UFDC Home  myUFDC Home  Help 



Full Text  
PAGE 1 1 SIGNAL TIMING OPTIMIZATION FOR RELI ABLE AND SUSTAINABLE MOBILITY By LIHUI ZHANG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 PAGE 2 2 2010 Lihui Zhang PAGE 3 3 To my family PAGE 4 4 ACKNOWLEDGMENTS First and foremost, I would like to express my gratitude to my superv isor, Dr. Yafeng Yin, for his guidance through my PhD study. Wit hout his support and encouragement, this dissertation would never have been finished. I am grateful to the transportation research center for providing me multiple resources for transportation study, including the wonderful cour ses, seminars, distinguished lectures and student conferences, which will be of great help to my future development, both academically and professionally. I would like to thank Drs. Siriphong Lawphongpa nich, Lily Elefteriadou, Scott Washburn and Sivaramakrishnan Srinivasan for serving as my committee members, and for their valuable comments and suggestions on this dissertation. I would also like to thank my colleagues in the transportation research center, especially Yingyan Lou, Ziqi Song, Di Wu for the discussi ons on a broad range of topics, and for their feedbacks to my research work. Finally, I want to thank my family for thei r love, and their support on any of the decisions I made. Special thanks go to my wife, Xiyuan Li for her company in this foreign country, and for the happiness and joy she brings to my life. PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES.........................................................................................................................8 ABSTRACT...................................................................................................................................10 CHAP TER 1 INTRODUCTION..................................................................................................................13 1.1 Background.......................................................................................................................13 1.2 Dissertation Outline....................................................................................................... ...15 2 LITERATURE REVIEW.......................................................................................................16 2.1 Traffic Signal Operation...................................................................................................16 2.1.1 FixedTime Control................................................................................................16 2.1.2 Actuated Control....................................................................................................16 2.1.3 Adaptive Control....................................................................................................18 2.2 Signal Timing Optimization............................................................................................. 18 2.2.1 Isolated Intersection................................................................................................18 2.2.2 Arterial....................................................................................................................23 2.2.3 Grid Network..........................................................................................................25 2.3 Summary...........................................................................................................................27 3 A STOCHASTIC PROGRAMMING APPROA CH FOR ROBUST SIGNAL TIMING OPTIMIZATION ....................................................................................................................29 3.1 Uncertainties in Traffic Modeling.................................................................................... 29 3.1.1 Supply Uncertainty................................................................................................. 29 3.1.2 Demand Uncertainty...............................................................................................30 3.2 Optimization under Uncertainty....................................................................................... 30 3.2.1 Simulation Optimization........................................................................................ 30 3.2.2 Stochastic Programming.........................................................................................31 3.2.3 Robust Optimization............................................................................................... 32 3.3 ScenarioBased Stochastic Programmi ng Approach for Signal Optimization................. 32 4 SIGNAL TIMING OPTIMIZATI ON UNDER DAYTODAY DEMAND VARIATIONS........................................................................................................................40 4.1 CellTransmission Model................................................................................................. 40 4.1.1 Model Introduction................................................................................................. 40 PAGE 6 6 4.1.2 Encapsulating CTM in Signal Timing Optimization.............................................. 42 4.2 Enhanced Deterministic Signal Optimization Model....................................................... 43 4.2.1 Objective Function.................................................................................................43 4.2.2 Constraints..............................................................................................................44 4.2.3 Model Formulation................................................................................................. 49 4.3 Stochastic Signal Optimization Model............................................................................. 50 4.4 Numerical Examples......................................................................................................... 50 4.4.1 SimulationBased Genetic Algorithm.................................................................... 50 4.4.2 Numerical Example I.............................................................................................. 54 4.4.3 Numerical Example II............................................................................................ 55 4.5 Summary...........................................................................................................................57 5 SIGNAL TIMING OPTIMIZATION WITH E NVIRONMENAL CONCERNS.................. 79 5.1 Emission and Roadside Pollutant Concentrations............................................................ 80 5.1.1 Emission Model...................................................................................................... 80 5.1.2 Pollution Dispersion Model....................................................................................82 5.1.3 Mean Excess Exposure........................................................................................... 83 5.2 Model Formulation and Solution Algorithm.................................................................... 86 5.2.1 BiObjective Optimization Model.......................................................................... 86 5.2.2 SimulationBased BiObj ective Genetic Algorithm............................................... 87 5.3 Numerical Example.......................................................................................................... 89 5.4 Summary...........................................................................................................................91 6 ROBUST SYNCHRONIZATION OF ACT UATED SIGNALS ON ARTERIALS ........... 101 6.1 Bandwidth Maximization for Arterial Signal Coordination........................................... 102 6.2 ScenarioBased Approach for Robust Synchronization of Actuated Signals................. 104 6.3 Numerical Example........................................................................................................ 107 6.3.1 Plan Generation....................................................................................................107 6.3.2 Plan Evaluation.....................................................................................................108 6.4 Summary.........................................................................................................................110 7 CONCLUSIONS AND FUTURE WORK ........................................................................... 123 7.1 Conclusions................................................................................................................123 7.2 Future Work...............................................................................................................124 LIST OF REFERENCES.............................................................................................................127 BIOGRAPHICAL SKETCH.......................................................................................................135 PAGE 7 7 LIST OF TABLES Table page 31 Delay associated with 15 scenarios.................................................................................... 39 41 CPF result under different 01 combinations..................................................................... 72 42 Traffic data for threenode network................................................................................... 73 43 Signal plans for threenode network..................................................................................74 44 CORSIM result for threenode network............................................................................ 75 45 Traffic data for El Camino Real arterial............................................................................ 76 46 Signal plans for El Camino Real arterial........................................................................... 77 47 CORSIM result for El Camino Real arterial...................................................................... 78 51 Emission factors for di fferent driving modes.................................................................... 98 52 30year wind probability data............................................................................................ 99 53 Pareto optimal signal plans unde r the NEMA phasing structure..................................... 100 61 Computation time and coor dination plan difference....................................................... 117 62 Robust plan and nominal plan.......................................................................................... 118 63 Monte Carlo simulation with normal distribution........................................................... 119 64 Critical values for uniform distribution...........................................................................120 65 Monte Carlo simulation w ith uniform distribution..........................................................121 66 Microscopic simulation re sults and hypothesis tests....................................................... 122 PAGE 8 8 LIST OF FIGURES Figure page 31 AMpeak hourly flow rates at one intersection in Gainesville, FL .................................... 37 32 Illustration of the concept of conditional valueatrisk...................................................... 38 41 Piecewise linear kq relationship...................................................................................58 42 Cell configurations........................................................................................................ .....59 43 Interpretation of the objective function..............................................................................60 44 Traffic intersection configurations..................................................................................... 61 45 NEMA phasing structure fo r a fourway intersection........................................................ 62 46 NEMA phasing structure for a Tintersection...................................................................63 47 Flow chart of the simulationbased GA............................................................................. 64 48 Configuration of the cell chromosome.............................................................................. 65 49 Cell representation of the threenode network................................................................... 66 410 Convergence of GA under both traffic conditions............................................................. 67 411 A snapshot of the threenode network in CORSIM........................................................... 68 412 Cell representation of El Camino Real arterial.................................................................. 69 413 Convergence of GA with different fitness functions......................................................... 70 414 A snapshot of the El Cami no Real arterial in CORSIM.................................................... 71 51 Driving mode of cell i ........................................................................................................92 52 Examples of pollutant dispersion....................................................................................... 93 53 An example of wind rose map (source: USDA)................................................................ 94 54 Pareto frontier from the GAbased algorithm.................................................................... 95 55 Average emission exposure across all scenarios............................................................... 96 56 Average emission exposure ac ross highconsequence scenarios....................................... 97 61 Geometry of green bands................................................................................................. 112 PAGE 9 9 62 Probability of earlyreturntogreen at two intersections ................................................. 113 63 Uncertain termination of green at two intersections........................................................ 114 64 A snapshot of the ActCtrl Example in CORSIM......................................................... 115 65 Computational time and plan difference.......................................................................... 116 PAGE 10 10 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy SIGNAL TIMING OPTIMIZATION FOR RELI ABLE AND SUSTAINABLE MOBILITY By Lihui Zhang August 2010 Chair: Yafeng Yin Major: Civil Engineering This dissertation develops a stochastic progr amming approach to proactively consider a variety of uncertainties associated with signa l timing optimization for fixedtime or actuated traffic signals. Representing the uncertain parameter of interest as a number of scenarios and the corresponding probabilities of occu rrence, the stochastic programm ing approach optimizes signal timings with respect to a set of highconsequence or worstcase scenarios. The resulting signal timing plans produce smaller delays and less vehi cular emissions under those scenarios, thereby leading to more reliable and sustainable mobility. To illustrate the stochastic programming appro ach, below are three applications in traffic signal timing. The first application is to optimize the settings of fixedtime signals along arterials under daytoday demand variati ons or uncertain future tra ffic growth. Based on a celltransmission representa tion of traffic dynamics, an integrat ed stochastic programming model is formulated to determine cycle length, green splits, phase sequences and offsets that minimize the expected delay incurred by highconsequence scenarios of traffic demand. The stochastic programming model is simple in structure but contains a larg e number of binary variables. Existing algorithms, such as branch and bound, are not able to solve it efficiently. Consequently, a simulationbased genetic algorithm is develope d to solve the model. The model and algorithm PAGE 11 11 are validated and verified using two networ ks, under congested and uncongested traffic conditions. The second application further considers tra ffic emissions, and deve lops a biobjective optimization model to make an explicit tradeo ff between traffic delays and roadside human emission exposure. Based on the celltransmission representation of traffic dynamics, a modal sensitive emission approach is used to estimat e the tailpipe emission rate for each cell of a signalized arterial. A cellbased Gaussian plum e air dispersion model is then employed to capture the dispersion of air pollutants and comput e the roadside pollutant concentrations. Given a stochastic distribution of the wind speed and direction of a co rridor, a scenariobased stochastic program is formulated to optimize the cycle lengt h, phase splits, offsets and phase sequences of signals along a corridor simultane ously. A genetic algorithm is further developed to solve the biobjective optimization problem for a set of Pare to optimal solutions. The solutions form an efficient frontier that presents explicit tradeo ffs between total delay of the corridor and the human emission exposure of the roadside ar ea incurred by highconsequence scenarios. The last application is to synchronize actuated signals al ong arterials for smooth and stable progression under uncertain traffic conditions, main ly addressing the issue of uncertain (not fixed) starts/ends of green of sync phases. The model developed is based on Littles mixedinteger linear programming (MIP) formulation, which determines, e.g., offsets and progression speed adjustment, to maximize the twoway bandwid th. By specifying scenarios as realizations of uncertain red times of sync phases, the regret associated with a coor dination plan is defined with respect to each scenario, and then a robust counterpart of Littles model is formulated as another MIP to minimize the average regret incurr ed by a set of highconsequence scenarios. The numerical example shows that the resulting robust coordination plan is able to increase the PAGE 12 12 worstcase and 90th percentile bandwidths by approxima tely 20% without compromising the average bandwidth. In summary, these three applications de monstrate that the proposed stochastic programming approach is valid for signal timi ng optimization under uncertainty. The resulting timing plans are expected to perform more robus tly and effectively in uncertain environments, thereby making the transportation syst em more reliable and sustainable. PAGE 13 13 CHAPTER 1 INTRODUCTION 1.1 Background Congestion has been a severe problem in ma ny metropolitan areas, and is getting worse. While the largest cities are the most congested, c ongestion occurs in regions of all sizes. Peak periods are becoming longer, stopa ndgo traffic conditions can last the entire day in some cities. From 1993 to 2003, travelers in the U.S. spent an extra seven hours per year in travel time; the percentage of congested freeway mileage increas ed from 51% to 60%, although the total mileage was increasing; trips in peak period averagely experienced 37% more time than a freeflow trip, a 9% increase. In 2007, congestion cost reached $ 87.2 billion, an increase of more than 50% over the previous decade (Schrank and Lomax, 2009). Traffic control devices are recognized as one of the major causes of traffic congestion. Disruption of traffic flow by control devices contributes mu ch to traffic congestion, and improperly timed traffic signals account for 5% of the total congestion (Schrank and Lomax, 2009). A recent survey (NTOC, 2007) on the qual ity of traffic signal operations in the United States concluded that the nation scored a D in terms of the overall quality of traffic signal operations. In another study, ORNL (2004) suggests th at 56% of the traffic signals in U.S. are in urgent need of reoptimization. Given a large segment of signal control systems in use today are still fixedtime or actuated, further improvements in their efficiency (e.g., dela y per vehicle) and robust ness (e.g., variance of delay per vehicle) of signal control syst ems can yield significant improvement in the management of traffic flows and level of conge stion. Studies around the c ountry have shown that improved signal timing typically requires little or no infrastructure cost s and produced a benefit to cost ratio of 40 (NTOC, 2007). The benefits include shorter commute times, improved air PAGE 14 14 quality, reduction in certain types and severity of crashes, and re duced driver frustration (FHWA, 2008). According to the 2009 Urban Mobility Report (Schrank and Lomax, 2009), signal coordination in 439 urban areas relieved a total of 19.6 million hours of delay in 2007, a value of $404 million. In the current practice, fixedtime or actuated signal systems typically segment a day into a number of time intervals, each of which is assigned a best suited signal timing plan as determined by applying Websters formula or us ing optimization tools such as TRANSYT7F. Typically, three to five signal ti ming plans are used in a given day. For such a system to work well, the traffic pattern within each interval should remain relatively constant. Unfortunately, travel demands and traffic arrivals at intersections in practice can vary significantly even for the same time of day and day of week. The signal timings obtained from traditional approaches are thus generally unstable and unreliable under such traffic conditions. On the other hand, because of its negative effects on health and living conditions, air pollution has long emerged as one of the most acute problems in many metropolitan areas. A major source of this pollution is the emissions from vehicular traffic. They contribute considerably to the level of carbon monoxide (CO), nitrogen oxides (NOx), and volatile organic compounds (VOCs) in the environment. They ar e also responsible for approximately 20% of greenhouse gas emissions (Poole, 2009). In orde r to achieve a more sustainable mobility, reducing traffic emissions has been an ongoing endeavor of ma ny governmental authorities over the past two decades. However, many emission reducing programs implemented thus far are passive in nature because they regulate only th e rates (e.g., grams per mile) at which a vehicle generates these hazardous air po llutants. Depending on the amount of driving or the mode of driving, cars meeting an emissi on standard can generate as much air pollution as cars not PAGE 15 15 meeting the standard. More proactive strategies that aim to reduce the mileage traveled by vehicles or improve the efficiency of the transportation network have recently gained more and more attention (e.g., Cambridge Systematics, 2009). Signal timing imposes a huge impact on traffic emissions because it interrupts traffic flow (for good reasons) and creates additional deceleration, idle and acceleration driving modes to the otherwise cruise driving mode. Traffic emissions are very sensitive to driving modes. For example, Co elho et al. (2005) reports that decelerating vehicles produce 125% more CO than those in the cruise driving mode, and Rakha et al. (2000) showed that an efficient signal coordination can re duce emissions up to 50%, in a highly simplified scenario. This dissertation proposes a stochastic pr ogramming approach to design signal timing under uncertain traffic conditions and determin e more sustainable signal timing to reduce harmful traffic emissions. 