Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UFE0024767/00001
## Material Information- Title:
- A Hierarchical Framework for Congestion Pricing of Transportation Networks
- Creator:
- Lou, Yingyan
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 2009
- Language:
- english
- Physical Description:
- 1 online resource (141 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Civil Engineering
Civil and Coastal Engineering - Committee Chair:
- Yin, Yafeng
- Committee Members:
- Elefteriadou, Ageliki L.
Washburn, Scott S. Srinivasan, Sivaramakrishnan Lawphongpanich, Siriphong - Graduation Date:
- 8/8/2009
## Subjects- Subjects / Keywords:
- Flow distribution ( jstor )
High occupancy toll lanes ( jstor ) Motor vehicle traffic ( jstor ) Pricing ( jstor ) Simulations ( jstor ) Tolls ( jstor ) Traffic congestion ( jstor ) Transportation ( jstor ) Travel time ( jstor ) Travelers ( jstor ) Civil and Coastal Engineering -- Dissertations, Academic -- UF adaptive, bounded, congestion, network, pricing, rationality, transportation - Genre:
- Electronic Thesis or Dissertation
born-digital ( sobekcm ) Civil Engineering thesis, Ph.D.
## Notes- Abstract:
- To further advance road pricing to be a more efficient and pragmatic tool for congestion mitigation, this dissertation proposes a hierarchical congestion pricing framework for urban transportation networks. Within the framework, toll determination is decomposed into two levels: network and facilities. Empirical studies have discovered that travelers have a strong preference for simple system-wide pricing structures. For example, dynamic network pricing models are not only difficult to implement, more importantly, their pricing signals are also too complicated for travelers to understand and consequently change their travel behaviors. On the other hand, time-varying tolls at particular facilities, such as managed lanes, are acceptable and effective. Therefore, tolls at these two levels should follow different strategies due to their distinctive purposes and travelers' different response abilities. At the network level, we propose a robust static or time-of-day pricing policy to avoid complex toll structures while ensuring the network to perform reasonably well against a variety of uncertainties. Sources of uncertainty in transportation networks consist of not only randomness in demand and supply, but also travelers' stochastic and irrational behaviors. This dissertation investigates one of the uncertainties resulting from boundedly rational route-choice behaviors. Users with bounded rationality seek for acceptable paths rather than a necessarily minimum one. Boundedly rational user equilibrium (BRUE) flow distribution is generally non-unique and can be characterized as a non-convex and non-empty path flow set. A more restrictive link-based representation is also presented. A robust pricing scheme is determined by solving a nonlinear mathematical program with complementarity constraints to minimize the system travel time of the worst-case tolled BRUE flow distribution. At some critical facilities, the toll scheme determined at the network level may be further adjusted in response to real-time traffic conditions. This dissertation focuses on developing pricing strategies for managed toll lanes. Adaptive tolls may be adopted in order to provide a superior free-flow travel service to the users of the toll lanes while maximizing the freeway's throughput. Two sensible and practically implementable approaches, one feedback and one self-learning, are proposed. The self-learning approach monitors conditions of the facility through both direct observation and real-time estimation, and learns recursively motorists' willingness to pay and short-term future demand by mining the traffic data from sensors. In determination of the tolls, a detailed modeling of drivers' lane-choice behavior and traffic dynamics is adopted to explicitly consider their impacts on the performance of the facility. In summary, based on practical considerations of pricing, robust time-of-day tolls are proposed for the entire network while adaptive tolls are advocated for special facilities. This composes a hierarchical congestion pricing framework for a transportation system. Feasibility of this framework is shown through discussions on its general inputs and concerns. Our investigation on the network-level robust pricing with boundedly rational user behavior and the facility-level traffic-responsive pricing of managed lanes demonstrates that the proposed hierarchical framework may be practical and promising for congestion mitigation. ( en )
- General Note:
- In the series University of Florida Digital Collections.
- General Note:
- Includes vita.
- Bibliography:
- Includes bibliographical references.
- Source of Description:
- Description based on online resource; title from PDF title page.
- Source of Description:
- This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2009.
- Local:
- Adviser: Yin, Yafeng.
- Electronic Access:
- RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2010-02-28
- Statement of Responsibility:
- by Yingyan Lou.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright Lou, Yingyan. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Embargo Date:
- 2/28/2010
- Resource Identifier:
- 489236233 ( OCLC )
- Classification:
- LD1780 2009 ( lcc )
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PAGE 1 1 A HIERARCHICAL FRAMEWORK FOR CONGE STION PRICING OF TRANSPORATION NETWORKS By YINGYAN LOU A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2009 PAGE 2 2 2009 Yingyan Lou PAGE 3 3 To my parents PAGE 4 4 ACKNOWLEDGMENTS First of all, I would like to thank my wonde rful parents for always being so supportive through each and every stage of my lif e, no matter how far away they are. I would also like to thank the faculty members at the University of Florida (UF), who have provided me not only huge amount of knowledge a nd technical skills, but also good work ethics and the spirit of teamwork during my Ph.D. study. I would like to thank my advisor Dr. Yafeng Yin for constantly being a source of inspira tion, an outstanding example as professional researcher and a good friend. I am grateful to the advices he has given me, both academic and otherwise. I would like to th ank Dr. Siriphong Lawphongpanich for his valuable suggestions to this dissertation, and the high standard of pr ofessionalism he has set up for me by his own earnest practice. I would like to thank Dr. Lily Elefteriadou for being one of my role models as women professors, Dr. Scott Washburn for his in fluence on a light and positive attitude towards life and work, and Dr. Sivaramakrishnan Srinivasan for being an example of diligent researchers. I would also like to thank them for their helpful comments from various perspectives, which helped me to place this research in a broader cont ext. I am also gratef ul to many other faculty members, such as Mr. William Sampson, Dr. Co le Smith, Dr. Ravindra Ahuja, and Dr. Panos Pardalos, for their valuable gui dance and help during my study. I would also like to thank all my friends and fellow students for making my UF experience pleasant and enjoyable, especially Lihui Zhang, Ziqi Song, Alexandra Kondyli, Dimitra Michalaka, Chao Cao and Jun Liu for their friend ship and support. Special thanks also go to Ines Aviles-Spadoni, coordinator of the Cent er for Multimodal Solutions for Congestion Mitigation at UF, for her help. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........8 LIST OF FIGURES................................................................................................................ .........9 ABSTRACT....................................................................................................................... ............11 CHAPTER 1 INTRODUCTION..................................................................................................................13 1.1 Background...................................................................................................................13 1.2 Problem Statement and Di ssertation Objective.............................................................16 1.2.1 Network Level Problem....................................................................................16 1.2.2 Facility Level Problem......................................................................................17 1.3 Dissertation Outline......................................................................................................17 2 LITERATURE REVIEW.......................................................................................................18 2.1 Economics Principle of Congestion Pricing.................................................................18 2.2 Canonical Equilibrium-Based Pricing Models..............................................................19 2.2.1 Static Network Modeling..................................................................................20 2.2.1.1 First-best pricing with fixed demand..................................................20 2.2.1.2 First-best pricing with elastic demand................................................23 2.2.1.3 Extensions to first-best pricing...........................................................24 2.2.1.4 Second-best pricing............................................................................25 2.2.1.5 Robust pricing.....................................................................................26 2.2.2 Time-Dependent Pricing Models......................................................................27 2.3 Traffic-Responsive Pricing Models..............................................................................29 2.3.1 Trial-and-Error Approach.................................................................................29 2.3.2 Day-to-Day Dynamics......................................................................................30 2.3.3 Adaptive Pricing Scheme for Managed Lanes..................................................30 2.3.3.1 Managed lanes....................................................................................31 2.3.3.2 Current operation of HOT lanes.........................................................32 2.4 Summary.......................................................................................................................34 3 HIERARCHICAL PRICING FRAMEWORK.......................................................................40 3.1 Robust Pricing at Network Level..................................................................................41 3.2 Adaptive Pricing at Facility Level................................................................................42 3.3 Summary.......................................................................................................................43 PAGE 6 6 4 ROBUST NETWORK PRICING WITH BOUNDEDLY RATIONAL USER EQUILIBRIUM.................................................................................................................... ..44 4.1 Bounded Rationality.....................................................................................................45 4.2 Boundedly Rational User Equilibr ium in a Static Network..........................................47 4.2.1 Definition of BRUE..........................................................................................47 4.2.2 Properties of BRUE Flow Distributions...........................................................49 4.2.2.1 Link-based represen tation of BRUE...................................................49 4.2.2.2 Existence, uniqueness and nonconvexity...........................................53 4.2.3 Finding Bestand Worst-Case BRUE Flow Distributions...............................54 4.2.4 Numerical Examples.........................................................................................55 4.3 Robust Pricing with Boundedly Rational User Equilibrium.........................................57 4.3.1 Model Formulations..........................................................................................57 4.3.2 Solution Algorithm...........................................................................................62 4.3.3 Numerical Examples.........................................................................................64 4.4 Summary.......................................................................................................................66 5 OPTIMAL ADAPTIVE TOLLING STRATEGI ES FOR HIGH OCCUPANCY/TOLL LANES: BASIC MODELS....................................................................................................82 5.1 Feedback Approach for HOT Pricing...........................................................................83 5.2 Self-learning Approach for HOT Pricing......................................................................84 5.2.1 General Framework...........................................................................................84 5.2.2 Calibration of Willingness-to-Pay.....................................................................85 5.2.3 Optimal Tolling Strategy with Point-Queue Model..........................................86 5.3 Simulation Study...........................................................................................................89 5.3.1 Design of Simulation Experiments...................................................................89 5.3.2 Comparison of Feedback and Self-learning Controllers...................................90 5.3.2.1 Performance of feedback controller....................................................91 5.3.2.2 Performance of self-learning contro ller with point-queue model.......92 5.4 Summary.......................................................................................................................93 6 OPTIMAL ADAPTIVE TOLLING STRATEGI ES FOR HIGH OCCUPANCY/TOLL LANES: A UNIFIED FRAMEWORK OF EXTENDED SELF-LEARNING APPROACH....................................................................................................................... ..104 6.1 Incorporating More Real istic Traffic Dynamics.........................................................105 6.1.1 Impacts of Lane-Changing Behaviors.............................................................105 6.1.2 Impacts of HOT Slip Ramp Design................................................................106 6.2 System Inference.........................................................................................................107 6.2.1 Traffic State Estimation..................................................................................107 6.2.2 Demand Learning............................................................................................110 6.3 Toll Determination......................................................................................................112 6.3.1 Deterministic Proactive Pricing Approach......................................................112 6.3.2 Robust Controller with Heterogeneous Motorists..........................................114 6.3.3 Robust Controller with Demand Uncertainties...............................................116 6.4 Simulation Study.........................................................................................................117 PAGE 7 7 6.5 Summary.....................................................................................................................120 7 CONCLUSION.....................................................................................................................127 LIST OF REFERENCES.............................................................................................................130 BIOGRAPHICAL SKETCH.......................................................................................................141 PAGE 8 8 LIST OF TABLES Table page 2-1 Current onsite-accessible HOT facilities in the U.S..........................................................39 4-1 Illustrative example of the restrictiven ess of the link-based BRUE representation..........78 4-2 Nonconvexity of the BRUE flow dist ribution set for the bridge network.........................79 4-3 Flow distributions fo r the nine-node network (10 0 ).................................................80 4-4 Pricing schemes and their perfor mances for the nine-node network.................................81 5-1 Performance of feedback controllers with different K values.........................................103 PAGE 9 9 LIST OF FIGURES Figure page 2-1 Economics Foundation of Marginal-Cost Pricing.............................................................36 2-2 Alternative HOT lane slip ramp configurations.................................................................37 2-3 I-394 toll rates on May 18, 2005 (data source: www.MnPASS.net).................................38 4-1 A bridge network.......................................................................................................... .....68 4-2 Relationship between th ree different flow sets..................................................................69 4-3 The nine-node network..................................................................................................... .70 4-4 Comparison of system performances of BRUE at the nine-node network........................71 4-5 Comparison of system performances of BRUE at the Sioux Falls network......................72 4-6 System performances with MC pricing scheme at the nine-node network.......................73 4-7 Three-parallel-link network...............................................................................................74 4-8 Comparison of worst-case total travel tim es with no toll, robust and MC tolls at the nine-node network.............................................................................................................75 4-9 System performances with MC and RT1 at the nine-node network (1)............................76 4-10 System performances with MC and RT1 at the nine-node network (2)............................77 5-1 System configuration for the feedback-control approach..................................................95 5-2 System sketch of the self-learning approach.....................................................................96 5-3 Throughputs, queues and toll rates resulte d by the feedback controller (Scenario 1, K =0.1).......................................................................................................................... ......97 5-4 Throughputs, queues and toll rates resulte d by the feedback controller (Scenario 2, K =0.1).......................................................................................................................... ......98 5-5 Calibrated 1 & by the self-learning cont roller (Scenario 1)........................................99 5-6 Throughputs, queues and toll rates resulte d by the self-learning controller (Scenario 1)............................................................................................................................. .........100 5-7 Performance comparison of two controllers under scenario 1.........................................101 PAGE 10 10 5-8 Throughputs, queues and toll rates resulte d by the self-learning controller (Scenario 2)............................................................................................................................. .........102 6-1 Discretized freeway representation..................................................................................122 6-2 Discretized freeway repres entation (barrier-separated)...................................................123 6-3 Simulation settings....................................................................................................... ....124 6-4 Calibration of Willingness to Pay....................................................................................125 6-5 Optimal toll rate and its performance..............................................................................126 PAGE 11 11 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A HIERARCHICAL FRAMEWORK FOR CONGE STION PRICING OF TRANSPORATION NETWORKS By Yingyan Lou August 2009 Chair: Yafeng Yin Major: Civil Engineering To further advance road pricing to be a more efficient and pragmatic tool for congestion mitigation, this dissertation proposes a hierarch ical congestion pricing framework for urban transportation networks. Within the framewor k, toll determination is decomposed into two levels: network and facilities. Empirical studies have discovered that travelers have a strong preference for simple system-wide pricing stru ctures. For example, dynamic network pricing models are not only difficult to implement, more importantly, their pricin g signals are also too complicated for travelers to understand and conseque ntly change their travel behaviors. On the other hand, time-varying tolls at particular facilities, such as managed lanes, are acceptable and effective. Therefore, tolls at these two levels should follow di fferent strategies due to their distinctive purposes and travelers different response abilities. At the network level, we propose a robust stat ic or time-of-day pricing policy to avoid complex toll structures while ensuring the netw ork to perform reasonabl y well against a variety of uncertainties. Sources of uncertainty in transportation networks consist of not only randomness in demand and supply, but also travelers stochastic and irrational behaviors. This dissertation investigates one of the uncertainties resulting fr om boundedly rational route-choice PAGE 12 12 behaviors. Users with bounded rationality seek for acceptable paths rather than a necessarily minimum one. Boundedly rational user equilibrium (BRUE) flow distribution is generally nonunique and can be characterized as a non-conve x and non-empty path flow set. A more restrictive link-based representati on is also presented. A robust pricing scheme is determined by solving a nonlinear mathematical program with complementarity constraints to minimize the system travel time of the worst-case tolled BRUE flow distribution. At some critical facilities, the toll scheme determined at the network level may be further adjusted in response to real-time traffic condi tions. This disserta tion focuses on developing pricing strategies for managed toll lanes. Adap tive tolls may be adopted in order to provide a superior free-flow travel service to the users of the toll lanes while maximizing the freeways throughput. Two sensible and practically implemen table approaches, one f eedback and one selflearning, are proposed. The self-l earning approach monitors conditions of the facility through both direct observation and real-t ime estimation, and learns recurs ively motorists willingness to pay and short-term future demand by mining the tra ffic data from sensors. In determination of the tolls, a detailed modeling of drivers lane-choice be havior and traffic dynamics is adopted to explicitly consider their impacts on the performance of the facility. In summary, based on practical considerations of pricing, robust time-of-day tolls are proposed for the entire network while adaptive to lls are advocated for spec ial facilities. This composes a hierarchical congesti on pricing framework for a transportation system. Feasibility of this framework is shown through discussions on its general inputs and concerns. Our investigation on the network-leve l robust pricing