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PAGE 1 1 PROACTIVE AND ROBUST DYNAMIC PRICING STRATEGIES FOR HIGH OCCUPANCY/TOLL LANES By DIMITRA MICHALAKA A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2009 PAGE 2 2 2009 Dimitra Michalaka PAGE 3 3 To my family for supporting me during my studies PAGE 4 4 ACKNOWLEDGMENTS I would like to thank my graduate advisor, Dr Y afeng Yin of the University of Florida for his insights and guidance throughout th is thesis and his valuable s upport. I, also, wish to thank the remaining members of the thesis committee, Dr. Siriphong Lawphonpanich and Dr. Lily Elefteriadou, for their assistance and their advi ces. Finally, I express my sincere thanks to graduate students, Yingyan Lou for her help wi th the coding and to Al exandra Kondyli for her valuable comments. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ............................................................................................................... 4 LIST OF TABLES ...........................................................................................................................7 LIST OF FIGURES .........................................................................................................................8 ABSTRACT ...................................................................................................................... .............10 CHAPTER 1 INTRODUCTION .................................................................................................................. 12 1.1 Background ...................................................................................................................12 1.2 Problem Statement ........................................................................................................ 13 1.3 Research Objective, Supporting Tasks and Validation .................................................14 1.4 Document Organization ................................................................................................ 14 2 LITERATURE REVIEW .......................................................................................................15 2.1 Introduction ...................................................................................................................15 2.2 Congestion Pricing ....................................................................................................... .15 2.3 HOT Lanes ....................................................................................................................18 2.3.1 HOT Lane Concept ........................................................................................... 18 2.3.2 Benefits of HOT Lanes .....................................................................................19 2.3.2.1 Benefits of HOT lanes on transportation networks ............................19 2.3.2.2 Benefits of HOT lanes to society ........................................................20 2.3.2.3 Implementation difficulties of HOT lanes ..........................................21 2.3.3 Implementation of HOT Lanes in the U.S. ....................................................... 21 2.4 Determination of Dynamic Pricing Strategies .............................................................. 22 2.4.1 Modeling Approaches ....................................................................................... 22 2.4.1.1 Bottleneck models .............................................................................. 22 2.4.1.2 Network models ..................................................................................28 2.4.2 Current Practice of Toll Determin ation in HOT Lanes Operations .................. 30 2.4.2.1 I394 HOT lanes in Minnesota ...........................................................30 2.4.2.2 I95 HOT lane project in South Florida .............................................. 32 2.4.3 SelfLearning Control Approaches for Dynamic Tolling ................................. 33 2.5 Conclusions That Can Be Draw n from the Literature Review ..................................... 34 3 RESEARCH APPROACH .....................................................................................................36 3.1 Introduction ...................................................................................................................36 3.2 Methodology Overview ................................................................................................36 3.3 Modeling Preparation ....................................................................................................37 3.3.1 WillingnesstoPay Learning ............................................................................ 38 PAGE 6 6 3.3.1.1 Basic concept ...................................................................................... 38 3.3.1.2 Calibration of willingnesstopay ....................................................... 39 3.3.2 Demand Learning ..............................................................................................42 3.3.3 Stochastic Capacity Determination ................................................................... 44 3.4 Robust Toll optimization .............................................................................................. 45 3.4.1 Deterministic Case ............................................................................................ 46 3.4.1.1 Modeling traffic dynamics .................................................................. 46 3.4.1.2 Model formulation ..............................................................................48 3.4.2 Scenariobased optimization .............................................................................53 4 DEMAND LEARNING RESULTS ....................................................................................... 57 4.1 Introduction ...................................................................................................................57 4.2 Empirical Analysis ....................................................................................................... .57 4.2.1 Analysis Using the Data of the Entire Period ...................................................58 4.2.2 Analysis Using the Data of the Peak Hour........................................................64 5 PROACTIVE AND ROBUST PRICING STRATEGIES ..................................................... 69 5.1 Introduction ...................................................................................................................69 5.2 Model Solution and Simulation Study ..........................................................................69 5.3 Numerical Results ........................................................................................................ .71 5.3.1 LowHighLow Demand Case .......................................................................... 71 5.3.1.1 Robust toll optimization ..................................................................... 71 5.3.1.2 Deterministic toll optimization ........................................................... 75 5.3.1.3 Comparison of the robust ve rsus the deterministic toll optim ization ........................................................................................ 78 5.3.2 LowMediumHighMediumLow demand case .............................................. 82 5.3.2.1 Robust toll optimization ..................................................................... 82 5.3.2.2 Deterministic toll optimization ........................................................... 85 5.3.2.3 Comparison of the robust ve rsus the deterministic toll optim ization ........................................................................................ 89 5.4 Conclusions ...................................................................................................................94 LIST OF REFERENCES ...............................................................................................................95 BIOGRAPHICAL SKETCH .........................................................................................................99 PAGE 7 7 LIST OF TABLES Table page 41 Data from SR91 ........................................................................................................... .....58 42 Estimated flows ........................................................................................................... .......61 43 Actual and predicted flows ................................................................................................62 44 Data from SR91 ........................................................................................................... .....65 45 Estimated flows ........................................................................................................... .......66 46 Actual and predicted flows ................................................................................................67 PAGE 8 8 LIST OF FIGURES Figure page 21 Optimal congestion toll (Morrison,1986). ......................................................................... 16 22 Diamond lane design (Scott, 2007). ................................................................................... 31 23 I394 toll rates on May 18, 2005 (source: www.MnPASS.net) .........................................32 31 Toll determination procedure ............................................................................................. 37 32 System configuration ...................................................................................................... ...39 33 Representation of the road with cells .................................................................................46 34 Flowdensity relationship.................................................................................................. .47 35 A loss function (Yin, 2007) ...............................................................................................54 41 SR91 ..................................................................................................................... ............57 4.2 Actual flows and weaker bounds. ...................................................................................... 64 43 Actual flows and weaker bounds during the peak hour. ....................................................68 51 FlowDensity Curves ....................................................................................................... ..70 52 Traffic demand profile. ......................................................................................................72 53 Optimal Toll Rates. ....................................................................................................... .....72 54 Freeway, HOT and GP throughputs...................................................................................73 55 Average densities along HOT and GP lanes. .....................................................................74 56 Queue length upstream of the bottleneck at HOT and GP lane. ........................................ 75 57 Optimal Toll Rates. ....................................................................................................... .....76 58 Freeway, HOT and GP throughputs...................................................................................76 59 Average densities along HOT and GP lanes. .....................................................................77 510 Queue length upstream of the bottleneck at HOT and GP lane. ........................................ 78 511 Optimal Toll Rates. ...................................................................................................... ......79 512 HOT throughput. .......................................................................................................... ......80 PAGE 9 9 513 Average density along HOT lane. ......................................................................................80 514 Queue length upstream of the bottleneck at HOT lane. ..................................................... 81 515 Traffic demand profile. ......................................................................................................82 516 Optimal Toll Rates. ...................................................................................................... ......83 517 Freeway, HOT and GP throughputs...................................................................................83 518 Average densities along HOT and GP lanes. .....................................................................84 519 Queue length upstream of the bottleneck at HOT and GP lane. ........................................ 85 520 Optimal Toll Rates. ...................................................................................................... ......86 521 Freeway, HOT and GP throughputs...................................................................................87 522 Average densities along HOT and GP lanes. .....................................................................88 523 Queue length upstream of the bottleneck at HOT and GP lane. ........................................ 89 524 Optimal Toll Rates. ...................................................................................................... ......90 525 HOT throughput. .......................................................................................................... ......91 526 Average density along HOT lane. ......................................................................................92 527 Queue length upstream of the bottleneck at HOT lane. ..................................................... 