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Integrating Adaptive Queue-Responsive Traffic Signal Control with Dynamic Traffic Assignment


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INTEGRATING ADAPTIVE QUEUE-RESPONSIVE TRAFFIC SIGNAL CONTROL WITH DYNAMIC TRAFFIC ASSIGNMENT BY LEE-FANG CHOW A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2003

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ii ACKNOWLEDGMENTS I wish to express my gratitude to my a dvisor, Dr. Gary Long, for his guidance and financial support over the course of this re search. Working for him is an invaluable experience for me as a graduate student at this institution. I would like to thank Professor Kenneth G. Courage, Dr. Bon A. Dewitt, Dr. Sherman X. Bai and Dr. Myron N. Chang for their constructive comments and recommendations. Finally, very special thanks go to my husband, Min-Tang Li, my son, Andrew, and my parents for their love and support through my study.

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iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS ................................................ii TABLE OF CONTENTS.................................................iii LIST OF TABLES......................................................vi LIST OF FIGURES....................................................vii ABSTRACT...........................................................ix CHAPTERS 1 INTRODUCTION .................................................1 1.1 Background ................................................1 1.2Problem Statement...........................................2 1.3 Goal and Objectives ..........................................3 1.4Scope and Limitations .......................................4 1.5 Dissertation Organization .....................................5 2REVIEW OF STATIC AND DYNAMIC TRANSPORTATION MANAGEMENT SYSTEMS.......................................................7 2.1 Introduction ................................................7 2.2 Static Traffic Control System ..................................7 2.3 Static Traffic Assignment .....................................8 2.3.1 User Equilibrium ......................................8 2.3.2 System Optimization ..................................14 2.3.3 Stochastic User Equilibrium ............................14 2.4 Combining a Static Traffic Signal Timing Plan and Traffic Assignment18 2.5 Real-Time Traffic-Responsive Signal Control System ..............20 2.5.1 Optimization Policies for Adaptive Control ................20 2.5.2 Non-Linear Program of Dynamic Traffic-Responsive Signal Control System.......................................21 2.5.3 Variational Inequality Model for Dynamic Traffic Control ....23 2.6 Dynamic Traffic Assignment ..................................24

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iv 2.6.1 Variational Inequality (VI) Formulation for DTA ............24 2.6.2 Route-Based Algorithms for DTA ........................27 2.7 Combining a Dynamic Traffic Signal Timing Plan and Traffic Assignment...............................................28 2.7.1 Game Theory Approach ................................28 2.7.2 Bilevel Programming ..................................29 2.8 Generalized Delay Model ....................................30 2.9 An Evaluation Tool: CORSIM 5.1 .............................31 3OPTIMIZATION MODEL FOR DYNAMIC TRAFFIC CONTROL ........34 3.1Introduction ...............................................34 3.2Optimization Model.........................................34 3.3Convexity .................................................37 3.4Solution Algorithm.........................................40 4DYNAMIC USER-OPTIMAL ROUTE CHOICE MODEL ................49 4.1 Introduction ..................................................49 4.2 Variational Inequality Models ....................................49 4.2.1 Static Transportation Network Equilibrium Model ............50 4.2.2 Dynamic Transportation Network Equilibrium Model .........52 4.3 Enhanced Solution Algorithm ....................................54 5COMBINED MODEL.............................................59 5.1 Introduction ..................................................59 5.2 Bilevel Model ................................................60 5.3 Iterative Optimization and Assignment Procedure ....................61 5.4 Detailed Procedure .............................................62 6IMPLEMENTATION .............................................66 6.1 Introduction ..................................................66 6.2 Numerical Example ............................................66 6.3 Software Implementation ........................................69 7RESULTS AND EVALUATION ....................................77 7.1 Introduction ..................................................77 7.2 Results for Static Signal Settings and Static Traffic Flows ..............77 7.3 Results for Adaptive Signal Settings and Dynamic Traffic Assignment ....77 7.4 Comparison of Static and Dynamic Transportation Management Systems.82

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v 8SUMMARY, CONCLUSIONS AND RECOMMENDATIONS ............87 8.1 Summary and Conclusions ......................................87 8.2 Recommendations .............................................89 REFERENCES........................................................92 BIOGRAPHICAL SKETCH..............................................96

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vi LIST OF TABLES Table Page 6-1O-D Demand Matrix for the Example Application.......................68 7-1Cumulative Network-Wide Vehicle Delays at 10-Minute Time Intervals for Static and Dynamic Cases.................................84

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vii LIST OF FIGURES Figure Page 3-1Flow Chart of the Solution Algorithm for the Real-Time Traffic-Responsive Signal Control Model........................................48 4-1Flow Chart of the Solution Algorithm for the DTA Model.................58 5-1Flow Chart of the Iterative Optimization Assignment Procedure............63 6-1Link-Node Structure of the Example Network ..........................67 6-2Initial Signal Settings for the Example Application ......................69 6-3Main Menu Screen................................................70 6-4Excerpt of the Initial Assignment Results for the Example Application .......71 6-5Excerpt of the Auxiliary Green Time Results ...........................73 6-6Excerpt of the Optimal Green Time Results at the Initial Iteration ...........74 6-7Excerpt of the Link Travel Time Results Based on the Simulation..........74 6-8Excerpt of the Link Travel Time Results from the BPR Formula............75 6-9Excerpt of the Comparison of Link Travel Times Based on Simulation Results and the BPR Formula........................................76 7-1Static Traffic Assignment Results....................................78 7-2Optimal Signal Settings for the Static Case.............................78 7-3Optimal Time-Dependent Signal Timing Settings for the Example Network...79 7-4Optimal Time-Dependent Traffic Flows for the Six Time Periods...........80

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viii 7-5Excerpt of the Cumulative Simulation Results at Elapsed Time of 40 Minutes for the Static Case................................................84 7-6Excerpt of the Cumulative Simulation Results at Elapsed Time of 40 Minutes for the Dynamic Case..............................................85 7-7Example of Animation Result for the Static Case........................86 7-8Example of Animation Result for the Dynamic Case.....................86

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ix Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy INTEGRATING ADAPTIVE QUEUE-RESPONSIVE TRAFFIC SIGNAL CONTROL WITH DYNAMIC TRAFFIC ASSIGNMENT by Lee-Fang Chow August 2003 Chairman: Gary Long Major Department: Civil and Coastal Engineering Recently, with their promise for Intellig ent Transportation Systems (ITS), dynamic traffic assignment (DTA) and adaptive traffic-responsive signal control have received greater attention. However, the mathematical models us ed to describe the interaction between these two systems are tenuous and considerable effort is still needed to improve the limitations described in the existing literature, especially regarding the solution capabilities, the oversaturated-queue phenomena, efficacious estimates of link travel disutilities, and a reliable evaluation method. The objectives of this research are to implement non-linear programming techniques to model dynamic traffic assignment and adaptive queueresponsive traffic control separately, and develop an iterative procedure to solve the components of traffic assignment and signal control by minimizing the overall system-wide signal delay.

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x For traffic signal control, total intersection delays in a network are minimized by allocating appropriate time splits. The genera lized intersection delay model in the Highway Capacity Manual was used to take the oversaturated queue problem into account. Improvements to DTA methods include form ulating the dynamic user-optimal route choice model as a variational inequality (VI) model, calculating the expected travel time on each link using a simulation model, applying a relaxation algorithm to produce an equivalent optimization formulation of the VI model, and developing a solution algorithm which can be implemented using existing traffic software. There is not much hope for developing exact solution algorithms to solve these two models simultaneously because of the comput ational complexity of the non-linear programs. Therefore, a heuristic procedure involving an iterative optimization assignment is used to solve the combined models. A computerized procedure was developed to implement the solution procedures and a numerical example of a traffic network was prepared to test the program. An accepted traffic simulation model, CORSIM 5.1, was used to validate the results for both static and dynamic optimizations. The test results showed that dynamic traffic assignment with adaptive traffic-responsive signal settings redu ced the network-wide delays by nearly 15%.

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1 CHAPTER 1 INTRODUCTION 1.1 Background Traffic congestion, especially during peak travel periods, is well-known to travelers in most urban areas. Most travelers evaluate the degree of congestion by the amount of time in which they are delayed. Consequently, mi nimizing delay is important to transportation engineers for setting the timing controls on traffic signals. Delay is related to both traffic volumes and the capacities of roadway links. Traffic volumes fluctuate with different levels of tr affic demand during a typical day, and also with diversions of traffic to alternative paths as mo torists strive to find the fastest paths between their origins and destinations to minimize their delays. When traffic congestion becomes intense with long queues of traffic waiting at tr affic signals such that portions of the queues must wait for the next signal cycle, known as oversaturated queues, minimizing delay becomes more complicated. In the traditional traffic management system, link capacities and traffic flow volumes are considered to remain fixed during a whol e analysis period. Therefore, the standard traffic management problem is to optimize tr affic network performance with given fixed demands for travel and a fixed supply of transportation facilities. Currently, Intelligent Transportation Systems (ITS) are being designed and developed to improve the efficiency of existi ng traffic network performance. Computer and

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2 electronic communication technologies are explo ited to provide real-time information and route guidance to motorists through Advance Traveler Information Systems (ATIS). Realtime information and route guidance are pr oduced by Dynamic Traffic Assignment (DTA) systems based on real-time traffic data, histor ical databases and dynamic traffic-responsive signal control strategies. Therefore, the in teraction in the Advanced Traffic Management Systems (ATMS) will become dynamic because the traffic flow on each link and control actions are time-dependent. 1.2 Problem Statement To date, the mathematical models used to describe the dynam ic interactions in transportation systems are still tenuous. Two si mulation-based, real-time computer systems have been developed under the DTA Project sponsored by the Federal Highway Administration (FHWA) with Oak Ridge Na tional Laboratories (ORNL) as the program manager [1]: 1.DynaMIT is the result of approximately 10 years of research and development at the Intelligent Transportation Systems Program of the Massachusetts Institute of Technology (MIT). 2.DYNASMART is the outgrowth of several y ears of research and development at the University of Texas, with the participati on of researchers currently at the University of Maryland, Purdue University, and Northwestern University. These DTA programs provide real-time computer systems for traffic estimation, prediction, and generation of traveler information and route guidance. They support the operation of ATIS and ATMS at Traffic Management Centers (TMC). However, they are simulationbased dynamic traffic assignment systems with micro-simulation of individual user decisions in response to traveler information and a macr oscopic traffic flow si mulation approach [2]. Although the analytical approach which formulates the entire system as a mathematical

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3 model has difficulties in formulating and solv ing the program, the solutions obtained can be proven to be optimal and satisfy the equilibrium conditions (i.e., the dynamic generalization of Wardrop’s principle of user-optimal network equilibrium [3]). Most current research addresses the deve lopment of DTA systems [4, 5] and realtime traffic-responsive control systems [6] indepe ndently. Allsop [7] was the first to address the interaction between traffic control and tr affic assignment, followe d by Smith [8], Sheffi and Powell [9], Smith and Vuren [10], and Yang and Yagar [11]. However, all of them only considered the static traffic situation. Some of the current development work utilizes frameworks for the integration of DTA and tr affic control systems into a combined model. Chen and Hsueh [12] proposed a combined model for a discrete-time dynamic trafficresponsive signal control system. A heuristic approach involving an iterative optimization and assignment (IOA) method was applied to solve a numerical example involving a small network. However, the traffic queue phenomenom was not considered in the traffic control model nor was a good estimate of travel disutility considered in the assignment model. Gartner and Stamatiadis [13] presented only a framework to integrate dynamic traffic assignment with real-time traffic adaptive control without any mathematical model or formulation. Chen and Ben-Akiva [14] formulated the combined dynamic traffic controlassignment system as a non-cooperative game be tween a traffic authority and users but did not give any algorithm to solve it. 1.3 Goal and Objectives The goal of this dissertation research is to develop a model to integrate the effects of queue-responsive signal timings on discretetime traffic flow patterns in dynamic traffic assignment systems. The specific objectives of this research are the following:

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4 1.To review the state-of-the-art in the areas of dynamic traffic assignment and dynamic traffic-responsive signal control. 2.To develop a combined model for dynamic traffic assignment and adaptive traffic control. In the traffic control part, the vehicle delays in the network are represented by the generalized delay model for signalized intersections in the Highway Capacity Manual [15]. In the traffic assignment part good estimates of link travel times are considered for motorists to find dynamic user-optimal paths over the network for given origin-destination (O-D) demands and signal timings from the traffic signal control model. 3.To propose an algorithm to solve this combined model. 4.To demonstrate how to program and apply existing traffic software to implement the solution algorithms and test by a numerical example of a traffic network. 5.To demonstrate the results of the combined model by simulation. 1.4 Scope and Limitations A state-of-the-art algorithm is devloped in this study to optimize total intersection delays in a given traffic network by consider ing the dynamic interaction effect between the traffic control system and traffic assignment model. The dynamic interaction is considered by adding discrete time periods as an additional dimension in the model structures and solution algorithm. Delays arising from oversaturated queues are included in optimizing signal settings and traffic flows. In this st udy, both of the cycle lengt h and signal phase plan at each intersection in a given traffic network are presumed fixed and are not altered during the optimization process. In addition, the tr avel demand between each traffic analysis zone pair is assumed fixed during the analysis period. To test the model, microscopic simulation was used as a surrogate for field data collecti on to enable controllabilty of input information and the due to limitations on resources available to the researcher. CORSIM 5.1 was chosen to perform the simulation because of its ava ilability to the researcher and because of its widespread acceptance and utilization in the USA.

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5 1.5 Dissertation Organization In Chapter 1, an introduction to the research topic and the needs for the research are presented. Improvements needed in previous research are identified and the specific objectives of the research are stated. A literature review on transportation management systems is presented in Chapter 2. The static transportation management syst em is first reviewed, followed by the dynamic transportation management system. An eval uation tool, the simulation software, and the generalized delay model which improves the oversaturated queue problem in dynamic traffic control are also reviewed. Chapter 3 presents an improved optimiza tion model for the dynamic traffic control system. A generalized intersection delay model is used to take the oversaturated queue problem into account. A strategy to improve the situation where the optimization program with a non-convex objective function may not r each the global minimum is developed. This chapter ends with an application of the Frank-Wolfe method to solve the optimization program. Chapter 4 deals with the dynamic traffic assignment system. A dynamic user-optimal route choice model is formulated as a vari ational inequality (VI) model. A simulation model, CORSIM 5.1, is used to estimate the travel time on each link. A relaxation algorithm is applied to produce an equivalent optimizati on formulation of the VI model when the travel time on each link is known. A solution algorithm which can be implemented by existing traffic software is developed. Chapter 5 presents strategies for combining the real-time traffic-responsive signal control model and the dynamic traffic assignmen t model. In Chapter 6, a computer program

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6 that was prepared to implement all the solution algorithms developed for dynamic transportation management is introduced. Seve ral existing traffic software systems are built into the program to simplify the input pro cesses for the transportation network, the O-D demand and the traffic control actions. A numerical example is used to test the program. Chapter 7 presents the comparison of the traffic performance between static and dynamic transportation management systems. Chapter 8 presents a summary and conclusions and recommends areas for further research.

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7 CHAPTER 2 REVIEW OF STATIC AND DYNAMIC TRANSPORTATION MANAGEMENT SYSTEMS 2.1 Introduction The performance of a transportation system is the result of a set of complex interactions between traffic operational contro ls and the demand for transportation services. This is the basis for the development of tra ffic control and assignment models for the design and planning of transportation management systems. In this chapter, the static transportation management system, which includes static traffic control, static traffic assignment and a “balancing” process to integrate these two systems together, is first reviewed, followed by recent research which extends these concepts to the dynamic case. Greater details are given for the algorithms more directly related to those needed in adaptive queue-responsive traffic signal control with dynamic traffic assignment as developed in this research. 2.2 Static Traffic Control Systems Traffic control systems deal with how to determine the optimal traffic signal timings to reduce fuel consumption, minimize delay a nd also improve safety. TRANSYT-7F [16] is the best-known software for determining the static network-wide optimal signal timing settings. With fixed-flow input, TRANSYT-7F us es a macroscopic, deterministic simulation and optimization procedure to find the best solution. The optimization procedure can be briefly described as follows:

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8 Step 1:An initial signal timing plan is si mulated by the traffic model and an initial Performance Index (PI) is calculated. Step 2:The signal timing setting is changed by a specified amount and the resulting traffic flow is re-simulated. A new PI is then calculated. Step 3:The new PI is compared with the prev ious value. If no further improvement can be made by varying the signal setting, the procedure will stop. Otherwise, go to Step 2. Although TRANSYT-7F is a powerful tool to determine static optimization, it is not feasible to enhance it to the dynamic case because it is too complicated to modify the macroscopic simulation procedure with time-dependent traffic flows. 2.3 Static Traffic Assignment The traffic assignment system can be stated as finding the link flows when given the O-D traffic volumes, the network, and the link pe rformance functions. To solve the problem, a rule of how motorists choose their routes has to be specified. The equilibrium traffic flows are determined according to the rule and link performance functions. Three definitions of equilibrium are user equilibrium, system optimization, and stochastic user equilibrium. 2.3.1 User Equilibrium User-equilibrium (UE) assumes that each motorist has full information about link performance relationships and he or she will choose the route with minimum travel impedance. The travel impedance can include many components. However, travel time is often used as the sole measure of link impeda nce [3]. For each origin-destination zone-pair, at user equilibrium, the travel disutilities on all used paths are equal and no traveler can improve his or her travel time by unilaterally changing routes [3]. Note that a network is defined mathematically as a set of nodes and a set of links connecting these nodes. A path is a sequence of directed links leading from one node to another.

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9 (1) In the mid-1950s, a formulation representing the UE condition was developed and is known as Beckmann's transformation [17]. Th e formulation can be expressed as follows: subject to k f rs k = qrs r, s(2) frs k 0 k, r, s(3) with the incidence relationship xm = rs k f rs k rs m,k m (4) where xm = traffic volume assigned to link m qrs = total traffic volume interchanging from origin r to destination s f rs k = traffic volume flowing along path k connecting origin r to destination s dm(xm) = travel disutility on link m related to traffic volume, xm, on link m rs m,k = 1, if link m is on path k between origin-destination pair r-s = 0, otherwise This program does not have any intuitive economic or behavioral interpretation. It is only a mathematical model that is utilized to solve the equilibrium problem. To prove the equivalence between the solution of Beckma nn's transformation and the UE condition, the method of Lagrange multipliers, which involves an auxiliary function known as the Lagrangian, is applied as follows. Given the incidence relationship, xm = xm( f ) in Eq. 4, the Lagrangian whose stationary point coincides with the minimum of the constrained optimization in Eq. 1 to Eq. 4 can be formulated as:

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10 (6) (7) (8) (9) L( f y ) = z[ x ( f )] + rs yrs(qrs k f rs k )(5) where yrs denotes the dual variable associated with the flow conservation constraint for O-D pair r-s in Eq. 2. The stationary point of the unconstrained Lagrangian can be found by solving for the root of the gradient, f L( f y ) = 0 and y L( f y ) = 0 y L( f y ) = 0 simply states the flow constraints. However, given the nonnegativity constraints of f 0 in Eq. 3, the following two conditions, associated with f L( f y ) = 0, have to hold because the stationary point of L( f y ) can occur either for positive f or it can be on the boundary of the feasible region where some f rs k = 0. Obviously, the condition f 0 has to hold as well. The above first-order condition can be obtained by the following calculation: where uk rs = travel disutility along path k connecting origin r to destination s.

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11 (14) The general first-order conditions for the minimization program in Eqs. 1 to 4 can be expressed as fk rs (uk rs yrs) = 0 k, r, s(10) uk rs yrs 0 k, r, s(11) k f rs k = qrs r, s(12) frs k 0 k, r, s(13) From Eq. 11, yrs is less than or equal to uk rs, the travel disutility along path k connecting origin r to destination s. Therefore, yrs equals the minimum path travel disutility between origin r and destination s. Eq. 10 holds at the point that minimizes the objective function for either fk rs = 0 and (uk rs yrs) > 0, or fk rs > 0 and (uk rs yrs) = 0. That means if the travel disutility, uk rs, on path k is greater than yrs, the minimum path travel disutility, the flow on this path is zero; or if uk rs = yrs, the flow fk rs is positive. This simply interprets the principle of UE. For the solution of the UE program to be uni que, it is sufficient only to prove that the objective function is convex with respect to li nk flow, since the conve xity of the feasible region is assured for linear equality and nonnegative constraints. To prove that z( x ) is convex, the matrix of the second derivatives of z( x ) with respect to x (the Hessian) has to be positive definite: where m and n are both link indexes. When m is not equal to n, the second derivatives would be zero.

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12 (15) The link travel disutility related to link traffic volume, d ( x ), is normally positive and increasing. Therefore, the above matrix is pos itive since it is a diagonal matrix with positive entries. Since the objective function is convex and constraints are all linear, the minimization model can be solved by the Frank-Wolfe (F W) method [18]. The FW method is based on finding a descent direction by minimizing a lin ear approximation of the objective function at the current solution point. This linear approximation is given by z( un) = z( xn) + z( xn) ( un xn) with respect to xn for the nth iteration. This linear function of un has to be minimized subject to the constraints of the original model: minz( un) = z( xn) + z( xn) ( un xn)(16) subject to k g rs k = qrs r, s(17) grs k 0 k, r, s(18) with the incidence relationship um n = rs k g rs k rs m,k m (19) where g denotes the path flows. The objective function can be simplified to min z( xn) unbecause z( xn) and z( xn) xn are constant when xn is known. Moreover, min z( xn) un = minm (z( xn)/xm) um n = min m dn m(xn m) um n. Since the travel times dn are fixed for a given

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13 xn, and the program calls for minimizing the to tal travel time over the network with flows independent of travel times, the solution of the above linear program, un which is an auxiliary flow representing the descent direction, can be solved by assigning all motorists to the smallest travel-disutility path connecting their origin and destination. After the descent direction is deci ded, the optimal move size between xn and un can be found by solving the following linear program. is between 0.0 and 1.0 and is chosen so that the link volumes are as close as possible to the user-optimized equilibrium loadings. The bisection method can be used to solve the linear program. minz[ xn + ( un xn)](20) subject to o 1(21) The initial solution can be determined by a pplying an all-or-nothing network loading procedure to an empty network. The remain ing algorithmic steps are to find the descent direction and the optimal move size iterativel y until the stopping criterion for solving the UE program is met. The assumption in the UE conditions that each motorist has full information about link performance relationships may be extended to the dynamic case. However, with the Advance Traveler Information System, motorist s could be provided with travel information for pre-trip planning (i.e., travel mode, depart ure time, and route) and guidance for en-route diversion. Therefore, with the assumption that ATIS could continuously provide all travelers with full information about all link disutilities, the UE conditions may be adoptable in the dynamic situation. Moreover, enhancing the model structures and solution algorithms to the dynamic case is feasible by adding discrete time periods as an additional dimension.

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14 2.3.2 System Optimization The system-optimization (SO) program minimizing the total travel time spent in the network while satisfying the flow conservation constraints is expressed as follows [19]: min z( x ) = m xm dm(xm)(22) subject tok f rs k = qrs r, s(23) frs k 0 k, r, s(24) with the incidence relationship xm = rs k f rs k rs m,k m (25) The general first-order conditions for the minimization of the SO program can be derived using a similar method as for UE and expressed as follows [3]: fk rs ( uk rs yrs) = 0 k, r, s(26) uk rs yrs 0 k, r, s(27) k f rs k = qrs r, s(28) frs k 0 k, r, s(29) where uk rs = m rs m,k [dm(xm) + xm d dm(xm)/ d xm] is the marginal total travel time on path k connecting O-D pair r-s. The marginal total travel times on all of the used paths connecting a given O-D pair are equal for the SO program at its optimal va lue which minimizes the total travel time for the network, not for the users. It does not repr esent an equilibria situation and is not stable. 2.3.3 Stochastic User Equilibrium Stochastic user equilibrium (SUE) models relax the assumption that motorists have full information about link travel times on all links in a network by assuming that each motorist may perceive a different travel time (disutility) but will still choose a path with the

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15 (30) (31) least perceived disutility path from his or her origin to his or her destination. At SUE, no motorist can improve his or her perceived disutility (travel time) by unilaterally changing routes. A minimization program was developed by Sh effi and Powell [20]. The solution of the program is the desired set of SUE flows. The unconstrained program is shown as follows: where Uk rs represents the perceived travel disutility on route k between origin r and destination s. Uk rs is a random variable with the mean equal to the actual travel disutility uk rswhich is measured at a given flow level x The first term of the objective function includes the expected perceived travel disutility function from origin r to destination s, E[min Uk rs| urs( x )]. The partial derivative of this function with respect to Uk rs is the probability of choosing path k between r and s, Pk rs, which is the probability that Uk rs is less than the disutility of any other route, Ui rs, between r and s. The various models for the probability of selecting each alternative route differ from each other in the assumed distri butions of the variance of Uk rs, rs k. If rs k are identically and independently distributed Gumbel variables, Pk rs can then be expressed as a logit model:

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16 The network loading approach (in which the link disutility function is not flow dependent) associated with this multinomial logit choi ce model, known as the STOCH method, includes a forward step to calculate the “link weights” according to the probabilities of selecting each reasonable alternative route (which includes only links that take the traveler further away from the origin and closer to the destinati on) and a backward step to assign the flows at destinations back to the origins [21]. If rs k is normally distributed, the joint density function of rs is the multivariate normal (MVN) function with mean vector 0 and variance matrix Vrs. The distribution of Urscan then be modeled as multivariate normal, too: Urs ~ MVN ( urs( x ), Vrs). However, Pk rscannot be expressed analytically since the cumulative normal distribution function cannot be evaluated in closed form. One of the approaches to compute Pk rs is based on a Monte Carlo simulation procedure which has no restriction on the number of alternative routes unlike other analytical approximation methods A network loading algorithm based on the Monte Carlo procedure was tested by Sheffi and Powell [22] and summarized by Sheffi [3] as: Step 0:Initialization. Set n = 1 where n is the number of iterations. Step 1:Sampling. Sample Dn m from Dm ~ N(dm, dm) for each link m. ( is variance of the perceived travel time over a road segment of unit travel time.) Step 2:All-or-nothing assignment. Based on Dn assign q to the shortest path connecting each O-D pair r-s. This step yields the set of link flows xn. Step 3:Flow averaging. Let xn m = [(n-1)xn-1 m + xn m]/n, m(32) Step 4:Stopping test. (a) Let

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17 (33) (35) (b) If maxm {n m/xn m} stop. The solution is xn. Otherwise, set n = n + 1 and go to step 1. To prove the equivalency between the a bove minimization program (Eq. (30)) and the SUE condition, the first-order conditions of this program have to coincide with the SUE conditions, which can be characterized as follows [3]: fk rs = qrs Pk rs, k, r, s (34) The partial derivative of z ( x ) with respect to a typical path flow, fk rs, can be written as Since the first-order condition for unconstrained minimizations requires only that the gradient vector of the objective function be equal to 0 the above derivatives have to equal zero for all k, r, and s, meaning that fk rs = qrs Pk rs k, r, s (36) which is the SUE condition. Moreover, the flow conservation constraints (k f rs k = qrs) are automatically satisfied at equilibrium since k Pk rs =1. The uniqueness conditions of z( x ) were also demonstrated by Sheffi [3] by showing that the Hessian matrix of z( x ) is positive definite. The core of any descent method for solv ing the unconstrained minimization model with a nonlinear convex function of several vari ables is to find the descent direction and minimize the objective function along that direc tion at each iteration. However, the iterative

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18 (37) process to find the descent direction and move size of Eq. 30 is difficult because the direction vector computed at a particular iteration could be random in some cases. An algorithm known as the method of successi ve averages (MSA) which is based on a predetermined move size along the descent direction was proposed to solve Eq. 30 and proved to converge to the minimum by Sh effi and Powell [22]. The algorithm is summarized in Sheffi [22]: Step 0:Initialization. Perform a stochastic ne twork loading based on a set of initial travel disutilities d0. This generates a set of link flows x1. Set n = 1. Step 1:Update. Set dn = d ( xn), n. Step 2:Direction finding. Perform a stochastic network loading procedure based on the current set of link travel disutilities, dn. This yields an auxiliary link flow pattern un. Step 3:Move. Find the new flow pattern by using the predetermined move size 1/n. xn+1 = xn + (1/n) ( un xn) Step 4:Convergence criterion. Further research is expected to enhance the complicated model structures of SUE to the dynamic case. However, the MSA method using the predetermined move size to guarantee convergence can be applied in the solution algorithm of the DTA model developed in this study. 2.4 Combining a Static Traffic Signal Timing Plan and Traffic Assignment Gartner and Improta [23] derived the fo llowing compound mathematical optimization formulation to describe the static traffic management system:

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19 (38) subject to flow conservation constraints: and non-negativity constraints: hp ; r(i,k) 0(39) where M= the set of management variables under the control of the traffic manager I= the set of all O-D pairs (i,k) P(i, k)= the set of all simple paths between O-D pair (i, k) F= the set of all link flows fj= the flow on link j r(i, k)= the rate of trip interchange (demand in vehicles) between origin node i and destination node k, with (i, k) I hp = the flow on path p, with p P(i, k)Z1 = j fj wj (fj, M) Z2 = jcj(x) dx cj (fj) = the average user-perceived travel cost function on link j wj (fj, M) = a more general performance function reflecting the multiplicity of objectives pursued by the traffic manager in the public’s interest The formulation consists of a primary optimization program (the systemoptimization), min Z1 (), and a secondary optimization program (the user-optimization), argmin Z2 (). The “argmin” of a mathematical program is the optimal solution of the program. Examples of computational procedur es for solving the whole problem were also given.

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20 The above framework can be extended to the dynamic case by changing the O-D demand and control actions to being time-dependent. However, there is no mathematical methodology that can solve the compound program simultaneously. If the system dynamics are slow and the process is relatively stable and predictable, the steady state models may be applied in consecutive time steps. If the dynamics of the system are more stochastic and thus less predictable, the steady-state approach is not applicable any more. 2.5 Real-Time Traffic-Responsive Control System The application of advanced technologies in ITS makes it possible for on-line traffic control systems to respond to real-time traffi c information. Three published research studies about real time traffic control systems, whic h take dynamic traffic demand into account, are optimization policies for adaptive control, non-linear program of dynamic traffic-responsive signal control system, and variational inequality model for dynamic traffic control. These are discussed in the following sections. 2.5.1 Optimization Policies for Adaptive Control Gartner [24] used a rolling horizon approach, which is used by operations research analysts in production-inventory control, to develop a demand-responsive strategy for traffic signal control. The basic steps in the process are as follows: Step 0:Determine the stage length, which is the project horizon consisting of k intervals, and the roll period r, which specifies the traffic condition is updated; the process is repeated every r intervals. Step 1:Obtain flow data for the first r interval s from detectors and calculate flow data for the next k-r intervals from a model. Step2:For a given switching sequence, the total delay on each approach was formulated as: d(t1, t2, t3) = i=1 k (Qo + Ai Di)(40) where Qo = initial queue Ai = arrivals during interval i

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21 (41) Di = departures during interval i t1, t2, t3 = possible switching times during this stage Calculate the optimal switching policy for the entire stage by an optimal sequential constrained search method in which the tota l delay is evaluated sequentially for all feasible switching sequences. Step 3:Implement the switching policy for the roll period only. Step 4:Shift the projection horizon by r units to obtain a new stage; repeat steps 1-4. The rolling horizon strategy provides an effective method for solving the dynamic traffic control problem in real time. Ho wever, one argument a bout the "rolling horizon" technology in traffic control is that this met hod minimizes the total delay at each roll period r using the predicted and thus unreliable inform ation about traffic flow for the future finite time period (the remaining horizon length) [13]. 2.5.2 Non-Linear Program of Dynamic Tr affic-Responsive Signal Control System Chen and Hsueh [12] formulated a nonlinear program for the dynamic trafficresponsive signal control system using Webster’s delay formula. Webster’s model uses an analytical method to replicate the delay time, which includes the time a vehicle is stopped while waiting to pass through the intersection and the time lost during acceleration and deceleration from/to a stop. Chen and Hsue h proposed a non-linear program as illustrated in Eq. 41 to minimize the total system delay. First, the approach delay is obtained by multiplying the traffic flow on an approach by the average delay per vehicle. The total system delay is then calculated by summing up the approach delay of each intersection in the network for every time interval. The proposed model is as follows:

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22 subject toHa m(t) = [va m (t) / Sa m] [CI(t) / gI m]I, a, m, t(42) Ha m(t) < 1a, m, t(43) gI m(t) min gI mI, m, t(44)m [gI m(t) + lI m] = CI(t)I, t(45) where a= link designation CI(t)= cycle length for intersection I during interval t gI m(t)= green time associated with phase m at intersection I during interval t Ha m(t)= degree of saturation over link “a” a ssociated with phase m during interval t lI m= lost time associated with phase m at intersection I Sa m= saturation flow on link “a” associated with phase m va m(t)= exit flow from link “a” in phase m during interval t The Frank-Wolfe method was used to solv e the model since the objective function was proved to be convex and all constraints are linear. The framework of the above nonlinear program is adopted in this research because of the intuitive interpretation of the mathematical model, the promise of the op timization model for finding the global minimum, and the amenability to computation with the application of computer programming. Although the concept of the above framework can ideally describe the environment of the real-time traffic control system, the delay f unction is the major key to locating the optimal signal timing settings. The delay function used in the above research over simplified the traffic congestion situation but not taking the spill-back queue problem into account.