1.2 Dissertation Outline The remainder of the dissertation is organized as follows. Chapter 2 reviews the literature of signal timing optimization under different netw ork configurations. Ch apter 3 introduces the uncertainties associated with traffic mode ling, and discusses the stochastic programming approach for optimizing signal timings. The next three chapters apply the approach to three different signal timing problems in practice: Chapter 4 optimizes the timings of fixedtime signals under daytoday demand variations, Chap ter 5 considers the tradeoff between delays and roadside human emission exposure to generate mo re sustainable signal timing plans, and Chapter 6 synchronizes actuated signals on arterials, addressing the issue of uncertain starts/ends of sync phases. Finally, conclusions and future work are given in Chapter 7. PAGE 16 16 CHAPTER 2 LITERATURE REVIEW Traffic signals have evolved considerably since they were first introduced to prevent collision in London in 1868. The first automatic tra ffic signal was implemented in Detroit, while the first vehicleactuated traffi c signal was installed in Baltim ore in 1928 (Hensher et al., 2001). Signal timing offers the opportunity to improve th e mobility of a transportation system and also prevent the environmental deterioration. 2.1 Traffic Signal Operation Traffic signals are generally operated in three modes: fixe dtime, actuated and adaptive control. 2.1.1 FixedTime Control Fixedtime control is the simplest, less expensive and easier to main tain. The signals assign rightofway at intersections according to predetermined schedules, i.e., timing plans. The phase sequence, phases splits, cycle length and (or) o ffset for each signal are fixed, and determined based on historical traffic patt ern. Because it does not account fo r any traffic demand variations, fixedtime signal control may cause additiona l delay (FHWA, 2008; Boillot et al., 1992). 2.1.2 Actuated Control Signal timing in actuated control, in contrast, consists of intervals that are called and extended in response to vehicle activations. The tr affic controller attempts to adjust green time continuously and in some cases, the sequence of phasing. These adjustments occur in accordance with realtime measures of traffic demand from vehicle detect ors placed at the intersection approaches. Depending on the settings of the c ontroller, the adjustments are constrained by necessary controller parameters. Actuated contro l usually reduces delay, increases capacity and PAGE 17 17 can be safer than the fixedtime control, though th ey are more expensive to implement and also require advanced training of practitioners to operate properly (FHWA, 2008; Boillot et al., 1992). Trafficactuated control can be of two types, semiactuated and fully actuated control, depending on the traffic approaches to be detected. In semiactuated control, the monitored phases include any protected leftturn phases and phases of the side streets. The major movements are called sync phase, se rved unless there is a conflicting call on a minor movement phase. Mino r movement phases receive green only after the syncphase yield point and are terminated on or before their respective forceoff points. These points occur at the same point during th e background signal cycle and ensure that the major road phase will be coordinated with the ad jacent signals. If there ar e no calls present at the yield point, the noncoordinated phases will be skipped for an entire cycle length. The major disadvantage of semiactuated control is that continuous demand on the minor phases can cause excessive delay to the major movements if the maximum green and passage time parameters are not set appropriately (FHWA, 2008). In fullyactuated control, vehicle detectors are installed on all traffic approaches. For each phase, there is a set of minimum and maximum gr een time. If there are no opposing vehicles that waiting for the rightofway, the moving traffic w ill receive additional green time. Fully actuated signals are mostly found at intersections that exhibit large fluctuati ons of traffic volumes from all of the approaches during the day. Much of the benefit of trafficactuated control is derived from the ability of the controllers proactively responding to the fluctuations in tr affic volume, which provides greater efficiency compared to fixedtime control by servicing cros sstreet traffic only wh en required. The primary disadvantage of fixedtime control is avoided as the main street traffic is not interrupted PAGE 18 18 unnecessarily (Jongenotter and Monsma, 2002). This is particularly beneficial during offpeak conditions, resulting in fewer stops and smaller de lays to the traffic on the major arterial, which ultimately leads to a decrease in fuel consump tion and pollutant emissions. However, actuated traffic signal can only respond to th e traffic flow fluctuation to a certain degree. A retiming is needed after a period of time to ensure its efficiency. 2.1.3 Adaptive Control Adaptive systems control traffic signals across an arterial or a large network systematically. These kinds of systems rely on advanced detec tion and information technologies and increasing computation speed to adjust the lengths of si gnal phases based on solving certain optimization problems in every few seconds. With such a mechanism, adaptive signal systems are obviously more capable of optimizing signal timings agains t fluctuating traffic co nditions, and generally can save up to 10% in total tr avel time (Boillot, 1992). On the other hand, these systems are expensive, beyond the budget of many agencies. 2.2 Signal Timing Optimization Since the introduction of automatic traffic signa ls, much work has been done to develop methodologies for timing signals. The configuration of the transportation network under consideration can have a significant impact on how its traffic signals are timed. 2.2.1 Isolated Intersection Isolated traffic signals can be timed without considering other adjacent signals, allowing the flexibility of setting timings that optimize different objectives for individual intersections. During the early stage, methods to determin e the fixed signal timings were developed assuming the traffic arrival from every intersec tion approach has a constant headway, which is not realistic in the field (Matson et al., 1955). We bster (1958) recognized th e uncertain nature of PAGE 19 19 the traffic arrivals and develope d the following delay equation to estimate the average delay of vehicles: )52(3/1 2 2 2)(65.0 )1(2)1(2 )1( x q c xq x x c d where, d is the average delay per vehicle; c is the cycle length; is the effective green in proportion of cycle length; q is the flow rate; s is the saturation flow rate; x is the degree of saturation, equal to q /( s ). The second term in the equation ca ptures the random nature of the traffic arrivals, for the average delay expe rienced by a Poisson stream of traffic. By further assuming that the effective green times of the phases are proportional to their respective degree of saturation, Webster (1958) de termined the cycle length using the following equation: i i ox tL C 0.1 55.1 where, Co is the optimal cycle length; tLi is total lost times per cycle; xi is the degree of saturation for phase i. Note that the equation is valid only when the sum of xi is less than 1.0, and all the approaches within the same phase will need to have the same degree of saturation, which are both easy to be violated under congested traffic conditions. Allsop (1971, 1972) presented a more rigorously complete treatment of the isolated signal optimization problem, still using Websters delay fo rmula, and explored the delay with respect to the traffic volume used to calculate signal timi ngs and concluded that the delay errors are especially serious when several volumes are measured incorrectly simultaneously. On the other hand, research was carried out to investigate sign al timing under oversaturated traffic conditions. Gazis (1964) first co nsidered oversaturated traffic condition and PAGE 20 20 optimized signals in a dynamic setting. His appr oach first allocates the maximum green to the major road and the minimum green to the minor road, then switches the green time between these two to balance the residual queues. The TRANSYT method (Robertson, 1969) is similar to Gaziss idea, and requires a deta iled input of network characteris tics and traffic flows. The central element of the TRANSYT model is flow profiles. Within a user specified modeling horizon, the model estimates the performance inde x as a combination of travel time and stops, and then gradually adjusts the current signal plan until no improvement of the performance index can be made. Michalopoulos and Stephanopoulos (1977) modified Gazis approach and optimized single intersections with state variable constraints. The optimal policy minimizes intersection delay subject to queue length constrai nts, and switches the signals as soon as the queues reach their limits to balance the input and output flows. Ch ang and Lin (2000) developed a discrete state space version of Gaziss model. Lan (2004) proposed a new optimal cycle lengt h methodology for nearsaturated or oversaturated conditions, where Websters equati on fails. A nonlinear programming problem was formulated with the delay equation adopted from the Highway Capacity Manual (TRB, 2000). The optimal simple cycle length formulation was fitted using field samples, which shows very small deviation from the analytical solution. Miller (1965) explored actuated control at an isolated intersection. The discrete model developed can determine the best time to switch pha se, taking into account the lost time due to changing phases. The system was demonstrated in a simulation study to reduce delay by a maximum of 40% compared with the standard vehicle actuated case. Many other studies also focus on actuated controllers. Head (2006) repr esented the actuated co ntrol logic with a PAGE 21 21 procedure graph, which can act as basis to suppo rt decision making like transit priority control. Liu and Bhimireddy (2009) developed an analyti cal model for vehicleactuated signal at an isolated intersection to study th e effects of one time green extension for vehicle platoon on major approach on the delays in subsequent cycles. The model exte nds the queuing model developed by Mirchandani and Zou (2007) to account for vehicle actuations. Comert et al. (2009) incorporated the queue length into actuated signal logic for an isol ated intersection to determine the maximum green in each cycle. The resulting timing plans were demonstrated to be efficient under fluctuated traffic conditions. AgbolosuAm ison and Park (2009) optimized timing plans under dynamic gapout feature using a stochastic optimization method a nd softwareintheloop simulation. The resulting timing plan was de monstrated using VISSIM to reduce vehicular delays by 12.5% compared with regular gapout feature. Viti and Zuylen (2009) presented a modeling approach for actuated signal timings based on the probabilistic theory, allowing the computation of variables like vehicle waiting tim e with any arrival distribution. HCM 2000 also provides a reasonable approximation of the operati on of actuated controllers for nearly all the conditions encountered in practice, and the results have correlated well with extensive simulation data, though the procedure assumes constant departure headway. The above studies have a central assumption th at the traffic flows at the intersection are given as constant average flow rate for each appr oach, which appears to be a great barrier to the effective design of traffic signal timings to achieve reasonable targets (Smith et al., 2001). Heydecker (1987) pointed out that once signal se ttings have been calculated, they would normally be used in a range of circumstances. For example, a single signal plan may be used throughout the morning peak period on each day of a week. If the parameters used in the calculation of the signal settings vary, then a number of realizations will be observed. If signal PAGE 22 22 settings are calculated to optimize some measure of signal performance using the mean values of these observations, then they will not be optim al for the observations themselves, because 1) most measures of signal performance vary in a nonlinear way with respect to some of the parameters; 2) the correlations between variations in the parameter values may give rise to further nonlinearity in the measure of junction performance. Heydecker investigated the consequences of variability in traffic flows and saturation flow s for the calculation of signal settings and then proposed an optimization formul ation that minimizes the mean rate of delay over the observed arrivals and saturation flow s. Following the same notion, Ribeiro (1994) proposed a novel technique called Grouped Networ k, using TRANSYT to calculate timing plans that are efficient even when demand is variable. Both studies focus on optimizing the average performance. Yin (2008b) developed three signa l optimization models: scenariobased meanvariance optimization, scenario based conditional valueatris k minimization, and minmax optimization to determine robust signal timing for isolated signals, which performs better against worst case scenarios. Much recent research has focused on adaptive signal controls. The work on single intersection can be traced back to Newell (1998), who proposed an adaptive traffic control strategy based on the queuing model. Mirchandani and Zou (2007) developed an approach to evaluate this adaptive system. Li and Prevedouros (2004) used a hybrid optimization and rulebased strategy to control an isol ated oversaturated intersection. The objective of the model is to maximize the throughput and control queue leng th. MOVA (1988) used a similar method for isolated signal control. In rec ognition of their capability in pr oviding uptodate signal timings, these adaptive models are able to better tolerate fluctuating traffic arrivals. PAGE 23 23 2.2.2 Arterial For intersections located along a major arteri al, isolated operati ons can be improved by considering coordination of the major movement s along the arterial. Common cycle lengths are often employed to facilitate this coordination. For most arterial streets with signal spacing between 500 feet and 0.5 mile, coordinated operation can often yield benefits by improving progression between signals. On ar terials with higher speeds, it can be beneficial to coordinate signals spaced a mile apart or even longer. Signals that are located very close together often require settings that manage queues rather th an progression as the dominant policy (FHWA, 2008). Morgan and Little (1964) first considered to maximize the ba ndwidth of green wave to increase the chance of vehicles traversing an ar terial without stop. Lit tles MILP formulation (Little, 1966) maximizes the tw oway bandwidth to synchron ize signals along arterials by determining offsets and progression speed adjustme nt etc. The model has been proved to be a flexible and robust approach for signal synchr onization and actually lays the foundation for MAXBAND. Gartner and Stamatiadis (2002) follo wed the similar concept and formulated a multiband/multiweight optimization program for arterials, which can potentially tolerate different traffic patterns. Rather than considering the green band, Allsop(1968) optimized offsets by minimizing the total system delay. The above studies deal with fixedtime signa l systems, while for actuated systems, the problem becomes more complex due to the fact that the starts of gr een of the sync phases (typically Phases 2 and 6) are not fixed. Several approaches have been proposed in the literature to address such a socalled early return to green problem in determining the offsets. Jovanis and Gregor (1986) sugge sted adjusting the end of gree n of the sync phases to the end of the throughband for noncr itical signals. Skabardonis (1996) proposed three methods for PAGE 24 24 determining offsets for actuated signals from th e optimal fixedtime splits and offsets. Although the three methods differ in the procedure and appli cable situation, the concepts are essentially the same: making best estimates of the average starti ng point of the sync phases and then optimizing the offset based on the estimates. Chang (1996) offered a similar suggest ion of obtaining the offsets from a second optimization run that uses the anticipated green times of the noncoordinated phases as constraint s on their maximum green times. Park et al. (2000) formulated an optimization problem to determine the cycle length, offsets, green split and phase sequence simultaneously. They implemented genetic algorithm to solve the problem, where the fitness function considers the queue length. The above studies have focused on determinatio n of appropriate offsets in the design stage of signal timing plans. Certainly after implementi ng the timing plans in the field, there are still opportunities for finetuning. Gartner (2002) proposed an adaptive control strategy for synchronizing traffic signals usi ng the virtualfixedcycle concep t. The strategy can continuously optimize signal settings in response to demand fluctuations, which is achieved via executing a distributed dynamic programming algorithm by means of a threelayer architecture. Shoup and Bullock (1999) examined a concep t of using the link travel times observed for the first vehicle in a platoon to adjust offsets. The concept could lead to an onlin e offset refiner, if vehicle identification technologies had been deployed in arterial corridors. Abbas et al. (2001) developed an online realtime offset transi tioning algorithm that continually adjusts the offsets with the objective of providing smooth progression of a platoon through an intersection. More specifically, the objective was achieved by moving th e green window so that more of the current occupancy actuation histogram is included in the new window. A greedy search approach was used to determine the optimal shift of the gr een window. In the ACSLite system (2003) a run PAGE 25 25 time refiner can modify the cycle, splits and offs ets in an incremental way, based on observation of traffic conditions. Gettman et al. (2007) elabor ated the datadriven algo rithm in the ACSLite for tuning offsets. The algorithm uses upstream dete ctors to construct cyclic profiles of traffic arrivals and then adjusts the offsets to maximi ze the number of vehicles arriving during the green phase. Yin et al. (2007) pr oposed an offline refiner to finet une signal offsets, making use of a large amount of archived signal status data fr om realtime signal operations. Based on a more realistic estimation of the distri butions of the starts/ends of green of the sync phases from the data, the refiner adjusts the offsets to mini mize the redmeeting proba bility of the leading vehicles as well as maximizing the average bandwidth. The systems developed so far have defici ency in considering the oversaturation phenomena such as the blockage due to an upstr eam queue spillback or the leftturning vehicles stored in a leftturn bay. The difficulties come fr om the limitations of the utilized traffic flow models, which are not able to adequately capture these oversaturation characteristics. 