93 PAGE 10 10 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science PROACTIVE AND ROBUST DYNAMIC PRICING STRATEGIES FOR HIGH OCCUPANCY/ TOLL LANES By Dimitra Michalaka May 2009 Chair: Yafeng Yin Major: Civil Engineering Congestion pricing is to reduce congestion in transportation infras tructure by charging motorists a certain amount of moneyknown as a tollfor the use of the roads. Congestion pricing has been promoted by economists and tr ansportation researchers as one of the most efficient means to mitigate traffic congestion because it employs the price mechanism with almost all the advantages of efficiency, unive rsality and clarity. When tolls implemented on highway lanes vary by the time of day, with higher values charged during peak traffic periods, it is called as dynamic tolling. The tolled lanes are High Occupancy/ Toll Lanes (HOT) if the high occupancy vehicles are allowed to use the lanes tollfree. As the literature review indicates, many studi es have been conducted to determine optimal dynamic tolls than can be implemented to roads with high congestion levels. However, most of these studies take into consider ation idealized and hypothetical s ituations in order to derive solutions. For instance, the travel demand is assumed to be known as well as motorists willingness to pay, i.e., how much money they are likely to pay for using the managed facility. In addition, there is not any model that takes into consideration uncertainty of demand or capacity for the determination of the to ll values. Therefore, this thesis develops a more robust and proactive approach to determine timevarying tolls for HOT lanes in response to realtime traffic PAGE 11 11 conditions. The toll rates are optim ized to provide freeflow condi tions to managed lanes while maximizing freeways throughput. The approach cons ists of several key components, including demand learning and scenariobased robust to ll optimization. Simulation experiments are conducted to validate and demons trate the proposed approach. PAGE 12 12 CHAPTER 1 INTRODUCTION 1.1 Background Traffic congestion has become one of the most important problems of modern life due to the increase in movement of both people and goods on highway faciliti es. People are demanding more goods and are engaging in more activities. Conge stion levels have risen in cities of all sizes since 1982, indicating that even the smaller areas are not able to keep pace with the rising demand (FHWA, 2004). According to Texas Trans portation Institute (2002), the average urban driver spent 62 hours sitting in traffic in 2000, co mpared to 16 in 1982, which is an increase of 288%. Furthermore, in 75 urban areas in 2000, rush hours lasted longer and were more extensive than the previous year and cost the country $68 b illion a year. These costs were due to 3.6 billion hours of delay and 5.7 billion gallons of wasted fuel. More than half of major roads are crowded during rush hour, up from a third in 1982. Two out of every five urban interstate miles are congested with traffic at volumes that result in significant delays. The proportion of urban interstate miles that are consid ered congested increased from 33 to 41 percent from 1996 to 2001 (FHWA, 2003). As the travel demand increas es, existing roads become c ongested and solutions are required to manage the flow of traffic. The cons truction of new facilities is very difficult due to environmental constraints and th e limited public funding. This has led transportation agencies to explore other alternatives to manage traffic flow. Typically, this has been done by using lane management strategies that regulate demand, sepa rate traffic streams to reduce turbulence, and utilize available and unused capacity on existing tr ansportation facilities. In recent years, such operational policies are called managed lanes The Federal Highway Administration (FHWA) defines managed lanes as highway facilities or a set of lanes in which ope rational strategies are PAGE 13 13 implemented and managed in real time in response to changing conditions. Managed lanes include highoccupancy vehicle (HOV) lanes, hi ghoccupancy/toll (HOT) lanes, priced and special use lanes such as express, busonly, or truckonly lanes (Obe nberger, 2004). HOT lanes are facilities that combine pricing and vehicle e ligibility to maintain freeflow conditions while maximizing the freeways throughput. People drivi ng alone in a vehicle can buy their way into the HOT lanes while the HOVs can use the HOT la nes for free. HOV and special use lanes have been used for decades while HOT lanes are much newer innovation. The first HOT express lanes opened at California State Route 91 in December 1995. Currently, there are six HOT lanes in the U.S. 1.2 Problem Statement Congestion pricing has been promoted by econo m ists and transportation researchers as one of the most efficient means to mitigate congest ion in highway faciliti es and in urban areas because it employs the price mechanism with almost all the advantages of efficiency, universality and clarity. Although th e basic theory to congestion pricing has been essentially unchanged since Pigou (1920) and Kn ight (1924) who were the first to analyze this policy, many extensions have been de veloped over the years. As the literature review indicates, many studi es have been conducted to determine optimal dynamic tolls than can be implemented to roads with high congestion levels. However, most of these studies take into consider ation idealized and hypothetical s ituations in order to derive solutions. For instance, in many papers the tr avel demand is assumed to be known and do not consider how much money users are likely to pa y for using the managed facility. In addition, no model takes into consideration un certainty of demand or capacity for the determination of the toll values. Therefore, there is a need to develop a mo re proactive approach that will give appropriate PAGE 14 14 toll rates with respect to incoming flow to main tain freeflow conditions on managed lane while maximizing roads throughput. 1.3 Research Objective, Supporting Tasks and Validation The objective of this res earch is to develop a procedure that determines timevarying tolls for HOT lanes in response to realtime traffic co nditions. The toll values must be appropriate in order to provide freeflow conditions for the users of the managed lanes while maximizing the freeways throughput. The tasks that will be conducted to achieve the objective are as follows: Review thoroughly the literature to identify existing methods and procedures for managed lanes. Learn the realtime demand based on Bayesian inference. Use the California SR91 data for the em pirical investigation of demand learning. Develop a scenariobased r obust toll optimization model. Validate the proposed approach via simulation. 1.4 Document Organization Chapter 2 presents the literatu re review on managed lanes, dynamic tolling and the methods and procedures used previously to de termine tolls. Chapter 3 describes the research approach including willingnesstopay learning, realtime demand learning, stochastic capacity determination and robust toll optimization. Chapte r 4 reports some results of demand learning by using empirical data from SR91 and Chapter 5 presents robust and proactive pricing strategies for different demand scenarios. PAGE 15 15 CHAPTER 2 LITERATURE REVIEW 2.1 Introduction This chapter reviews current practice of HO T lane operations and existing me thodologies for determining dynamic pricing strategies. 2.2 Congestion Pricing Road pricing is not a new innovation. The first toll roads in the Unites States and in ma ny other countries date back to the late 18th century. In these cases, to ll rates were fixed and everybody paid the same price for using the same ro ad. The purpose of tolling was to recover the construction cost or simply to raise revenue. In the early 1920s, econom ists and transportation researchers started to consider tolling as a policy measure to manage demand and address congestion that has started to increase in ma ny cities throughout the world (Morrison, 1986). The basic theoretical approach to congestion pricing has been developed by Pigou (1920) and Knight (1924) who were the first persons who analyzed this policy as a measure to alleviate congestion. Morrison (1986) further developed the theory of optimal congestion tolls based on the works of Pigou (1920) and Knight (1924). He used the speedflow curve that is very important in highway engineering to derive the relationship between flow and cost per user. The most significant effect of congestion was considered as the cost associated with increased trip times with an assumed value of travel time. If sp eed is inverted at the speedflow curve, time per mile is obtained. Multiplying the value of tim e by the time per mile and adding operational vehicle costs gives the cost relationship that is the average variable cost (AVC). The extra cost of adding a vehicle to the flow is th e shortrun marginal cost (SRMC). The cost curves as well as a demand curve, that represents the willingness to pa y for various quantities of trips, are shown in figure 21. In this figure the backward bending portion of the AVC curve is not illustrated PAGE 16 16 because the optimal flow will never occur in this region due to the f act that the same flow can be achieved at a lower cost. Figure 21. Optimal conges tion toll (Morrison,1986). If there is not any toll (t), equilibrium will occur at Q0 which is the intersection point of demand and AVC curves. In this case, users who valu e the trip more than they can afford, travel. At this point, the additional cost to society from, for instance, considering other users exceeds the benefit derived by the last traveler. This is happening for all trips beyond point Q*. The amount by which the extra cost of these Q0 Q* exceeds the additional benefits represents the welfare loss from nonoptimal pricing. In order to have e quilibrium a toll, t, must be added to optimal quantity Q*. This toll is equal to cong estion externality, that is, the difference between the cost a traveler affords (AVC) and the cost s/ he imposes on society (SRMC). In other words, externality is the congestion cost that each additional user of a congested road or other facility imposes on other users, by slowing down, incr easing inconvenience, risk of accidents, etc. (Carey and Srinivasan, 1993). PAGE 17 17 Kraus et al (1976) and Keeler and Small (1977) further analyzed in two independent papers the congestion problem in a longrun perspectiv e. They argue that, in a longrun analysis, optimal tolls depend on highway cap acity cost due to the fact that optimal highway capacity and congestion depend on the cost of the additional ca pacity. Kraus et al estimated longrun peak tolls using pseudoempirical analysis for a ge neric U.S. urban expressway. They found that tolls vary according to the capital cost and to location of the expre ssway. Keeler and Small (1977) conducted their research taking into consideration speedflow relationships for uninterrupted flow conditions and highway construction costs for the San Francisco Bay Area and their results were that peak tolls vary very much among the types of roads.Congestion pricing or value pr icingas it is sometimes call edis a tool for mitigating traffic congestion because it has been observed that people tend to make socially efficient choices when they are faced with the costs of th eir actions and the social benefits (Lindsey and Verhoef, 2000). Congestion pricing leads the rushhour travel to shift to offpeak periods or to other transportation modes. By removing even a sma ll percentage such as 5% of the vehicles that travel during the peak periods from a congested facility by implementi ng pricing, the system performs much better (FHWA, 2006). Congestion pricing entails setting tolls depending on how severe the congestion is. This implies that toll s must vary according to time, location, current circumstances such as bad weather, accidents and special events, type of vehicle and occupancy of the vehicle. Congestion pricing except for traffic alleviation is used also in other sections of the economy such as to telephone rates, air fares, electricity rates, hotels and other public utilities. FHWA (2006) refers four types of vari able pricing strategies. The first involves priced lanes with variable tolls on separated lanes within a highway, such as express toll lanes and HOT lanes while the second includes variable tolls on en tire roadways, that is, on toll roads, bridges, PAGE 18 18 and existing tollfree ro ads during rush hours. The third pricing strategy is cordon charges to drive into a congested area with in a city and the fourth is the areawide charges which are charges per mile on all roads within an area th at vary by level of c ongestion. Our literature review focuses on HOT lanes operations and associated pricing strategies. 2.3 HOT Lanes In this section the general principles of HOT lanes including definition, purpose, objectives, motivation and the cu rrent practice are summarized. 2.3.1 HOT Lane Conce pt HOT lanes are facilities that combine pricing and vehicle eligibility to m aintain freeflow conditions while still providing a travel timesaving incentive for highoccupancy vehicles (Obenberger, 2004). This allows additional HOV la ne capacity to be used while acting as a stimulant for mode shifting. HOT lanes were first advocated by those who believed that congestion pricing can reduce the congestion levels on freeways because drivers have to pay an amount of money in order to use the congested facility. Advocates support HOT lanes as first step toward more widespread pricing of congested roads (Dahlgren, 1999). Dahlgren (2002) investigated when to im plement HOT, HOV and General Purpose (GP) lanes, and suggested that HOT lane seemed to perform as well as or even better than an HOV lane in any circumstance. More specifically, HOT lanes may offer a solution to the issue of underutilization of HOV lanes. The concept of HOT lanes combines two very effective highway management tools: lane management and value pricing. More precisel y, lane management includes limitedaccess to designated highway lanes that depend on the vehi cles occupancy and type, and other objectives. The desirable level of traffic service is maintained by limiting the number of vehicles on the PAGE 19 19 designated lanes. Lane management can promote a range of policies such as car pools and transit vehicles to encourage higher occupancy or low emission vehicles to im prove air quality or vehicles equipped for electronic toll collection to improve operational efficiency (FHWA, 2006). Dynamic tolling includes the introdu ction of road user charges that vary over the time of day and according to congestion level. During the peak periods where the volumes are high, even the shift of a small number of vehicles can reduce si gnificantly the overall congestion levels and to lead to more reliable travel times (FHWA, 2006). 2.3.2 Benefits of HOT Lanes HOT lanes are an important m anagement tool w ith a variety of benefits to transportation networks and to society. 2.3.2.1 Benefits of HOT lanes on transportation networks The implementation of HOT lanes is app ealing because it h as many advantages to individuals in a variety of traffi c conditions on road facilities. First of all, the HOT lanes can ensure free flow conditions due to management of the demand during the periods of peak congestion (FHWA, 2006) and mitigate congestion by giving drivers a financial incentive to seek alternative modes of tran sportation, such as carp ooling and transit or to drive during offpeak hours (Replogle, 2006). This keeps the practical tr affic capacity of these lanes from shrinking as usually occurs in periods of high congestion. That is, managing traffic with tolls adds new road capacity without building more lanes (Replogle, 2006). Secondl y, the congestion pricing can lead to savings in the total travel time compared to the situation without tolls and can minimize the total travel time in the ne twork (Jaksimovic et al., 2005). Thirdly, tolling lanes offer travel options for saving time and enhance the reliabil ity of travel times (O benberger, 2004). In addition, dynamic tolling improves transit speeds and the reliability of transit service and prevents the loss of vehicle th roughput that comes from a brea kdown of traffic flow (FHWA, PAGE 20 20 2006). Toll lanes often reduce traffic in the free la nes. This happens especially in cases where dynamic toll lanes are created by converting HOV lanes that are underused (Replogle, 2006). HOT lanes can, also, reduce turbulence among the vehi cles because they separate traffic streams (Sisiopiku and Sullivan, 2007). More over, lanes with tolls improve safety on the road and the performance of freeways (Obenberger, 2004). Anothe r important benefit of congestion tolling is that it can take into considerati on not only congestion at traffic rush hour but also congestion that is caused by special events such as accidents, sport events, parades, construction, maintenance and severe weather conditions (hurricanes floods, etc) (Halem et al, 2007). 