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23 (49) 2.5.3 Variational Inequality Model for Dynamic Traffic Control The characteristics of the VI formula have been extensively studied in economics. A mathematical program often may be more intuitive for representing a real situation. However, the VI formulation is much broader than a mathematical program. Chen and Ben-Akiva [14] defined a dynami c system optimum principle, which can be expressed in Eq. 46 48 for dynamic traffi c control, as: for each intersection, any phase with a positive green time must have an equal and minimal marginal delay. gi m(t) (ci m(t) -i(t)) = 0 i, m(46) ci m(t) -i(t) 0 i, m(47) gi m(t) 0 i, m(48) where gi m(t) = green time for phase m of intersection i at time t ci m(t) = marginal cost for phase m of intersection i at time ti(t) = minimal marginal phase delay for intersection i at time t The equivalent formulation for Eqs. 46-48 was then stated as follows: where gi m*(t) is the dynamic system optimal setting, ci m*(t) is the marginal phase delay when the timing is gi m*(t) and is the feasible set for green time splits. The VI formula of the real-time traffic-responsive signal control system is broader than the mathematical program. However, no solution algorithm was developed in the above research although the equivalent mathematical formulation for the VI model was provided.

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24 2.6 Dynamic Traffic Assignment Static traffic assignment distributes on OD traffic flow based on the assumption that the traffic flow on a network is static, whereas dynamic traffic assignment models can present traffic situations on a network in infi nitesimal short time periods. Therefore, static traffic assignment is sufficiently effective to predict the traffic flow for a long-term period but dynamic traffic assignment is powerful enough to analyze the traffic state over a specific time such as a peak hour or shorter period. Merchant and Nemhauser [25], who proposed a discrete time, nonlinear and systemoptimal model for the case of single-destinati on networks, were among the first researchers to address the dynamic traffic assignment syst em. The techniques available for DTA have progressed since then. Two kinds of formulations published in existing research studies about DTA are VI formulations and route-ba sed algorithms. These are discussed in the following sections. 2.6.1 Variational Inequality (VI) Formulation for DTA Drissi-Kaitouni [4] proved three theorems about the VI formulation for DTA, which is equivalent to the equilibrium DTA condition: Sk( h ) = ut pq if hk > 0 p, q, t, k(50) or Sk( h ) ut pq if hk = 0 p, q, t, k(51) where Sk(h) is the travel cost on a path k and ut pq is the travel cost on the shortest path from origin node p to the destination node q at period t, given traffic flow h on the network. These three theorems are summarized as follows: Theorem 1. Let h be the set of feasible path flows. Then the above equilibrium condition, Eq. 50, may be rewritten as a variational inequality:

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25 Find h h such that pqtk (Sk( h ) ut pq ) (hk hk ) 0 h h. Theorem 2. The variational inequality in Theorem 1 is equivalent to the following variational inequality: Find h h such that pqtk Sk( h ) (hk hk ) 0 h h. Theorem 3. The variational in equality in Theorem 2 has a link variational inequality formulation: Find f f such that a Sa( f ) (fa fa ) 0f f. where f is the set of feasible link traffic patterns over the network. He also pointed out that according to the symmetry principle, the above link-based VI program is equivalent to the UE mathem atical programming model in Eq. 1 if Sa( f ) is equal to Sa(fa). Chen and Hsueh [12] proposed a nested diagonalization algorithm to solve the following dynamic user-optimal route choice m odel which was formulated as a VI model. c* [ u u*] 0, u (52) where denotes the feasible region that is delineated by the following constraints: Flow conservation constraints:P hp rs (k) = qrs (k) r, s, k(53) Flow propagation constraints: uapk rs (t) = rs p k hp rs (k) apk rs (t) a, t(54) uapk rs (t) = ubpk rs (t-b(t)) apk rs (t) r, s, p, k, t, ap, bp, aB(j), bA(j)(55) Nonnegativity constraints: hp rs (k) 0 r, s, p, k(56)

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26 where A(j)= set of links whose tail node is j B(j) = set of links whose head node is j hp rs(k)= flows from origin r departing during in terval k over route p toward destination s qrs(k)= departure flows between origin-destination rs during interval k ua(t)= inflow entering link “a” during interval t uapk rs(t) = portion of inflows entering link “a” during interval t which departs origin r during interval k over route p toward destination sa(t)= travel time on link “a” during interval tapk rs(t) = 1, if flows departing origin r over route p during interval k entering link “a” during interval t = 0, otherwise Since the Jacobian matrix of the travel time function was asymmetric, two relaxation (or diagonalization) techniques were needed to transform the VI model to a non-linear optimization problem. One is to estimate the li nk travel time and the other is to temporarily fix the flow on each link other than on the subject time-space link. The solution algorithm which combines diagonalization and the Frank-Wolfe method can be summarized as follows: Step 0:Initialization. Step 1:“First Loop” Operation. Update the estimated link travel times based on the initial traffic flow conditions. Construct the corresponding feasible time-space network based on the estimated link travel times. Step 2:“Second Loop” Operation. Modify th e initial feasible solution based on the timespace network constructed by the estimated link travel time from Step 1. Fix the flows on all links to transform the VI model to a non-linear optimization model. Step 3:Solve the optimization program in Step 2 using the FW method.

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27 Step 4:Convergence check for the “Second Loop” operation. Step 5:Convergence check for the “First Loop” operation. The Variational Inequality is a useful technique to formulate the DTA system. Applying the relaxation algorithm to relax the link interactions to find the equivalent optimization model is the key action to solve th e VI model. However, Chen and Hsueh [12] could not define a good estimate of link travel tim es to relax the link interactions in the DTA model when they derived the above soluti on algorithm. The VI formulation and the relaxation algorithm are applied in this research to solve the DTA system. Greater effort is devoted to finding reasonable estimates of the link travel times. 2.6.2 Route-Based Algorithms for DTA To solve the dynamic user-optimal route c hoice problem, Chen et al. [26] compared three route-based algorithms with the linkbased algorithm using the Frank-Wolfe method. The three route-based algorithms were the desegregate simplicial decomposition of Larsson et al., desegregate simplicial decomposition of von Hohenbalken, and a gradient projection method. They found that the link-based algorith m is inferior to the route-based algorithms in terms of execution time but superior in te rms of memory requirements because the results from the route-based algorithms for the DT A system include not only the link traffic volumes but also the information related to turning-movement volumes. The traffic volume for each turning movement at every intersection is essential to feed into a traffic control model. With this information, a traffic control model can determine the optimal green time for each signal phase which allows only specific traffic movements to move. However, CORSIM’s assignment feature using a link-based algorithm to solve the assignment problem also provides turning movement information using extra

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28 memory to record the route information at each iteration. Therefore, the link-based algorithm is adopted in this research with th e application of CORSIM’s assignment feature. 2.7 Combining a Dynamic Traffic Signal Timing Plan and Traffic Assignment Most research addressing the interaction between traffic control and assignment only considers the static traffic situation. Two approach techniques for integrating DTA with real-time traffic-responsive control systems, which were proposed by recent studies, are game theory and bilevel programming. These are summarized in the following sections. 2.7.1 Game Theory Approach Game theory provides a framework for mode ling a decision-making process in which more than one player is involved and each individual’s actions determine the outcome jointly. Fisk [27] analyzed the characteristics of a Stackelberg game in which one player knows how the other players will respond to any decision he or she may make, and the characteristics of a Nash noncooperative game in which each player is trying to minimize his or her performance function without prio r knowledge of the other players’ functions. The user equilibrium condition can be stated identically as a Nash noncooperative game with each traveler considered as a player. The signal optimization model can be formulated as a Stackelberg game in which the traffic author ity tries to minimize the network performance function and motorists choose static user-optimal routes. Chen and Ben-Akiva [14] combined the dynamic traffic control system and the dynamic traffic assignment system as three game theories. First the combined controlassignment system is formulated as a Cournot game, in which the players, i.e., the traffic authority and the users, choose their strategies simultaneously. In this game, each player makes his or her move independently without knowing the strategy of the other. Second, the

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29 combined control-assignment system is formul ated as a Stackelberg game in which the authority sets the signal timing first by anticipa ting the traffic flow and motorists then choose their best routes accordingly. In the end, the combined control-assignment system is formulated as a monopoly game in which both control and assignment solutions are systemoptimal since the traffic authority controls both the signal setting and traffic assignment. However, only frameworks, not solution algorithms, were discussed for these game theories. 2.7.2 Bilevel Programming Bilevel programming involves two levels of mathematical programming that can be viewed as a particular case of Stackelberg game s. At the upper level, decision makers are bound by the decisions of the lower levels a nd maximize their own profit accordingly, taking into account the reactions of the lower levels. Marcotte [28] presented the static network design problem as a bilevel programming model. Two of the four heuristic procedur es analyzed to solve the bilevel model were iterative optimization assignment procedures with user-optimized and system-optimized equilibrium models. The numerical experiments showed that the iterative optimization assignment method with the use r-optimal equilibrium model yi elded a near-optimal solution. Chen and Hsueh [12] combined the dynamic traffic control and the traffic assignment systems into a bilevel model. A heuristic procedure, involving an iterative optimization and assignment method, was used to solve the model for a near-optimal solution. Since it does not appear possible to find a polynomial algorithm to solve the combined model because of the non-linear progra mming structures in both the traffic control and assignment models, the concept of bile vel programming and the iterative optimization and assignment method are pursued in this research.

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30 (57) 2.8 Generalized Delay Model A generalized delay model was proposed by Fambro and Rouphail [29], and validated by Rouphail et al. [30]. The model is a much improved version over the previous HCM model in estimating delay at vehicle-actua ted traffic signals. Moreover, it includes a term to account for the effects of queues not being cleared during a signal cycle, a condition referred to as having oversaturated queues during variable demand conditions. This generalized delay model for Chapter 16 of the HCM is as follows: There are three cases for estimating d3: (1) If no oversaturated queue exists at the start of the analysis period, d3 = 0. (2) If an oversaturated queue exists at the start, but not at the end of the analysis period, d3 = (3600Ni/c) [0.5Ni/Tc(1-X)]. (3) If an oversaturated queue exists at both the start and the end of the analysis period, d3 = (3600Ni/c) -1800T[1 min(X, 1.0)]. where C= average cycle length g= average effective green time X= degree of saturation for a subject lane group PF= progression adjustment factor T= analysis period k= parameter for given arrival and service distributions I= parameter for variance-to-mean ratio of arrivals from upstream signal c= capacity of the lane group Ni= initial queue at the start of the analysis period

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31 This generalized delay model for signalized intersections can provide good estimates of intersection delays in a network for allocating appropriate green times at intersections because it takes the effects of oversaturated queues into account. 2.9 An Evaluation Tool: CORSIM 5.1 There are several simulation software packages that are currently utilized by transportation professionals to evaluate diffe rent traffic scenarios. For example, the TRansportation ANalysis SIMulation System (TRANSIMS) is designed to simulate the detailed interaction between individuals' activ ity plans and congestion on the transportation system [31]. The program is capable of simu lating the movements of individuals across the network, including mode selection, on a secondby-second basis. However, intensive data collection and extreme computer execution tim e are currently obstacles to large-scaled applications of TRANSIMS. CORSIM 5.1 [32], on the other hand, is a comprehensive microscopic traffic simulation computer program using commonly accepted vehicle and driver behavior models. Therefore, it is usually used as an evaluation tool for traffic signal timings although it also provides a user-equilibri um platform to perform a traffic assignment. CORSIM’s traffic assignment feature as well as its evaluation function are adopted in this research although there are not many reports ev aluating CORSIM’s effectiveness for traffic assignment. It is applied because the rou ting logic in CORSIM 5.1 will convert the O-D tables into turning percentages for each inters ection, which can then be fed into the traffic control model. In each iteration of CORSIM’s assignment procedure, an intermediate solution is obtained using link travel times produced by the previous iteration (direction finding). An iterative line search is then applied to the range between the current intermediate solution and the previous iterati on solution to obtain an optimal solution for

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32 each iteration (optimal moving size). The assignment process will terminate when the change in the objective functi on between two successive iterati ons is less than a threshold value (convergence test). The Bureau of Public Roads (BPR) and the modified Davidson link impedance functions are available to evaluate the travel time on a link. The BPR formula is as follows: T = T0 [1 + a(V/C)b](58) The modified Davidson impedance function is as follows: T = T0 [1 + aV/(S-V)]if V bS(59) orT = T0 [1 + ab/(1-b)] + aT0 (V-bS)/[S(1-b)2] if V > bS(60) where T= travel time on link T0 = free-flow travel time on link a, b= parameters to be estimated for each class of roadway V = volume on link C= capacity on link S= saturation rate on link The BPR formula increases travel times even after a link's traffic flow is higher than its capacity. The modified Davidson function, using a linear extension at a volume close to capacity, has been shown to reduce the error in traffic assignment results relative to actual volume counts. However, there is very lim ited estimation experience reported in the literature for the Davidson func tion. Instead, the BPR formula is frequently adopted in practice and the model parameters which appeared in the original publication are usually employed.

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33 To estimate the capacity used in the impedance functions, the discharge rates for turns are held constant, and are estimated initially for free-flow conditions. These estimates could be calibrated after the assignment of turn movements, then applied to the next assignment process, if requested. The following equation is used for capacity calibration: Cn = [rCc+ (100-r)Cp] where Cn = new estimate of capacity (for the next assignment iteration) r= capacity smoothing (in a percentage) Cc= calculated capacity using previously assigned volumes Cp= previous estimate of capacity

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34 CHAPTER 3 OPTIMIZATION MODEL FOR DYNAMIC TRAFFIC CONTROL 3.1 Introduction This chapter presents the techniques applied for developing a mathematical model and the solution algorithm for optimizing dynamic traffic controls. In the mathematical model, the total intersection delay in a network is minimized by allocating optimal time splits of each traffic si gnal. The HCM generalized delay model is used to estimate the average delay for vehicles arriving during a di screte time interval. However, the objective function of the optimization m odel Eq. 61, cannot be proven to be convex. An approach which can deal with this situation is presented. The last section in this chapter presents a solution algorithm for solving the optimization model. 3.2 Optimization Model As described in Section 2.8, three cases corresponding to the oversaturation situations presented in the HCM delay model are formulated for the dynamic traffic control system by adding time as an additional dimension: 1.No oversaturated queue exists: the degree of the saturation at the previous time period, Xp m(t-1), is less than one. 2.An oversaturated queue exists at the start, but not at the end of the analysis period: the degree of the saturation at previous time period, Xp m(t-1), is greater than or equal to one but the degree of the satu ration at current time period, Xp m(t), is less than one. 3. An oversaturated queue exists at both the start and the end of the analysis period: both of the degrees of the satu ration, at previous time period, Xp m(t-1), and at current time period, Xp m(t), are greater than or equal to one.

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35 (61) (62) (63) (64) The model for the dynamic traffic control system is expressed as follows: If Xp m(t-1) < 1 and Xp m(t) < 1, then If Xp m(t-1) 1 and Xp m(t) < 1, then

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36 (65) If Xp m(t-1) 1 and Xp m(t) 1, then subject to gI p (t) I, p, t(66) P [gI p (t) + l I p] = CI (t) I ,t(67) where m= link index B(I)= set of links entering intersection I I= intersection index p= phase index vm p(t) = vehicle flow on link m during phase p at time t dm p(t) = vehicle delay on link m during phase p at time t sm= saturation flow on link m gI p(t)= green time associated with phase p at intersection I during interval t CI(t)= cycle length for intersection I at interval t Xm p(t)= degree of saturation associated with phase p at intersection I during interval t = minimum green time associated with phase p at intersection I l I p= lost time associated with phase p at intersection I

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37 PF = progression adjustment factor T = length of time interval k = parameter for given arrival and service distributions R = parameter for variance-to-mean ratio of arrivals from upstream signal The objective function minimizes the total intersection delay, which is represented by the sum of the products of the traffic flow and the average vehicle delay for each signal phase at each intersection during each tim e period. The average delay model, dm p(t), is essentially the HCM generalized delay model wh ich has either two or three terms depending on the degree of saturation during the previous time period. There are two constraints. The first one restricts the green time assigned to each phase to be not less than the minimum green time which is usually determined based on the green time needed for pedestrians to cross streets safely. The second constraint rest ricts the sum of green tim es and lost times for all signal phases at each intersection to be equal to the cycle length. 3.3 Convexity The above optimization model is a nonlinear programming model with linear constraints for conserving cycle length and mini mum green time. Since the constraints are all linear, this nonlinear programming model can be solved by the Frank-Wolfe method. The Frank-Wolfe method is a feasible direction me thod and its direction finding procedure is a descent method, meaning that the objective func tion value decreases at every iteration. This method converges to a local minimum, which would naturally be a global minimum for convex objective functions. The objective function of the above optimization model is the sum of nonlinear functions associated with vehicle delay at each intersection. Without considering the

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38 (68) (69) incremental delay due to oversaturated queues, the objective function, Z(g), can be proven to be a convex function by showing that the matrix of the second derivatives of Z( g ) with respect to g (the Hessian) is positive definite. Assuming there are I intersections in the analysis network and each intersection has p phases, then g = [g1 1 (1), g1 1 (2),..., g1 1 (t), g2 1 (1),..., gp I (t)]. Let n = I p t, and g = [g1, g2,..., gn]. The Hessian is calcula ted by using a representative term of the matrix. The derivative of Z( g ) is therefore taken with respect to green time on the mth and nth elements in g so This means that all of the off-diagonal elements of the Hessian, 2Z( g ), are zero and all of the diagonal elements are given by 2Z( g )/ 2gn. In other words, If 2Z( g )/ 2gi 0, this matrix is positive definite b ecause it is then a diagonal matrix with positive entries. However, for the oversaturated queue situation, the third term in the generalized delay model, which estimates dela y due to oversaturated queues for time interval

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39 (70) (71) t, is dependent on the green time at time interval t-1 because the initial queue at the start of interval t is a variable of the green time at interval t-1. Therefore, This means that not all of the off-diagonal elem ents of the Hessian are zero. In other words, Even if 2Z( g )/ gi gj 0, this matrix cannot be proven to be positive definite because it is not a diagonal matrix. Although the objective function of the above model cannot be proven to be a convex function, there are some methods which can fi nd a near-optimal local minimum. One of these methods is the multistart approach which chooses several starting points and then compares the local minimums determined from those starting points. Since a good starting point is important for finding the global optimum, the static network-wide optimal signal settings, which can be obtained from TRANSYT -7F, are used as the initial signal timings for a near-optimal minimum.

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40 3.4 Solution Algorithm A procedure using the Frank-Wolfe method is used to solve the real-time traffic control model. To initialize, the link-base d traffic volume on each link, which is obtained from the traffic assignment model, is put into TRANSYT-7F to obtain the static networkwide optimal signal settings. Since the Frank-Wolfe method is a feasible direction method and its direction finding procedure is a descent method, the descent direction at each iteration is then found by assigning a green time equal to the preset minimum green time for each phase (feasible direction) and then assigning the remaining green time to the phase with the highest vehicle delay (descent direction). Moreover, with th e linear constraints for conserving cycle length and minimum green time (Eq. 66 and Eq. 67), the preset cycle length has to be greater than the sum of minimum green times and lost times for all signal phases at each intersection. An auxiliary green time setting, g, is obtained to determine the descent direction. After the descent direction is decided, the optimal move size between the current solution g and the auxiliary solution g can be found by solving the following linear program: minz[ gn + ( gn gn)](72) subject to 0 1(73) Solving the above linear program is equivalent to finding the value of that satisfies d Z() / d = 0. The objective function is differentiable with respect to However, the derivative is complicated since the average delay model is quite complex itself. The derivative of the objective function is given as follows:

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41 (74) (75) If Xp m(t-1) < 1 and Xp m(t) < 1, then

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42 If Xp m(t-1) 1 and Xp m(t) < 1, then

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43 (76) If Xp m(t-1) 1 and Xp m(t) 1, then

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44 (77)

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45 The bisection method can be used to find an approximate value of which sets the above function equal to zero. The optimal solutions are found when the above algorithm runs iteratively until the stopping criterion is met. There are several stopping criteria that can be applied while implementing the FrankWolfe method. For instance, the convergen ce criterion can be based on the marginal contribution of successive iterations. Alterna tively, the algorithm can be terminated if the elements of the gradient vector are close to zero. In some cases, criteria that are based on the change in the variables between successive iterations are used. Because calculating the relative change of the objective function involves more computation and longer computer running time, the convergence criterion based on the change in the variables between successive iterations is used in this research, that is, maxi {|gi n gi n-1|} The notation maxi {.} stands for the maximum, over all po ssible values of i, of the arguments in the braces. In other words, the iterations terminate if the maximum difference between the green times of each phase at the previous and current iterations for all intersections during all time periods is less than or equal to The procedure for solving the real time traffic control model is summarized as follows: Step 0:Obtain initial solutions from TRANSYT-7F. Step 1:Find the descent direction. For every intersection and each time period, assign the green time g = min g to each phase. Assign the remaining green time to the phase with the highest vehicle delay. Step 2:Determine the optimal move size. min Z (g +(gg))(78) subject to 0 1(79)

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46 Use the bisection method of iterative interval reduction to find the value of which can satisfy d Z() / d = 0 Step 3: Update the green times. g = g +(gg)(80) Step 4: Test convergence. If max {|gn gn+1|} stop. Otherwise, go to Step 1. A signal phase plan comprises a complete specification of phasing sequence, splits (green time for each phase) and the length of amber intervals. The choice of phasing sequence depends primarily on the treatment of le ft turn traffic. The length of a cycle and its splits are determined by a number of factors, the main ones are traffic demand patterns and delay to traffic imposed by signals. While a series of intersections along an arterial streets are treated as a single system and their timing plans are developed together to provide good vehicl e progression along the arterial, the reasonable 2-way progression between traffic signals generally requires the same cycle lengths and similar phase patterns at all signals in a network. Cycle offsets for progression between signals must usually be in half-cycles. Although the cycle length and signal phase plan can be varied at each inte rsection, they have to be specified before initialization and are not modified during the optimization. The same restriction is also applied to the number of time periods a nd the duration of each time period. The methodology developed in the study may allow constructing additional external looping to optimize cycle length and signal phase sequence. However, such optimization is out of the scope of this study and its performance has not been investigated. Figure 3-1 shows the flow chart of the above solution algorithm. In Chapter 5, the optimization program fo r dynamic traffic-responsive signal control is combined with the DTA model describe d in Chapter 4. A Visual Basic program

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47 developed to implement this flow chart to determine the optimal signal settings, which are then imported into the DTA model, is presented in Chapter 6.

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48 Get initial g, min g, cycle length, saturation flow rate, and traffic volumes Read .TRF file Find descent direction Find optimal move size Update g Stop Testconvergence Yes For every I and t, assign the auxiliary green time g’ to each phase g’ = min g Assign the remaining green time to the phase with the highest value of v d 0 1 min z (g + (g’-g)) Using the bisection method to find the approximated value of satisfying dz/d = 0 No Figure 3-1Flow Chart of the Solution Algor ithm for the Real-Time Traffic-Responsive Signal Control Model.

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49 CHAPTER 4 DYNAMIC USER-OPTIMAL ROUTE CHOICE MODEL 4.1 Introduction This chapter presents the techniques applied for developing a variational inequality formulation for a dynamic user-optimal route c hoice model. A variational inequality is a broader formulation for equilibrium problems co mpared to a mathematical program. The static formulation and the equivalent optimization program are presented first. The formulation is then extended for the dynamic case. Subsequently, the relaxation algorithm for relaxing the inseparable link cost functions caused by time-dependent link flow is discussed. A good travel time estimate is introduced to relax the link interaction. The equivalent optimization program is presented after the link interaction is relaxed. The last section presents the solution algorithm developed to solve the equivalent optimization program based on the improvement in the travel time estimation. 4.2 Variational Inequality Models A variational inequality model is a general model formulation that encompasses a set of mathematical models, including nonlinear equations, optimization models and fixed point models [33]. Variational inequalities were or iginally developed as a tool for the study of certain classes of partial differential equations such as those that arise in mechanics. However, with its capability of formulating and analyzing more general models for

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50 (85) equilibrium conditions than the constrained op timization approach, the basic concept of VI theory has received increasing attention from transportation network modelers during the last decade [5]. The definition and proposition presented in the following section focus on variational inequality models suitable for the analysis of transportation network equilibrium models. 4.2.1 Static Transportation Network Equilibrium Model Let x = (x1, x2, ... xm) be a vector of li nk traffic volumes and d [ x ] = [d1( x ), d2( x ), ... dm( x )] be a vector of disutility functions. The equivalent variational inequality formulation of a discrete link-based user-optimal route choice model can be summarized as follows from existing literature [4]: the variational inequality model is to determine a control vector x* G such that d [ x* ] [ x x* ] 0 x G (81) where G denotes the feasible region that is delineated by the following three constraints: 1.flow conservation constraints:k f rs k = qrs r, s (82) 2.non negativity constraints:frs k 0 k, r, s(83) 3.flow propagation constraints:xm = rs k f rs k rs m,k m (84) According to the Symmetry Principl e [34], if the Jacobian matrix d(x) is symmetric, then the variational inequality above has an equi valent optimization model. In this particular case, where the link travel disutility functions are separable, the above VI formulation is equivalent to the following mathematical programming model:

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51 This program is exactly the same optimi zation program used to represent the UE condition discussed in Section 2.3. Therefore, the solution algorithm presented in Section 2.3 can be used to solve it. The solution algor ithm with the application of the Frank-Wolfe method is summarized in the following: Step 0: Find initial volumes. Step 1: Update disutility estimates. Step 2: Find descent direction. Perfor m an all-or-nothing traffic assignment. Step 3: Perform line search for optimal move size. Step 4: Calculate new solution volumes. Step 5: Test convergence. As discussed in Section 2.3, at some point in step 2 involving the procedure for finding the descent direction, the solution algor ithm will call for minimizing the total travel time over a network without flow-dependent trav el times on each link. The total travel time spent in the network will be minimized by assigni ng all motorists to the shortest travel time paths connecting their origins to their destina tions. Such an assignment is performed by the all-or-nothing network loading procedure. The core of the all-or-nothing procedure is the determination of the shortest paths between all origins and destinations. An efficient method to find these paths between all network nodes can be easily found in existing literature [16]. However, since the traffic assignment feature built into CORSIM 5.1 is used to implement the above algorithm, the methods to locate the shortest path are not introduced here. The procedure of step 3, performing a line search for optimal move size between the current solution and the auxiliary solution from step 2, involves the bisection method to find

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52 the value of which can set the derivative of th e objective function with respect to equal to zero. This procedure is also discussed in Section 2.3. 4.2.2 Dynamic Transportation Network Equilibrium Models Unlike the static network system, the dynami c network system concerns a vector of control variables x ( t ) = [x1( t ), x2( t ), ... xm( t )]. The equivalent variational inequality formulation was defined as follows in existing literature [5]: the variational inequality model for the dynamic transportation network equilibri um model is to determine a control vector x* ( t ) G ( t ) such that d [ x* ( t )] [ x ( t ) x* ( t )] 0 x ( t ) G ( t ) (86) where G ( t ) denotes the feasible region that is delineated by the three sets of constraints, which are the same as for the static condition, and another set of flow propagation constraints: in-flow xmk rs (t) = exit-flow xmk rs (t + travel time on link m)(87) The link interactions caused by the above flow propagation constraints result in an asymmetrical Jacobian matrix of the dynamic travel disutility functions such that the Symmetry Principle cannot be applied to the re formulated VI format as an optimization program. However, there is an iterative method (relaxation algorithm) that can solve the problem. Assuming that there exists a vector of smooth auxiliary functions g ( x y ) with suitable properties, Ran and Boyce [5] proved that at iteration n, solving the following variational inequality model: g(x(n), x(n-1)) ( x x(n)) 0 x G (88) is equivalent to solving the corresponding mathematical programming model: min Z ( x x(n-1)) x G(89)

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53 (90) Chen and Hsueh [12] applied the above relaxation algorithm and suggested relaxing the link interactions by calculating the link tr avel times based on the traffic volumes from the assignment result at the previous iteration. This procedure resulted in a model with a positive-valued diagonal Jacobian of d [ x ( t )]. The following mathematical programming model is then equivalent to the relaxed variational inequality model: A nested diagonalization al gorithm applying the relaxati on (diagonalization) method is then: Step 0:Find an initial feasible solution. Step 1:Estimate link travel times Tm (t). Step 2:Solve the optimization model (Eq. (90)) based on Tm(t). Step 2.1: Update the xm(t) based on Tm(t). Step 2.2: Solve the optimization model (Eq. (90)) by the FW method. Step 2.3: Test convergence. Step 3:If dm(t) Tm(t), stop. Otherwise, set Tm(t) n+1 = Tm(t) n + (1/n) (dm(t) n Tm(t) n), and go to step 2. The optimization program in Eq. 90 has many similarities with the one representing the static case. Therefore, step 2 is the typi cal procedure for solving the static route choice model with asymmetric travel time functions which are dependent on the traffic flows from the previous iteration. In step 2.1, the initial so lution is updated every time. In other words, the traffic patterns resulting from the previous iteration in step 2.2 cannot be carried over into the next iteration because different li nk travel times constr uct different subproblem

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54 feasible regions. In step 3, the estimated link travel times are updated using the method of successive averages to predetermine the move size of 1/n. The estimated link travel times were defined using the following setting: Tm(t) = NINT [dm(t)]. The notation NINT indicates the round-off arithmetic operation: set Tm(t) = i, if i 0.5 Tm(t) i + 0.5. Although finding the link travel time is a critical issue for solving the above algorithm, Chen and Hsueh [12] simply used the above round-off operation to estimate it and claimed to leave more precise estimation to further research. Improvements for this issue are discussed in the following section. 4.3 Enhanced Solution Algorithm One of the most important analytical t ools of traffic engineering is computer simulation. Computer simulation is more practi cal than a field experiment because it is less costly and the results are obtained quickly. CORSIM 5.1, the standard simulation model, uses a microscopic stochastic simulation model to represent the movements of individual vehicles, which includes the influences of driver behavior. Therefore, CORSIM 5.1 is an efficient tool to simulate the utilization of transportation resources and develop precise measures to a transportation system’s operational performance. In this study, CORSIM 5.1 is used to simulate the traffic conditions at each iteration given the traffic volumes assigned in the previous iteration. The actual travel time on each link is then calculated from the simulation results. Moreover, since CORSIM 5.1 provides a user-e quilibrium platform to perform traffic assignments, and the program internally translat es the origin-and-destination data into a form suitable for use by its simulation model, it is adopted not only to calculate the link travel times but also to perform the traffic assignm ent procedure. Adopting CORSIM 5.1 into the

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55 solution algorithm improves both the computational efficiency and the quality of the solution. Although CORSIM 5.1 can only deal with the static case, an important feature of CORSIM 5.1 is that characteristics that change over time, such as signal timings and traffic volumes, can be represented by dividing the simulation into a sequence of user-specified time periods during which the traffic flows, tra ffic controls and geometry are held constant. This feature is especially useful for solvi ng the optimization program in Eq. 90 because the procedure for solving the static route choice model with asymmetric travel time functions has to be performed for each time period separately in each iteration. The solution algorithm adopting CORSIM 5.1 is illustrated and explained as follows. Step 0:Assign traffic initially by user-equilib rium assignment using CORSIM 5.1 to obtain traffic volumes xm (0). Step 1:Simulate the traffic conditions based on xm (n)(t) and calculate the link travel times Tm (n)(t). Step 2:Solve the optimization model in Eq. 90 based on Tm (n)(t). Step 2.1: Calculate congested speeds based on Tm (n)(t). Step 2.2: Assign traffic usi ng CORSIM 5.1 to obtain new xm (n)(t) for each time period sequentially. Step 2.3: Compute dm (n)(t) based on new xm (n)(t) Step 3:Test Convergence. If max {| dn Tn|} stop. Otherwise, update xm (n)(t) and then go to Step 1. In step 0, the initial assignment is perf ormed by user-equilibrium assignment using CORSIM 5.1. The analysis time interval is divi ded into user-specified time periods but each time period is assumed to have the same link volumes initially.