with boundedly ra tional user behavior and the facility-level traffic-responsiv e pricing of managed lanes dem onstrates that the proposed hierarchical framework may be practical and promising for congestion mitigation. PAGE 13 13 CHAPTER 1 INTRODUCTION 1.1 Background The continuously-growing traffic congestion re mains one of the most severe problems many societies are facing today. In the United States, congestion has grown in cities of every size in the past ten years in extent, duration and intensity. The peak-period trips are on average seven percent longer; delay per tr aveler per year has increased from 40 hours to 47 hours; and the percent of congested freeway mileage has risen from 51 to 60 (FHWA, 2005). The need for congestion mitigation strategi es has never been greater. Three categories of strategies may be applie d as congestion reliefs. While the traditional response of adding capacity is re stricted by various constraints, the combination of advanced traffic operations (ATO) and tr avel demand management (TDM ) has gained an increasing attention as a more effective a nd cost-efficient solution to cong estion (FHWA, 2005). One of the key components of ATO and TDM is congestion pricing. Pricing of road networks were common in Br itain and United States during most of the nineteenth century (Lindsey, 2006). But the id ea of employing pricing as an approach to improve efficiency did not initiate until the seminal work by Pigou (1920) and Knight (1924). This school of thoughts originated from the public goods theory and the key idea is to internalize the externality of a trip by charging up to the full price that the trip imposes to the society. Though it has long been advocated, congestion pr icing was not adopted until recently perhaps due to technical and institutional barriers. As an approach of TDM, conge stion pricing has been proved to be efficient in reshaping travel patterns. For example, Singapore implem ented its Area Licensing Scheme to restrict vehicular traffic into the centr al area of the city in 1975. Later in 1988 an electronic toll PAGE 14 14 collecting system, namely the Electronic Road Pricing (ERP), was installed to replace the original manual pricing scheme. During its first phase, the pricing scheme reduced the traffic entering the central area by 31% despite the growth of employment and driver population in the same period. Employment of ERP further re duced the congestion by 10-15% (Chin, 2002). More recently, the city of London introduced in Fe bruary 2003 a five-pound (later increased to eight) daily fee on cars entering its city center. Compared to the traffic condition in 2002, traffic entering the charged zones in 2006 was 21% lower (Transport for London, 2007). Congestion pricing has gained more relevance in the United States as an approach of advanced highway traffic operations. A prevalent form is managed lanes, more specifically, high occupancy/toll (HOT) lanes. By allowing lower o ccupancy vehicles to pay to gain access to the facility, the aim of HOT lanes is to make better use of the existing capacit y of the original high occupancy vehicle (HOV) lanes and reduce the to tal delay of the highway. Since the first managed lane was implemented in 1995 on State Route 91 in Orange County, California, HOT lanes have become popular among governors and transp ortation officials, in state legislatures and the media (Orski, 2006). Evaluation of the existi ng HOT lanes indicates th at they significantly reduced traffic delay, improved travel time reliabi lity and the overall cost-benefit ratio is greater than one (e.g., Burris and Sullivan, 2006, Sullivan and Burris, 2006, Supernak et al., 2003a). In May 2006, the U.S. Department of Transportati on (USDOT) further launched a new national congestion relief initiative promoting congestio n pricing and variable tolling (USDOT, 2006). Despite of its favorable developments and the potential as a promising solution to congestion problem, road pricing still faces ma ny challenges. Though pricing strategies in practice are often effective, the ma gnitude of benefits depends largel y on the details of the tolling structure (Santos and Fraser, 2006). However, tolls are mostly determined in an ad-hoc manner PAGE 15 15 in practice. For inst ance, the Land Transport Authority ( LTA) of Singapore simply doubled the preset initial ERP charge and rate increment recen tly in response to drivers insensitiveness to the original pricing scheme (L TA, 2008); the Transport of L ondon set the toll rate without estimating the marginal congestion cost (Santos and Fraser, 2006); and th e tolls on Minnesota I394 HOT lane are adjusted according to a look-up table created based on static traffic assignment with assumptions regarding travel de mand and value of travel time (Halvorson et al., 2006). This suggests that the current practice of road pric ing can be further improved. Yet, most pricing models in the literature do not meet the practical needs well. On one hand, network-level models are either too simple to represent the real-w orld situation or too complex to implement. Canonical static m odels (e.g., Beckmann et al., 1956, Walters, 1961, and works in the same school) rely on assumptions of users perfect rationality and macroscopic network performance functions. These models are ideal as they simplify many human and traffic factors. Some dynamic models employ dynamic traffic assignment with assumptions on the time-dependent origin-destinati on (OD) demand (e.g., Wie and T obin, 1998). But the resulting pricing schemes are often so complex and dramatic ally fluctuating that they are intractable for real operations in general networ ks. Moreover, Bonsall et al. ( 2007) discovered that people have a strong preference for simple netw ork-level pricing structures. It is not reasonable to expect users to predict and respond to the prices if th e tolling signals across th e entire network are very complicated and change quite often. On the othe r hand, it is believed that time-varying tolls at facility-level are accepta ble to the users. One good exampl e is HOT lanes (Bonsall et al., 2007), whose performance can often be improved thro ugh dynamic pricing (Sup ernak et al., 2003b). However, studies on HOT pricing are limited. Re lated single-bottleneck models (e.g. Vickery, 1963; Arnott et al., 1998, etc.) are mostly idealized and cannot be applied to HOT lanes directly. PAGE 16 16 These challenges provide many opportunities fo r researchers to develop sensible and practical pricing schemes. In this dissertati on, we propose a hierarchical pricing framework for congestion mitigation. Within the framework, toll determination is decomposed into two levels: network and facility. At the ne twork level, robust static or tim e-of-day pricing policy will be developed to avoid complex toll structures while ensuring the ne twork will perform reasonably well under various circumstances. A detailed desc ription of traffic dynamics is avoided at the network level. At the facility level, tolls can be further adjusted to meet local objectives with a detailed modeling of drivers behavior and tr affic dynamics. We now state the problem more specifically. 1.2 Problem Statement and Dissertation Objective 1.2.1 Network Level Problem Being strategic, toll rates at the network leve l will not be revisited frequently and often extend over a certain period of time. A static tim e-of-day pricing structure will be easy for the general public to understand and for transportation authorities to implement. Nevertheless, we still expect this simple pricing scheme to function well in the pres ence of demand and supply uncertainties. Therefore, robus tness becomes an important me rit and consideration in the network-level toll optimization. We seek for robust pricing strategies against uncertain equilibrium flow distributi ons due to volatilities in both demand and supply side. There are many sources of uncertainty in tr ansportation networks. This dissertation primarily focuses on examining one of the uncerta inties associated with users route choice behavior, namely the bounded rationality. This problem contains three tasks: 1) Propose rigorous definition of BRUE; 2) Develop mathematical model to characterize BRUE flow patterns; 3) Determine robust tolls to accommod ate the resulting uncertain equilibrium flow distribution. PAGE 17 17 1.2.2 Facility Level Problem We mainly focus on HOT lanes at the facility le vel. Consistent with the current HOT lane operating policies (FHWA, 2003) a nd the recent trend of variable pricing (USDOT, 2006), this dissertation aims to build adaptiv e tolling strategies that can provi de a superior free-flow traffic service on the HOT lanes while maximizing the th roughput rate of the entire freeway segment. Note that between these two object ives, the operators often give higher priority to the former, because HOV lanes are designed first and foremo st to provide less congested conditions for carpoolers and transit users (M unnich, 2006). The pricing mode l should take into account drivers reaction to the toll rates, i.e. their willingness-to-pay (WTP) to use the HOT lane, the impact of lane changing at HOT lane entrance on traffic dynamics, and be capable of handling other practical issues. 1.3 Dissertation Outline This dissertation is organized as follows. Ch apter 2 reviews the existing pricing models in the literature, including canonical static and dynamic equilibrium-based models and traffic responsive models. The proposed ge neral hierarchical pricing stru cture is presented in Chapter 3. Chapter 4 is devoted to robust network pric ing models against user s boundedly rational route choice behaviors. Basic models for HOT lane operation are discussed in Chapter 5, followed by a unified framework of extensions in Chapter 6 that addresses seve ral critical prac tical issues. Conclusion with guidelines of future research is provided in Chapter 7. PAGE 18 18 CHAPTER 2 LITERATURE REVIEW There is an extensive litera ture on congestion pricing in both fields of economics and transportation engineering. Early studies in economics have built up the theoretical foundation of marginal-cost (MC) pricing while more practic al settings are taken in to account by researchers in transportation engineering. Based on different concepts and assumptions adopted, congestion pricing models can be coarsely categorized into two types: canonical equilibrium-based and traffic-responsive. This chapter mainly reviews various modeling approaches in the field of transportation engineering. We will briefly summarize the eco nomics principle of congestion pricing and examine more carefully the canonical equilibr ium-based and the traffic-responsive models respectively. 2.1 Economics Principle of Congestion Pricing The standard approach in Walters (1961) may se rve as a landmark of MC road pricing. With a number of assumptions, th e model suggests that in order to maximize the social benefit, road users should pay the differe nce between the marginal soci al cost and private cost. Figure 21 illustrates this model. Let Q denote the flow along a given unifo rm stretch of road. Without pricing, equilibrium occurs at the in tersection of the inverse demand curve ID and the average cost curve AC at which users WTP is equal to the av erage cost. However, social optimum is achieved at the point B where ID and marginal social cost curve MSC intersect. This point can be achieved with a toll whose magnitude is represented by BC and can be mathematically calculated as: 0 0) ( ) ( ) ( ) ( ) (0 0 0 0 Q Q Q QdQ Q dAC Q Q AC dQ Q AC Q d Q AC Q MSC PAGE 19 19 With this MC pricing, total soci al surplus is increased by area ABD The travelers lose their surplus by area ABBE while the government coll ects a toll revenue of BCCB Total net social benefit is thus CEEC ABE which is equal to area ABD Such a pricing scheme is often referred to as first-best pricing because there are no constraints on the pricing instrument and it is assumed that the government and other markets are perfectly efficient so that the collected toll revenue can be optimally transf erred (Small and Verhoef, 2007). Though the standard approach received many cr itiques, mainly on its over-simplification, it is nonetheless a good starting point of congest ion pricing. Debates on how to cover common costs, how to allocate excess revenues, whethe r and how to compensate tolled-off users, and whether to privatize highways have never stopped among economists (Lindsey, 2006). These issues are beyond the scope of this dissertation. We will assume an efficient society and focus on how to set up practically implementable a nd effective tolls within this context. 2.2 Canonical Equilibrium-Based Pricing Models Canonical equilibrium-based models usually assume that travel demand (functions) and network performance functions are known. The pricing problems were first studied in a relatively ideal situation where demand for each OD pa ir is fixed and travel time function of each link is separable (travel time of a link is only affected by the flow on that particular link). Users are assumed to be homogeneous, perfectly inform ed and rational in making their travel choices (route, departure time and mode, etc.), i.e. each user attempts to minimize his own general travel cost. Many relaxations to these assumptions, in cluding elastic demand, nonseparable link travel time functions, imperfect information and hete rogeneity among users can be found in the literature. Canonical models often try to dete rmine congestion tolls th at maximize the total social benefit in the short-run equilibrium (transportation and activity systems do not change), PAGE 20 20 although the measure of social benefit may differ in different contents, pa rticularly when other externalities such as environmental, fina ncial and maintenance cost are considered. Canonical models can be further classified in to static and time-dependent ones. Static models only consider users rout e choice, relying on the standard traffic assignment to describe the tolled equilibrium flow distribution. Another dimension of choice, namely departure time, is included in the time-dependent models. Dynami c traffic assignment models are employed to predict tolled dynamic equilibrium flows. 2.2.1 Static Network Modeling To facilitate the presentation, let N and L denote the sets of nodes and links in a road network, and W the set of OD pairs. A link is represen ted by its head and tail nodes, i.e., an element of L is denoted as ( i j ). For OD pair w w ijx represents th e flow on link ( i j ). The vector xw denotes the flow vector for OD pair w with w ijx as its element, and v = w wx is the associated aggregate link flow vector. For each link ( i, j ) L ) (ij ijv t denotes the separable link travel time function that is continuous and monotonica lly increasing with the aggregate link flow, vij. Let ) ( w o and ) ( w d represent the origin and destination of OD pair w the node-link incidence matrix and wD the demand vector satisfying that w w w oq D ) (, w w w dq D ) ( and 0 w iD for all other node i where wq is the given demand of OD pair w 2.2.1.1 First-best pricing with fixed demand Wardrop (1952) introduced two behavior pr inciples, which was later termed user equilibrium (UE) and system optimum (SO) respec tively in the work of Dafermos (Patriksson, 1994). Wardrops first principle st ates that travel time on all used routes are equal and less than those on any unused routes. Note that this principl e is equivalent to the assumption that users are PAGE 21 21 perfectly rational. Under this condition, no user can further mini mize his travel cost, thus users will have no incentive to change their route choi ces. Wardrops second principle is limited to the fixed demand case and suggests the total system tr avel time is a minimum. SO principle can be generalized to total social benefit maximization wh en elastic demand is considered. Static traffic assignment models are formulated to find link flow distributions such that UE or SO principle is satisfied. Accordingly, the first-best pricing pr oblem is to determine toll schemes such that the tolled UE is SO. Beckmann et al. (1956) formulated the static traffic assignment problems as mathematical programs, and proved the equivalence between op timality conditions of the proposed UE model and Wardrops first principle. The UE flow pattern can be obtained by solving the following mathematical program: x v ,min L ij v ijijdv v t0) ( ( 2.1) s.t. w wD x W w ( 2.2) w wx v ( 2.3) 0 x ( 2.4) Similarly, SO traffic assignment can be formulated as: x v,min L ij ij ij ijv v t) ( ( 2.5) s.t. Conditions ( 2.2)-( 2.4) Note that L ij v ij ij L ij ij ij ijijdv dv v dt v v t v v t0) ( ) ( ) (, suggesting the SO problem is equivalent to a UE problem with modified travel cost ij ij ij ij ij ij ij ijdv v dt v v t v t ) ( ) ( ) ( ~ which is exactly the marginal social cost of having one more traveler on link ( i, j ). Further more, PAGE 22 22 following the Karush-Kuhn-Tucker (KKT) first-or der optimality conditions of the UE and SO formulations, it can be shown that link-based MC to ll is able to drive the system from UE to SO. Mathematically, the MC link toll ij can be expressed as SO ij ijv v ij ij ij SO ijdv v dt v ) (, where SO ijv is the SO link flow for ( i, j ). This can be viewed as a networ k generalization of the marginal cost pricing model developed in Walters (1961). For decades, the only first-best pricing scheme discussed in the litera ture is the above MC toll. However, first-best tolls that can drive UE to SO are not necessa rily unique (Hearn and Ramana, 1998). Consider the tolled UE condition: 0 ) ( w j w i ij ij ij w ijv t x W w L ij ( 2.6) 0 ) ( w j w i ij ij ijv t W w L ij ( 2.7) Conditions ( 2.2)-( 2.4) where w i represents the node potential of node i for OD pair w. This set of conditions is exactly the KKT condition of tolled UE problem and is equi valent to Wardrops first principle with the presence of link tolls (see e.g., Patriksson, 1994). Since the tolled UE with first-best pricing is SO, a set of link tolls is first be st if and only if ther e exists a set of node potentials such that conditions ( 2.2)-( 2.4) and ( 2.6)-( 2.7) are satisfied with SO ij ijv v and w ijx equal to some corresponding commodity link flow. More concisely, summing equation ( 2.6) over all links and OD pairs yields ( 2.8), where SOv, ) ( v t, and ware vectors with SO ijv, ) (ij ijv t, ij and w i as their respective components. When the SO flow pattern is unique, the first-best toll set can be characterized as a polyhedr on described in conditions ( 2.8)-( 2.9). PAGE 23 23 0 ) ( w w T w SO T SOD v v t ( 2.8) 0 ) ( w T SOv t W w ( 2.9) Since there exist multiple first-best toll patterns that can force the system from UE to SO, it becomes possible to consider a secondary objective within the first-best toll set. Secondary objectives may be minimizing, e.g., total toll re venue, the largest link to ll, or the number of tolled links (Hearn and Ramana, 1998). Specific algorithms have been developed for finding the first-best toll that also minimizes toll re venue (Dial, 1999a, 2002 and Bai et al., 2004). 2.2.1.2 First-best pricing with elastic demand The fixed-demand models can be easily exte nded to the case of elastic demand. The equilibrium OD demand is now a function of th e equilibrium general OD travel cost, which includes both the travel time and the toll. Let ) (w wq d denote the inverse demand function for OD pair w. The UE problem with elastic demand is formulated as follows: x v q ,max L ij v ij w q wij wdv v t dq q d0 0) ( ) ( ( 2.10) s.t. Conditions ( 2.2)-( 2.4) while the SO problem is in the following form (Beckmann et al., 1956): x v q ,max L ij ij ij ij w q wv v t dq q dw) ( ) (0 ( 2.11) s.t. Conditions ( 2.2)-( 2.4) The first-best toll set with elastic demand (Hearn and Yildirim, 2002) can then be described as w SO w SO w w SO T SOq q d v v t) ( ) ( ( 2.12) 0 ) ( w T SOv t W w ( 2.13) w w o w w d SO w wq d) ( ) () ( W w ( 2.14) PAGE 24 24 where SO wq is the SO demand for OD pair w Similar to the case of fixed demand, the MC toll belongs to the first-best toll set as well. Furthermore, any toll ve ctor in the firstbest toll set can be interpreted as a MC pricing scheme, as the total price charge d along any utilized path for any toll pattern are equal, and matc hes the difference between the ma rginal social cost and the average cost along this path (Yin and Lawphonpanich, 2009). 2.2.1.3 Extensions to first-best pricing The literature offers several extensions to th e basic first-best prici ng models from various perspectives, such as incorporating users pe rception errors of travel time, travelers heterogeneity, and different performance meas ures other than system surplus, etc. The assumption that travelers are perfectly in formed are relaxed in the stochastic user equilibrium (SUE) models, where eac h traveler is assumed to have a random perception error of link travel times (Sheffi, 1985). Smith et al. ( 1994) provided the conditions of the existence of link tolls that can support SO as tolled SUE, wher e SO is considered as minimizing total system travel time. Yang (1999) explor ed the feasibility of MC pricing when social benefit is considered as the SO objective instead. First-be st pricing with secondary objectives under SUE is further studied in Stewart and Maher (2006). User heterogeneity has been taken into account in forms of multiple classes of travelers. These classes may differ in travel cost functions, values of travel times, vehicle types, and travel modes (Dafermos, 1973; Dial, 1999b; Yin and Yang, 2004; Chu and Tsai, 2004; and Hamdouch et al., 2007). Some models try to determine tolls for objec tives different from the simple measure of system surplus. These problems can be form ulated as bi-level programming models or mathematical programs with equilibrium constraint s (MPEC) with the UE or SUE models as the PAGE 25 25 lower-level problem or embedded equilibrium c onstraints (see, e.g., Ferrari, 2002 and Chiou, 2004). When the alternative obj ective reduces to a performan ce requirement depending only on the link flows, a simple UE traffic assignment model with additional constrai nt is able to provide information on the link tolls (Larsson and Patriksson, 1999). Examples of other extensions are non-separab le link travel cost (Yang and Huang, 2005) and capacity constraints on link flows (see, e.g., Ferrari, 1995; Yang and Bell, 1997). 