2.3.2.2 Benefits of HOT lanes to society HOT lane operations benefit drivers by reduc ing delays and stress and by increasing the predictability of trip times. Pricing roads guarant ees vehicles a reliable speed and redu ced travel time in comparison with roads without toll. Also, the HOT lanes can help sellers to deliver more products per hour to the market (FHWA, 2006) and leads to re ductions in money lost when sitting in traffic (Replogle, 2006). Additionally, HOT lanes benefit the society as a whole due to reduction of fuel consumption and vehicle emissions and to allowa nce of more efficient land use decisions (FHWA, 2006). New toll lanes created from existing capacity do not have negative impact on the environment and if the tolls are appr opriate they lead to less air pollution. This has been proved from studies of the toll roads in U.S.A. and the cordon pricing program in London. Another significant benefit of the implement ation of tolling to congested roads is the improvement of the quality of transportation servi ces without increase of taxes or large capital expenditures by providing addi tional revenues for funding tran sportation (FHWA, 2006). PAGE 21 21 2.3.2.3 Implementation diffi culties of HOT lanes The operation of HOT lanes has m any advantages to both motorists and society. However it also has some disadvantages with respect to their implementation. Many transportation agencies recognize that there is a risk associated with inaccurate traffic forecasts on toll roads. If inaccurate traffic models are used in predicting the tolls, the expectations of toll roads will not be met (Replogle, 2006). Traffic forecasts that were not accurate were used, for example, at Dulles Greenway in Virginia (U.S. DOT, 2006). In add ition, the application of tolling to highway lanes requires funding from the government or compan ies for the toll collec tion infrastructure. Moreover, although the payment of a toll is not actually a cost but an economic transfer from travelers to the toll authority, consumers feel as having an extra financial cost for paying the toll (Smith, 2007). How much consumers are affected by this transfer depends on the value of the time savings and the redistributed revenues. 2.3.3 Implementation of HOT Lanes in the U.S. Currently, six HOT lanes (more precisely, m ana ged lanes) are in operation in the United States: 1) The State Route 91 in Orange County Califor nia opened in December 1995. It is a four lane toll facility in the median of one of the most congested highways in the U.S. The tolls vary over the day to ensure that the toll lane s operate with free flow conditions. Priced lanes are separated with plastic pylons from free lanes on SR91. Sullivan (2000) conducted a study to evaluate th e impacts of the value prici ng on SR91. Some interesting results from his study are that th e toll lanes attracted a substan tial share of the traffic using the corridor SR91, the congestion in the fr ee lanes and the new trips induced by better travel conditions were for nonwork purposes. Furthermore, the number of accidents in the express lanes decreased significantly after th e implementation of the tolling. 2) I15 lanes at San Diego opened in December 1996. I15 lanes were initially HOV lanes and then a pricing program was implemented for so lo drivers. Singleoccupancy vehicles were allowed to use these lanes after the payment of a toll. Tolling lanes are separated with a barrier from general purpose lanes. Tolls in th is case vary dynamically according to traffic conditions in the HOV lanes (FHWA, 2006). The express lanes are better utilized and the number of HOVs increased and users tr avel times are reduced (Smith, 2007). PAGE 22 22 3) In Houston, tolls were introduced for the twoperson carpoolers on existing HOV lanes at corridor I10 in January 1998. In November 2000, tolling was implemented and in US 290 HOV lanes in Houston. The average number of trips on the HOT lanes increased and the main source of the travelers on toll lanes were th ose who used to travel in SOVs in regular lanes (FHWA, 2006). 4) In May 2005, the MnPass program converted the HOV lanes on I394 at Minneapolis into HOT lanes. The tolls vary dynamically and change value even every three minutes depending on the traffic density. 5) In June 2006 opened the I25/US36 managed lane s in Denver. Solo drivers have to pay a toll to use these lanes while ca rpools, buses, and motorcycles not The usage of this facility increased after the first month of its operati on more than 46% (Colorado Department of Transportation, 2006). 6) Recently, in August 2008, the 95 Express Prog ram converted the HOV lanes on I95 at Miami into HOT lanes. There are two HOT lanes separated from the local traffic lanes with pylons. The tolls depend on the traffic condi tions and they are expected to be between $0.25 and $2.65 during the average traffic conditions During the peak hours, tolls can be up to $6.20 in order to provide operating speeds between 45 and 50 mph at the HOT lanes. More managed lanes are planned in Miami, Wa shington D.C., Northern Virginia, Seattle, Maryland, Austin, Dallas, Atlanta, the San Francisco Bay Area, RaleighDurham and Portland. 2.4 Determination of Dynamic Pricing Strategies In this sectio n, we summarize the models that have been developed over the years for the determination of dynamic pricing strategies a nd the current practice for the HOT lanes. 2.4.1 Modeling Approaches This section discusses the me thodologies th at have been developed for determining dynamic pricing strategies. More specifically, it focuses on bottleneck and network models. 2.4.1.1 Bottleneck models The bottleneck model wa s introduced by Vickre y (1969) and further developed by Arnott, de Palma and Lindsey (1993). It focuses on the time at which motorists wa nt to travel and the principal innovation of it was to homogenize travelers departure ti mes. Motorists in this model travel along a single road with a bottleneck or bottlene cks downstream of certain flow capacity. PAGE 23 23 Bottleneck model doesnt take into consideration route choice because there is one single route every time. Vickrey (1969) was the first to consider trip costs with respect on desired arrival times (i.e. travelers arriving later or earlier than the desired time experien ce not only their travel time costs but also schedule costs). The simple bottleneck model is dynamic, deri ving the time pattern of congestion over the peak hour (Ar nott et al, 1995; Arnott, 1998). It assumes that every morning a fixed number, N of individuals living in suburbs would like to travel from home, which is the origin, O to work, which is the destination, D at the same time t Each person travels by his/her own car along a single road connecting the origin and destination which has a bottleneck downstream. Traffic conditions are uncongested except at the bottleneck that has a deterministic capacity of scars per unit time. Due to bottlenecks cap acity, it is not possible for all the motorists to be at the same time at their destination. As a result, tr avelers arrive at work different times, some early and others late that entails a cost of delay. Furthe rmore, travelers incur expenses including a fixed component which can be set equal to zero for computational simplifications and a variable component which depends on the time spent waiting at the bottleneck. The bottleneck model addressed individua ls decisions with regard to the time depart from their homes. The basic insight is that the tota l trips cost including the travel cost, the delay cost and the toll must be constant over the depa rture interval under e quilibrium. For analysis simplification, in the model total trip cost is assumed linear in its components: ()()()()tCqueuingtimetimeearlytimelatetoll where, tC = the trips cost when an individual arrives at time t = is the shadow value of queuing time PAGE 24 24 = is the shadow value of time early = is the shadow value of time late It is assumed that because all the individuals have the some official starting time, *t(Small, 1982). Let f t denote the time at which the first traveler arrives to work, lt the time at which the last traveler arrives to work and Tt the variable travel time. During the peak hour the capacity throughput of the bottleneck must be used because in any other case, a traveler could depart in the interior of the peak hour having zero queuing cost and less delay cost than either the first or the last person to arrive for the equilibrium consistency. This implies that /lfttNs the first individual to arrive, he/ she doesnt face queue and experiences only delay cost equal to f tt and the last individual to arri ve who, also, doesnt face queue experiences only delay cost equal to ltt Under equilibrium, delay costs of the first and last person to arrive must be equal so the equi librium price, p is equal to / Ns Therefore, the travel cost function without toll is cNN s If a dynamic toll is intr oduced that equals th e queuing cost component in the equilibrium without toll, the queue will be eliminated withou t changing the rushhour interval. In this case, every traveler has the same trip cost as before (equilibrium without toll ) and as the trip costs have become equal, no one wants to change his/ her behavior. Imposing this toll the social optimum is decentralized, the delay cost is mini mized because all travelers face the same delay cost, the queuing costs are eliminated and the bottleneck is us ed to capacity dur ing the peak hour. PAGE 25 25 The amount of travel doesnt chan ge while the total social cost is reduced by reallocating the traffic over the peak hour. The bottleneck model presented above is simp le and theoretical so many extensions are needed in order to make it more realistic. Ov er the years, many studies, considering elastic demand, heterogeneous individua ls, stochastic capacity and de mand, simple networks and alternative congestion treatments, have been conducted to improve the simple bottleneck model. Arnott et al (1993) used, also the bottleneck a pproach to determine the timevarying tolls. They assumed a high number of individuals who travel on a network and want to be at work at the same time. However, Arnott et al (1993) also assumed that there is a bottleneck somewhere in the network with limited capacity so not all the tr avelers can reach destination on the desired time. In this model, the social op timum and the distribution of travel delays, the scheduling costs at the peak pe riod, and the duration of the p eak are determined endogenously. The optimal toll depends on time and has its maximu m value when drivers arriving at the desired arrival time. The most important thing in this appro ach is that private costs of the use of the road which include the toll, the travel time cost and the delay cost, should be constant over the peak period. Iryo and Kuwahara (2000) consider travelers that can choose their departure times for the minimization of their travel cost that incl udes the queuing delay on a bottleneck and schedule delay at the destination. They used a mathematical model to analyze the travelers decision assuming one bottleneck between a residential area and a working ar ea that must be used from all the commuters with constant capacity and FIFO service. At first, they derived the model without considering congestion toll and then they applied it to ev aluate a dynamic road pricing. The purpose of Iryo and Kuwahara (2000) was to cr eate a tool that can evaluate policies that PAGE 26 26 have been proposed to mitigate congestion such as Traffic Demand Management (TDM) policies. Specifically, Iryo and Kuwahara (2000 ) considered a policy th at disperses travel demand over time because individual variation in time is very important when a road pricing scheme is analyzed. Their conclu sions after the applic ation of the method to road pricing were that dynamic congestion tolls that reduce the waiting time are not proportional to waiting time without the existence of the to lls. Moreover, commuters are likely to change their departure times and that can cause different travel costs for them. This case will not be true if all travelers have the same willingnesstopay. Finally, Iry o and Kuwahara (2000) concluded that although the individual variation of time values, ther e is a dynamic congestion toll which can reduce queuing delay to zero. In this case, toll change s the travelers behavior and their costs. Although the single bottleneck model that Vick rey (1969) used gives a good insight into the travel times, the optimal toll and its benef its, it has an important deficiency. It doesnt consider the spatial extent of queues which is a significant aspect in the analysis of extended networks because it gives more realistic patterns to avoid co ngestion. Yperman et al (2005) followed the same procedure of Vickrey (1969) but replaced the simple bottleneck model by a traffic flow model in order to take into consid eration the road space that is occupied by the queues. Specifically, they used the LWR traffic fl ow model which considers the spatial extent of queues and in the same time is not a very complicated model. The LWR model assumes that traffic is behaving as in a kinematic wave m odel. Yperman et al (2005) used a simple multidestination network in their st udy and tried to determine the advantages of congestion pricing and understanding the congestions mechanisms. Th ey used the traffic demand from Vickreys bottleneck model and determined user equilib rium and system optimum network conditions using the LWR traffic model. After the analysis, they concluded that congestion can be mitigated PAGE 27 27 if an optimal toll is imposed and that the benefi ts of introducing this ti mevarying optimal toll are higher than those expected by conventional point queuing bottleneck mode ls. The toll must be equal to the delay costs that commuters would experience if there is not any toll. After the imposing of the toll, commuters that want to tr avel through the bottleneck have the same travel costs as in the case without toll but their travel time is reduced while the commuters that dont want to use the bottleneck but e xperience queue that spill over fr om the bottleneck have reduced trip costs. Thus, traveling demand will increas e without increase in congestion. Therefore, optimal tolling can lead to reduced trip costs fo r the travelers, to more efficient use of the transportation network and even to extra revenues for the government. Verhoef (1997) considered a dynamic model of road congestion for the determination of timevarying tolls. The elastic demand for the morning p eak road usage and a congestion technology used being flow congestion and not bottleneck congest ion but the model is based on the bottleneck approach. Such elasticity of demand could come from the presence of different transport modes. In this case, the optimal timevarying toll should include a timeinvariant component when drivers share the same desired a rrival time. This is very important in reality because it means that the regulator should have information of the distribution of travelers desired arrival times in order to set the optimal tolls, because the underlying reason of the timeinvariant component is the assumption that desire d travel times are equal among the users. This timeinvariant component is relevant only in st udies of road traffic congestion with elastic demand. In this approach, the optim al toll is not zero even in the case that congestion, in terms of travel time delays, has gone to zero. Although bottleneck models taking into consid eration realworld complications give a good insight about the tolls to introduced for mitig ating congestion levels they dont incorporate PAGE 28 28 route choice and they always consider a bottlene ck at the road. Therefore, bottleneck models cannot be applied to la rge networks. That led researchers to develop models called network models that can include more parameters with re gard to individual choices and can be used to determine pricing strategies to networks. 