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56 In step 1, CORSIM 5.1 is used to simulate the traffic conditions for all of the time periods sequentially given assigned traffic volum es either from step 0 (each time period has the same traffic volumes) or from step 3. CORSIM’s simulation output can present the average speed of travel on each link. The av erage travel time on each link can be obtained by dividing the link length by the average speed. In step 2.1, the congested speed on each link, based on the link travel times obtained from step 1, is updated. As discussed in the Section 4.2.2, the travel time function in the above optimization program is dependent on the traffic volumes assigned in the prior iteration. Therefore, while performing the static traffic assignment, the travel time function has to apply traffic volumes from the prior iteration. CORSIM 5.1 provides two travel time functions, the BPR and the modified Davidson link impedance functions, which were explained in Section 2.9. Users are allowed to choose one of the impedance functions, but are not allowed to modify them. To deal with this shortcoming, the CORSIM 5.1 “free-flow speed ” on each link has to be adjusted based on traffic volumes from the previous iteration to consider the transportation network as not being empty for the assignment performed in step 2.2. Step 2.2 performs a traffic assignment using CORSIM 5.1 to obtain new traffic volumes for each time period sequentially. As mentioned above, minimizing the optimization program in Eq. 90 is to minimize the static route choice model with asymmetric travel time functions for each time period separately. In step 2.3, the link travel times from the travel time function used in the traffic assignment model based on traffic volumes from step 2.2 are computed. The relaxation algorithm relaxes the link interactions by assumi ng the link travel times based on the traffic

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57 volumes from the previous iteration are known fr om the first loop (outside steps 2.12.3) of the solution procedure. Therefore, the link travel times calculated from the travel time function built into the assignment model in the second loop operation have to converge to the estimated link travel times in the first loop. This convergence is tested in step 3. Since CORSIM’s assignment output includes the aver age link speed according to the travel time function, the average travel time based on the travel time function can be obtained also by dividing the link length by the average speed. The remaining algorithmic step is to perform updating and convergence testing. The solution procedure is performed iteratively until the stopping criterion is met. Figure 4-1 shows the flow chart of the above solution algorithm. In Chapter 5, the above DTA model is combined with the optimization program for the real-time traffic control model develope d in Chapter 3. The Visual Basic program presented in Chapter 6 is developed to implemen t the process in this flow chart to determine the user-equilibrium traffic flows on each link for each time period given the signal settings from the real-time traffic control model.

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58 Simulate & calculate link travel times Tm (n)(t) Assign traffic initially using CORSIM & get xm (0)(t) Calculate congested speeds based on Tm (n)(t) Assign traffic using CORSIM to get new xm (n)(t) dm (n)(t) Tm (n)(t) Stop Compute dm (n)(t) yesno Update xm (n)(t)Figure 4-1 Flow Chart of the Solution Algorithm for the DTA Model.

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59 CHAPTER 5 COMBINED MODEL 5.1 Introduction This chapter presents a theo ry to deal with the traffic flow equilibrium which involves traffic-responsive signal control policies. This situation has been extensively investigated in two different ways [11]: global optimization models and the iterative optimization assignment (IOA) procedure. However, global optimization models have major difficulty in finding an efficient solution al gorithm for calculating the optimal signal settings in traffic networks while anticipating user-optimum equilibrium flows. Some authors [9, 12, 27] have used a bilevel programming method to combine the traffic control and the transportation assignm ent models. This presumes that decision makers at two levels act in a hierarchical manner. At the upper level, decision makers, bound by the decisions of the lower level, try to maximize the their own profit, taking into account the reactions of the lower level accord ingly. The iterative optimization assignment procedure can be used to solve a bilevel program. The procedure of the IOA is to update th e signal settings for fixed flows and solve the traffic equilibrium assignment for fixed signal settings sequentially until the solutions of the two models are considered to be c onsistent. Yang [11] claimed that the IOA procedure has the advantages that the tr aditional traffic assignment and signal setting techniques can be employed to solve the problem and can be applied to a large network.

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60 (91) (92) Marcotte [28] used the IOA procedure to so lve for optimal signal settings based on capacity variables corresponding to user-optimized equilibrium traffic flows. Marcotte hoped that the convergence toward a ‘good’ solution woul d be obtained. He concluded that the numerical experiments tended to show the IOA procedure yielded near-optimal solutions. In the following sections, the bilevel progr am is presented first. The iterative optimization assignment procedure is then addressed. 5.2 Bilevel Model The bilevel programming model described fo rmally and completely by Bard and Falk [35] can be formulated as: subject to where g and v represent the decision vectors associated with the upper and lower levels, respectively. G is the feasible set of the g-variables and V(g) is the feasible set, possibly dependent on g, of the v-variables [28]. The “argmin” of a mathematical program is the optimal solution of the program; in other words, Z1() has to be evaluated at the optimal solution of Z2() Since the objective of traffic management system s is to calculate equilibrium flows which are consistent with a given traffic-responsive control policy, this framework can be applied to integrate the traffic control policy and the transportation assignment procedure. In the upper level, Z1() represents the optimization model for dynamic traffic control in which total intersection delay in a network is minimized by allocating appropriate green times, g, given the user-optimal traffic flows, v, which are

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61 (93) obtained from the lower level model. In the lower level, Z2() represents the dynamic traffic assignment model formulated as a variational inequality formula in which the link traffic flows, z, reach user equilibrium, v, given the link travel times gathered from the upper model. The mathematical formulation of the dynamic transportation management system, which combines the models developed in Chapter 3 and Chapter 4, is given as follows: subject to T [ v ( t )] [ z ( t ) v ( t )] 0 z ( t ) V ( g ) (94) where G denotes the feasible region that is delineated by the two constraints for conserving the cycle length and the minimum green times shown earlier in Eqs. 66 and 67. The vehicle delay function, d, is dependent on the variables of green times, g and traffic flows, v To calculate the total delay happening at each intersection, the traffic flows, v in the upper level have to be specified as traffic volumes moving on each link during a specific phase, at a specific time, that is, vm p(t). However, these traffic flows have to satisfy the VI formula in the lower level in which the tota l volumes on each link are represented in vector form, v ( t ). T [ v ( t )] denotes the travel disutility functi ons which are dependent on link traffic flows whereas V ( g ) is the feasible traffic flow set delineated by the flow constraints presented in Eqs. 82-84 in Chapter 4. One of the flow constraints, the flow propagation constraint, is dependent on the link travel tim es which are affected by the green time splits. 5.3 Iterative Optimization and Assignment Procedure Since there is not much hope to develop exact solution algorithms for large or even medium-size networks [28], the IOA procedure is used to solve the above bilevel program.

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62 The iteration of the algorithm consists of solving sequentially an optimization model involving the signal timing variables, with th e flow variables fixed, and a user-optimized equilibrium model corresponding to this new gr een time vector. The procedure for iterative optimization and assignment is illustrated in Figure 5-1. Although there are limitations, Smith and Vu ren [36] proved the convergence of the iterative optimization and assignment algorithm. Using numerical experiments, Marcotte [28] also showed the IOA heuristic can yield near-optimal solutions. 5.4 Detailed Procedure The two submodels included in Figure 5-1, the adaptive traffic-responsive signal control submodel and the dynamic traffic assignm ent submodel, can be replaced by the flow charts shown in Figure 3-1 and Figure 4-1, respectively. To initialize the procedure, users need the information of the link-node network structure and the O-D demand matrix, and must decide on the number of time periods, the duration of each time period and signal settings. The detailed procedure for implemen ting the IOA method to optimize the dynamic transportation management system is described as follows: Step 1: Initialize. Step 1.1: Prepare input data. Prepare the link-node netw ork connection structure, initial travel speed estimates, link lengt h measurements, link capacities, the O-D trip matrix, the number of time periods, the duration of each time period, and for each intersection, the signal cycle length, cycle offset, and signal phase plan. Step 1.2: Perform static traffic assignment The traditional user-equilibrium traffic assignment method is used to assign the O-D trip matrix to the network using CORSIM 5.1 based on initial estimates of travel times. Step 1.3: Determine initial signal settings and intersection delays The static networkwide optimal signal timing settings and delays for the user-specified phase plans based on the traffic flows obtained from Step 1.2 are determined by TRANSYT-7F.

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63 Initialization Adaptive TrafficResponsive Signal TimingSubmodel Dynamic Traffic Assignment Submodel Convergence Test End Yes No Figure 5-1 Flow Chart of the Iterative Optimization Assignment Procedure.

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64 Step 1.4: Perform initial flow simulation The total analysis time interval is split into user-specified time periods. Each time period has the same signal timing settings obtained from Step 1.3. With the link traffic flows obtained from Step 1.2 at the initial time period, the CORSIM 5.1 simulation model simulates the traffic conditions and obtains the link traffic flows for each time period. The queue buildups from each time period will carry into the next time period. The signal timings and intersection delays can then be calculated for the revised traffic volumes. Step 2: Execute the adaptive traffic-responsive signal control submodel. Step 2.1: Obtain input data. For the first iteration, read the signal timing settings and delays from TRANSYT-7F (Step 1.3) and the link traffic flows from the CORSIM 5.1 simulation results (Step 1.4) for each time period. Starting from the second iteration, read the op timal signal timing settings and delays resulting from the previous iteration and read the link traffic flows for each time period from the dynamic traffic assignment submodel. Step 2.2: Find descent direction. Assign the minimum gr een time to each phase at each intersection for each time period. Then assign the remaining green time from the cycle length of the intersection to the phase with the highest vehicle delay. Step 2.3: Optimize move size. Perform a line search to find the optimal extent value to move from the last solution of gr een times toward the auxiliary solutions obtained form Step 2.2. Step 2.4: Test convergence. If the convergence criteria for green times are not satisfied, return to Step 2.2; otherwise, proceed to Step 3. Step 3 Execute the dynamic traffic assignment submodel. Step 3.1: Perform simulation. The optimal signal timing settings for each time period are updated using the results from the adaptive traffic-responsive control submodel (Step 2). Using the link traffic flows obtained from Step 1.2 for the initial time period at the first itera tion (or using the link traffic flows obtained from the previous iteration for the second iteration and beyond), the simulation model simulates traffic conditions based on dynamic trafficresponsive signal timings to obtain the link traffic flows for each time period. Step 3.2: Calculate link travel times. Calculate the link travel times for each time period by dividing the link length by the average speed from the CORSIM 5.1 simulation results of Step 3.1. Step 3.3: Modify free-flow speeds. Modify the free-flow speed on each link for each time period to transfer the effects of the traffic flows resulting from the

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65 previous iteration to the travel time f unctions built into the static traffic assignment software (CORSIM 5.1). Step 3.4: Perform static traffic assignment for each time period separately. The static traffic assignment is performed for each time period separately with congestion speeds represented as modified free-flow speeds. Step 3.5: Compute estimated link travel times based on the travel time functions. The estimated link travel times based on the travel time functions built into the traffic assignment software are calculated. Step 3.6: Test convergence. Compare the links travel times resulting from Step 3.2 and Step 3.5. If the convergence criteria are not satisfied, return to Step 3.1; otherwise, proceed to Step 4. Step 4 Test convergence. Compare the traffic flows between two iterations. If the convergence criteria, max {| v n v n+1|} are not satisfied, return to Step 2; otherwise, stop.

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66 CHAPTER 6 IMPLEMENTATION 6.1 Introduction Software implementation is a necessary step in optimizing the transportation management system. The objective of this ch apter is to develop computer software to implement the detailed procedure presented in Section 5.4. A numerical example is used to illustrate the procedure for implementing the IOA method to optimize the dynamic transportation management system. 6.2 Numerical Example Figure 6-1 shows the link-node structure of a traffic network illustrated by the ITRAF display. ITRAF is a preprocessor for CORSIM which allows users to create traffic networks, enter traffic volumes and timing data on a map window and then transfer all information into input files for CORSIM. There are eight roadways with eleven intersections in the example network. Most numerical examples used in the exis ting literature were small traffic networks because of the complicated computation and the limitation of computer memory to store large link-node structures. The computer progr am developed in this research was written in Visual Basic for Windows 2000. With th eir increasing power, microcomputers have become a suitable hardware platform for rather large network processing applications. However, there are size limitations for CORSIM 5.1 network characteristics.

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67 Figure 6-1 Link-Node Structure of the Example Network. In CORSIM 5.1, the entry and/or exit nodes must be numbered between 8000 and 8999 while the internal nodes can be numbered from 1 to 6999. In other words, the latest version of CORSIM 5.1 allows a maximum num ber of 1,000 entry and/or exit nodes. Entry and/or exit nodes were used in this study to repr esent the centroids of traffic analysis zones (TAZ). The TAZ is the basic analysis unit for calculating trip productions and attractions. These trips are assumed to start or end at the centroid of a zone and are distributed throughout the network to create the O-D matrix. In Florida, the numbers of TAZs for Palm Beach County, Broward County and Miami-Dade County are 1,118, 892 and 1,466, respectively. Therefore, for urban areas with less than 1,000 TAZs, CORSIM 5.1 can accommodate the origin and destination zones without adjustment. However, for urban areas with more than 1,000 TAZs, some TAZs have to be merged to reduce the number of zones to less than 1,000.

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68 The maximum number of internal nodes is 6,999 in CORSIM 5.1. Internal nodes are used to represent intersections of a give n street network. The numbers of non-centroid nodes (representing intersections of streets) in the 1999 networks for Palm Beach County, Broward County and Miami-Dade County are 2,861, 3,378 and 5,969, respectively. For a street network with more intersections than 6,999, some minor streets have to be excluded to reduce the total number of intersection to be less than the maximum number of internal nodes allowed in CORSIM 5.1. Table 6-1 shows the O-D demand matrix for the numerical example. The number of trips per hour for a specific trip purpose (e.g., working or shopping) shown in the cells were defined for demonstration purposes. For example, in Table 6-1, there were 100 shopping trips per hour from TAZ 8021 to TAZ 8019. Table 6-1O-D Demand Matrix for the Example Application Destination Origin80218015801980238025801380148024801680208012 8021050100301106010010020050110 8015200120150505050505060110 80199050040140100150200150100150 802350100120011011012040503040 80255010010050050501001005050 801380100100130700100601106050 8014501001501001501500801008050 80245010050202001105002004050 8016301301001508013050130050100 80202012015013015012050120900100 8012607060120110130908070500 Note that nodes 8017, 8018, 8022, 8026 and 8027 show n in Figure 6-1 do not have volumes in the O-D matrix. These nodes were treated like dummy nodes and can be connected to new nodes in the future.

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69Cycle Length = 104 seconds N-S Left-Turn N-S Through E-W Left-Turn E-W Through 15 sec37 sec 15 sec 37 sec Figure 6-2 Initial Signal Settings for the Example Application. Figure 6-2 shows the pre-timed signal settings which were defined at each intersection. It consists of four phases for each signal cycle with a protected left-turn phase and a through movement phase for both north-sout h and east-west directions. As discussed in Section 3.4, the signal settings can be different at each intersection. However, to simplify the implementation procedure, the signal settings for all of the intersections in the example network were the same. In addition, the pha se durations for the north-south and east-west directions were originally set to be equal. 6.3 Software Implementation A comprehensive, menu-driven software pr ogram was developed in Visual Basic to implement the detailed procedure presented in the previous chapter. Figure 6-3 shows the program’s main menu. A copy of the program can be downloaded from the following web site: http://www.fiu.edu/~chowl/ The application of the software is illustrated in the following procedure following the steps in Section 5.4:

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70 Figure 6-3 Main Menu Screen. Step 1 Initialization: Step 1.1: Prepare input data 1.The ITRAF button on the menu bar launches the application of ITRAF and allows users to create traffic networks, enter traffic volumes and timing data on a map window and then transfer this informati on into input files for CORSIM 5.1. The required input information at this step includes: (1) link-node geometric data (2) length of each link (3) number of lanes on each link (4) mean value for number of vehicles in initial queues and start-up lost time (5) free-flow speed on each link (6) number of approaches and signal phase sequences for each node (7) durations of signal timing intervals for each node (8) cycle offset for each node 2.The O-D Input button on the menu bar serves as an O-D matrix file manager and allows users to create or modify O-D matrix files in Excel spreadsheets. 3.The Transfer button on the menu bar transfers th e O-D demand matrix, as created or modified using the O-D Input option, into CORSIM 5.1 input files. Users can

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71 Traffic Assignment Results Link Left Turn Through Right Turn (8025, 25) 0 700 0 ( 25, 5) 106 594 0 ( 1, 6) 230 460 141 ( 1, 2) 118 685 95 ( 1, 21) 0 500 0 (8021, 21) 0 910 0 ( 21, 1) 357 553 0 (8024, 24) 0 870 0 ( 24, 8) 0 493 377 ( 6, 8) 410 480 0 ( 6, 13) 0 1010 0 ( 6, 7) 185 317 281 ( 6, 1) 0 173 540 (8013, 13) 0 860 0 ( 13, 6) 288 352 221 (8014, 14) 0 1010 0 ( 14, 7) 289 423 298 Figure 6-4 Excerpt of the Initial Assignment Results for the Example Application. specify the following assignment parameters through ITRAF if the default values provided by the software are not satisfactory: (1) acceptable threshold of objective function (default value: 0.1%) (2) maximum number of iterations (default value: 5) (3) impedance function type (default type: BPR formula) (4) parameters of impedance function (5) capacity smoothing factor (default value: 0) (6) line-search accuracy threshold (default value: 0.1%) Step 1.2: Perform static traffic assignment The Browse button at the Initial Assignment window invokes the Open File dialog box. With this dialog box, users can select the desired CORSIM 5.1 input file, with the O-D information embedded, and then click the Start button to activate the assignment model provided by CORSIM 5.1. The static user-equilibrium traffic is applied to the network using the specified origin-destination information. For the first iteration, the link impedances are evaluated for free-flow speed conditions throughout the entire network. An intermediate solution for each iteration is obtained using link impedances produced by the previous iteration. To obtain an optimal solution for each iteration, an iterative line search is applied to the range between the current intermediate solution and the previous iteration solution. The search terminates when the contribution of the current iteration is less than the accuracy threshold value. The traffic assignment process terminates when the relative change of the objective function between two successive iterations is less or equal to the threshold value. The Visual Basic (VB) program extracts a nd formats part of CORSIM’s output file and shows the link turning volumes on the screen when the procedure is completed. Figure 6-4 shows an excerpt of initial assignment results for the example.

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72 Step 1.3: Determine initial signal settings The TRANSYT option on the menu bar launches the application of T7F9, an interface shell program for TRANSYT-7F, and allows users to input the initial input data specified in Step 1.1.1 and the assignm ent results from Step 1.2 into the input files of TRANSYT-7F. Th e T7F9 shell program can also be used to access TRANSYT-7F to get the static optimal green times. The timing data from TRANSYT-7F can then be imported into CORSIM’s format by the ITRAF button. Step 1.4: Perform initial flow simulation The Browse button at the Initial Simulation command invokes the Open File dialog box. With this dialog box, users can select the CORSIM 5.1 input file from Step 1.3. After clicking the Start button, users are asked to input the number of time periods and the duration of each time period before th e simulation model is activated. In the example, an analysis time interval of one hour was selected and separated into six time periods with a duration of ten mi nutes for each time period. CORSIM 5.1 allows users to partition the simulation time into a series of time periods (up to 19 time periods) of varying duration. The VB program modifies the CORSIM 5.1 input file to specify the number of time periods as well as the duration of each period. With the link traffic flows obtained from Step 1.2 at the initial time period, the CORSIM 5.1 simulation model simulates the traffic conditions and obtains the link traffic flows for each time period. Step 2 Execute the initial dynamic traffic-responsive signal control submodel. Step 2.1 & Step 2.2: Obtain input data and find descent direction. The Init and Find Descent Direction button activates the VB program to read the signal timing settings from Step 1.3 and the link traffic flows for each time period from Step 1.4 after users input the minimum green time for each intersection. The VB program then calculates delay for each phase at each intersection at each time period according Eqs. 63-65. The descent direction is obtained by assigning the minimum green time to each phase at each intersection for each time period and then assigning the remaining green time from the cycle length of the intersection to the phase with the highest vehicle delay. Figur e 6-5 shows an excerpt of the results of the auxiliary green times. Step 2.3: Optimize move size. The Determine Optimal Move Size button activates the VB program to find the optimal move size from the last set of green times to the auxiliary set of green times. This is achieved by using the bisection method to find an approximate value of which sets the complicated derivative of the objective function (Eq. 74) equal to zero. Step 2.4: Test convergence. The Test Convergence button executes a VB command to examine the criteria for assessing convergence. A message box on the screen shows whether the criteria are satisfied or not.

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73Time Period: 1 AUXILIARY GREEN TIME AT EACH PHASE NODE N-S LT N-S TH E-W LT E-W TH 1 22 21.6 33.6 42.8 6 14.8 40.4 14.4 50.4 8 16.4 19.2 16.8 67.6 2 25.6 25.6 22 46.8 7 14.4 56.8 17.6 31.2 9 25.2 43.2 12.4 39.2 3 20.8 41.6 20 37.6 10 12.4 43.2 21.2 43.2 4 14.4 37.4 49.8 18.4 5 8 89.2 9.6 13.2 11 11.2 83.6 8.8 16.4 Time Period: 2 NODE 1 26.8 20.8 29.2 43.2 6 21.6 42 20.8 35.6 8 21.2 18 16.8 64 2 22.4 28.4 34.4 34.8 7 16.4 48.4 19.2 36 9 21.6 43.6 15.2 39.6 Figure 6-5 Excerpt of the Auxiliary Green Time Results. The Update g(t) and Re-iterate button automatically runs the traffic-responsive signal control submodel (Step 2.1 to Step 2.3) repeatedly until the criteria for convergence are satisfied. Figure 6-6 shows an excerpt of the results for the optimal green times at the initial iteration. Step 3: Execute the dynamic traffic assignment submodel. Step 3.1: Perform simulation. The Perform Simulation button activates the VB program to update the signal timing settings for each intersection at each time period using the results from the dynamic traffic-responsive signal control submodel (Step 2). The simulation model (CORSIM 5.1) is then activated to simula te the traffic conditions for the network with the optimal signal settings and the tra ffic flows from the initial assignment (Step 1) or the last iteration. Step 3.2: Calculate link travel times. The Calculate T(t) button calculates the link travel times from the simulation results. The VB program calculates the link travel times for each time period by dividing the link lengths by the average speeds from th e CORSIM 5.1 simulation results of Step 3.1. Figure 6-7 shows an excerpt of results of the link travel times for each time period. Steps 3.3 and Step 3.4: Modify free-flow speeds and perform static traffic assignment for each time period separately.

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74Time Period: 1 GREEN TIME AT EACH PHASE NODE N-S LT N-S TH E-W LT E-W TH 1 26 24 25 45 6 16 43 18 43 8 22 26 20 52 2 26 31 24 40 7 17 49 20 35 9 20 44 15 41 3 22 34 22 43 10 15 45 17 43 4 18 49 32 21 5 12 60 21 26 11 14 58 12 36 Time Period: 2 NODE 1 27 24 24 45 6 17 44 19 40 8 24 25 20 50 2 25 32 25 37 7 16 47 21 36 9 19 44 16 41 Figure 6-6 Excerpt of the Optimal Green Time Results at the Initial Iteration.TIME PERIOD 2 Link LENGTH (ft) AVERAGE SPEED (MPH) TRAVEL TIME (sec) ( 25, 5) 1277 20 43 ( 1, 6) 1533 13 80 ( 1, 17) 272 24 7 ( 1, 2) 1552 10 105 ( 1, 21) 1269 25 34 ( 21, 1) 1269 10 86 ( 24, 8) 1256 11 77 ( 6, 8) 1268 11 78 ( 6, 13) 1689 26 44 ( 6, 7) 1552 10 105 ( 6, 1) 1533 10 104 ( 13, 6) 1689 10 114 ( 14, 7) 1552 10 105 ( 8, 24) 1256 26 32 ( 8, 18) 255 25 6 ( 8, 9) 1552 10 105 ( 8, 6) 1268 10 86 ( 18, 8) 255 10 17 Figure 6-7 Excerpt of the Link Travel Time Results Based on the Simulation. The Adjust Free Flow Speed and Assign button activates the VB program to replace the free-flow speeds (specified on Entry 25 of Record 11 in a CORSIM 5.1

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75TRAVEL TIME, d(t), ON THE PATH-LINK USING FHWA IMPEDANCE FUNCTION TIME PERIOD 1 Link LENGTH (ft) AVERAGE SPEED (MPH) TRAVEL TIME (sec) ( 25, 5) 1277 24.6 35 ( 1, 6) 1533 10.9 95 ( 1, 17) 272 10 18 ( 1, 2) 1552 15.9 66 ( 1, 21) 1269 29.7 29 ( 21, 1) 1269 10 86 ( 24, 8) 1256 11.5 74 ( 6, 8) 1268 10 86 ( 6, 13) 1689 29.1 39 ( 6, 7) 1552 13.3 79 ( 6, 1) 1533 10 104 ( 13, 6) 1689 16.4 70 ( 14, 7) 1552 12.7 83 ( 8, 24) 1256 28.6 29 ( 8, 18) 255 10 17 ( 8, 9) 1552 13.9 75 ( 8, 6) 1268 13.8 62 Figure 6-8 Excerpt of the Link Travel Time Results from the BPR Formula. input file) with the average speeds from the CORSIM 5.1 simulation results of Step 3.1 for each link at each time period. The VB program then creates six CORSIM 5.1 input files, each for a specific time period, with the effects of the traffic flows resulting from the initial assignment or pr evious iteration, and activates CORSIM 5.1 to perform the static user-equilibrium traffic assignment using these files as input separately. Step 3.5: Compute estimated link travel times based on the travel time functions. The Compute d(t) command activates the VB program to retrieve the link travel times estimated based on the travel time functions built into the static traffic assignment model in CORSIM 5.1 from the si x output files generated at Step 3.4 for each time period. Figure 6-8 shows an excerpt of the results of the estimated link travel times from the BPR formula. Step 3.6: Test convergence. The Test Convergence: d(t) = T(t) ? button compares the link travel times based on the simulation results and the BPR formula. A message box on the screen shows whether the criteria are satisfied or not. Figure 6-9 shows an excerpt of the comparison of link travel times based on simulation results and the BPR Formula. If the convergence criteria are not satisfied, clicking the Update X(t) and Re-iterate button will automatically run the dynamic traffic assignment procedure iteratively until the criteria are satisfied. After the criteria are satisfied, clicking the TrafficResponsive Signal Control button will run Steps 2.1 through 2.4 of the dynamic traffic-responsive signal control submodel but using the traffic flows resulting from the dynamic traffic assignment submodel.

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76 LINK TRAVEL TIME (sec) TRAVEL TIME (sec) BPR SIMULATION ( 25, 5) 71 43 ( 1, 6) 104 80 ( 1, 17) 18 7 ( 1, 2) 72 105 ( 1, 21) 34 34 ( 21, 1) 86 86 ( 24, 8) 85 77 ( 6, 8) 86 78 ( 6, 13) 45 44 ( 6, 7) 105 105 ( 6, 1) 104 104 ( 13, 6) 114 114 ( 14, 7) 105 105 ( 8, 24) 34 32 ( 8, 18) 17 6 ( 8, 9) 79 105 ( 8, 6) 86 86 ( 18, 8) 17 17 ( 12, 11) 85 65 ( 2, 7) 103 51 Figure 6-9Excerpt of the Comparison of Link Travel Times Based on Simulation Results and the BPR Formula. Step 4 Test convergence The Iterative Optimization Assignment button runs the dynamic traffic-responsive signal control submodel and the dynamic traffic assignment submodel sequentially until the solutions converge. The final results for the traffic-responsiv e signal timings and dynamic traffic flows are presented in the next Chapter.

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77 CHAPTER 7 RESULTS AND EVALUATION 7.1 Introduction The objective of this chapter is to evalua te the time-dependent traffic flows and the signal settings produced by the optimal dynamic transportation management system developed in this research using the numerical example presented in Chapter 6. The results for both static and dynamic cases are presented and are followed by a comparison and assessment. 7.2 Results for Static Signal Settings and Static Traffic Flows Static optimal traffic flows determined by the traditional user-equilibrium methodology for the sample network and the given O-D matrix are shown in Figure 7-1. From the traffic flows shown in Figure 71, TRANSYT-7F found that the traffic signal settings shown in Figure 7-2 should produce minimum delays during the analysis time interval. There are eleven intersections a nd each intersection has four phases for each signal cycle. In the static case, traffic flows a nd signal settings remain the same throughout the analysis time interval. 7.3 Results for Adaptive Signal Settings and Dynamic Traffic Assignment In the dynamic case, traffic flows and signal setting are time-dependent. In the example, the analysis time interval, one hour, was separated into six time periods with a duration of ten minutes for each time period.

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78 Link LT TH RT Link LT TH RT (8025, 25) 0 700 0 ( 9, 8) 509 0 223 ( 25, 5) 146 554 0 ( 9, 10) 317 562 273 ( 1, 6) 222 446 226 ( 9, 7) 322 367 240 ( 1, 2) 69 622 139 (8015, 15) 0 710 0 ( 1, 21) 0 500 0 ( 15, 2) 370 297 43 (8021, 21) 0 910 0 (8016, 16) 0 950 0 ( 21, 1) 362 548 0 ( 16, 9) 155 530 265 (8024, 24) 0 870 0 ( 3, 10) 194 247 345 ( 24, 8) 0 501 369 ( 3, 2) 396 570 432 ( 6, 8) 419 451 0 ( 3, 4) 0 1037 600 ( 6, 13) 0 1010 0 ( 3, 19) 0 1050 0 ( 6, 7) 299 276 264 ( 10, 20) 0 570 0 ( 6, 1) 0 176 468 ( 10, 9) 321 561 400 (8013, 13) 0 860 0 ( 10, 11) 277 644 0 ( 13, 6) 255 396 209 ( 10, 3) 317 403 341 (8014, 14) 0 1010 0 (8019, 19) 0 1170 0 ( 14, 7) 280 395 336 ( 19, 3) 364 399 407 ( 8, 24) 0 960 0 (8020, 20) 0 1050 0 ( 8, 9) 0 540 248 ( 20, 10) 342 543 165 ( 8, 6) 210 293 221 ( 4, 23) 0 920 0 (8012, 12) 0 840 0 ( 4, 3) 218 674 230 ( 12, 11) 0 591 249 ( 4, 5) 0 946 276 ( 2, 7) 294 369 168 (8023, 23) 0 770 0 ( 2, 1) 346 0 324 ( 23, 4) 586 0 184 ( 2, 3) 417 932 169 ( 5, 11) 166 0 255 ( 2, 15) 0 920 0 ( 5, 4) 320 536 0 ( 7, 9) 347 550 16 ( 5, 25) 0 1170 0 ( 7, 6) 215 574 96 ( 11, 10) 50 595 201 ( 7, 14) 0 810 0 ( 11, 12) 0 810 0 ( 7, 2) 58 418 526 ( 11, 5) 302 0 224 ( 9, 16) 0 1120 0 Figure 7-1 Static Traffic Assignment Results.Intersection Green Time for Each Phase Cycle N-S LT N-S TH E-W LT E-W TH Length 1 23 21 19 41 104 2 22 28 20 34 104 3 18 28 18 40 104 4 15 47 24 18 104 5 9 50 20 25 104 6 12 40 15 37 104 7 13 43 16 32 104 8 19 23 17 45 104 9 15 40 12 37 104 10 12 41 12 39 104 11 11 48 9 36 104 Figure 7-2 Optimal Signal Settings for the Static Case.

