet¨ara utgifter, inom EU uppskattas till ungef¨ar 1% av BNP. Tr¨angselavgifter ¨ar ett ekonomiskt styrmedel som kan anv¨andas f¨or att f¨or¨andra resen¨arernas val av f¨ardmedel, destination, resv¨ag och avre- setidpunkt, genom att p˚averka kostnaden som ¨ar relaterad till varje alternativ. Genom att p˚averka resen¨arernas val kan ett mer effektivt utnyttjande av befintlig infrastruktur uppn˚as, och de negativa effek- terna relaterade till tr¨angsel kan d¨arigenom minskas. Att inf¨ora ett tr¨angselavgiftssystem ¨ar dock kostsamt och det ¨ar d¨arf¨or viktigt att utforma tr¨angselavgiftssystemet s˚aattdetf˚ar ¨onskad effekt. Trans- portsystemet ¨ar komplext, och det ¨ar inte sj¨alvklart vilken effekt ett tr¨angselavgiftssystem f˚ar p˚atrafiken.D¨arf¨oranv¨ands transportmodeller f¨or att ber¨akna hur inf¨orandet av tr¨angselavgifter kommer att p˚averka transportsystemet. Transportmodeller ger en f¨orenklad bild av delar av transportsystemet och m¨ojligg¨or att m˚anga alternativa utformningar av ett tr¨angselavgiftssystem kan utv¨arderas. F¨oratt utv¨ardera nyttan med ett tr¨angselavgiftssystem anv¨ands ett samh¨allsekonomiskt v¨alf¨ardsm˚att som i monet¨ara termer v¨arderar f¨or¨andringar i b˚ade restid och faktiska kostnader. Den h¨ar avhandlingen behandlar utformningen av tr¨angselavgifts- system baserade p˚av¨agtullar. Matematisk optimering anv¨ands f¨or att best¨amma placering av tullportalerna, samt niv˚an p˚a avgiften som beta- las vid tullportalen, s˚a att den samh¨allsekonomiska v¨alf¨arden maximeras. I avhandlingen visas potentialen med att anv¨anda metoder som bygger p˚a matematisk optimering f¨or att justera b˚ade avgiftsniv˚aer samt lokalis- era tullportaler i ett tr¨angselavgiftssystem. v Acknowledgements This thesis marks the end of my PhD-studies at Link¨oping University, under the supervision by Jan Lundgren and Clas Rydergren. I am very grateful for the support, encouragement and guidance from Jan and Clas throughout my thesis work. Clas introduced me to the field of transportation modeling, and I would especially like to thank him for always finding time for discussions and for giving feedback on my work. In the beginning of 2009, Agachai Sumalee at the Hong Kong Poly- technic University (PolyU) gave me the opportunity to work under his supervision within the department of Civil and Structural Engineering (CSE) during four months, and later followed up by an additional four month period in 2011. The time in Hong Kong let me grow as a re- searcher, and I am very grateful that I got this opportunity. Thank you Agachai for your encouragement and support! I would like to acknowledge the support and encouragement from Leonid Engelson at KTH, during my work within the DORIS-project, and from Hong K. Lo at the Hong Kong University of Science and Tech- nology, whom I have had the pleasure of writing a paper together with. Thanks also to Torbj¨orn Larsson, for reading and giving much valued comments on my work. I would also like to acknowledge VINNOVA, for financing parts of the work presented in this thesis, and the PolyU re- search student attachment program and Norrk¨opings Polytekniska F¨oren- ing, for financially supporting my visits to Hong Kong. The division of Communication and Transportation Systems is a dynamic and stimulating place to work, and for this I sincerely thank my colleagues. I would also like to thank my fellow doctoral students at CSE for their comradeship during my visit to PolyU; thank you Paramet, without you I would have been lost in Hong Kong. Finally, I would like to thank my family and friends for all their en- couragement. Last but not least, thank you Ida for your support. Norrk¨oping, March 2012 Joakim Ekstr¨om vii Contents 1 Introduction 1 1.1Congestionpricing...................... 1 1.2Modelingthetransportationsystem............ 3 1.3Outline............................ 4 2 The static user equilibrium model 5 2.1Modelingtraveldemand................... 5 2.2 Wardrop equilibrium . .................... 6 2.3Mathematicalformulations................. 6 2.3.1 Thetrafficnetwork.................. 6 2.3.2 Anoptimizationformulation............ 7 2.3.3 Alternativeformulations............... 9 2.3.4 Extensionsofthestandardmodel.......... 10 3 Economics of congestion pricing 13 3.1 The standard analysis of congestion pricing . 13 3.2Thewelfaremeasure..................... 15 3.3First-bestoptimalcongestionpricing............ 17 3.4Second-bestoptimalcongestionpricing........... 18 4 The toll design problem 21 4.1 Bilevel formulation of the toll design problem . 21 4.2 Alternative formulation of the toll design problem . 23 4.3Solvingthetolldesignproblem............... 25 4.3.1 Solutionapproaches................. 25 4.3.2 Globaloptimalapproaches............. 25 4.3.3 Localoptimalapproaches.............. 26 4.3.4 Heuristicsearchapproaches............. 28 5 The thesis 31 5.1Motivation.......................... 31 5.2Contributions......................... 32 ix 5.3Delimitations......................... 33 5.4Summaryofpapers...................... 34 5.5Futureresearch........................ 40 Bibliography 41 Paper I 49 Paper II 77 Paper III 109 Paper IV 155 1 Introduction 1.1 Congestion pricing Road congestion is one of the major negative effects of road transport. For 2010, Schrank et al. (2011) estimates that congestion in urban areas in the USA incurred a total of 4.8 billion hours in travel delay, and in- creased fuel consumption by 7.2 billion liters, resulting in a total cost, in terms of social welfare loss, of $101 billion which is approximately 0.7% of the GDP. Compared with historical data, this is an increase of 28% in the congestion cost during the last ten years, and of 481% during the last 28 years. For the EU, the cost of congestion was estimated to reach 1% of the GDP in 2010 (European Commission, 2001), which makes the problem of reducing the cost of congestion an important one. This prob- lem has commonly been addressed by increasing the capacity of the road infrastructure. Increasing road capacity is not only expensive, but will in itself lead to an increased demand for road traffic, a relationship which is well established1, and cannot be considered as a sustainable solution. Road pricing can be used for charging the road users a fee for using the road infrastructure, and provides a tool for achieving a more efficient usage of the road capacity, without building new road infrastructure. The users of the transportation system make travel choices based on their individual costs and benefits; for example, travel time, fuel cost, comfort of the transportation alternatives and attractiveness of the destinations, associated with different travel alternatives. While the choices associated with making a trip are based on the individual costs and benefits, there are also other, external, costs associated with making a trip. Limiting the discussion to cars, such costs include increased travel costs for fellow road users and the emission of pollutants. These are negative external effects of making a trip, which are not experienced 1This is commonly referred to as induced demand.