2.4.1.2 Network models In the recent literature, in traffic engineer ing and transport economics, network models, also, have been examined to find policies to allevi ate the problem of conges tion that is present in the most of the transportation facilities. Networ k models in contrast with bottleneck models encompass the mode, departure time and route ch oice and longerrun choices. The traffic models must be as realistic as possible in order to derive logi cal and effective policies. In the literature, there are many types of network models, some of them consider fixed departure times, others variable demand, others many alternative modes, others destination choice or route choice and others combination of the factors mentioned. Some of them are presented in the next paragraphs. Palma and Lindsey (1997) focused on the efficien cy of use of private toll roads assuming a simple network with two parallel routes that can have differe nt freeflow travel times and capacity, connecting one or igin and one destination. In addi tion, they assumed that congestion takes the form of queuing and that every traveler has three options: whether to travel by car, and if choose to drive to decide th e route and the time of travelin g. For the analysis, Palma and Lindsey (1997) considered three cases, firstly, a free access on the one of the routes and a private road with tolls on the other, secondly, private roads on both routes and thir dly a public road with tolls competing with a private road. In each case, they measured the efficiency gain by determining the social surplus relatively to efficien cy gain if both routes had firstbest optimal tolls. The conclusion of the study wa s that the efficiency gain is much higher if tolls are imposed dynamically than using a fixed toll in each assumed case. PAGE 29 29 Timedependent tolls on a general network are determined by Joksinovic et al (2005a) using a dynamic traffic model to describe the netw ork performance. Joksinovic et al determined the timevarying road prices that minimize the total travel time in the network, taking into account the time changes of the route and departure as a respon se to the prices with the formulation of a network design problem. For the analysis, they considered dynamic traffic flows, dynamic road pricing strategies; they formatted the problem as a Mathematical Programming with Equilibrium C onstraints and analyzed a small and simple network. In their research, they demonstrated that dynamic pricing can lead to savings in the total travel time in the network. Finally, they concl uded that it is very difficult to find any simple solution to the dynamic toll design problem in real size dynamic traffic networks because the objective function is nonlinear and non convex so it is hard to find a global optimal solution, that is, the optimal toll values and they suggested that in order to find a global solution to large networks, a global search algorithm should be deve loped. That was the reason th at they considered a small hypothetical network in order to analy ze the uniform and varying pricing. Joksinovic et al (2005b) considered elastic demand a nd applied secondbest tolling scenarios only to a subset of links on the network. They used the same methodology with Joksinovic et al (2005a) in order to determine the optimal toll. Friesz et al (2006) introduced a dynamic optimal toll problem with user equilibrium constraints (DOTPEC). They furt her presented and tested two al gorithms for solving the optimal control representations of the DOTPEC. Firstly, Friesz et al (2006) studied a dynamic efficient toll problem by employing a form of dynamic user equilibrium model to compare the efficient tolls with DOTPEC which is not equivalent to dy namic generalization of the static efficient toll problem. Then, they formulated DOTPEC with tw o different ways and solved it using both a PAGE 30 30 descent in Hilbert spa ce without time discretization and a finite dimensional approximation solved as a nonlinear program algor ithm. The mathematical representation, in their approach, is detailed enough to capture the behavioral and technological considerations about dynamic tolling and it is referred as a computable theory. The studies about network mode ls for pricing strategies mentioned are some among a large number. Nowadays, in each tolling facility differe nt models are used for determining the tolls. 2.4.2 Current Practice of Toll Determin ation in HOT Lanes Operations Currently, the approaches are us ed to determine dynamic tolls depend on the availability of data, modeling software, model structure and objec tive of the study. In th e following, the current operation practices on Interstate I3 94 and Interstate I95 will be described as an example of how tolls are determined on HOT lanes. 2.4.2.1 I394 HOT lanes in Minnesota The I394 is an eastwest oriented facility that links I94 on th e east end to I494 and downtown Minneapolis and the Twin Cities western suburbs. It runs for 11 miles and has three lanes in each direction. It has 3 miles reversible lanes separated with barriers and 8 miles of diamond lanes. The diamond lanes are designed as in Figure 22. The facility opened with HOV and generalpurpose lanes in 1992 but significant congestion of GP lanes and the underutilization of the HOV lanes led to requests that the MnDOT open HOV lanes to solo drivers. The transportation service in Mi nnesota called MnPASS conducted a major study, which was completed in 2001, that conclude d that opening the HOV lanes to so lo drivers would lead to a congested facility so the conversion of these lanes to HOT lanes will be the most effective solution (Halvorson and Buckeye, 2006). PAGE 31 31 Figure 22. Diamond lane design (Scott, 2007). The MnPass program which was implemente d in May 2005, which converted the HOV lanes on I394 into HOT lanes. The SOVs are charged to use the lanes while HOVs and motorcyclists can use for free. The toll rates vary from $0.25 to $4.00 per trip, with the highest value at $8.00, adjusted as often as every three minutes based on the detect ed traffic density. The objective of the HOT lane operation is to mainta in free flow conditions at all times (speeds greater than 50 mph). The MnPASS project intr oduced dynamic tolling on multiple consecutive roadway segments. Separate pricing to segments is necessary for managing the demand to this corridor. To maintain freeflow travel speed fo r all vehicles, the algorithm that adjusts the toll rates dynamically is based on the total number of vehi cles per mile in the lane, the rate of change of traffic conditions and on the speed of the vehicl es. When a change in the density is detected by the roadway sensors, the to ll is adjusted upward or downwar d, determined from a lookup table (Halvorson et al., 2006). The toll rates are ca lculated at each entry point according to the PAGE 32 32 maximum traffic density downstream of each entry point. The rate calculation interval is adjusted so that traffic conditions that change rapidly to be measured. High differences between the toll rates in a single calcula tion interval are avoided. The to ll rates for one hour of one day on I394 are presented in Figure 23. 0 0.5 1 1.5 2 2.5 3 3.57:09 7:12 7:15 7:18 7:21 7:24 7:27 7:30 7:33 7:36 7:39 7:42 7:45 7:48 7:51 7:54 7:57 8:00 8:03 8:06 8:09Time of dayToll rate ($) Figure 23. I394 toll rates on Ma y 18, 2005 (source: www.MnPASS.net) A comprehensive evaluation plan is being implemented to thoroughly assess conditions and public attitudes before and during the project operations (FHWA, 2006a). Preliminary performance data of this new facility were obtained for the first six months: the average number of toll trips per week is around 15,918 and the average re venue per week is around $12,484 (FHWA, 2006a). 2.4.2.2 I95 HOT lane project in South Florida I95 is one of the main highways on the East Coast of the United States serving some of the largest urban areas in th e United States. In summer 2008, Express Toll lanes opened to PAGE 33 33 reduce congestion on I95 from I395 in downtown Miami to Br oward Boulevard in Fort Lauderdale. The modeling approach to determin e the tolls depend on the existing demand which is obtained by data collection (t raffic counts and on the amount of money that people are willing to pay to use the tolling facility. More precisely, the procedure for the dynamic pricing on I95 is as follows. Firstly, real time traffic data are collected and the level of HOT lane operation is examined. Next, if the HOT lanes operate over the capacity threshold the toll rate will be increased by a preset margin of 25 cents. After the toll change, if the coming volumes decrease the toll remains the same until the new traffic data obtained. However, if the coming volumes increase or remain the same, the toll rate will be increased by the specific preset margin. In case of special event or an incident, the incident ma nagement manual override will be considered. If the HOT lanes operate with very low volumes a nd the minimum toll is in effect, no changes are required but if the toll is higher than the mini mum the toll rate will be reduced by the preset margin and then according to the change in volum es after the toll reduction, the toll remains the same or changes again. The range of toll rates on I95 is going to be from $0.25 to $10.00 according to operational conditions ( normal or single lane operation). 2.4.3 SelfLearning Control Approaches for Dynamic Tolling Yin and Lou (2006) proposed two practical ap proaches for the determination of pricing strategies for operating managed lanes. They cons idered dynamic tolls that change according to traffic conditions in order to maximize the throug hput of a freeway and keep superior free flow conditions to travelers. The first approach is a feedbackcontrol approach where one loop detector station located downstream is required to detect the tr affic condition along the facilitys segment. The second approach is a reactive self learning approach that calibrates the willingness to pay using revealed preference information and then determines the optimal toll rate using the approaching flow rates, the estimated travel time and the calibrated willingness to pay. For the PAGE 34 34 implementation of this approach, two loop detectors are required, one before the toll entrance to detect approaching flows and one after the entr ance to detect the flows on both the managed and the regular lanes. For the two approaches valid ation, simulation experiments were conducted. The conclusions of this research are that al though both approaches ar e simple and easy to implement, the results of the two approaches ma y lead to wildly varying toll pattern than can cause unstable traffic conditions. Moreover, the toll price is determined for each entry point without considering other entries, which may create inequality among motorists that enter to the managed lane from different points. Lou et al (2007) further expa nded the approach proposed by representing traffic dynamics more realistically and with an explicit formula tion to optimize tolls. The impacts or the lanechanging before the entrance of the toll lanes on the freeway throughput a nd the travel time are considered using the multilane hybrid traffi c flow model that was proposed by Laval and Daganzo (2006). The optimal tolls are determined for specific time intervals solving a nonlinear optimization model in order to maximize the freeway s throughput and to ensu re that the critical density of HOT lanes will not be exceeded from the density of the corridor. Lou et al (2007), also, examined further the conversion of HOV lanes to HOT lanes and presented some extensions to the approach that they proposed co nsidering more realistic cell representations for differences to HOT lane slip ramp configuratio ns. For the validation and demonstration of the approach, simulation experiments were c onducted using data fr om loop detectors. 2.5 Conclusions That Can Be Draw n from the Literature Review The studies presented in this li terature review are some of a large number of studies that have already been conducted for applying dynamic pricing to congested road s. The high interest for dynamic tolling all over the world as a c ongestion mitigation policy and its implementation PAGE 35 35 on many networks show that dynamic pricing is a viable mean of combating traffic congestion and not just a topic for academic discussion. Although a large number of studies considering different aspects of dynamic tolling have been conducted, there are issues th at have not been solved yet a nd deficiencies to some of the existing studies. The literature does not offer a practical and sensible approach for determining dynamic toll rates for HOT lanes. Many of the studies (see, e.g., Morrison, 1986; Palma and Lindsey, 1997; Arnott et al., 1998; Liu and McDona ld, 1999) consider hypothetical and idealized situations in which analytical solutions can be derived. For example, the travel demand function or travel demand is usually assume d to be known. Therefore, there is need for the use of real data for the determination of more accu rate toll rates. In contrast, Yi n and Lou (2006) and Lou et al (2007) proposed two readilyimplementable appro aches for determining timevarying tolls in response to the traffic arrival. The first approach adjusts the toll rate ba sed on the concept of the feedback control, while the sec ond approach is a reactive selflea rning approach and it provides first the willingness to pay using previous pref erence information and th en it determines the optimal toll rate using the approaching flow rates. However, a more proactive approach will give better amount of tolls according to the incoming flow. In addition, there is not any robust model that takes into consideration diffe rent scenarios of demand for the determination of the toll rates. Moreover, none of the studies that have been conducted for dynamic tolling takes into consideration that the capacity of a facility is not a constant value but fluctuates over the time. PAGE 36 36 CHAPTER 3 RESEARCH APPROACH 3.1 Introduction This chapter describes the proposed methodol ogy for the research about dynamic tolling with realtime demand and willi ngnesstopay learning for HOT lanes. The research approach includes the determination of the willingness to pay of travelers using a recursive Kalman filter, the realtime demand learning based on Bayesian inference and the empiri cal investigation of demand using data for SR91 which is located at California. Furthermore, a scenariobased robust toll optimization w ith linear representation of celltransmission model, stochastic demand and stochastic capacity will be addressed. 