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79 Time Period: 1 Time Period: 2 Time Period: 3 N-S N-S E-W E-W N-S N-S E-W E-W N-S N-S E-W E-W LT TH LT TH LT TH LT TH LT TH LT TH 1 22 20 20 41 1 23 21 19 41 1 23 20 20 41 6 12 40 14 38 6 13 40 15 36 6 13 40 15 37 8 18 22 17 47 8 19 22 17 47 8 18 22 17 47 2 22 27 20 35 2 22 28 21 34 2 23 27 20 34 7 13 44 16 31 7 13 43 16 32 7 13 43 16 32 9 15 40 12 38 9 15 40 12 37 9 15 40 12 37 3 19 28 18 40 3 19 29 17 39 3 18 28 18 40 10 12 41 13 39 10 12 41 12 39 10 12 39 13 40 4 14 45 27 18 4 15 46 25 19 4 14 46 26 18 5 8 54 18 23 5 8 54 19 23 5 8 54 19 23 11 11 52 8 33 11 10 52 8 33 11 10 52 8 34 Time Period: 4 Time Period: 5 Time Period: 6 N-S N-S E-W E-W N-S N-S E-W E-W N-S N-S E-W E-W LT TH LT TH LT TH LT TH LT TH LT TH 1 23 21 20 41 1 23 22 19 41 1 24 20 18 42 6 12 41 15 37 6 12 40 15 37 6 12 39 15 37 8 18 21 18 46 8 19 24 17 45 8 18 23 18 45 2 22 27 21 34 2 23 29 19 33 2 22 27 20 34 7 13 43 16 32 7 13 43 16 32 7 13 44 16 32 9 16 38 13 36 9 14 41 11 38 9 14 41 11 38 3 18 29 18 40 3 17 29 19 38 3 18 29 18 40 10 12 41 12 39 10 12 39 13 40 10 12 41 12 39 4 14 46 26 18 4 15 46 24 19 4 15 46 25 18 5 8 54 19 23 5 8 54 19 23 5 8 53 19 24 11 10 51 8 34 11 13 50 8 33 11 13 50 8 33 Figure 7-3 Optimal Time-Dependent Signa l Timing Settings for the Example Network. Figure 7-3 shows the optimal time-dependent signal settings for those six time periods. Unlike the static signal settings shown in Figure 7-2, the dynamic traffic signal settings shown in Figure 7-3 are different in different time periods. In the dynamic case, travelers are assumed to receive real-time tra ffic information based on delays associated with the signal settings, and change their paths to mi nimize their travel times. The transportation management system then modifies the signa l settings according the updated traffic flows at each time period to minimize the system-wide de lay. The optimal dynamic traffic flows are shown in Figure 7-4. For the example network, the run-time on a computer with 900 MHz CPU was about 6 minutes for each iteration. It took the program less than an hour to reach the optimal

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80 TP=1 TP=2 TP=3 LINK L T R L T R L T R (8025, 25) 0 117 0 0 117 0 0 117 0 ( 25, 5) 14 103 0 14 102 0 22 95 0 ( 1, 6) 42 73 40 42 73 48 42 66 48 ( 1, 2) 21 125 13 33 100 11 19 131 13 ( 1, 21) 0 83 0 0 83 0 0 83 0 (8021, 21) 0 152 0 0 152 0 0 152 0 ( 21, 1) 74 78 0 74 77 0 78 74 0 (8024, 24) 0 145 0 0 145 0 0 145 0 ( 24, 8) 0 84 62 0 86 59 0 79 67 ( 6, 8) 56 84 0 69 82 0 61 72 0 ( 6, 13) 0 168 0 0 168 0 0 168 0 ( 6, 7) 38 60 44 52 53 37 34 54 45 ( 6, 1) 0 48 84 0 43 69 0 44 84 (8013, 13) 0 143 0 0 143 0 0 143 0 ( 13, 6) 33 71 39 26 70 48 34 70 40 (8014, 14) 0 168 0 0 168 0 0 168 0 ( 14, 7) 50 79 39 56 64 48 56 72 40 ( 8, 24) 0 160 0 0 160 0 0 160 0 ( 8, 9) 6 75 36 0 80 48 14 73 40 ( 8, 6) 29 56 28 35 51 30 44 48 21 (8012, 12) 0 140 0 0 140 0 0 140 0 ( 12, 11) 0 99 41 0 94 47 0 94 46 ( 2, 7) 47 53 40 45 43 29 44 46 29 ( 2, 1) 78 0 35 86 0 40 83 0 40 ( 2, 3) 70 155 40 58 147 43 65 152 49 ( 2, 15) 0 153 0 0 153 0 0 153 0 ( 7, 9) 48 99 1 55 80 1 53 89 5 ( 7, 6) 27 99 44 30 85 37 32 77 46 ( 7, 14) 0 135 0 0 135 0 0 135 0 ( 7, 2) 2 69 65 5 58 74 5 68 62 ( 9, 16) 0 187 0 0 187 0 0 187 0 ( 9, 8) 76 0 29 78 0 30 83 0 34 ( 9, 10) 57 83 40 60 97 40 69 76 34 ( 9, 7) 51 59 29 59 36 38 54 60 37 (8015, 15) 0 118 0 0 118 0 0 118 0 ( 15, 2) 75 34 10 75 28 16 74 31 14 (8016, 16) 0 158 0 0 158 0 0 158 0 ( 16, 9) 27 75 57 24 73 62 26 80 53 ( 3, 10) 39 47 44 49 45 63 48 51 57 ( 3, 2) 94 101 64 78 105 62 75 104 67 ( 3, 4) 0 184 102 0 165 89 0 172 90 ( 3, 19) 0 175 0 0 175 0 0 175 0 ( 10, 20) 0 95 0 0 95 0 0 95 0 ( 10, 9) 52 78 58 58 84 60 58 87 57 ( 10, 11) 46 104 0 71 108 0 57 103 0 ( 10, 3) 51 67 65 45 69 55 48 70 61 Figure 7-4 Optimal Time-Dependent Tr affic Flows for the Six Time Periods. results. Although this cannot be used to procla im the feasibility in terms of computer runtime for applications with real networks, whic h are usually in large in scale, the program should be suitable for rather large network processing applications.

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81 TP=1 TP=2 TP=3 LINK L T R L T R L T R (8019, 19) 0 195 0 0 195 0 0 195 0 ( 19, 3) 65 64 66 53 80 62 49 84 62 (8020, 20) 0 175 0 0 175 0 0 175 0 ( 20, 10) 53 94 28 54 84 36 53 85 36 ( 4, 23) 0 153 0 0 153 0 0 153 0 ( 4, 3) 26 142 38 30 138 49 23 136 40 ( 4, 5) 0 160 50 0 155 37 0 159 41 (8023, 23) 0 128 0 0 128 0 0 128 0 ( 23, 4) 102 0 26 101 0 27 101 0 28 ( 5, 11) 31 0 32 27 0 25 32 0 31 ( 5, 4) 52 104 0 64 116 0 64 98 0 ( 5, 25) 0 195 0 0 195 0 0 195 0 ( 11, 10) 8 91 31 10 84 24 10 90 25 ( 11, 12) 0 135 0 0 135 0 0 135 0 ( 11, 5) 52 0 35 78 0 40 67 0 36 TP=4 TP=5 TP=6 LINK L T R L T R L T R (8025, 25) 0 117 0 0 117 0 0 117 0 ( 25, 5) 27 89 0 26 90 0 21 96 0 ( 1, 6) 51 68 46 49 66 46 47 69 42 ( 1, 2) 28 114 19 22 125 12 25 111 9 ( 1, 21) 0 83 0 0 83 0 0 83 0 (8021, 21) 0 152 0 0 152 0 0 152 0 ( 21, 1) 79 73 0 80 72 0 74 77 0 (8024, 24) 0 145 0 0 145 0 0 145 0 ( 24, 8) 0 89 56 0 78 67 0 81 64 ( 6, 8) 56 79 0 53 81 0 50 80 0 ( 6, 13) 0 168 0 0 168 0 0 168 0 ( 6, 7) 47 59 46 34 61 45 45 60 55 ( 6, 1) 0 46 82 0 48 79 0 47 71 (8013, 13) 0 143 0 0 143 0 0 143 0 ( 13, 6) 26 81 37 47 60 37 31 81 31 (8014, 14) 0 168 0 0 168 0 0 168 0 ( 14, 7) 52 71 45 57 62 50 50 76 43 ( 8, 24) 0 160 0 0 160 0 0 160 0 ( 8, 9) 0 72 40 5 72 44 0 73 40 ( 8, 6) 43 59 20 50 35 31 40 42 32 (8012, 12) 0 140 0 0 140 0 0 140 0 ( 12, 11) 0 98 42 0 93 47 0 95 46 ( 2, 7) 42 48 32 49 42 35 43 49 34 ( 2, 1) 91 0 38 89 0 36 81 0 36 ( 2, 3) 58 166 40 69 153 42 58 151 44 ( 2, 15) 0 153 0 0 153 0 0 153 0 ( 7, 9) 53 86 8 56 83 6 58 89 8 ( 7, 6) 30 80 43 30 73 45 30 86 46 ( 7, 14) 0 135 0 0 135 0 0 135 0 ( 7, 2) 4 65 72 6 69 65 0 71 71 ( 9, 16) 0 187 0 0 187 0 0 187 0 ( 9, 8) 81 0 33 79 0 39 80 0 33 ( 9, 10) 52 91 37 53 87 41 56 95 41 ( 9, 7) 50 49 34 51 55 26 51 54 32 (8015, 15) 0 118 0 0 118 0 0 118 0 Figure 7-4. Continued

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82 TP=4 TP=5 TP=6 LINK L T R L T R L T R ( 15, 2) 78 26 14 74 32 12 71 39 9 (8016, 16) 0 158 0 0 158 0 0 158 0 ( 16, 9) 21 81 56 25 80 54 19 77 62 ( 3, 10) 41 47 53 46 46 53 48 44 67 ( 3, 2) 77 111 61 82 107 62 79 108 57 ( 3, 4) 0 169 98 0 161 100 0 155 95 ( 3, 19) 0 175 0 0 175 0 0 175 0 ( 10, 20) 0 95 0 0 95 0 0 95 0 ( 10, 9) 61 86 52 60 88 48 57 86 59 ( 10, 11) 63 104 0 59 110 0 67 111 0 ( 10, 3) 47 65 56 49 61 58 45 68 55 (8019, 19) 0 195 0 0 195 0 0 195 0 ( 19, 3) 46 83 67 50 80 65 45 85 65 (8020, 20) 0 175 0 0 175 0 0 175 0 ( 20, 10) 51 90 34 60 79 36 52 89 35 ( 4, 23) 0 153 0 0 153 0 0 153 0 ( 4, 3) 18 135 53 22 154 45 29 134 49 ( 4, 5) 0 160 37 0 155 34 0 159 26 (8023, 23) 0 128 0 0 128 0 0 128 0 ( 23, 4) 101 0 27 101 0 27 99 0 29 ( 5, 11) 32 0 32 25 0 35 24 0 23 ( 5, 4) 55 104 0 54 103 0 58 113 0 ( 5, 25) 0 195 0 0 195 0 0 195 0 ( 11, 10) 11 94 25 8 83 37 10 84 24 Figure 7-4. Continued 7.4 Comparison of Static and Dynamic Transportation Management Systems CORSIM 5.1 was used to simulate both static and dynamic cases. Again, because CORSIM 5.1 simulates traffic conditions of a network with the feature that traffic characteristics can change with time, the tim e-varying signal timing plans (shown in Figure 7-3) and traffic flows (shown in Figure 7-4) in the dynamic case were specified in a sequence of time periods in a CORSIM 5.1 input file for simulation and animation. For the static case, cumulative results were reported by CORSIM 5.1 every ten minutes. For the dynamic case, CORSIM 5.1 produced its cumulativ e simulation results at the completion of each time period. Figures 7-5 and 7-6 show the excerpts of the cumulative simulation results, e.g.,vehicle miles and vehicle delay time, at the elapsed simulation time of 40 minutes for the static and dynamic cases, respectively.

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83 To measure quantitatively and compare the quality of traffic service (i.e., frequency, expediency, smoothness and safety) for both st atic and dynamic transportation management systems, a Measure of Effectiveness (MOE) ha s to be used. The available MOEs include delay, average vehicle speed, degree of satura tion, number of stops, fuel consumption, etc. Among these useful MOEs, delay time, which includes increased travel time from reduced speed and time added due to traffic signal contro l, is a primary MOE used to evaluate the performance of transportation systems. Table 7-1 summarizes the cumulative networkwide vehicle delays at the end of each 10-minute time period for both the static and dyna mic cases. The table shows the traffic conditions were improved at the fourth time period for the dynamic case. At end of the onehour analysis period, for the sample network and the given traffic demands, the networkwide delay was 1,102 vehicle-hours for the sta tic case. For the dynamic case, CORSIM 5.1 simulated the six 10-minute time periods sequentially and the network-wide delay during the one hour analysis period was 925 vehicle-hours. The fifth column in Figures 7-5 and 7-6 shows the vehicle delay for each link at the end of the fourth period for both cases. The results show that the traffic conditions on so me links were improved in the dynamic case but some became worse. For example, for Link (12, 11), the total delay was 1,453 vehicleminutes in the static case and was improved to 338 vehicle-minutes in the dynamic case. For Link (6,8), opposite results were observed. However, by summing up the vehicle delay on each link, the network-wide delay in the dynamic case was improved by 43 vehicle-hours at the end of the fourth time period and 150 vehicle-hours for the complete one-hour analysis period.

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84 CUMULATIVE NETSIM RESULTS AT TIME 12:40: 0 ELAPSED TIME IS 0:40: 0 ( 2400 SECONDS) VEHICLE MINUTES VEHICLE MOVE DELAY TOTAL LINK MILES TRIPS TIME TIME TIME --------------------------------( 25, 5) 112.70 466 150.3 229.8 380.1 ( 1, 6) 161.43 556 322.9 652.0 974.9 ( 1, 2) 142.69 489 190.3 354.7 544.9 ( 1, 21) 98.80 415 197.6 30.1 227.7 ( 21, 1) 136.27 567 272.5 1375.6 1648.1 ( 24, 8) 112.28 472 224.6 2418.8 2643.3 ( 6, 8) 106.84 447 213.7 443.6 657.3 ( 6, 13) 157.08 493 314.2 49.5 363.6 ( 6, 7) 176.50 603 353.0 1196.6 1549.6 ( 6, 1) 128.86 446 257.7 1272.0 1529.7 ( 13, 6) 184.25 576 368.5 773.6 1142.2 ( 14, 7) 182.54 621 365.1 1634.9 2000.0 ( 8, 24) 106.57 448 213.1 46.6 259.7 ( 8, 9) 105.38 361 210.8 382.0 592.8 ( 8, 6) 115.46 484 230.9 389.6 620.5 ( 12, 11) 123.99 520 248.0 1452.5 1700.5 ( 2, 7) 152.29 529 227.1 791.7 1018.9 ( 2, 1) 136.36 464 181.8 192.2 374.0 ( 2, 3) 500.44 840 667.3 1467.5 2134.8 ( 2, 15) 65.96 296 98.4 17.2 115.6 ( 7, 9) 136.31 566 203.3 921.5 1124.8 ( 7, 6) 135.46 462 270.9 837.1 1108.0 ( 7, 14) 182.56 623 365.1 61.2 426.3 ( 7, 2) 122.48 426 182.7 1020.2 1202.9 Figure 7-5 Excerpt of the Cumulative Simu lation Results at Elapsed Time of 40 Minutes for the Static Case. Table 7-1Cumulative Network-Wide Vehicle Delays at 10-Minute Time Intervals for Static and Dynamic Cases Elapsed Simulation Time Total Delay (vehicle-hours) Static CaseDynamic Case 10 min.98.93101.22 20 min.235.88241.67 30 min.409.56407.22 40 min.617.55574.50 50 min.846.67763.45 60 min.1,101.98952.07

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85 CUMULATIVE NETSIM RESULTS AT TIME 12:40: 0 ELAPSED TIME IS 0:40: 0 ( 2400 SECONDS) VEHICLE MINUTES VEHICLE MOVE DELAY TOTAL LINK MILES TRIPS TIME TIME TIME --------------------------------( 25, 5) 111.01 459 148.0 352.5 500.5 ( 1, 6) 156.78 540 313.6 1045.9 1359.4 ( 1, 2) 148.76 510 198.4 357.9 556.3 ( 1, 21) 66.76 281 133.5 14.4 147.9 ( 21, 1) 143.00 595 286.0 937.7 1223.8 ( 24, 8) 117.27 493 234.5 1956.2 2190.7 ( 6, 8) 116.30 487 232.6 1046.7 1279.3 ( 6, 13) 153.26 481 306.5 51.6 358.1 ( 6, 7) 154.91 529 309.8 769.1 1079.0 ( 6, 1) 106.06 366 212.1 319.7 531.8 ( 13, 6) 183.93 575 367.9 754.8 1122.7 ( 14, 7) 155.79 530 311.6 2122.0 2433.5 ( 8, 24) 103.00 433 206.0 42.1 248.1 ( 8, 9) 109.11 374 218.2 582.7 800.9 ( 8, 6) 91.69 383 183.4 304.0 487.4 ( 12, 11) 114.22 479 228.4 337.6 566.0 ( 2, 7) 140.03 487 208.9 626.7 835.6 ( 2, 1) 118.53 404 158.0 194.7 352.7 ( 2, 3) 474.88 797 633.2 776.3 1409.4 ( 2, 15) 100.15 450 149.4 29.5 178.8 ( 7, 9) 120.32 500 179.5 859.1 1038.5 ( 7, 6) 111.97 382 223.9 327.9 551.9 ( 7, 14) 127.90 436 255.8 40.0 295.8 ( 7, 2) 116.05 404 173.1 2201.1 2374.2 Figure 7-6 Excerpt of the Cumulative Simu lation Results at Elapsed Time of 40 Minutes for the Dynamic Case. Figures 7-7 and 7-8 show the animations fo r a portion of the sample network at the end of the one-hour period for the static and dynamic cases, respectively. Figure 7-7 shows Link (6,7) in the static case had more serious spillback problems. Based on the results from the numerical example, the dynamic traffic mana gement system performed more efficiently, by modifying the traffic conditions based on the signal settings and then adjusting the signal timing settings according to the updated traffic fl ows six times during an hour than the static traffic management system did for the same traffic facilities.

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86 Figure 7-7 Example of Animation Result for the Static Case. Figure 7-8 Example of Animation Result for the Dynamic Case.

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87 CHAPTER 8 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS 8.1 Summary and Conclusions Intelligent Transportation Systems (ITS), which have been a topic of substantial research during the past decade, are being developed to improve the efficiency and productivity of existing transportation facil ities. The success of ITS depends on two important sub-systems, Advanced Traffic Management Systems (ATMS) and Advanced Traveler Information Systems (ATIS). Real-time traffic control and dynamic tr affic assignment are two major support technologies of ATMS because the traditional st atic network equilibrium models which were developed for long-term transportation planning, and the traditional traffic control management systems which were developed to set traffic signals by assuming fixed flow patterns, are not suitable for analyzing and solving transportation problems in real time. A combined model has been developed to in tegrate the real-time traffic control model and the dynamic traffic assignment model. For the real-time traffic control model, the framework of the traditional traffic contro l model, which minimi zes total delay, was extended to the dynamic case by adding time as an additional dimens ion. However, the conventional delay model, Webster’s delay formula, which was originally derived through theoretical queuing analysis for isolated intersec tions, predicts infinite values of delay when flows approach capacity. Yet, in realistic situ ations, a queue will not grow infinitely because

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88 drivers will change routes to avoid large queue s. The generalized delay model for signalized intersections in the 2000 Highway Capacity Manual was used in the real-time traffic control model to take the oversaturated queue probl em into account. A solution algorithm applying the Frank-Wolfe method has been developed to implement the real-time traffic control model. It uses TRANSYT-7F’s static optimal signal settings as the initial solution to increase the chance for the non-convex objective function to converge to a near-optimal solution. The dynamic traffic assignment model is formulated as a Variational Inequality (VI) formula. Using the simulation model to calcula te link travel times, a relaxation algorithm is used to relax the asymmetric link travel time function resulting from the flow propagation constraints. A solution algorithm has been developed to implement the equivalent optimization model of the relaxed VI formula. The iterative optimization assignment (IOA) procedure consisting of solving sequentially an optimization model involving the signal timing variables, with the flow variables fixed, and a user-optimized equilib rium model corresponding to the new green time settings, is used to solve the combined model. Applying the IOA procedure, a solution algorithm with operational capability to solve dynamic network management models corresponding to the above technological concepts in ITS has been developed. A comp rehensive computer program has been prepared to implement and test all algorithms associated with the dynamic transportation management model. CORSIM 5.1, the simulation software which a llows traffic characteristics to be timevarying, was used to demonstrate the performa nces of both static and dynamic cases using

PAGE 99

89 a sample network and a given O-D demand matrix. Because of the limitations on resources available to this study, a sample network instead of the real-world data was used to test the procedure. It is a common practice that multiple simu lation runs, each specified with different sets of random seeds, are performed when CO RSIM or other simulation computer packages are used. However, since the variation in the simulation results obtained by changing random number seeds are generally much less than 13.6%, only one CORSIM simulation run was performed in this study. The test results indicate that dynamic traffic assignment with adaptive traffic-responsive signal settings reduced the network-wide delays about 13.6% by periodically modifying the traffic fl ows based on the traffic conditions and then adjusting the signal timing settings according to time-dependent traffic flows. This study applied mathematical programming methodologies to model traffic assignment and traffic signal control systems as well as the dynamic interactions between them. The dynamic transportation management system was modeled more realistically than those under a static state. In addition, the ma thematical solutions are proven to be optimal and satisfy the equilibrium conditions in comparison with those developed solely based upon simulation techniques. Incorporating time-dependent optimal signal settings into an existing traffic control system can increase system efficiency by better assessing network conditions and achieving better traffic control. 8.2 Recommendations The travel time function is very important. The same traffic assignment policy considered under different disutility assumptions may possess completely different theoretical properties and thus may be expected to produce completely different traffic flow

PAGE 100

90 results. The BPR formula is known to have the limitation of an ambiguous definition of capacity in the function because the BPR formul a increases travel times even after a link's traffic flow is higher than its capacity. Further investigation into enhancements of the BPR formula is recommended. The dynamic queue-responsive tr affic signal control with dynamic traffic assignment technique developed in this research consists of a number of parameters, including the cycle length, signal phase plan and O-D demand, th at require input from users and were not modified during the optimization. Unlike the simulation-based, real-time computer systems which can simulate the traveler’s behavior of changing departure time in response to congestion conditions, the O-D demand is presum ed fixed during the analysis period in the mathematical programming model developed in th is study. In other words, travelers would not be able to change their departure time b ecause of traffic conditions. This assumption is practical for peak hour traffi c since most roadway users do not have the privilege of arbitrarily altering the time they begin work or complete their jobs for the day. Further research is recommended to consider varied cycle lengths and signal phase sequences by programming additional external looping into the optimization process. In addition, the traffic persisting for an hour under the static state was separated into six discrete 10-min time intervals for the exam ple. The duration of each time period may change the performance of a dynamic transpor tation management syst em. Intuitively, the smaller the duration of each time period, the be tter the performance of the transportation system since the optimal signal settings and traffic flows can be updated more frequently. However, the marginal benefits for dividing hourly demand into shorter time periods may be limited since it takes time for road users to respond to the traveler information they are

PAGE 101

91 given and for new signal timing plans to become effective. Further research is thus recommended to determine the robustness of the duration of discrete time interval. While it is desirable to reduce the burden of input from users, it is also desirable to have some control over the IOA procedure through some parameter settings such as the convergence criteria and the maximum number of iterations. This is because the impact of the values of these parameters involve some trade-offs between computer running time and the quality of the solution. It might be benefici al to investigate the possibility of calibrating the optimal values for these parameters. Although the IOA procedure has been demonstrated empirically in previous research as producing a near optimal solution, the appro ach is not theoretically proven to minimize total travel time. It might be more practical to change different parameters and run the procedure again if the result leads to a dec line in network performance rather than an improvement. A criterion can be added at the convergence test to monitor if there is a decline in network performance. To simply the process, the Visual Ba sic program was designed to only handle intersections without diagonal traffic, and with pre-timed signals with protected left-turning movements and no permitted left-turning movements during other signal phases. Ideally, the program should handle any signal control type allowed in CORSIM 5.1 because the program reads the CORSIM 5.1 input files. Future work should include developing modules to process different signal control types and more complex geometries.

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92 REFERENCES 1.Federal Highway Administration, A Roadmap for the Research, Development and Deployment of Traffic Estimation and Pr ediction Systems for Real-Time and OffLine Applications (Treps, Treps-P) Office of Operations Research and Development, July, 2001. 2.Mahmassani, H.S., H.A. Sbayti, and Y.-C. Chiu, DynaSMART-P User’s Guide Version 0.930.0, U.S. Department of Transportation, Federal Highway Administration, October 2002. 3.Sheffi, Y., Urban Transportation Networks: Equilibrium Analysis with Mathematical Programming Methods Prentice-Hall, Inc., Englewood Cliffs, N.J., 1985. 4.Drissi-Katouni, O., “A Variational Ine quality Formulation of the Dynamic Traffic Assignment Problem,” European Journal of Operational Research Vol. 71, pp. 188204, 1993. 5.Ran, B. and D. Boyce, Modeling Dynamic Transportation Networks: an Intelligent Transportation System Oriented Approach Springer, Berlin, 1996. 6.Gartner, N.H., C. Stamatiadis, and P.J. Tarnoff, “Development of Advanced Traffic Signal Control Strategies fo r ITS: A Multi-Level Design,” Transportation Research Record 1494 Transportation Research Board, National Research Council, Washington, DC, pp. 98-105, 1995. 7.Allsop, R.E., “Some Possibilities for Usi ng Traffic Control to Influence Trip Distribution and Route Choice,” in Proceedi ngs of the 6th International Symposium on Transportation and Traffic Theor y, Elsevier, Amsterdam, 1974, pp. 345-374. 8.Smith, M.J., “Properties of a Traffic Control Policy Which Ensure the Existence of Traffic Equilibrium Consis tent with the Policy,” Transportation Research Vol. 15B, No. 6, pp.453-462, 1981. 9.Sheffi, Y. and W.B. Powell, “Optim al Signal Settings over Transportation Networks,” Journal of Transportation Engineering Vol. 109, No. 6, pp. 824-839, 1983.

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93 10.Smith, M.J. and T.V. Vuren, “Traffic Equ ilibrium with Responsive Traffic Control,” Transportation Science Vol. 27, No. 2, pp. 118-132, 1993. 11.Yang, H. and S. Yagar, “Traffic Assignment and Traffic Control in a Saturated Road Network,” Transportation Research Vol. 29A, No. 2, pp. 125-139, 1995. 12.Chen, H.-K. and C.-F. Hsueh, “Combining Signal Timing Plan and Dynamic Traffic Assignment,” a paper presented at the 76t h Annual Transportation Research Board Meeting, Washington, D.C., 1997. 13.Gartner, N.H. and C. Stamatiadis, “Int egration of Dynamic Traffic Assignment with Real Time Traffic Adaptive Control,” a paper presented at the 76th Annual Transportation Research Board Meeting, Washington, D.C., 1997. 14.Chen, O.J. and M.E. Ben-Akiva, “D ynamic Traffic Control and Assignment: A Game Theoretic Approach,” a paper presented at the 77th Annual Transportation Research Board Meeting, Washington, D.C., 1998. 15.Transportation Research Board, Highway Capacity Manual National Research Council, Washington, D.C., 2000. 16.Wallace, C.E., K.G. Courage, M.A. Ha di, and A. Gan,“Methodology for Optimizing Signal Timing: M|O|S|T Volume 4, TRANSYT-7F Users Guide,” FHWA, Office of Traffic Operations and Intelligent Vehicle/Highway Systems, 1997. 17.Beckmann, M.J., C.B. McGuire, and C.B. Winston, Studies in the Economics of Transportation Yale University Press, New Haven, Conn, 1956. 18.Leblanc, L.J., E.K. Morlok, and W. Piersk alla, “An Efficient Approach to Solving the Road Network Equilibrium Traffic Assignment Problem,” Transportation Research Vol. 9B, No. 5, pp. 309-318, 1975. 19.Potts, R.B., and R.M. Oliver, Flows in Transportation Networks Academic Press, New York, 1972. 20.Sheffi, Y., and W.B. Powell, “An Al gorithm for the Equilibrium Assignment Problem with Random Link Times”, Networks Vol. 12, No. 2, pp. 191-207, 1982. 21.Dial, R.B., “A Probabilistic Multipath Traffic Assignment Algorithm Which Obviates Path Enumeration,” Transportation Research Vol. 5A, No. 2, pp. 83-111, 1971. 22.Sheffi, Y., and W.B. Powell, “A Comparis on of Stochastic and Deterministic Traffic Assignment Over Congested Networks,” Transportation Research Vol. 15B, No. 1, pp. 53-64, 1981.

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94 23.Gartner, N.H. and G. Improta, Urban Traffic Networks Springer Verlag, Berlin, 1995. 24.Gartner, N.H., “OPAC: A Demand-Respons ive Strategy for Traffic Signal Control,” Transportation Research Record 906 Transportation Research Board, National Research Council, Washington, DC, pp. 75-81, 1983. 25.Merchant, D.K. and G.L. Nemhauser, “A Model and an Algorithm for the Dynamic Traffic Assignment Problem,” Transportation Science Vol. 12, pp.183-199, 1978. 26.Chen, H.-K, C.-W. Chang, M.-S. Chang, and C.-Y. Wang, “A Comparison of LinkBased versus Route-Based Algorithms with the Dynamic User-Optimal Route Choice Problem,” a paper presented at the 77th Annual Transportation Research Board Meeting, Washington, D.C., 1998. 27.Fisk, C.S., “Game Theory and Transportation Systems Modeling,” Transportation Research Vol. 18B, No. 4/5, pp. 301-313, 1984. 28.Marcotte, P., “Network Design Problem with Congestion Effects: A Case of Bilevel Programming,” Mathematical Programming Vol. 34, No. 1, pp.142-162, 1986. 29.Fambro, D.B. and N.M. Rouphail, “Generalized Delay Model for Signalized Intersections and Arterial Streets,” Transportation Research Record 1572 Transportation Research Board, Nationa l Research Council, Washington, DC, pp. 112-121, 1996. 30.Rouphail, N.M., M. Anwar, D.B. Fambro, P. Sloup, and C.E. Perez, “Validation of Generalized Delay Model for Vehicle-Actuated Traffic Signals,” Transportation Research Record 1572 Transportation Research Board, National Research Council, Washington, DC, pp. 105-111, 1996. 31. TRANSIMS-2.1 User Documents Los Alamos National Laboratory, July 2001. 32.ITT Systems & Sciences Corporation, CORSIM User’s Manual Version 5.1, Colorado Springs, CO, January 2002. 33.Nagurney, A., Network Economics: A Variational Inequality Approach Boston, Massachusetts, Kluwer Academic Publishers, 1993. 34.Ortega, J.M. and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables New York, Academic Press, 1970. 35.Bard, J.F. and J.E. Falk, “An Explicit Solution to the Multilevel Programming Problem,” Computers and Operations Research Vol. 9, pp. 77-100, 1982.

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95 36.Smith, M.J. and T.V. Vuren, “Traffic Equ ilibrium with Responsive Traffic Control,” Transportation Science Vol. 27, No. 2, pp. 118-132, 1993.

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96 BIOGRAPHICAL SKETCH Lee-Fang Chow was born in Fung-Sang, Ta iwan, on July, 4, 1966. She received her B.B.A. degree in transportati on engineering and management from the National Chiao-Tung University at Taiwan in 1988. Before coming to the United States in 1992, she had worked as a traffic engineer in two major consulting companies in Taiwan for four years. She received her M.S. degree in civil engineering from the University of Florida in 1994 and continued to pursue doctoral studies in 1996. Ms. Chow worked as a graduate research assistant with the Un iversity of Florida Transportation Research Center from 1993 to 1994 and completed a sponsored research project with Dr. Mohammed Ha di and Dr. Joseph Wattleworth. She worked as a graduate teaching assistant with Dr. Gary Long and Dr. Albert Gan for the courses of transportation engineering and civil engineering systems from 1996 to 2000. She was also involved in the test of EVIPAS and other tra ffic computer packages develope d under the sponsorship of the Federal Highway Administration. Ms. Chow has worked as a research associate with the Florida International University Transportation Research Center since 2000. She won the Best Student Paper Award of District 10 Institute of Transportation Engineers in 1997. Ms. Chow is married to Min-Tang Li.