2.2.1.4 Second-best pricing So far we have discussed models for firs t-best pricing proble ms where there is no constraint on the toll. However, a first-best toll might not be pr actical in real-world implementation. For instance, every link has to be charged under the MC pricing scheme, which may not be practical. In the cas e of heterogeneous users, discriminatory tolls may be the only solution to first-best pricing problem (Chu and Tsai, 2004). But it might be difficult to distinguish different groups of travelers to apply discriminatory tolls. Therefore, pricing problems with constraints on the toll itself are of practical im portance. These problems are known as second-best pricing problems. Second-best pricing models are often formulat ed as bi-level program ming (Yang and Lam, 1996; Brotcorne et al., 2001; Chen et al., 2004) or MPEC (Lawphongpanich and Hearn, 2004) where explicit constraints on the pricing instrume nt are employed. The constraints on the tolls can originate from either the network sp atial or the user heterogeneity aspect. One problem from the network sp atial aspect is to select a limited set of toll points and determine the optimal toll rate. Verhoef ( 2002a) derived a general an alytical solution for second-best problem where only a subset of the links can be toll ed. Heuristics for integrated toll location and rate problem were developed based on the optimal solution fo r a given tollable subnetwork (Verhoef, 2002b; May et al., 2002; Shepherd et al., 2004) More generally, Han and PAGE 26 26 Yang (2008) provided a coarse effi ciency evaluation of any given second-best pricing scheme. Another problem important in practice is the co rdon design. Since the cordon design is more restrictive in that a cordon has to be a closed toll ring where users are charged each time they cross the ring, many adopted meta-heuristic algorithms (Yang and Zhang, 2002; Zhang and Yang, 2004; Sumalee, 2004) and others proposed simplified or approximation models (Mum et al., 2003 and Ho et al., 2005). A third problem is for area-based pricing, where trip-chain might be involved. Area-based pricing is similar to cordon pricing but a traveler may have unlimited access to the controlled zone after paying the toll. Lawphongpanich a nd Yin (2008a) developed the link-based equilibrium condi tion under the area-based prici ng and formulate the pricing problem as a MPEC. Comparison between cord on and area-based pricing can be found in Murayama and Sumalee (2007). From a different perspective, economists demonstrated that accounting for user heterogeneity is important in evaluating cons trained pricing policies (Small and Yan, 2001; Verhoef and Samll, 2004). However, the literatu re on constructing second-best pricing scheme where user groups cannot be differentiated is very limited. Verhoef et al. (1995) derived for a simple case the second-best toll as a weighted average of the first-best tolls for each group. 2.2.1.5 Robust pricing A recent trend of static networ k-level pricing models is to address uncertainties of the transportation system. This is meaningful as the system may change within the planning horizon. For example, Li et al. (2007) develope d a bi-level program model to improve travel time reliability with the presence of both random demand and capacity. Considering only demand uncertainty, Gardner et al (2008) proposed three methods to minimize the weighted sum of the mean and the variance of total system travel time. PAGE 27 27 It should be noted that al though robust pricing models are not extensively studied, numerous studies have been done in networ k design problems under demand uncertainties. Some studies apply the notion of reliability-based design (e.g., Yin and Ieda, 2002; Lo and Tung, 2003; Chootinan et al., 2005 and Sumalee et al., 2006) while others seek a robust design that functions reasonably well in ex treme cases (Chen et al., 2003; Ukkusuri et al., 2007; Chen et al., 2007; Yin and Lawphongpanich, 2007; and Yin et al., 2008). The m odels and solution methods in these robust network design problems can be readily applied to r obust pricing problems. 2.2.2 Time-Dependent Pricing Models In this sub-section, we will review time-d ependent equilibrium models in which another dimension of travel decision, namely departur e time choice, is incorporated. Under the timedependent setting, the UE condition can be interpre ted as that no user has incentive to change his combined departure time and route choice. To demonstrate the dispersion in departure time choice, the seminal work by Vickrey (1969) assumed that there is only a single road connecting the origin and destination with a bottleneck of fixed capacity. Traffic congestion wa s assumed to take the form of a point queue. An analytical solution was developed for the sing le bottleneck model with the travel cost as a linear function of travel time, time early and time late disutility. A ccordingly, the first-best timedependent toll takes a triangular form rising linearly with time up to some threshold and declining linearly back to zero. Under this pricing scheme, travelers will not experience any queuing delay but a toll equal to the queuing delay under UE is charged in order to guarantee the tolled equilibrium condition. Newell (1987) provided a general analysis of single bottle neck model considering heterogeneous users. By limiting the heterogene ity only in the values of time, Arnott et al. (1988, 1992 and 1994) further developed a close-form first-best pricing scheme for the case with PAGE 28 28 two groups of users. Their analysis implies th at under UE, users order themselves according to their relative values of travel time to sche dule delay, while under SO the self-assorting only depends on the value of the schedule delay becaus e the queuing delay is eliminated by tolling. In addition to the spatial and us er-class restrictions, there is one more consideration for the second-best pricing in time-depende nt models: the toll may be rest ricted not to vary continuously with time. Both uniform toll and step tolls (Braid, 1989; Laih, 1994; Chu, 1999 and De Palma et al., 2005) are examined in the literature. Though not as effective as first-be st tolls in eliminating queues, second-best tolls can reduc e congestion to a certain level. Extending single-bottleneck model to networks of bottlenecks is not very straightforward. Similar analytical solutions can only be derived for limited simple cases (Arnott et al., 1998). Some studies focused on modeling a more r ealistic interaction between bottlenecks by considering physical queues (Mun, 1999; Lo a nd Szeto, 2005). For general networks, dynamic traffic assignment models are incorporated to obtain optimal pricing schemes (Carey and Srinivasan, 1993; Huang and Yang, 1996; Wie a nd Tobin, 1998; Joksimovic et al., 2005; Wie 2007). However, these models are extremely difficu lt to solve due to th e ill-behaved properties of dynamic traffic assignment, such as non-convexi ty and non-differentiab ility, arising from the need to realistically model flow dynamics (Peet a and Yang, 2003). Moreover, instead of solving for departure time jointly, some of the models require a full knowledge of dynamic OD demand, a type of information difficult to obtain. Robust pricing models in time-dependent sett ing are limited. Nagae and Akamatsu (2006) developed a stochastic singular control model for dynamic revenue management under demand uncertainty for a private road. The resulting model is complicated and no efficient solution approach is provided for netw orks of a realistic size. PAGE 29 29 2.3 Traffic-Responsive Pricing Models We have discussed canonical e quilibrium-based models (static and time-dependent) in the previous section. These models usually a ssume known demand and link performance functions, and ignore the evolution of the system to the stati onary equilibrium state. In contrast, in trafficresponsive models, travel demand is usually unknown. Moreover, this approach features an observation-adaption feedback loop which takes in to account travelers learning process in response to the toll charges. The objective in traffic-responsive models is not necessarily system-wide social benefit but some local target. 2.3.1 Trial-and-Error Approach All the equilibrium-based models require the knowledge of travel demand functions, which is usually unknown and difficult to estimate in pr actice. Trial-and-error approach is proposed particularly to handle the situat ion where travel demand is unknown. Vickery (1993) and Downs (1993) first argued that MC congestion pricing can be applied with the absence of demand functi on on a trial-and-error basis. Li (1999, 2002) materialized this concept and developed a bisection implementation. The MC toll is calculat ed in the same way as in the static model with an initial target flow leve l. After applying this to ll and the realized flow is observed, the new target flow is set as the av erage of the originally intended and the realized flows. The procedure then proceeds back and fo rth until the realized flow coincides with the target flow. Yang et al. (2004) further enhanced the procedure by modifying the updating rule of the target flow and provided a theoretical foundation of the feasibility of the trial-and-error approach. However, both models assumed the obs erved flow is the stationary state under the current toll. This assumption ag ain ignores the evolution of syst em state to the equilibrium and consequently the loss of system benefit within this process. PAGE 30 30 2.3.2 Day-to-Day Dynamics Day-to-day dynamics of travelers response to the change of travel cost and the planners follow-on adjustments to the tolls are considered important in the implementation of congestion pricing. Whether and how the sy stem will achieve SO is not a concern in the equilibrium-based models but is the main subject of several studies on modeling pricing stra tegies under day-to-day dynamics. Sandholm (2002) was among the first to study e volutionary implementation of congestion pricing. A model based on potential game th eory (Sandholm, 2001) was developed. It is assumed that the planners only have limited know ledge of travelers pr eference and behavior, and update the pricing scheme with new observations of users re sponse. Friesz et al. (2004) developed a disequilibrium prici ng scheme to maximize the net social benefit. Travelers adjustment to experienced travel time is assume d proportional to excess travel cost, known as the proportional-switch adjustment pr ocess originally proposed by Smith (1984). The resulting dayto-day dynamic tolls do not necessarily settle the system at SO but some other tolled UE flow pattern. On the other hand, Yang and Szeto (200 6) proposed a model that guarantees the final stationary condition to be SO, regardless of users underlying adjustment process. More recently, Yang et al. (2007) suggested a new appr oach of maximizing the decreasing rate of total system travel time at each particular day. Th is approach not only ensu res the convergence to SO but also expedites the evol ution process substantially. 2.3.3 Adaptive Pricing Scheme for Managed Lanes Traffic-responsive pricing strate gies are more widely applied to facility-level problems than network-level ones. One pl ausible reason is that implementa tion of traffic-responsive tolls at facility-level is much easie r. Another reason is that thr ough traffic-responsive tolls, local objectives can be better achieved (Supernak et al., 2003b). The concern th at traffic-responsive PAGE 31 31 tolls may confuse the travelers is barely an issue at the facility level (Bonsall et al., 2007). The most common traffic-responsiv ely-priced facility in the U.S. is managed lanes. 2.3.3.1 Managed lanes The FHWA defines managed lanes as a lane or lanes designed and operated to achieve stated goals by managing access via user group, pr icing, or other criteria. A managed lane facility typically provides improved travel c onditions to eligible users. (FHWA, 2003) Although the concept of managed lanes has not ga ined significant interest until recently, the practice started decades ago. Since the first bus exclusive lane was opened in Washington, D.C. in 1969 (FHWA, 1990), HOV lanes became widely acc epted and implemented in the U.S as the most prevalent form of managed lanes. The early projects have demonstrated the potential of managed lanes in improving the efficiency of th e transportation system. Nowadays, managed lanes are considered as a cost-efficient operationa l strategy to effectively address the mobility needs under limited ability of capacity expansion. Besides various exclusive/bypass /reversible facilities, the ma jority of existing managed lanes are still HOV/HOT lanes. There are over 2600 miles of HOV/HOT fac ilities in U.S. and Canada by the end of 2005 (see Fuhs, 2005 for a mo re detailed inventor y), and more HOV/HOT projects are upcoming these years (see, e.g. FH WA, 2008 for a summary of new HOT projects). Based on their physical design, managed lanes can be classified into th ree general categories: barrier-separated, wide-buffer-separated and non-se parated. The ingress and egress designs of managed lanes depend largely on both its phys ical separation, location and management strategies. A variety of differe nt weaving and buffering configura tions of the ingress/egress are possible. Figure 2-2 illustrates the three most typical weaving designs for HOT facilities (FHWA, 2003) and can be generalize d to other managed lanes as well. PAGE 32 32 2.3.3.2 Current operation of HOT lanes While in a broad sense the concept of managed lane covers various management strategies as discussed above, it refers only to HOT facilities in a na rrower sense (Obenberger, 2004). These facilities have the ability to adapt their management stra tegies (in a real-time manner) to the changing traffic conditions. This dynamic ch aracteristic is what differentiates HOT lanes with other facilities as a recent trend of the latest managed-lane technology (Obenberger, 2004). Currently there are seven HOT f acilities in operation in the U. S. All the facilities allow eligible vehicles to pay onsite to gain access ex cept the I-15 express lane in Utah, which provides a limited number of permits on a monthly basis (Utah DOT, 2009). Table 2-1 summarizes other six HOT facilities. These managed lanes have diffe rent time periods of operation and the pricing schemes vary from facility to facility. Note th at the pricing schemes ar e categorized into three classes: fixed tolling scheme charges a constant usage fee throughout the entire operation period; time-dependent tolls vary with time (usually ch ange every 30 minutes to one hour) but the rates are pre-determined; and in adaptive pricing, tolls are adjusted real time in response to the current traffic condition and may vary every several minut es. For example, tolls on I-394 in Minnesota vary from $0.25 to $4.00. When changes in density are detected, tolls can be adjusted as often as every three minutes based on a lookup table (Halvorson et al., 2006). Figure 2-3 illustrates the toll rates for one hour of a particular day on I-394. Similarly, tolls on I-15 vary from $0.50 to $4.00 with the highest value at $8.00, and can be adjusted up to every six minutes based on traffic counts. Recently, a new distance-based pricing system was launched on I-15. The total toll rate depends on the distance the user travel s, while the per-mile toll rate is calculated responsively to the current tra ffic condition (San Diego Associ ation of Governments, 2002). Another newly opened HOT facility on I-95, Florida also has the capability of adaptive tolling. PAGE 33 33 The proposed adaptive scheme responds to vol ume changes and adjusts toll rates each time interval by a preset margin (Wilbur Smith, 2007). Although HOT has been launched for more than ten years and achieved visible benefits, there is still a heated debate on when and where to implement HOT, HOV and general purpose lanes in the transportation community. While Dahlgren (1998) concluded that adding a HOV lane is not always more effect ive than general purpose lanes, Da hlgren (2002) suggested that an added ideal (in the sense that toll can be correctly set such that th e HOT lane can be fully utilized but not congested) HOT lane performs (in terms of reducing delay) as well as or better than an HOV lane in all circumstances. Pr actical tools are deve loped to assess the conversion of HOV to HOT as well (e.g., Eisele et al., 2006). Besi des original conditions on HOV and general-purpose lanes, tolling structure, driver s behaviours and performance of HOT, factors considered in implementation also include facility cross section and access design, enforcement and institutional considerations (E isele et al., 2006 and Ungemah and Swisher, 2006). Same as in general road pricing, equity issue has always been a concern for HOT projects. Equity among different socio-demographic groups and users of different modes, and other equity issues caused by location and access of the HOT ha ve attracted a lot of atten tion among all equity concerns (Weinstein and Sciara, 2006). Regardless of all the debates about HOT projects, there is no doubt the tolling scheme plays a key role in the success of HOT lane ope rtations. Often time operation policies of HOT lanes are to provide a superior free-flow traffic service on the toll lanes while maximizing the throughput rate of the freeway, i.e., the comb ined throughput of both re gular and toll lanes (FHWA, 2003). To achieve the objectives effici ently, tolls should be ad justed real time in response to changes in traffic conditions. Empiri cal studies also show that adaptive tolling PAGE 34 34 structure outperforms fixed ones (Supernak et al., 2003b and Minneso ta Department of Transportation, 2006). Supernak et al. (2003b) found the dynamic traffic-responsive tolls on I15 significantly increase the utilization of the f acility, create the desirable redistribution of peakhour traffic and maintain a level of service C vi rtually at all times, whil e early phase of fixed tolls were not able to achieve the same performa nce. However, the lite rature on adaptive tolling strategies is very limited. Previous studies on time-varying tolls for bottlenecks (e.g., Arnott et al., 1998; Chu, 1995; Liu and McDonald, 1999; Yang and Huang, 1997) only considered hypothetical and idealized situations where syst em performance under equilibrium condition is the major concern, thus do not fit in the adaptive control category. It is worth mentioning that although the idea of adaptive control is rarely us ed in the content of pricing, it has been successfully applied to another freeway manage ment strategy, ramp metering (see Papageorgiou and Kotsialos, 2002 for a review). 2.4 Summary This chapter has briefly reviewed the advant ages, limitations and applicability of the pricing models in the literature. The models are coarsely classified into three categories. Canonical equilibrium-based stat ic models provide a genera l guidance for planning level pricing applications. Most of th e static models are developed ba sed on the firstand second-best pricing concept in economics. Although some models attempted to incorporate more sophisticated human and network factors such as heterogeneity and uncertainty, most of them have oversimplified the r eal-world situation. For instance, ideal assumption of users perfect rationality and employment of simple macrosc opic network performance functions are common. Also, sources of network uncertainties are li mited only to variable demand and capacity. Time-dependent equilibrium-based models e nhance the description of traffic dynamics by applying dynamic traffic assignment models wi th known time-dependent OD demands. More PAGE 35 35 comprehensively, some models combine the depa rture time and route choice together as the underlying behavioral representation One major drawback of these models is that the resulting pricing schemes are usually so complicated that they are not suitable for real operations in general networks. Traffic-responsive pricing models fill the gap from the implem entation of tolls to reaching equilibrium by taking into account the evolutionary pro cess in between. Mo re importantly, the observation-adaption approach is applicable to the cases where travel demand function is not known. Traffic-responsive models are particular ly useful in managed-lane operations. The feedback feature also makes it possible to achi eve other local objectives more effectively. However, studies on developing practical adaptive pricing models for managed lanes are limited. We conclude that in order to be applicable to a general urba n network, system-level pricing strategies should be static or at least static during each time period (an hour or a few hours) of a day. The current static equilib rium-based pricing models can be further enhanced by improving the representation of travelers behaviors and ne twork performances to provide more efficient and robust pricing schemes for system-wide implemen tation. At certain critical facilities, the pricing can be traffic-responsive for better local performance. The proliferation of managed lanes has imposed a pressing need for more efficient traffic-responsive pricing models. PAGE 36 36 Figure 2-1. Economics Foundati on of Marginal-Cost Pricing PAGE 37 37 Median Barrier Shoulder Shoulder Regular Lane Weave Lane HOT Lane E x i tE n t r a n c eTag Reader Barrier-Separated Option Median Barrier Wide Buffer Regular Lane Weave Lane HOT Lane ExitEnt r an c eTag Reader Narrow Buffer-Separated Option Wide Buffer Median Barrier Regular Lane HOT Lane Tag Reader Reduced Narrow Buffer-Separated Option (No Weave Lane) Figure 2-2. Alternative HOT la ne slip ramp configurations PAGE 38 38 0 0.5 1 1.5 2 2.5 3 3.57:09 7:12 7:15 7:18 7:21 7:24 7:27 7:30 7:33 7:36 7:39 7:42 7:45 7:48 7:51 7:54 7:57 8:00 8:03 8:06 8:09Time of dayToll rate ($) Figure 2-3. I-394 toll rates on May 18, 2005 (data source: www.MnPASS.net) PAGE 39 39 Table 2-1. Current onsite-accessi ble HOT facilities in the U.S. PAGE 40 40 CHAPTER 3 HIERARCHICAL PRICING FRAMEWORK This chapter elaborates the hierarchical pr icing framework we proposed to advance road pricing to be a more efficient and pragmatic tool for congestion mitigation. The essential idea is to decompose the decision of toll rates into two levels: network and facility. Tolls at these two levels serve distinctive purposes and address different objectives. Tolls at the network level aim to manage th e travel demand of the system. These tolls should be simple and logical so that they can be well understood by the travelers, which is a necessary condition for them to respond to the char ges and shift their travel pattern towards the direction desired by the planners. Complicated network-level time -dependent tolls are not only difficult to calculate and implement but also hard for the travelers to follow, thus less likely to affect users travel pattern (Bons all et al, 2007). In addition, netw ork-level tolls are expected to accommodate to a certain extent th e uncertain nature of the system A system-level toll strategy should be robust enough against demand surge, ra ndom network failures ( due to incidents or adverse weather) and uncertainties arising from travelers stoc hastic and irrational travel behaviors. These issues have not been adequately addressed in the literature. Facility-level tolls, especially when applied to managed lanes, serve as the tickets of an extra travel option. Travelers are able to buy-in based on their overall evaluation of the managed facilities. Tolls for a certain f acility can be more adaptive becau se travelers are usually able to adjust their lane choice locally to respond to the varyi ng toll rates. The tol ling objective at the facility level are often operati onal, and thus a dynamic pricin g scheme may serve the purpose better through iterativ ely learning information from bot h the demand and the supply side, including the demand function of the managed lanes and the traffi c condition along the facility. PAGE 41 41 The literature does not offer sensible and practi cal pricing models for managed lanes, leaving a gap to fill. In the following two sections, we briefly identify the key issues that may need to be addressed to materialize the propos ed hierarchical pricing concept. General data requirements and implementation concerns of both the networkand the facility-level pricing schemes are described. 