3.2 Methodology Overview The present methodology provides a robust and proactive appro ach to determine toll rates for HOT lanes. The goal is to adjust the toll rates according to traffic demand and freeways prevailing traffic conditions in a smooth manner to maximize the throughput of the freeway and to maintain freeflow conditions. For the toll determination, historical data (arrival flows and toll rates) are used in order to predict the shortterm inflows and the motorists willingnesstopay. The flows and densities of toll and GP will be described by propagating the inflow according to the celltransmission model. According to flows, willingnesstopay and capacity, toll rates will be optimized (see Figure 31). In the proactive and robust to ll optimization approach, stoc hastic demand and stochastic capacity are considered. More precisely, dema nd scenarios are generated using the negative binomial distribution and capacity scenarios are generated using the Weibull distribution. Toll optimization will be done with respect to each sc enario. However, before taking into account the demand and capacity scenarios, we consider de terministic demand and capacity to simply PAGE 37 37 illustrate the methodology. In both cases, traffic dynamics will be modeled using the celltransmission model. Finally, simulation experi ments will be conducted to validate and clearly illustrate the proposed methodology. Willingness to pay learning Demand learning Stochastic Capacity determination Data (arrival flows, toll) TOLL Cell Transmission Model Optimization Figure 31. Toll dete rmination procedure The methodologies for the will ingnessto pay learning, the de mand learning, the stochastic capacity and the scenariobased robust toll optimi zation that will be used in this research are described below. 3.3 Modeling Preparation This section addresses three major component s of toll optimization including willingnesstopay learning, the demand learning and the stochastic capacity determination. PAGE 38 38 3.3.1 WillingnesstoPay Learning 3.3.1.1 Basic concept Willingnesstopay is the amount of money that individuals are willing to pay to obtain a good. It refers to the maximum monetary amount that a person would pay for a good. In the present research the good is the use of the HOT lanes, more precisely, travel time savings. Motorists willingness to pay can be graduall y learned by mining data from loop detectors. The gained knowledge can be appl ied afterwards to determine optimal tolls to maximize the freeways throughput rate while ma intaining freeflow conditions. More precisely, it is assumed that the number of the travelers who are willing to pay in orde r to use the HOT lane can be formulated as an aggregate Logit model with unk nown parameters if a specific toll is given. However, the Logit models parameters can be estimated recursively using revealedpreference information that includes the flow rates before and after the lane choice which can be measured directly from the loop detector s and the travel times on the re gular purpose and the toll lanes which can be estimated or directly measured by installing additional tolltag readers along the freeway. The optimal tolls can be determined explicitly to achieve the operating objectives using the estimated Logit model. The present approach decomposes the pricing st rategies into two consecutive steps. First, the motorists willingness to pay learning using the previous revealedp reference information and second, the determination of the optimal to ll rate based on the approaching flows (here different demand scenarios will be developed), the calibrated willingness to pay and the estimated travel times. For the implementation of this approach two set of detectors are required as it shown at figure 32. The first set of detectors is installed before the tolltag reader for detecting the approaching traffic flows and the second set is in stalled after the reader for dete cting the flows PAGE 39 39 on the toll and the general purpose lanes. Without loss of general ity, we assume that there are one HOT lane, one general purpose lane and a recurrent bottleneck downstream (Figure 32). Furthermore, there are only one entrance and one exit to the toll lane and no on/off ramps between. In figure 32, t is the total approaching flow during time interval, Rt is the flow on regular lane during time interval t and Rt is the flow on regular lane during time interval t. Figure 32. Syst em configuration 3.3.1.2 Calibration of willingnesstopay To formulate the motorists decision on wh ether to choose the HOT lane or not, an aggregate Logit model whose parameters are unknow n is adopted as in L ou et al. (2007) and Yin and Lou (2006), given a specific value of toll t that changes over the time. Making the assumptions that motorists are homogeneous and they have the same willingness to pay, the relationships between the flow rates on HOT and regular lanes and the approaching flow rates are presented below: TRttt (31) 121 1expT TRtt ctctt (32) Where, t = total approaching flow at time interval t PAGE 40 40 Tt = flow on HOT lane at time interval t Rt = flow on regular lane at time interval t Tct = (average) travel time on the HOT lane at time interval t Rct = (average) travel time on the regular lane at time interval t t = toll at time interval t 1 = marginal effect of travel time on motorists utility 2 = marginal effect of toll on motorists utility = parameter that encapsulates other factors that affect motorists willingness to pay The ratio 1 2 represents the motorists value of time, in other words, their tradeoff between time savings an d tolls. The variables t Tt and Rt can be obtained directly from loop detectors. The variables Tct and Rct can be estimated using traffic flow models or directly measured and the variable t is set by the operator. It mu st be mention that for the approximation of the lane choices probabilities, the volume splits Tt t and Rt t are used. A discrete Kalman filtering or a recursive leastsquares technique can be used to estimate the constant parameters 1 2 and in realtime operation. The equation (2) which is actually the demand function of the HOT lane can be written as follows: 12ln1TR Tt ctctt t (33) Let ln1Tt yt t 1 2t x tt t and 1TRHtctctt the system/ observation equation (33 ) can be written as below: yytHtxte (34) PAGE 41 41 Where, yt: the observed system output ye: a random measurement error with zero mean and a known variance of y The value of y can vary according to the type of sensor. If we apply the Kalman filtering technique to estimate the parameters, we obtain: 11 1 22 2 1 1l n11 1 1 11TR T TRTR TRytt t t ttGtctcttt t tt t ctct ctct GtPttctcttPtt 111TRPtIGtctcttPt (35) Where, ^: indicate the variab les which are estimates P: the expected covariance matr ix of the estimation errors G: the Kalman gain Equation (35) can update the estimates of 1 2 and in real time each time that a new information is obtained with an initialization of 10, 20, 0 and 0P. As time involves, accurate estima tes of the impact of 1 2 and are expected and the initialization will not have much impact on them. In this research, homogeneous motorists are assumed with same willingness to pay. However, this procedure can be applied and even if the motorists are heterogeneous with different willingness to pay. In this case, there will be distributions associated 1 2 and across the population. PAGE 42 42 3.3.2 Demand Learning This paragraph addresses the methodology of realtime demand learning based on Bayesian Inference. First of all, traffic arrival is assumed that follows a Poisson process whose average arrival rate is unknown, denoted as Therefore, during 0,t, the number of vehicles that arrived, N, follows the distribution below: !n tet PNnt n (36) Since is unknown, from historical data, we ma y estimate its prior distribution as gamma distribution with parameters ,k and the following density function: 1 ke f k (37) Where, 1 0 xkkexdx a gamma function When k is an integer, 1!kk. Note that 22/k and 2/a An integer k can be selected to simplify the computation. The arrival rate is assumed to follow a gamma distribution. A normal distribution has at least two disadvantages in this case. Firstly, the normal distribution can give negative valu es while the arrival rate can ne ver be negative and, secondly, as the pricing moves forward, doesnt have a closed dist ribution form and this is mathematically intractable. Moreover, with the appropriate choice of the gamma distributions parameters its shape can be like th e bell shape of normal distribution. The prior distribution of may be updated in real time dur ing operations, as demonstrated in Lin (2006). Assuming that during 0,t, i number of vehicles have been observed from loop PAGE 43 43 detectors. After the observation that i vehicles have passed, the re searcher has a better sense about the true arrival rate. If i is relatively large, is more likely to be large too. As a consequence, the posterior probability density function of can be updated by using the Bayes theorem: 1 1 0 0! !ki t ki Nti teet fPNti ki f eet fPNtid d ki (38) 1 ki attet ki From above, it can be detected that the posteri or distribution is another gamma distribution with parameters tki this time. With it, the number of vehicles that may arrive during time interval ttt can be estimated. It turns out that vehicles follow a negative binomial distribution shown below: 1 0!ki n t ttet et PNttNtnNti d nk i !ki nkin tt nkitttt (39) 1ki nkin tt n tttt (k is integer) The mean and the variance for the above binomial distribution are: bt t tt kiki t t tt (310) PAGE 44 44 2 22 bt ttt tt kiki t t tt (311) The result that the number of vehicles follo ws the negative binomial distribution seems logical because the future vehicles come one im mediately after another and not in a continuoustime setting. The negative binomial distribution is a discrete probability distribution. 3.3.3 Stochastic Capacity Determination The capacity of a freeway is not a constant va lue but varies according to traffic, roadway, control, and weather conditions. Th e concept of stochastic capacity is more realistic and gives better insight for the maximum flow that can pass from a point. In the present paragraph, the approach for stochastic capacity determina tion based on the methodology proposed by Brilon et al. (2005) is addressed. The capacity distribution function is difficult to be estimated because the capacity itself cannot be measured directly. The capacity di stribution function is defined as follows: cFqpcq (312) Where, cFq = capacity distribution function c = capacity q = traffic volume To find an estimate that will be appropriate for the whole capacity distribution function, more information about the mathematical type of the distribution function cFq must be known. Brilon and Zurlinden (2003) examined various distribution functions like Normal, Gamma and Weibull. A maximum likelihood technique was us ed for the estimation of the distribution functions parameters. The likelihood function used is: PAGE 45 45 1 11i in cici iLfqFq (313) Where, ci f q = statistical density function of capacity c ci F q = cumulative distribution function of capacity c n = number of intervals i = 1, if uncensored (the observed volume causes a breakdown) i = 0, elsewhere The parameters of the distribution function ar e calibrated by maximizing the likelihood function or its natural logarithm. Based on the value of the lik elihood function, it show ed that the capacity of a freeway section follows the Weibull distribu tion with a nearly constant shape parameter. The function of the Weibull di stribution is shown below: 10xFxeforx (314) Where, = shape parameter = scale parameter The random capacity is considered because it le ads to more robust toll optimization and more realistic roll rates. 3.4 Robust Toll optimization Robust Toll Optimization is an approach to op timize the amount of toll that motorists have to pay to use the HOT lanes. This approach is called robust because it has to deal with uncertainty. Some of the parameters are random variables and the problem data are described with possible realizations called as scenarios. PAGE 46 46 3.4.1 Deterministic Case To build a basis for robust toll optimization, we consider deterministic case first. In the deterministic case, the traffic demand of the pl anning horizon, the willin gnesstopay and the capacity are considered determinis tic and given. In this case the toll optimization is done by solving the maximization problem presented in paragraph 3.4.1.2. 3.4.1.1 Modeling traffic dynamics Modeling traffic dynamics is the key for the to ll optimization. In the present research, traffic dynamics will be modeled using the lin ear representation of celltransmission model (CTM) (Daganzo, 1994, 1995). The CTM is chosen because it covers all the fundamental flowdensity, speeddensity, speedflow relationships and has been valid ated by field data (Lin and Daganzo, 1994; Lin and Ahanotu, 1995). In additi on, because it is a dynamic formulation, time variant demands can be modeled easily. The evolution of traffic is examined over a oneway road with one access and one egress. Actually, the traffic is modeled for both the to ll and GP lanes. The tra ffic conditions are updated every few seconds. To model traffic, the road is divided to cells, i.e. homogeneous sections (Figure 33). Figure 33. Representation of the road with cells In Figure 34, the flowdensity diagram is shown. More precisely, there are two triangles in Figure 34, the big one is for the lanes upstream of the bottleneck and the small is for the lanes in PAGE 47 47 the downstream bottleneck. In the diagram, maxq and max bq are the maximum flows at the lanes upstream and in the bottleneck, respectively, while ok and bokare the optimum densities at the lanes upstream and in the bottleneck, respectively. The jam density (jk) is the same for all the lanes. q (vphpl) Figure 34. Flowdensity relationship In celltransmission model, the road is divide d into cells (Figure 33). The length of the cells is set equal to the distance traveled by a ty pical vehicle in freeflow conditions in one time interval. Therefore, under uncongested conditions, all the vehicles in a cell will travel to the next cell in one time interval so the syst em evolution obeys (Danganzo, 1994): 11iintnt (315) Where, int = number of vehicles in cell i at time interval t PAGE 48 48 The equation (315) holds for all flows, unless qu eues are developed. To incorporate queuing in the model, two constants are introduced: iNt which is the maximum number of vehicles that can be presented in cell i at time interval t and iQt which is the maximum number of vehicles that can flow into cell i at time interval t. Thus, the number of ve hicles that can flow from cell i into cell 1i at time interval t is the smallest of the three values: int, 1iQt and 11 iiW Ntnt V which is the empty space in cell i at time interval t, Wis the backward wave speed and Vis the freeflow speed. The celltran smission model is based on equation (316) (flow conservation): 11iiiintntytyt (316) Where 11 1 1min,,ii iiiW y tntQtNtnt V indicating that flows depend on the current conditions at time t The cell occupancies will be updated with each time interval. The boundary conditions are specified by means of the first (input) and the last (output) cell. The first cell (cell 0) has infinite size 0Nt and the inflow 0Qt is equal to the desired link input flow. The last cell has, also, infinite size 1iNt and a suitable, maybe timevarying, capacity. 