2.2.3 Grid Network Intersections to be considered are often located in grid ne tworks with either crossing arterials or a series of intersecting streets with comparable function and traffic volumes. In these situations, the entire network is often timed t ogether. Those grid netw orks with short block spacing, particularly in downtown environments, are frequently timed using fixed settings and no detection (FHWA, 2008). Many optimization approaches described in th e previous section for arterial signal optimizations can be further extended to deal w ith the grid network problems. For examples, the multiband/multiweight approach that Gartner an d Stamatiadis (2002) developed was further expanded for the networkwide coordination and a decomposition method was proposed to solve the mixed integer problem. PAGE 26 26 Recent advancements in technology allow a dire ct linkage between simulation models and the actual signal controllers or th eir software emulations, known as hardwareintheloop (HITL) or softwareintheloop (SITL) (Martinez et al., 2003). Th ese intheloop simulations allow actual controllers and/or their algorithms to replace the approximation used in a simulation model to more accurately reflect how a controller operates. In these systems, a simulation model first generates traffic flows and sends vehicles to the controllers via a detection design implemented in the simulation. The controller re ceives the calls as if it were operating in the field and then uses its own internal algorithms to adjust signal timings based on the calls received and the current signal timing. Fina lly, the signal displays are passe d back to the simulation model, to which traffic responds. Many commercial signal optimization packages are capable of optimizing signal plans for grid networks. Each system uses one particular algorithm or procedure to evaluate a variety of combinations of cycle length, splits, and offsets with respect to one or more performance indices and then attempts to find an optimal combination that minimize or maximize those performance indices. These systems can provide a quick response to the temporary changes in traffic pattern. SCOOT and SCATS are two examples, described as follows: SCOOT (e.g., Hunt et al., 1981) is the abbr eviation of Split Cycle Offset Optimization Technique, which is an adaptive signal optimizati on system. The offset selection is similar to TRANSYT, but the split at each intersection is optimized to mi nimize the maximum of degree of saturation at each intersection. SCOOT is co mputationally efficient and thus it can be implemented online, seeking small and fre quent modification in the signal settings. SCATS (Lowrie, 1982) is the abbreviation of Sydney Coordinated Adaptive Traffic System, which has been widely implemented wo rldwide. The system provides two levels of PAGE 27 27 optimization: strategic and tactical. Strategic co ntrol continuously optimizes the signal settings for the whole network, while the tactical control provides rapid re sponse to the traffic conditions at the intersection level. One issue with the adaptive signal control systems is that when demand increases, they always seek longer cycle lengt h, which may not be preferre d when maximizing throughput or minimizing queue length. Practitioners generally do not think adaptive control to be effective in congested conditions (FHWA, 2009). 2.3 Summary This chapter has briefly reviewed the si gnal timing optimization literature for signal control at isolated intersections, along arterials, and in grid netw orks. For each type of network, previous studies are discussed according to the c ontroller types, in the sequence of fixedtime, actuated and adaptive control. Timing an isolated signal only n eeds to consider the traffic conditions at the particular intersection. Much research ha s been carried out for both uncongested and congested conditions, among which some have explored the me thodology for timing signals under demand uncertainties. To time signals along arterials or in grid networks, another layer of complexity is to coordinate among the signals. Therefore the flow propagation between signals needs to be taken into account, which, to a gr eat extent, influences the perfor mance of the final signal plans. So far, few studies have been reported to di rectly solve the problem, let alone to consider uncertain traffic conditions. Although adaptive systems can continuously optim ize or adjust the signal setting s and provide uptodate signal plans to deal with fluctuating traffi c conditions, they are still too expensive to be widely implemented. Therefore a more proactive and cost efficient approach PAGE 28 28 needs to be developed for fixedti me and actuated signals to explicitly deal with the uncertainties associated with timing optimizations at the arterial or network level. PAGE 29 29 CHAPTER 3 A STOCHASTIC PROGRAMMING APPROA CH FOR ROBUST SIGNAL TIMING OPTIMIZATION 3.1 Uncertainties in Traffic Modeling Uncertainties arise from many aspects in planning, operating and managing a transportation system. Sources of uncertainties mainly include the randomness in demand and supply, and also travelers st ochastic and irrational behavi ors (Lou et al., 2009). Addressing uncertainty in modeling transportation systems ha s received more and more attention recently. 3.1.1 Supply Uncertainty Physically, a transportation system is composed of a variety of transportation facilities, whose main characteristics are re presented as critical parameters in traffic modeling, such as capacity, the saturation flow rate and freeflow tr avel time. These parameters are stochastic in nature. Take capacity as an example. In tradit ional procedures for asse ssing of the level of service for highway facilities, cap acity is treated as a constant value. For example, the design capacities of freeway f acilities are derived using some gi ven guidelines based on speedflow diagrams like in Highway Capac ity Manual (Geistefeldt, 2008). Ponzlet (1996) demonstrated that highway capacities vary accordi ng to external conditions and have to be regarded as random variables. Among others, the differences in the individual driver behaviors, vehicle characteristics, changing road and whether condi tions, all contribute to variations in capacity values. A few models are available to es timate a single value of the capacity or its distribution, like bimodal distribution method, selected maxima method, expected extreme value method, fundamental diagram method and such (Minde rhoud et al., 1997; Br ilon et al., 2005). To PAGE 30 30 estimate the capacity distribution, both nonparame tric and parametric methods can be used, among which Weibull distribution is validated by many real data. 3.1.2 Demand Uncertainty Demand uncertainty is one of the major uncer tainties in traffic modeling. The travel demands and traffic arrivals to intersections can vary significantly even for the same time of day and day of week. As an example, Figure 31 displa ys the hourly arrivals at two crossing streets, 34th Street and University Avenue, in Gainesville, Florida, during an AM peak on weekdays over a period of four months. The flows pres ent significant day to day variations. The uncertainty of the demand data used in modeling can arise from two aspects: 1) the stochastic nature of the demand, 2) the detection or prediction erro rs. Like most of the studies in the literature, this dissertation mainly considers the uncertainty from the demand side. The uncertainty from the supply side, like stochastic capacity, can be incorporated in a similar way into the modeling framework. 3.2 Optimization under Uncertainty Problems of optimization under uncertainty a ppear in many disciplines like finance and manufacturing, other than tran sportation engineering. Many tec hniques have been developed to address uncertainty, and this section introduces three widelyused ones: simulation optimization, stochastic programming, and robust optimization. 3.2.1 Simulation Optimization Simulation optimization assumes almost nothing about the structure of the model to be optimized: uncertain parameters may depend on decision variables and the objective and constraints may be nonlinear, nonconvex or nonsmooth functions of the variables. At each iteration, the simulationbased optimizer proposes values for the decision variables, followed by PAGE 31 31 simulation runs to evaluate the objective and constr aints. Since the derivative information is hard to obtain, many trial values are needed. For each set of trial values, a thousand or more simulation runs may be conducted. In some appli cations, the optimization model is truly a black box with no discoverable structure for the prob lem. The simulation optimization seems the best choice for those applications (Rockafellar, 2001). Simulation optimization is often slow, even on relatively small problems involving tens to hundreds of decision variables and uncertain paramete rs. Often such applications are left to run overnight or over days. 3.2.2 Stochastic Programming Stochastic programming models ta ke advantage of the fact that the uncertain elements in a problem can be modeled as random variables to wh ich the theory of probability can be applied. Thus, the probability distributions of the random elements must be available (Rockafellar, 2001). Moreover, the probability distributions remain unaffected by any of the decisions taken during the optimization process. Stochastic programming typically seeks to find some policy or pl an that is feasible for all (or almost all) the possible data instances and minimizes expected values of the optimal cost function based on the known probabilistic constraints, which are often approximated via a finite scenario approach. The technique has been successfully applied in many areas, however it still remains challenging to implement for several reas ons: first, the probability distributions of uncertain parameters are hard to obtain (or guess, or estimate); second, the scenario approach becomes quickly computationally intr actable as the problem size grows (S ahinidis, 2004; Rockafellar, 1991). PAGE 32 32 3.2.3 Robust Optimization While applications of stochastic programmi ng have been reported over many years in the literature, robust optimization has gain more recent attention. Robust optimization describes uncertainties via bounded sets, as opposed to the probability distributions for stochastic programming. W ith few clue to obtain the whole probability distributions, the uncertain parameters are assu med to belong to some bounded sets. The robust counterpart usually minimizes the worstcase value of the cost function to obtain a solution to the problem. Just as the stochastic programming approach, robust optimization often leads to computationally demanding problems. However, the use of sets rath er than probability distributions makes it easier to use the arsenal of convex analysis to derive approximation to the problem, as well as bounds on the quality of the approximation. (BenTal et al., 2002; Bertsimas, 2003) 3.3 ScenarioBased Stochastic Programmi ng Approach for Signal Optimization As the advancement of portablesensor and telecommunications tec hnologies make highresolution traffic data more readily available, the probability distributions of the uncertain parameters of interest become easier to estimate. Thus the stochastic programming approach can be applied to address the uncertainties associated with signal timing optimization. Current traffic signal control strategies assume the traffic pattern will not change, which is reasonable if the solution being sought is just fo r next few seconds or minutes. Though clearly it is not reasonable to assume tomorrow will be just as same as today, considering the huge variance shown in Figure 31. A consequent issue that traffi c engineers may be confronted with is to determine what flows to use to optimize signal timi ngs. Use of the average flows (i.e.,0q in Figure 31) may not PAGE 33 33 be a sensible choice. Heydecker (1987) pointed out that if the degree of variabili ty of traffic flows is significant, optimizi ng signal timing with respect to the average flows may incur considerable additional delay, compared with th e timing obtained by taking this variability into account. If the degree of variability is small, use of the average flows in conventional timing methods will only lead to small losses in average performance (efficiency). However, as observed in a preliminary investigation (Yin, 2008b) it may still cause cons iderable losses in the performance against the worstca se scenarios or th e stability of perf ormance (robustness), thereby causing motorists travel time to be high ly variable. On the other hand, if the highest observed flows are used instead, the resul ting timing plans may be overprotective and unjustifiably conservative. The average performance is very likely to be inferior. Smith et al. (2002) suggested using 90th percentile volumes as the represen tative volumes to generate optimal timing plans and further noted that if time perm its, other percentile volumes should be used to compare the results. However, it is well known in the statistical literatu re that extreme value estimates can be easily biased and highly unre liable when not computed properly (Reiss and Thomas, 2007). This dissertation is to deve lop a methodology to fully utilize the field data, e.g., the collected traffic flow data here to design a robust optimal sign al timing plan. The performance of such timing plan is optimal against worstcas e traffic conditions and also stable under any realization of uncertain traffic flows in arterials. As a prelim inary investigation, Yin (2008b) developed two robust timing approaches for isolated fixedtime signalized intersections. The first approach assumes specific probabilistic distributions of traffic flows and then formulates a stochastic programming model to minimize the mean of the delays exceeding the percentile (e.g., 90th percentile) of the entire de lay distribution. In contrast, the second approach assumes PAGE 34 34 uncertain traffic flows to be unknown but bounde d by a likelihood region, and then optimizes signal timing against the worstcase scenario reali zed within the region. It has been demonstrated that, when compared with traditional timings, robust timings may reduce the worstcase delay per vehicle by 4.9% and 11.3% resp ectively as well as the standard deviation of delay per vehicle by 12.0% and 16.3% respectively, without adversel y affecting the average performance at a realworld intersection. This dissertation extends the fi rst approach, i.e., stochastic programming approach, to a general setting, optimizing the timings of fixedtime or actuated signals along arterials. It fully recognizes the uncertainty of traffic flows and assumes that they follow certain probability distributions. To represent the uncertainty of traffic flows, a set of scenarios K ,,3,2,1 is introduced. For each scenario k, the probability of occurrence is k p With these random generated scenarios, it is feasible to formulat e a stochastic program to maximize the average performance of the robust signal plan across all scenarios. Howe ver, practically travelers and system managers may be more concerned with the adverse system performance, and are less likely to complain if system performs better than expected. To address such a riskaverse attitude and avoid being too conservative, we attempt to determine a robust timing plan that performs better against highconsequence or worstcase sc enarios. More specifically, we minimize the expected loss (to be defined. Loss may represent di fferent things in different settings, e.g., delay) incurred by those highconsequence scenarios w hose collective probability of occurrence is 1, where is a specified confidence level (say, 80%). In financial engineering, the performance measure is known as conditional valueatrisk (CVaR), or mean excess loss (Rockafellar and Uryasev, 2000). See Figure 32A for an illustration of the concept. The probability density function of a continuous regret and the probability mass function for discrete PAGE 35 35 case are shown in the figure. The right tail area has an area size 1, which contains relatively higher losses. And the CVaR is simply the mean of the losses in this area. By minimizing the CVaR, it can be claimed that the losses incu rred by the highconsequence scenarios are minimized. The following is an example to further illu strate the CVaR con cept: 15 scenarios with equal probability of occurrence (1/15) are generated to repres ent the demand uncertainty. The corresponding delays resulted from the same si gnal settings are listed in Table 31. Choose confidence level = 80%, then 3)1(*15 scenarios with the largest delays are regarded as the highconsequence scenarios, which are scenar ios 5, 13 and 1, with corresponding delays 0.89, 0.92 and 0.95. Then the CVaR va lue can be easily calculated as 92.0)15/1*95.015/1*92.015/1*89.0( 8.01 1 Generally, for each scenario k and one particular feasible signal plan, the loss can be computed according to the problem settings, which we denote as kL. Consider all the scenarios and order the loss askLLL ...21, let k be the unique index such that: 1 1 1 k k k k k kp p In words, kL is the maximum loss that is exceeded only with probability 1, called as valueatrisk. Consequently, the expected loss exceeding the valueatrisk, i.e., conditional valueatrisk is: ] )[( 1 11 1 K kk kk k k k kLpLp (31) PAGE 36 36 The second component in the bracket is simply to compute the mean value, and the first is to split the probability atom at the delay point kL to make the collective probability of scenarios considered in the bracket exactly equal to 1. See Figure 32B for an illustration of the concept. It can be seen that the proba bility mass function has a jump at the point kL due to the associated probability of kp, which makes k k kp1. To make the collective probability of scenarios exactly equal to 1, we need to split the probability of delay kL. Note that if kpmakes k k kp1, then split is not needed, and Equation (31) reduces to )( 1 11 K kk kkLp For each feasible signal plan, Equation (31 ) can be used to compute the resulting conditional valueatrisk and our intention is to find a signal pl an that leads to the minimum conditional valueatrisk. Rockaf ellar and Uryasev (2002) showed that minimizing Equation (31) is equivalent to minimizing the following equation: K k k k ttLp Z1 ,0,)(max 1 1 min where is a free decision variable. Subjective to a se t of specific constraint s, the optimal value of the objective function is the minimum cond itional valueatrisk and the optimal solution **,t represents the robust signal timing plan and valueatrisk respectively. The proposed stochastic programming framew ork is implemented in the next three chapters. Noted that, owing to a large number of notations required in each application, the notations defined in each chapter only apply to that particular chapter. PAGE 37 37 Figure 31. AMpeak hourly flow rates at one intersection in Gainesville, FL 0 500 1000 1500 2000 2500 400600800100012001400160018002000 University Avenue34th Street 0q Q PAGE 38 38 100% Conditional Valueatrisk 0% Loss Probability Probability density function of the loss Probability mass function of the loss Area = 1A 100% 0% Loss Probability Probability density function of the loss Probability mass function of the loss sL s S ss s s s s sLpLp1 11 1 B Figure 32. Illustration of the concept of conditional valueatris k. A) A continuous loss function. B) A discrete loss function PAGE 39 39 Table 31. Delay associated with 15 scenarios 1 2 3 45 6789101112 13 1415 0.95 0.23 0.61 0.490.89 0.760.450.010.820.440.610.79 0.92 0.730.17 PAGE 40 40 CHAPTER 4 SIGNAL TIMING OPTIMIZATION UNDER DAYTODAY DEMAND VARIATIONS Since the seminal work of Webster (1958), significant efforts have been devoted to improving signal timing for saturated isolated in tersections, coordinate d arterials and grid networks etc. However, only a fe w studies have been conducted in the literature to directly address signal timing under flow fluctu ations for fixedtime control systems. This chapter applies the stochastic program ming approach to optimize the timings of signals along arterials under daytoday demand variations or un certain traffic future growth. Based on a celltransmission representation of traffic dynamics, a stochastic programming model is formulated to determine cycle length, green splits, phase sequences and offsets to minimize the expected delay incurred by hi ghconsequence scenarios of traffic demand. The stochastic programming model is simple in structure but contains a larg e number of binary variables. Existing algorithms, such as branch and bound, are not able to solve it efficiently. Consequently, a simulationbased genetic algorithm is developed to solve the model. 