3.1 Robust Pricing at Network Level The following technical issues are to be solv ed in order to develop a simple but wellfunctioning network-level pricing scheme. Determine times-of-day for pricing. The di spersion of congestion during the day needs to be analyzed for a complete understanding of the current traffic patt ern and the congestion problem. Since many problems of transportati on systems occur during the peak period, differentiation of peak and off-peak periods is necessary in developi ng appropriate pricing strategies. In many situations the congestion externality in the off-peak periods may not be severe and thus the pricing may not be necessary. Estimate OD travel demand for each period. OD tr avel demand is one critical data input to network-level pricing models. Time-of-day OD information can either be predicted from the planning models of metropolitan pla nning organizations (e.g., FHWA, 1997) or estimated using prior OD data combined with ob served link flows (e.g., Chen et al., 2004). To obtain a robust pricing scheme, demand uncerta inty should be proactively considered in the pricing models. The uncerta inty set or likelihood region of OD demand may need to be specified, as discussed in Yin et al. (2008). Select a primary system perfor mance measure. The performance measure of major concern may vary among different agencies, which will sh ape the toll scheme they may implement. Total system travel time, total social benefit, reliability or some other environmental measures may all serve as valid performance measures. Identify and model uncertainties in the system. The sources of uncertainty for a transportation system may reside in various aspects. Common sources are travel demand, incidents, infrastructure condi tions and travelers behaviors etc. It is important to investigate which one(s) has ma jor effects on the performance of the network so that the modeling effort can be correctly pu t into those that need the most. Toll determination. The pricing scheme should be robust in the presen ce of uncertainties in order to perform well in most circumstances. Practical and physical constraints on tolls should be considered when determin ing where and how much to charge. PAGE 42 42 Communicate with general public and implemen t the pricing plan. The importance of public acceptance of a pricing scheme cannot be overly emphasized. It is the key to the success of the pricing strategy. Before ac tual launch of the pricing program, public outreach in many ways and a trial period will be necessary. The above are among the major issues of designi ng a network-level toll This dissertation will focus on the most technically-challenging issue, i.e., modeling the uncertainties and determining robust tolls. 3.2 Adaptive Pricing at Facility Level In order to better achieve the local objectives at the facility level, tolls obtained from the network level can be further adjusted. More microscopic details of traffic dynamics and travelers choice behaviors should be considered in order to adjust the tolls in a real-time manner to effectively respond to the changes in traffic condition and demand. Some of the major issues are as follows. Define local objectives. Local objectives of a particular facility often depend on its design functionality. As for HOT lane s, the objective is usually to provide a superior free-flow traffic service on the toll lane s while maximizing the throughput rate of the freeway, i.e., the combined throughput of both regu lar and toll lanes (FHWA, 2003). Estimate traffic state. Difference between th e current and the desire d traffic c onditions of the facility governs what toll should be charge d for managed lanes. It is crucial to have a complete image of the freeways traffic state at the current time interval based on a limited amount of measurement data at sensor lo cations (e.g., Wang a nd Papageorgiou, 2005). Learn users WTP. Travelers choice of whet her to use the optional to lled facility depends largely on their WTP. This information is usually unknown in advance because it varies among different locations, time of day and driv er populations. During the operation, this information can be gradually l earned by mining the loop detector data, and then be applied to determine optimal tolls for achieving control objectives. Predict short-term future demand. The demand of the tolled facility also depends on the total traffic demand, which often fluctuates and is one of the uncertainties in facility level pricing. It is important to predict traffic demand such that toll determination can be more proactive. The forecast is instrumented by Bayesian learning based on archived and realtime loop data (e.g., Lin, 2006). Toll determination. Based on the estimated current traffic conditi on, calibrated users WTP, and the predicted demand, optimal adaptive tolls can be determined to achieve local PAGE 43 43 objectives. The tolls determined from the ne twork level should be the basis for deciding the feasible ranges of f acility tolls. When optimizing th e tolls, detailed traffic dynamics can be modeled to predict the effect of pricing. Coordination of tolls in time and space. In order to avoid charges that fluctuate dramatically in time, tolls can be determined proactively considering the future traffic demand. A rolling horizon framework may be a dopted to coordinate the tolls in time. Coordination of several segments of the manage d facility may help reduce the inequity that may arise among users who access the facility from different entrances. Enforcement. Managed lanes are easier for th e general public to accep t, because instead of tolling off travelers from original travel routes, they provide an additional option. Compared with network-level tolls, institutional difficulties in implementation may be reduced. But violations of managed lane pol icies may occur due to ineffective design of the facility or insufficient enforcement. Th is issue is among the mo st important practical concerns (Ungemah and Swisher, 2006). For facility-level adaptive pricing, this disse rtation will focus on developing a practical and sensible pricing scheme for HOT lanes. 3.3 Summary This chapter has presented a hierarchical pricing framework for system-wide congestion pricing. Based on practical need s at the network and facility le vels respectively, robust time-ofday tolls are proposed for the network-wide impl ementation while adaptive tolls are advocated for critical facilities. Genera l inputs and technical issues of this hierarchical framework are discussed. We believe the propos ed framework is feasible and has great potential for actual implementation. In the following chapters of this dissertation, we will address main technical issues at both levels to further materialize the framework. PAGE 44 44 CHAPTER 4 ROBUST NETWORK PRICING WITH BOUN DEDLY RATIONAL US ER EQUILIBRIUM This chapter presents a static model for dete rmining optimal pricing scheme at the network level. Since network-level tolls need to be simp le and logical for users to follow, it may be less frequently revisited and often ex tends over a certain period of time (e.g., one year). Therefore, robustness becomes a more importa nt merit and concern in the toll optimization because we expect the pricing scheme to function reasonabl y well under various circumstances. Sources of uncertainty in transportation networks consist of not only randomness in demand and supply, but also travelers stochastic and ir rational behaviors. Note that uncertainties in demand and supply can be easily incorporated using similar methods that have been applied to robust network design problems (see Section 2.2.1.5). This chapter wi ll instead focus on one particular uncertainty arising from travelers bounded rational route choice behavior. Under such behavior, users do not necessarily choose a shortest or cheapest r oute when doing so does not reduce their travel time by a significant amount. The objective is to study the implication of BRUE in transportation systems planning a nd develop an analytical framew ork to proactively account for boundedly rational travel behaviors in the context of congestion pricing. In Section 4.1, we will first review the literature on bounded rationality. Section 4.2 then mathematically defines BRUE and discusses the property of the set of all possible BRUE flow patterns. The problems of finding the besta nd worst-case system travel times among the possible BRUE flow patterns are formulated as mathematical programs with complementarity constraints (MPCC) and then illustrated with examples. Section 4.3 formulates congestion pricing models that minimizes the worst-case system travel time and develops a heuristic solution algorithm to the models. Numerical ex amples are then presented to demonstrate and validate the models and solution algorithm. Summary of this chapter is pr ovided in Section 4.4. PAGE 45 45 4.1 Bounded Rationality Congestion pricing models in th e literature (see Chapter 2) t ypically assume that users are perfectly or unboundedly rational, i.e ., they always choose shortest (o r cheapest) routes possible. However, when doing so only lead to a small or negligible improvement in their travel times, some users may not be sufficiently encouraged to change routes in practice. Behaving in this manner, users are said to have a bounded ra tionale. The literature in psychology and economics has provided a wide range of evidence that bounded rationality is important in many contexts, particularly in the context of day-to-day choices (see Conlisk, 1996 for a recent review on this topic). In transportation, the number of models that allow for bounded rationality is minuscule. Mahmassani and Chang (1987) study the travel and trip timing deci sions by boundedly rational travelers in an idealized setti ng that consists of one OD pair and a single route. Later, Jayakrishnan et al. (1994), Hu and Mahma ssani (1997), Mahmassani and Liu (1997), and Mahmassani (2000) apply the concept and similar m odels to study, e.g., the effects of advanced traffic information and management systems on road systems. Similar to Conlisk (1996), Nakayama et al. (2001) concludes that their experimental study indi cates a need to evaluate the validity of the perfectly rationa l assumption in the traffic e quilibrium analysis. Although referred to as tolerance-based, Szeto a nd Lo (2006) are forced to consider bounded rationality in their dynamic traffic assignment pr oblem because the problem may be infeasible when travelers are perfectly rational. In their paper, Szeto and Lo compare and contrast queuing paradigms in a dynamic setting, a setting under which many problems in transportation can be massively large, cannot be solved analytically, a nd often have to rely on computer simulation for solutions. PAGE 46 46 In contrast to previous researches on BRU E in dynamic setting, this chapter addresses boundedly rational route choice beha viors in static networks wh ere BRUE arises whenever all users travel costs are not sufficiently larger than the best available ones and no user has an incentive to switch his or her route. The focus is to study the implication of BRUE in transportation systems planning a nd develop an analytical framew ork to proactively account for boundedly rational travel behaviors in the context of congestion pricing. We also note that there are other network equilibrium models in the literature that attempt to describe more realistically users route choice behaviors, su ch as stochastic user equili brium models (e.g., Sheffi, 1985), risk-taking models (e.g., Mirchandani and S ourosh,1987; Yin and Ieda, 2001), travel time budget models (e.g., Lo et al. 2006; Shao et al ., 2006). The purpose of studying BRUE is to offer another alternative model that enriches the lite rature and to fully ex plore the properties and consequences of bounded rati onality in route choices. As Mahmassani and Chang (1987) pointed ou t, BRUE flow distribut ions in a static network may not be unique. In one interpretati on, the flow distributi on we observe at one particular day is just a partic ular realization from the set of multiple feasible BRUE flow patterns. Such an uncertainty incurred by bounde dly rational travel behaviors may deteriorate the effectiveness of traditional congestion pricing strategies. For example, the link-based MC pricing scheme commonly advocated in the cong estion pricing literature may not necessarily reduce congestion to its minimum level because us ers may not switch to routes with the least generalized cost. In this chapte r, we seek a tolling pattern that minimizes the worse-case system travel delay among all the possible BRUE flow pa tterns based on generalized costs (time plus tolls). PAGE 47 47 4.2 Boundedly Rational User Equilibrium in a Static Network 4.2.1 Definition of BRUE As explained in Mahmassani and Chang (1987) and Chen et al. (1997), travelers with bounded rationale still follow the behavior that ex hibits a tendency toward utility maximization, but not necessarily to the absolute maximum le vel. We thus define travelers with bounded rationality as those who (a) alwa ys choose routes with no cycle and (b) do not necessarily switch to the shortest (cheapest) routes when the diffe rence between the travel times (or costs) on their current route and the shortest one is no larger th an a threshold value. Using the terminology in Ahuja et al. (1992), all utilized routes under bounded rationality must correspond to paths. (In the literature, others refer to pa ths with no cycle as simple paths. Ahuja et al., 1992, refer to routes with cycles as walks.) The above c onsideration motivates to the following definitions: Definition 4.1: A path is considered acceptable if the difference between its travel time or cost and that of the shortest or least-cost path is no larger th an a pre-specified threshold value. Definition 4.2: A path flow distribution is at BRUE if it is feasible or compatible with the travel demands and every user uses an acceptable path. Unlike the conventional or perf ectly rational user equilibrium (PRUE) the flow districution under BRUE as defined above may not utiliz e any shortest or least-cost path. To mathematically define BRUE, we follow the notation in Chapter 2. We further let Pw be the set of paths for OD pair w. For each path r Pw, w rf and w rc denote the corresponding path flow and path travel time. We use wf to represent a vector of path flows for OD pair w with w rf as its elements, and T W wf f f f ,1 where denotes the cardinality of a set. We may similarly define wc and c for the vectors of path travel times. Assuming that users of PAGE 48 48 the same OD pair have the same threshold value, the result below describes a BRUE flow distribution algebraically. Theorem 4.1: Assume that the threshold value for OD pair w is w. Then, a feasible path flow distribution f is at BRUE if and only if there exist Rw for each w such that the following conditions hold: w w rc W w P rw ( 4.1) w w w rc r wP { r : w w rP r f 0} ( 4.2) w P r w rq fw W w ( 4.3) 0w rf W w P rw ( 4.4) Proof: For a given feasible pa th flow distribution wf, denote the corresponding minimum OD travel time as wc. Necessity: If f is a BRUE flow distribution, it is feasible by definiti on, thus conditions ( 4.3) and ( 4.4) are satisfied. Conditions ( 4.1) and ( 4.2) can be satisfied by setting w wc Sufficiency: If f is a flow distribution with wsatisfying the above conditions, then conditions ( 4.3) and ( 4.4) guarantee that the path flow distribution is feasible. Condition ( 4.2) implies w is the lower bound of w rc. Together with condition ( 4.2), we have w w w w r P r w w w rc c cw min for every utilized path r, suggesting that the travel time of a ny utilized path may differ at most the threshold value from the minimum OD travel time. The above conditions requiring knowing utilized pa ths. As an alternative, the ones below use slack variables w r and do not require the uti lized-path information. PAGE 49 49 0 w r w w rc W w P rw ( 4.5) 0 ) ( w r w w rf W w P rw ( 4.6) w P r w rq fw W w ( 4.7) 0w rf W w P rw ( 4.8) 0w r W w P rw ( 4.9) Conditions ( 4.5)-( 4.9) are equivalent to conditions ( 4.1)-( 4.4), thus equiva lent to the BRUE definition. We therefore refer equations ( 4.5)-( 4.9) as the BRUE conditions. Note that when 0 w for all w, that the BRUE conditions reduce to the PRUE conditions. To illustrate, combining conditions ( 4.6), ( 4.8) and ( 4.9) yields 0w r w rf, which together with condition ( 4.5) leads to 0 ) ( w w r w rc f, W w P rw ,. On the other hand, a PRUE flow distribution al ways satisfies conditions ( 4.5)-( 4.9) with 0w r and w as the equilibrium O-D travel time. Thus, a PRUE distribution is al ways a BRUE flow distribution. 4.2.2 Properties of BRUE Flow Distributions When users are perfectly rational, it is wellknown (see, e.g., Patriksson, 1994) that there exist equivalent link-based equilibrium conditions and the set of all PRUE flow patterns is convex. Below, we show that these propertie s may not hold when user s are boundedly rational. More specifically, the link-based representation of BRUE proposed in this dissertation is not equivalent to the path-based definition a nd the set of BRUE flow s may not be unique. 4.2.2.1 Link-based representation of BRUE Let the notations follow those defined in Chapter 2. Moreover, we assume 0 ) (ij ijv t when 0ijv. The following conditions ( 4.10)-( 4.15) provide a link-based representation of BRUE, analogous to those under the perfectly rational assumption: PAGE 50 50 0 ) ( w ij w j w i ij ijv t L j i W w ) ( ( 4.10) 0 w j w i w ij w ijx L j i W w ) ( ( 4.11) w w o w w w d) ( ) ( W w ( 4.12) w wD x W w ( 4.13) v = w wx ( 4.14) 0 x ( 4.15) where N wR, L wR and N wR are vectors of node potentials associated with travel time, .travel times in excess of the minimum, a nd node potential associated with the excess travel times respectively. Using a similar argument previously for the path -based BRUE definition, it can be verified that when 0w, conditions ( 4.10)-( 4.15) reduce to PRUE conditions in the literature when users are perfectly rational. When 0 w, we prove in the following that a flow distribution satisfying conditions ( 4.10)-( 4.15) is a BRUE flow (see Defi nition 4.2). However, as a counterexample below illustrates, the converse is not true. Lemma 4.1: Define a sub-network wG with respect to each OD pair w, which consists of all the links with strictly positive OD flows w ijx. If there exist a link flow distribution v, vectors N wR, L wR and N wR satisfying conditions ( 4.10)-( 4.15), then the sub-network wG is acyclic. Proof: Suppose that there is a cycle in wG. By definition, 0w ijx for each link wG j i) (. Condition ( 4.11) then requires 0 w j w i w ij Thus, summing conditions ( 4.10) and ( 4.11) along the cycle leads to 0 ) ( ij ij ij ij ijv t ( 4.16) PAGE 51 51 However, since 0 w ij ijx v, we have 0 ) (ij ijv t which contradicts condition ( 4.16). Therefore wG is acyclic. Note that Lemma 4.1 applies to PRUE as we ll, under which the flow-carrying sub-network is acyclic if the assumption of link travel time functions holds. Theorem 4.2: For a link flow distribution v if there exist N wR, L wR and N wR such that conditions ( 4.10)-( 4.15) hold, then any of its underlying path flow patterns satisfies Definition 4.2. Proof: Assume that a link flow distribution v satisfies conditions ( 4.10)-( 4.15). As v satisfies conditions ( 4.13)-( 4.15), it is a feasible fl ow distribution. According to Lemma 4.1, the flowcarrying sub-networks are acyclic, implying the u nderlying utilized routes are all paths. Let r ij (equals 0 or 1) indi cates whether link ( i, j ) is on path r It follows from conditions ( 4.10) and ( 4.15) that for any path between O-D pair w ij ij ij r ij ij w ij r ij ij ij ij r ij w w o w w dv t v t ) ( ) () ( ) ( ( 4.17) Therefore, w w o w w d ) ( ) ( is the lower bound of wc, the minimum travel time between a given OD pair w. For any utilized path wP rwith respect to the given link flow distribution, the flow on each link along the path must be positive, i.e., 0w ijx if 1r ij. Thus summing together condition ( 4.11) for links on a utilized path wP r yields: w w d w w o ij w ij r ij ) ( ) ( ( 4.18) Essentially w w d w w o ) ( ) ( is the upper bound on the amount in exce ss of the shortest travel time. Combining conditions ( 4.12), ( 4.17) and ( 4.18) for a utilized path r leads to: w w w d w w o ij w ij r ij w w o w w d ij ij ij r ij w ij ij ij r ijv t c v t ) ( ) ( ) ( ) () ( ) ( ) ( PAGE 52 52 which implies that for any utilized path, its travel time may differ at most the threshold value from the minimum OD travel time. Ther efore, any underlying path pattern of v is at BRUE. On the other hand, it is possi ble that a BRUE flow distribu tion that satisfies conditions ( 4.5)-( 4.9) does not satisfy conditions ( 4.10)-( 4.15). This implies that the link-based representation of BRUE is more restrictive than the path-based definition. Consider the bridge network in Figure 4-1 in which there is only one OD pair (1, 4) with a demand of 3 and a threshold value of 4. The link tr avel time functions are reported in Table 4-1 together with a BRUE path flow pattern. The minimum OD travel time is 9 and all the ut ilized paths have travel times of no more than 9 + 4 = 13. Therefore, the path flow pattern is a valid BRUE flow. However, the corresponding link flows do not satisfy conditions ( 4.10)-( 4.15) as the following system of equations does not admit a solution. 0 4 4 4 4 0 4 0 1 0 6 0 1 0 8 0 534 13 24 32 13 34 23 12 24 12 4 3 34 2 3 32 4 2 24 3 2 23 2 1 12 3 1 13 ij The nonequivalence between the two sets of conditions originates from the relationship between path and link flows. Under BRUE, it is possible that a path flow pattern results in a cyclic flow-carrying sub-network (the original network in the above example) while with the link-based BRUE definition, all the flow-carrying sub-networks must be acyclic (Lemma 4.1). PAGE 53 53 Moreover, for a non-utilized path th at only consists of links with positive flows (path 1-2-4 in the above example), the link-based BRUE definition s till requires its travel time to be within the indifference band even that the pa th is not utilized (conditions ( 4.10) and ( 4.11)). Note that both issues do not arise in the PRUE because (a) the flow-carrying sub-networks are acyclic under UE; and (b) all links with positive flows must be on shortest paths. For a non-utilized path that consists of only positive-flow links, its trav el time will be minimum OD travel time and conditions ( 4.10) and ( 4.11) alike will be satisfied automatically. In this sub-section, we provided a more re strictive link-based re presentation of BRUE flow. Figure 4-2 illustrates the relati onship between the feasible flow set, the path-based and the link-based BRUE flow set. Note that all th ree sets contain the PR UE flow distribution. 