3.4.1.2 Model formulation The formulation of the toll optimi zation problem is shown below: 11 (,) 00 1max min ,0Tt n TG TT nn i o p ti yn tj iyjyjktnt (317) s..t. 1 121 0,..., 1expT TG y tqt tT ctctt (318) PAGE 49 49 1/ /1 0,...,n TTTT ii ictlWNtnt tT (319) 1/ /1 0,...,n GGGG ii ictlWNtnt tT (320) 11()() 0,...,TGytytqt tT (321) 11 0,...,1,...,TTTT iiiintntytyt tTandin (322) 10,...,2,...,1TT iiytnt tTandin (323) () 0,...,1,...,TT iiytQt tTandin (324) 0,...,1,...,T TTT iii TW ytNtnttTandin V (325) 11 0,...,1,...,GGGG iiiintntytyt tTandin (326) 10,...,2,...,1GG iiytn tTandin (327) 0,...,1,...,GG iiytQt tTandin (328) 0,...,1,...,G GGG iii GW ytNtnttTandin V (329) max0 0,..., tt T (330) 0 0,...,1,...,T iyt tTandin (331) 0 0,...,1,...,G iyt tTandin (332) 0 0,...,1,...,T int tTandin (333) 0 0,...,1,...,G int tTandin (334) Where, i = refers to a cell i PAGE 50 50 T iyt= actual inflow to cell i at time interval ,1tt at HOT lane G iyt= actual inflow to cell i at time interval ,1tt at GP lane = penalty parameter T ioptkt = optimal (or critical) density of cell i at time interval t of the HOT lane qt = demand at time interval t Tct = (average) travel time on the HOT lane at time interval t Rct = (average) travel time on the GP lane at time interval t l = length of each cell T iNt = maximum number of vehicles that can be presented in cell i at time interval t at HOT lane G iNt = maximum number of vehicles that can be presented in cell i at time interval t at GP lane t = toll at time interval t max = maximum toll that can be set 1 = marginal effect of travel time on motorists utility 2 = marginal effect of toll on motorists utility = parameter that encapsulates other factors that affect motorists willingness to pay T int = number of vehicles in cell i at time interval t at HOT lane G int = number of vehicles in cell i at time interval t at GP lane T iQt = maximum number of vehicles that can flow into cell i at time interval t at HOT lane G iQt = maximum number of vehicles that can flow into cell i at time interval t at GP lane TW = backward wave speed at HOT lane PAGE 51 51 TV = freeflow speed at GP lane GW = backward wave speed at HOT lane GV = freeflow speed at GP lane The objective function (317) is to maximize the sum of the freeways throughput at the downstream bottleneck while ensuring that the density of cell i of the HOT lane does not exceed the critical density. Equations (318)(320) indicate how many ve hicles are going to choose the HOT lane prescribed by a Logit model. Constraint (321) assures that all the incoming flow goes either to HOT or GP lane. Constraint (322) a nd (326) are the flow conservation constraints for HOT and GP lane, respectively. Constraints (323) (325) and (327)(329) anticipate that the actual inflow at HOT and GP lane, respectively, will take up the maximum allowed. Actually, constraints (322)(329) model the traffic dyna mics prescribed by CTM. Constraint (330) doesnt allow toll rate to take values lower than zero and greater than a specific maximum value. Constraints (331)(334 ) ensure no negativity. The above formulation has some drawbacks. First of all, the problem is not linear so it may require more computational time. Although, in the deterministic case, this is not very important, after the development of demand and capacity scenar ios is very critical because this problem will be solved many times. Moreover, the planning horiz on and the time interval that toll changes must be the same. In this case traffic propagate s every some seconds while toll changes every some minutes. Tolls cannot be set every some seconds because this is not practical and the traffic modeling cannot be every some minutes because the results wont be accurat e. This issue can be solved by adding more constraints to the formulation ensuring that the toll will be constant every some minutes but this will increas e the computational complexity. PAGE 52 52 The reasons mentioned above lead to the need of a two step formulation. In the first step, the flows to each lane (general purpose or toll ) will be determined given a demand and in the second step, the toll for each time interval will be computed. The two step formulation is as follows: Stage 1: 11 (,) 00 1max min ,0Tt n TG TT nn i o p ti yn tj iyjyjktnt (317) s.t. 11()() 0,...,TGytytqt tT (321) 11 0,...,1,...,TTTT iiiintntytyt tTandin (322) 10,...,2,...,1TT iiytnt tTandin (323) () 0,...,1,...,TT iiytQt tTandin (324) 0,...,1,...,T TTT iii TW ytNtnttTandin V (325) 11 0,...,1,...,GGGG iiiintntytyt tTandin (326) 10,...,2,...,1GG iiytn tTandin (327) 0,...,1,...,GG iiytQt tTandin (328) 0,...,1,...,G GGG iii GW ytNtnttTandin V (329) min1 max()()() 0,...,Tqtpytqtp tT (335) 0 0,...,1,...,T iyt tTandin (331) 0 0,...,1,...,G iyt tTandin (332) PAGE 53 53 0 0,...,1,...,T int tTandin (333) 0 0,...,1,...,G int tTandin (334) Stage 2: 2 1 1 121 min() 1expT T TG tqt yt ctct (336) s.t. max 00,..., tT (330) Where, 1 / /1 0,...,n TTTT ii ictlWNtnt tT (319) 1 / /1 0,...,n GG G G ii ictlWNtnt tT (320) In the above formulation, min p and max p are the minimum and maximum probability, respectively, of the incoming vehicles to choose the HOT lane. The objective function of the stag e 1, (317), as mentioned be fore, is to maximize the sum of the freeways throughput at the downstream bottleneck while ensuring that th e density of cell i of the HOT lane does not exceed the critical de nsity. The objective function of stage 2, (336), is to minimize the difference between the actual in flow at HOT lane and the predicted inflow at the same lane. Constraint (335) ensures that th e vehicles choose to travel to HOT lane is a percentage of the entire incoming flow. 3.4.2 Scenariobased optimization Scenariobased optimization, as mentioned above, is an approach to optimize the tolls considering different scenarios for some parameters. The purpose is to determine a toll that PAGE 54 54 performs reasonably well across all the scenarios. In other words, the approach to determine the toll rates is pro active and robust. In this part, different scenarios for the approaching flows (demand) and the available capacity will be generated, using the Weibull distribution for capacity and the updated negative binomial distribution for demand. Each scenario will be associated with positive probability of occurrence and it will specify how many vehicl es may come during th e planning horizon and what the available capacity of the downstream bo ttleneck is. Due to the fact that designing for the worst case may be very conservative, a to ll against highconsequence scenarios is determined. The performance measure for examin ing the toll is the conditional valueatrisk, mean shortfall, mean excess loss or tail valueat risk. This performance measure allows handling large number of scenarios and offers computati onal efficiency (Rockafellar and Uryasev, 2002). Figure 34 illustrates the probability density function and the mass function of a loss. 100% Conditional Valueatrisk 0% Loss Probability Probability density function of the loss Probability mass function of the loss Area = 1Figure 35. A loss function (Yin, 2007) PAGE 55 55 Each scenario, as mentioned above, indicates the number of vehicles that may arrive on each time interval and the available capacity at downstream bottleneck at each time interval. We generate a set of scenarios, S, and each scenario is indicated by the superscript s. Therefore, the robust toll optimization model is formulated as follows: Stage 1: 11 (,) 00 1max min ,0Tt n TsGs TT n n iopti yn tj iyjyjktnt (337) s.t. 11()() 0,...,TsGssytytqt tT (338) () 0,...,1,...,TT s iiytQt tTandin (339) 0,...,1,...,Gs Gs iiytQt tTandin (340) min1 max()()() 0,...,sTsqtpytqtp tT (341) (321) (329), (331) (335) Stage 2: ,1 minmax,0 1s s sS sSZpL a (342) s.t. max 00,..., tT (330) Where, 2 1 1 121 () 1expT Ss Ts Ts Gs tLqt yt ctct (343) s = each scenario = specified confidence level PAGE 56 56 s p = probability of occurrence for scenario s sqt = demand for scenario s 1sT y t = demand at toll lane at time interval t with respect to each scenario s ()Ts iQt = capacity at cell i at time interval t at HOT lane Gs iQt = capacity at cell i at time interval t at GP lane The scenariobased robust toll optimizati on is another two step model like in the deterministic case. At stage 1, the objective function, (337), is to maximize the freeways throughput considering the scenarios by decidi ng the demand split between toll and regular lanes. The constraints are the same as in the stage 1 of the deterministic case [constraints (321) (329), (331)(335)] but will be respect to each scenario. The problem of stage 2 is to minimize the conditional valueatris k by deciding a robust toll. PAGE 57 57 CHAPTER 4 DEMAND LEARNING RESULTS 4.1 Introduction This chapter implements the realtime de mand learning based on empirical data from California SR91. The methodology proposed in the previous chapter will be applied and the results will be presented. Last, the tasks which are to be achieved to accomplish the objective are discussed. 4.2 Empirical Analysis The approach proposed in chapter 3 for the de mand learning has been exercised using flow data from the SR91 (Figure 41). State Route 91 is a major eastwest freeway located entirely within Southern California and serving several regions of the Los Angeles metropolitan area. Specifically, it runs from Vermont Avenue in Gardena, just west of the junction with the Harbor Freeway (Interstate 110), east to Ri verside at the junction with the Pomora (State Route 60 west of 91), Moreno Valley (State Route 60 east of the 91), and Esc ondido (Interstate 215). Figure 41. SR91 PAGE 58 58 The data of total flow at SR91 have been collected every five minutes during the morning period from 5:00a.m. to 10:00a.m. in September 2001. For the analysis, the data of five days are used. The flows of the first four days are used in order to calculate the parameters of the gamma distribution k and a. The flows of the fifth day are pred icted using the approach presented above. Then the results of th e presented approach are compared with the actual data. The analysis conducted in two ways. First, the data of the entire period between 5:00 a.m. to 10:00 a.m. are used to predict the flows of the fifth day. Then the analysis is conducted only with the data of the peak hour. 4.2.1 Analysis Using the Data of the Entire Period The data collected are presented in tabl e 41 and the analysis in table 42. Table 41. Data from SR91 Total arrival flow (veh/5min) Total arrival flow (veh/5min) Total arrival flow (veh/5min) Total arrival flow (veh/5min) Time (a.m.) 9/12/2001 9/18/2001 9/19/2001 9/25/2001 5:00 8664 7968 8400 8613 5:05 8916 8616 8736 7908 5:10 8076 9384 8496 7740 5:15 8052 8352 8304 8013 5:20 7608 7920 8052 8076 5:25 7416 8352 8316 8316 5:30 7680 7968 8340 8184 5:35 7932 7956 7932 7812 5:40 8040 8064 8387 8172 5:45 7800 7680 7335 7980 5:50 7416 7512 6216 7680 5:55 7668 7968 5244 7572 6:00 7596 7245 5040 7752 6:05 7356 7932 6696 8016 6:10 7488 7356 7128 7740 6:15 7536 7668 6888 8664 6:20 7176 8088 6264 8196 6:25 7404 8064 6456 7884 6:30 7476 7968 6960 8244 6:35 7212 7824 6156 8136 PAGE 59 59 Table 41. Continued Time (a.m.) 9/12/2001 9/18/2001 9/19/2001 9/25/2001 6:40 8076 7920 6720 8892 6:45 7920 7896 6612 8208 6:50 7752 7548 6960 8364 6:55 7620 7668 7755 8400 7:00 7536 8080 7613 7740 7:05 7848 7980 8508 7848 7:10 7620 8136 8244 8175 7:15 7704 8160 8400 7980 7:20 7248 7680 7776 8076 7:25 7908 8292 6888 7596 7:30 7848 8100 7212 7653 7:35 7896 7944 7080 7812 7:40 7440 8496 7164 8088 7:45 7428 7920 6900 8124 7:50 7464 8388 7480 7488 7:55 7380 7860 8016 7548 8:00 7347 7973 8328 7560 8:05 7680 7644 7812 7272 8:10 7416 7440 7860 7464 8:15 7320 7872 8220 8052 8:20 7824 7404 7800 7236 8:25 7656 7608 8205 7404 8:30 7512 7824 7980 7956 8:35 7740 7704 7656 7227 8:40 7524 7704 7827 7644 8:45 7200 7464 7320 7932 8:50 7164 7884 7935 6876 8:55 8100 7200 8124 7344 9:00 7596 7980 8100 7613 9:05 7152 7368 7836 6192 9:10 7044 7733 7452 8100 9:15 7470 7752 7656 7788 9:20 7476 7476 8028 7284 9:25 7764 7680 7836 8052 9:30 7740 7560 7596 7212 9:35 7596 7764 7704 7140 9:40 7584 7716 7872 7524 9:45 7632 7260 7392 7368 9:50 7920 7416 7668 7476 9:55 7452 7320 7128 7092 10:00 7575 7188 7404 6732 Average 7634 7834 7531 7774 PAGE 60 60 From table 41, the average and the standard deviation of the average flows are calculated. 