4.1 CellTransmission Model 4.1.1 Model Introduction Modeling traffic dynamics is particularly impor tant for signal timing optimization because realistic evaluation of each feas ible timing plan cannot be perform ed without a realistic traffic flow model. At the same time, the evaluation should be efficient such that it can be incorporated into an optimization procedure. For these reas ons, we select the macroscopic celltransmission model (CTM) proposed by Daganzo (1994, 1995) in order to fully capture traffic dynamics, such as shockwaves, and queue formation and dissipation. CTM is a finite difference solution scheme for the hydrodynamic theory of traffic flow or the LighthillWhithamRichards (LWR) models. Mathem atically the theory can be stated as the following equations: PAGE 41 41 0 x q t k (41) ) ,,(txkQq (42) Where the q and k are two macroscopic variables: flow and density. Equation (41) is the flow conservation equation and Equati on (42) defines the traffic flow (q), at location x and time t, as a function of the density (k). For a homogeneous roadway, Daganzo (1994, 1995) suggested using the timeinvariant flowdensity relationship: )}(,,min{kkWQVkqjam where V = the free flow speed; Q = the inflow capacity; jamk = the jam density; W = the backward wave speed. Figure 41 shows the flowdensity relati onship in a piecewise linear diagram. By dividing the whole network into homogeneous cells with the cell length equal to the duration of time step multiplied by the free flow speed, the results of the LWR model can be approximated by a set of recursive equations: ) ()()()1(1tytytntni i i i (43) )]}( [),(),(min{)(1 max,1tnNtQtntyi i ii i (44) where ) (tni = the number of vehicles in cell i during time step t; )(tyi = the number of vehicles that leave cell i during time step t; max, iN = the maximum number of vehicl es that can be accommodated by cell i; )(tQi = the minimum of the capacity flows of cell i and 1 i; = VW/. Equation (43) ensures the flow conservati on that the number of vehicles in cell i during time step 1t equals to the number of vehicles in cell i during time step t plus the inflow and PAGE 42 42 minus the outflow. Equation (44) determines th e outflow for each cell during each time step, which is a piecewise linear function. 4.1.2 Encapsulating CTM in Signal Timing Optimization Lin and Wang (2004), Lo (1999) and Lo et al. (2001) have successfully incorporated CTM in their signal timing optimization formulations. Lin and Wang (2004) formulated a 01 mixed integer linear program, considering the number of stops and fixed or dynamic cycle lengt h. In the model, cells in the network are categorized into four groups: ordinary, intersecti on, origin and destinatio n cells. The objective is to minimize a weighted sum of total delay and to tal number of stops. In their model, Equation (44) is replaced by three linear inequalities, wh ich do not accurately replicate flow propagation and may suffer the socalled vehicle holding problem. To address this issue, one additional penalty term is added to the objective function. The authors demonstrated the model capable of capturing traffic dynamic using an emergency vehicle problem. However, the model is developed only for oneway streets and neither merge nor diverge of traffic is considered. Lo (1999, 2001) and his colleagues (2001) deve loped dynamic signal control formulations based on CTM. By introducing binary variables, Equation (44) is equivale ntly converted into a linear system. The models proposed are able to generate dyna mic or fixed timing plan and optimize cycle length, phase splits and offsets explicitly. Unfortunately, the models are again proposed for oneway streets. This section expands Los models to a more general and realistic setting, including modeling twoway traffic, phase sequence optimization and applying new technique to equivalently transforming CTM for a general signalcontrolled netw ork to be a linear system of equalities and inequalities with integer variables. PAGE 43 43 4.2 Enhanced Deterministic Signal Optimization Model Assuming deterministic constant or timevaria nt demands within the optimization horizon, this section presents a CTMbased determinis tic signal timing optimization model. The model extends Los models in the following aspects: Modeling twoway traffic: this extension not only increases the size of the problem but also introduces another layer of complexity in representing signalized intersections and signal settings. For example, as the number of traffic movements increases, the number of phase combinations and sequences increase significantly; Optimization of phase sequence: the leftt urn leading or lagging control is modeled explicitly; New formulation: we transform the CTM of a general signalcontrolled arterial to be an equivalent linear system of equalities a nd inequalities with integer variables using a technique recently proposed by Pavlis and Recker (2009) and formulate a mixedinteger linear program to optimize cycle length, green splits, offsets and phase sequences. It is assumed that every in tersection along the arterial is signalized, and all the cells comprising the network can be categorized into six groups: ordinary, origin, destination, nonsignalized diverge, signalized diverge and signaliz ed merge cells, as shown in Figure 42 from A to F. Each group has a different conf iguration to be discussed below. 4.2.1 Objective Function In the deterministic setting, we aim to optim ize signal timing to minimize the total system delay of an urban arterial. The objective is to minimize the total area (as in Figure 43) between the cumulative arrival curves of the origin cel ls and the cumulative departure curves of the destination cells, expressed as the following linear function: Di T t t j i Oi T t t j ijy jd L11 11))( ))( min( where, O is the set of origin cells and D is the set of destination cells; T is the duration of the optimization horizon; ) (jdi is the demand at origin cell i during time step j It is straightforward to observe that if the demands at origin cells are given, the objective function is PAGE 44 44 equivalent to maximizing the second component i.e., the area under the cumulative departure curves. 4.2.2 Constraints 4.2.2.1 Constraints for Ordinary Cells The ordinary cells are those with onl y one inflow and one outflow as cell i in Figure 42A. According to the cell transmission model, the flow constraints are as follows: )()()()1(1tytytntni i i i )]}( [),(),(),(min{)(1 max,1 max,1 max,tnNtQtQtntyi i i ii i )(tyi is determined by the min function, which is essentially a linear conditional piecewise function (CPF). Pavlis and Recker (2009) provided a scheme to transform this kind of CPF into mixed integer constraints with the least number of integer variables. By introducing two binary variables, i.e., 1 and 2 and a sufficiently large negative constant, i.e., U, the CPF can be equivalently translated into the following constraints: 0)()()(21tntyUii 0)()()1(max, 21tQtyUi i 0)()()1(max,1 21 tQtyUi i 0)]( [)()2(1 max,1 21 tnNtyUi i i To see the equivalence, Table 41 enumerates all possible 01 combinations and evaluates the value of ) ( tyi. 4.2.2.2 Constraints for Origin Cells The origin cells, as shown in Figure 42B, ha ve the same structure as the ordinary cells, except that the inflow is fixed as the correspon ding demand input. These cells perform as valves PAGE 45 45 that control the traffic volume flowing into th e network. The above c onstraints are slightly changed to incorporate the demand: )()()()1( tytdtntni i i i 0)()()(43tntyUii 0)()()1(max, 43tQtyUi i 0)()()1(max,1 43 tQtyUi i 0)]( [)()2(1 max,1 43 tnNtyUi i i 4.2.2.3 Constraints for Destination Cells The destination cells, as in Figure 42C, are those with outflow unlimited, implying that all the vehicles currently reside in th e cells are able to flow out of the system at the next time step. The constraints are as follows: )()()()1(1tytytntni i i i )()( tntyi i 4.2.2.4 Constraints for NonSignalized Diverge Cells Nonsignalized diverge occurs at certain roadway segments where the geometry or capacity changes and traffic diverse to different la nes for their respective destinations. Figure 42D is a typical configuration for nonsignalized diverge: traffic in cell i diverges to cells 1j and 2j according to proportion parameters 1j and 2j The constraints can be stated as follows: ij jy y 1 11 ij jy y 2 21 12 1 jj 0)()() (765tntyUii 0)()() 1(max, 765tQtyUi i 0/)()() 1(1 1max, 765 j j itQtyU PAGE 46 46 0/)()() 1(2 2max, 765 j j itQtyU 0]/)]( [)() 2(1 1 1max, 765 j j j itnNtyU 0]/)]( [)() 2(2 2 2max, 765 j j j itnNtyU 2765 176 4.2.2.5 Constraints for Signalized Diverge Cells Signalized diverge is the diverge that happens within a signalized intersection, when traffic from one direction enters the intersection during a corresponding green phase and leaves the intersection while diverging into two or more bounds of traffic. Figure 42E sketches a configuration of the signaliz ed diverge, where the sign S indicates a traffic signal. The constraints are as follows: ij jy y 1 11 ij jy y 2 21 12 1 jj 0)()() (1098tntyUii 0)()() 1(1098tQtyUi i 0/)()() 1(1 1max, 1098 j j itQtyU 0/)()() 1(2 2max, 1098 j j itQtyU 0]/)]( [)() 2(1 1 1max, 1098 j j j itnNtyU 0]/)]( [)() 2(2 2 2max, 1098 j j j itnNtyU 21098 1109 The set of constraints is identical to those for nonsignalized diverge cells except that )(max,tQi is replaced by ) ( tQi. The value of ) ( tQi depends on the status of the signal phase associated with cell i and will be discussed later. PAGE 47 47 4.2.2.6 Constraints for Signalized Merge Cells Figure 42F is an example of traffic merge under signal control. A ccording to the signal settings, these three streams of traffic enteri ng the intersection are associated with three individual signal phases that c onflict with each other. Therefore, practically there is only one stream of traffic entering the intersection at one time step. The constraints are thus as follows: )()()()(3 2 1 1tytytytyi i i j Approach 1: 0)()()(1 11211tntyUi i 0)()() 1(1 11211tQtyUi i 0)()() 1(max, 12111tQtyUj i 0)]( [)() 2(max, 12111tnNtyUj j i Approach 2: 0)()()(2 21413tntyUi i 0)()() 1(2 21413tQtyUi i 0)()() 1(max, 14132tQtyUj i 0)]([)() 2(max, 14132tnNtyUj j i Approach 3: 0)()()(3 31615tntyUi i 0)()() 1(3 31615tQtyUi i 0)()() 1(max, 16153tQtyUj i 0)]( [)() 2(max, 16153tnNtyUj j i where ) ( tQi is to be discussed next. 4.2.2.7 Constraints for Connection between Signal and Flow At signalized intersections, the capacity flow of a cell depends on the status of the corresponding signal phase, because only when this phase turns green can the traffic propagates forward or makes a turn. The capacity fl ow satisfies the following statement: If ),()( petpb then stQi ) (; otherwise, 0 )( tQi, PAGE 48 48 where s is saturation flow rate; )( pb is the beginning of green phase p ; )( pe is the end of green phase p The above ifthen relationship can be translated into a system of equalities and inequalities by introducing two binary variables),(1tpzand),(2tpz. The system is stated as follows: )],(1[)(),(1 1tpzUpettpzU )],(1[)(),(2 2tpzUtpbtpzU 1),(),(),(2 1 tpztpztpz stpztpztQi )1),(),(()(2 1 ptpztpz 2)),(),((2 1 where Uis a sufficiently large positive number and is an arbitrary small number. The last constraint ensures that there ar e at most two phases that can be green at the same time. 4.2.2.8 Constraints for Signal Phase Sequence The model intends to explicitly optimize phase sequences under the NEMA phasing structure. Two types of intersections are cons idered as shown in Figure 44: A Fourway intersection and B Tintersection. FourWay Intersection. Figure 45A illustrates the sta ndard NEMA phasing for a fourway intersection. With the barrier in the middle, the structure can be divi ded into four portions and phase sequence is determined within each portion. A binary variable is introduced for each por tion as shown in Figure 45B. Consider phase 1 and 2 as an example. Let g denote the green time duration; o denote the offset point of each signal; l be the cycle length; k indicate the signal identification number; c represent the cycle identification number and h be the barrier time point. The fo llowing constraints are included for determining the phase sequence for phase 1 and 2: PAGE 49 49 ),"2",()1()1()(),"1",(1 1 1cme clmocmb )"1",(),"1",(),"1",( mgcmbcme ),"1",()1()1()()1(),"2",(1 1 1cmecl mo cmb )"2",(),"2",(),"2",( mgcmbcme )()"2",()"1",( mhmgmg It can be seen that when 1 equals 1, phase 1 starts at the offset point of the signal and phase 2 follows phase 1. It is al so true reversely. Similarly, by introducing another three binary variables 2 3 and 4 respectively, constraints can be constructed to determine phase sequence for the pairs of phase 3 and 4, 5 and 6, 7 and 8. Once the value of (1 2 3 4 ) is determined, the leftturn leading and lagging information can be obtained explicitly. Figure 45B presents a particular phase sequence corresponding to 11 02 03 and 04 TIntersection. Tintersection can be modeled the same way as the fourway intersection but is much simpler. Figure 46 illustrates th e NEMA phasing structur e of a particular Tintersection where the only phase sequence needs to be determin ed is between phase 5 and 6. Therefore one binary variable 3 is introduced for the whole intersection. Correspondingly, only one set of constraints is needed for th e entire structure, listed as follows: ),"6",()1()1()(),"5",(3 3 3cke clkockb )"5",(),"5",(),"5",( kgckbcke ),"5",()1()1()()1(),"6",(3 3 3ckecl ko ckb )"6",(),"6",(),"6",( kgckbcke )()"6",()"5",( khkgkg 4.2.3 Model Formulation Given a particular network, the cell representa tion should be first constructed according to the geometry and signal setting. The cells are then classified into six categories and the corresponding set of constraints can be writte n for each cell as previously presented. The constraints comprise a linear system with intege r variables. With the linear objective function to minimize total system delay, the optimization pr oblem is a mixedinteger linear program. One PAGE 50 50 portion of the optimal solution to the program specifies the signal timing, denoted as a vector Tgol****,,,, where *l *o ,* and *g are vectors of optimal cycle length, offsets, phase sequences and green splits. 4.3 Stochastic Signal Optimization Model In the above deterministic case, traffic de mand is assumed to be fixed within the optimization horizon. However in reality, demand ma y vary significantly, like in Figure 31. We assume that the demand at each origin cell follow s a certain stochastic di stribution. To capture the joint stochastic distribution of traffic demands, also a set of scenarios K ,,3,2,1 is introduced. A typical scenario c onsists of demand realizations at all origin cells. More specifically, a scenario is a vector OrddddTk r kk k ,),,(21. For each demand scenario k and one particular feasible signal plan ),,,( gol the total system delay can be computed, as described in th e previous section 3.2. We denote the resulting delay as ) ,,,( golLk By applying the scenariobased appr oach, the robust signal coordination plan can be obtained by minimi zing the following equation: K k k k golgolL Z1 ,,,,0,),,,(max 1 1 min Each demand scenario requires a set of the constraints as discu ssed in the previous section, so the final stochastic optimization problem will include multiple sets of such constraints depending on the number of scenarios generated. 4.4 Numerical Examples 4.4.1 SimulationBased Genetic Algorithm The stochastic programming model formulated a bove is simple in structure but contains a large number of binary variables. Therefore, ex isting algorithms, such as branch and bound, are not able to solve it efficiently, particularly wh en the optimization horizon is long and the network PAGE 51 51 size is large. We thus develop a simulationba sed binary genetic algorithm (GA) to solve the model. Here the simulationbased means that the fitness function in the GA is evaluated through macroscopic simulation using CTM. GAs have been widely used in different fiel ds such as engineering, economics and physics to solve problems that are not analytically solv able or cannot be solved by traditional search methods. In the transportation literature, re searchers have developed GAbased solution algorithms to solve problems including equilibr ium network design, dynamic traffic assignment and secondbest congestion prici ng and traffic control problems. In Lo et al. (2001), GA was used to solve the signal optimization problem. GA is a global search technique. It starts fr om an initial group of randomly generated feasible solutions, and then em ploys operations like crossover and mutation to generate the new solution pools. The iteration continues until so me criterion is satisfied, e.g., the maximum number of generation. The simulationbased GA proposed in this report follows the general framework of GA, and Figure 47 presents the flow chart of the algorithm. There are two loops: the outer loop for counting the number of generations while the inner is to track the number of individuals within each generation. Other core components of the algorithm will be discussed next. 4.4.1.1 Chromosome Configuration The chromosome is defined according to the decision variables, which include the cycle length, offsets, phase sequences and phase splits Each chromosome defines a solution, and only the feasible solutions can be selected as the indi viduals in each generation. Figure 48 presents an example of the chromosome that represents a threesignal arterial. There are in total 144 bits, of which the first six bits 16 represent the cycl e length. A length of six binary numbers can PAGE 52 52 represent a decimal value from 0 to 63. If tr affic dynamic is modeled at two seconds per time step, then the cycle length can vary from 0 to 126 seconds. The rest bits are equally divided into three portions, 46 bits for each signal. Consider the first signal. Bits 7, 8, 9, and 10 define the phase sequences for the signal as discussed in Section 3.2.8. Generally, if a b it has a value of 0, then the corresponding odd phase is activated before the even phase. Otherwis e, the even phase comes first. A fourway intersection will require determining values in all four bits, while a Tintersection only needs one bit information. The next seven bits, i.e., 1117, represent the offset for the signal. A sevenbit binary number can represent a decimal number from 0 to 127. Because an offset is expressed as a percentage of the cycle length in this report, one additional constrai nt on the binary number is in place to ensure the feasibility of the offset. Bits 1824 represent the barrier point, which is also expressed as a percentage of the cycle length. Anot her additional constraint is required as well to ensure that the newly generated barrie r point stays in the current cycle. The next four clusters of b its represent four green times 1G,2G,3G and4G(see Figure 45B), which are the green durat ions of the phases that lead in the respective portions.1G and 3G are in percentage of the barrier time while 2Gand 4G are in percentage of the difference between cycle length and barrier time. 4.4.1.2 Fitness Evaluation For each generation, every individual needs to be evaluated so as to decide its priority to breed the next generation. Here each individual represents a specific signal plan and the fitness evaluation is to determine its corresponding mean excess delay. More specifically, for each individual signal plan, we run a macrosc opic simulation based on CTM with all demand PAGE 53 53 scenarios and calculate the corresponding system control delays. The mean excess delay can be computed and will be used to determine its priority for breading the next generation. 