4.2.2.2 Existence, uniqueness and nonconvexity In a static network with con tinuous travel time functions, a BRUE flow distribution always exists since a PRUE distribution is a BRUE flow distribution with 0wand the existence of the former is ensured by the continuity of travel time functions (e.g., Patriksson, 1994). If the threshold value, w, equals zero for each OD pair, BRUE redu ces to PRUE. In this case, BRUE link flow distribution is unique as we have assumed that travel time functions are monotonically increasing (e.g., Patriksson, 1994). In general, however, BRUE link flow distributions are not unique. For example, if the threshold value is sufficiently large, every feasible path flow distribution is a BRUE distributi on and the set of BRUE flows is a polyhedron. We further use the following example to illustrate that in general the BRUE set is not convex in path flows. Consider the same network in Figure 4-1 with different trav el time functions reported in Table 4-2 with one OD pair (1, 4) as before. However, other pr oblem parameters are new. The PAGE 54 54 travel time functions are as shown in Table 4-2. The demand and a threshold value are 6 and 15, respectively. Table 4-2 displays two BRU E flow distributions, BRUE-1 and BRUE-2. The maximum differences between the lengths of utilized paths and the shortest paths are 13 for BRUE-1 and 14 for the other, i.e., both differences are with in the threshold value. However, the convex combination of these two flow distributions 0.5(BRUE-1 + BRUE-2) is not a BRUE flow distribution because the maximum difference now is 24. This implies that the BRUE flow set is not convex. Indeed, the non-convexity of the BRU E flow set is largely due to the nonlinear condition ( 4.2) in the BRUE definition. Similarly, the restrictive link-based BRUE set is not convex either due to the complementarity condition ( 4.7). 4.2.3 Finding Bestand Worst-Case BRUE Flow Distributions In practice, traffic assignment models in the four-step process are based on PRUE and typically provide a unique equilibrium solution. In other words, PRUE m odels typically provide a point estimate of traffic flow distribution. On the other hand, there are generally many BRUE flow distributions and models ba sed on BRUE provide an interval estimate instead. The problem of finding best-case BRUE flow distribut ion can be formulated as follows: BC/WC-BRUE: ,max / minf f f cT) ( s.t. Conditions ( 4.5)-( 4.9) Clearly, the above two problems identify BRUE fl ow distributions that minimize and maximize the total travel time in the system. To avoid generating paths, we henceforth work with the more restrictive link-based BRUE representation and define the analogous best and worse-case problems as follows. The formulation contains auxiliary variables, w ijz, and additional constraints to highlight the complementarity structure in the problems. PAGE 55 55 BC/WC-ABRUE: ,max / minx v t ( v )Tv s.t. 0 ) ( w ij w j w i ij ijv t L j i W w ) ( 0w ij w ijz x L j i W w ) ( ) (w j w i w ij w ijz L j i W w ) ( 0w ijz L j i W w ) ( Conditions ( 4.12)-( 4.15) As formulated, both BCand WC-ABRUE are MPCC, a class of optimization problems difficult to solve because the problems are generally non-convex and violate the MangasarianFromovitz constraint qualification (MFCQ) at any feasible point (see, e.g., Scheel and Scholtes, 2000). In the literature, many (s ee, e.g., Luo et al. 1996 and references cited therein) have proposed special algorithms to solve MP CC. Lawphongpanich and Yin (2008b) applied concepts from manifold sub-optimization and proposed a new algorithm that converges to a strongly stationary solution in a finite number of iterations. The algorithm can be applied with multiple initial solutions to solve both form ulations for good strongly stationary solutions. 4.2.4 Numerical Examples This sub-section presents two numerical exampl es to demonstrate the relevance of BRUE. BC-ABRUE and WC-ABRUE are solved on two ne tworks: nine-node (see, e.g., Hearn and Ramana, 1998) and Sioux Falls (see, e.g., LeBlan c et al., 1975) using GAMS (Brook et al, 2003) and CONOPT (Drud, 1995). Nine-node problem: Figure 4-3 displays the underlyi ng network. The network data can be found in Hearn and Ramana (1998). Ther e are four OD pairs and the demands are 103 1q, 204 1q, 303 2q and 404 2q. The threshold value is assumed to be w UE wt* where is a parameter and w UEt* is the PRUE travel time between a given OD pair w. PAGE 56 56 Results for the nine-node problem are presented below. Table 4-3 compares the PRUE, SO, bestand worst-case BRU E flow distributions with 10 0 Note that the SO flow distribution in a network with boundedly rational behaviors is the same as standard SO that minimizes total system travel time. Figure 4-4 displays the system travel times of SO, bestand worst-case BRUE flow distributions with varying from 0 to 0.20. It can be observed that when 0 the BRUE flow distribution reduces to the unique PRUE flow and there is no difference between the bestand worst-case performances. As increases, the difference becomes more substantial. For example, th e difference rises from 9.2% to 21.0% when changes from 0.10 to 0.20. As the SO flow di stribution is still the one that minimizes total system travel time, it does not change with the threshold value. Sioux Falls problem: The Sioux Falls network (see, Le Blanc et al., 197 5) contains 76 links, 24 nodes, and 528 OD pairs. The original network parameters in Le Blanc et al (1975) are used while original demands are divided by 11. We assume the same threshold value for users departing from the same origin, i.e., o UE ot ~ where o UEt ~ is the maximum PRUE travel times among the OD pairs that share the same origin o Figure 4-5 presents the discrepancy between the best and worst system performances of BRUE flow distributions for Sioux Falls with varying from 0 to 0.20. Although the Sioux Falls network is not as congested as the nine-node network, a similar pattern can be observed and the difference increases up to 6.7%. The above two examples reveal a difficulty of evaluating or designi ng improving strategies to highway networks as the resu lting performances are likely to be uncertain (Mahmassani and Chang, 1987). Such uncertainty should be acco mmodated proactively in the decision making process. Note that the true ra nge of possible total system travel times may be larger than those PAGE 57 57 shown in Figure 4-4 and Figure 4-5, for two reasons. First, the link-based conditions ( 4.10)-( 4.15) only characterize a subset of BRUE flows. Second, the al gorithms we implemented, like most algorithms for MPCC, do not guarantee local or global optimality, regardless of the fact that we initiated or warm-started the al gorithms with many init ial solutions, randomly generated and otherwise. 4.3 Robust Pricing with Boundedly Rational User Equilibrium 4.3.1 Model Formulations To motivate the models, we implement a tradi tional MC pricing scheme on the nine-node network. Under MC pricing, tolls are in the form of SO ij SO ij ijv v t) (' where SO ijv is the SO flow. Imposing such a pricing scheme may not necessarily evolve the system to SO, because the tolled BRUE flow distributions may not be unique. Figure 4-6 presents the best and worst performances that MC pricing scheme may achieve as compared with those without pricing. As expected, MC pricing may result in a range of system performance and the difference can be as high as 21.3% when 20 0 Although MC pricing achieves SO at its best performance (i.e., under the scenario with perfect ra tionality), it may potentially make the traffic condition worse because users are not perfectly rational. More specifically, it can be observed from Figure 4-6 that when is greater than 0.08, the worst-case syst em travel time with MC pricing will be greater than the best-case tr avel time without tolling. Although Figure 4-6 does not show a case that MC pricing leads to an inferior worse-case performance, we do observe it when is large. This suggest s that MC pricing does not necessarily ensure an improvement in the worst-case performance. To further illustrate, consider a network with three parallel li nks connecting one O-D pair in Figure 4-7. Assume that the O-D demand is 2 and the threshold valu e is 3. It is easy to verify that the worst-case BRUE flow PAGE 58 58 distribution is ) 0 0 2 (v with a total system travel time of 8. The SO flow for the problem is ) 1 0 1 (SOv This yields the MC toll ) 1 0 2 (MC. The worst-case BRUE flow distribution with MC tolls is ) 0 2 0 (and has a system travel time of 10, an amount larger than the one without any toll. In real-world applications, deci sion makers and planners tend to be risk averse and may be more concerned with the worst-case performan ce. Following the notion of robust optimization (e.g., Ben-Tal and Nemirovski, 2002), we attempt to determine a pricing scheme to minimize the worst-case system travel time among all the to lled BRUE distributions. Then, the problem of finding an optimal pricing scheme can be fo rmulated as the following min-max program: RTP-1: ) , (max min x v v v tT) ( s.t. 0 ) ( w j w i w ij ij ij ijv t L j i W w ) ( ( 4.19) max0 ( 4.20) Conditions ( 4.11)-( 4.15) where, denotes a toll vector with a link toll ij as its element and max is the maximum toll vector that can be possibly charged to each link. For convenience, we represent the feasible regi on of the inner problem or the maximization part of RTP-1 as ) ( the decision variable ) , ( x v as and thus the objective function as ) ( The inner problem can then be written as: RTP-1-IN: )} ( : ) ( { max ) ( ( 4.21) It then follows that RTP-1 is essentially in a form of) ( minmax0 which is a generalized semiinfinite min-max problem (see, e.g., Polak and Royset, 2005), because the feasible set of the inner problem depends on the decision variable s of the outer problem. A generalized semiinfinite optimization problem is more difficult to solve than its ordinary counterpart (e.g., Lopez PAGE 59 59 and Still, 2007). There are only a few studies dealing with numerical algorithms for such a generalized problem (see, e.g., St ill, 1999), and the methods presen ted may not be applicable to RTP-1 as its inner problem is a MPCC and the MFCQ is not satisfied at any feasible point of ) ( The optimum solution to RTP-1 is expected to reduce the worst-case system travel time but without any guarantee for its be st-case performance. Alternativ ely, it may be meaningful to find a robust toll whose best-case pe rformance is guaranteed. To th is aim, recall that Hearn and Ramana (1998) derived a valid toll set for PRUE, which consists of an infinite number of valid tolls and each valid toll can evolve the system from PRUE to SO when users are perfectly rational. Clearly, any toll vector in Hearn and Ramanas valid toll se t will achieve SO as its bestcase performance under the BRUE setting. However, other toll vectors not in the valid toll set may also admit SO as their tolled best-case performance. We denot e all these tolls as optimistic tolls. We now derive the optimistic toll set. Let V v v v t ST | ) ( min arg : where 0 and :v D v v Vw w denotes the feasible aggr egate link flow set, and W w x D x v x x v Xw w w w w, 0 and : ) (. Lemma 4.2: Let be a toll vector, v be a feasible aggregate flow and ) ( v X x. Then x is a tolled (link-based) BR UE flow distribution with being the toll vector if and only if there exist vectors N wR L wR and N wR such that the following holds: PAGE 60 60 0 w j w i w ij ij ij ijv t L j i W w ) ( ( 4.22) 0 w j w i w ij w ijx L j i W w ) ( ( 4.23) w w o w w w d) ( ) ( W w ( 4.24) 0 w ij L j i W w ) ( ( 4.25) In the above, the triplet (, ) plays the role of (, ) in the link-based BRUE conditions ( 4.10)-( 4.15). More specifically, are vectors of node potential s associated with travel time, travel times in excess of the minimum, and node potentials associated with the excess travel times respectively in the tolled BRUE. Theorem 4.3: Let ) ( x T denote the part of the polyhedron ( 4.22)-( 4.25), then the optimistic toll set is equivalent to S vv X xx T) ( ) ( : T Proof: Necessity: If is an optimistic toll, then there must exist some feasible by-commodity (OD pair) link flow x in the tolled BRUE flow set whose aggregate link flow v is SO. By Lemma 4.2, ) ( x T Together with that S v and ) ( v X x we have T Sufficiency: If T must belong to some set ) ( x T where the aggregate link flow corresponding to x is the SO flow. By Lemma 4.2, x is a tolled link-based BRUE flow with being the toll vector. Therefore, is an optimistic toll since the tolled best-case BRUE can achieve SO. The optimistic toll set T is not necessarily convex although all its component sets T are polyhedrons. Even when the set S is singleton under the assumption that the link travel time function is strict monotone, set T is still a union of an infinite number of subsets T because the by-commodity link flow will not be unique. On ly under the extreme case where there is only PAGE 61 61 one commodity, the optimistic toll set T reduces to a polyhedron. As previously indicated, Hearn and Ramanas valid toll set is a subset of the optimistic toll set T. All optimistic tolls achieve SO at their be st-case performance, but their worse-case performances are not necessarily the same. We then formulate an alternative model to find the most robust toll vector from the optimistic toll set. The solution will achieve SO at its best-case performance but probably lead to a higher worstcase travel time compared with the solution to RTP-1. Since T is a complicated set, we instead use its subset, the valid toll set in Hearn and Ramana (1998), in formulating the robust toll problem for a proof of concept. With the assumption of strictly monotonic link travel time functions, S is singleton. Th e alternative robust pricing problem can be formulated as follows: RTP-2: ) (min ) ( s.t. 0 w w T w SO T SOD v t v ( 4.26) 0 w j w i ij SO ij ijv t L j i W w ) ( ( 4.27) As formulated, RTP-2 is another generalized semi-infinite min-max problem, which is of the same complexity as RTP-1. The inner prob lems are the same, defining the worst-case tolled BRUE flow distribution, as shown in equation ( 4.21). RTP-2 incorporates some additional linear constraints ( 4.26)-( 4.27) in its outer problem, which are separable from its inner problem. The linear constraints define the valid toll set (Hea rn and Ramana, 1998), and ensure that SO flow satisfies the tolled PRUE conditions with being the toll vector and the corresponding node potentials. Consequently, SO flow will be a tolled BRUE flow distribution induced by the toll vector i.e., the toll achieves SO at its best-case performance. PAGE 62 62 4.3.2 Solution Algorithm We propose a heuristic procedure to solve RT P-1 and RTP-2. The key idea is to use a differentiable penalty function to remove the co nstraints of the inner problem involving the decision variables of the outer problem, thereb y transforming a generalized semi-infinite minmax problem into an ordinary semi-infinite optimization problem. The procedure then applies a cutting-plane scheme (e.g., Lawphongpanich and H earn, 2004) to solve a sequence of ordinary finite optimization problems, each one better approximating the original problem than its predecessors. In the following, we use solving RT P-1 as an example to describe the procedure. We formulate a penalized inner problem as follows: P-WS( ): ) , (max x v wij w j w i w ij ij ij Tt M v v t2) ( ) ( s.t. Conditions ( 4.11)-( 4.15) where M is a sufficiently-large penalty parameter. For convenience, we further denote the feasible region of the above problem as and the objective function as ) ( With a finite penalty, P-WS( ) provides an upper bound to the origin al inner problem. Given a toll vector assume that is the optimal solution to the original problem RTP-1-IN, and let denote the optimal solution to the above penalized problem for some M > 0. It follows that ) ( ) ( ) ( The first equality is due to that no penalty is associated with and the second inequality is due to the fact that is feasible to the penalized problem. As the penalty coefficient M goes to infinity, P-WS( ) reduces to the original inner problem. We then formulate a penalized versio n of robust toll problem RTP-1 as ) ( max minmax0 Because the objective is to minimize the upper bo und of total system travel time, the solution may reduce the worst-case system travel time as the upper bound is pretty tight when the penalty PAGE 63 63 parameter is sufficiently large. Note that the penalized robust optimization problem is an ordinary semi-infinite min-max problem because th e feasible region of its inner problem doesnt depend on the decision variables of the outer problem. Moreover, it is equivalent to the following ordinary semi-infinite optimization problem: P-RTP-1 ,min s.t. max0 We now discuss a cutting-plane scheme to solve P-RTP-1. Assume that 1, 2, n are elements of Then, a relaxed penalized robust toll problem (RP-RTP-1) can be written as: RP-RTP-1 ,min s.t. i n i ..., 1 max0 The RP-RTP-1 problem stated above is si mply the original P-RTP-1 problem with approximated by the discrete set ={1, n}. Let ~ ~ be a global optimal solution to the above relaxed toll problem. If ~ ~ is feasible to the original P-RTP-1 problem, it is an optimal solution. To check its feasibility, one may solve P-WS() ~ i.e., ) ~ ( max If ~ a global optimal solution to P-WS() ~ is such that ~ ) ~ ~ ( then ~ ~ is an feasible and optimal solution to P-RTP-1. On the other hand, if ~ ) ~ ~ ( then an improved solution may be obtained by solving the RP-RTP-1 pr oblem with an expanded discrete set } ~ { Below is a detailed description of the procedure we outlined above: Step 0: Solve the WC-BRUE problem without to lls to obtain an initial solution 1. Set n = 1 and 1 1 PAGE 64 64 Step 1: Solve the RP-RTP-1 problem with the discrete set n and let n n denote the resulting optimal solution. Step 2: Solve P-WS(n) and let ) 1 ( n denote the resulting optimal solution. Step 3: If n n n ) ( ) 1 (, stop and n is an optimal robust pricing scheme. Otherwise, set ) 1 ( ) 1 ( n n n and n = n + 1. Go to Step 1. Assume that both n n and ) 1 ( n globally solve the RP-RTP and P-WS(n) problems in Steps 1 and 2, respectively. If the abov e algorithm stops at some finite iteration n it is easy to see that n n is an optimal solution to the original P-RTP problem. When the algorithm generates an infinite sequence, any of its subsequential limits is optimal to the P-RTP-1 problem under a set of strong conditions discussed in Yin and Lawphongpanich (2007). Because it is difficult in practice to verify those conditions an d ensure global optimality for solving RP-RTP-1 and P-WS(n) problems, the above algorithm should be regarded as heuristic in general. 4.3.3 Numerical Examples The above algorithm was implemented using GAMS to solve both RTP-1 and RTP-2 for the nine-node network. We set the value of M to be 10000 based on various preliminary tests and the upper bound for the toll on each link is set as the minimum integer number that is greater than the MC toll. Other settings are the same as in Section 4.2.4. RTP-1 was solved for four scenarios with 05 0 0.10, 0.15 and 0.20 while RTP-2 was solved with 20 0 for comparison purpose. Since the outer problem is a nonconvex quadratic program and the inner problem is a MPCC, multiple (10 and 4 respectiv ely) initial solutions were used in Step 2 and 3 in order to obtain better local optimal solutions. The comp utation time thus increased significantly. Although not particularly efficient, the cutting-plane algorithm generated some reasonable solutions. When 05 0 the robust toll from solving RTP-1 was the same as the MC toll PAGE 65 65 (reported in Table 4-4) while different tolls were obtained for the other three scenarios. Table 44 reports the robust toll vector for 20 0 denoted as RT1. Figure 4-8 compares the worst-case total trav el time with robust tolls (RT in the figure), the MC toll and no toll (NT in the figure). Wh en compared to the situation with no toll, the figure shows that robust tolls reduce the worst-ca se system travel time significantly, e.g., the reduction is approximately 10.2% when 20 0 When compared to MC tolls, robust tolls have superior worst-case performances when is large, in particular when 05 0 For smaller BRUE does not deviate much from PRUE thereby implying that the worst-case performances of robust and MC tolls are similar. The solution to RTP-2 with 20 0 is also reported in Table 4-4, denoted as RT2. Table 4-4 also presents the bestand worst-case sy stem travel times of MC, RT1 and RT2 with 20 0 From the table, RT2 achieves SO at its best-case performance while reducing the worst-case system travel time from 2793 to 2662, a 4.7% reduction. If the full set of optimistic tolls is considered, a higher reduction rate can be expected. RT2 also dominates MC in the sense that both tolls have the same best-case performances while the former performs better against the worst-case scenarios. The parameter reflects the level of rationality in travelers route choice decision making, which may be estimated by conducting a st ated preference survey. However, given the fact that the estimate of is likely to be biased or inaccurate, it makes sense to examine how a robust toll designed for one particular level of ra tionality performs at other levels of rationality. In Figure 4-9, tolls from RT1with = 0.20 reduces both the bestand worst-case system travel times significantly for all values compared with the no-toll case. Compared with the MC toll, the same toll leads to a superi or worst-case performance with 08 0 but inferior otherwise. PAGE 66 66 Intuitively, the smaller is, the more rational travelers route choices are. Thus the first-best tolls are expected to perform well with small values. We further note that although robust tolls are designed to guard against the worst-case scenarios, implementation of robust tolls actually leads to a more stable system performance in the sense that the span betw een the best and worst performances is much narrower. Figure 4-10 displays the difference between the worseand bestcase travel time as a percentage (or fraction) of the latter. In all three cases (no toll, MC and RT1 tolls), the difference increases as increases. However, the difference for the case w ith the robust toll (see the bottom most graph) is significantly smaller than the other two for large This suggests that there is less variability in travel time under robust tolls, i.e., the tolls are more effective and relia ble at inducing travelers to use the road network more efficiently, especia lly when the threshold value is relatively large. 