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INTEGRATING ADAPTIVE QUEUE-RESPONSIVE TRAFFIC
SIGNAL CONTROL WITH DYNAMIC TRAFFIC ASSIGNMENT
















BY

LEE-FANG CHOW


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


2003















ACKNOWLEDGMENTS

I wish to express my gratitude to my advisor, Dr. Gary Long, for his guidance and

financial support over the course of this research. Working for him is an invaluable

experience for me as a graduate student at this institution.

I would like to thank Professor Kenneth G. Courage, Dr. Bon A. Dewitt, Dr. Sherman

X. Bai and Dr. Myron N. Chang for their constructive comments and recommendations.

Finally, very special thanks go to my husband, Min-Tang Li, my son, Andrew, and

my parents for their love and support through my study.















TABLE OF CONTENTS

Page

ACKN OW LEDGM EN TS ................................................ ii

TABLE OF CONTENTS ............... ............................. iii

LIST OF TABLES ......... ............................................ vi

LIST OF FIGURES ................. ................................ vii

ABSTRACT ......... ................................................. ix

CHAPTERS

1 INTRODUCTION .................................................1

1.1 Background ............ ...................... ....... 1
1.2 Problem Statement ...................................... 2
1.3 Goal and Objectives ............... ........................ 3
1.4 Scope and Limitations .................................... 4
1.5 Dissertation Organization ................................... 5

2 REVIEW OF STATIC AND DYNAMIC TRANSPORTATION MANAGEMENT
SYSTEMS ........ .............................................7

2.1 Introduction .................................... .......... 7
2.2 Static Traffic Control System ................. ................. 7
2.3 Static Traffic Assignment ..................................... 8
2.3.1 User Equilibrium ................. .................. 8
2.3.2 System Optimization .............................. 14
2.3.3 Stochastic User Equilibrium ......................... 14
2.4 Combining a Static Traffic Signal Timing Plan and Traffic Assignment 18
2.5 Real-Time Traffic-Responsive Signal Control System .............. 20
2.5.1 Optimization Policies for Adaptive Control .............. 20
2.5.2 Non-Linear Program of Dynamic Traffic-Responsive Signal
Control System ................................... 21
2.5.3 Variational Inequality Model for Dynamic Traffic Control .... 23
2.6 Dynamic Traffic Assignment .............................. 24









2.6.1 Variational Inequality (VI) Formulation for DTA ............ 24
2.6.2 Route-Based Algorithms for DTA ................... ..... 27
2.7 Combining a Dynamic Traffic Signal Timing Plan and Traffic
A ssignm ent ....... ............................. ......... 28
2.7.1 Game Theory Approach ........................... ... 28
2.7.2 Bilevel Programming ............... ............... 29
2.8 Generalized Delay M odel .................................. 30
2.9 An Evaluation Tool: CORSIM 5.1 ............................. 31

3 OPTIMIZATION MODEL FOR DYNAMIC TRAFFIC CONTROL ........ 34

3.1 Introduction ............................................ 34
3.2 Optimization Model ................ ...................... 34
3.3 Convexity ........................... ................. 37
3.4 Solution Algorithm ..................................... 40

4 DYNAMIC USER-OPTIMAL ROUTE CHOICE MODEL ................ 49

4.1 Introduction ............. ................ ... ....... 49
4.2 Variational Inequality Models ................................ 49
4.2.1 Static Transportation Network Equilibrium Model ............ 50
4.2.2 Dynamic Transportation Network Equilibrium Model ......... 52
4.3 Enhanced Solution Algorithm ......................... ... ... 54

5 COMBINED MODEL .................................. ......... 59

5.1 Introduction ............................................. 59
5.2 Bilevel M odel .................................... .. ....... 60
5.3 Iterative Optimization and Assignment Procedure .................. 61
5.4 Detailed Procedure ............... .......................... 62

6 IMPLEMENTATION .............. ...........................66

6.1 Introduction .................... ................ ......... 66
6.2 Numerical Example ......................................... 66
6.3 Software Implementation ..................................... 69

7 RESULTS AND EVALUATION ................................. 77

7.1 Introduction ............... ... ... ..... ... .......... 77
7.2 Results for Static Signal Settings and Static Traffic Flows .............. 77
7.3 Results for Adaptive Signal Settings and Dynamic Traffic Assignment .... 77
7.4 Comparison of Static and Dynamic Transportation Management Systems 82









8 SUMMARY, CONCLUSIONS AND RECOMMENDATIONS ............ 87

8.1 Summary and Conclusions .................................... 87
8.2 Recommendations ............... .......................... 89

REFERENCES ....... ............................................... 92

BIOGRAPHICAL SKETCH ........................................... 96















LIST OF TABLES

Table Page

6-1 O-D Demand Matrix for the Example Application ....................... 68

7-1 Cumulative Network-Wide Vehicle Delays at 10-Minute Time Intervals
for Static and Dynamic Cases ................................. 84















LIST OF FIGURES


Figure Page

3-1 Flow Chart of the Solution Algorithm for the Real-Time Traffic-Responsive
Signal Control Model ..................................... 48

4-1 Flow Chart of the Solution Algorithm for the DTA Model ................. 58

5-1 Flow Chart of the Iterative Optimization Assignment Procedure ............ 63

6-1 Link-Node Structure of the Example Network .......................... 67

6-2 Initial Signal Settings for the Example Application ................... .. 69

6-3 Main Menu Screen ............................................... 70

6-4 Excerpt of the Initial Assignment Results for the Example Application ....... 71

6-5 Excerpt of the Auxiliary Green Time Results ........................... 73

6-6 Excerpt of the Optimal Green Time Results at the Initial Iteration ........... 74

6-7 Excerpt of the Link Travel Time Results Based on the Simulation .......... 74

6-8 Excerpt of the Link Travel Time Results from the BPR Formula ............ 75

6-9 Excerpt of the Comparison of Link Travel Times Based on Simulation Results
and the BPR Formula .................................... 76

7-1 Static Traffic Assignment Results .................................. 78

7-2 Optimal Signal Settings for the Static Case ............................. 78

7-3 Optimal Time-Dependent Signal Timing Settings for the Example Network ... 79

7-4 Optimal Time-Dependent Traffic Flows for the Six Time Periods ........... 80









7-5 Excerpt of the Cumulative Simulation Results at Elapsed Time of 40 Minutes
for the Static Case ......... ..... ...................... ....... 84

7-6 Excerpt of the Cumulative Simulation Results at Elapsed Time of 40 Minutes
for the Dynamic Case ........................................... 85

7-7 Example of Animation Result for the Static Case ........................ 86

7-8 Example of Animation Result for the Dynamic Case ................... 86















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


INTEGRATING ADAPTIVE QUEUE-RESPONSIVE TRAFFIC
SIGNAL CONTROL WITH DYNAMIC TRAFFIC ASSIGNMENT


by

Lee-Fang Chow

August 2003

Chairman: Gary Long
Major Department: Civil and Coastal Engineering

Recently, with their promise for Intelligent Transportation Systems (ITS), dynamic

traffic assignment (DTA) and adaptive traffic-responsive signal control have received greater

attention. However, the mathematical models used to describe the interaction between these

two systems are tenuous and considerable effort is still needed to improve the limitations

described in the existing literature, especially regarding the solution capabilities, the

oversaturated-queue phenomena, efficacious estimates of link travel disutilities, and a

reliable evaluation method. The objectives of this research are to implement non-linear

programming techniques to model dynamic traffic assignment and adaptive queue-

responsive traffic control separately, and develop an iterative procedure to solve the

components of traffic assignment and signal control by minimizing the overall system-wide

signal delay.









For traffic signal control, total intersection delays in a network are minimized by

allocating appropriate time splits. The generalized intersection delay model in the Highway

Capacity Manual was used to take the oversaturated queue problem into account.

Improvements to DTA methods include formulating the dynamic user-optimal route

choice model as a variational inequality (VI) model, calculating the expected travel time on

each link using a simulation model, applying a relaxation algorithm to produce an equivalent

optimization formulation of the VI model, and developing a solution algorithm which can

be implemented using existing traffic software.

There is not much hope for developing exact solution algorithms to solve these two

models simultaneously because of the computational complexity of the non-linear programs.

Therefore, a heuristic procedure involving an iterative optimization assignment is used to

solve the combined models.

A computerized procedure was developed to implement the solution procedures and

a numerical example of a traffic network was prepared to test the program. An accepted

traffic simulation model, CORSIM 5.1, was used to validate the results for both static and

dynamic optimizations. The test results showed that dynamic traffic assignment with

adaptive traffic-responsive signal settings reduced the network-wide delays by nearly 15%.















CHAPTER 1
INTRODUCTION

1.1 Background

Traffic congestion, especially during peak travel periods, is well-known to travelers

in most urban areas. Most travelers evaluate the degree of congestion by the amount of time

in which they are delayed. Consequently, minimizing delay is important to transportation

engineers for setting the timing controls on traffic signals.

Delay is related to both traffic volumes and the capacities of roadway links. Traffic

volumes fluctuate with different levels of traffic demand during a typical day, and also with

diversions of traffic to alternative paths as motorists strive to find the fastest paths between

their origins and destinations to minimize their delays. When traffic congestion becomes

intense with long queues of traffic waiting at traffic signals such that portions of the queues

must wait for the next signal cycle, known as oversaturated queues, minimizing delay

becomes more complicated.

In the traditional traffic management system, link capacities and traffic flow volumes

are considered to remain fixed during a whole analysis period. Therefore, the standard

traffic management problem is to optimize traffic network performance with given fixed

demands for travel and a fixed supply of transportation facilities.

Currently, Intelligent Transportation Systems (ITS) are being designed and

developed to improve the efficiency of existing traffic network performance. Computer and







2

electronic communication technologies are exploited to provide real-time information and

route guidance to motorists through Advance Traveler Information Systems (ATIS). Real-

time information and route guidance are produced by Dynamic Traffic Assignment (DTA)

systems based on real-time traffic data, historical databases and dynamic traffic-responsive

signal control strategies. Therefore, the interaction in the Advanced Traffic Management

Systems (ATMS) will become dynamic because the traffic flow on each link and control

actions are time-dependent.

1.2 Problem Statement

To date, the mathematical models used to describe the dynamic interactions in

transportation systems are still tenuous. Two simulation-based, real-time computer systems

have been developed under the DTA Project sponsored by the Federal Highway

Administration (FHWA) with Oak Ridge National Laboratories (ORNL) as the program

manager [1]:

1. DynaMIT is the result of approximately 10 years of research and development at the
Intelligent Transportation Systems Program of the Massachusetts Institute of
Technology (MIT).

2. DYNASMART is the outgrowth of several years of research and development at the
University of Texas, with the participation of researchers currently at the University
of Maryland, Purdue University, and Northwestern University.

These DTA programs provide real-time computer systems for traffic estimation, prediction,

and generation of traveler information and route guidance. They support the operation of

ATIS and ATMS at Traffic Management Centers (TMC). However, they are simulation-

based dynamic traffic assignment systems with micro-simulation of individual user decisions

in response to traveler information and a macroscopic traffic flow simulation approach [2].

Although the analytical approach which formulates the entire system as a mathematical







3

model has difficulties in formulating and solving the program, the solutions obtained can be

proven to be optimal and satisfy the equilibrium conditions (i.e., the dynamic generalization

of Wardrop's principle of user-optimal network equilibrium [3]).

Most current research addresses the development of DTA systems [4, 5] and real-

time traffic-responsive control systems [6] independently. Allsop [7] was the first to address

the interaction between traffic control and traffic assignment, followed by Smith [8], Sheffi

and Powell [9], Smith and Vuren [10], and Yang and Yagar [11]. However, all of them only

considered the static traffic situation. Some of the current development work utilizes

frameworks for the integration of DTA and traffic control systems into a combined model.

Chen and Hsueh [12] proposed a combined model for a discrete-time dynamic traffic-

responsive signal control system. A heuristic approach involving an iterative optimization

and assignment (IOA) method was applied to solve a numerical example involving a small

network. However, the traffic queue phenomenon was not considered in the traffic control

model nor was a good estimate of travel disutility considered in the assignment model.

Gartner and Stamatiadis [13] presented only a framework to integrate dynamic traffic

assignment with real-time traffic adaptive control without any mathematical model or

formulation. Chen and Ben-Akiva [14] formulated the combined dynamic traffic control-

assignment system as a non-cooperative game between a traffic authority and users but did

not give any algorithm to solve it.

1.3 Goal and Objectives

The goal of this dissertation research is to develop a model to integrate the effects

of queue-responsive signal timings on discrete-time traffic flow patterns in dynamic traffic

assignment systems. The specific objectives of this research are the following:







4

1. To review the state-of-the-art in the areas of dynamic traffic assignment and dynamic
traffic-responsive signal control.

2. To develop a combined model for dynamic traffic assignment and adaptive traffic
control. In the traffic control part, the vehicle delays in the network are represented
by the generalized delay model for signalized intersections in the Highway Capacity
Manual [15]. In the traffic assignment part, good estimates of link travel times are
considered for motorists to find dynamic user-optimal paths over the network for
given origin-destination (O-D) demands and signal timings from the traffic signal
control model.

3. To propose an algorithm to solve this combined model.

4. To demonstrate how to program and apply existing traffic software to implement the
solution algorithms and test by a numerical example of a traffic network.

5. To demonstrate the results of the combined model by simulation.

1.4 Scope and Limitations

A state-of-the-art algorithm is developed in this study to optimize total intersection

delays in a given traffic network by considering the dynamic interaction effect between the

traffic control system and traffic assignment model. The dynamic interaction is considered

by adding discrete time periods as an additional dimension in the model structures and

solution algorithm. Delays arising from oversaturated queues are included in optimizing

signal settings and traffic flows. In this study, both of the cycle length and signal phase plan

at each intersection in a given traffic network are presumed fixed and are not altered during

the optimization process. In addition, the travel demand between each traffic analysis zone

pair is assumed fixed during the analysis period. To test the model, microscopic simulation

was used as a surrogate for field data collection to enable controllabilty of input information

and the due to limitations on resources available to the researcher. CORSIM 5.1 was chosen

to perform the simulation because of its availability to the researcher and because of its

widespread acceptance and utilization in the USA.







5

1.5 Dissertation Organization

In Chapter 1, an introduction to the research topic and the needs for the research are

presented. Improvements needed in previous research are identified and the specific

objectives of the research are stated.

A literature review on transportation management systems is presented in Chapter

2. The static transportation management system is first reviewed, followed by the dynamic

transportation management system. An evaluation tool, the simulation software, and the

generalized delay model which improves the oversaturated queue problem in dynamic traffic

control are also reviewed.

Chapter 3 presents an improved optimization model for the dynamic traffic control

system. A generalized intersection delay model is used to take the oversaturated queue

problem into account. A strategy to improve the situation where the optimization program

with a non-convex objective function may not reach the global minimum is developed. This

chapter ends with an application of the Frank-Wolfe method to solve the optimization

program.

Chapter 4 deals with the dynamic traffic assignment system. A dynamic user-optimal

route choice model is formulated as a variational inequality (VI) model. A simulation

model, CORSIM 5.1, is used to estimate the travel time on each link. A relaxation algorithm

is applied to produce an equivalent optimization formulation of the VI model when the travel

time on each link is known. A solution algorithm which can be implemented by existing

traffic software is developed.

Chapter 5 presents strategies for combining the real-time traffic-responsive signal

control model and the dynamic traffic assignment model. In Chapter 6, a computer program







6

that was prepared to implement all the solution algorithms developed for dynamic

transportation management is introduced. Several existing traffic software systems are built

into the program to simplify the input processes for the transportation network, the O-D

demand and the traffic control actions. A numerical example is used to test the program.

Chapter 7 presents the comparison of the traffic performance between static and

dynamic transportation management systems. Chapter 8 presents a summary and

conclusions and recommends areas for further research.















CHAPTER 2
REVIEW OF STATIC AND DYNAMIC TRANSPORTATION MANAGEMENT
SYSTEMS

2.1 Introduction

The performance of a transportation system is the result of a set of complex

interactions between traffic operational controls and the demand for transportation services.

This is the basis for the development of traffic control and assignment models for the design

and planning of transportation management systems.

In this chapter, the static transportation management system, which includes static

traffic control, static traffic assignment and a "balancing" process to integrate these two

systems together, is first reviewed, followed by recent research which extends these concepts

to the dynamic case. Greater details are given for the algorithms more directly related to

those needed in adaptive queue-responsive traffic signal control with dynamic traffic

assignment as developed in this research.

2.2 Static Traffic Control Systems

Traffic control systems deal with how to determine the optimal traffic signal timings

to reduce fuel consumption, minimize delay and also improve safety. TRANSYT-7F [16]

is the best-known software for determining the static network-wide optimal signal timing

settings. With fixed-flow input, TRANSYT-7F uses a macroscopic, deterministic simulation

and optimization procedure to find the best solution. The optimization procedure can be

briefly described as follows:







8

Step 1: An initial signal timing plan is simulated by the traffic model and an initial
Performance Index (PI) is calculated.

Step 2: The signal timing setting is changed by a specified amount and the resulting traffic
flow is re-simulated. A new PI is then calculated.

Step 3: The new PI is compared with the previous value. If no further improvement can be
made by varying the signal setting, the procedure will stop. Otherwise, go to Step
2.

Although TRANSYT-7F is a powerful tool to determine static optimization, it is not

feasible to enhance it to the dynamic case because it is too complicated to modify the

macroscopic simulation procedure with time-dependent traffic flows.

2.3 Static Traffic Assignment

The traffic assignment system can be stated as finding the link flows when given the

O-D traffic volumes, the network, and the link performance functions. To solve the problem,

a rule of how motorists choose their routes has to be specified. The equilibrium traffic flows

are determined according to the rule and link performance functions. Three definitions of

equilibrium are user equilibrium, system optimization, and stochastic user equilibrium.

2.3.1 User Equilibrium

User-equilibrium (UE) assumes that each motorist has full information about link

performance relationships and he or she will choose the route with minimum travel

impedance. The travel impedance can include many components. However, travel time is

often used as the sole measure of link impedance [3]. For each origin-destination zone-pair,

at user equilibrium, the travel disutilities on all used paths are equal and no traveler can

improve his or her travel time by unilaterally changing routes [3]. Note that a network is

defined mathematically as a set of nodes and a set of links connecting these nodes. A path

is a sequence of directed links leading from one node to another.







9

In the mid-1950s, a formulation representing the UE condition was developed and

is known as Beckmann's transformation [17]. The formulation can be expressed as follows:


min z(x) = f xm dm(O)dO (1)


subject to

fsk = qrs "r, s (2)

frk O k, r, s (3)

with the incidence relationship

Xm rs ~ k frskmk *m, m (4)

where

Xm = traffic volume assigned to link m

q,s = total traffic volume interchanging from origin r to destination s

frsk = traffic volume flowing along path k connecting origin r to destination s

dm(xm) = travel disutility on link m related to traffic volume, Xm, on link m

im,k = 1, if link m is on path k between origin-destination pair r-s
= 0, otherwise

This program does not have any intuitive economic or behavioral interpretation. It

is only a mathematical model that is utilized to solve the equilibrium problem. To prove the

equivalence between the solution of Beckmann's transformation and the UE condition, the

method of Lagrange multipliers, which involves an auxiliary function known as the

Lagrangian, is applied as follows.

Given the incidence relationship, xm = Xm(f) in Eq. 4, the Lagrangian whose

stationary point coincides with the minimum of the constrained optimization in Eq. 1 to Eq.

4 can be formulated as:







10

L(f, y)= z[x(f)] + r Yrs(qrs f"k ) (5)

where y. denotes the dual variable associated with the flow conservation constraint for O-D

pair r-s in Eq. 2. The stationary point of the unconstrained Lagrangian can be found by

solving for the root of the gradient, 0 ?L(f, y) = 0 and o L(f, y) = 0. 0* L(f, y) = 0 simply

states the flow constraints. However, given the nonnegativity constraints off 40 in Eq. 3,

the following two conditions, associated with L(f, y) = 0, have to hold because the

stationary point of L(f, y) can occur either for positive f or it can be on the boundary of the

feasible region where some fsk = 0. Obviously, the condition f '0 has to hold as well.


frs dL(f, y) 0 L(f, y) > Vk,r, s
fk afk(6)


The above first-order condition can be obtained by the following calculation:


dL(f, y) z[x(f)] + yr(qrs fk) (7)
af4" afks f k

dZLX(I)] Z(X) /Am
Sf.rs m dXm rs


= fdm(O)do ) 8M,k = dmm,k = U (8)
m axm m m


8 CYr(qs f ) E "Yk(-C Yrk)
y.s k ) = ) = -Yrs (9)
f.rs rs k rs k f f.r


where uks = travel disutility along path k connecting origin r to destination s.







11

The general first-order conditions for the minimization program in Eqs. 1 to 4 can be

expressed as

fkrs (Ukrs- Yr) = 0 k, r, s (10)

Ukrs- yrs O* k, r, s (11)

k fnk qrs fr, (12)

frk O *k, r, s (13)

From Eq. 11, y,,rs is less than or equal to ukrs, the travel disutility along path k

connecting origin r to destination s. Therefore, y,,rs equals the minimum path travel disutility

between origin r and destination s. Eq. 10 holds at the point that minimizes the objective

function for either fkrs = 0 and (ukrs rs) > 0, or fkrs> 0 and (ukrs rs) = 0. That means if the

travel disutility, ukrs, on path k is greater than y,,rs, the minimum path travel disutility, the flow

on this path is zero; or if ukrs = rs, the flow fkrs is positive. This simply interprets the

principle of UE.

For the solution of the UE program to be unique, it is sufficient only to prove that the

objective function is convex with respect to link flow, since the convexity of the feasible

region is assured for linear equality and nonnegative constraints. To prove that z(x) is

convex, the matrix of the second derivatives of z(x) with respect to x (the Hessian) has to be

positive definite:



since 2Z() Odm(xm) ddn(xn) for m = n
since dx (14)
XmdXn x 0 otherwise


where m and n are both link indexes. When m is not equal to n, the second derivatives

would be zero.










dd (xl)
0 0 ...
dxI1

dd2(x2)
0 0
Therefore, V2z(x) = 2 (15)
0 0
ddA(xA)
dxA
^A


The link travel disutility related to link traffic volume, d(x), is normally positive and

increasing. Therefore, the above matrix is positive since it is a diagonal matrix with positive

entries.

Since the objective function is convex and constraints are all linear, the minimization

model can be solved by the Frank-Wolfe (FW) method [18]. The FW method is based on

finding a descent direction by minimizing a linear approximation of the objective function

at the current solution point. This linear approximation is given by z(u") = z(x") + 2(x")

(u" x")" with respect to x" for the nth iteration. This linear function of u" has to be

minimized subject to the constraints of the original model:

min z(u") = z(xn) + t(xn) (u" xn)" (16)

subject to g sk g rs "r, s (17)

grsk 0 k, r, s (18)

with the incidence relationship
n= k* *s r g *"m (19)
Um rs k k m,k (19)

where g denotes the path flows. The objective function can be simplified to min 2(xn) u"n1

because z(x") and t(xn)xn are constant when x" is known. Moreover, min 2(xn) u"n* min

* (* 2(x")/* xm) um = min m* dm(xm) un. Since the travel times d" are fixed for a given







13

x", and the program calls for minimizing the total travel time over the network with flows

independent of travel times, the solution of the above linear program, u" which is an

auxiliary flow representing the descent direction, can be solved by assigning all motorists

to the smallest travel-disutility path connecting their origin and destination.

After the descent direction is decided, the optimal move size between x" and u" can

be found by solving the following linear program. is between 0.0 and 1.0 and is chosen

so that the link volumes are as close as possible to the user-optimized equilibrium loadings.

The bisection method can be used to solve the linear program.

min z[x" + (u x")] (20)

subject to o *1 (21)

The initial solution can be determined by applying an all-or-nothing network loading

procedure to an empty network. The remaining algorithmic steps are to find the descent

direction and the optimal move size iteratively until the stopping criterion for solving the UE

program is met.

The assumption in the UE conditions that each motorist has full information about

link performance relationships may be extended to the dynamic case. However, with the

Advance Traveler Information System, motorists could be provided with travel information

for pre-trip planning (i.e., travel mode, departure time, and route) and guidance for en-route

diversion. Therefore, with the assumption that ATIS could continuously provide all travelers

with full information about all link disutilities, the UE conditions may be adoptable in the

dynamic situation. Moreover, enhancing the model structures and solution algorithms to

the dynamic case is feasible by adding discrete time periods as an additional dimension.









2.3.2 System Optimization

The system-optimization (SO) program minimizing the total travel time spent in the

network while satisfying the flow conservation constraints is expressed as follows [19]:

min z(x) = xm dm(xm) (22)

subject to k frsk =s "-, s (23)

fk 0 *k4, r, s (24)

with the incidence relationship

Xm =* r0 k fskk s m (25)

The general first-order conditions for the minimization of the SO program can be

derived using a similar method as for UE and expressed as follows [3]:

fkrs (Ukrs- Yr) = 0 k, r, s (26)

U1krs yrs 0 *k, r, s (27)

k fnk = qrs "r, s (28)

fk "* 0 *k, r, s (29)

where ukrs = sm,k [dm(xm) + Xm ddm(xm)/dxm] is the marginal total travel time on path k

connecting O-D pair r-s.

The marginal total travel times on all of the used paths connecting a given O-D pair

are equal for the SO program at its optimal value which minimizes the total travel time for

the network, not for the users. It does not represent an equilibria situation and is not stable.

2.3.3 Stochastic User Equilibrium

Stochastic user equilibrium (SUE) models relax the assumption that motorists have

full information about link travel times on all links in a network by assuming that each

motorist may perceive a different travel time (disutility) but will still choose a path with the







15

least perceived disutility path from his or her origin to his or her destination. At SUE, no

motorist can improve his or her perceived disutility (travel time) by unilaterally changing

routes.

A minimization program was developed by Sheffi and Powell [20]. The solution of

the program is the desired set of SUE flows. The unconstrained program is shown as

follows:



min z(x) = -y q, E[min Uk lurS(x) ] + E Xmdm(xm) xm dm()d (30)
x rs k m m 0



where Ukrs represents the perceived travel disutility on route k between origin r and

destination s. Ukrs is a random variable with the mean equal to the actual travel disutility uk"

which is measured at a given flow level x.

The first term of the objective function includes the expected perceived travel

disutility function from origin r to destination s, E[min Uks u"r(x)]. The partial derivative

of this function with respect to Ukr is the probability of choosing path k between r and s, Pkrs,

which is the probability that Ukr is less than the disutility of any other route, Ui~, between

r and s.

The various models for the probability of selecting each alternative route differ from

each other in the assumed distributions of the variance of Ukr, k. If a k are identically and

independently distributed Gumbel variables, Pkr, can then be expressed as a logit model:


-"k
Pkrs e
e -pu (31)
Ce-







16

The network loading approach (in which the link disutility function is not flow dependent)

associated with this multinomial logit choice model, known as the STOCH method, includes

a forward step to calculate the "link weights" according to the probabilities of selecting each

reasonable alternative route (which includes only links that take the traveler further away

from the origin and closer to the destination) and a backward step to assign the flows at

destinations back to the origins [21].

If ** is normally distributed, the joint density function of "* is the multivariate

normal (MVN) function with mean vector 0 and variance matrix V". The distribution of U"

can then be modeled as multivariate normal, too: Urs" MVN (urs(x), Vs). However, Pkrs

cannot be expressed analytically since the cumulative normal distribution function cannot

be evaluated in closed form. One of the approaches to compute Pkrs is based on a Monte

Carlo simulation procedure which has no restriction on the number of alternative routes

unlike other analytical approximation methods. A network loading algorithm based on the

Monte Carlo procedure was tested by Sheffi and Powell [22] and summarized by Sheffi [3]

as:

Step 0: Initialization. Set n = 1 where n is the number of iterations.

Step 1: Sampling. Sample Dnmfrom Dm N(dm, dm) for each link m. (* is variance of the
perceived travel time over a road segment of unit travel time.)

Step 2: All-or-nothing assignment. Based on D" assign q to the shortest path connecting
each O-D pair r-s. This step yields the set of link flows x".

Step 3: Flow averaging.
Let xnm = [(n-1)xn-1m + xn]/n, 0 *m (32)

Step 4: Stopping test.
(a) Let










S 1 -[xi m]2 V m (33)
M 1 n(n- 1) i =1


(b) If maxm,{ fm/Xnm} stop. The solution is x". Otherwise, set n = n + 1 and go
to step 1.

To prove the equivalency between the above minimization program (Eq. (30)) and

the SUE condition, the first-order conditions of this program have to coincide with the SUE

conditions, which can be characterized as follows [3]:

fkrs = rs Pkrs *k, r, s (34)

The partial derivative of z (x) with respect to a typical path flow, fkrs, can be written as



dz[x(f)] (-_rsp + frrs ddm(xm) rs
^r k6 (35)
ars m dx m,k (35)


Since the first-order condition for unconstrained minimizations requires only that the

gradient vector of the objective function be equal to 0, the above derivatives have to equal

zero for all k, r, and s, meaning that

fkrs = rs Pks k, r, s (36)

which is the SUE condition. Moreover, the flow conservation constraints ( fk = q,rs) are

automatically satisfied at equilibrium since PPk" =1. The uniqueness conditions of z(x)

were also demonstrated by Sheffi [3] by showing that the Hessian matrix of z(x) is positive

definite.

The core of any descent method for solving the unconstrained minimization model

with a nonlinear convex function of several variables is to find the descent direction and

minimize the obj ective function along that direction at each iteration. However, the iterative







18

process to find the descent direction and move size of Eq. 30 is difficult because the

direction vector computed at a particular iteration could be random in some cases.

An algorithm known as the method of successive averages (MSA) which is based on

a predetermined move size along the descent direction was proposed to solve Eq. 30 and

proved to converge to the minimum by Sheffi and Powell [22]. The algorithm is

summarized in Sheffi [22]:

Step 0: Initialization. Perform a stochastic network loading based on a set of initial travel
disutilities do. This generates a set of link flows x1. Set n = 1.

Step 1: Update. Set dn = d(xn), *n.

Step 2: Direction finding. Perform a stochastic network loading procedure based on the
current set of link travel disutilities, dn. This yields an auxiliary link flow pattern u".

Step 3: Move. Find the new flow pattern by using the predetermined move size 1/n. x" =
Xn + (1/n) (u" xn)

Step 4: Convergence criterion.

Further research is expected to enhance the complicated model structures of SUE to

the dynamic case. However, the MSA method using the predetermined move size to

guarantee convergence can be applied in the solution algorithm of the DTA model developed

in this study.

2.4 Combining a Static Traffic Signal Timing Plan and Traffic Assignment

Gartner and Improta [23] derived the following compound mathematical optimization

formulation to describe the static traffic management system:

min ZIM, argminZ2F) (37)









subject to flow conservation constraints:


hP = r(ik) with (i,k) I (38)
P E P(ik)


and non-negativity constraints:

hp ; r(i,k) (39)

where

M = the set of management variables under the control of the traffic manager

I = the set of all O-D pairs (i,k)

P(i, k) = the set of all simple paths between O-D pair (i, k)

F = the set of all link flows

fj = the flow on link j

r(i,k) = the rate of trip interchange (demand in vehicles) between origin node i and
destination node k, with (i, k) *I

h, = the flow on path p, with p *P(i, k)

Z1 =* fj wj (fj, M)

Z2 = j cj(x) dx

cj (f) = the average user-perceived travel cost function on link j

wj (f, M)= a more general performance function reflecting the multiplicity of
objectives pursued by the traffic manager in the public's interest

The formulation consists of a primary optimization program (the system-

optimization), min Z, (), and a secondary optimization program (the user-optimization),

argmin Z2 (). The "argmin" of a mathematical program is the optimal solution of the

program. Examples of computational procedures for solving the whole problem were also

given.







20

The above framework can be extended to the dynamic case by changing the O-D

demand and control actions to being time-dependent. However, there is no mathematical

methodology that can solve the compound program simultaneously. If the system dynamics

are slow and the process is relatively stable and predictable, the steady state models may be

applied in consecutive time steps. If the dynamics of the system are more stochastic and thus

less predictable, the steady-state approach is not applicable any more.