7693.5 136.80 The above parameters refer to 5 minutes period so the average and the standard deviation for 1 minute period are: 7693.5/51538.70 136.80/527.36 Therefore, 22/ k 22 2/3163 /2.0555 k a The number of vehicles that ma y arrive during time interval ttt can be estimated using the negative binomial distribution shown below (t =5 min): 316331631 2.05555 5 2.055552.05555inin t PNtNtnNti n tt The mean and the variance for the above binomial distribution are: 5 3163 2.0555bi t 252.05555 3163 2.0555bt i t The analysis and the estimated flows per 5 minutes are presented in table 42. PAGE 61 61 Table 42. Estimated flows Cumulative arrivals Actual Time Time t t Probability Mean (Estimated flows) Variance Standard deviation 0 8440 5:00 AM 0 5 7693.65 26407.62 162.50 16885 5:05 AM 5 5 8222.56 14049.54 118.53 25225 5:10 AM 10 5 8314.81 11763.35 108.46 32905 5:15 AM 15 5 8322.20 10761.93 103.74 40801 5:20 AM 20 5 8176.61 10030.25 100.15 49345 5:25 AM 25 5 8124.75 9626.25 98.11 57373 5:30 AM 30 5 8190.15 9467.64 97.30 64693 5:35 AM 35 5 8168.27 9270.43 96.28 72637 5:40 AM 40 5 8067.42 9026.55 95.01 80029 5:45 AM 45 5 8054.30 8910.13 94.39 87565 5:50 AM 50 5 7990.69 8758.20 93.59 94921 5:55 AM 55 5 7950.84 8647.60 92.99 103201 6:00 AM 60 5 7902.91 8539.67 92.41 110773 6:05 AM 65 5 7931.03 8522.41 92.32 118789 6:10 AM 70 5 7906.12 8454.73 91.95 126529 6:15 AM 75 5 7913.25 8426.72 91.80 134437 6:20 AM 80 5 7902.69 8384.24 91.57 142225 6:25 AM 85 5 7903.00 8356.90 91.42 150601 6:30 AM 90 5 7896.75 8325.66 91.25 158461 6:35 AM 95 5 7921.44 8329.53 91.27 166609 6:40 AM 100 5 7918.43 8306.38 91.14 174517 6:45 AM 105 5 7929.15 8299.48 91.10 182521 6:50 AM 110 5 7928.21 8281.97 91.01 190513 6:55 AM 115 5 7931.45 8270.23 90.94 198126 7:00 AM 120 5 7933.93 8258.94 90.88 206226 7:05 AM 125 5 7921.31 8233.04 90.74 213918 7:10 AM 130 5 7928.08 8228.25 90.71 221310 7:15 AM 135 5 7919.46 8208.38 90.60 229182 7:20 AM 140 5 7900.90 8178.99 90.44 236814 7:25 AM 145 5 7899.91 8168.52 90.38 244134 7:30 AM 150 5 7891.11 8150.59 90.28 252270 7:35 AM 155 5 7872.92 8123.56 90.13 259842 7:40 AM 160 5 7881.04 8124.20 90.13 267462 7:45 AM 165 5 7871.79 8107.39 90.04 275154 7:50 AM 170 5 7864.47 8093.02 89.96 283290 7:55 AM 175 5 7859.60 8081.56 89.90 291426 8:00 AM 180 5 7867.19 8083.26 89.91 299226 8:05 AM 185 5 7874.38 8084.86 89.92 307122 8:10 AM 190 5 7872.44 8077.39 89.87 314742 8:15 AM 195 5 7873.04 8072.81 89.85 322350 8:20 AM 200 5 7866.78 8061.45 89.79 PAGE 62 62 Table 42. Continued Cumulative arrivals Actual Time Time t t Probability Mean (Estimated flows) Variance Standard deviation 330030 8:25 AM 205 5 7860.53 8050.35 89.72 337458 8:30 AM 210 5 7856.27 8041.51 89.67 344766 8:35 AM 215 5 7846.41 8027.15 89.59 351942 8:40 AM 220 5 7834.28 8010.69 89.50 359574 8:45 AM 225 5 7819.79 7991.99 89.40 367164 8:50 AM 230 5 7815.74 7984.14 89.35 374856 8:55 AM 235 5 7810.98 7975.73 89.31 381924 9:00 AM 240 5 7808.52 7969.82 89.27 389364 9:05 AM 245 5 7793.54 7951.26 89.17 396408 9:10 AM 250 5 7786.52 7940.98 89.11 404400 9:15 AM 255 5 7772.08 7923.26 89.01 412164 9:20 AM 260 5 7776.28 7924.65 89.02 419919 9:25 AM 265 5 7776.05 7921.63 89.00 427407 9:30 AM 270 5 7775.66 7918.57 88.99 435171 9:35 AM 275 5 7770.47 7910.70 88.94 443511 9:40 AM 280 5 7770.35 7908.10 88.93 450651 9:45 AM 285 5 7780.28 7915.79 88.97 458631 9:50 AM 290 5 7769.31 7902.33 88.90 465903 9:55 AM 295 5 7772.86 7903.69 88.90 473439 10:00 AM 300 5 7764.57 7893.10 88.84 The actual and the predicted flows are illustrated in table 43. Table 43. Actual and predicted flows Actual Flow Probability Mean Standard deviation mean3* mean+3* 9/26/2001 8440 7694 162.50 7206 8181 8445 8223 118.53 7867 8578 8340 8315 108.46 7989 8640 7680 8322 103.74 8011 8633 7896 8177 100.15 7876 8477 8544 8125 98.11 7830 8419 8028 8190 97.30 7898 8482 7320 8168 96.28 7879 8457 7944 8067 95.01 7782 8352 7392 8054 94.39 7771 8337 7536 7991 93.59 7710 8271 7356 7951 92.99 7672 8230 8280 7903 92.41 7626 8180 7572 7931 92.32 7654 8208 8016 7906 91.95 7630 8182 7740 7913 91.80 7638 8189 PAGE 63 63 Table 43. Continued Actual Flow Probability Mean Standard deviation mean3* mean+3* 7908 7903 91.57 7628 8177 7788 7903 91.42 7629 8177 8376 7897 91.25 7623 8170 7860 7921 91.27 7648 8195 8148 7918 91.14 7645 8192 7908 7929 91.10 7656 8202 8004 7928 91.01 7655 8201 7992 7931 90.94 7659 8204 7613 7934 90.88 7661 8207 8100 7921 90.74 7649 8194 7692 7928 90.71 7656 8200 7392 7919 90.60 7648 8191 7872 7901 90.44 7630 8172 7632 7900 90.38 7629 8171 7320 7891 90.28 7620 8162 8136 7873 90.13 7603 8143 7572 7881 90.13 7611 8151 7620 7872 90.04 7602 8142 7692 7864 89.96 7595 8134 8136 7860 89.90 7590 8129 8136 7867 89.91 7597 8137 7800 7874 89.92 7605 8144 7896 7872 89.87 7603 8142 7620 7873 89.85 7603 8143 7608 7867 89.79 7597 8136 7680 7861 89.72 7591 8130 7428 7856 89.67 7587 8125 7308 7846 89.59 7578 8115 7176 7834 89.50 7566 8103 7632 7820 89.40 7552 8088 7590 7816 89.35 7548 8084 7692 7811 89.31 7543 8079 7068 7809 89.27 7541 8076 7440 7794 89.17 7526 8061 7044 7787 89.11 7519 8054 7992 7772 89.01 7505 8039 7764 7776 89.02 7509 8043 7755 7776 89.00 7509 8043 7488 7776 88.99 7509 8043 7764 7770 88.94 7504 8037 8340 7770 88.93 7504 8037 7140 7780 88.97 7513 8047 7980 7769 88.90 7503 8036 7272 7773 88.90 7506 8040 7536 7765 88.84 7498 8031 PAGE 64 64 The following figure (figure 42) shows the plot of the actual flows measured every five minutes and the plot of the predicted (probabi lity) mean after subtra cting three standard deviations and after adding three times the standa rd deviation of the probability. According to the Chebyshev's inequality, within 6 standard deviations from the mean, there are at least 97% of the values predicting from the distribution. From the plot, one can conclude that the majority of the flows that have been observed are within this area, so the use of this approach in order to determine the demand learning is reasonable. Figure 4.2. Actual flows and weaker bounds. 4.2.2 Analysis Using the Data of the Peak Hour. The data collected are presented in tabl e 43 and the analysis in table 44. PAGE 65 65 Table 44. Data from SR91 Total arrival flow (veh/5min) Total arrival flow (veh/5min) Total arrival flow (veh/5min) Total arrival flow (veh/5min) Time (a.m.) 9/12/2001 9/18/2001 9/19/2001 9/25/2001 5:00 8664 7968 8400 8613 5:05 8916 8616 8736 7908 5:10 8076 9384 8496 7740 5:15 8052 8352 8304 8013 5:20 7608 7920 8052 8076 5:25 7416 8352 8316 8316 5:30 7680 7968 8340 8184 5:35 7932 7956 7932 7812 5:40 8040 8064 8387 8172 5:45 7800 7680 7335 7980 5:50 7416 7512 6216 7680 5:55 7668 7968 5244 7572 6:00 7596 7245 5040 7752 Average 7913 8076 7600 7986 From table 44, the average and the standard deviation of the average flows are calculated. 7894 206.88 The above parameters refer to 5 minutes period so the average and the standard deviation for 1 minute period are: 7894/51578.80 206.88/541.37 Therefore, 22 2/1456 /0.9221k a PAGE 66 66 The number of vehicles that ma y arrive during time interval ttt can be estimated using the negative binomial distribution shown below (t =5 min): 145614561 0.92215 5 0.922150.92215inin t PNtNtnNti n tt The mean and the variance for the above binomial distribution are: 5 1456 0.9221bi t 250.92215 1456 0.9221bt i t The analysis and the estimated flows per 5 minutes are presented in table 44. Table 45. Estimated flows Cumulative arrivals Actual time Time t t Probability Mean (Estimated flows) Variance Standard deviation 0 8440 5:00 AM 0.00 5.00 7894.67 50700.88 225.17 16885 5:05 AM 5.00 5.00 8355.09 15409.20 124.13 25225 5:10 AM 10.00 5.00 8396.25 12239.93 110.63 32905 5:15 AM 15.00 5.00 8378.58 11009.70 104.93 40801 5:20 AM 20.00 5.00 8211.64 10174.06 100.87 49345 5:25 AM 25.00 5.00 8150.75 9722.91 98.60 57373 5:30 AM 30.00 5.00 8214.34 9542.57 97.69 64693 5:35 AM 35.00 5.00 8188.40 9328.15 96.58 72637 5:40 AM 40.00 5.00 8082.30 9069.82 95.24 80029 5:45 AM 45.00 5.00 8067.24 8945.60 94.58 87565 5:50 AM 50.00 5.00 8000.94 8786.55 93.74 94921 5:55 AM 55.00 5.00 7959.37 8671.02 93.12 103201 6:00 AM 60.00 5.00 7909.85 8559.03 92.52 The actual and the predicted flows are illustrated in table 45. PAGE 67 67 Table 46. Actual and predicted flows Actual Flows Probability Mean Standard deviation mean3* mean+3* 9/26/2001 8,440.00 7894.671 225.17 7219.17 8570.18 8,445.00 8355.086 124.13 7982.69 8727.49 8,340.00 8396.248 110.63 8064.34 8728.15 7,680.00 8378.584 104.93 8063.80 8693.37 7,896.00 8211.636 100.87 7909.04 8514.24 8,544.00 8150.754 98.60 7854.94 8446.57 8,028.00 8214.341 97.69 7921.28 8507.40 7,320.00 8188.404 96.58 7898.66 8478.15 7,944.00 8082.3 95.24 7796.59 8368.01 7,392.00 8067.241 94.58 7783.50 8350.99 7,536.00 8000.94 93.74 7719.73 8282.15 7,356.00 7959.37 93.12 7680.01 8238.72 8,280.00 7909.85 92.52 7632.31 8187.40 The following figure (figure 43) shows the plot of the actual flows measured every five minutes during the peak hour and the plot of the predicted (probability) mean after subtracting three standard deviations and after adding three times the standard deviat ion of the probability. According to the Chebyshev's inequality, within 6 standard deviations from the mean, there are at least 89% of the values predicting from the distribution. PAGE 68 68 Figure 43. Actual flows and weaker bounds during the peak hour. PAGE 69 69 CHAPTER 5 PROACTIVE AND ROBUST PRICING STRATEGIES 5.1 Introduction This chapter implements both the deterministic and the scenariobased toll optimization approach in order to validate and compare th em. The methodology, which provides a robust and proactive approach to determine toll rates for the HOT lanes, pr oposed in the third chapter is applied and the results are presented. 5.2 Model Solution and Simulation Study The toll optimization model is solved using GAMS 22.6 (GAMS Development Corporation, 2003) for pricing strategies for a freeway facility 3 miles long with a bottleneck downstream. The operation horizon is 90 minutes a nd other parameters assumed are as follows: Freeflow speed: 60 mph (or 88 ft/sec) Backward wave speed: 30 mph (or 44 ft/sec) Jam density: 120 vpm Saturation flow along the freeway: 2400 vphpl (or 3.33 vehicles/ 5sec) Saturation flow at the bottleneck: 1800 vphpl (or 2.50 vehicles/ 5sec) Toll values can vary from $0.00 to $8.00. The propagation of traffic flow is based on th e fundamental flowdensity curves that are illustrated in Figure 51. As previously discussed in the paragraph 3.4.1.2, toll optimization consists of two stages. In the first stage, given traffic demand, the optimal volume split between HOT and GP lanes is computed by maximizing freeways throughput us ing the cell transmission model for the flow propagation. In this stage, the optimization ho rizon is 10 minutes while traffic propagation is done with a time step of 5 seconds. In the sec ond stage, the optimal toll is determined for matching the optimal volume. In this stage, a va lue of toll is determined every 3 minutes which is a reasonable amount of time that the toll value can change in practice. PAGE 70 70 q (vphpl) Figure 51. FlowDensity Curves Both the deterministic and scenariobased toll optimization models are solved. In the former, there is only one demand scenario wh ile 20 demand scenarios are generated for the latter. After the toll optimization, simulation ex periments are conducted in order to demonstrate and validate the proposed robust to ll optimization. A selfdeveloped simulation platform is used, which consists of a monitor, a controller and a simulator. The monitor is collecting information at each time interval including th e total arrival flow, the flows at GP and HOT lanes, th e densities, and the travel time s. The controller solves toll optimization models using the information of th e densities and flows from the monitor. The simulator attempts to replicate motorists behavior with respect to which lane they will choose to travel. At each time interval, based on the comi ng flow, the instantaneous travel times provided from the monitor and the toll value obtained from the controller, the Logit model is applied by the simulator to compute how ma ny motorists are going to choose the HOT lane versus the GP lane. PAGE 71 71 The simulation site is a 3 mile freeway segm ent as mentioned above. This segment has two lanes, one HOT lane and one GP lane that each of them obeys the big triangular flowdensity curve shown in Figure 51. At the end of the se gment, it is assumed that there is a bottleneck where the lanes obey the small triangular flowdensity curve in Figure 51. All the relevant parameters are given in the Figure 51. The simulation time step is 5 sec and both lane s are partitioned into small cells of 440 ft each. The values of 1 2 and used in the simulator are 0.5, 1, 0.2, respectively, as proposed by Lou et al (2007). Two experi ments are conducted, one with lowhighlow demand profile and the other lowmediumhighmediumlow demand profile. 5.3 Numerical Results The numerical results from the simulation expe riments are presented in Figures 52 525. 5.3.1 LowHighLow Demand Case In the lowhighlow demand case, traffi c demand varies from 1200 vphpl to 2100 vphpl. More precisely, traffic demand is uniformly distributed from 2400 vph to 3200 vph for the first 15 min, from 3200 vph to 4200 vph for the next 30 min and from 2400 vph to 3200 vph for the rest of the simulation horizon (Figure 52). 5.3.1.1 Robust toll optimization The Robust toll optimization is conducted by generating 20 di fferent demand scenarios. Each scenario is associated with a probability of 5%. Figure 53 presents the optimal toll rates determined by the controller and Figure 54 reports the resulting freeway, HOT and GP throughputs. PAGE 72 72 Traffic demand2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 4200 4400 123456789101112131415161718192021222324252627282930 Time interval (3min)Traffic Demand (vph) Lowerbound demand Average demand Upperbound demand Figure 52. Tra ffic demand profile. Optimal Toll Rates0.0 0.5 1.0 1.5 2.0 2.5 3.0 123456789101112131415161718192021222324252627282930Time Interval (3min)Toll Rate ($) Figure 53. Optimal Toll Rates. PAGE 73 73 Throughputs0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600 123456789101112131415161718192021222324252627282930 Time interval (3min)Throutput (vph) Freeway throughput HOT throughput GP throughput Figure 54. Freeway, HOT and GP throughputs. It can be seen that the contro ller is able to apply a toll va lue to maintain a high throughput of the freeway while toll value is changing in a smooth manner. For an incoming demand from 1200 vph to 2100 vph for each lane, tolls are taking va lues from $0.00 to $1.62. When there is no difference in travel time between the HOT lane and the GP lane, toll implementation is not necessary so the toll takes th e value of $0.00. The averag e throughput is 3084, 1442 and 1643 vph for the freeway, the HOT and the GP lane, re spectively. When the demand is high, between time intervals 6 and 15, the average of the freeways throughput is 3464 vph, HOT lanes throughput 1693 vph and GP lanes throughput is 1771 vph. This indicates that the freeway can be managed well with the proposed approach. Figure 55 shows the average densities al ong the HOT and GP lanes, and figure 56 presents the queue length upstream of the bottleneck at HOT and GP lane. PAGE 74 74 Average densities0 10 20 30 40 50 60 123456789101112131415161718192021222324252627282930 Time interval (3min)Average density (vpmpl) Average density along HOT Average density along the GP Critical density Figure 55. Average densiti es along HOT and GP lanes. The average density for the HOT lane is 23.29 vpmpl while for the GP lane is 33.47 vph which means that the HOT lane operates with hi gher speed for the vehicles. In addition, from figure 55, it can be observed that the average density of the GP lane is over the critical density for 9 time intervals or 27 min while the average density of the HOT lane never exceeds the critical density. This is important because if the density of one lane exceeds the critical value, it does not operate under free flow speed anymor e but with lower speed depending on the flowdensity curve (Figure 51). PAGE 75 75 Queue length upstream of the bottleneck0.0 0.5 1.0 1.5 2.0 2.5 3.0 123456789101112131415161718192021222324252627282930 Time Interval (3min)Overcritical density (miles) HOTQueue length GPQueue Length Figure 56. Queue length upstream of the bottleneck at HOT and GP lane. As the above Figure illustrates, the proposed approach achieves very well the desired operating conditions at the HOT lane. Although the demand is very high for half hour, only a small queue is developed at HOT lane upstream the bottleneck, due to the random demand. The maximum queue length is 0.3 miles. However, the queue in the GP lane may extend up to 2.7 miles because the tota l arrival is beyond the freeways capacity. 