4.4 Summary This chapter investigates static or time-of-d ay robust network-level pricing problem with bounded rationality in travelers route choi ces. As an alternative source of network uncertainties, BRUE is not fully explored in prev ious literatures. This chapter enriches the literature by providing a rigorous model of BRUE, a discussion on the implication of BRUE and a robust pricing scheme with BRUE. More specifically, a general path-based definition of BRUE is presented, followed by a more restri ctive link-based defin ition. The BRUE flow distributions are characterized as non-empty and usually non-convex sets. Robust pricing models have then been formulated to minimize the maximum system travel time realized from the tolled BRUE set. Numerical experiments demonstrate that system performance may vary substantially within the BRUE set and the proposed robust pricing models are able to protect against the worst-case scenario effectively. PAGE 67 67 Our future research will develop more efficient solution approaches, e.g., based on metaheuristic algorithms. Moreover, path-based algor ithms may be developed to solve the path-based counterparts of the proposed formulations. We will further investigate how to price under other more realistic network settings with boundedly ra tional behaviors, such as elastic demand or combined route and departure time choices. PAGE 68 68 Figure 4-1. A bridge network 3 4 2 1 PAGE 69 69 Figure 4-2. Relationship between three different flow sets PAGE 70 70 Figure 4-3. The nine-node network 1 2 8 7 6 5 4 3 9 PAGE 71 71 2200 2300 2400 2500 2600 2700 2800 00.020.040.060.080.10.120.140.160.180.2AlphaSystem Travel Time WC-BRUE BC-BRUE SO Figure 4-4. Comparison of system performa nces of BRUE at the nine-node network PAGE 72 72 1950 2000 2050 2100 2150 00.020.040.060.080.10.120.140.160.180.2AlphaSystem Travel Time WC-BRUE BC-BRUE SO Figure 4-5. Comparison of system performa nces of BRUE at the Sioux Falls network PAGE 73 73 2200 2300 2400 2500 2600 2700 2800 00.020.040.060.080.10.120.140.160.180.2AlphaSystem Travel Time WC-BRUE BC-BRUE SO WC-BRUE_MC BC-BRUE_MC Figure 4-6. System performances with MC pricing scheme at the nine-node network PAGE 74 74 Figure 4-7. Three-parallel-link network D O 1 12v t 52 t 23 3 v t PAGE 75 75 2200 2300 2400 2500 2600 2700 2800 0.050.10.150.2AlphaSystem Travel Time WC-BRUE WC-BRUE_RT WC-BRUE_MC Figure 4-8. Comparison of worst-ca se total travel times with no to ll, robust and MC tolls at the nine-node network PAGE 76 76 2200 2300 2400 2500 2600 2700 2800 00.020.040.060.080.10.120.140.160.180.2AlphaSystem Travel Time WC-BRUE WC-BRUE_MC WC-BRUE_RT1 BC-BRUE BC-BRUE_MC BC-BRUE_RT1 Figure 4-9. System performances with MC and RT1 at the nine-node network (1) PAGE 77 77 0.00 0.05 0.10 0.15 0.20 0.25 00.050.10.150.2 AlphaPerformance Difference BRUE MC RT1 Figure 4-10. System performances with MC and RT1 at the nine-node network (2) PAGE 78 78 Table 4-1. Illustrative example of the restri ctiveness of the link-based BRUE representation BRUE Flow Time Link Travel time function (1, 3) t13 = 3 + v13 2 5 (1, 2) t12 = 7 + v12 1 8 (2, 3) t23 = v23 1 1 (2, 4) t24 = 5 + v24 1 6 (3, 2) t32 = v32 1 1 (3, 4) t34 = 2 + v34 2 4 Path 1-3-4 1 9 1-3-2-4 1 12 1-2-3-4 1 13 1-2-4 0 14 PAGE 79 79 Table 4-2. Nonconvexity of the BRUE flow distribution set for the bridge network BRUE-1 BRUE-2 0.5(BRUE-1+BRUE-2) Flow Time Flow Time Flow Time Link Travel time function (1, 3) t13 = 10 v13 3 30 4 40 3.5 35 (1, 2) t12 = 50 + v12 3 53 2 52 2.5 52.5 (2, 3) t23 = v23 0 0 2 2 1 1 (2, 4) t24= 10 v24 3 30 4 40 3.5 35 (3, 2) t32 = 10 + v32 0 10 4 14 2 12 (3, 4) t34 = 50 + v34 3 53 2 52 2.5 52.5 Path 1-3-4 3 83 0 92 1.5 87.5 1-3-2-4 0 70 4 94 2 82 1-2-3-4 0 106 2 106 1 106 1-2-4 3 83 0 92 1.5 87.5 PAGE 80 80 Table 4-3. Flow distributions for the nine-node network (10 0 ) Link UE WC-BRUE BC-BRUE SO (1, 5) 8.2 4.0 11.5 9.4 (1, 6) 21.8 26.0 18.5 20.6 (2, 5) 47.4 53.2 43.0 38.3 (2, 6) 22.6 16.8 27.0 31.7 (5, 6) 0.0 0.0 0.0 0.0 (5, 7) 27.8 29.2 26.4 21.3 (5, 9) 27.7 28.0 28.0 26.4 (6, 5) 0.0 0.0 0.0 0.0 (6, 8) 44.5 42.8 45.6 39.5 (6, 9) 0.0 0.0 0.0 12.8 (7, 3) 38.2 34.0 31.7 29.6 (7, 4) 17.4 23.2 22.8 20.8 (7, 8) 0.0 0.0 0.0 0.0 (8, 3) 1.8 6.0 8.3 10.4 (8, 4) 42.6 36.8 37.2 39.2 (8, 7) 0.0 0.0 0.0 0.0 (9, 7) 27.7 28.0 28.0 29.1 (9, 8) 0.0 0.0 0.0 10.2 PAGE 81 81 Table 4-4. Pricing schemes and their performances for the nine-node network Link No Toll MC Toll RT1 RT2 (1, 5) 1.13 0.00 2.00 (1, 6) 6.16 4.43 6.00 (2, 5) 2.59 0.00 0.00 (2, 6) 3.62 4.00 0.00 (5, 6) 0.00 1.00 1.00 (5, 7) 16.88 17.00 15.81 (5, 9) 5.13 4.37 4.61 (6, 5) 0.00 0.00 1.00 (6, 8) 7.37 2.42 7.81 (6, 9) 0.11 1.00 0.61 (7, 3) 3.54 0.00 4.00 (7, 4) 2.01 0.00 3.00 (7, 8) 0.00 1.00 1.00 (8, 3) 0.02 0.00 0.00 (8, 4) 2.50 0.00 3.00 (8, 7) 0.00 1.00 1.00 (9, 7) 3.75 4.00 3.20 (9, 8) 0.06 1.00 0.00 Best-case system travel time 2309 2254 2306 2254 Worst-case system travel time 2793 2735 2509 2662 PAGE 82 82 CHAPTER 5 OPTIMAL ADAPTIVE TOLLING STRATEGIES FOR HIGH OCCUPANC Y/TOLL LANES: BASIC MODELS This chapter is devoted to the facility-level pricing problem. Different from network-wide pricing, it is found that travelers are able to re spond to dynamic tolls charged at some critical facilities, e.g., HOT facilities (Bons all et al., 2007). Literature review in Chapter 2 also reveals that several agencies in the U.S. have applied traffic-responsive tolls on their HOT facilities. Although empirical studies shows that adaptive tolling outperforms fixed time-of-day tolling on HOT facilities (e.g., Burris and Sullivan, 2006, Sullivan and Burris, 2006, Supernak et al., 2003a, b), most of the adaptive tolling schemes implemented are simple and heuristic, a fact that provides many opportunities for improvements. This chapter aims to enhance adaptive pricing for HOT lanes such that they are more effec tive and more attractive to both transportation authorities and motorists. Two sensible and pragmatic approaches are pr oposed for determining time-varying tolls in response to the detected traffic condition. Th e first approach adjusts the toll rate based on the concept of the feedback control, while the second approach would learn in a sequential fashion both the demand and supply information of the HOT facility and then determine pricing strategies to explicitly achieve the operating objectives. This chapter presents the basic models of these two approaches in Section 5.1 and 5. 2 respectively. A simulation experiment is provided in Section 5.3 to demons trate and compare the proposed approaches. In the prototype model of the self-learning approach presented in this chapter, only one pa rticular component of information learning, namely learning users WT P, and a simple point-queue model for traffic dynamics are employed for a proof of concept. A couple of extensio ns within a unified framework of the self-learning pricing approach will be discussed in the next chapter. PAGE 83 83 Without loss of generality and to facilitate the presentation of the essential idea, this dissertation considers a simple setting of HOT facilities shown in Figure 5-1 and Figure 5-2 where there are one HOV/HOT lane and one re gular lane, and a bottleneck is activated downstream. Moreover, it is a single segment of HOT lane with one entry, and there is no on/off ramp in between. We assume that HOVs have sh ifted to the HOT lane be fore the toll gate and each motorist on regular lane must pay to access the toll lane. Also note that starting from this chapter, a set of notations different than that in Chapter 2 and 4 will be adopted. 5.1 Feedback Approach for HOT Pricing The first approach is based on the concept of feedback control and requires one loopdetector station (or other types of sensors) located downstream of the toll tag reader, as illustrated in Figure 5-1 (in a real-world implementation, several loop-detector stations need be installed downstream in order to better de tect the traffic condition along the segment.) In the feedback control, the toll rate at time interval t+ 1 depends on the toll at interval t and the occupancy detected by the downstream loop detectors. Mathematically the control logic can be stated as follows: *(1)()() ttKoto where () t and ( t +1) are the toll rates at interval t and t+ 1 respectively; K is a regulator parameter for adjusting the distur bance of the feedback control; *o is the desired occupancy of the toll lane, which is typically set equal to or slightly less than the critical occupancy (corresponding to the critical density), and () ot is the measured occupancy. Note that a similar concept has been succe ssfully applied to ramp metering known as ALINEA or Asservissement Line aire d'Entree Autoroutiere (Papageorgiou et al., 1997). Therefore, we anticipate that th is approach will perform reas onably well and thus provide a PAGE 84 84 benchmark for evaluating the other approach. Moreov er, this approach is cost efficient and easy to implement. 5.2 Self-learning Approach for HOT Pricing 5.2.1 General Framework The objective of HOT lane operation is to maximize the total freew ay throughput while maintaining a free-flow condition on the toll lanes. In other words, it is to attract the just right amount of paid users to make full use of the HO T lane without making it congested. To do so, information from both the demand and supply side is needed to determine the right amount of toll. The essential idea of the self-learning appr oach is that in the operation, the information needed can be gradually learned by mining th e loop detector data, and the tolls can be determined based on the updated information to better serve the local operation objectives. More specifically, the proposed framework of the se lf-learning approach consists of two steps: System Inference. From mining real-time traffic data (such as speed, flow or occupancy) collected at a regular intervals from loop detectors (often at limited locations and inaccurate), the first step learns travelers WT P in order to predict motorists reaction to tolls, composes a full picture of current traffic condition for the entire freeway, and forecasts short-term traffic demand. Toll Determination. The attained knowledge will then be used in the second step to maximize the throughput rate for the entire fr eeway while maintaining a superior service on the toll lanes. Given that future demand is likely uncertain, a stochastic programming approach is adopted to determine robust toll s whose performance is not very sensitive to different realizations of un certain traffic demand. Moreover, a robust optimization approach can be applied to acco mmodate users heterogeneity. Figure 5-2 sketches the framework for pr oactive self-learning dynamic pricing. In the following sub-sections, we demonstrate the framework of the self-learning approach by building a prototype model which employs the WTP learning component of system inference and a point-queue model in describing the traffic dynamics for toll determination. PAGE 85 85 5.2.2 Calibration of Willingness-to-Pay Understanding motorists WTP is critical for toll determination. While stated-preference surveys often overestimate motorists WTP, the data collected during the tolling operation reflect actual lane choice behaviors when travelers are facing the tolls be ing charged. Such revealedpreference information can be used to estimate more accurately motorists WTP. Assuming homogeneous motorists with the same WTP whose decision on whether to pay to gain access to the HOT lanes follows a logit model, the relation ship between the approaching flow rates and the flows on HOT and regular lanes can be stated as follows: ) ( ) ( ) ( exp 1 1 ) ( ) ( ) (2 1t t c t c t t tR T R T T ( 5.1) where ) ( tT and ) ( tR represent the approach ing flow rates on HOV and regular lanes during time interval t respectively; ()Tt and) ( tR are the flow rates afte r the lane choice, and ()Tct and()Rct are the (average) travel times on HOT and regular lanes at time interval t In equation ( 5.1), there are three parameters to be estimated: 1 ,2 and where 1 and 2 indicate respectively the marginal eff ect of travel time and toll on motorists utility, and encapsulates other factors affecting motorists WTP. Note that 2 1 represents motorists trade-off between time savings and tolls, i.e., the value of travel time. Other variables in equation ( 5.1) can either be obtained directly from loop de tectors or estimated using traffi c flow models, and the toll rate () t is set by the operator. It should be pointed out that here we use volume splits ) ( ) ( ) ( t t tR T T to approximate the probabilities of lane choices. In real-time operation, a recursive least-sq uares technique or discrete Kalman filtering (KF) (Kalman, 1960) can be used to estimate the constant parameters, 1 ,2 and To do so, equation ( 5.1) can be reformulated as follows: PAGE 86 86 ) ( ) ( ) ( 1 ) ( ) ( ) ( ln2 1t t c t c t t tR T T T R ( 5.2) Applying discrete KF technique to estimate the parameters, we have: ) 1 ( 1 ) ( ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( ) 1 ( 1 ) ( ) ( ) ( 1 ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( 1 ) ( ) ( ) ( 1 ) ( ) ( ) ( ln ) ( ) ( ) ( ) ( ) 1 ( ) 1 ( ) 1 ( 1 2 1 2 1 2 1t P t t c t c t G I t P t t c t c t P t t c t c t t c t c t P t G t t t t t c t c t t t t G t t t t t tWTP R T WTP WTP WTP R T R T R T WTP WTP R T T T R WTP ( 5.3) where the variables with ^ are estimates. WTP is the variance of the left-hand side of equation ( 5.2) computed from the known variance of random detector measurement error (the value could vary with different types of sensors), WTPP is the expected covariance matrix of the estimation errors and WTPG is the Kalman gain. With an initialization of ) 0 (1 ,) 0 (2 (0) and ) 0 (WTPP equation ( 5.3) can update estimates of 1 ,2 and real time with newly-obtained information. As time evolves, the impact of the initializatio n will be diminishing, and accurate estimates of1 2 and are expected. 5.2.3 Optimal Tolling Strategy with Point-Queue Model The calibrated motorists WTP at interval t provides a basis for determining a toll for interval t+ 1. Consistent with the prevailing operation policies of HOT lanes, we now attempt to specify toll rates to maximize the throughput of the corridor while ensuring a superior traffic condition on the toll lane. Assume in rush hours when the total demand is much higher than the capacity of the freeway segment, the queuing delay dominates other factors that may affect the travel time of the freeway segment, such as lane-changing maneuver s. Therefore, we adopt a simple point-queue PAGE 87 87 model to describe traffic dynamics in this prototype model of the self-learning approach. The objectives of maintaining free-flow condition on the HOT lanes and maximizing the throughput of the entire freeway are approximately equiva lent to operating the HOT lane at a throughput close to its capacity while keep ing it from being congested. This is because general purpose lanes are assumed to be congested and their disc harge rates are equal to the capacity of the downstream bottleneck. Consequent ly, based on the measured condi tion on the managed lane at interval t the desired inflow at interval t+ 1, denoted as ) 1 ( ~ tT, can be determined such that the managed lane is uncongested but efficiently utilized. With homogeneous motorists, it is straight forward to determine the toll rate for time interval t +1 by using the following relationship: ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ~ ) 1 ( ~ ) 1 ( ln ) 1 (2 1 t t t c t c t t t t tR T T T R ( 5.4) where (1)Tct and (1)Rct are estimated using traffic flow models, and ) 1 ( tR is obtained from loop detectors. The above toll determination is flexible in in corporating various models of flow dynamics to estimate travel times as long as lane-changing behavior is neglectable. For the purpose of the proof of concept, we employ the point-queu e concept (Kuwahara and Akamatsu, 1997) to describe traffic dynamics, determining ) 1 ( ~ tT and calculating ) 1 ( t cT and ) 1 ( t cR at each time interval. It is assumed that the average travel time during time interval t can be decomposed into a flow-independent cruise time and an average queuin g delay at the end of the segment, written as: s t v t v c t c 2 ) ( ) 1 ( ) (0 ( 5.5) PAGE 88 88 where c0 is the flow-independent cruise time, and ) 1 ( t v and ) ( t v are the numbers of the queuing vehicles by the end of the time interval t -1 and t respectively and s is the capacity of the bottleneck. The evolution of queue length can be captured by the following equation: 0 ) ( ) 1 ( max ) ( t s t t v t v ( 5.6) where t is the duration of the time interval. Huang and Lam (2002) proved that equation ( 5.6) is consistent with the first-in-first-out discipline, which ensures that a vehicle must leave the segment in the same order as it arrived at that segment. Equations ( 5.5) and ( 5.6) apply to both the managed and regular lanes. Again, at current interval t ()Rt and ()Tt are obtained directly from loop detectors, and then ) ( t v can be estimated from ( 5.6). Consequently, ()Tct and ()Rct are estimated from equation ( 5.5). Up to this point, we are ready to apply equation ( 5.3) to estimate the moto rists willingness to pay. To maintain a superior fr ee-flow traffic condition on th e managed lane, the desired inflow) 1 ( ~ tT should be regulated as follows: Others 0 ) ( ) ( 0 ) ( ) 1 ( ~ t s t v t t v s t v s tT T T T T T T ( 5.7) Note that the case of 0 ) 1 ( ~ tT implies that the managed la ne is already congested and the toll rate will be set to its predetermined maximum value. Given) 1 ( ~ tT and the estimated willingness to pay, the toll can be determined according to equation ( 5.4). PAGE 89 89 5.3 Simulation Study 5.3.1 Design of Simulation Experiments Given a test environment on real facilities is not readily available, we conducted simulation experiments to validate and compare the pr oposed approaches. The developed simulation platform consists of three major components: si mulator, monitor and controller. The simulator attempts to replicate the motorists lane c hoice behaviors and traffic dynamics. The monitor serves as a surveillance system, collecting info rmation of traffic condi tion at each interval including total arrival, flow rate s before and after lane choices, and travel times. The controller implements the operations of the proposed a pproaches, namely the feedback and the selflearning approaches. Simulator. A Poisson arrival is gene rated based on the given averag e arrival flow rate. At each interval, based on the toll rate specified by the controller and the instantaneous travel times from the monitor, the simulator applie s the Logit model with the true values of 1 ,2 and to compute the percentages of the motorists for choosing the HOT lane. Moreover, the simulator applies traffic flow models to replicate traffic dynamics. With application of point-queue model, equations ( 5.5) and ( 5.6) are adopted to track the queue evolution and estimate the tr avel times on both lanes. Monitor. At each time interval, the monitor coll ects the total arrival, flows entering the managed and general purpose lanes, the travel conditions and travel times generated from the simulator. In a real-world implementation, this information could be directly measured, although not necessarily adequate or accurate. In this case, the monitor will generate perturbed measurements in the simula tion to mimic the real-world situation, and it is then important to estimate the actual traffic conditions th rough the partial or inaccurate measurements. Traffic state estimation will be discussed in Chapter 6. Controller. The controller implements the operations of the proposed approaches. For the feedback controller, the occupancy is not ava ilable in the simulation when the point-queue model is used. Instead, the inflow rate of HOT lane will be used to adjust the toll rate. More specifically, )) ( ~ ) ( ( ) ( ) 1 ( t t K t tT T For the self-learning approach, the controller first uses information of flows and travel times from the monitor to calibrate the value of 1 ,2 and according to equation ( 5.3). Toll rate is then determined based on equations ( 5.4)-( 5.7) with point-queue model. PAGE 90 90 5.3.2 Comparison of Feedback and Self-learning Controllers This sub-section examines the performan ces of the feedback and the self-learning controllers, and compares them under the same se ttings. Note that as described above, the simulator mimics individual motorists lane c hoice behavior based on the Logit model. Since actual lane-choice behaviors are much more comp licated, the simulator may not replicate the real-world situation. Moreover, since the cont roller and the simulator adopt the same behavior assumption, i.e. the simulator represent the ac tual traffic dynamics accurately, the performance of the self-learning approach ma y be overestimated. However, there seems no better way unless we do a real-world experiment or a human-in-the-loop simulation. The downstream bottleneck has a capacity of 1800 vehicles per hour per lane. We simulate a peak 20-minutes period and the duration is e qually divided into discrete time intervals with two minutes each. For the computation simplicity, we assume in the Logit model ( 5.1) that 2 is known as one. Hence, only two parameters are le ft to be estimated for motorists utility of lane choice. The true values of 1 and used in the simulator are 1 and 0.2. We represent the travel time difference in the unit of one time interval (2 minutes), hence, 12 1 means a value of travel time of one dollar per two minutes, i.e., 30 dollars per hour. ) 0 (1 (0) and P (0) are initialized as 0.8, 0.1 and an identity matrix respectively. The upper bound of the toll rate is set to eight dollars. A variety of demand scenarios are created to test the performance of the two controllers. In the following, we report the simulation results for two selected scenarios. Scenario 1 assumes an average ar rival rate of 3600 vehicles per hour for the regular lane and 300 vehicles per hour for the HOT la ne throughout the simulated ten in tervals, and scenario 2 has arrival rates of 3600 and 3000 vehicl es per hour at first and second five intervals respectively. PAGE 91 91 5.3.2.1 Performance of fe edback controller We first examine the performance of the fee dback controller with homogenous motorists. Figure 5-3 displays the time -varying throughputs and queues on bot h the HOT and regular lanes, resulted by the feedback controller with K =0.1 under Scenario 1. Since the arrival flow is far beyond the capacity of the bottleneck, it is inev itable that queues constantly increase on the regular lane. At the same time, the feedback contro ller is able to adjust the toll to avoid severe queuing on the managed lane as well as maintain a fairly high and stable throughput. In fact, the average throughput for the corri dor is 1642 vehicles per hour pe r lane, and the average queue length on the managed lane is 7.7 vehicles, a descent performance for operating the managed lane. Figure 5-4 illustrates the perf ormance of the same controlle r under demand scenario 2. Although the average total arri val (both HOV and lower-occupanc y-vehicle flows) during the simulation period is equal to the capacity of the bottleneck, the queues can not be cleared due to the random arrival. Again, the controller is able to achieve the operating objectives fairly well, resulting in an average thr oughput of 1602 vehicles per hour pe r lane and an average queue length on the managed lane of 5.27 vehicles. The sensitivity and performan ce of the feedback controller may depend on the setting of the regulator parameter K By adjusting the value of K we examine the dependency under demand scenario 1. For each K 30 runs (a sufficient sample size determined from preliminary ten runs with an error tolerance of one) of si mulation are conducted and Table 5-1 reports the average queue length and averag e throughput of the managed lane. Within our expectation, when K is small, the toll adjustment may not be ad equate enough to adapt to the traffic condition, thereby leading to severe queui ng on the managed lane but a highe r throughput (par tially due to the use of the point-queue model). When K gets larger, we see a reverse tendency with fewer PAGE 92 92 queues and less throughput. However, a large K does not necessarily lead to fewer queues on the managed lane. In fact, in our simulation experi ments with this particular demand scenario, the controller with K =0.12 outperforms the others with K >0.12 in both operating objectives. 