2.5 Real-Time Traffic-Responsive Control System

The application of advanced technologies in ITS makes it possible for on-line traffic

control systems to respond to real-time traffic information. Three published research studies

about real time traffic control systems, which take dynamic traffic demand into account, are

optimization policies for adaptive control, non-linear program of dynamic traffic-responsive

signal control system, and variational inequality model for dynamic traffic control. These

are discussed in the following sections.

2.5.1 Optimization Policies for Adaptive Control

Gartner [24] used a rolling horizon approach, which is used by operations research

analysts in production-inventory control, to develop a demand-responsive strategy for traffic

signal control. The basic steps in the process are as follows:

Step 0: Determine the stage length, which is the project horizon consisting ofk intervals, and
the roll period r, which specifies the traffic condition is updated; the process is
repeated every r intervals.

Step 1: Obtain flow data for the first r intervals from detectors and calculate flow data for the
next k-r intervals from a model.

Step2: For a given switching sequence, the total delay on each approach was formulated
as:
d(tl, t2, t3)= ik (Qo+ A- Di) (40)
where Qo = initial queue
A, = arrivals during interval i







21

Di = departures during interval i
t1, t2, t3 = possible switching times during this stage
Calculate the optimal switching policy for the entire stage by an optimal sequential
constrained search method in which the total delay is evaluated sequentially for all
feasible switching sequences.

Step 3: Implement the switching policy for the roll period only.

Step 4: Shift the projection horizon by r units to obtain a new stage; repeat steps 1-4.

The rolling horizon strategy provides an effective method for solving the dynamic

traffic control problem in real time. However, one argument about the "rolling horizon"

technology in traffic control is that this method minimizes the total delay at each roll period

r using the predicted and thus unreliable information about traffic flow for the future finite

time period (the remaining horizon length) [13].

2.5.2 Non-Linear Program of Dynamic Traffic-Responsive Signal Control System

Chen and Hsueh [12] formulated a non-linear program for the dynamic traffic-

responsive signal control system using Webster's delay formula. Webster's model uses an

analytical method to replicate the delay time, which includes the time a vehicle is stopped

while waiting to pass through the intersection and the time lost during acceleration and

deceleration from/to a stop. Chen and Hsueh proposed a non-linear program as illustrated

in Eq. 41 to minimize the total system delay. First, the approach delay is obtained by

multiplying the traffic flow on an approach by the average delay per vehicle. The total

system delay is then calculated by summing up the approach delay of each intersection in

the network for every time interval. The proposed model is as follows:



mi = 9Z mt CI(t)[1-gIm(t)/CI(t)]2 [Hm(t)]2
m t Z=va m 1 ()- 1 (t) + 2v t)[ (41)
9 t a m 10 2(1-vam(t)/Sm) 2vm(t)[l-Hm(t)]







22

subject to Ham(t) = [va (t) / Sam] [Ci(t) / gim] t, a, m, t (42)

Ham(t) < 1 &, m, t (43)

g1m(t) min g1m m, t (44)

[gim(t) + 1im] = C(t) t, t (45)

where

a = link designation

Ci(t) = cycle length for intersection I during interval t

g1m(t) = green time associated with phase m at intersection I during interval t

Ham(t) = degree of saturation over link "a" associated with phase m during interval
t

Iim = lost time associated with phase m at intersection I

Sam = saturation flow on link "a" associated with phase m

vam(t) = exit flow from link "a" in phase m during interval t

The Frank-Wolfe method was used to solve the model since the objective function

was proved to be convex and all constraints are linear. The framework of the above non-

linear program is adopted in this research because of the intuitive interpretation of the

mathematical model, the promise of the optimization model for finding the global minimum,

and the amenability to computation with the application of computer programming.

Although the concept of the above framework can ideally describe the environment of the

real-time traffic control system, the delay function is the major key to locating the optimal

signal timing settings. The delay function used in the above research over simplified the

traffic congestion situation but not taking the spill-back queue problem into account.







23

2.5.3 Variational Inequality Model for Dynamic Traffic Control

The characteristics of the VI formula have been extensively studied in economics.

A mathematical program often may be more intuitive for representing a real situation.

However, the VI formulation is much broader than a mathematical program.

Chen and Ben-Akiva [14] defined a dynamic system optimum principle, which can

be expressed in Eq. 46 48 for dynamic traffic control, as: for each intersection, any phase

with a positive green time must have an equal and minimal marginal delay.

gim(t) (cim(t) -* t)) = 0 i, m (46)

cim(t) -* t) 0 .i, m (47)

gim(t) )0 i, m (48)

where

gim(t) = green time for phase m of intersection i at time t

cim(t) = marginal cost for phase m of intersection i at time t

~t) = minimal marginal phase delay for intersection i at time t

The equivalent formulation for Eqs. 46-48 was then stated as follows:



T Ci(t)(gim(t) gim(t)) dt > 0 Vgim(t) E (49)
0 im


where gim*(t) is the dynamic system optimal setting, cim*(t) is the marginal phase delay when

the timing is gim*(t) and is the feasible set for green time splits.

The VI formula of the real-time traffic-responsive signal control system is broader

than the mathematical program. However, no solution algorithm was developed in the above

research although the equivalent mathematical formulation for the VI model was provided.







24

2.6 Dynamic Traffic Assignment

Static traffic assignment distributes on O-D traffic flow based on the assumption that

the traffic flow on a network is static, whereas dynamic traffic assignment models can

present traffic situations on a network in infinitesimal short time periods. Therefore, static

traffic assignment is sufficiently effective to predict the traffic flow for a long-term period

but dynamic traffic assignment is powerful enough to analyze the traffic state over a specific

time such as a peak hour or shorter period.

Merchant andNemhauser [25], who proposed a discrete time, nonlinear and system-

optimal model for the case of single-destination networks, were among the first researchers

to address the dynamic traffic assignment system. The techniques available for DTA have

progressed since then. Two kinds of formulations published in existing research studies

about DTA are VI formulations and route-based algorithms. These are discussed in the

following sections.

2.6.1 Variational Inequality (VI) Formulation for DTA

Drissi-Kaitouni [4] proved three theorems about the VI formulation for DTA, which

is equivalent to the equilibrium DTA condition:

Sk(h) = utpq if hk > 0 p, q, t, k (50)

or Sk(h) e.utp,, ifhk = 0 9p, q, t, k (51)

where Sk(h) is the travel cost on a path k and utpq is the travel cost on the shortest path from

origin node p to the destination node q at period t, given traffic flow h on the network.

These three theorems are summarized as follows:

Theorem 1. Let be the set of feasible path flows. Then the above equilibrium condition,

Eq. 50, may be rewritten as a variational inequality:







25

Find h h such that ;* P* ?_ (Sk(h) tpq ) (hk*t- hk ) 0 0* h .

Theorem 2. The variational inequality in Theorem 1 is equivalent to the following variational

inequality:

Find h h such that ;* P* q* Sk(h) (hk*- hk ) 0 ee -.

Theorem 3. The variational inequality in Theorem 2 has a link variational inequality

formulation:

Find f t such that Sa(f) (fa,- fa ) @0 f*- 7

where fis the set of feasible link traffic patterns over the network.

He also pointed out that according to the symmetry principle, the above link-based VI

program is equivalent to the UE mathematical programming model in Eq. 1 if Sa(f) is equal

to Sa(fa).

Chen and Hsueh [12] proposed a nested diagonalization algorithm to solve the

following dynamic user-optimal route choice model which was formulated as a VI model.

c* [u- u*] 0, (52)

where *denotes the feasible region that is delineated by the following constraints:

Flow conservation constraints:

hp" (k) = qr (k) or, s, k (53)

Flow propagation constraints:

Uapk" (t)= s k hps (k) apkrs (t) 0* a, t (54)

uapk" (t)= Ubpkrs (t- bt)) krs (t) r, s, p, k, t, a* p, b* p, a B(j), b A(j) (55)

Nonnegativity constraints:

hp (k) 0 r, s, p, k (56)









where

A(j) = set of links whose tail node isj

B(j) = set of links whose head node isj

hp'(k) = flows from origin r departing during interval k over route p toward
destination s

q"(k) = departure flows between origin-destination rs during interval k

u,(t) = inflow entering link "a" during interval t

uapk"(t)= portion of inflows entering link "a" during interval t which departs origin
r during interval k over route p toward destination s

(t) = travel time on link "a" during interval t

pk(t) = 1, if flows departing origin r over route p during interval k entering link "a"
during interval t

= 0, otherwise

Since the Jacobian matrix of the travel time function was asymmetric, two relaxation

(or diagonalization) techniques were needed to transform the VI model to a non-linear

optimization problem. One is to estimate the link travel time and the other is to temporarily

fix the flow on each link other than on the subject time-space link. The solution algorithm

which combines diagonalization and the Frank-Wolfe method can be summarized as follows:

Step 0: Initialization.

Step 1: "First Loop" Operation. Update the estimated link travel times based on the initial
traffic flow conditions. Construct the corresponding feasible time-space network
based on the estimated link travel times.

Step 2: "Second Loop" Operation. Modify the initial feasible solution based on the time-
space network constructed by the estimated link travel time from Step 1. Fix the
flows on all links to transform the VI model to a non-linear optimization model.

Step 3: Solve the optimization program in Step 2 using the FW method.







27

Step 4: Convergence check for the "Second Loop" operation.

Step 5: Convergence check for the "First Loop" operation.

The Variational Inequality is a useful technique to formulate the DTA system.

Applying the relaxation algorithm to relax the link interactions to find the equivalent

optimization model is the key action to solve the VI model. However, Chen and Hsueh [12]

could not define a good estimate of link travel times to relax the link interactions in the DTA

model when they derived the above solution algorithm. The VI formulation and the

relaxation algorithm are applied in this research to solve the DTA system. Greater effort is

devoted to finding reasonable estimates of the link travel times.

2.6.2 Route-Based Algorithms for DTA

To solve the dynamic user-optimal route choice problem, Chen et al. [26] compared

three route-based algorithms with the link-based algorithm using the Frank-Wolfe method.

The three route-based algorithms were the desegregate simplicial decomposition ofLarsson

et al., desegregate simplicial decomposition of von Hohenbalken, and a gradient projection

method. They found that the link-based algorithm is inferior to the route-based algorithms

in terms of execution time but superior in terms of memory requirements because the results

from the route-based algorithms for the DTA system include not only the link traffic

volumes but also the information related to turning-movement volumes.

The traffic volume for each turning movement at every intersection is essential to

feed into a traffic control model. With this information, a traffic control model can

determine the optimal green time for each signal phase which allows only specific traffic

movements to move. However, CORSIM's assignment feature using a link-based algorithm

to solve the assignment problem also provides turning movement information using extra







28

memory to record the route information at each iteration. Therefore, the link-based

algorithm is adopted in this research with the application of CORSIM's assignment feature.

2.7 Combining a Dynamic Traffic Signal Timing Plan and Traffic Assignment

Most research addressing the interaction between traffic control and assignment only

considers the static traffic situation. Two approach techniques for integrating DTA with

real-time traffic-responsive control systems, which were proposed by recent studies, are

game theory and bilevel programming. These are summarized in the following sections.

2.7.1 Game Theory Approach

Game theory provides a framework for modeling a decision-making process in which

more than one player is involved and each individual's actions determine the outcome

jointly. Fisk [27] analyzed the characteristics of a Stackelberg game in which one player

knows how the other players will respond to any decision he or she may make, and the

characteristics of a Nash noncooperative game in which each player is trying to minimize

his or her performance function without prior knowledge of the other players' functions.

The user equilibrium condition can be stated identically as a Nash noncooperative game with

each traveler considered as a player. The signal optimization model can be formulated as

a Stackelberg game in which the traffic authority tries to minimize the network performance

function and motorists choose static user-optimal routes.

Chen and Ben-Akiva [14] combined the dynamic traffic control system and the

dynamic traffic assignment system as three game theories. First the combined control-

assignment system is formulated as a Cournot game, in which the players, i.e., the traffic

authority and the users, choose their strategies simultaneously. In this game, each player

makes his or her move independently without knowing the strategy of the other. Second, the







29

combined control-assignment system is formulated as a Stackelberg game in which the

authority sets the signal timing first by anticipating the traffic flow and motorists then choose

their best routes accordingly. In the end, the combined control-assignment system is

formulated as a monopoly game in which both control and assignment solutions are system-

optimal since the traffic authority controls both the signal setting and traffic assignment.

However, only frameworks, not solution algorithms, were discussed for these game theories.

2.7.2 Bilevel Programming

Bilevel programming involves two levels of mathematical programming that can be

viewed as a particular case of Stackelberg games. At the upper level, decision makers are

bound by the decisions of the lower levels and maximize their own profit accordingly, taking

into account the reactions of the lower levels.

Marcotte [28] presented the static network design problem as a bilevel programming

model. Two of the four heuristic procedures analyzed to solve the bilevel model were

iterative optimization assignment procedures with user-optimized and system-optimized

equilibrium models. The numerical experiments showed that the iterative optimization

assignment method with the user-optimal equilibrium model yielded a near-optimal solution.

Chen and Hsueh [12] combined the dynamic traffic control and the traffic assignment

systems into a bilevel model. A heuristic procedure, involving an iterative optimization and

assignment method, was used to solve the model for a near-optimal solution.

Since it does not appear possible to find a polynomial algorithm to solve the

combined model because of the non-linear programming structures in both the traffic control

and assignment models, the concept of bilevel programming and the iterative optimization

and assignment method are pursued in this research.







30

2.8 Generalized Delay Model

A generalized delay model was proposed by Fambro and Rouphail [29], and

validated by Rouphail et al. [30]. The model is a much improved version over the previous

HCM model in estimating delay at vehicle-actuated traffic signals. Moreover, it includes

a term to account for the effects of queues not being cleared during a signal cycle, a

condition referred to as having oversaturated queues during variable demand conditions.

This generalized delay model for Chapter 16 of the HCM is as follows:


d = 0.5C[1 (g/C)]2 PF + 900T[(X-1)+v(X-1)2+8kIX/Tc] + d3 (57)
ri -(R/C)min(X. 1.0)]

There are three cases for estimating d3:

(1) If no oversaturated queue exists at the start of the analysis period, d3 = 0.

(2) If an oversaturated queue exists at the start, but not at the end of the analysis period,
d3 = (3600Ni/c) [0.5Ni/Tc(1-X)].

(3) If an oversaturated queue exists at both the start and the end of the analysis period,
d3 = (3600Ni/c) -1800T[1 min(X, 1.0)].

where
C = average cycle length

g = average effective green time

X = degree of saturation for a subject lane group

PF = progression adjustment factor

T = analysis period

k = parameter for given arrival and service distributions

I = parameter for variance-to-mean ratio of arrivals from upstream signal

c = capacity of the lane group

N, = initial queue at the start of the analysis period







31

This generalized delay model for signalized intersections can provide good estimates

of intersection delays in a network for allocating appropriate green times at intersections

because it takes the effects of oversaturated queues into account.

2.9 An Evaluation Tool: CORSIM 5.1

There are several simulation software packages that are currently utilized by

transportation professionals to evaluate different traffic scenarios. For example, the

TRansportation ANalysis SIMulation System (TRANSIMS) is designed to simulate the

detailed interaction between individuals' activity plans and congestion on the transportation

system [31]. The program is capable of simulating the movements of individuals across the

network, including mode selection, on a second-by-second basis. However, intensive data

collection and extreme computer execution time are currently obstacles to large-scaled

applications of TRANSIMS. CORSIM 5.1 [32], on the other hand, is a comprehensive

microscopic traffic simulation computer program using commonly accepted vehicle and

driver behavior models. Therefore, it is usually used as an evaluation tool for traffic signal

timings although it also provides a user-equilibrium platform to perform a traffic assignment.

CORSIM's traffic assignment feature as well as its evaluation function are adopted in this

research although there are not many reports evaluating CORSIM's effectiveness for traffic

assignment. It is applied because the routing logic in CORSIM 5.1 will convert the O-D

tables into turning percentages for each intersection, which can then be fed into the traffic

control model. In each iteration of CORSIM's assignment procedure, an intermediate

solution is obtained using link travel times produced by the previous iteration (direction

finding). An iterative line search is then applied to the range between the current

intermediate solution and the previous iteration solution to obtain an optimal solution for







32

each iteration (optimal moving size). The assignment process will terminate when the

change in the objective function between two successive iterations is less than a threshold

value (convergence test). The Bureau of Public Roads (BPR) and the modified Davidson

link impedance functions are available to evaluate the travel time on a link. The BPR

formula is as follows:

T = To [1 + a(V/C)b] (58)

The modified Davidson impedance function is as follows:

T = To [1 + aV/(S-V)] ifV *bS (59)

or T = To [1 + ab/(1-b)] + aT0 (V-bS)/[S(1-b)2] ifV > bS (60)

where

T = travel time on link

To = free-flow travel time on link

a, b = parameters to be estimated for each class of roadway

V = volume on link

C = capacity on link

S = saturation rate on link

The BPR formula increases travel times even after a link's traffic flow is higher than

its capacity. The modified Davidson function, using a linear extension at a volume close to

capacity, has been shown to reduce the error in traffic assignment results relative to actual

volume counts. However, there is very limited estimation experience reported in the

literature for the Davidson function. Instead, the BPR formula is frequently adopted in

practice and the model parameters which appeared in the original publication are usually

employed.







33

To estimate the capacity used in the impedance functions, the discharge rates for

turns are held constant, and are estimated initially for free-flow conditions. These estimates

could be calibrated after the assignment of turn movements, then applied to the next

assignment process, if requested. The following equation is used for capacity calibration:

C. = [rC,+ (100-r)Cp]

where

Cn = new estimate of capacity (for the next assignment iteration)

r = capacity smoothing (in a percentage)

Cc = calculated capacity using previously assigned volumes

C, = previous estimate of capacity















CHAPTER 3
OPTIMIZATION MODEL FOR DYNAMIC TRAFFIC CONTROL

3.1 Introduction

This chapter presents the techniques applied for developing a mathematical model

and the solution algorithm for optimizing dynamic traffic controls. In the mathematical

model, the total intersection delay in a network is minimized by allocating optimal time

splits of each traffic signal. The HCM generalized delay model is used to estimate the

average delay for vehicles arriving during a discrete time interval. However, the objective

function of the optimization model Eq. 61, cannot be proven to be convex. An approach

which can deal with this situation is presented. The last section in this chapter presents a

solution algorithm for solving the optimization model.

3.2 Optimization Model

As described in Section 2.8, three cases corresponding to the oversaturation

situations presented in the HCM delay model are formulated for the dynamic traffic control

system by adding time as an additional dimension:

1. No oversaturated queue exists: the degree of the saturation at the previous time
period, XPm(t-1), is less than one.

2. An oversaturated queue exists at the start, but not at the end of the analysis period:
the degree of the saturation at previous time period, XPm(t-1), is greater than or equal
to one but the degree of the saturation at current time period, XPm(t), is less than one.

3. An oversaturated queue exists at both the start and the end of the analysis period:
both of the degrees of the saturation, at previous time period, XPm(t-1), and at current
time period, XPm(t), are greater than or equal to one.

34










The model for the dynamic traffic control system is expressed as follows:


vP(t) dP(t)
vM/"mL


(61)


I p meB(I)


X (t) = v mP(t) CI(t)
Xs (t) -'
Sm giP(t)


If Xm(t-1)


VI, m, p, t


1 and XPm(t) < 1, then

0.5C,(t)[ gI'(t)/C,(t)]2
df(t) PF
v P(t)
1-
Sm


+ 900T[(XP(t)- l) + (Xm(t)


If Xm(t-1) *1 and XPm(t) < 1, then

S 0.5C (t)[1-giP(t)/C(t)]2
df(t) = PF
v (t)
1 -
Sm


2+ p() SmgIP(t)
1)2+8kRXm (t)/(T sm--(t)) ]
Ci(t)


(63)


+ 900T[(XP(t)- 1) + (XP(t)- 1)2 +kRXP(t)/(T S ]
S Ci(t)


+ [3600(v(t-1)-sm g(t1) Sm gP(t)
Ci(t- 1) Ci(t)



g (t-1) sm g1p(t)
[0.5(vp(t- )-sm g 1)/T( (tt))]
Cl(t- 1) Ci(t)


min Z(g)=E
t


(62)


(64)


I









If XPm(t-1) *1 and XPm(t) *1, then

0.5C,(t)[l-g (t)/C,(t)]2 SmIPt )
dmP(t) = -(t)C(t) + 900T[(XP(t)-1)+ (XP(t)- 1)2 + 8kRXP(t)/(T smgIt)
Sgi (t) CI(t)
Ci(t)

[3600(v 1 1(t-1)-sg SP( m g)(() (65)

Mc- _i- 1) (65)
CL(t- 1) C(t)



subject to

gi (t)** gP **I, p, t (66)

[gip(t) +lp] = CI (t) I,t (67)

where

m = link index

B(I) = set of links entering intersection I

I = intersection index

p = phase index

VmP(t) = vehicle flow on link m during phase p at time t

dmP(t) = vehicle delay on link m during phase p at time t

sm = saturation flow on link m

gP(t) = green time associated with phase p at intersection I during interval t

Ci(t) = cycle length for intersection I at interval t

Xmn(t) = degree of saturation associated with phase p at intersection I during interval
t

gP = minimum green time associated with phase p at intersection I

1 I = lost time associated with phase p at intersection I







37

PF = progression adjustment factor

T = length of time interval

k = parameter for given arrival and service distributions

R = parameter for variance-to-mean ratio of arrivals from upstream signal

The objective function minimizes the total intersection delay, which is represented

by the sum of the products of the traffic flow and the average vehicle delay for each signal

phase at each intersection during each time period. The average delay model, dmP(t), is

essentially the HCM generalized delay model which has either two or three terms depending

on the degree of saturation during the previous time period. There are two constraints. The

first one restricts the green time assigned to each phase to be not less than the minimum

green time which is usually determined based on the green time needed for pedestrians to

cross streets safely. The second constraint restricts the sum of green times and lost times for

all signal phases at each intersection to be equal to the cycle length.

3.3 Convexity

The above optimization model is a nonlinear programming model with linear

constraints for conserving cycle length and minimum green time. Since the constraints are

all linear, this nonlinear programming model can be solved by the Frank-Wolfe method. The

Frank-Wolfe method is a feasible direction method and its direction finding procedure is a

descent method, meaning that the obj ective function value decreases at every iteration. This

method converges to a local minimum, which would naturally be a global minimum for

convex objective functions.

The objective function of the above optimization model is the sum of nonlinear

functions associated with vehicle delay at each intersection. Without considering the







38

incremental delay due to oversaturated queues, the objective function, Z(g), can be proven

to be a convex function by showing that the matrix of the second derivatives of Z(g) with

respect to g (the Hessian) is positive definite.

Assuming there are I intersections in the analysis network and each intersection has

p phases, then go- [gl\ (1), g\l (2),..., g\ (t), g21 (1),..., g P (t)]. Let n = I x p x t, and go-

[g1, g2,..., gJ]. The Hessian is calculated by using a representative term of the matrix. The

derivative of Z(g) is therefore taken with respect to green time on the mth and nth elements

in g*so


2Z~g)
a2Z(g)2Z(g) for m = n
Wn= a2g (68)
n 0 otherwise


This means that all of the off-diagonal elements of the Hessian, Z(g), are zero and all of

the diagonal elements are given by Z(g)/ gn. In other words,



a2Z(g) 0 0
a2g

0 a2Z(g) 0
V2Z(g) = a282 (69)
0 0
a2Z(g)
a2gn


If Z(g)/ *'gi *0, this matrix is positive definite because it is then a diagonal matrix with

positive entries. However, for the oversaturated queue situation, the third term in the

generalized delay model, which estimates delay due to oversaturated queues for time interval







39

t, is dependent on the green time at time interval t-1 because the initial queue at the start of

interval t is a variable of the green time at interval t-1. Therefore,


a2Z(g) for m = n

02Z(g) a2gn
= < 82Z(g) (70)
agmgn 2Z(g) for m = n-1 m = n+ (70)
agmagn
0 otherwise

This means that not all of the off-diagonal elements of the Hessian are zero. In other words,


2Zg) (g) 0Z() O
a2g1 ag1ag2

a2Z(g) a2Z(g) a2Z(g)
8g2 g1 02g2 2g2g3
V2Z(g) = (71)
o a2Z(g)
ag3ag2

a2Z(g)
a2g.

Even if Z(g)/ gi gj* 0, this matrix cannot be proven to be positive definite because it is

not a diagonal matrix.

Although the objective function of the above model cannot be proven to be a convex

function, there are some methods which can find a near-optimal local minimum. One of

these methods is the multistart approach which chooses several starting points and then

compares the local minimums determined from those starting points. Since a good starting

point is important for finding the global optimum, the static network-wide optimal signal

settings, which can be obtained from TRANSYT-7F, are used as the initial signal timings

for a near-optimal minimum.







40

3.4 Solution Algorithm

A procedure using the Frank-Wolfe method is used to solve the real-time traffic

control model. To initialize, the link-based traffic volume on each link, which is obtained

from the traffic assignment model, is put into TRANSYT-7F to obtain the static network-

wide optimal signal settings.

Since the Frank-Wolfe method is a feasible direction method and its direction finding

procedure is a descent method, the descent direction at each iteration is then found by

assigning a green time equal to the preset minimum green time for each phase (feasible

direction) and then assigning the remaining green time to the phase with the highest vehicle

delay (descent direction). Moreover, with the linear constraints for conserving cycle length

and minimum green time (Eq. 66 and Eq. 67), the preset cycle length has to be greater than

the sum of minimum green times and lost times for all signal phases at each intersection.

An auxiliary green time setting, g* is obtained to determine the descent direction.

After the descent direction is decided, the optimal move size between the current

solution g and the auxiliary solution g"can be found by solving the following linear

program:

min z[gn"+ 9tg_ gn)] (72)

subject to 0* *1 (73)

Solving the above linear program is equivalent to finding the value of* *that satisfies

dZ( ) Id* *- 0. The objective function is differentiable with respect to However, the

derivative is complicated since the average delay model is quite complex itself. The

derivative of the objective function is given as follows:









dZ[a]
da


P(t) ddm(t)
v(t) da
da,


(74)


t -EE
t I p meB(I)


If XPm(t-1) < 1 and XPm(t) < 1, then



ddmP(t) 0.5CI(t)PF 1 (gP(t) +aI(g(t)-gP(t)) gp(t)-g(t)
da VP- t) Ci(t) Ci(t)
1-
Sm


vm(t)C (t) glp(t)-g I'(t) 1
+ 900T( m ((X (t)-
Sm (gi'(t) + a(g /P(t)- g'(t)))2 2


8KRXP(t)


Sm(g(t) + a (g (t)-gP(t)))
CI(t)


glP(t)- g /(t)


Sm (gIP(t) + a (g (t) g(t)))


+8KR vmO(t)C(t)(gp(t)-g/~(t))
T m(gP(t) + a (g '- gP(t)))2


Ci(t)
Sm(glP(t) + a (g i- gP(t)))


sm(gl(t)-g / (t))(75)
Ct ))) (75)
Ci(t)


-)-1/2


Ci(t)2XP(t)
Sm2(gI(t)+ a(g I gIP(t)))2


vmP(t)CI(t)








If XPm(t-1) *1 and XPm(t) < 1, then


ddg(t) 0 .5CI(t)PF 1 (glP(t) + a(g 'P(t)-gP(t)) glp(t)-g'p(t)
da vP(t) Ci(t) Ci(t)
1-
Sm


vm(t)CI(t) g g(t) g (t) 1 P(t) 1)2
Sm (gi (t)+a(g '(t)-giP(t)))2 2


v P(t)Ci(t)
* (2(Xm(t)- 1)


8KRXm(t)


s m(gI(t)+ a(g '(t)-g (t)))
CI(t)


glP(t) g (t)


Sm (glP(t)+ a(g I(t)- giP(t)))2


+8KR vm(t)CI(t)(g1P(t)-g ())
T s(gP(t) + a (g -gl(t)))2


Ci(t)2X P(t)
2(gP(t) + a(g/P-g P(t)))2


+ 3600(


Ci(t)
Sm(giP(t) + a(g giP(t)))


sm(giP(t)- g (t))
C,(t)


Ci(t)
C,(t- 1)


+ 900T(


-)-1/2


vP(t- 1)Ci(t)


g,(t- l)+a(g /P(t- 1) gP(t- 1))
g,(t) + a (g /O(t) -gP(t))


I-


sm(gi(t) + a (g/(t)- gP(t)))









v (t)C1(t)


sm(gIP(t) + a (g /(t) giP(t)))



Ci(t) gP(t -1)- g(t -1) (g
Ci(t- 1) giP(t) + a(g '(t) gP(t))


v(P(t- 1)C g(t)( -g (T))
sm(gIP(t) + a (g '(t)- gIP(t)))2


+ 1800(


sm(gi(t) + a (g 'P(t) g,(t)))


Ci(t)
Ci(t- 1)


g,(t- 1) + a (g '/(t- 1) gP(t- 1))2
g,(t) + a(g P(t)- gip(t))


g /P(t) gi(t)


T(1 -


vP(t)Ci(t)
sm(giP(t) + a (g '[(t)- gIP(t)))


Sm (gP(t) + a (g '/(t)- gP(t)))2


If XPm(t-1) 1 and XPm(t) 1, then


0.5CI(t)PF 2 (1


g1P(t) + a (gi(t) g 'P(t)) gip(t) g t)
Ci(t) Ci(t)


SgiP(t) + a (g (t) gip(t))
C1(t)
C,(t)


v (t)c1(t)


(76)


dd P(t)
da


'(t)- gP(t))(giP(t- 1)+ a(g P(t- 1)- gi(t 1)))
(gIP(t) + a (g '/(t) giP(t)))2


v P(t- 1)C,(t)










0.5CI(t)PF(1 g'P(t) + a (glP(t) g /' (t))
+ O.C(tCPF(1(t)
C,(t)


vm(t)C(t) gp(t)-g'i(t) (t) g 1 -1)2
+ 900T( +-((X (t)-1)2
Sm (g'P(t)+ (g 'P gP(-g,(t)))2 2
I I 2


8KRXP(t)


sm(gIP(t)+ a(g '(t)-gi(t)))
Ci(t)


SvP(t)C(t)
*(2(X Z(t)- 1)


glP(t)- g i(t)


Sm (gP(t) + a (g I(t) gP(t)))2


8KR Vm(t)C,(t)(gIP(t)-g /(t)) C(t) CI(t)2X(t) sgI(t)-g i(t))
T sm(gP(t) + a(g '/ gIP(t)))2 Sm(gP(t) + a(g/P gIP(t))) sm(giP(t) + a(g gP(t)))2 C,(t)


v+(t- 1)C,(t- 1)(g1P(t)-g '(t))
+ 3600[a (
sm(gI(t)+ a(g/'(t) -gP(t)))2


Ci(t) gl(t-1)-g/i(t-1)
SCi(t-1) glP(t)+a(g/ l(t)-g P(t))


(giP(t- 1) + a(g 'l(t- 1)- gi(t- 1)))(gP(t) g (t))
(gP(t) + a (g '(t)- gP(t)))2


(-1)


1gi(t) gP(t)
CI(t)


g(1 (t) + a(gi(t)-g /(t))2
Ci(t)


)-1/2


(77)







45

The bisection method can be used to find an approximate value of *which sets the

above function equal to zero. The optimal solutions are found when the above algorithm

runs iteratively until the stopping criterion is met.

There are several stopping criteria that can be applied while implementing the Frank-

Wolfe method. For instance, the convergence criterion can be based on the marginal

contribution of successive iterations. Alternatively, the algorithm can be terminated if the

elements of the gradient vector are close to zero. In some cases, criteria that are based on

the change in the variables between successive iterations are used.

Because calculating the relative change of the objective function involves more

computation and longer computer running time, the convergence criterion based on the

change in the variables between successive iterations is used in this research, that is,

maxi { gin- gi"-li} --

The notation maxi {.} stands for the maximum, over all possible values of i, of the arguments

in the braces. In other words, the iterations terminate if the maximum difference between

the green times of each phase at the previous and current iterations for all intersections

during all time periods is less than or equal to **

The procedure for solving the real time traffic control model is summarized as

follows:

Step 0: Obtain initial solutions from TRANSYT-7F.