4.4.1.3 Probability Assignment Generally, the smaller the mean excess delay is, the larger probabi lity the corresponding signal plan will be chosen to breed the next generation. To calculate these probabilities, we have tested a variety of fitness f unctions and the following two generally show good performance: 5)( 1 )ln( 1 CVAR CVAR The crossover probabilities are calculated proportionally to the f itness function value. 4.4.1.4 Crossover Crossover is the main procedure to generate new chromosomes. To increase diversity, we use multipoint crossover, developed according to the chromosome structure, other than using onepoint crossover. After the selection of two parents according to the crossover probability, several crossover points will be randomly generated but ensure that one is among the first six bits, which influences the cycle leng th, and one for each signal, which may change the setting for each signal. Therefore, if there are n signals, there will be 1 n crossover points in total. 4.4.1.5 Mutation Each crossover operation will generate two offspring and the mutation operation is subsequently conducted. Mutation randomly ch anges the value of the bit value in the chromosomes to increase the diversity in the pop ulation, so that the GA will have the chance to find a better solution rather than stop at one local optimum. The muta tion rate to be used is 5. PAGE 54 54 4.4.1.6 Individual Validation New individual produced through crossove r and mutation operations may not be appropriate, i.e., the correspondi ng timing plan may not be technically feasible. Therefore, additional constraints need be set to ensure the validity of each individual. Our algorithm mainly checks the followings: The individual newly added will not repeat any individual contained in the population to maintain the diversity of the population; An individual with a cycle length smaller th an a certain value will not be considered to ensure the cycle length to be in a reasonable range; The offsets must be smalle r than the cycle length; The barrier points must be in the corresponding cycle; And each signal phase maintains a certain minimum green. 4.4.2 Numerical Example I 4.4.2.1 Test Network and Demand Data The first numerical experiment is carried out on an artificial arterial with three intersections, whose cell representation is shown in Figure 49. The design speed limit is 35mph, which is approximately equivalent to 50 feet/s econd. Since traffic dynamics is modeled in a resolution of two seconds per time step, 100 feet is the cell lengt h for all 117 cells. We implement the stochastic signal timing mode l under both uncongested and congested traffic conditions. The low demand in Table 42 is for the uncongested situation while the high demand is for the congested cases. The turning percentage s at each signal are al so given in the table 4.4.2.2 Plan Generation For the comparison purpose, two plans are generated under both uncongested and congested conditions: one is called as robust plan, which is generated by solving the stochastic signal timing model with the demand scenarios crea ted from the uniform distributions shown in Table 42; the other is called nominal plan, de rived by solving the deterministic signal timing model with the mean demand. Because the two fitness functions have similar convergence speed PAGE 55 55 and generate timing plans with si milar performance, we only repor t the result from using the logform fitness function. According to the convergence performance of the algorithm, we set the maximal number of generations to 600 and 1000 for the uncongested and congested case respectively. Figure 410 presents the convergence tendency of the algorithm in both cases. The algorithm converges faster in the uncongested case, particularly in the early stage of the iterations. Table 43 presents the resulting signal plans. Th e phase sequence is given in form of binary vector while others are decimal numbers in the unit of second. The minimum green for each phase is set as four seconds. P1 to P8 stand for the phases in NEMA phasing. 4.4.2.3 Plan Evaluation We compare the robust and nominal signa l plans using the microscopic CORSIM simulation, and the system delay is selected as the performance measure. Figure 411 is a snapshot of the CORSIM network. In the simu lation, demand scenarios are obtained by sampling the uniform distributions provide d in Table 42. Table 44 summa rizes the simulation result. It can be seen that the robust timing plans reduce the mean excess delay by 28.68% in the uncongested case and 7.46% in the congested case In both cases, it also improves the average delay across all demand scenarios by over 20%. 4.4.3 Numerical Example II 4.4.3.1 Test Network and Demand Data The second numerical experiment is carried out on a stretch of El Camino Real in the San Francisco Bay Area of California, starting from Crystal Springs Rd to 5th Ave. Figure 412 is the cell representation of the arterial. The speed lim it on the major street is 35 mph or 50 feet per second while 25 mph or 36 feet per second on the side streets. Because traffic dynamics is modeled second by second, the cell length for th e major and side streets is 50 and 36 feet PAGE 56 56 respectively. Traffic demand data were collected from loop detectors for peak hours in a duration of 10 working days in July 2008. Table 45 provid es a summary of the flow data and the turning proportions at the intersections. 4.4.3.2 Plan Generation The observed flow rates are used directly as demand scenarios with equal probability of occurrence, to generate the robust plans by solv ing the stochastic programming model using the simulationbased GA approach. For comparison, a nominal plan is generated by solving the deterministic model with the mean demands pres ented in Table 45. Both robust and nominal plans are generated after 600 generations. Figur e 413 shows the convergence of the GA with both 5thform and logform fitness func tions. It can be observed the 5thform fitness function converges faster than th e logform counterpart. Table 46 presents the signal plans genera ted from both fitness functions, and their performance will be compared next. The minimum green for each phase is set as eight seconds. 4.4.3.3 Plan Evaluation The comparison is also conducted via microscopic simulation with demand profiles randomly generated based on Table 45 assuming truncated normal distributions. Figure 414 is a snapshot of the CORSIM network for the corrid or. Table 47 presents the CORSIM simulation result. The traffic condition is ve ry congested through the whole simu lation period. It can be seen that the robust plans outperfo rm the corresponding nominal plan with the mean delay reduced by 23.69% and 17.82%, and the mean excess delay reduced by 22.80% and 17.34%. It demonstrates that the robust plans perform much better against highconsequence scenarios. As a side effect, the average performance is also improved. Although the 5thform fitness function PAGE 57 57 leads to a faster convergence, it does not impr ove the performance as much as the logform fitness function does. 4.5 Summary This chapter has presented a stochastic m odel to optimize traffic signals on arterials considering daytoday demand variations or uncertain further demand growth. The model optimizes the cycle length, green splits, offset points and phase sequences in an integrated manner. The resulting robust timing plans have been demonstrated to perform better against highconsequence scenarios without losing optimality in the average sense. Considering a large number of binary variables in the formulation, we have developed a simulationbased GA to solve the problem. It should be mentioned that the setting of the GAbased algorithm, such as the fitness function, may influence the quality of the final plan and the convergence speed. Numerical experiments are needed to finetune the setting. We also note that the simulationbased model is broadly applicable particularly when the objective function is difficult or timeconsuming to evaluate. PAGE 58 58 )( k Density)( q Flowjamk Figure 41. Piecewise linear kq relationship PAGE 59 59 i+ 1 nini+ 1yi1yi i ninj1 S S AB DE F C i+ 1 iyinini+ 1di ininiyi1 i j1 j2yi1yiyi1yi inj1nj2 j1 j2 i2 jyj1 ni2 nj yi2 yj11yj21yj11yj21ninj2 Figure 42. Cell configurations PAGE 60 60 TimeVehicles Arrival Departure Cumulative difference Figure 43. Interpretation of the objective function PAGE 61 61 A 2 5 6 47 B Figure 44. Traffic intersection configurations. A) Fourway in tersection. B) Tintersection. PAGE 62 62 2 1 3 4 5 6 7 8 Offset Point Barrier Cycle End A 2 1 5 6 3 4 7 8Offset Point Barrier Cycle End 1= 1 2= 0 3= 0 4= 0 G1G2G3G4B Figure 45. NEMA phasing structure for a fourway intersection. A) NEMA phasing. B) Transformation of NEMA phasing. PAGE 63 63 A B 2 4 5 6 7 Offset Point Barrier Cycle End 5 6Offset Point Barrier Cycle End 3= 0 G1G2G3G4 2 4 7 Figure 46. NEMA phasing structure for a Tintersect ion. A) NEMA phasing. B) Transformation of NEMA phasing. PAGE 64 64 Figure 47. Flow chart of the simulationbased GA PAGE 65 65 25 24 18 17 7 6 5 11 10 9 8 Cycle Length Bits: 1 6 Signal 1 Phase Sequence Bits: 7 10 Offset Bits: 11 17 Barrier Bits: 18 24 G1 Bits: 25 31 G2 Bits: 32 38 3 2 1 4 G3 Bits: 39 45 32 45 39 38 31 52 46 G4 Bits: 46 52 71 70 64 63 53 57 56 Signal 2Phase Sequence Bits: 53 56 Offset Bits: 57 63 Barrier Bits: 64 70 G1 Bits: 71 77 G2 Bits: 78 84 G3 Bits: 85 91 78 91 85 84 77 98 92 G4 Bits: 92 98 117 116 110 109 99 103 102 Signal 3 Phase Sequence Bits: 99 102 Offset Bits: 103 109 Barrier Bits: 110 116 G1 Bits: 117 123 G2 Bits: 124 130 G3 Bits: 131 137 124 137 131 130 123 144 138 G4 Bits: 138 144 Figure 48. Configuratio n of the cell chromosome PAGE 66 66 Figure 49. Cell representati on of the threenode network PAGE 67 67 1700000 1900000 2100000 2300000 2500000 2700000 2900000 3100000 01002003004005006007008009001000GenerationDelay inde x Robust Plan Nominal Plan A 1100000 1200000 1300000 1400000 1500000 1600000 0100200300400500600GenerationDelay inde x Robust Plan Nominal Plan B Figure 410. Convergence of GA under both traffic conditions. A) Uncongested case. B) Congested Case. PAGE 68 68 Figure 411. A snapshot of the threenode network in CORSIM PAGE 69 69 Connectors Connectors Connectors Connectors Connectors Crystal Springs Rd 2nd Avenue 3rd Avenue 4th Avenue 5th Avenue Figure 412. Cell representati on of El Camino Real arterial PAGE 70 70 600000 1100000 1600000 2100000 2600000 3100000 0100200300400500600 GenerationDelay inde x Robust Plan Nominal Plan A 600000 1100000 1600000 2100000 2600000 3100000 0100200300400500600 GenerationDelay inde x Robust Plan Nominal Plan B Figure 413. Convergence of GA with different fitness functions. A) 5thform fitness function. B) Logform fitness function. PAGE 71 71 Figure 414. A snapshot of the El Camino Real arterial in CORSIM PAGE 72 72 Table 41. CPF result under different 01 combinations 01combination (1 ,2 ) Constraint representations (0, 0) )()( tntyi i )()(max,tQtyi i )()(max,1tQtyi i )]( [)(1 max,1tnNtyi i i (0, 1) )()( tntyi i )()(max,tQtyi i )()(max,1tQtyi i )]( [)(1 max,1tnNtyi i i (1,0) )()( tntyi i )()(max,tQtyi i )()(max,1tQtyi i )]( [)(1 max,1tnNtyi i i (1,1) )()( tntyi i )()(max,tQtyi i )()(max,1tQtyi i )]( [)(1 max,1tnNtyi i i PAGE 73 73 Table 42. Traffic data for threenode network Traffic volume Westbound Northbound Eastbound Southbound Low demand 162 *1223 125 High demand 462 1523 325 Left 0.14450.28760.0291 0.6637 Through 0.57720.13730.8960 0.2389 Signal 1 Right 0.27830.57520.0750 0.0973 Low demand 117 169 High demand 317 269 Left 0.00610.76640.0778 0.1923 Through 0.97030.09350.7243 0.0577 Signal 2 Right 0.02370.14020.1979 0.7500 Low demand 75 400 High demand 1075 400 Left 0.6000 Through 0.8000 1.0000 Signal 3 Right 0.2000 0.4000 *: a b means that demand is uniformly distributed in the interval (ab, a+ b) vehicles per hour. PAGE 74 74 Table 43. Signal plans for threenode network Uncongested case Cycle length Phase sequence Offset P1P2P3P4 P5 P6 P7P8 S1 80 (1, 0, 1, 1)0458414 4 58 810 S2 80 (0, 1, 1, 0)76658412 4 60 610 Robust plan S3 80 (1, 0, 0, 1)26*5030 32 18 30S1 80 (1, 1, 0, 1)0462410 22 44 410 S2 80 (1, 0, 0, 1)16446624 6 44 426 Nominal plan S3 80 (0, 1, 0, 0)285030 22 28 30Congested case Cycle length Phase sequence Offset P1P2P3P4 P5 P6 P7P8 S1 108 (1, 1, 1, 1)0878814 18 68 814 S2 108 (0, 0, 0, 0)981472814 28 58 814 Robust plan S3 108 (1, 0, 1, 0)166840 32 36 40S1 80 (1, 1, 0, 1)046646 20 50 46 S2 80 (1, 0, 1, 0)10446426 4 46 624 Nominal plan S3 80 (0, 0, 1, 1)4764 34 42 4*: phase not applicable. PAGE 75 75 Table 44. CORSIM result for threenode network Traffic condition Index measure R obust plan Nominal plan Change Mean delay 13.15*17.70 25.69% Uncongested case Mean excess delay 14.0219.66 28.68% Mean delay 79.1698.97 20.06% Congested case Mean excess delay 106.58115.18 7.46% *: in vehicle hours. PAGE 76 76 Table 45. Traffi c data for El Camino Real arterial Traffic volume Westbound Northbound Eastbound Southbound Demand mean *179 1112 Demand SD 13 44Left 0.06030.7205 Through 0.93970.8369 Crystal Spring Right 0.2795 0.1631 Demand mean 174Demand SD 18Left 0.6647 0.1118 Through 0.7818 0.8882 2nd Ave Right 0.33530.2182 Demand mean 270 238 Demand SD 22 18 Left 0.45450.06000.1911 0.0247 Through 0.25570.86790.4837 0.8983 3rd Ave Right 0.28980.07220.3252 0.0770 Demand mean 528 101 Demand SD 32 10 Left 0.33510.02370.1272 0.1193 Through 0.29900.87320.6301 0.8593 4th Ave Right 0.36600.10310.2428 0.0214 Demand mean 2191443 184 Demand SD 10 82 18 Left 0.38530.04470.2889 0.0312 Through 0.43910.94070.5804 0.8982 5th Ave Right 0.17560.01470.1307 0.0706 *: means data not applicable or available. PAGE 77 77 Table 46. Signal plans fo r El Camino Real arterial 5thform fitness function Cycle length Phase sequence Offset P1P2P3P4 P5 P6 P7P8 S1 90 (0, 1, 1, 1)01158813 47 22 129 S2 90 (0, 0, 1, 0)1516342614 34 16 931 S3 90 (0, 1, 1, 1)1324332310 38 19 1716 S4 90 (0, 0, 0, 0)6732184050 40 Robust plan S5 90 (0, 0, 0, 0)03555 15 20 55S1 112 (0, 0, 0, 0)01476148 78 12 913 S2 112 (1, 0, 0, 1)343255178 70 17 1015 S3 112 (1, 0, 0, 1)3826432518 57 12 2419 S4 112 (0, 0, 1, 1)8329216250 62 Nominal plan S5 112 (1, 1, 0, 0)436844 21 47 44Logform fitness function Cycle length Phase sequence Offset P1P2P3P4 P5 P6 P7P8 S1 94 (0, 0, 0, 0)0863158 62 9 1013 S2 94 (0, 0, 1, 1)241635358 35 16 2716 S3 94 (1, 0, 0, 1)6021323011 26 27 2813 S4 94 (1, 1, 1, 0)1329105539 55 Robust plan S5 94 (1, 0, 0, 0)695539 44 11 39S1 112 (1, 0, 1, 1)0880168 79 9 1410 S2 112 (0, 1, 1, 0)121160333 45 26 3110 S3 112 (1, 1, 1, 0)2626451328 62 9 1922 S4 112 (1, 1, 1, 1)1032225854 58 Nominal plan S5 112 (0, 0, 1, 1)502785 19 8 85 PAGE 78 78 Table 47. CORSIM result for El Camino Real arterial Fitness function Index measure Robust plan Nominal plan Change Mean delay 234.36307.13 23.69% Logform Mean excess delay 240.92312.08 22.80% Mean delay 228.02277.46 17.82% 5thform Mean excess delay 232.30281.02 17.34% PAGE 79 79 CHAPTER 5 SIGNAL TIMING OPTIMIZATION WI TH ENVIRONMENAL CONCERNS Previous studies have deve loped signal timing optimization models to minimize both congestion and emissions. For example, Li et al. (2004) optimized the cycle length and green splits for an isolated intersection to minimize a weighed sum of the delay, fuel consumption and emission. Park et al. (2009) developed a signal timing optimization model to minimize the fuel consumption and vehicle emission based on micr oscopic simulation. Th ese previous studies consider tailpipe emissions inst ead of roadside air pollution con centrations, which decide the air quality. In other words, dispersi on of air pollutants is not captured in their models. The tradeoff between delays and emissions in street canyons should be very diffe rent from that in open rural environments, because air pollutants are difficult to disperse in the former, resulting in higher pollution concentrations and worse air quality in the area adjacent to the streets. Moreover, previous studies either use a highly simplified a pproach to compute traffic emissions or rely on microscopic drivingcyclebased emission models. Th e former approach fails to capture spatially and temporally varying traffic state and may underestimate the emissions due to accelerations and decelerations (Lin and Ge, 2006). The latter approach results in microscopicsimulationbased optimization models that are computationally intractable for real network optimization. In contrast, this chapter tries to apply the scenariobased stochast ic programming approach to determine traffic signal timings to minimize tr affic delay and the risk associated with human exposure to traffic emissions. The model is macr oscopic and computationally tractable. At the PAGE 80 80 same time, it is sufficiently ac curate in modeling traffic dynami cs and capturing the impact of timedependent traffic char acteristics on emissions. 5.1 Emission and Roadside Pollutant Concentrations 5.1.1 Emission Model Existing emission modeling systems can estimate and predict traffic emission at regional, facility and individualvehicle levels. See, e.g., Yu et al. (2010), for a recent review on the emission models. For the analysis of signa lized intersection emissions, the modal and instantaneous emission models are more app licable because they predict secondbysecond tailpipe emissions as a function of the vehicles operating mode (e.g., Cernuschi et al., 1995; Barth et al., 2000 and Ra kha et al., 2004). In this research, emission factors are es timated using the modal emission approach proposed by Frey et al. (2001, 2002) where a vehi cles operating mode is divided into four modes of idle, acceleration and d eceleration and cruise, and the av erage emission factors for each mode are determined. As demonstrated below, the resolution and data requirement of this approach are very compatible with the CTM repr esentation of traffic dynamics. Moreover, it is efficient enough to be incorporated into the signal timing optimization procedure. Frey et al. (2001) defined the idle mode as zero speed and zero acceleration. For the acceleration mode, the vehicle speed must be greater than zero with an acceleration rate averaging at least 1 mph/s for 3 seconds or more Deceleration is defined in a similar manner as acceleration but with a negative ac celeration rate. The cruising mode is approximately steadyspeed driving, but some drifting of speed is allowed. In the CTM implementation, the traffic state PAGE 81 81 at each cell is determined at each time interval and the driving mode of the vehicles in the cell1 can then be identified through comparing the de nsities of the current and the immediately downstream cells. Figure 51 illustrates how the dr iving mode of an ordinary cell is identified. The current cell is i and the downstream cell is i +1 and their densities are denoted as ik and 1 ik. The fundamental diagram of the signaliz ed arterial is also presented in the figure to facilitate the presentation, where jamk is the jam density and ck is the critical density. In Figure 51, the relationship between ik and 1 ik falls into one of the four areas, i.e., C, A, D and I. Area C stands for the cruise mode and consists of tw o portions: the first portion is a rectangle for the situations when cikk and cikk 1, i.e., the vehicles in both cell s are traveling at the freeflow speed; the other is the line separati ng the areas of A and D, where jam iikkk 1, implying that vehicles in both cells are traveling at the sa me positive speed. The end of that line is the area (point) of I, representing th e idle mode. At this point, jam iikkk 1, and all vehicles have a zero speed. Albeit a point, it occu rs frequently in the cells near signalized intersections during the red intervals. The lower area of D represents the deceleration mode where 1 iikk, and cikk 1. This implies that vehicles in the current cell are traveling faster than the downstream vehicles. Because these vehicles will move into the downstream cell, they have to decelerate to adapt to the prevailing downstream speed. Simila rly, Area A represents the acceleration mode where1 iikk and cikk implying that vehicles in the curr ent cell are traveling slower than 1 In CTM, vehicles in one cell are treated homogenous. PAGE 82 82 the downstream vehicles, and are accelerating. In summary, by simply comparing the cell density with that of the immediately downstream cell, one can easily identify the driving mode of the vehicles in each cell. As an example, Figur e 51 shows a case that the vehicles in cell i are in the deceleration mode. Figure 51 describes the determination of driv ing mode of an ordinary cell. Necessary amendments are introduced to dete rmine the driving modes for othe r type of cells. For example, the driving mode of signalized cells will be idle when the co rresponding signal phase is red, irrelevant of the density of the downstream cell. 5.1.2 Pollution Dispersion Model Frey et al. (2001, 2002) developed the aver age emission factors for each driving mode. Applying these factors, the cell emission rate can be calculated as follows: )()()( tEFtntERi i i where )( tERi denotes the emission rate of cell i at time interval t ; )( tni is the number of vehicles in the cell; )( tEFi is the emission factor for a driving mode in units of mg/s or g/s. With the cell emission rates, an atmospheric dispersion model can be used to estimate the roadside air pollutant concentrations. To facilit ate the presentation of the modeling framework, this research adopts the simplest steadystate Gaussian plume dispersion model (e.g., Turner, 1994). Other more advanced disper sion models can be applied in a similar way, particularly those nonsteadystate models to capture the timevarying emission rates computed above. However, we note that the dispersion model should be computationally tractab le so that it can be incorporated into the signal timing optimization procedure. PAGE 83 83 To apply the Gaussian plume dispersion model, it is assumed that the pollution source of each cell is located in the cell cen ter. Within the modeling horizon, e.g., 1 hour, the average cell emission rate iER is estimated as TtERERT t i i/)(1. Subsequently, at any point in space ),,( zyx, the incremental pollutant concentration ),,( zyxCi from the source cell i can be approximately estimated as follows: 2 2 2 2 2 22 )( exp 2 )( exp 2 exp 2 ),,(z z y zy i ihz hz y U ER zyxC where U is the wind speed at the pollutant release height h; x is the distance along the wind direction; y is the distance along the crosswind direction; z is the vertical distance; y is the crosswind dispersion coefficient and z is the vertical dispersion coefficient. For numerical computation, the roadside area is cut into grids and the pollutant concentrations of each grid are computed with the coordinates of the center of the grid. Figure 52 presents two concentr ation contour maps for one particular signalized arterial. The maps are generated using the same wind speed but different directions. The wind direction in Figure 52A is 4/ clockwise to the north while in Figure 52B it is the north. These two examples demonstrate the need for considering pollutant dispersion as the impacts of traffic emissions will be substantially different in these two cases. 