5.3.1.2 Deterministic to ll optimization The deterministic toll optimization is conducte d by assuming that the incoming flow will be 1400 vphpl for the first 15 minutes, 1850 vphpl for the next 30 minutes and then 1400 vphpl for the rest of the experiment horizon, whic h are the averages of random traffic demands assumed (Figure 52). PAGE 76 76 Figure 57 presents the optimal toll rates de termined by the controller and Figure 58 shows the freeway, HOT and GP throughputs. Optimal Toll Rates0.0 0.5 1.0 1.5 2.0 2.5 3.0 123456789101112131415161718192021222324252627282930Time Interval (3min)Toll Rate ($) Figure 57. Optimal Toll Rates. Throughputs0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600 123456789101112131415161718192021222324252627282930 Time interval (3min)Throutput (vph) Freeway throughput HOT throughput GP throughput Figure 58. Freeway, HOT and GP throughputs. PAGE 77 77 As demonstrated in Figure 57, the controller is able to maintain a high throughput of the freeway with a toll varying from $0.00 to $1.71. The average throughput is 3095, 1458 and 1637 vph for the freeway, the HOT and the GP lane, respectively (Figure 58). Figure 59 illustrates the average densities along the HOT and GP lanes. The average density for the GP lane is 32.84 vpmpl while fo r the HOT lane is 24.10 vph. However, GP has density over the critical for 24 min out of the 90 min experiment horizon, suggesting that it is congested and vehicles travel with less speed than the free flow speed. Average densities0 10 20 30 40 50 60 123456789101112131415161718192021222324252627282930 Time interval (3min)Average density (vpmpl) Average density along HOT Average density along the GP Critical density Figure 59. Average densiti es along HOT and GP lanes. Figure 510 shows the queues developed at HOT and GP lanes when the demand is over the capacity. In this case, the HOT lane operates really well comparatively to GP lane that has long queues because no managing strategy is applied to it. PAGE 78 78 Queue length upstream of the bottleneck0.0 0.5 1.0 1.5 2.0 2.5 3.0 123456789101112131415161718192021222324252627282930 Time Interval (3min)Overcritical density (miles) HOTQueue length GPQueue Length Figure 510. Queue length upstream of the bottleneck at HOT and GP lane. 5.3.1.3 Comparison of the robust versus th e deterministic toll optimization The results presented above show some perf ormance differences between the two proposed approaches. To better compare the results, diag rams that merge the results from the two approaches for the toll rates and the HOT throughput, density a nd queue length are provided as follows. Figure 511 compares the optimal toll rates. PAGE 79 79 Optimal Toll Rates0.0 0.5 1.0 1.5 2.0 2.5 3.0123456789101112131415161718192021222324252627282930 Time Interval (3min)Toll Rate ($) Tolls (Robust optimization) Tolls (Deterministic Optimization) Figure 511. Optimal Toll Rates. At robust optimization, the toll rates are more stable, change in smoother manner. They reach gradually a maximum value that is less than the maximum value under the deterministic case. In addition, the profit at the robus t case is $18,835 for the 90 min under analysis while it is $14,844 for the deterministic case. This means that at robust optimization the profit is larger. Figure 512 shows the HOT lanes throughput for the two approaches. During the 30 min that the coming demand is over the capacity, the r obust approach maintains a slightly more stable and higher HOT throughput than the deterministic one. This can be seen by computing the standard deviation of HOT throughput for the time intervals 6 to 15. The standard deviation for the robust approach is 175.35 while for the deterministic case is 178.41 vph. The average throughput is 1458 vph and. 1441 vph respectively. However, those differences are not statistically significant. PAGE 80 80 HOT Throughput0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600 123456789101112131415161718192021222324252627282930 Time interval (3min)Throutput (vph) HOT throughput (Robust Optimization) HOT throughput (Deterministic Optimization) Figure 512. HOT throughput. Figure 513 provides the density along the HOT lane. In this case the standard deviation for the robust approach is 2.51 vs. 3.95 for the deterministic approach. If the density does not fluctuate much, the performance of the HOT lane can be better. Average density along HOT lane0 10 20 30 40 50 60 123456789101112131415161718192021222324252627282930 Time interval (3min)Average density (vpmpl) Average density(Robust Optimization) Average density (Deterministic Optimization) Critical density Figure 513. Average density along HOT lane. PAGE 81 81 Figure 514 illustrates the queues developed at HOT lane. At the deterministic case, the queue is longer which indicates that the lane does not perform so well as in the robust case. Queue length upstream of the bottleneck0.0 0.5 1.0 1.5 2.0 2.5 3.0 123456789101112131415161718192021222324252627282930 Time Interval (3min)Overcritical density (miles) HOTQueue length (Robust Optimization) HOTQueue Length (Deterministic Optimization) Figure 514. Queue length upstream of the bottleneck at HOT lane. To summarize, the results of the tw o proposed approaches showed that: The robust approach leads to more smooth and stable toll pattern. The profit at the robust case is larger. There is not much difference at the throughput of the HOT lane. The average density is less than the critical density in both approaches but at the scenariobased case it varies less. The queue is longer at the deterministic case. Overall, the robust approach give s slightly better results but bo th approaches perform really well in preventing the HOT lane for being congest ed while keeping the th roughput in high levels. PAGE 82 82 5.3.2 LowMediumHighMediumLow demand case In the lowmediumhighmediumlow demand case, traffic demand is uniformly distributed from 1200 vph to 1500 vph for th e first 15min, from 1500 vph to 1800vph for the next 15 min, from 1800 vph to 2100 vph for the next 15 min, from 1500 vph to 1800vph for the following 15 min, and from 1200 vph to 1500 vph fo r the last 30 min of the simulation horizon (Figure 515). Traffic demand2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 4200 4400 123456789101112131415161718192021222324252627282930 Time interval (3min)Traffic Demand (vph) Lowerbound demand Average demand Upperbound demand Figure 515. Traffic demand profile. 5.3.2.1 Robust toll optimization Twenty different demand scenarios with a proba bility of 5% are generated for the robust approach. Figure 516 shows the optimal to ll rates determined by the controller. PAGE 83 83 Optimal Toll Rates0.0 0.5 1.0 1.5 2.0 2.5 3.0 123456789101112131415161718192021222324252627282930Time Interval (3min)Toll Rate ($) Figure 516. Optimal Toll Rates. Figure 517 presents the freew ay, HOT and GP throughputs. Throughputs0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600 123456789101112131415161718192021222324252627282930 Time interval (3min)Throutput (vph) Freeway throughput HOT throughput GP throughput Figure 517. Freeway, HOT and GP throughputs. PAGE 84 84 From the Figures 516 and 517, it can be observed that the controller is able to apply a toll value to maintain a high throughput of the freeway while toll values are changing smoothly. For a coming flow from 1200 vph to 2100 vph for each lane, the lowest toll value is $0.00 when the flow is less than the capacity for both lanes while the highes t value is $1.78 when the demand is very high. The average throughput after a pplying the toll rates is 3086, 1417 and 1668 vph for the freeway, the HOT and the GP lane, respectively. Figure 518 presents the average densities along the HOT and GP. Average densities0 10 20 30 40 50 60 123456789101112131415161718192021222324252627282930 Time interval (3min)Average density (vpmpl) Average density along HOT Average density along the GP Critical density Figure 518. Average densities along HOT and GP lanes. The average density for the HOT lane is 22.91 vpmpl while for the GP lane is 35.70 vph. Furthermore, Figure 518 illustrates that the averag e density of the GP lane is over the critical density for 33 min. At the same time, the average density of the HOT lane never exceeds the PAGE 85 85 critical density. This is important because when the average density of one lane exceeds the critical value means that this lane does not ope rate under free flow speed in more than the half time. Figure 519 shows the queue length upstream of the bottleneck at HOT and GP lane Queue length upstream of the bottleneck0.0 0.5 1.0 1.5 2.0 2.5 3.0 123456789101112131415161718192021222324252627282930 Time Interval (3min)Overcritical density (miles) HOTQueue length GPQueue Length Figure 519. Queue length upstream of the bottleneck at HOT and GP lane. The queue at HOT lane does not exceed th e 0.32 miles upstream the bottleneck although the total demand is beyond the total capacity for 30 min. On the other hand, the queue in the GP lane reached 2.83 miles. 5.3.2.2 Deterministic to ll optimization The deterministic toll optimizati on in this case is conducted by assuming that the traffic demand will be 1350 vphpl for the first 15 min, 1650 vphpl for the next 15 min, 1850vphpl for the following 15 min, 1650 vphpl for the other 15 min and 1350 vphpl for the last 30 min for the PAGE 86 86 rest of the experiment horizon, with are the averages of th e random traffic demands assumed (Figure 515). Figure 520 reports the optimal toll values give n by the controller and Figure 521 presents the freeway, HOT and GP throughputs. Optimal Toll Rates0.0 0.5 1.0 1.5 2.0 2.5 3.0 123456789101112131415161718192021222324252627282930Time Interval (3min)Toll Rate ($) Figure 520. Optimal Toll Rates. PAGE 87 87 Throughputs0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600 123456789101112131415161718192021222324252627282930 Time interval (3min)Throutput (vph) Freeway throughput HOT throughput GP throughput Figure 521. Freeway, HOT and GP throughputs. The controller applies a toll value to maintain a high thr oughput of the freeway while keeping HOT lane uncongested. Toll values, in th is case, are taking values from $0.00 to $1.79 and the average throughput is 3091, 1420 and 1671 vph for the freeway, the HOT and the GP lane, respectively. Figures 522 provides the aver age densities along the HOT and GP lanes. The average density for the GP lane is 36.78 vpmpl while for the HOT lane is 23.28 vpmpl. However, GP has average density over the critical density for 12 tim e intervals or in other words for 36 min out of the 90 min experimental horizon. PAGE 88 88 Average densities0 10 20 30 40 50 60 123456789101112131415161718192021222324252627282930 Time interval (3min)Average density (vpmpl) Average density along HOT Average density along the GP Critical density Figure 522. Average densities along HOT and GP lanes. Figure 523 illustrates the extent of the HOT and GP lane queues upstream of the bottleneck. As in the robust optimization, HOT la ne has a short queue and GP experiences long queues. This shows that the HOT lane operates under good traffic conditions. PAGE 89 89 Queue length upstream of the bottleneck0.0 0.5 1.0 1.5 2.0 2.5 3.0 123456789101112131415161718192021222324252627282930 Time Interval (3min)Overcritical density (miles) HOTQueue length GPQueue Length Figure 523. Queue length upstream of the bottleneck at HOT and GP lane. 5.3.2.3 Comparison of the robust versus th e deterministic toll optimization The results illustrated above show some performance differences between the two proposed approaches. To better compare the resu lts, diagrams that merge the results from the approaches for the toll rates and the HOT throughput, density a nd queue length upstream of the bottleneck are given as follows. Figure 524 presents the optimal to ll rates for the two approaches. PAGE 90 90 Optimal Toll Rates0.0 0.5 1.0 1.5 2.0 2.5 3.0123456789101112131415161718192021222324252627282930Time Interval (3min)Toll Rate ($) Tolls (Robust optimization) Tolls (Deterministic Optimization) Figure 524. Optimal Toll Rates. At robust optimization, the toll rates increase progressively and do not fluctuate much as at the deterministic case. However, the profit in this case is $22,493 whic h is less than $25,605 which is the profit at the deterministic case. Figure 525 provides the HOT lanes throughput for the two approaches. PAGE 91 91 HOT Throughput0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300 3600 123456789101112131415161718192021222324252627282930 Time interval (3min)Throutput (vph) HOT throughput (Robust Optimization) HOT throughput (Deterministic Optimization) Figure 525. HOT throughput. In this case, the average throughput as well as the standard deviation of the HOT lane is almost the same. Figure 526 shows the density along the HOT lane In both approaches the average density is less than the critical and there is not grea t difference on how the average density changes. PAGE 92 92 Average density along HOT lane0 10 20 30 40 50 60 123456789101112131415161718192021222324252627282930 Time interval (3min)Average density (vpmpl) Average density(Robust Optimization) Average density (Deterministic Optimization) Critical density Figure 526. Average density along HOT lane. Figure 527 illustrates the queues developed at HOT lane. At the deterministic case, the queue is present more time than in the robust case but the average is 0. 31 miles while at the robust case is 0.26 miles which is slightly worse. PAGE 93 93 Queue length upstream of the bottleneck0.0 0.5 1.0 1.5 2.0 2.5 3.0 123456789101112131415161718192021222324252627282930 Time Interval (3min)Overcritical density (miles) HOTQueue length (Robust Optimization) HOTQueue Length (Deterministic Optimization) Figure 527. Queue length upstream of the bottleneck at HOT lane. From the above diagrams, one can conclude that: The tolls fluctuate more in the deterministic case. The profit is more in the deterministic case. The throughput of the HOT lane is almost the same in both cases. The average density varies, also, almost in the same manner in both approaches. The queue lasts longer and it is slightly bigger at the deterministic case. All in all, in the lowmediumhighmediumlow case, the two approaches yield similar performances. Both the approaches perform pretty well in meeting the assumptions made for the operation of the HOT lane and the freeway. PAGE 94 94 5.4 Conclusions In this Chapter the results of the two proposed approaches pr esented and compared in their effectiveness of providing free flow conditions at the HOT lane while utilizing the freeways capacity. It can be concluded that both approaches ar e effective and can be used to manage a freeway segment. However, the scenariobased or robust approach produces a smoother toll pattern and generally be tter performance of the HOT lane. Al so, it is observed that the robust approach respond more adaptively to a sudde n demand surge. On the other hand, the deterministic approach requires much less com putational time and yields fairly good results. 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The 86th Annual Meeting of the Transportation Research Boar d, Compendium of Pa pers CDROM, No 071806, January 21 25, 2007. Yperman, I., Logghe, S., and Immers, B., 2005. Dynamic congestion pricing in a network with queue spillover. In Proc. 12th World Congress on Intelligent Transportation Systems, San Francisco. Ziliaskopoulos, T. 2000 A linear programming model for the single destinat ion system optimum dynamic traffic assignment problem. Transportation Science, 34, 112. PAGE 99 99 BIOGRAPHICAL SKETCH Di m itra Michalaka was born in Lesv os, Greece, in 1984. In 2001, after she passed the national general exams, she enrolled as a student at the National Technical University of Athens, where she received the Bachelor of Science in ivil ngineeri ng in 2006. In 2009 she earned the Master of Science in ivil ngineering from Univ ersity of Florida. During her graduate studies, she was a research assistant for her advisor, Dr. Yafeng Yin. 