Step 1: Find the descent direction.
For every intersection and each time period, assign the green time g** min g to each
phase. Assign the remaining green time to the phase with the highest vehicle delay.

Step 2: Determine the optimal move size.
min Z (g +* t(g, g)) (78)
subject to 0 *1 (79)







46

Use the bisection method of iterative interval reduction to find the value of which
can satisfy dZ(* ) Id* 0

Step 3: Update the green times.
g = g +* (g, g) (80)

Step 4: Test convergence.
If max {|g" g" n} ** stop. Otherwise, go to Step 1.

A signal phase plan comprises a complete specification of phasing sequence, splits

(green time for each phase) and the length of amber intervals. The choice of phasing

sequence depends primarily on the treatment of left turn traffic. The length of a cycle and

its splits are determined by a number of factors, the main ones are traffic demand patterns

and delay to traffic imposed by signals.

While a series of intersections along an arterial streets are treated as a single system

and their timing plans are developed together to provide good vehicle progression along the

arterial, the reasonable 2-way progression between traffic signals generally requires the same

cycle lengths and similar phase patterns at all signals in a network. Cycle offsets for

progression between signals must usually be in half-cycles. Although the cycle length and

signal phase plan can be varied at each intersection, they have to be specified before

initialization and are not modified during the optimization. The same restriction is also

applied to the number of time periods and the duration of each time period. The

methodology developed in the study may allow constructing additional external looping to

optimize cycle length and signal phase sequence. However, such optimization is out of the

scope of this study and its performance has not been investigated. Figure 3-1 shows the flow

chart of the above solution algorithm.

In Chapter 5, the optimization program for dynamic traffic-responsive signal control

is combined with the DTA model described in Chapter 4. A Visual Basic program







47

developed to implement this flow chart to determine the optimal signal settings, which are

then imported into the DTA model, is presented in Chapter 6.












Read .TRF file


____ v__4_


- Find descent direction


Find optimal move size


M__ \_
Update g




No Yes
Test convergence Stop


Figure 3-1 Flow Chart of the Solution Algorithm for the Real-Time Traffic-Responsive
Signal Control Model.


Get initial g, min g,
cycle length,
saturation flow rate,
and traffic volumes


For every I and t, assign the
auxiliary green time g' to each
phase g' = min g

Assign the remaining green
time to the phase with the
highest value of v d


0 a <1 min z (g + a(g'-g))

Using the bisection method to
find the approximated value of
a satisfying dz/d a = 0















CHAPTER 4
DYNAMIC USER-OPTIMAL ROUTE CHOICE MODEL

4.1 Introduction

This chapter presents the techniques applied for developing a variational inequality

formulation for a dynamic user-optimal route choice model. A variational inequality is a

broader formulation for equilibrium problems compared to a mathematical program. The

static formulation and the equivalent optimization program are presented first. The

formulation is then extended for the dynamic case.

Subsequently, the relaxation algorithm for relaxing the inseparable link cost

functions caused by time-dependent link flow is discussed. A good travel time estimate is

introduced to relax the link interaction. The equivalent optimization program is presented

after the link interaction is relaxed. The last section presents the solution algorithm

developed to solve the equivalent optimization program based on the improvement in the

travel time estimation.

4.2 Variational Inequality Models

A variational inequality model is a general model formulation that encompasses a set

of mathematical models, including nonlinear equations, optimization models and fixed point

models [33]. Variational inequalities were originally developed as a tool for the study of

certain classes of partial differential equations such as those that arise in mechanics.

However, with its capability of formulating and analyzing more general models for







50

equilibrium conditions than the constrained optimization approach, the basic concept of VI

theory has received increasing attention from transportation network modelers during the last

decade [5]. The definition and proposition presented in the following section focus on

variational inequality models suitable for the analysis of transportation network equilibrium

models.

4.2.1 Static Transportation Network Equilibrium Model

Let x = (x1, x, ..., Xm) be a vector of link traffic volumes and d[x] = [dl(x), d2(x),..

, dm(x)] be a vector of disutility functions. The equivalent variational inequality formulation

of a discrete link-based user-optimal route choice model can be summarized as follows from

existing literature [4]: the variational inequality model is to determine a control vector x* *

G such that

d [x*] [x -x*] 0 x G (81)

where G denotes the feasible region that is delineated by the following three constraints:

1. flow conservation constraints: k fsk = qrs "r, s (82)

2. non negativity constraints: fsk 0 *k, r, s (83)

3. flow propagation constraints: xm = r k frsk sm,k m (84)

According to the Symmetry Principle [34], if the Jacobian matrix d(x) is symmetric,

then the variational inequality above has an equivalent optimization model. In this particular

case, where the link travel disutility functions are separable, the above VI formulation is

equivalent to the following mathematical programming model:


min z(x) = Jxm dm(o)do (85)
m JO







51

This program is exactly the same optimization program used to represent the UE

condition discussed in Section 2.3. Therefore, the solution algorithm presented in Section

2.3 can be used to solve it. The solution algorithm with the application of the Frank-Wolfe

method is summarized in the following:

Step 0: Find initial volumes.

Step 1: Update disutility estimates.

Step 2: Find descent direction. Perform an all-or-nothing traffic assignment.

Step 3: Perform line search for optimal move size.

Step 4: Calculate new solution volumes.

Step 5: Test convergence.

As discussed in Section 2.3, at some point in step 2 involving the procedure for

finding the descent direction, the solution algorithm will call for minimizing the total travel

time over a network without flow-dependent travel times on each link. The total travel time

spent in the network will be minimized by assigning all motorists to the shortest travel time

paths connecting their origins to their destinations. Such an assignment is performed by the

all-or-nothing network loading procedure. The core of the all-or-nothing procedure is the

determination of the shortest paths between all origins and destinations. An efficient method

to find these paths between all network nodes can be easily found in existing literature [16].

However, since the traffic assignment feature built into CORSIM 5.1 is used to implement

the above algorithm, the methods to locate the shortest path are not introduced here.

The procedure of step 3, performing a line search for optimal move size between the

current solution and the auxiliary solution from step 2, involves the bisection method to find







52

the value of *which can set the derivative of the objective function with respect to equal

to zero. This procedure is also discussed in Section 2.3.

4.2.2 Dynamic Transportation Network Equilibrium Models

Unlike the static network system, the dynamic network system concerns a vector of

control variables x(t) = [x,(t), x,(t), ... Xm(t)]. The equivalent variational inequality

formulation was defined as follows in existing literature [5]: the variational inequality model

for the dynamic transportation network equilibrium model is to determine a control vector

x* (t) G* G (t) such that

d [x* (t)] *x (t)- x* (t)] 0 ( (t) G (t) (86)

where G (t) denotes the feasible region that is delineated by the three sets of constraints,

which are the same as for the static condition, and another set of flow propagation

constraints:

in-flow x"rs (t) = exit-flow x"rs (t + travel time on link m) (87)

The link interactions caused by the above flow propagation constraints result in an

asymmetrical Jacobian matrix of the dynamic travel disutility functions such that the

Symmetry Principle cannot be applied to the reformulated VI format as an optimization

program. However, there is an iterative method (relaxation algorithm) that can solve the

problem. Assuming that there exists a vector of smooth auxiliary functions g (x, y) with

suitable properties, Ran and Boyce [5] proved that at iteration n, solving the following

variational inequality model:

g(n), x(n-1)) t(x x(n)) 0 *x G (88)

is equivalent to solving the corresponding mathematical programming model:

min Z (x, x(n-1)) *x G (89)







53

Chen and Hsueh [12] applied the above relaxation algorithm and suggested relaxing

the link interactions by calculating the link travel times based on the traffic volumes from

the assignment result at the previous iteration. This procedure resulted in a model with a

positive-valued diagonal Jacobian of d[x (t)]. The following mathematical programming

model is then equivalent to the relaxed variational inequality model:


M (n- (n-1l)(- 1) \n 1
min z = (t) dm(xm 1(1), x -1(2), ... x (t- 1), )do (90)
tm O



A nested diagonalization algorithm applying the relaxation (diagonalization) method

is then:

Step 0: Find an initial feasible solution.

Step 1: Estimate link travel times Tm (t).

Step 2: Solve the optimization model (Eq. (90)) based on Tm(t).

Step 2.1: Update the xm(t) based on Tm(t).

Step 2.2: Solve the optimization model (Eq. (90)) by the FW method.

Step 2.3: Test convergence.

Step 3: If dm(t)* 'Tm(t), stop. Otherwise, set Tm(t) n+1 = Tm(t) n+ (1/n) (dm(t) n Tm(t) n), and
go to step 2.

The optimization program in Eq. 90 has many similarities with the one representing

the static case. Therefore, step 2 is the typical procedure for solving the static route choice

model with asymmetric travel time functions which are dependent on the traffic flows from

the previous iteration. In step 2.1, the initial solution is updated every time. In other words,

the traffic patterns resulting from the previous iteration in step 2.2 cannot be carried over

into the next iteration because different link travel times construct different subproblem







54

feasible regions. In step 3, the estimated link travel times are updated using the method of

successive averages to predetermine the move size of 1/n.

The estimated link travel times were defined using the following setting: Tm(t) =

NINT [dm(t)]. The notation NINT indicates the round-off arithmetic operation: set Tm(t) =

i, if i 0.5 *Tm(t) *i + 0.5. Although finding the link travel time is a critical issue for

solving the above algorithm, Chen and Hsueh [12] simply used the above round-off

operation to estimate it and claimed to leave more precise estimation to further research.

Improvements for this issue are discussed in the following section.

4.3 Enhanced Solution Algorithm

One of the most important analytical tools of traffic engineering is computer

simulation. Computer simulation is more practical than a field experiment because it is less

costly and the results are obtained quickly. CORSIM 5.1, the standard simulation model,

uses a microscopic stochastic simulation model to represent the movements of individual

vehicles, which includes the influences of driver behavior. Therefore, CORSIM 5.1 is an

efficient tool to simulate the utilization of transportation resources and develop precise

measures to a transportation system's operational performance.

In this study, CORSIM 5.1 is used to simulate the traffic conditions at each iteration

given the traffic volumes assigned in the previous iteration. The actual travel time on each

link is then calculated from the simulation results.

Moreover, since CORSIM 5.1 provides a user-equilibrium platform to perform traffic

assignments, and the program internally translates the origin-and-destination data into a form

suitable for use by its simulation model, it is adopted not only to calculate the link travel

times but also to perform the traffic assignment procedure. Adopting CORSIM 5.1 into the







55

solution algorithm improves both the computational efficiency and the quality of the

solution.

Although CORSIM 5.1 can only deal with the static case, an important feature of

CORSIM 5.1 is that characteristics that change over time, such as signal timings and traffic

volumes, can be represented by dividing the simulation into a sequence of user-specified

time periods during which the traffic flows, traffic controls and geometry are held constant.

This feature is especially useful for solving the optimization program in Eq. 90 because the

procedure for solving the static route choice model with asymmetric travel time functions

has to be performed for each time period separately in each iteration.

The solution algorithm adopting CORSIM 5.1 is illustrated and explained as follows.

Step 0: Assign traffic initially by user-equilibrium assignment using CORSIM 5.1 to obtain
traffic volumes xm(0)

Step 1: Simulate the traffic conditions based on Xm()(t) and calculate the link travel times
Tm(n)(t).

Step 2: Solve the optimization model in Eq. 90 based on Tm(n)(t).

Step 2.1: Calculate congested speeds based on Tm(n)(t).

Step 2.2: Assign traffic using CORSIM 5.1 to obtain new Xm(n)(t) for each time period
sequentially.

Step 2.3: Compute dm(n)(t) based on new xm(n)(t)

Step 3: Test Convergence. If max { |dn Tn } "*stop. Otherwise, update xm(n)(t) and then
go to Step 1.

In step 0, the initial assignment is performed by user-equilibrium assignment using

CORSIM 5.1. The analysis time interval is divided into user-specified time periods but each

time period is assumed to have the same link volumes initially.







56

In step 1, CORSIM 5.1 is used to simulate the traffic conditions for all of the time

periods sequentially given assigned traffic volumes either from step 0 (each time period has

the same traffic volumes) or from step 3. CORSIM's simulation output can present the

average speed of travel on each link. The average travel time on each link can be obtained

by dividing the link length by the average speed.

In step 2.1, the congested speed on each link, based on the link travel times obtained

from step 1, is updated. As discussed in the Section 4.2.2, the travel time function in the

above optimization program is dependent on the traffic volumes assigned in the prior

iteration. Therefore, while performing the static traffic assignment, the travel time function

has to apply traffic volumes from the prior iteration.

CORSIM 5.1 provides two travel time functions, the BPR and the modified Davidson

link impedance functions, which were explained in Section 2.9. Users are allowed to choose

one of the impedance functions, but are not allowed to modify them. To deal with this

shortcoming, the CORSIM 5.1 "free-flow speed" on each link has to be adjusted based on

traffic volumes from the previous iteration to consider the transportation network as not

being empty for the assignment performed in step 2.2.

Step 2.2 performs a traffic assignment using CORSIM 5.1 to obtain new traffic

volumes for each time period sequentially. As mentioned above, minimizing the

optimization program in Eq. 90 is to minimize the static route choice model with asymmetric

travel time functions for each time period separately.

In step 2.3, the link travel times from the travel time function used in the traffic

assignment model based on traffic volumes from step 2.2 are computed. The relaxation

algorithm relaxes the link interactions by assuming the link travel times based on the traffic







57

volumes from the previous iteration are known from the first loop (outside steps 2.1- 2.3) of

the solution procedure. Therefore, the link travel times calculated from the travel time

function built into the assignment model in the second loop operation have to converge to

the estimated link travel times in the first loop. This convergence is tested in step 3. Since

CORSIM's assignment output includes the average link speed according to the travel time

function, the average travel time based on the travel time function can be obtained also by

dividing the link length by the average speed.

The remaining algorithmic step is to perform updating and convergence testing. The

solution procedure is performed iteratively until the stopping criterion is met. Figure 4-1

shows the flow chart of the above solution algorithm.

In Chapter 5, the above DTA model is combined with the optimization program for

the real-time traffic control model developed in Chapter 3. The Visual Basic program

presented in Chapter 6 is developed to implement the process in this flow chart to determine

the user-equilibrium traffic flows on each link for each time period given the signal settings

from the real-time traffic control model.










Assign traffic initially
using CORSIM & get xm(0)(t)




Simulate &
calculate link
travel times Tm(n)(t)




Calculate congested speeds
based on Tm(n)(t)


Compute dm(n)(t)




yes
Stop dm(n)(t) Tm(n)t)


Sno

Update xm("n)(t)


Figure 4-1 Flow Chart of the Solution Algorithm for the DTA Model.


Assign traffic
using CORSIM to
get new xm(n)(t)















CHAPTER 5
COMBINED MODEL

5.1 Introduction

This chapter presents a theory to deal with the traffic flow equilibrium which

involves traffic-responsive signal control policies. This situation has been extensively

investigated in two different ways [11]: global optimization models and the iterative

optimization assignment (IOA) procedure. However, global optimization models have maj or

difficulty in finding an efficient solution algorithm for calculating the optimal signal settings

in traffic networks while anticipating user-optimum equilibrium flows.

Some authors [9, 12, 27] have used a bilevel programming method to combine the

traffic control and the transportation assignment models. This presumes that decision

makers at two levels act in a hierarchical manner. At the upper level, decision makers,

bound by the decisions of the lower level, try to maximize the their own profit, taking into

account the reactions of the lower level accordingly. The iterative optimization assignment

procedure can be used to solve a bilevel program.

The procedure of the IOA is to update the signal settings for fixed flows and solve

the traffic equilibrium assignment for fixed signal settings sequentially until the solutions

of the two models are considered to be consistent. Yang [11] claimed that the IOA

procedure has the advantages that the traditional traffic assignment and signal setting

techniques can be employed to solve the problem and can be applied to a large network.







60

Marcotte [28] used the IOA procedure to solve for optimal signal settings based on capacity

variables corresponding to user-optimized equilibrium traffic flows. Marcotte hoped that

the convergence toward a 'good' solution would be obtained. He concluded that the

numerical experiments tended to show the IOA procedure yielded near-optimal solutions.

In the following sections, the bilevel program is presented first. The iterative

optimization assignment procedure is then addressed.

5.2 Bilevel Model

The bilevel programming model described formally and completely by Bard and Falk

[35] can be formulated as:


Min Z1(g,v) (91)
geG


subject to

v c { argmin Z2(g, z) (92)
Ze V (g) (92)

where g and v represent the decision vectors associated with the upper and lower levels,

respectively. G is the feasible set of the g-variables and V(g) is the feasible set, possibly

dependent on g, of the v-variables [28]. The "argmin" of a mathematical program is the

optimal solution of the program; in other words, Zi,( has to be evaluated at the optimal

solution of Z2(" Since the objective of traffic management systems is to calculate

equilibrium flows which are consistent with a given traffic-responsive control policy, this

framework can be applied to integrate the traffic control policy and the transportation

assignment procedure. In the upper level, Zi(' represents the optimization model for

dynamic traffic control in which total intersection delay in a network is minimized by

allocating appropriate green times, g, given the user-optimal traffic flows, v, which are







61

obtained from the lower level model. In the lower level, Z2(* represents the dynamic traffic

assignment model formulated as a variational inequality formula in which the link traffic

flows, z, reach user equilibrium, v, given the link travel times gathered from the upper

model. The mathematical formulation of the dynamic transportation management system,

which combines the models developed in Chapter 3 and Chapter 4, is given as follows:


min Z(g, v)= vP(t) dmP(t) (93)
geG t I p meB(I)




subject to T [v (t)] -[z (t) v (t)] *0 z (t) *V(g) (94)

where G denotes the feasible region that is delineated by the two constraints for conserving

the cycle length and the minimum green times shown earlier in Eqs. 66 and 67. The vehicle

delay function, d, is dependent on the variables of green times, g, and traffic flows, v.

To calculate the total delay happening at each intersection, the traffic flows, v, in the

upper level have to be specified as traffic volumes moving on each link during a specific

phase, at a specific time, that is, vmP(t). However, these traffic flows have to satisfy the VI

formula in the lower level in which the total volumes on each link are represented in vector

form, v(t). T [v(t)] denotes the travel disutility functions which are dependent on link traffic

flows whereas V(g) is the feasible traffic flow set delineated by the flow constraints

presented in Eqs. 82-84 in Chapter 4. One of the flow constraints, the flow propagation

constraint, is dependent on the link travel times which are affected by the green time splits.

5.3 Iterative Optimization and Assignment Procedure

Since there is not much hope to develop exact solution algorithms for large or even

medium-size networks [28], the IOA procedure is used to solve the above bilevel program.







62

The iteration of the algorithm consists of solving sequentially an optimization model

involving the signal timing variables, with the flow variables fixed, and a user-optimized

equilibrium model corresponding to this new green time vector. The procedure for iterative

optimization and assignment is illustrated in Figure 5-1.

Although there are limitations, Smith and Vuren [36] proved the convergence of the

iterative optimization and assignment algorithm. Using numerical experiments, Marcotte

[28] also showed the IOA heuristic can yield near-optimal solutions.

5.4 Detailed Procedure

The two submodels included in Figure 5-1, the adaptive traffic-responsive signal

control submodel and the dynamic traffic assignment submodel, can be replaced by the flow

charts shown in Figure 3-1 and Figure 4-1, respectively. To initialize the procedure, users

need the information of the link-node network structure and the O-D demand matrix, and

must decide on the number of time periods, the duration of each time period and signal

settings. The detailed procedure for implementing the IOA method to optimize the dynamic

transportation management system is described as follows:

Step 1: Initialize.

Step 1.1: Prepare input data. Prepare the link-node network connection structure,
initial travel speed estimates, link length measurements, link capacities, the
O-D trip matrix, the number of time periods, the duration of each time
period, and for each intersection, the signal cycle length, cycle offset, and
signal phase plan.

Step 1.2: Perform static traffic assignment. The traditional user-equilibrium traffic
assignment method is used to assign the O-D trip matrix to the network using
CORSIM 5.1 based on initial estimates of travel times.

Step 1.3: Determine initial signal settings and intersection delays. The static network-
wide optimal signal timing settings and delays for the user-specified phase
plans based on the traffic flows obtained from Step 1.2 are determined by
TRANSYT-7F.


















































Figure 5-1 Flow Chart of the Iterative Optimization Assignment Procedure.









Step 1.4: Perform initialflow simulation. The total analysis time interval is split into
user-specified time periods. Each time period has the same signal timing
settings obtained from Step 1.3. With the link traffic flows obtained from
Step 1.2 at the initial time period, the CORSIM 5.1 simulation model
simulates the traffic conditions and obtains the link traffic flows for each
time period. The queue buildups from each time period will carry into the
next time period. The signal timings and intersection delays can then be
calculated for the revised traffic volumes.

Step 2: Execute the adaptive traffic-responsive signal control submodel.

Step 2.1: Obtain input data. For the first iteration, read the signal timing settings and
delays from TRANSYT-7F (Step 1.3) and the link traffic flows from the
CORSIM 5.1 simulation results (Step 1.4) for each time period. Starting
from the second iteration, read the optimal signal timing settings and delays
resulting from the previous iteration and read the link traffic flows for each
time period from the dynamic traffic assignment submodel.

Step 2.2: Find descent direction. Assign the minimum green time to each phase at
each intersection for each time period. Then assign the remaining green time
from the cycle length of the intersection to the phase with the highest vehicle
delay.

Step 2.3: Optimize move size. Perform a line search to find the optimal extent value
to move from the last solution of green times toward the auxiliary solutions
obtained form Step 2.2.

Step 2.4: Test convergence. If the convergence criteria for green times are not
satisfied, return to Step 2.2; otherwise, proceed to Step 3.

Step 3 Execute the dynamic traffic assignment submodel.

Step 3.1: Perform simulation. The optimal signal timing settings for each time period
are updated using the results from the adaptive traffic-responsive control
submodel (Step 2). Using the link traffic flows obtained from Step 1.2 for
the initial time period at the first iteration (or using the link traffic flows
obtained from the previous iteration for the second iteration and beyond), the
simulation model simulates traffic conditions based on dynamic traffic-
responsive signal timings to obtain the link traffic flows for each time period.

Step 3.2: Calculate link travel times. Calculate the link travel times for each time
period by dividing the link length by the average speed from the CORSIM
5.1 simulation results of Step 3.1.

Step 3.3: Modify free-flow speeds. Modify the free-flow speed on each link for each
time period to transfer the effects of the traffic flows resulting from the









previous iteration to the travel time functions built into the static traffic
assignment software (CORSIM 5.1).

Step 3.4: Perform static traffic assignment for each time period separately. The static
traffic assignment is performed for each time period separately with
congestion speeds represented as modified free-flow speeds.

Step 3.5: Compute estimated link travel times based on the travel time functions. The
estimated link travel times based on the travel time functions built into the
traffic assignment software are calculated.

Step 3.6: Test convergence. Compare the links travel times resulting from Step 3.2
and Step 3.5. If the convergence criteria are not satisfied, return to Step 3.1;
otherwise, proceed to Step 4.

Step 4 Test convergence. Compare the traffic flows between two iterations. If the
convergence criteria, max {|v v n+l} **are not satisfied, return to Step 2; otherwise,
stop.















CHAPTER 6
IMPLEMENTATION

6.1 Introduction

Software implementation is a necessary step in optimizing the transportation

management system. The objective of this chapter is to develop computer software to

implement the detailed procedure presented in Section 5.4. A numerical example is used to

illustrate the procedure for implementing the IOA method to optimize the dynamic

transportation management system.

6.2 Numerical Example

Figure 6-1 shows the link-node structure of a traffic network illustrated by the ITRAF

display. ITRAF is a preprocessor for CORSIM, which allows users to create traffic

networks, enter traffic volumes and timing data on a map window and then transfer all

information into input files for CORSIM. There are eight roadways with eleven

intersections in the example network.

Most numerical examples used in the existing literature were small traffic networks

because of the complicated computation and the limitation of computer memory to store

large link-node structures. The computer program developed in this research was written

in Visual Basic for Windows 2000. With their increasing power, microcomputers have

become a suitable hardware platform for rather large network processing applications.

However, there are size limitations for CORSIM 5.1 network characteristics.
































Figure 6-1 Link-Node Structure of the Example Network.
In CORSIM 5.1, the entry and/or exit nodes must be numbered between 8000 and

8999 while the internal nodes can be numbered from 1 to 6999. In other words, the latest

version of CORSIM 5.1 allows a maximum number of 1,000 entry and/or exit nodes. Entry

and/or exit nodes were used in this study to represent the centroids of traffic analysis zones

(TAZ). The TAZ is the basic analysis unit for calculating trip productions and attractions.

These trips are assumed to start or end at the centroid of a zone and are distributed

throughout the network to create the O-D matrix. In Florida, the numbers of TAZs for Palm

Beach County, Broward County and Miami-Dade County are 1,118, 892 and 1,466,

respectively. Therefore, for urban areas with less than 1,000 TAZs, CORSIM 5.1 can

accommodate the origin and destination zones without adjustment. However, for urban

areas with more than 1,000 TAZs, some TAZs have to be merged to reduce the number of

zones to less than 1,000.







68

The maximum number of internal nodes is 6,999 in CORSIM 5.1. Internal nodes

are used to represent intersections of a given street network. The numbers of non-centroid

nodes (representing intersections of streets) in the 1999 networks for Palm Beach County,

Broward County and Miami-Dade County are 2,861, 3,378 and 5,969, respectively. For a

street network with more intersections than 6,999, some minor streets have to be excluded

to reduce the total number of intersection to be less than the maximum number of internal

nodes allowed in CORSIM 5.1.

Table 6-1 shows the O-D demand matrix for the numerical example. The number

of trips per hour for a specific trip purpose (e.g., working or shopping) shown in the cells

were defined for demonstration purposes. For example, in Table 6-1, there were 100

shopping trips per hour from TAZ 8021 to TAZ 8019.

Table 6-1 O-D Demand Matrix for the Example Application
Destination

Origin 8021 8015 8019 8023 8025 8013 8014 8024 8016 8020 8012
8021 0 50 100 30 110 60 100 100 200 50 110
8015 20 0 120 150 50 50 50 50 50 60 110
8019 90 50 0 40 140 100 150 200 150 100 150
8023 50 100 120 0 110 110 120 40 50 30 40
8025 50 100 100 50 0 50 50 100 100 50 50
8013 80 100 100 130 70 0 100 60 110 60 50
8014 50 100 150 100 150 150 0 80 100 80 50
8024 50 100 50 20 200 110 50 0 200 40 50
8016 30 130 100 150 80 130 50 130 0 50 100
8020 20 120 150 130 150 120 50 120 90 0 100
8012 60 70 60 120 110 130 90 80 70 50 0

Note that nodes 8017, 8018, 8022, 8026 and 8027 shown in Figure 6-1 do not have volumes

in the O-D matrix. These nodes were treated like dummy nodes and can be connected to

new nodes in the future.







69

Figure 6-2 shows the pre-timed signal settings which were defined at each

intersection. It consists of four phases for each signal cycle with a protected left-turn phase

and a through movement phase for both north-south and east-west directions. As discussed

in Section 3.4, the signal settings can be different at each intersection. However, to simplify

the implementation procedure, the signal settings for all of the intersections in the example

network were the same. In addition, the phase durations for the north-south and east-west

directions were originally set to be equal.




N-S N-S E-W E-W
Lef-Turn Through Left-Turn Through








15 sec 37 sec 15 sec 37 sec


Cycle Length = 104 seconds


Figure 6-2 Initial Signal Settings for the Example Application.

6.3 Software Implementation

A comprehensive, menu-driven software program was developed in Visual Basic to

implement the detailed procedure presented in the previous chapter. Figure 6-3 shows the

program's main menu. A copy of the program can be downloaded from the following web

site: http://www.fiu.edu/-chowl/. The application of the software is illustrated in the

following procedure following the steps in Section 5.4:











III--I .L II I II- -I IL I-- I I


INITIAL 4ASIGNMENT

INITIAL SIMULATION I

INITIAL TRAFFIC-RESPONSI'vE SIGNAL CONTROL
'lep 1 C[T[. .ANT. FM DLSCEIN4T DL'FI,'TI:'N I SIEp,


DETEFAJIRNE PTIM\LX RIt\ :j' E I


DYNAMIC TRAFFIC ASSIGNMENT


.IF 'p -4


P'EP- '-PA SIMBIATI- *N

'.-, Ll. ATE T(t


K.... I -,,lii I

li"Su


'F:ST '.-'i:'N I ?'P N'- E

I.i'D.It +..) -lJl.' FE-ITEB+. it



,+.'. I)IPFT. -IL ) 1


' le t- TEST -:'NIP, C E-NCE. (,t = T()

r ;1+1:' I Pi 1.1" 1 .'i.',1I F-ii'FI-P I'F


T': FFIC-RESP O,. I E H S."I F-I' 'l) '.SSI iN


TRAFFIC-RESPONSIVE SIGNAL CONTROL 'rmi


ITERATIVE OPTIMIZATION AND ASSIGNMENT


GO...


Figure 6-3 Main Menu Screen.

Step 1 Initialization:

Step 1.1: Prepare input data.
1. The ITRAF button on the menu bar launches the application of ITRAF and allows
users to create traffic networks, enter traffic volumes and timing data on a map
window and then transfer this information into input files for CORSIM 5.1. The
required input information at this step includes:
(1) link-node geometric data
(2) length of each link
(3) number of lanes on each link
(4) mean value for number of vehicles in initial queues and start-up lost time
(5) free-flow speed on each link
(6) number of approaches and signal phase sequences for each node
(7) durations of signal timing intervals for each node
(8) cycle offset for each node

2. The O-D Input button on the menu bar serves as an O-D matrix file manager and
allows users to create or modify O-D matrix files in Excel spreadsheets.

3. The Transfer button on the menu bar transfers the O-D demand matrix, as created
or modified using the O-D Input option, into CORSIM 5.1 input files. Users can


Srep 1

- ep 2










specify the following assignment parameters through ITRAF if the default values
provided by the software are not satisfactory:
(1) acceptable threshold of objective function (default value: 0.1%)
(2) maximum number of iterations (default value: 5)
(3) impedance function type (default type: BPR formula)
(4) parameters of impedance function
(5) capacity smoothing factor (default value: 0)
(6) line-search accuracy threshold (default value: 0.1%)

Step 1.2: Perform static traffic assignment.
The Browse button at the Initial Assignment window invokes the Open File dialog
box. With this dialog box, users can select the desired CORSIM 5.1 input file, with
the O-D information embedded, and then click the Start button to activate the
assignment model provided by CORSIM 5.1. The static user-equilibrium traffic is
applied to the network using the specified origin-destination information. For the
first iteration, the link impedances are evaluated for free-flow speed conditions
throughout the entire network. An intermediate solution for each iteration is
obtained using link impedances produced by the previous iteration. To obtain an
optimal solution for each iteration, an iterative line search is applied to the range
between the current intermediate solution and the previous iteration solution. The
search terminates when the contribution of the current iteration is less than the
accuracy threshold value. The traffic assignment process terminates when the
relative change of the objective function between two successive iterations is less or
equal to the threshold value.
The Visual Basic (VB) program extracts and formats part of CORSIM's output file
and shows the link turning volumes on the screen when the procedure is completed.
Figure 6-4 shows an excerpt of initial assignment results for the example.


Traffic Assignment Results

Link Left Turn Through Right Turn

(8025, 25) 0 700 0
25, 5) 106 594 0
1, 6) 230 460 141
1, 2) 118 685 95
1, 21) 0 500 0
(8021, 21) 0 910 0
(21, 1) 357 553 0
(8024, 24) 0 870 0
24, 8) 0 493 377
6, 8) 410 480 0
6, 13) 0 1010 0
6, 7) 185 317 281
6, 1) 0 173 540
(8013, 13) 0 860 0
(13, 6) 288 352 221
(8014, 14) 0 1010 0
14, 7) 289 423 298

Figure 6-4 Excerpt of the Initial Assignment Results for the Example Application.









Step 1.3: Determine initial signal settings.
The TRANSYT option on the menu bar launches the application of T7F9, an
interface shell program for TRANSYT-7F, and allows users to input the initial input
data specified in Step 1.1.1 and the assignment results from Step 1.2 into the input
files of TRANSYT-7F. The T7F9 shell program can also be used to access
TRANSYT-7F to get the static optimal green times. The timing data from
TRANSYT-7F can then be imported into CORSIM's format by the ITRAF button.