5.1.3 Mean Excess Exposure We define a surrogate measure to represent the potential negative impacts of roadside pollutant concentrations. The soc alled total human emission exposur e is calculated as follows: PAGE 84 84 dxdydzzyxCzyxp TEI i zyx i 1),,(),,( (51) where T E is the total human emission exposure; I is the total number of cells of the signalized corridor and ),,( zyxp is the population density func tion beside the corridor. The above definition is written for one partic ular pollutant for illustration purposes. If multiple pollutants are considered, Equation (51) can be applied to each pollutant and the total human emission exposure is the weighted sum of the exposures to all pollutants. The weighting factors are determined based on, e.g., the social cost of each pollutant (Delucchi, et al., 2002). The above human emission exposure is computed for a certain wind speed U and direction WD. These two parameters change constantly while the signal plan implemented in the field remains the same for a certain period of time, e.g., years. Therefore, the changing wind speed and direction should be proactively consid ered. To represent such wind uncertainty, we introduce a set of wind scenarios S ,,3,2,1 A scenario consists of a certain wind speed and direction, i.e., ),(sssWDU and its probability of occurrence is sp. The wind scenarios can be specified using the wind ro se of the study area. Figure 53 is the 30year wind rose map for the month of July obtained from a wind station in San Francisco bay area. Given the set of wind scenarios, it is strai ghtforward to estimate the mean of the human emission exposures across all scenarios, and th en optimize signal timings to minimize it. However, decision makers may be more concerne d with worstcase scenarios where substantial emission exposure may occur. To address such a riskaverse attitude and avoid being too conservative at the same time, we optimize the signal timings against a set of worstcase PAGE 85 85 scenarios. More specifically, we minimize th e expected emission exposure incurred by those highconsequence scenarios whose collective probability of occurrence is 1, where is a specified confidence level, e.g., 80%. In financ ial engineering, the performance measure is known as conditional valueatrisk (CVAR) (Rockafe llar and Uryasev, 2000) and we name it as mean excess (emission) exposure. Mathematically, for a wind scenario s, the total emission exposure can be computed, denoted as sTE. Consider all wind scenarios and order the emission exposure as STE TETE ...21. Let s be the unique index such that: 1 1 1 s s s s s sp p In words, sTE is the maximum emission exposure that is exceeded only with probability 1. Consequently, the expected emission exposure exceeding sTE, i.e., the mean excess exposure, can be computed as follows: S ss ss s s s sTEp TEp MEE1 11 1 (52) The second component in the bracket is simply to compute the mean value, and the first is to split the probability atom at the emission exposure point sTEto make the collective probability of scenarios considered in the bracket exactly equal to 1. Rockafellar and Uryasev (2002) showed that minimizing Equa tion (52) is equivalent to minimizing the following equation: PAGE 86 86 )0, max( 1 1 min1 s S s sTEp Z (53) where is a free decision variable. 5.2 Model Formulation and Solution Algorithm 5.2.1 BiObjective Optimization Model Given a particular network, the cell representa tion should be first c onstructed based on the geometry and signal setting. The cells are th en classified into six categories and the corresponding constraints for flow propagation can be written for each cell as previously described in Chapter 4. The constraints comprise a linear system with integer variables. Other constraints include those for signal timings su ch as minimum and maximum green times, and relationships of phase sequences. For nume rical computation of roadside pollutant concentrations, the roadside area is cut into gr ids and the constraints for computing pollutant concentrations can be written. All these constraint s are essentially linear (with respect to decision variables) but involve integer variables. With the linear objective functions to minimize total system delay as described in Figure 43 (denoted as TD in this chapter ) and the mean excess exposure (Equation (43)), the op timization problem is a biobj ective mixedinteger linear program. One portion of the optimal solution to th e program specifies the signal timing, denoted as a vector Tgol****,,,, where *l, *o,* and *g are vectors of optimal cycle length, offsets, phase sequences and green splits. Conceptually, the biobjective signal optimi zation model is summarized as follows: MEETDgol;min ),,,( PAGE 87 87 s.t. Constraints for traffic dynamics and re lating signal timing to flow propagation Constraints for feasible signal settings Constraints for computing pollutant c oncentrations and human emission exposure There may not exist an unambiguous optimal so lution that minimizes both the total delay and mean excess exposure simultaneously. Hence, a set of Pareto optimal solutions or nondominated solutions are sought in stead. Those solutions are optimal in the sense that no improvement can be achieved in any objective without degradation in the other. All these solutions form a Pareto frontier. Based on the de cision makers considerati on of social costs of delays and emissions, an optimal timing plan can be selected. 5.2.2 SimulationBased BiObj ective Genetic Algorithm The biobjective model formulated above contai ns a large number of binary variables, particularly when the optimization horizon is long and the network size is large. A geneticalgorithmbased solution approach (Y in, 2002) is adopted to solve th e problem for a set of Pareto optimal solutions. The proposed GA follows the general framework of GA. For each individual signal plan, we run a macroscopic simulation based on CTM with all wind scenarios and calculate the corresponding system delays and m ean excess exposure. The fitness value of each signal plan is then determined using the m odified distance method proposed by Osyczka and Kundu (1996). The basic idea is that if a newly ge nerated Pareto optimal plan is farther away from the existing Pareto set, it will have a grea ter finiteness value. The detailed algorithm is listed as follows: PAGE 88 88 Step 1 Let = 1 and j = 1, where is the generation index and j is the individual index within each generation. Generate the first generation randomly a nd make the first solution as a starting Pareto solution with an arbitrarily assigned value 1 called the starting latent potential. Step 2 For each generated solution X calculate the relative distances from all existing Pareto solutions using the following formula: 2 1)( )( I i l i i l i lf Xff Xd for l = 1, 2, prl Where prl is the number of existing Pareto optimal solutions and if is the ith objective function in the biobjective optimization model. Step 3 Find the minimum value from the set )} ({ Xdl, using the formula: )}({min...2,1*Xd dl ll lp Where *l is the nearest Pareto optimal point to solution X Step 4 Compare the newly generated solution X with the existing Pareto solutions in the following way: If it is a new Pareto solution, dominating so me of the existing Pareto solutions, then calculate its fitness value using *max ld F and let F max Update the Pareto set by removing the solutions dominated by th e newly generated solution, then go to Step 5. If it is a Pareto solution, dominating none of the existing Pareto solutions, then calculate the fitness value using *lldF and add this solution to the Pareto set with the latent potential value F If max F let F max Go to Step 5. If it is not a Pareto solution, then update the fitness value as ) 0, max(* *lld F Go to Step 5. Step 5 Substitute max for all existing Pareto solutions, i.e. max l, for l = 1, 2, pl Step 6 Let j = j + 1. If Jj where J is the predetermined population size. go to Step 2, otherwise go to Step 7. PAGE 89 89 Step 7 Let = + 1. If R where R is the predetermined generation number, perform GA operators: crossover and mutation to generate a new popul ation with size of J. Otherwise, terminate the iterations. 5.3 Numerical Example The numerical experiment was also carried out on the stretch of El Camino Real described in Chapter 4, between Crystal Springs Rd and 5th Ave. Figure 412 is th e cell representation of the arterial. The speed limit remains the same, thus the cell length for the major and side streets is 50 and 36 feet respectively. Without considering the demand variations, the mean values in Table 45 were used as input demand. The popul ation distribution besi de the arterial was assumed to be the following: )30000()100(108.3),,(11x z zyxp where 100z, and 30000 30000x. In our coordinate system, the y axis is located along the arterial centerline. The above implies that the population density is less for a location farther away from the arterial and the gr ound level, i.e., with a larger x and z To facilitate the presentation, we considered one pollutant only in this example. Table 51 presents the emission factors of Nitric Oxide under different driving m odes (Frey et al., 2002). Using the 30year historical wind data from USDA, we specified the wind scenarios. Table 52 reports the probabilities of occu rrence for all the scenarios we considered. The wind speed is categorized into five bins while the wind direction is captured every 8/ In total, there are 80 scenarios, but only 37 of them have positive probability of occurrence. PAGE 90 90 The simulationbased GA was applied to solv e the biobjective signal timing optimization model. The resulting Pareto frontier is show n in Figure 54 where one point represents a particular signal timing plan. For example, Plan A minimizes the mean excess exposure and Plan B minimizes the total system delay. The plans in between are all other Pareto optimal solutions, not dominated by any other timing plan. It can be observed from Figure 54 that the mean excess exposure resulting from these timing plans varies from 5108 to 2585 persongrams, a 49% reduction. Correspondingly, the total system delay changes from 1325 to 1949 vehiclehours, a 47% increase. The frontier presents the trad eoff between congestion and emissions, allowing decision makers to learn about the problem befo re committing to a final decision of an optimal timing plan. As an example, Table 53 presents the optimized timing plans A and B where signal S1 is the one at 5th Avenue and S5 at Crystal Spring. To demonstrate the impacts of these two substantially different timing plan, Figures 55 and 56 present their corresponding emission exposure contour maps. The former is for the av erage emission exposure across all the possible scenarios at different height levels while the latter focuses on the 20% highconsequence scenarios. Although the contour maps for Plan s A and B show similar shapes, the emission exposure under Plan B is slightly larger than th at of Plan A. For side streets, the emission exposure under Plan B is visibly larger than that under Plan A. PAGE 91 91 5.4 Summary This chapter has formulated a biobjective traffic signal optimization model to make an explicit tradeoff between delays and roadside air pollution concentrations. The modal emission approach is integrated with the cell transmission model to compute the pollutant emission rates at each time step. A cellbased Gaussian plume di spersion model is encapsulated to capture the pollutant dispersion process to estimate the human emission exposure. A risk measure, namely, the mean excess exposure, is then defined to represent the mean of the human emission exposures against highconsequence wind scenario s. The signal timing optimization optimizes the cycle length, green splits, offset points and phase sequences to minimize the total system delay and the mean excess exposure simultaneously. A GA based algorithm is employed to solve the bilevel mixedinteger problem. Both the model and the solution algorithm are demonstrated in a numerical example using field data from a selected realworld arterial network. A set of Pareto optimal signal timing plans are generated that form an efficient frontier. The frontier exhibits an obvious trad eoff between these two objective functions, providing a fou ndation for a sound decision making. PAGE 92 92 Figure 51. Driving mode of cell i PAGE 93 93 A B Figure 52. Examples of pollutant dispersion PAGE 94 94 NORTH SOUTH WEST EAST 7% 14% 21% 28% 35% Figure 53. An example of wind rose map (source: USDA) PAGE 95 95 2000 2500 3000 3500 4000 4500 5000 5500 6000 1200130014001500160017001800190020002100 Total System Delay (vehiclehour)Mean Excess Exposure (persongram) A B Figure 54. Pareto frontie r from the GAbased algorithm PAGE 96 96 A (Plan A, Z = tailpipe height) B (Plan B, Z = tailpipe height) C (Plan A, Z = 6 feet) D (Plan B, Z = 6 feet) E (Plan A, Z = 50 feet) F (Plan B, Z = 50 feet) Figure 55. Average emission exposure across all scenarios PAGE 97 97 A (Plan A, Z = tailpipe height) B (Plan B, Z = tailpipe height) C (Plan A, Z = 6 feet) D (Plan B, Z = 6 feet) E (Plan A, Z = 50 feet) F (Plan B, Z = 50 feet) Figure 56. Average emission exposu re across highconsequence scenarios PAGE 98 98 Table 51. Emission factors for different driving modes Driving mode Idle (g/s) Accelerate Decelerate Cruise Emission rate 0.040.310.060.17 PAGE 99 99 Table 52. 30year wind probability data Windspeed anticlock 1.803.34 (m/s) 3.345.40 (m/s) 5.408.49 (m/s) 8.4911.06 (m/s) >11.06 (m/s) 0 (north) 0 0.0010870.00084200 8/ 0 0.0074610.0133020.0020290.0001148/2 0.018862 0.0798370.114560.0783480.0094558/3 0.031752 0.0942640.1207490.0715190.018798/4 0.007061 0.0507430.0852330.0382610.0121538/5 0.002639 0.0236770.0203740.0046240.0008838/6 0 0.0047020.0040190.00104608/7 0 00.0002910.00042808/8 0 00008/9 0 00008/10 0 00008/11 0 00008/12 0 00008/13 0 0.0014010008/14 0.001069 0.0098130.000186008/15 0 0.0055810.00029400 PAGE 100 100 Table 53. Pareto optimal signal pl ans under the NEMA phasing structure Timing plans (secs) Cycle length Phase sequence Offset P1P2P3P4P5 P6 P7P8 S1 120 (1, 0, 0, 0)0781617918 76 818 S2 120 (1, 0, 1, 0)382325611123 25 5319 S3 120 (1, 1, 1, 1)442440175629 35 488 S4 120 (1, 1, 0, 1)4051472298 22 Plan A S5 120 (1, 0, 1, 0)40309015 15 90S1 106 (1, 0, 0, 0)01266181045 33 1117 S2 106 (0, 1, 0, 1)521329105423 19 5410 S3 106 (1, 0, 1, 0)02830331549 9 2523 S4 106 (1, 0, 0, 1)49118620 86 Plan B S5 106 (0, 1, 1, 0)14565028 28 50 PAGE 101 101 CHAPTER 6 ROBUST SYNCHRONIZATION OF ACTUATED SIGNALS ON ARTERIALS An increasing number of traffic signal contro llers used in the United States are trafficactuated. It has been a common pr actice to operate these controllers in coordinated systems to provide progression for major traffic movements along arterials and networks. Compared with fixedtime coordinated systems, these semiactuated coordinated systems offer additional flexibility in responding to fluc tuations in traffic demand. Un der signal coordination, traffic actuated signals operate under a common backgr ound cycle length. Coordination is provided through a fixed reference point, which defines the st art of the controller local clock, and can be set to the start of green, end of green (yield point) or other time in terval for the sync phases (e.g., beginning of the flashing dont walk interval). No te that the controller local clock definition varies among signal controller manufacturers (FHWA, 1996). To ensure operation efficiency of coordinated actuated systems, attention should be paid to determining appropriate signal settin gs, particularly offsets due to the fact that the start of green of the sync phases (typically Phases 2 and 6) is not fixed. This chapter attempts to use the scenariobased approach to s ynchronize actuated signals along co rridors. The model developed is based on Littles mixedinteger linear pr ogramming (MILP) formulation (Little, 1966) maximizing the twoway bandwidth to synchronize signals along arterials by determining offsets and progress speed adjustment etc. By specifying scenarios as realizations of uncertain red times of sync phases, we define the regret associated with a coordination plan with respect to each scenario, and then formulated a robust counterpart of Littles formulatio n as another MILP to minimize the average regret incurred by a set of highconsequence scenarios. PAGE 102 102 6.1 Bandwidth Maximization for Ar terial Signal Coordination Generally speaking, there are two approach es of generating c oordination plans to synchronize signals along arterials and grid networks. One aims at bandwidth maximization, e.g., MAXBAND (Little et al., 1981) and PASSERII (Chang et al., 1988) while the other is performancebased optimization, synchronizing signals to mini mize the performance measures such as control delay and corridor travel time, e.g., TRANSY7F (Wallace et al., 1998). To facilitate the presentation of our robust approach, this study bases the model development on Littles MILP formulation (Little, 1966). which maximizes the twoway bandwidth to synchronize signals along arteri als by determining offsets and progress speed adjustment etc. The model has been proven to be a flexible and robust approach for sign al synchronization and actually lays the foundation for MAXBAND It has been later extended to consider variable bandwidth, phase sequencing and grid network s ynchronization (Gartner and Stamatiadis, 2002; Gartner and Stamatiadis, 2004 and Messer et al., 1987). Below, we summarize Littles MILP formulation. Given a twoway arterial with an arbitrary number of signals, a common backgrou nd cycle length, and the split information for each signal, the formulation attempts to synchr onize the signals to produce a maximum sum of the inbound and outbound bandwidths. Let hS and iS be any pair