5.3.2.2 Performance of self-learning co ntroller with point-queue model We then examine the performance of the self -learning controller under the same setting. Figure 5-5 is a proof of concep t that motorists WTP can be gradually learned. The figure presents the estimates of parameters 1 and with homogenous motorists. As time evolves, the estimates converge from the initial values of 0.8 for 1 and 0.1 for to the true values of 1 and 0.2 respectively. Figure 5-6 presents the performance of the self-learning controller under scenario 1, a counterpart to Figure 5-3. The self-learning controlle r results in an av erage throughput of 1764 vehicles per hour per lane and a queue length of 4.1. Compared with the feedback controller, it reduces the queue on the managed lane by 47% a nd increases the throughput by 7%. To further visualize the performance difference, Figure 5-7 plots the resu lting average throughput and queue length over ten runs (sampl e size determined similarly as for the feedback controller) by the self-learning controller versus averages over 30 runs by the f eedback controller with different values of K The latter forms a frontier in terms of the two control objectives, and the selflearning controller actua lly outperforms the feed back controller with05 0 K. When K <0.05, the feedback controller results in higher throughput but longer queues compared to the selflearning controller. Figure 5-8 further reports the simulation results of the self-l earning controller under scenario 2, a counterpart to Figure 5-4. The average through put is 1773 vehicles per hour per PAGE 93 93 lane and the average queue length is 2.8 vehi cles on the managed la ne, superior to the performance of the feedback controller. In summary, the self-learning controller is able to achieve a high throughput as well as prevent the managed lane from being congested. In view of the priority that the HOT lane operator put is to maintain a free-flow traffic condition, we conclude that the self-learning controller generally outperforms the feedback controller. 5.4 Summary This chapter has investigated the facility-level pricing problem particularly for HOT lane operations. Two approaches, feedback and se lf-learning, are proposed to determine dynamic traffic-responsive pricing strategies for HOT lane s to provide superior fr ee-flow travel services and efficiently utilize the capacity. The proposed a pproaches are feasible a nd easy to implement. Simulation experiments in this chapter have validated both models and verified their effectiveness. It is concluded that the pr ototype model of the se lf-learning controller outperforms the feedback controller. However, the latter could be more cost efficient, and easier to implement. The basic model of the self-learning approach developed in this chapter has simplified some practical concerns in HOT lane operation. It only employs the WTP-learning component of system inference, while assuming the true traffic condition along the freeway and the future inflow are known. In the second step of toll determination, the underlying point-queue model does not capture the complicated characteristi cs of traffic flow caused by lane-changing behaviors. Also, the assumption that travelers are homogeneous is not very rea listic. Another shortcoming from which the basic models of both the feedback and the se lf-learning approaches suffer is that the toll determination only focuses on the time interval t +1 and responds reactively to measured flows from interval t Although the controllers are able to achieve high throughput PAGE 94 94 rates, they may lead to a wildly fluctuating toll pattern that frustrates motorists and cause unstable traffic condition on the regular lanes. In Chapter 6, we will discuss a couple of possibilities for improving the self-learning approach. Other components in its general fr amework will be developed, including more realistic description of traffic dynamics, integrat ed traffic state estimati on and WTP learning, and proactive robust toll optimization in coupled with demand prediction. These enhancements addressing several critical practical issues will advance the self-l earning approach to be more realistic and efficient. PAGE 95 95 Figure 5-1. System configuration for the feedback-control approach Toll tag reader Regular Regular HOT Downstream Bottlenec k HOV Traffic sensors Occupancy measure Feedback toll calculation PAGE 96 96 Figure 5-2. System sketch of the self-learning approach Toll tag reader Traffic sensors Regular Regular HOT Downstream Bottlenec k HOV Traffic measurements at sensor locations Traffic state estimate WTP learning Demand learning Toll optimization Toll rate Minimum sensors required System Inference PAGE 97 97 1 2 3 4 5 6 7 8 9 10 0 1000 2000 ThroughputThroughput (veh/hr/ln) 0 1 2 3 4 5 6 7 8 9 10 0 20 40 Queue# vehicles in queue (Toll Lane) 0 1 2 3 4 5 6 7 8 9 10 0 200 400 # vehicles in queue (Regular Lane) 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 Toll Time Interval (2 min)($) Toll Lane Regular Lane HOV Inflow Regular Lane Toll Lane Figure 5-3. Throughputs, queues an d toll rates resulted by the feedb ack controller (Scenario 1, K =0.1) PAGE 98 98 1 2 3 4 5 6 7 8 9 10 0 1000 2000 ThroughputThroughput (veh/hr/ln) 0 1 2 3 4 5 6 7 8 9 10 0 10 20 Queue# vehicles in queue (Toll Lane) 0 1 2 3 4 5 6 7 8 9 10 0 50 100 # vehicles in queue (Regular Lane) 0 1 2 3 4 5 6 7 8 9 10 0 1 2 Toll Time Interval (2 min)($) Toll Lane Regular Lane HOV Inflow Regulare Lane Toll Lane Figure 5-4. Throughputs, queues an d toll rates resulted by the feedb ack controller (Scenario 2, K =0.1) PAGE 99 99 0 1 2 3 4 5 6 7 8 9 10 0.6 0.8 1 1 Time Interval (2 min) 0 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 Time Interval (2 min) Calibrated Value Actual Value Calibrated Value Actual Value Figure 5-5. Calibrated 1 & by the self-learning cont roller (Scenario 1) PAGE 100 100 1 2 3 4 5 6 7 8 9 10 0 1000 2000 ThroughputThroughput (veh/hr/ln) Toll Lane Regular Lane Toll Lane Inflow 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 Queue# vehicles in queue (Toll Lane) 0 1 2 3 4 5 6 7 8 9 10 0 100 200 300 # vehicles in queue (Regular Lane) 1 2 3 4 5 6 7 8 9 10 0 2 4 6 toll Time Interval (2 min)($) Toll Lane Regular Lane Figure 5-6. Throughputs, queues and toll rates re sulted by the self-learnin g controller (Scenario 1) PAGE 101 101 1440 1500 1560 1620 1680 1740 1800 0 5 10 15 20 25 30 Throughput (veh/hr/ln)Avg. # Vehicles in Queue Feedback Control Self-learning Capacity = 1800 veh/hr/ln K = 0.02 K = 0.05 K = 0.03 K = 0.1 K = 0.3 K = 0.2 Figure 5-7. Performance comparison of two controllers under scenario 1 PAGE 102 102 1 2 3 4 5 6 7 8 9 10 0 1000 2000 ThroughputThroughput (veh/hr/ln) 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 8 10 Queue# vehicles in queue (Toll Lane) 0 1 2 3 4 5 6 7 8 9 10 0 20 40 60 80 100 # vehicles in queue (Regular Lane) 1 2 3 4 5 6 7 8 9 10 0 1 2 3 Toll Time Interval (2 min)($) Toll Lane Regular Lane HOV Inflow Regular Lane Toll Lane Figure 5-8. Throughputs, queues and toll rates re sulted by the self-learnin g controller (Scenario 2) PAGE 103 103 Table 5-1. Performance of fee dback controllers with different K values K Avg. Queue Length ( # vehicle) Throughput (vehicle per hour per lane) 0.02 29.47 1776.78 0.03 21.81 1773.27 0.05 14.64 1756.56 0.10 8.33 1659.99 0.11 7.96 1635.21 0.12 7.86 1632.81 0.13 8.07 1608.39 0.14 7.87 1608.12 0.15 8.12 1592.64 0.16 8.04 1592.37 0.17 8.43 1584.57 0.18 8.73 1569.63 0.19 9.39 1554.48 0.20 10.68 1536.45 PAGE 104 104 CHAPTER 6 OPTIMAL ADAPTIVE TOLLING STRATEGIES FOR HIGH OCCUPANCY/TOLL LANES: A UNIFIED FRAMEWORK OF EXTEN DED SELF-LEARNING APPROACH Two approaches, one feedback and one self-lear ning, have been presented in Chapter 5 for traffic-responsive pricing on HOT lanes. Betwee n those two, the self-l earning approach takes into consideration many human and traffic factors that are ignored by the simple feedback logic. Chapter 5 has demonstrated that the prototype re active self-learning dynamic pricing approach is able to learn motorists WTP in a sequential fash ion and explicitly optimizes the toll rate for the next rolling horizon. In this chapter, we will fu rther advance this promising method to be a wellpolished and readily-implementable approach. The general framework of the self-learning approach consists of two steps, system inference and toll determination (s ee Section 5.2.1). The first st ep reveals the information from both demand and supply side by mining the real-tim e traffic data, and the second step adjusts the toll rate based on the updated information. Chapter 5 focuses on l earning users WTP and reactive toll determination. Within the framework, we will further address several critical issues of the self-learning approach in this chapter, in cluding more realistic description of the traffic dynamics, integrated traffic state estimation and WTP learning, and proactive robust toll optimization in coupled with demand learning. Section 6.1 explores the possibili ty of incorporating more realistic traffic dynamics into the unified self-learning approach framework. Im pacts of the lane-chang ing behaviors and the physical design of the HOT lane slip ramp are considered. Two other components of the systeminference step in the framework, the traffic state estimation and the demand prediction, are investigated in Section 6.2. Section 6.3 fu rther presents several improvements of the tolldetermination step in the framework, including a proactive pricing approa ch to provide smoothly changing toll rates, a robust approach to acco mmodate users heteroge neity, and a stochastic PAGE 105 105 method to incorporate demand uncertainty. A si mulation study is conducted in Section 6.4 to demonstrate that the extended framework is feasible effective and efficient. Finally, Section 6.5 summarizes this chapter. 6.1 Incorporating More Realistic Traffic Dynamics In Chapter 5, the point-queue model is adopted to describe the traffic dynamics. However, the true characteristics of the traffic flow are more complicated due to the lane-changing behaviors of the lower-occupancy vehicles. Th ese vehicles shifting to the HOT lane from regular lanes usually have a velocity less than th at of the prevailing traffic on the HOT lane, and they are referred to as moving bottlenecks as they may cause queues to form behind them. Impact of moving bottlenecks and lane-changing behaviors on traffic conditions has attracted much attention in the literatur e (see, e.g., Newell, 1998 and Lava l and Daganzo, 2006). In order to describe the traffic dynamics more realistical ly, a multi-lane hybrid tra ffic flow model will be incorporated in the framework to improve the toll determination. Moreover, the physical design of the HOT lane has a non-neglecta ble impact on the traffic characteristics as well. To take into account this factor, alternative HOT lane slip ramp design will be modeled. The enhanced traffic flow model serves as a basis for both the sy stem inference and toll determination in the framework of the self-learning approach. The deta ils will be discussed in the Sections 6.2 and 6.3. 6.1.1 Impacts of Lane-Changing Behaviors Motivated by the postulate that the lane-chang ing behaviors before the entry points to the HOT lane may create voids in traffic streams a nd reduce the throughput of the lane, we adopt the multi-lane hybrid cell transmission model (CTM ) proposed by Laval and Daganzo (2006) for a more realistic representation of traffic dynamics. The multi-lane hybrid CTM is able to capture both the mandatory and the discretionary lane-c hanging behaviors, and to treat lane changing PAGE 106 106 vehicles as moving bottlenecks. This way, th e impacts of the lane-c hanging behaviors on the freeway throughput and travel time can be explicitly considered. In Laval and Daganzos multi-lane hybrid model, each lane is modeled as a separate kinematic wave (KW) stream interrupted by lane -changing particles that completely block the traffic. The flow transfers are predicted by using the incremental-transfer (IT) principle (Daganzo et al., 1997) for multiple-lane KW prob lems, coupled with a one-parameter model for discretionary lane-changing demand. In this di ssertation, the lane changes made by the loweroccupancy vehicles that want to pay to access th e HOT lane are considered as mandatory lanechanging. The lane-changing demand is given by equation ( 5.1). If the demand plus the arrival flow on the HOV lane exceeds available capacity of the HOT lane, the IT principle is used to prorate that available capacity1. To implement the hybrid model, all lanes are partitioned into small cells of length x in addition to discretizing the time into time intervals t See Figure 6-1 for a discretized freeway representation. 6.1.2 Impacts of HOT Slip Ramp Design It is known that the lane-changing behaviors be fore the entrance of the HOT lanes will be largely affected by the physical conf iguration of access of the HOT lane. Figure 2-2 presents three typical designs of HOT lane slip ramp (FHWA, 2003). The cell representation scheme presented in Section 6.1.1 is relatively si mple, and may reflect the reduced narrow bufferseparated (no weave lane) design but with a strong assumption that all the lane changes will be made at one location right before the end of the access segment. More realistic cell representation may be used, and the modeling of lane changes will be updated accordingly. 1 In this case, the number of lane changes estimated from the loop detector data does not necessarily represent the lane-changing demand. Therefore, when the HOT lane is congested, the detected flows will not be use to update the estimates of motorists willingness to pay. PAGE 107 107 Figure 6-2 shows a cell represen tation scheme for the barrie r-separated design option. The additional cells between the regular and the HOT la ne represent the entrance (can be viewed as an acceleration lane). The weaving area may not need to be modeled since of more interest is the effect of the lane-changing behaviors on the HOT la ne rather than the regular lane. The demand for the HOT lane is still calculated based on equation ( 5.1). However, the lane changes from the acceleration lane to the HOT lane will then be modeled as discretionary behaviors, using the approach proposed in Laval and Daganzo (2006). Lou et al. (2007) conducted prelim inary numerical experiments ba sed on this more realistic HOT slip ramp cell representation and the co rresponding multi-lane hybrid CTM. It is found that the total freeway throughput increases with the barrier-sepa rated HOT slip ramp design. This is due to the fact that vehicles are able to accelerate before they merge into the HOT lane, and thus the voids created in the HOT lane flow are significantly reduced. These results suggest that HOT lane slip ramp configurations have substantial impacts on the performance of HOT lanes and the proposed mode ling approach is able to capture those impacts. 6.2 System Inference The first step in the framework of the self -learning approach, system inference, is to analyze traffic data to estimate the freeway sy stem state and learn char acteristics of travel demand and supply. The step includes three comp onents, WTP learning, traffic state estimation and demand forecast. Discussion on WTP learning can be found in Section 5.2.2. With the enhanced traffic flow model pres ented in Section 6.1, this se ction elaborates the other two components in the system inference step. 6.2.1 Traffic State Estimation Traffic state estimation is to deliver a comple te image of freeway traffic conditions based on available traffic data at limited locations. It is the basis for facility performance monitoring PAGE 108 108 and providing initial conditions for the toll optimi zation in the unified fram ework. Data obtained from the detectors serves as direct measur es with random measurement errors, while the predicted states from traffic flow models are viewed as an indi rect measure which might not be very accurate either. The basic idea is to r ecover the full image by establishing a state that balances both the direct and indirect measuremen ts. Various filtering techniques can be applied based on the different traffic fl ow models adopted (e.g., Wang and Papageorgiou 2005; Antoniou et al., 2007; Mihayl ova et al. 2007). To more realistically capture traffic dynamics along the tolling facility, Section 6.1 discusses the use of the multi-lane hybrid CT M proposed by Laval and Daganzo (2006) in the framework. However, the model is a non-lin ear non-differentiable tr ansformation from one traffic state to another, particularly with the lane-choice probability invo lved. We thus apply unscented Kalman Filter (UKF) (Julier et al., 1995) to predict the new state with unscented sampling. The state-space model of this filtering problem can be written as: z avg avge t d h t z e t d f t d e t d f t d ) 1 ( ) 1 ( ) ( ) 1 ( ) ( ) 1 (2 2 1 1 ( 6.1) In model ( 6.1), ) ( t d is the vector of the densities alon g the freeway for both HOT and regular lanes at time point t ) (1 f represents the transformation from ) ( t d to ) 1 ( t d following the multi-lane hybrid CTM. Note that the measurements from the traffic detectors are usually aggregated, i.e., directly measured density is an average value across a certain time period. We then let ) ( t davg represents the average densities for both lanes between tw o data-pulling time points 1 t and t computed from function ) (2 f based on the same multi-lane hybrid CTM. PAGE 109 109 ) ( t z is the vector of limited direct measurements, and h denote the matrix indicating locations of the detectors. 1e and 2e are the corresponding process errors, and ze is the measurement error. While ze can be safely assumed to be Gaussian white noises with known variance matrix z further investigation (empirical study) of the traffic flow model is required to make reasonable assumptions on 1e and 2e. In model ( 6.1), the parameters of the multi-lane hybrid CTM, such as the free flow speed, jam density and capacity, as well as the WTP are not treated as unknown states for better effici ency. Instead, the parameters fo r the traffic flow model can be calibrated offline, and the WTP can be estimate d independently with the minimum requirements of detectors (see Figure 5-2). The first two state equa tions can be combined by augmenting the state variables to T T avg Tt d t d t x )] ( ), ( [ ) ( grouping two CTM transforma tions and process errors to T T Tf f f )] ( ), ( [ ) (2 1 and T T T xe e e ] [2 1 and expanding h to H accordingly. A more concise form of the state-space model ( 6.1) becomes: z xe t x H t z e t x f t x ) 1 ( ) 1 ( ) ( ) 1 ( Given the estimates of the mean and the covariance of state variable x at time t as ) ( t x and ) (t Px, the following xn2 points with the same weight xn2 1 can be generated, where xn is the dimension of the state variable x and i x xt P n) ( is the ith row or column of the matrix square root of ) (t P nx x: x i x x i i x x in i t P n t x t x t P n t x t x ,..., 2 1 ) ( ) ( ) ( ) ( ) ( ) (2 1 2 ( 6.2) Equation ( 6.2) is called unscented sampling becaus e these points have the same sample mean and covariance as ) ( t x and ) ( t Px. Applying UKF, the new traffic state estimates are: PAGE 110 110 ) 1 ( ) 1 ( ) 1 ( ) 1 ( ~ ) 1 ( ) 1 ( ~ ) 1 ( ~ ) 1 ( ~ ) 1 ( ~ 2 1 ) 1 ( ~ ) 1 ( ~ ) 1 ( ~ ) 1 ( ~ ) 1 ( ~ 2 1 ) 1 ( ) 1 ( ~ ) 1 ( ~ ) 1 ( ~ ) 1 ( ~ 2 1 ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ~ ) 1 ( ) 1 ( ) 1( ~ ) 1 ( ) 1 ( ~ 2 1 ) 1 ( ~ 2 ,..., 2 1 ) ( ) 1 ( ~ 2 1 2 1 2 1 1 2 1t G t P t G t P t P t x t x t x t x n t P t x H t x H t x H t x H n t P t x H t x H t x t x n t P t P t P t G t x H t z t G t x t x t x n t x n i t x f t xT x z x x x x n i T mean i mean i x x z n i T mean i mean i x z n i T mean i mean i x xz z xz x mean x mean n i i x mean x i ix x x x ( 6.3) where x is the covariance matrix of the process error xe Two most time-consuming steps in model ( 6.3) are applying multi-lane CTM to each sample point ) ( t xi and the matrix inversion. Efficiency of the former depends on the dimension of the traffic state variable, and the latter the number of detectors installed. In order to improve the efficiency of the traffic state estimation, a co arser discretization of th e site or simplification of traffic state estimation is necessary. Tw o possible simplified traffic state estimation procedures are: to generate less sample points in UKF, and use linear interpolation. Note that in the former simplified procedure, although the vari ance of the sample point s after transformation may not be preserved, the sample mean remains unscented. 6.2.2 Demand Learning Another component of system inference is to forecast short-term traffic demand such that toll determination can be more proactive. The forecast is instrumented by Bayesian learning. It is assumed that future traffic arrival follows a Poisson process, with unknown average arrival rate, denoted as As time evolves, the detected arrival rates are used to adju st the estimate of the average arrival rate and thus the distribution of the ar rival rate. The operator can then PAGE 111 111 make use of the short-term de mand forecast to determine tolls to better achieve the HOT operating objectives. Lin (2006) applies the a bove Bayesian learning co ncept to dynamically price a product to maximize the expected profit of a firm. Based on the assumption, the number of vehicles arrived during time ] 0 ( t denoted as N will follow the Poisson distribution: ) ( ) ( n t e t n N Pn t Since is unknown, we may estimate its prior distribution from histor ical data as the follo wing gamma distribution: ) ( ) ( ) (1k a ae pdfk a where dx x e kk x 0 1) ( is the gamma function, and parameters a and k are selected to fit the historical sample mean and variance of : ) ( ) ( Var E a and ) ( ) (2 Var E k To update the prior distribution of during operations, assume that during ] 0 ( t, i number of vehicles have been observed from loop detectors. By Bayes theorem, posterior probability density function of is: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ) ( ( ) ( ) ) ( ( ) ( ) (1 ) ( 0 1 1 0 ) (i k t a e t a d i t e k a ae i t e k a ae d i t N P pdf i t N P pdf pdfi k t a i t k a i t k a i t N which is another gamma distribution with parameters i k t a ,. With the updated distribution of the number of vehicles that ma y arrive during time interval ] ( t t t can be estimated as below: PAGE 112 112 n i k i k t a n tt t a t t t a t a i k n n i k d i k t a e t a n t e i t N n t N t t N P ) ( ) ( ) ( ) ( ) ( ) ( ) ) ( ) ( ) ( (1 ) ( 0 Note that when k is integer, the above reduces to a negative binomial distribution: n i kt t a t t t a t a n n i k i t N n t N t t N P 1 ) ( ) ( ) ( 6.3 Toll Determination The first step in the framework, system inference, has revealed the current system condition and predicted future tra ffic demand. With this information, the second step is to determine toll rates to maximize the total throughput of the en tire freeway while ensuring the HOT lane operate in free flow condition. The multi-lane hybrid CTM can be incorporated in a rolling horizon scheme for toll determination. In this section, the toll determination is improved in several aspects. In order to avoid dramatically fluctuating toll rates resulting from the prototype reactive pricing model, Section 6.3.1 proposes a proactive approach Since robustness in traffic-ad aptive setting is as important as in the static setting, Section 6.3.2 and 6.3.3 investigate the two major sources of uncertainty, users heterogeneity and random demand, in the HOT lane operations respectively. Two approaches of robust modeling are employed. 