Step 1.4: Perform initialflow simulation.
The Browse button at the Initial Simulation command invokes the Open File dialog
box. With this dialog box, users can select the CORSIM 5.1 input file from Step 1.3.
After clicking the Start button, users are asked to input the number of time periods
and the duration of each time period before the simulation model is activated. In the
example, an analysis time interval of one hour was selected and separated into six
time periods with a duration of ten minutes for each time period. CORSIM 5.1
allows users to partition the simulation time into a series of time periods (up to 19
time periods) of varying duration. The VB program modifies the CORSIM 5.1 input
file to specify the number of time periods as well as the duration of each period.
With the link traffic flows obtained from Step 1.2 at the initial time period, the
CORSIM 5.1 simulation model simulates the traffic conditions and obtains the link
traffic flows for each time period.

Step 2 Execute the initial dynamic traffic-responsive signal control submodel.

Step 2.1 & Step 2.2: Obtain input data and find descent direction.
The Init and Find Descent Direction button activates the VB program to read the
signal timing settings from Step 1.3 and the link traffic flows for each time period
from Step 1.4 after users input the minimum green time for each intersection. The
VB program then calculates delay for each phase at each intersection at each time
period according Eqs. 63-65. The descent direction is obtained by assigning the
minimum green time to each phase at each intersection for each time period and then
assigning the remaining green time from the cycle length of the intersection to the
phase with the highest vehicle delay. Figure 6-5 shows an excerpt of the results of
the auxiliary green times.

Step 2.3: Optimize move size.
The Determine Optimal Move Size button activates the VB program to find the
optimal move size from the last set of green times to the auxiliary set of green times.
This is achieved by using the bisection method to find an approximate value of *
which sets the complicated derivative of the objective function (Eq. 74) equal to
zero.

Step 2.4: Test convergence.
The Test Convergence button executes a VB command to examine the criteria for
assessing convergence. A message box on the screen shows whether the criteria are
satisfied or not.



































Figure 6-5 Excerpt of the Auxiliary Green Time Results.


The Update g(t) and Re-iterate button automatically runs the traffic-responsive
signal control submodel (Step 2.1 to Step 2.3) repeatedly until the criteria for
convergence are satisfied. Figure 6-6 shows an excerpt of the results for the optimal
green times at the initial iteration.

Step 3: Execute the dynamic traffic assignment submodel.

Step 3.1: Perform simulation.
The Perform Simulation button activates the VB program to update the signal
timing settings for each intersection at each time period using the results from the
dynamic traffic-responsive signal control submodel (Step 2). The simulation model
(CORSIM 5.1) is then activated to simulate the traffic conditions for the network
with the optimal signal settings and the traffic flows from the initial assignment (Step
1) or the last iteration.

Step 3.2: Calculate link travel times.
The Calculate T(t) button calculates the link travel times from the simulation results.
The VB program calculates the link travel times for each time period by dividing the
link lengths by the average speeds from the CORSIM 5.1 simulation results of Step
3.1. Figure 6-7 shows an excerpt of results of the link travel times for each time
period.

Steps 3.3 and Step 3.4: Modifyfree-flow speeds and perform static traffic assignment
for each time period separately.


Time Period: 1
AUXILIARY GREEN TIME AT EACH PHASE

NODE N-S LT N-S TH E-W LT E-W TH
1 22 21.6 33.6 42.8
6 14.8 40.4 14.4 50.4
8 16.4 19.2 16.8 67.6
2 25.6 25.6 22 46.8
7 14.4 56.8 17.6 31.2
9 25.2 43.2 12.4 39.2
3 20.8 41.6 20 37.6
10 12.4 43.2 21.2 43.2
4 14.4 37.4 49.8 18.4
5 8 89.2 9.6 13.2
11 11.2 83.6 8.8 16.4

Time Period: 2
NODE
1 26.8 20.8 29.2 43.2
6 21.6 42 20.8 35.6
8 21.2 18 16.8 64
2 22.4 28.4 34.4 34.8
7 16.4 48.4 19.2 36
9 21.6 43.6 15.2 39.6











Time Period: 1
GREEN TIME AT EACH PHASE

NODE N-S LT N-S TH E-W LT E-W TH
1 26 24 25 45
6 16 43 18 43
8 22 26 20 52
2 26 31 24 40
7 17 49 20 35
9 20 44 15 41
3 22 34 22 43
10 15 45 17 43
4 18 49 32 21
5 12 60 21 26
11 14 58 12 36

Time Period: 2
NODE
1 27 24 24 45
6 17 44 19 40
8 24 25 20 50
2 25 32 25 37
7 16 47 21 36
9 19 44 16 41

Figure 6-6 Excerpt of the Optimal Green Time Results at the Initial Iteration.



TIME PERIOD 2

Link LENGTH (ft) AVERAGE SPEED (MPH) TRAVEL TIME
(sec)

25, 5) 1277 20 43
1, 6) 1533 13 80
1, 17) 272 24 7
1, 2) 1552 10 105
1, 21) 1269 25 34
21, 1) 1269 10 86
24, 8) 1256 11 77
6, 8) 1268 11 78
6, 13) 1689 26 44
6, 7) 1552 10 105
6, 1) 1533 10 104
13, 6) 1689 10 114
14, 7) 1552 10 105
8, 24) 1256 26 32
8, 18) 255 25 6
8, 9) 1552 10 105
8, 6) 1268 10 86
18, 8) 255 10 17

Figure 6-7 Excerpt of the Link Travel Time Results Based on the Simulation.



The Adjust Free Flow Speed and Assign button activates the VB program to
replace the free-flow speeds (specified on Entry 25 of Record 11 in a CORSIM 5.1










input file) with the average speeds from the CORSIM 5.1 simulation results of Step
3.1 for each link at each time period. The VB program then creates six CORSIM 5.1
input files, each for a specific time period, with the effects of the traffic flows
resulting from the initial assignment or previous iteration, and activates CORSIM 5.1
to perform the static user-equilibrium traffic assignment using these files as input
separately.

Step 3.5: Compute estimated link travel times based on the travel time functions.
The Compute d(t) command activates the VB program to retrieve the link travel
times estimated based on the travel time functions built into the static traffic
assignment model in CORSIM 5.1 from the six output files generated at Step 3.4 for
each time period. Figure 6-8 shows an excerpt of the results of the estimated link
travel times from the BPR formula.


TRAVEL TIME, d(t), ON THE PATH-LINK USING FHWA IMPEDANCE FUNCTION
TIME PERIOD 1

Link LENGTH (ft) AVERAGE SPEED (MPH) TRAVEL TIME
(sec)

25, 5) 1277 24.6 35
1, 6) 1533 10.9 95
1, 17) 272 10 18
1, 2) 1552 15.9 66
1, 21) 1269 29.7 29
21, 1) 1269 10 86
24, 8) 1256 11.5 74
6, 8) 1268 10 86
6, 13) 1689 29.1 39
6, 7) 1552 13.3 79
6, 1) 1533 10 104
13, 6) 1689 16.4 70
14, 7) 1552 12.7 83
8, 24) 1256 28.6 29
8, 18) 255 10 17
8, 9) 1552 13.9 75
8, 6) 1268 13.8 62

Figure 6-8 Excerpt of the Link Travel Time Results from the BPR Formula.


Step 3.6: Test convergence.
The Test Convergence: d(t) = T(t) ? button compares the link travel times based on
the simulation results and the BPR formula. A message box on the screen shows
whether the criteria are satisfied or not. Figure 6-9 shows an excerpt of the
comparison of link travel times based on simulation results and the BPR Formula.
If the convergence criteria are not satisfied, clicking the Update X(t) and Re-iterate
button will automatically run the dynamic traffic assignment procedure iteratively
until the criteria are satisfied. After the criteria are satisfied, clicking the Traffic-
Responsive Signal Control button will run Steps 2.1 through 2.4 of the dynamic
traffic-responsive signal control submodel but using the traffic flows resulting from
the dynamic traffic assignment submodel.











TRAVEL TIME
BPR


(sec)


TRAVEL TIME (sec)
SIMULATION


25, 5) 71 43
1, 6) 104 80
1, 17) 18 7
1, 2) 72 105
1, 21) 34 34
21, 1) 86 86
24, 8) 85 77
6, 8) 86 78
6, 13) 45 44
6, 7) 105 105
6, 1) 104 104
13, 6) 114 114
14, 7) 105 105
8, 24) 34 32
8, 18) 17 6
8, 9) 79 105
8, 6) 86 86
18, 8) 17 17
12, 11) 85 65
2, 7) 103 51


Figure 6-9


Excerpt of the Comparison of Link Travel Times
Based on Simulation Results and the BPR Formula.


Step 4 Test convergence.
The Iterative Optimization Assignment button runs the dynamic traffic-responsive
signal control submodel and the dynamic traffic assignment submodel sequentially
until the solutions converge.


The final results for the traffic-responsive signal timings and dynamic traffic flows
are presented in the next Chapter.


LINK















CHAPTER 7
RESULTS AND EVALUATION

7.1 Introduction

The objective of this chapter is to evaluate the time-dependent traffic flows and the

signal settings produced by the optimal dynamic transportation management system

developed in this research using the numerical example presented in Chapter 6. The results

for both static and dynamic cases are presented and are followed by a comparison and

assessment.

7.2 Results for Static Signal Settings and Static Traffic Flows

Static optimal traffic flows determined by the traditional user-equilibrium

methodology for the sample network and the given O-D matrix are shown in Figure 7-1.

From the traffic flows shown in Figure 7-1, TRANSYT-7F found that the traffic signal

settings shown in Figure 7-2 should produce minimum delays during the analysis time

interval. There are eleven intersections and each intersection has four phases for each signal

cycle. In the static case, traffic flows and signal settings remain the same throughout the

analysis time interval.

7.3 Results for Adaptive Signal Settings and Dynamic Traffic Assignment

In the dynamic case, traffic flows and signal setting are time-dependent. In the

example, the analysis time interval, one hour, was separated into six time periods with a

duration of ten minutes for each time period.













Link LT
(8025, 25) 0
25, 5) 146
1, 6) 222
1, 2) 69
1, 21) 0
(8021, 21) 0
( 21, 1) 362
(8024, 24) 0
24, 8) 0
6, 8) 419
6, 13) 0
6, 7) 299
6, 1) 0
(8013, 13) 0
( 13, 6) 255
(8014, 14) 0
14, 7) 280
8, 24) 0
8, 9) 0
8, 6) 210
(8012, 12) 0
12, 11) 0
2, 7) 294
2, 1) 346
2, 3) 417
2, 15) 0
7, 9) 347
7, 6) 215
7, 14) 0
7, 2) 58
9, 16) 0


TH
700
554
446
622
500
910
548
870
501
451
1010
276
176
860
396
1010
395
960
540
293
840
591
369
0
932
920
550
574
810
418
1120


Link
9, 8)
9, 10)
9, 7)
(8015, 15)
( 15, 2)
(8016, 16)
16, 9)
3, 10)
3, 2)
3, 4)
3, 19)
10, 20)
10, 9)
10, 11)
10, 3)
(8019, 19)
( 19, 3)
(8020, 20)
20, 10)
4, 23)
4, 3)
4, 5)
(8023, 23)
23, 4)
5, 11)
5, 4)
5, 25)
11, 10)
11, 12)
11, 5)


Figure 7-1 Static Traffic Assignment Results.




Intersection Green Time for Each Phase Cycle
N-S LT N-S TH E-W LT E-W TH Length

1 23 21 19 41 104
2 22 28 20 34 104
3 18 28 18 40 104
4 15 47 24 18 104
5 9 50 20 25 104
6 12 40 15 37 104
7 13 43 16 32 104
8 19 23 17 45 104
9 15 40 12 37 104
10 12 41 12 39 104
11 11 48 9 36 104


Figure 7-2 Optimal Signal Settings for the Static Case.


TH
0
562
367
710
297
950
530
247
570
1037
1050
570
561
644
403
1170
399
1050
543
920
674
946
770
0
0
536
1170
595
810
0











Figure 7-3 shows the optimal time-dependent signal settings for those six time

periods. Unlike the static signal settings shown in Figure 7-2, the dynamic traffic signal

settings shown in Figure 7-3 are different in different time periods. In the dynamic case,

travelers are assumed to receive real-time traffic information based on delays associated with

the signal settings, and change their paths to minimize their travel times. The transportation

management system then modifies the signal settings according the updated traffic flows at

each time period to minimize the system-wide delay. The optimal dynamic traffic flows are

shown in Figure 7-4.


Figure 7-3 Optimal Time-Dependent Signal Timing Settings for the Example Network.


For the example network, the run-time on a computer with 900 MHz CPU was about

6 minutes for each iteration. It took the program less than an hour to reach the optimal











results. Although this cannot be used to proclaim the feasibility in terms of computer run-


time for applications with real networks, which are usually in large in scale, the program


should be suitable for rather large network processing applications.


LINK
(8025, 25)
( 25, 5)
( 6)
( 2)
( 21)
(8021, 21)
( 21, 1)
(8024, 24)
( 24, 8)
( 6, 8)
( 6, 13)
( 6, 7)
( 6, 1)
(8013, 13)
( 13, 6)
(8014, 14)
( 14, 7)
( 8, 24)
( 8, 9)
( 8, 6)
(8012, 12)
12, 11)
2, 7)
2, 1)
2, 3)
2, 15)
7, 9)
7, 6)
7, 14)
7, 2)
9, 16)
9, 8)
9, 10)
9, 7)
(8015, 15)
( 15, 2)
(8016, 16)
16, 9)
3, 10)
3, 2)
3, 4)
3, 19)
10, 20)
10, 9)
10, 11)
10, 3)


TP=1
L T R
0 117 0
14 103 0
42 73 40
21 125 13
0 83 0
0 152 0
74 78 0
0 145 0
0 84 62
56 84 0
0 168 0
38 60 44
0 48 84
0 143 0
33 71 39
0 168 0
50 79 39
0 160 0
6 75 36
29 56 28
0 140 0
0 99 41
47 53 40
78 0 35
70 155 40
0 153 0
48 99 1
27 99 44
0 135 0
2 69 65
0 187 0
76 0 29
57 83 40
51 59 29
0 118 0
75 34 10
0 158 0
27 75 57
39 47 44
94 101 64
0 184 102
0 175 0
0 95 0
52 78 58
46 104 0
51 67 65


TP=2
L T R
0 117 0
14 102 0
42 73 48
33 100 11
0 83 0
0 152 0
74 77 0
0 145 0
0 86 59
69 82 0
0 168 0
52 53 37
0 43 69
0 143 0
26 70 48
0 168 0
56 64 48
0 160 0
0 80 48
35 51 30
0 140 0
0 94 47
45 43 29
86 0 40
58 147 43
0 153 0
55 80 1
30 85 37
0 135 0
5 58 74
0 187 0
78 0 30
60 97 40
59 36 38
0 118 0
75 28 16
0 158 0
24 73 62
49 45 63
78 105 62
0 165 89
0 175 0
0 95 0
58 84 60
71 108 0
45 69 55


TP=3
L T R
0 117 0
22 95 0
42 66 48
19 131 13
0 83 0
0 152 0
78 74 0
0 145 0
0 79 67
61 72 0
0 168 0
34 54 45
0 44 84
0 143 0
34 70 40
0 168 0
56 72 40
0 160 0
14 73 40
44 48 21
0 140 0
0 94 46
44 46 29
83 0 40
65 152 49
0 153 0
53 89 5
32 77 46
0 135 0
5 68 62
0 187 0
83 0 34
69 76 34
54 60 37
0 118 0
74 31 14
0 158 0
26 80 53
48 51 57
75 104 67
0 172 90
0 175 0
0 95 0
58 87 57
57 103 0
48 70 61


Figure 7-4 Optimal Time-Dependent Traffic Flows for the Six Time Periods.













LINK
(8019, 19)
( 19, 3)
(8020, 20)
( 20, 10)
( 4, 23)
( 4, 3)
( 4, 5)
(8023, 23)
23, 4)
5, 11)
5, 4)
5, 25)
11, 10)
11, 12)
11, 5)



LINK
(8025, 25)
( 25, 5)
( 6)
( 2)
( 21)
(8021, 21)
( 21, 1)
(8024, 24)
( 24, 8)
( 6, 8)
( 6, 13)
( 6, 7)
( 6, 1)
(8013, 13)
( 13, 6)
(8014, 14)
( 14, 7)
( 8, 24)
( 8, 9)
( 8, 6)
(8012, 12)
12, 11)
2, 7)
2, 1)
2, 3)
2, 15)
7, 9)
7, 6)
7, 14)
7, 2)
9, 16)
9, 8)
9, 10)
9, 7)
(8015, 15)


TP=1
L T R
0 195 0
65 64 66
0 175 0
53 94 28
0 153 0
26 142 38
0 160 50
0 128 0
102 0 26
31 0 32
52 104 0
0 195 0
8 91 31
0 135 0
52 0 35


TP=4
L T R
0 117 0
27 89 0
51 68 46
28 114 19
0 83 0
0 152 0
79 73 0
0 145 0
0 89 56
56 79 0
0 168 0
47 59 46
0 46 82
0 143 0
26 81 37
0 168 0
52 71 45
0 160 0
0 72 40
43 59 20
0 140 0
0 98 42
42 48 32
91 0 38
58 166 40
0 153 0
53 86 8
30 80 43
0 135 0
4 65 72
0 187 0
81 0 33
52 91 37
50 49 34
0 118 0


TP=2
L T R
0 195 0
53 80 62
0 175 0
54 84 36
0 153 0
30 138 49
0 155 37
0 128 0
101 0 27
27 0 25
64 116 0
0 195 0
10 84 24
0 135 0
78 0 40


TP=5
L T R
0 117 0
26 90 0
49 66 46
22 125 12
0 83 0
0 152 0
80 72 0
0 145 0
0 78 67
53 81 0
0 168 0
34 61 45
0 48 79
0 143 0
47 60 37
0 168 0
57 62 50
0 160 0
5 72 44
50 35 31
0 140 0
0 93 47
49 42 35
89 0 36
69 153 42
0 153 0
56 83 6
30 73 45
0 135 0
6 69 65
0 187 0
79 0 39
53 87 41
51 55 26
0 118 0


TP=3
L T R
0 195 0
49 84 62
0 175 0
53 85 36
0 153 0
23 136 40
0 159 41
0 128 0
101 0 28
32 0 31
64 98 0
0 195 0
10 90 25
0 135 0
67 0 36


TP=6
L T R
0 117 0
21 96 0
47 69 42
25 111 9
0 83 0
0 152 0
74 77 0
0 145 0
0 81 64
50 80 0
0 168 0
45 60 55
0 47 71
0 143 0
31 81 31
0 168 0
50 76 43
0 160 0
0 73 40
40 42 32
0 140 0
0 95 46
43 49 34
81 0 36
58 151 44
0 153 0
58 89 8
30 86 46
0 135 0
0 71 71
0 187 0
80 0 33
56 95 41
51 54 32
0 118 0


Figure 7-4. Continued








82


TP=4 TP=5 TP=6
LINK L T R L T R L T R
15, 2) 78 26 14 74 32 12 71 39 9
(8016, 16) 0 158 0 0 158 0 0 158 0
16, 9) 21 81 56 25 80 54 19 77 62
3, 10) 41 47 53 46 46 53 48 44 67
3, 2) 77 111 61 82 107 62 79 108 57
3, 4) 0 169 98 0 161 100 0 155 95
3, 19) 0 175 0 0 175 0 0 175 0
10, 20) 0 95 0 0 95 0 0 95 0
10, 9) 61 86 52 60 88 48 57 86 59
10, 11) 63 104 0 59 110 0 67 111 0
10, 3) 47 65 56 49 61 58 45 68 55
(8019, 19) 0 195 0 0 195 0 0 195 0
(19, 3) 46 83 67 50 80 65 45 85 65
(8020, 20) 0 175 0 0 175 0 0 175 0
20, 10) 51 90 34 60 79 36 52 89 35
4, 23) 0 153 0 0 153 0 0 153 0
4, 3) 18 135 53 22 154 45 29 134 49
4, 5) 0 160 37 0 155 34 0 159 26
(8023, 23) 0 128 0 0 128 0 0 128 0
23, 4) 101 0 27 101 0 27 99 0 29
5, 11) 32 0 32 25 0 35 24 0 23
5, 4) 55 104 0 54 103 0 58 113 0
5, 25) 0 195 0 0 195 0 0 195 0
11, 10) 11 94 25 8 83 37 10 84 24


Figure 7-4. Continued

7.4 Comparison of Static and Dynamic Transportation Management Systems

CORSIM 5.1 was used to simulate both static and dynamic cases. Again, because

CORSIM 5.1 simulates traffic conditions of a network with the feature that traffic

characteristics can change with time, the time-varying signal timing plans (shown in Figure

7-3) and traffic flows (shown in Figure 7-4) in the dynamic case were specified in a

sequence of time periods in a CORSIM 5.1 input file for simulation and animation. For the

static case, cumulative results were reported by CORSIM 5.1 every ten minutes. For the

dynamic case, CORSIM 5.1 produced its cumulative simulation results at the completion of

each time period. Figures 7-5 and 7-6 show the excerpts of the cumulative simulation

results, e.g.,vehicle miles and vehicle delay time, at the elapsed simulation time of 40

minutes for the static and dynamic cases, respectively.







83

To measure quantitatively and compare the quality of traffic service (i.e., frequency,

expediency, smoothness and safety) for both static and dynamic transportation management

systems, a Measure of Effectiveness (MOE) has to be used. The available MOEs include

delay, average vehicle speed, degree of saturation, number of stops, fuel consumption, etc.

Among these useful MOEs, delay time, which includes increased travel time from reduced

speed and time added due to traffic signal control, is a primary MOE used to evaluate the

performance of transportation systems.

Table 7-1 summarizes the cumulative network-wide vehicle delays at the end of each

10-minute time period for both the static and dynamic cases. The table shows the traffic

conditions were improved at the fourth time period for the dynamic case. At end of the one-

hour analysis period, for the sample network and the given traffic demands, the network-

wide delay was 1,102 vehicle-hours for the static case. For the dynamic case, CORSIM 5.1

simulated the six 10-minute time periods sequentially and the network-wide delay during the

one hour analysis period was 925 vehicle-hours. The fifth column in Figures 7-5 and 7-6

shows the vehicle delay for each link at the end of the fourth period for both cases. The

results show that the traffic conditions on some links were improved in the dynamic case but

some became worse. For example, for Link (12, 11), the total delay was 1,453 vehicle-

minutes in the static case and was improved to 338 vehicle-minutes in the dynamic case. For

Link (6,8), opposite results were observed. However, by summing up the vehicle delay on

each link, the network-wide delay in the dynamic case was improved by 43 vehicle-hours

at the end of the fourth time period and 150 vehicle-hours for the complete one-hour analysis

period.











Table 7-1


Figure 7-5 Excerpt of the


Cumulative Simulation Results at Elapsed Time of 40 Minutes


for the Static Case.


84

Cumulative Network-Wide Vehicle Delays at 10-Minute Time Intervals for
Static and Dynamic Cases

Elapsed Total Delay (vehicle-hours)

Simulation Time Static Case Dynamic Case

10 min. 98.93 101.22

20 min. 235.88 241.67

30 min. 409.56 407.22

40 min. 617.55 574.50

50 min. 846.67 763.45

60 min. 1,101.98 952.07


CUMULATIVE NETSIM RESULTS AT TIME 12:40: 0
ELAPSED TIME IS 0:40: 0 ( 2400 SECONDS)

VEHICLE MINUTES
VEHICLE MOVE DELAY TOTAL
LINK MILES TRIPS TIME TIME TIME

25, 5) 112.70 466 150.3 229.8 380.1
1, 6) 161.43 556 322.9 652.0 974.9
1, 2) 142.69 489 190.3 354.7 544.9
1, 21) 98.80 415 197.6 30.1 227.7
21, 1) 136.27 567 272.5 1375.6 1648.1
24, 8) 112.28 472 224.6 2418.8 2643.3
6, 8) 106.84 447 213.7 443.6 657.3
6, 13) 157.08 493 314.2 49.5 363.6
6, 7) 176.50 603 353.0 1196.6 1549.6
6, 1) 128.86 446 257.7 1272.0 1529.7
13, 6) 184.25 576 368.5 773.6 1142.2
14, 7) 182.54 621 365.1 1634.9 2000.0
8, 24) 106.57 448 213.1 46.6 259.7
8, 9) 105.38 361 210.8 382.0 592.8
8, 6) 115.46 484 230.9 389.6 620.5
12, 11) 123.99 520 248.0 1452.5 1700.5
2, 7) 152.29 529 227.1 791.7 1018.9
2, 1) 136.36 464 181.8 192.2 374.0
2, 3) 500.44 840 667.3 1467.5 2134.8
2, 15) 65.96 296 98.4 17.2 115.6
7, 9) 136.31 566 203.3 921.5 1124.8
7, 6) 135.46 462 270.9 837.1 1108.0
7, 14) 182.56 623 365.1 61.2 426.3
7, 2) 122.48 426 182.7 1020.2 1202.9












CUMULATIVE NETSIM RESULTS AT TIME 12:40: 0
ELAPSED TIME IS 0:40: 0 ( 2400 SECONDS)

VEHICLE MINUTES


VEHICLE
MILES TRIPS

111.01 459
156.78 540
148.76 510
66.76 281
143.00 595
117.27 493
116.30 487
153.26 481
154.91 529
106.06 366
183.93 575
155.79 530
103.00 433
109.11 374
91.69 383
114.22 479
140.03 487
118.53 404
474.88 797
100.15 450
120.32 500
111.97 382
127.90 436


MOVE DELAY TOTAL
TIME TIME TIME


LINK

25,
1,
1,
1,
21,
24,
6,
6,
6,
6,
13,
14,
8,
8,
8,
12,
2,
2,
2,
2,
7,
7,
7,


500.5
1359.4
556.3
147.9
1223.8
2190.7
1279.3
358.1
1079.0
531.8
1122.7
2433.5
248.1
800.9
487.4
566.0
835.6
352.7
1409.4
178.8
1038.5
551.9
295.8


148.0
313.6
198.4
133.5
286.0
234.5
232.6
306.5
309.8
212.1
367.9
311.6
206.0
218.2
183.4
228.4
208.9
158.0
633.2
149.4
179.5
223.9
255.8


( 7, 2) 116.05 404 173.1 2201.1 2374.2


Figure 7-6 Excerpt of the Cumulative Simulation Results at Elapsed Time of 40 Minutes
for the Dynamic Case.


Figures 7-7 and 7-8 show the animations for a portion of the sample network at the


end of the one-hour period for the static and dynamic cases, respectively. Figure 7-7 shows


Link (6,7) in the static case had more serious spillback problems. Based on the results from


the numerical example, the dynamic traffic management system performed more efficiently,


by modifying the traffic conditions based on the signal settings and then adjusting the signal


timing settings according to the updated traffic flows six times during an hour than the static


traffic management system did for the same traffic facilities.


352.5
1045.9
357.9
14.4
937.7
1956.2
1046.7
51.6
769.1
319.7
754.8
2122.0
42.1
582.7
304.0
337.6
626.7
194.7
776.3
29.5
859.1
327.9
40.0
































Figure 7-7 Example of Animation Result for the Static Case.


Figure 7-8 Example of Animation Result for the Dynamic Case.















CHAPTER 8
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

8.1 Summary and Conclusions

Intelligent Transportation Systems (ITS), which have been a topic of substantial

research during the past decade, are being developed to improve the efficiency and

productivity of existing transportation facilities. The success of ITS depends on two

important sub-systems, Advanced Traffic Management Systems (ATMS) and Advanced

Traveler Information Systems (ATIS).

Real-time traffic control and dynamic traffic assignment are two major support

technologies of ATMS because the traditional static network equilibrium models which were

developed for long-term transportation planning, and the traditional traffic control

management systems which were developed to set traffic signals by assuming fixed flow

patterns, are not suitable for analyzing and solving transportation problems in real time.

A combined model has been developed to integrate the real-time traffic control model

and the dynamic traffic assignment model. For the real-time traffic control model, the

framework of the traditional traffic control model, which minimizes total delay, was

extended to the dynamic case by adding time as an additional dimension. However, the

conventional delay model, Webster's delay formula, which was originally derived through

theoretical queuing analysis for isolated intersections, predicts infinite values of delay when

flows approach capacity. Yet, in realistic situations, a queue will not grow infinitely because







88

drivers will change routes to avoid large queues. The generalized delay model for signalized

intersections in the 2000 Highway Capacity Manual was used in the real-time traffic control

model to take the oversaturated queue problem into account. A solution algorithm applying

the Frank-Wolfe method has been developed to implement the real-time traffic control

model. It uses TRANSYT-7F's static optimal signal settings as the initial solution to

increase the chance for the non-convex objective function to converge to a near-optimal

solution.

The dynamic traffic assignment model is formulated as a Variational Inequality (VI)

formula. Using the simulation model to calculate link travel times, a relaxation algorithm is

used to relax the asymmetric link travel time function resulting from the flow propagation

constraints. A solution algorithm has been developed to implement the equivalent

optimization model of the relaxed VI formula.

The iterative optimization assignment (IOA) procedure consisting of solving

sequentially an optimization model involving the signal timing variables, with the flow

variables fixed, and a user-optimized equilibrium model corresponding to the new green time

settings, is used to solve the combined model.

Applying the IOA procedure, a solution algorithm with operational capability to

solve dynamic network management models corresponding to the above technological

concepts in ITS has been developed. A comprehensive computer program has been prepared

to implement and test all algorithms associated with the dynamic transportation management

model.

CORSIM 5.1, the simulation software which allows traffic characteristics to be time-

varying, was used to demonstrate the performances of both static and dynamic cases using







89

a sample network and a given O-D demand matrix. Because of the limitations on resources

available to this study, a sample network instead of the real-world data was used to test the

procedure.

It is a common practice that multiple simulation runs, each specified with different

sets of random seeds, are performed when CORSIM or other simulation computer packages

are used. However, since the variation in the simulation results obtained by changing

random number seeds are generally much less than 13.6%, only one CORSIM simulation

run was performed in this study. The test results indicate that dynamic traffic assignment

with adaptive traffic-responsive signal settings reduced the network-wide delays about

13.6% by periodically modifying the traffic flows based on the traffic conditions and then

adjusting the signal timing settings according to time-dependent traffic flows.

This study applied mathematical programming methodologies to model traffic

assignment and traffic signal control systems as well as the dynamic interactions between

them. The dynamic transportation management system was modeled more realistically than

those under a static state. In addition, the mathematical solutions are proven to be optimal

and satisfy the equilibrium conditions in comparison with those developed solely based upon

simulation techniques. Incorporating time-dependent optimal signal settings into an existing

traffic control system can increase system efficiency by better assessing network conditions

and achieving better traffic control.

8.2 Recommendations

The travel time function is very important. The same traffic assignment policy

considered under different disutility assumptions may possess completely different

theoretical properties and thus may be expected to produce completely different traffic flow







90

results. The BPR formula is known to have the limitation of an ambiguous definition of

capacity in the function because the BPR formula increases travel times even after a link's

traffic flow is higher than its capacity. Further investigation into enhancements of the BPR

formula is recommended.

The dynamic queue-responsive traffic signal control with dynamic traffic assignment

technique developed in this research consists of a number of parameters, including the cycle

length, signal phase plan and O-D demand, that require input from users and were not

modified during the optimization. Unlike the simulation-based, real-time computer systems

which can simulate the traveler's behavior of changing departure time in response to

congestion conditions, the O-D demand is presumed fixed during the analysis period in the

mathematical programming model developed in this study. In other words, travelers would

not be able to change their departure time because of traffic conditions. This assumption is

practical for peak hour traffic since most roadway users do not have the privilege of

arbitrarily altering the time they begin work or complete their jobs for the day. Further

research is recommended to consider varied cycle lengths and signal phase sequences by

programming additional external looping into the optimization process.

In addition, the traffic persisting for an hour under the static state was separated into

six discrete 10-min time intervals for the example. The duration of each time period may

change the performance of a dynamic transportation management system. Intuitively, the

smaller the duration of each time period, the better the performance of the transportation

system since the optimal signal settings and traffic flows can be updated more frequently.

However, the marginal benefits for dividing hourly demand into shorter time periods may

be limited since it takes time for road users to respond to the traveler information they are