of adjacent signals, and iS follows hS in the outbound direction. Figure 61 pres ents the geometry of the green bands between iS and hS The horizontal lines indicat e when the sync phases are red, and the zigzag lines represent the vehicle trajectories. PAGE 103 103 Notations to be used together with those in Figure 61 are introduced as follows: ir red time of signal i on the corridor (cycles) () bb Outbound (inbound) bandwidth (cycles). ),(),(hitiht Outbound (inbound) travel time from signal () hi to () ih(cycles) ),(),( hiih time from the center of red at hS to the center of a particular red atiS The two reds are chosen so that each is immediately to the left (right) of the same outbound (inbound) green band (cycles) ()iiww time from the right (left) side of iS s red to the green band (cycles) ),( ihm ),(),( ihih According to Figure 61, (,) mhi must be integer (cycles) 1T 2T lower and upper bounds on cycle length (s) z signal frequency (cycles/s), the inverse of cycle length (,) dhi distance from hS to iS (m). (,1)iddii ()iivv speed between iS and 1iS outbound (inbound) (m/s) ,(,)iiiiefef lower and upper bounds on outbound (inbound) speed (m/s) Note that in Littles formulat ion, in addition to offsets, cycle length, travel speed, and change in speed between street segments are al so decision variables, constrained by upper and lower limits. The bandwidth maximization problem is mathematically written as follows: bbBmttwwzbb),,,,,,,(max s.t. 121/1/ TzT (61) 1iiwbr ni ,...,1 (62) 1iiwbr ni ,...,1 (63) 1 11()()()()i iiiiiiiiwwwwttmrr 1 ,...,1 ni (64) im integer 1 ,...,1 ni (65) (/)(/)iiiiidfztdez 1 ,...,1 ni (66) PAGE 104 104 (/)(/)iiiiidfztdez 1 ,...,1 ni (67) 11(/)(/)(/)iiiiiiiidhzddttdgz 2 ,...,1 ni (68) 1 1(/)(/)(/)ii i ii ii idhzddttdgz 2 ,...,1 ni (69) ,,,0i ibbww ni ,...,1 In the objective function, ttwwbb ,,,, and mrepresent vectors whose elements are the scalar decision variables ttwwbb ,,,, and im Equation (61) is the c onstraint on cycle length. Equations (62) and (63) are th e constraints on green bandwidth. Equations (64) and (65) are the integer constraints due to the fact that ),(),( ihih must be integer. Equations (66) and (67) are the constraints associated with travel sp eed. Equations (68) and (6 9) are the constraints on speed changes between adjacent street segmen ts. All of these constraints and the objective function are linear, and (64) and (65) are integer, therefore the problem is an MILP, which can be efficiently solved by using, e.g., the bran chandbound algorithms. From the solutions, offsets can be easily obtained as follows accord ing to the geometry in Figure 61: )](2/1 [1 1 1 ii iii irrtww 1,...,1 ni where 1 i is the offset of signal i +1 with respect to i ; ][x is defined as )int(xx and )int(x represents the largest in teger not greater than x Hereafter, we denote the vector of offsets as 6.2 ScenarioBased Approach for Robust Sy nchronization of Actuated Signals Littles formulation assumes that the durations of minor phases (red times of sync phases) are deterministic. Such an assumption does not hold for actuated signal control, where the phase durations of minor phases vary between zero (s kipped) and the corresponding maximum greens. That is the reason for a socalled early return to green problem. To illustrate how prevailing the problem of early return to green is, Figure 62 depicts the histograms for starts of green of Phase 2 and 6 at two selected in tersections, Page Mill and Stanfo rd, along El Camino, Palo Alto, PAGE 105 105 CA. The data were collected from 11:00 am 3:00 pm between February 6 and March 7, 2005, for a total of 14 weekdays with more than 1,600 cy cles. Page Mill is a criti cal intersection for the corridor with almost equal amounts of mainline and crossstreet tra ffic. Still, the probabilities of early return to green are 61% for Phase 2 and 39% for Phase 6. Stanford has low volume of minorphase traffic, thus the probabilities are as pretty high as 92% for Phase 2 and 94% for Phase 6 respectively. The histograms confirm the assertion made in the previous studies that the problem of early return to green should be recognized and explicitly addressed in actuated signal synchronization. Note that in addition to uncertainty of start of green, end of green is also uncertain, especially under leadlag phase sequence, due to skip or gapout of the leftturn phase. Figure 63 presents the histograms for green term inations of Phase 2 and 6 at Page Mill and Stanford. It can be seen that compared with st arts of green, terminations of green have much narrower spans. Under many circumstances, th e termination is the forceoff point. The above empirical data shows that for actuated signals, the red times, ir in Littles formulation should follow random distributions with supports between zero and the sum of maximum greens of the conflicting minor phases. To represent the uncertainty, a set of scenarios K,,3,2,1 in introduced, and each individual scenario is composed of the red time of the sync phases2 at all the signals, which for example at intersection i, is denoted as k ir We now define the regret function. For each scenario k, we solve Littles MILP formulation to obtain the maximu m twoway bandwidth, denoted as *kB Consequently, for any other feasible coordination plan (offsets and cycle frequency z ) that may not be optimal for scenario k, the regret or loss can be defined as: 2 Normally Phases 2 and 6. For simplicity, we assume he re that both phases start and end at the same time. PAGE 106 106 ) ,(*zBBLkkk k where kL is the regret or loss of the coordination plan ) ,( z with respect to red time scenario k, and kB is the twoway bandwidth resulted by ) ,( z under scenario k Then the robust signal coordination plan can be obtained by minimizing the following equation: K k k k zzLp Z1 ,,0,),(max 1 1 min Therefore, the scenariobased robust synchr onization model for actuated signals can be written as follows: K k kk LUmttwwzbbUp Zkk k k k k k k k1 ),,,,,,,,,,,(1 1 min s.t. kkLU Kk ,...,1 0 kU k )(* kk kkbbBL k 121/1/ TzT k i kk irbw 1 ni ,...,1 k k i kk irbw 1 ni ,...,1 k )()()()(1 1 1 k i k i k i k i k i k i k i k i k irrmttwwww ni ,...,1 k k im =integer ni ,...,1 k ) (2/11 11 1 k i k i k i k i k iirrtww 1 ,...,1 ni k zedtzfdii k iii)/()/( 1,...,1 ni ,k zedtzfdi i k i i i)/()/( 1,...,1 ni ,k zgdttddzhdii k i k iii ii)/()/()/(11 2,...,1 ni ,k zgdttddzhdi i k i k i ii i i)/()/()/(1 1 2,...,1 ni ,k 0,,, k k k kwwbb k where kU is an auxiliary decisi on variable, equal to ) 0,),(max( zLk. Note that kB is predetermined by solving Littles formulation for each scenario. As formulated, the problem is PAGE 107 107 another MILP and can be efficiently solved. As demonstrated in pervious studies (Krokhmal et al., 2002; Chen et al., 2006 and Yin, 2008a), su ch a formulation can offer computational advantages and allow handling a large number of scenarios. Our numerical example below also shows that the problem can be solved in polynomial time. 6.3 Numerical Example 6.3.1 Plan Generation We solve the robust synchronization formulation for an arterial with six signals, which is included in CORSIM as one of the tutorial examples, named ActCtrl Example, shown in Figure 64. The outbound signals, starting from signal 1, are located at 0, 314, 554, 759, 1012, 1317m respectively. The timing plan for each isolat ed signal is determined using the Websters equation. The red times of the sync phases are a ssumed to be independently normally distributed with means of 0.27, 0.24, 0.50, 0.35, 0.35, 0.43 cycles respectively and the same standard deviation of 0.05 cycles across all the signals. The scenarios are specified by random sampling and are assumed to have equal probability to occur. The upper and lower limits of the cycle length are 100 s and 45 s. Limits on travel speed fo r different street segments are set to be the same with the upper limit of 17.9 m/s (40 mph) and the lower limit of 13.4 m/s (30 mph). Changes in reciprocal speed s across all the segments ar e limited between 0.0121 and 0.0121(m/s)1, corresponding to a maximum possible change in speed of .3 m/s (.9 mph) at the lower limit of the speed and .9 m/s (. 8 mph) at the upper lim it. The confidence level is selected to be 0.90. We use an algebraic modeling system ca lled GAMS (Brooke et al., 1992) and CPLEX solver (CPLEX, 2004) to solve the robust sync hronization formulation with the number of scenarios varying from 10 to 250. The com putation times (in CPU seconds) and the plan differences are presented in Table 61, plotted in Figure 65. PAGE 108 108 The plan difference is defined as 2 250 2 250/K, where Kis the plan generated with K scenarios. Within expectation, the difference tends to decrease as the number of scenarios increases. However, it can be observed that relatively small number of scenarios is already enough to produce similar robust plans. The reported computation times include the times needed to solve Lttiles formulation for each scenario to obtain kB Regressing the computational times against the number of scenarios, we obtain the following equation, su ggesting that the problem may be solved in polynomial time: 2Time = 0.3286*(number of s cenario) 2.6556 R = 0.9962. For the comparison purpose, we also generate a nominal pl an following the procedure of deterministic signal coordination, in which we use the mean red times as our estimates, and then use Littles formulation described in part two of the s ection to obtain the nominal plan.. Both the robust plan (with 250 scenarios) and the nomin al plan are reported in Table 62. 6.3.2 Plan Evaluation To evaluate the performance of both nomina l and robust plans, we conduct macroscopic Monte Carlo simulation and micros copic simulation in CORSIM. In the Monte Carlo simulation, 2000 samples of red times are drawn from the sa me normal distributions previously used to generate scenarios for solving the robust sync hronization formulation. With each sample, the bandwidths resulted by both the nominal and robus t plans are computed. Consequently, several performance measures, including the average, worstcase (minimum) and 90th percentile minimum bandwidths, and the 90% conditional valu eatrisk, are calculated and reported in Table 63. The results indicate that the robust coordi nation plan performs better against highconsequence scenarios, with the worstca se bandwidth increasing by 23.1%, and the 90th PAGE 109 109 percentile bandwidth by 23.9%, and the conditional valueatris k decreasing by 17.3%. At the same time, the average bandwidth also increa ses by 20%. However, since the robust plan is designed to guard against highconsequence s cenarios, an improvement of the average performance is not what we should expect and will not be necessarily obtained. To further validate the robust synchronization formulation, we evaluate the plans with 2000 samples randomly drawn from independent uniform distributions with the minimum and maximum values described in Table 64. The re sulting performance measures are summarized in Table 65. The robust plan still outperforms the nominal plan, with the average, worstcase and 90th percentile bandwidths increasing by 16.7%, 22.1% and 22.5%, and the regret decreasing by 16.5%. This examination suggests that the robus t formulation is not overly sensitive to the specification of scenarios and usi ng distorted distributions to gene rate scenarios may still result in robust timing plans. We recognize the limitation of the bandwidthb ased synchronization that traffic flows and intersection capacities are not considered in the optimization criterion (Gartner, 1991), and thus bandwidth maximization does not necessarily optimize other delayrelated performance measures. To examine how the robust plan affects those measures, we conduct a microscopic simulation using CORSIM. The robust and nominal plans are implemented respectively in the semiactuated corridor of ActCtrl Example in CORSIM. We select the control delay, corridor travel time, and vehicle stop ra tio as the performance measures where vehicle stop ratio is defined as the total number of stops (when speed is lower than 3 mph) divided by the total number of vehicles served by the corridor within the simulation period. The means and standard deviations calculated from ten simulation runs are reported in Table 66. We conduct t tests and F tests to examine whether those performance meas ures are statistically different. As shown in PAGE 110 110 Table 66, the three t values are all greater than the critical t value of 2.552 at the significance level of 1%, suggesting that the corridor has better average pe rformance with the robust plan. The F values are all smaller than the critical F value of 2.44 at the significance level of 10%, indicating that we can no t reject the hypothesis that the variances are the same. It should be pointed out that the intention of the CORSIM simulation is to examine whether the robust plan makes the delayrelated performance measures worse off. Although the simulation results actually suggest otherwise, we do not expect to always obtain such improvements, since it is not what the robust plan is designe d for. The only conclusion we dr aw from the CORSIM simulation is that the robust plan seems unlikely to wors en the delayrelated performance measures. 6.4 Summary This chapter presented a robust approach to synchronize actuated si gnals along arterials. The formulation is an MILP, a class of mathema tical programs easy to solve using the stateoftheart solvers. The computational time only in creases polynomially as the number of scenarios increases. The resulting robust co ordination plan has been demons trated in a numerical example to perform better against highc onsequence scenarios w ithout losing optimalit y in average. The approach can be used to either design a new coordination plan for implementation or finetune the plan offline after implementation. In the latter case, the specification of scenarios is an easy task with the archived signal status data One may randomly select 50 to 200 red time realizations from the data and assume equa l probability of occurrence. To design a new coordination plan where the di stributions of red times are normally unknown, we suggest specifying 50 to 200 scenarios as the points that equally divide the redtime intervals into K +1 segments and assume equal probability of occurrence of 1/ K The suggestion is based on our observation from the numerical experiments that th e robust formulation is not overly sensitive to PAGE 111 111 the specification of scenarios. Even with bias ed scenarios, the formulation may still produce meaningful robust plans. PAGE 112 112 (,) hi (,) hi (,) hi b b (,) thi (,) thi Figure 61. Geometry of green bands PAGE 113 113 0 20 40 60 80 100 120 0 50 100 150 200 250 300 Local ClockFrequencyStart of Phase 2 at Page Mill 0 20 40 60 80 100 120 0 100 200 300 400 500 600 700 Local ClockFrequencyStart of Phase 6 at Page Mill 0 20 40 60 80 100 120 0 50 100 150 200 250 Local ClockFrequencyStart of Phase 2 at Stanford 0 20 40 60 80 100 120 0 200 400 600 800 1000 1200 Local ClockFrequencyStart of Phase 6 at Stanford 61% 39% 92% 94% Mean = 99 Mean = 80 Mean = 70 Mean = 51 Forceoff of preceding phase = 108 Forceoff of preceding phase = 83 Forceoff of preceding phase = 81 Forceoff of preceding phase = 58 Figure 62. Probability of earlyret urntogreen at two intersections PAGE 114 114 0 20 40 60 80 100 120 0 200 400 600 800 1000 1200 Local ClockFrequencyEnd of Phase 2 at Page Mill 0 20 40 60 80 100 120 0 200 400 600 800 1000 1200 1400 1600 Local ClockFrequencyEnd of Phase 6 at Page Mill 0 20 40 60 80 100 120 0 100 200 300 400 500 Local ClockFrequencyEnd of Phase 2 at Stanford 0 20 40 60 80 100 120 0 500 1000 1500 2000 Local ClockFrequencyEnd of Phase 6 at Stanford Forceoff = 19 Forceoff = 0 Forceoff = 0 Forceoff = 25 Minimal green of Phase 5 Figure 63. Uncertain termination of green at two intersections PAGE 115 115 Figure 64. A snapshot of th e ActCtrl Example in CORSIM PAGE 116 116 0 10 20 30 40 50 60 70 80 90 2050100150200250Number of ScenariosComputational Tim e 0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00%Plan Difference Computational Time Plan Difference Figure 65. Computational time and plan difference PAGE 117 117 Table 61. Computation time a nd coordination plan difference Number of scenario Time*(sec) Plan difference 10 2.88 169.3% 20 5.11 2.9% 50 11.67 0.7% 100 28.58 1.0% 150 45.67 1.1% 200 61.97 0.4% 250 81.81 0.0% *: including the times for solving Littles formulation for each scenario PAGE 118 118 Table 62. Robust plan and nominal plan Offset (sec) Intersection 2 3456 Cycle length (sec) Robust plan 33 22354 80 Nominal plan 53 74724768 77 PAGE 119 119 Table 63. Monte Carlo simu lation with normal distribution Change Bandwidth (sec) Mean Worst case 90th percentile 90% CVaR Mean Worst case 90th percentile 90% CVaR Nominal plan 28.717.0 23.840.1Robust plan 35.821.8 30.634.524.7%28.2%28.6% 14.0% PAGE 120 120 Table 64. Critical values for uniform distribution Red time Signal Minimum duration (cycles) Maximum duration (cycles) # 1 0*0.42 # 2 0 0.25 # 3 0 0.56 # 4 0 0.53 # 5 0 0.56 # 6 0 0.45 *: Minor phases skipped. PAGE 121 121 Table 65. Monte Carlo simula tion with uniform distribution Change Bandwidth (sec) Mean Worst case 90th percentile 90% CVaR Mean Worst case 90th percentile 90% CVaR Nominal plan 36.6 17.8 24.354.1Robust plan 44.3 22.6 31.046.921.0%27.0%27.6% 13.3% PAGE 122 122 Table 66. Microscopic simula tion results and hypothesis tests Signal plan Performance measure Control delay (sec) Travel time (sec) Stop ratio Mean 96.2318.40.148 Robust plan Std. deviation 12.212.70.008 Mean 122.6346.80.176 Nominal plan Std. deviation 9.912.40.010 t value 5.315.067.08 Hypothesis test F value 1.541.051.32 PAGE 123 123 CHAPTER 7 CONCLUSIONS AND FUTURE WORK 7.1 Conclusions This dissertation has presented a stochastic programming approach to deal with a variety of uncertainties associated with signal timing op timization for fixedtime and actuated traffic signals. Representing the uncertain parameter of interest as a number of scenarios and the corresponding probabilities of occu rrence, the stochastic programm ing approach optimizes signal timings with respect to a se t of highconsequence scenario s. The proposed approach is demonstrated in three applications. The first application optimizes the timing of fixedtime signals along arterials under daytoday demand variations or uncertain future traffic growth. By specifying scenarios as realizations of uncertain demand, a general signal optimization model is formulated to optimize the cycle length, green splits, offsets and phase se quences in an integrated manner. Considering a large number of binary variables in the formulat ion, a simulationbased GA is developed to solve the problem. It should be mentioned that the se tting of the GAbased al gorithm, such as the fitness function, may influence the quality of the final plan and the convergence speed. Numerical experiments are needed to finetune the setting. The simulationbased model is broadly applicable, particularly when the objec tive function is difficult or timeconsuming to evaluate. The second application expands the first one by further considering the impact of signal timing on vehicle emissions. A biobjective signal optimization model is developed to make an explicit tradeoff between delays and the human emission exposure, considering the uncertainty of wind speed and direction. The modal emi ssion approach is integrated with the cell PAGE 124 124 transmission model to compute the pollutant em ission rates at each modeling step, followed by a cellbased Gaussian plume dispersi on model to capture the polluta nt dispersion to estimate the human emission exposure. A GAbased solution appr oach is employed to solve the large mixedinteger problem to come by a set of Pareto optimal signal timing plans, minimizing the total system delay and the mean excess exposure simu ltaneously. The solutions form an efficient frontier that represents explicit tradeoffs be tween these two objective functions. Based on the decision makers consideration of social costs of delays and emissions, an optimal timing plan can be selected. The last application synchroni zes actuated signals along arteri als (the offsets and cycle length) for smooth and stable progression under un certain traffic conditions, addressing the issue of uncertain starts/ends of green of sync phase s. By specifying scenario s as realizations of uncertain red times of sync phases, a robust counte rpart of Littles formulation is formulated as another MILP. The model can be easily solved using the stateofthea rt solvers, and the computational time only increases polynomially as the number of scenarios increases. The approach can be used to either design a new coordination plan for implementation or finetune the plan offline after implementation. These three applications validate the stoc hastic programming approach for optimizing signal timings under uncertainty. Th e resulting timing plans have b een demonstrated to perform more robustly and effectively in an uncertain en vironment, thereby leading to more reliable and sustainable mobility. 