6.3.1 Deterministic Proactive Pricing Approach The prototype model of the self-learning approa ch presented in Section 5.2.3 determines a toll rate for each time interval in response to the inflows measured at that particular time. The tolls may thus fluctuate dramatically, which may cause safety issues in reality. If a driver who decides to access the toll lane observes a sudden price jump, he or she may become reluctant and make abrupt maneuvers that possi bly lead to a rear-end accident. A proactive approach may be PAGE 113 113 adopted to address the issue. Assume for now that future traffic arrivals are known and deterministic. This knowledge can be incorporated in advance into the rolling horizon framework, which simulates the traffic for a period longer than the to ll interval to allow appropriate flow propagation, obtain a more accurate estimate of the resulting control objectives, and provide some lead time for adjusting to ll rates in a smoother manner to respond to the changes in traffic demand a nd maintain a high throughput. Suppose the length of the rolling horizon is set to V tolling intervals. Within the horizon, the decision is now a vector of toll rates, which has V components. Let Tq and Rq represent the average throughput of HOT and regular lanes at the downstream end of the freeway segment respectively during the rolling horizon, and Tv the average speed of HOV/HOT lane. The control objectives can now be more specifically stated as maximizing the sum of the average throughputs at the downstream end while attempti ng to keep the average speed of HOV/HOT lane as free flow speed fv These measurements of performance can be computed from the hybrid CTM with mandatory lane-changing demand at the HOT entrance calculated by equation ( 5.1). With being the toll vector, T and R the singleton inflow va lue or the vector of future inflows, computing the objectiv e measures is essentially a mapping : R T T R Tv q q , ,2 1 ( 6.4) Consequently, the toll optimizati on problem can be written as: max 0 minf T R Tv v q q ( 6.5) s.t. max0 i i1, 2, V ( 6.6) 1 i i i1, 2, V ( 6.7) Condition ( 6.4) PAGE 114 114 where is a penalty parameter, 0 is the toll rate of the time interval previous to the rolling horizon, and max are the maximum margin of toll vari ation for two consecutive intervals (due to safety consideration) and the absolu te maximum toll rate specified by the tolling authority. The second term in the objective f unction is a penalty for speed reduction. If is selected appropriately, this term should be equa l to zero with the optimal solution, ensuring that the average speed of HOT lane is free flow speed Note that the objectiv e function is continuous and bounded above, and the feasibility set is co mpact, therefore optimal solution exists. A variety of numerical algorithms, such as the gold en-section method, can be used to search for a local optimum. To illustrate this proactive concept, Lou et al. (2007) conducts a numerical example upon the simplest cell representation. The numeri cal study compares the reactive model, which determines the toll rates interval by interval in response to the current de tected inflow, with the proactive model that computes a toll vector each time based on the predicted future inflows. The study suggests that the proactive approach leads to a smoother toll patter n without sacrifice the control objectives. 6.3.2 Robust Controller with Heterogeneous Motorists Previous studies have identified substantial he terogeneity in motorists values of travel time and its important implications for road prici ng policy (e.g. see Small et al., 2005). In this section, heterogeneous motorists with differe nt WTP are considered, i.e., there will be distributions associated with 1 ,2 and across the population. To accommodate the heterogeneity of motorists, we seek for a robust pricing strategy th at makes the control performance insensitive to the vari ations of the three WTP paramete rs. Since the distributions of the WTP parameters are difficult to obtain, we apply the notion of robus t optimization (e.g., Ben- PAGE 115 115 Tal and Nemirovski, 2002; El-Ghaoui, 2003) which does not require any di stribution information on the uncertain parameters. This approach se eks a robust solution mini mizing the worst-case performance against an ellipsoidal uncertainty se t that reflects decision makers attitudes toward risk. It has been proved that the robust solu tion with ellipsoidal uncertainty set is less conservative than, e.g., the polyhedral uncer tainty set, and has properties that are computationally advantageous. With the presence of user heterogeneity, the le arning procedure of user s WTP discussed in Section 5.2.2 is still able to estimate approximately the means and variances of 1 ,2 and Without requiring the knowledge abou t the underlying distributions of 1 ,2 and we assume these three parameters vary within an ellipsoida l likelihood region. In other words, motorists WTP is confined by the following ellipsoid: } 1 ) ( ) ( ), ( ), ( | { ) 1 (2 2 1 3 u u t M t t t y R y t YT WTP Here M ( t ) is a diagonal matrix with standard deviations of 1 ,2 and as its elements, and is the multiplier, adjusting the size of the ellipsoi d to represent the operators trade-off between efficiency and robustness. We then determine a toll vector that maximizes the minimum control objective (improves the worst-case performance) in curred by the WTP parameters in the set. Mathematically, the robust optimal toll rate solves the following max-min problem (a special case of the least-square problem s with uncertain data discusse d in El-Ghaoui and Lebret, 1997): ) 1 ( ,2 1min max t YWTP 0 minf T R Tv v q q s.t.Constraints ( 6.4), ( 6.6)-( 6.7) Efficient iterative procedures can be deve loped to solve the problem. A cutting plane scheme similar to the one in S ection 4.3.2 can be applied. Th e algorithm solves a sequence of relaxed max-min formulations that approximate the ellipsoidal uncertainty set with a set of PAGE 116 116 discrete parameter scenarios. The algorithm starts from an initial parameter scenario set that only contains the estimates at current time interval, and optimizes toll against the set. It then finds the worst-case parameters 1 ,2 and under the toll rate, a nd then includes those parameters into the scenario set, against which the toll rate is then optimized again. This procedure will repeat until the optimal solution converges. Yin and Lou (2009) conducts numerical examples using the point-queue model to illustrate the concept of robust pricing. It is demonstrated that the robust controller is able to produce WTP estimates that are satisfactorily close to the actual values, and deliver more stable performance. 6.3.3 Robust Controller with Demand Uncertainties Given the future traffic demand is likely uncerta in, this sub-section further addresses this issue in addition to the user heterogeneity. Similar robust optimization with ellipsoidal uncertainty set can be applied to model random fu ture demand. However, while the distributions of WTP parameters are difficult to estimation, the distribution of the future arrival can be updated in real time using Bayesi an learning discussed in Secti on 6.1.2. Therefore, another stream of optimization methods under uncertainty namely stochastic programming (Birge and Louveaux, 1997), can be employed to seek for a to ll rate that performs well under most of the possible scenarios. With the demand learning, the probability dens ity function of the ar rival distribution can be approximated by a discrete set of possible scenarios, S s ,..., 2 1 where each scenario specifies the traffic arrival rates for each in terval in the rolling horizon, denoted as ) ( ts T and ) ( ts R, with a probability of occurrence as sp Since most decision makers are risk averse, a scenario-based stochastic program is formulated to minimize the expected loss (penalty minus PAGE 117 117 throughput) incurred by high-consequence scenario s, which is called co nditional value-at-risk (CVaR) in financial engineering (Rockafellar an d Uryasev, 2000). The decision makers attitude toward risk can be controlled by a parameter More specifically, only scenarios with loss greater than the -percentile (say, 90%) value are consid ered, and the expected loss of those scenarios is called -CVaR. The scenario-based stochastic toll optimization problem is written as follows: sL ,min S s s sL p1) 0 max( 1 1 s.t. 0 maxs T f s T s T sv v q q L s R s T s T s R s Tv q q , ,2 1 Constraints ( 6.6)-( 6.7) It can be proved that the optimal value of the objective function is the minimum -CVaR, and the solution is the -percentile loss (Rockafellar and Uryasev, 2002). Note that the above stochastic formulation is flexible and can be extended to incorporate other sources of uncertainty. For example, if user heterogeneity can be approximated by a number of discrete scenarios, it is possible to integrate it with the scenarios of traffic demand, and the above stochastic formulati on will be able to consider bot h types of the uncertainty. On the other hand, the robust formulation can also be expanded to incorporate demand uncertainty by modifying the uncertainty set correspondingly. 6.4 Simulation Study We conduct simulation experiments to demons trate and validate the proposed approach. Multiple components of the system inference and toll determination are ac tivated to illustrate that they can be integrated to a unified framewor k. Within the simulation platform, the simulator PAGE 118 118 attempts to replicate the motorists lane-choi ce behaviors and traffic dynamics; the controller implements the proposed framework of system in ference and toll optimization; and the monitor mimics a real-world surveillance system, genera te perturbed measurements of actual traffic conditions simulated with actual parameters. In th e following, to facilitate the key concepts of the proposed framework, we implemented the sy stem inference components of WTP learning and traffic state estimation, and the deterministic proactive version of toll optimization. The simulation site is a freeway segment shown in Figure 6-3a. The site is 2 miles long, with a barrier-separated HOT slip ramp. It is assumed that each lane obeys the same triangular fundamental diagram. The releva nt parameters are reported in Figure 6-3b, and the small triangle is for the downstream bottleneck. Additi onally, all lane-changin g vehicles are assumed to have an acceleration rate of 12.22 ft/s2, i.e., it will take a vehicle 7.2 seconds to accelerate from zero speed to free-flow speed. The true values of 1 2 and used in the simulator are 0.5, 1 and 0.2. 5 02 1 indicates a value of travel time of 15 dollars per hour since the travel time difference is represented in th e unit of one tolling interval (2 minutes). The entire simulation duration is 20 minutes. The discrete simulation interval in hybrid CTM model is 0.6 seconds, and all lanes are parti tioned into small cells of 0.01 mile, leading to a vector with a total number of 420 elements in the tr affic state estimation. To be consistent with the practice, the monitor reports aggregated tr affic measurements every single minute, and the traffic state estimation is conducted once new me asurements are received. Moreover, the toll rate varies every two minutes, and the rolling hor izon for toll optimization is 4 minutes (2 time intervals). The weighting factor in model ( 6.5) is set as one and the toll optimization problem is solved using the golden-section method. In th e simulation, random arrivals are generated from a virtual source with an average rate of 2400 vph for the regular lane, 600 vph and 1200 vph PAGE 119 119 during the first and last 10 minutes for the HOV lane. ) 0 (1 ) 0 (2 ) 0 ( and ) 0 (WTPP are initialized as 1, 2, 0 and an identity matrix resp ectively. Initial values of traffic conditions are generated randomly, and ) 0 (xP is set to identity matrix. Figure 6-4 shows that travelers WTP can be gradually learned within the unified framework during the tolling operation. The calibrated values of parameters 1 2 and are able to converge from the initial values to the true values within 6 minutes. The calibration time for each interval is less than 0.003 seconds. Since the size of the traf fic state vector is very large, we do not present the detailed comp arison between the estimated and the actual values directly. Instead, we will compare the control results to th e base case where actual traffic condition is known precisely. The resulting toll rates and the performances of the freeway segment are presented in Figure 6-5. It is demonstrated th at the controller is able to adjust the toll rate in response to the estimated traffic condition in real time. The pro active toll optimization ensures that the toll rate changes smoothly from interval to interval. Figure 6-5 also compares the resulting average speed and throughputs from two cases where HOT la ne is operated respectively under optimal toll rate and without any control. The latter case means that LOV can use the HOT lane without paying any toll. It can be observed that the c ontroller is able to maintain a high and stable throughput while preventing the HOV/HOT lane fr om being congested. The minimum average speed along the HOV/HOT lane is 55.41 mph under optimal toll rates; while in the no-control case, the average speed can be as low as 34 mph. On the other hand, the average throughput is 3291 vph under optimal toll rates, which is 91% of the downstream bottleneck capacity, and only slightly lower than the throughput of 3350 vph wit hout any control, while 50% of the capacity would be wasted if LOV is not allowed to acce ss the HOT lane. One of the reasons that the PAGE 120 120 HOT lane throughput does not reach the capacity of the downstream bottleneck is that lanechanging vehicles act as movi ng bottlenecks while accelerating to the speed prevailing on the HOT lane. Those vehicles create gaps in the fl ow in front of them that propagate forward, thereby reducing the throughput. In the simulation, additional loss of throughput is due to initial inaccurate estimates of motorists WTP. In the above numerical example, the comput ation time of the toll rates for each time interval is less than 9 seconds. However, updati ng the estimates of traffic conditions once every minute takes more than 90 seconds due to the large number of unscented samples required in the UKF method. Therefore, the two simplified traf fic state estimation procedures proposed in Section 6.2.1 are tested. The impacts of different traffic state estimators on the control objectives are investigated by comparing them to the base case where actual traf fic condition is known precisely. We conducted multiple runs and our statis tical tests indicate that different versions of traffic state estimation do not affect the resul ting average speed of HOT lane and the freeway throughput significantly. One plausibl e reason is that the simplified traffic state estimators are still able to capture the prev ailing traffic conditions along the freeway because the multilane hybrid CTM is mainly a first-order traffic flow model. On the other hand, the two simplified versions are far less demanding in computation w ith an average of less than 8 and 0.001 seconds respectively. Their computational efficiency is a desirable attribute in real-time operation. 6.5 Summary In this chapter, the unifie d framework of proactive self -learning dynamic pricing for HOT lanes is further elaborated. This framework inco rporates two major steps, system inference and toll optimization, and is able to generate robus t and proactive dynamic pr icing strategies The system inference applies a variety of techniques such as regular KF, UKF and Bayesian learning to mine traffic data to gain better understanding of motorists WTP, traffic state and short-term PAGE 121 121 traffic demand. The attained knowledge contribu tes in the second step in determining optimal toll rates that lead to efficient utilization of freeway capacity and superi or travel services for HOT lanes. Three toll optimization formulat ions have been proposed to enhance the proactiveness and the robustness of the self-learni ng controller. Simulation experiments confirm that the integrated framework is feasible, efficient and effective. The proposed unified framework can be extended in multiple ways to take into account more considerations. For example, another more advanced discrete choice model, such as a mixed logit model, can be incorporated as an alte rnative to describe the travelers heterogeneous lane-choice behaviors. Our future research w ill also integrate ramp metering with dynamic pricing, and develop coordinated pricing strategies for freeways with multiple HOT segments. We also plan to enhance the capability of the microscopic traffic simulation software CORSIM in simulating HOT lane operations, and then co nduct simulation experiments to evaluate the proposed framework in a microscopic simulation platform. PAGE 122 122 Figure 6-1. Discretized freeway representation Regular Downstream Bottleneck HOV/HOT Toll tag reader PAGE 123 123 Figure 6-2. Discretized freeway representation (barrier-separated) Regular Downstream Bottleneck HOV/HOT PAGE 124 124 0.5 mi 1.5 miHOV Regular HOT Downstream Bottleneck Toll tag reader 0.2 mi q (vphpl) 2400 1800 60 30 -30 a b Figure 6-3. Simulation settings PAGE 125 125 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 Time Interval (2min)1 Calibrated Value Actual Value 0 1 2 3 4 5 6 7 8 9 10 0.5 1 1.5 2 Time Interval (2min)2 Calibrated Value Actual Value 0 1 2 3 4 5 6 7 8 9 10 -0.5 0 0.5 Time Interval (2min) Calibrated Value Actual Value Figure 6-4. Calibrati on of Willingness to Pay PAGE 126 126 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 Time Interval (2min)Toll rate ($) 1 2 3 4 5 6 7 8 9 10 30 35 40 45 50 55 60 65 Time Interval (2min)Average speed along HOV/HOT (mph) Opt. toll No controll Free flow speed 1 2 3 4 5 6 7 8 9 10 0 600 1200 1800 2400 3000 3600 Time Interval (2min)Freeway throughput (vph) Opt. toll No controll Figure 6-5. Optimal toll rate and its performance PAGE 127 127 CHAPTER 7 CONCLUSION This dissertation develops a hierarchical congest ion pricing framework for transportation network in view of the fact that travelers have limited response capabilitie s to the tolling signals at different spatial levels. Toll determination is hence decomposed into two levels: network and facility. Robust static or timeof-day pricing scheme is proposed at the network level to avoid complex toll structures while ensuring the network will perform reasonably well against various uncertainties. At the facility level, tolls determined from the network level can be further adjusted based on real-time traffic conditions to meet local objectives w ith a detailed modeling of drivers behavior and traf fic dynamics. Through systematica lly reviewing the general inputs and technical issues of this hierarchical fr amework, it is demonstrated that the proposed framework is feasible and has gr eat potential for actual implementation. Moreover, the proposed pricing approaches for the two levels take in to account the correspond ing key features, and complete the missing pieces of robust pricing with users boundedly rational behaviors and adaptive pricing for managed lane operations in the literature. Simulation studies lead to promising results, suggesting that the proposed hierarchical fra mework may be efficient and effective for congestion mitigation in practice. At the network level, this di ssertation explores static or time-of-day robust network-level pricing problem with bounded rationality in traveler s route choices. The static or time-of-day feature makes the tolling signals easy for th e travelers to understand and follow. The investigation of bounded rationality, an important alternative source of network uncertainties, enriches the literature by providi ng a rigorous definition, models of traffic assignment with bounded rationality, implications of the resulting fl ow distribution, and a robust pricing scheme. It is found that system performance may vary su bstantially within the po ssible equilibrium flow PAGE 128 128 set under bounded rationality and the proposed robust pricing models are able to protect against the worst-case scenario effectively. More efficient solution approaches will be developed in the future. Moreover, further study will investigate how to determine a robust network-level tolling scheme under other more realistic network settings w ith boundedly rational behaviors, such as elastic demand or combined rout e and departure time choices. At the facility level, this di ssertation develops one feedback and one self-learning approach particularly for HOT lane operations. Both approaches are dynamic and traffic-adaptive, providing superior free-flow travel services for HOT lanes while efficientl y utilizing the capacity of the freeway. The self-learning approach inco rporates several human and traffic factors and has the potential to address seve ral critical issues in practical HOT operations. Through mining the real-time data of traffic measurements, this approach is able to learn information from both the demand and supply side of the HOT facility and calculate proactive and robust toll rates based on the updated information. Components of system inference, such as travelers willingness-to-pay learning, traffic state estima tion, and short-term demand prediction, are integrated in a unified framework to serve as a basis of the toll determination. These enhancements help the self-learn ing approach become more realis tic, efficient and effective. Both of the proposed approaches are feasible and easy to implement. Simulation experiments demonstrate that the self-learni ng approach to be a more prom ising method for HOT operations. It can be further extended in multiple ways to take into account more considerations as well. For example, more advanced models can be adopted to describe the travelers lane-choice behaviors more realistically. Future research may also examine the possibility of integrating ramp metering with dynamic pricing, and develop coor dinated pricing strategi es for freeways with multiple HOT segments. 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ASCE Journal of Transportation Engineering, Vol. 135, No. 2, 45-52. Yin, Y., Madanat, S. and Lu, X. (2008) Robust improvement schemes for road networks under demand uncertainty European Journal of Operational Research, Vol. 198, No. 2, 470479. Yin, Y. and Yang, H. (2004). Optimal tolls with a multiclass, bicriterion traffic network equilibrium. Transportation Research Record, 1882, 45. PAGE 141 141 BIOGRAPHICAL SKETCH Yingyan Lou received two bachelors degrees in both engineering sc ience and economics from Peking (Beijing) University, China, in 200 5; and earned her masters degree in civil engineering from the Univer sity of Florida in 2007. Yingyan Lous primary research interest is transportation systems modeling and optimization, with applications on system-wide congestion pricing, traffic-responsive tolling for managed-lane operations, dynamic origin-destina tion demand estimation, robust transportation network design, freeway incident response planni ng, and infrastructure asset management. She also has a keen interest in traffic flow theory and operations. During her Ph.D. study at the Tr ansportation Research Center in University of Florida, Yingyan Lou has co-authored seven papers and made eight presentations at various conferences. She won the second prize of stude nt paper competition at the 2nd International Symposium of Freeway and Tollway Operations in 2009, the best poster award at the 2007 Institute of Transportation Engineers Florida District A nnual Meeting, and was awarded the graduate scholarship of Womens Transportation Seminar in 2008. She also won numerous travel grants from various sources and was awarded the title of Outstanding Internat ional Students by the University of Florida twice. |