7.2 Future Work With the same stochastic programming framew ork, an immediate followup work would be to develop an integrated model to directly optimize the settings of actuated signals, including not only the cycle length and offs ets as in Chapter 6, but also other general settings like PAGE 125 125 minimum and maximum greens, gr een extensions and phase sequences. Developing an exact or approximated mathematical representation of the actuated control logic ca n be a big challenge. It is also noted that the pr oposed stochastic programming appr oach is quite flexible to consider other types of uncertainties, e.g., the sa turation flow rates. Future work may expand the current models to simultaneously address a variet y of uncertainties from both the demand side and supply side. For example, by specifying a numb er of scenarios as realizations of certain saturation flow rates and uncertain demands simu ltaneously, another scen ariobased stochastic program can be formulated to minimize some se lected performance meas ure to obtain a robust timing plan. Another potential future work is to expand the proposed models for more sophisticated network configurations, such as grid networks. The major issue to be resolved will be the coordination among multiple signals in the complex networks. The embedded traffic flow model will need to be enhanced for this purpose. The simulationbased genetic algorithm proposed in the dissertation is broadly applicable, particularly when the objective f unction or constraints are difficul t to evaluate. However, as the problem size increases, the algorithm will easily become time consuming. Future work may explore how to further improve the efficiency of the algorithm through, e.g., finetuning the fitness function used in the algorithm. Given the promising experiment results in these three applications, future work may also be a field operation test A software package can be also deve loped to include these three models, with necessary user interfaces for parameter input and result output. To apply these models, users should provide the necessary data for both the su pply side and the demand side. At this stage, users will also have to code the network into the model, and estimate th e distributions of the PAGE 126 126 uncertain parameters of interest from the raw da ta. In the future, two modules can be developed to facilitate the usage of the models: one is to estimate the probability distributions of the parameters from the input raw data with multiple c hoices of distribution types, and the other is to automatically generate network representations. The latter is particularly useful if the cell transmission model is used as th e underlying traffic flow model. PAGE 127 127 LIST OF REFERENCES Abbas, M., Bullock, D., Head, L., 2001. Realtime offset transitioning algorithm for coordinating traffic signals. Transportation Research Record 1748, 26. AgbolosuAmison, S.J., Park, B., 2009. Performa nce evaluation of dynamic gapout feature at actuated traffic signal controller using a simulation based optimization method. The 88th Annual Meeting of the Transportation Re search Board, (Compendium of Papers DVDRom), No.092802. Allsop, R.E., 1968. Selection of offsets to minimi ze delay to traffic in a network controlled by fixedtime signals. Transporta tion Science 2(1), 1. Allsop, R.E., 1971. Delayminimizing settings for fixedtime traffic signa ls at a single road junction. IMA Journal of Applied Mathematics 8(2), 164. Allsop R. E., 1972. Sensitivity of delay at a fixe d time traffic signal to small errors in the observations used for calculating signal se ttings. In Traffic Flow and Transportation (Edited by G. F. Newell), 253, Elsevier, London. Barth, M., An, F., Younglove, T., Scora, G ., Levine, C., Ross, M., Wenzel, T., 2000. Comprehensive Modal Emission Model (CMEM), Version 2.0 User's Guide, Riverside, CA. BenTal, A., Nemirovski, A., 2002. Robust optimizationmethodology and applications. Mathematical Programming, Series B, 92, 456. Bertsimas D., Sim, M., 2003. Robust discrete op timization and network flows. Mathematical Programming, Series B, 98, 49. Boillot, F., Blosseville, J.M., Lesort, J. B., Motyka, V., Papageorgiou, M., Sellam, S., 1992. Optimal signal control of urban traffic networks. In Proceedings of the 6th IEEE International Conference on Road Tr affic Monitoring and Control, 75. Brilon, W., Geistefeldt, J., Regl er, M., 2005. Reliability of freeway traffic flow: a stochastic concept of capacity. In Proceedings of the 16th Symposium on Transportation and Traffic Theory, 125. Brooke, A., Kendirck, D., Meeraus, A., 1992. GAMS : A Users Guide. The Scientific Press, South San Francisco, California. Cambridge Systematics, 2009. Moving Cooler: An Analysis of Transportation Strategies for Reducing Greenhouse Gas Emissions. Urban Land Institute, Washington, D.C. Cernuschi, S., Giugliano, M., Cemin, A., Giovannini, I., 1995. Modal analysis of vehicle emission factors. The Science of the Total Environment, 169, 175. PAGE 128 128 Chang, E.C.P., 1996. Guidelines for actuated controllers in coordinated systems. Transportation Research Record, 1554, 61. Chang, E., Lei, J.C., Messer, C.J., 1988. Ar terial Signal Timing Optimization using PASSERII87 Microcomputer Users Guide. TTI Res earch Report 4671, Texas A&M University, College Station, Texas. Chang, T.H., Lin, J.T., 2000. Optimal signal timing for an oversaturated intersection Transportation Research, Part B, 34, 471. Chen, G., Daskin, M.S., Shen, Z.J.M., Uryasev, S., 2006. The reliable meanexcess regret model for stochastic facility location modeling. Naval Re search Logistics, 53, 617. Coelho, M.C., Farias, T.L., Rouphail, N.M., 2005. Impact of speed contro l traffic signals on pollutant emissions. Transportation Research, Part D, 10, 323. Comert, G., Cetin, M., Nichols, A. P., 2009. In corporating queue length measurements into actuated signal control: eval uation of efficiency benefits at an intersection. The 88th Annual Meeting of the Transportation Res earch Board, (Compendium of Papers DVDRom), No. 092566. CPLEX, Version 9.0., 2004. CPLE X Optimization Inc., Nevada. Daganzo, C.F., 1994. The cell transmission mode l: A dynamic representation of highway traffic consistent with the hydrodynamic theory. Tr ansportation Research, Part B, 28, 269. Daganzo, C.F., 1995. The cell transmission model, part II: network traffic. Transportation Research, Part B, 29, 79. Delucchi, M.A., Murphy, J.J., McCubbin, D.R., 2002. The health and visibility cost of air pollution: a comparison of estimation methods Journal of Environmental Management, 64, 139. Federal Highway Administration, 1996. Traffic Control Systems Handbook. U.S. Department of Transportation. Publicati on Number: FHWASA95032. Federal Highway Administration, 2008. Traffic Signal Timing Ma nual. U.S. Department of Transportation. Publicati on Number: FHWAHOP08024 Federal Highway Administration, 2009. Signal Timing under Saturated Conditions. http://ops.fhwa.dot.gov/publica tions/fhwahop09008/chapter2.htm Flyvbjerg, B., Holm, M.K.S., Buhl, S.L., 2005. How (in)accurate are demand forecasts in public works projects: the cas e of transportation. Journa l of the American Planning Association, 71 (2), 131. PAGE 129 129 Flyvbjerg, B., Holm, M.K.S., Buhl, S.L., 2006. In accuracy in traffic forecasts. Transport Reviews, 26 (1), 1. Frey, H.C., Rouphail, N.M., Unal, A., Coly ar, J.D., 2001. Emission Reduction Through Better Traffic Management: An Empi rical Evaluation Based Upon OnRoad Measurements. Report No. FHWA/NC/2002001, North Carolina Department of Transportation, USA. Frey, H.C., Unal, A., Chen, J., 2002. Recommended Strategies for OnBoard Emission Data Analysis and Collection for the New Generation Model, Prepared by NC State University for Office of Transportation and Air Quality U.S. Environmental Protection Agency, Ann Arbor, Michigan. Gartner, N.H., 2002. Development and implementa tion of an adaptive control strategy in a traffic signal network: th e virtualfixedcycle approach. In Proceeding of 15th International Symposium on Trans portation and Traffic Theory, 137. Gartner, N.H., Assmann, S.F., Lasaga, F., Hou, D.L., 1991. A multiband approach to arterial traffic signal optimization. Tran sportation Research, Part B, 25, 55. Gartner, N.H., Stamatiadis, C., 2002. Arterialb ased control of traffic flow in urban grid networks. Mathematical and Co mputer Modeling, 35, 657. Gartner, N.H., Stamatiadis, C., 2004. Progressi on optimization featuri ng arterialand routebased priority signal networks. ITS Journal, 8, 77. Gazis, D.C., 1964. Optimum control of a system of oversaturated in tersections. Operation Research, 12(6) 815. Geistefeldt, J., 2008. Empirical relation between stocha stic capacities and capacities obtained from the speedflow diagram. In Proceedings of the Symposium on the Fundamental Diagram: 75 years, Woods Hole, MA. Gettman, D., Shelby, S.G., Head, L., Bullock, D.M., Soyke, N., 2007. Datadriven algorithms for realtime adaptive tuning of offsets in coordinated traffic signal systems. Transportation Research Record, 2035, 1. Head, L., Gettman, D., Wei, Z., 2006. Decision model for priority c ontrol of traffic signals. Transportation Research Record, 1978, 169 177. Hensher, D., Button, K., Sperling D., 2001. Environm ent Protection. In Handbook of Transport Systems and Traffic Control. Instit ute of Transportation Studies, University of California, Davis, Research Report UCDITSRR0119. Heydecker, B., 1987. Uncertainty and vari ability in traffic signal calculations Transportation Research, Part B, 21, 79. PAGE 130 130 Hunt, P.B., Robertson, D.I., Bretherton, R.D., Winton, R.I ., 1981. SCOOT a Traffic Responsive Method of Coordinating Si gnals. TRL Laboratory Report 1014. Jongenotter, E., Monsma, S., 2002. Green logic a nd the benefits for trafficactuated control the Dutch experience. Traffic E ngineering & Control, 43(9), 351. Jovanis, P.P., Gregor, J.A., 1986. Coordination of actuate d arterial traffic signal systems. Journal of Transportation Engineering, 112(4), 416. Krokhmal, P., Palmquist, J., Uryasev, S., 2002. Portfolio optimization with conditional valueatrisk objective and constraints. Journal of Risk, 4, 46. Lan C.J., 2004. New optimal cycle length formul ation for pretimed si gnals at isolated intersections. Journal of Trans portation Engineering, 130(5), 637. Li, H., Prevedouros, P.D., 2004. Traffic adap tive control for oversaturated isolated intersections: model development and simulation testing. Journal of Transportation Engineering 130(5), 594. Li, X., Li, G., Pang, S., Yang, X., Tian, J ., 2004. Signal timing of intersections using integrated optimization of traffic quality, emissions and fuel consumption: a note. Transportation Research, Part D, 9, 401. Lin, J., Ge, Y.E., 2006. Impacts of traffi c heterogeneity on roadside air pollution concentration. Transportation Research, Part D, 11, 166. Lin, W., Wang, C., 2004. An enhanced 01 mixe d integer LP formulation for traffic signal control. IEEE Transactions on Intellig ent Transportation Systems, 5, 238. Little, J.D., 1966. The synchronization of traffic signals by mixedintege r linear programming. Operations Research, 14(4), 568. Little, J.D., Kelson, M.D., Gartner, N.H., 1981. MAXBAND: A program for setting signals on arteries and triangular networks. Transportation Research Record, 795, 40. Liu, H., Bhimireddy, S., 2009. Evaluation of inte grated platoonpriority and advance warning flasher system at highsp eed intersections. At The 88th Annual Meeting of the Transportation Research Board, (Compendium of Papers DVDRom) 092111 Lo, H., 1999. A novel traffic signal control formul ation. Transportation Re search, Part A, 44, 436. Lo, H., 2001. A cellbased traffic control formulation: strategies and benefits of dynamic timing plan. Transportation Science, 35, 148. PAGE 131 131 Lo, H., Chang, E., Chan. Y. C., 2001. Dyna mic network traffic c ontrol. Transportation Research, Part A, 35, 721. Lou, Y., Yin, Y., Lawphongpanich, S., 2009. Robust congestion pricing with boundedly rational user equilibrium. Transpor tation Research, Part B, 44(1), 15. Lowrie, P., 1982. The sydney coordinate adaptiv e traffic system principles, methodology, algorithms. In Proceedings of the Internati onal Conference on Road Traffic Signaling. Institution of Electrical Engineers, London, 67. Luyanda, F., Gettman, D., Head, L., Shelby, S., Bullock, D., Mirchandani, P., 2003. ACSlite algorithmic architecture: applying adaptive control system technology to closedloop traffic signal control systems. design guide lines for deploying cl osed loop systems. Transportation Research Record 1856, 175. Martinez, R., Wu, W., McNeill, K., Deal, J., Ha ynes, T., Bradford, D., 2003. Hardware and softwareintheloop techniques using the OP NET modeling tool for JTRS developmental testing. IEEE Military Communications Conference, 1, 469. Matson, T. M., Smith, W.S., Hurd, F.W., 1955. Traffic Engineering. MacGrawHill, New York. Messer, C.J., Hogg, G.L., Chaudhary, N.A., Chan g, E., 1987. Optimization of Left Turn Phase Sequence in Signalized Networks using M AXBAND 86, Vol. 1 Summary Report. Report Number: FHWA/RD87/109, U.S. De partment of transportation. Michalopoulos, P.G., Stephanopoulos G., 1977. Ov ersaturated signal sy stems with queue length constraintsI : single intersection. Transportation Research, 11(6), 413. Miller, A.J., 1965. A computer control system fo r traffic networks. In Proceedings of the 2nd International Symposium on the Theory of Road Traffic Flow, Paris, OECD. Minderhoud, M.M., Botma H., Bovy P.H.L ., 1997. Assessment of roadway capacity estimation methods. Transportation Research Record 1572, 59. Mirchandani, P. B., Zou, N., 2007. Queuing mode ls for analysis of traffic adaptive signal control. IEEE Transactions on Intellig ent Transportation System, 8(1), 50. Morgan J.T., Little, J.D., 1964. Synchronizi ng traffic signals for maximal bandwidth. Operation Research, 12(6), 896. Newell, G.F., 1998. The rolling horizon scheme of traffic signal control. Transportation Research, Part A, 32(1), 39. NTOC, 2007. 2007 National Traffic Signal Report Card Technical Report. PAGE 132 132 Oak Ridge National Laboratory, 2004. Temporary Losses of Highway Capacity and Impacts on Performance: Phase 2. Report Number: ORNL/TM2004/209, Oak Ridge, TN, US. Osyczka, A., Kundu, S., 1996. A modified dist ance method for multicriteria optimization, using genetic algorithms. Computers and Industrial Engineering, 30(4), 871. Park, B., Messer, C.J., Urbanik, T., 2000. Enhanced genetic algorithm for signaltiming optimization of oversaturated intersections Transportation Res earch Record 1727, 32. Park, B., Yun, I., Ahn, K., 2009. Stochastic optim ization of sustainable tr affic signal control. International Journal of Sustainable Transportation, 3, 263. Pavlis, Y., Recker, W.A., 2009. Mathematical logic approach for the transformation of the linear conditional piecewise functions of dispersionandstore and cell transmission traffic flow models into mixedintege r form. Transportation Science, 43, 98. Ponzlet, M., 1996. Dynamik der leistungsfhi gkeiten von autobahnen (dynamics of freeway capacity). Schriftenreihe des Lehrstuhls fuer Verkehrswesen der RuhrUniversitaet Bochum, 16. Bochum. Poole, R., 2009. Reducing greenhouse gas emi ssions without reducing mobility. Reason Foundation. http://reason.org/news/show/1007715.html Rakha, H., Ahn, K., Trani, A., 2004. A developm ent of VTmicro model for estimating hot stabilized light duty vehicle and truck emi ssions. Transportation Research, Part D, 2004, 9, 49. Rakha, H., Van Aerde, M., Ahn, K., Trani, A., 2000. Requirements for evaluating traffic signal control impacts on energy and emi ssions based on instantaneous speed and acceleration measurements. Transporta tion Research Record, 1738, 56. Reiss, R.D., Thomas, M., 2007. Statistical Analysis of Extreme Values, 3rd edition. Birkhauser. Ribeiro, P.C.M., 1994. Handling traffic fluctu ation with fixedtime plans calculated by TRANSYT. Traffic Engineer ing and Control, 35, 365. Robertson, D.I., 1969. TRANSYT a Tra ffic Network Study Tool. Report Number: TRRLLR253, Transport and Ro ad Research Laboratory. Rockafellar, R.T., 2001. Optimi zation under Uncertainty: Lectur e Notes. University of Washington Rockafellar, R.T., Uryasev, S., 2000. Optimization of conditional valueatrisk. Journal of Risk, 2, 21. PAGE 133 133 Rockafellar, R.T., Uryasev, S., 2002. Conditiona l valueatrisk for general loss distribution. Journal of Banking and Finance, 26, 1446. Rockafellar, R.T., Wets, R.J.B., 1991. Scenarios and policy aggregation in optimization under uncertainty. Mathematics of Op erations Research, 16(1), 119. Sahinidis, N.V., 2004. Optimization under uncerta inty: state of the ar t and opportunities. Computation and Chemical Engineering, 28, 971 983. Schrank, D., Lomax, T., 2009. Urban Mobility Re port. Texas Transportati on Institute, Texas A&M University. Shoup, G.E. Bullock, D., 1999. Dynamic offset tuning procedure using travel time data. Transportation Research Record 1683, 84. Skabardonis, A., 1996. Determination of timings in signal systems with trafficactuated controllers. Transportation Re search Record 1554, 18. Smith, M., Clegg, J., Yarrow R., 2001. Modelin g traffic signal control. In Handbook of Transport Systems and Traffic Control (edited by: Button K.J., Hensher, D.A.) Elsevier. Sm ith, B.L., Scherer, W.T., Hauser, T.A., Park., B.B., 2002. Datadriven methodology for signal timing plan development: a computationa l approach. ComputerAided Civil and Infrastructure Engineering, 17, 387. Transportation Research Board, 2000. Highway Ca pacity Manual. National Research Council, Washington, D.C. Turner, D.B., 1994. Workbook of Atmospheric Dispersion Estimates: an Introduction to Dispersion Modeling, CRC Press. USDA Natural Resources Conservation Service, NRCS National Water & Climate Center: Climate Products: Climate Data: Wind Rose Data. ftp://ftp.wcc.nrcs.usda.gov/downloads/climate/windrose/california/san_francisco/ Vincent, R.A., Peirce, J.R., 1988. 'MOVA': Tr affic Responsive, SelfOptimising Signal Control for Isolated Intersections. Report number 170, Transport and Road Research Laboratory. Viti, F., Van Zuylen, H.J., 2009. A probabilistic model for traffic at actuated control signals. Transportation Research, Part C, in Press. Wallace, C.E., Courage, K.G., Hadi, M.A., Gan, A.G., 1998. TRANSYT7F Users Guide. University of Florida, Gainesville, FL. PAGE 134 134 Webster, F.V., 1958. Traffic Signal Settings. Road Research Technical Paper, 39, Her Majesty's Stationary Office, London, U.K.. Yin, Y., 2002. Multiobjective bilevel optimization for tr ansportation planning and management problems. Journal of Adva nced Transportation, 36(1), 93. Yin, Y., 2008a. A scenariobased model for fleet allocation of freeway service patrols. Network and Spatial Economics, 8(4), 407. Yin, Y., 2008b. Robust optimal traffic signal timi ng. Transportation Research, Part B, 42, 911 Yin, Y., Lawphongpanich, S., 2006. Internalizing emission externality on road networks. Transportation Research, Part D, 11, 292. Yin, Y., Li, M., Skabardonis, A., 2007. An offlin e offset refiner for coordinated actuated signal control system. Journal of Trans portation Engineering, 133(7), 426. Yu, L., Jia, S.C., Shi, Q.Y., 2009. Research on transportationrelate d emissions: current status and future directions. Journal of Th e Air and Waste Management Association, 59, 183. PAGE 135 135 BIOGRAPHICAL SKETCH Lihui Zhang was born in Zhoushan, China, in 1984. He studied stru cture engineering in Tsinghua University and received his bachelor s degree in 2006. Then, he joined the PhD program in transportation engineering in the University of Florida, under the supervision of Dr. Yafeng Yin. During his PhD study, Lihui Zhang worked on projec ts related to travel demand modeling, signal timing optimization and transp ortation network modeling. He was also the teaching assistant for two major transportation courses. Lihui Zhang has coauthored four papers in leading transportation journals and conference proceedings, and has delivered six presentations in various conferences. His main research interests include transportation system modeling and optimization, traffi c control and operation, traffic simulation, and travel demand modeling. 