# An adaptive control algorithm for traffic-actuated signalized networks

TABLE OF CONTENTS

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LIST OF FIGURES……….…………………………………………….………………vi LIST OF TABLES……….…………………………………………...…………………ix ACKNOWLEDGEMENT………..……………………………….……………………..x CURRICULUM VITAE……………………………………….….……………………xi ABSTRACT………………………………………….....……………………………...xiii CHAPTER 1 INTRODUCTION….…………………………………………………….1 1.1 Overview…………………………………………………………………………..1 1.2 Research Objective and Contribution……………………………………………..4 1.3 Dissertation Outline……………………………………………………………….7 CHAPTER 2 BACKGROUND…………………..……………………………………...8 2.1 General Issues regarding Online Optimal Control………………………………...8 2.1.1 Literature review of signal-controlled street models……………...………8 2.1.2 Literature review of intersection control delay models…………….……12 2.1.3 Literature review of on-line optimization techniques……………………21 2.2 Examples of Online Signal Control Algorithms…………………………………24 2.2.1 Centralized traffic-responsive systems…………………………………..25 2.2.2 Distributed traffic-adaptive systems……………………..........................28 2.3 Description of Traffic-Actuated Control………………………………………...32 2.3.1 Phasing parameters of traffic-actuated controllers………………………32

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2.3.2 T iming parameters of traffic-actuated controllers……………………….36 2.3.3 Timing logic of traffic-actuated controllers……………………………...42 2.3.3.1 Timing logic of each phase………………………………………42 2.3.3.2 Timing logic of the signal control cycle…………………………47 CHAPTER 3 METHODOLOGY……………….……………………………………..53 3.1 Introduction………………………………………………………………………53 3.2 Consideration of General Issues regarding Online Optimal Control…………….55 3.2.1 Modeling the traffic flow between signalized intersections……………..55 3.2.2 Specifying the real-time optimal control objective………………………60 3.2.3 Designing the on-line optimization technique…………………………...64 3.3 Dynamically Recursive Optimization Procedure……………………………...…67 3.3.1 Overview………………………………………………………………....67 3.3.2 Data processing…………………………………………………………..68 3.3.3 Flow prediction…………………………………………………………80 3.3.4 Parameter optimization…………………………………………………..84 3.3.4.1 Determining optimal phase sequence……………………………84 3.3.4.2 Determining optimal minimum green…………………………....94 3.3.4.3 Determining optimal unit extension……………………………...95 3.3.4.4 Determing optimal maximum green..............................................97 CHAPTER 4 TESTING AND EVALUATION………………………………………98 4.1 Introduction………………………………………………………………………98 4.2 Simulation………………………………………………………………………..99 4.2.1 Study network……………………………………………………………99

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4.2.2 P arameter specification and adjustment………………………………...100 4.2.3 Scenario setup…………………………………………………………..101 4.2.4 Specification of measure of performance………………………………103 4.2.5 Performance analysis…………………………………………………104 4.3 Conclusion……………………………………………………………………...112 CHAPTER 5 CONCLUSION AND FUTURE WORK…………………………..…114 5.1 Conclusion of the Research…………………………………………………….114 5.2 Future Work…………………………………………………………………….116 REFERENCES………………………………………………………………………...125 APPENDIX…………………………………………………...………………………..134

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LIST OF FIGURES

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F igure 2-1 Uniform Delay Estimation..…………………………………………………14 Figure 2-2 Uniform and Overflow Delay Estimation…………………………………...15 Figure 2-3 Shock Wave Analysis under Under-Saturated Conditions………………….17 Figure 2-4 Shock Wave Analysis under Over-Saturated Conditions…………………...18 Figure 2-5 Concept of Time-Dependent Stochastic Models…………………………….20 Figure 2-6 Concept of Rolling Horizon Scheme………………………………………..23 Figure 2-7 Detector Configuration of SCATS…………………………………………..26 Figure 2-8 Detector Configuration of SCOOT………………………………………….28 Figure 2-9 Detector Configuration of OPAC……………………………………………30 Figure 2-10 Detector Configuration of RHODES………………………………………31 Figure 2-11 Designation of Phase Numbers…………………………………………….33 Figure 2-12 Dual-Ring Control Structure……………………………………………….34 Figure 2-13 Definition of Floating Cycle……………………………………………….35 Figure 2-14 Phase Sequence…………………………………………………………….35 Figure 2-15 Detector Configuration of Traffic-Actuated Control………………………37 Figure 2-16 Maximum Green Start Moment……………………………………………43 Figure 2-17 Unit Extension Start Moment………………………………………………44 Figure 2-18 Gap Seeking Logic…………………………………………………………46 Figure 2-19 Timing Process of Signal Control Cycle…...………………………………48

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Figure 2-20 Concept of Dual-Entry Operation………………………………………….49 Figure 2-21 Concept of Simultaneous Gap-Out………………………………………...51 Figure 2-22 Concept of Conditional Service……………………………………………52 Figure 3-1 Description of Approaching Flow…………………………………………...56 Figure 3-2 Distribution of the Approaching Vehicles based on Groups………………...58 Figure 3-3 Distribution of the Approaching Vehicles based on Directions……………..60 Figure 3-4 Queue Accumulation Curves………………………………………………..62 Figure 3-5 Combined Queue Accumulation Curves…………………………………….64 Figure 3-6 Modified Rolling Horizon Scheme………………………………………….66 Figure 3-7 Recursive Optimization Procedure…………..………………………………68 Figure 3-8 Phase State…………………………………………………………………...69 Figure 3-9 Queue Accumulation Diagram………………………………………………70 Figure 3-10 Gap-Out Situations…………………………………………………………75 Figure 3-11 Max-Out Situations………………………………………………………...79 Figure 3-12 Expression of Red Duration………………………………………………..89 Figure 4-1 Study Network……………………………………………………………...100 Figure 4-2 Set-back Extension Detector Configuration………………………………..101 Figure 4-3 Study Intersection…………………………………………………………..107 Figure 4-4 Flow Profile at Study Intersection………………………………………….109 Figure 4-5 Vehicle Spillover Profile…………………………………………………...110 Figure 5-1 Detector Configuration of Semi-Actuated Control………...………………118 Figure 5-2 Detector Configuration of Volume-Density Control………………………119 Figure 5-3 Timing Logic of Volume-Density Control…………………………………120

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Figure 5-4 Behavior of Vehicle Group along the Street………………………………122 Figure 5-5 Turning Fractions from Different Vehicle Groups………………………...123

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LIST OF TABLES

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T able 4-1(a) Simulation results of the network (Scenario 1)…………………………..105 Table 4-1(b) Simulation results of the network (Scenario 2)…………………………..105 Table 4-1(c) Simulation results of the network (Scenario 3)…………………………..105 Table 4-2 Performance of the network…………………………………………………106 Table 4-3 Parameters for the study intersection………………………………………..107 Table 4-4 Simulation results of the intersection (Scenario 1)………………………….108 Table 4-5 Performance of the intersection (Scenario 1)……………………………….112

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ACKNOWLEDGEMENT

I sincerely appreciate my advisor and study leader, Professor Wilfred W. Recker, for his academic guidance, mentorship and contribution to this work. This work would not have been completed without his exemplary motivation and encouragement. I also appreciate my committee members, Professor Michael G. McNally and Professor R. Jayakrishnan, for their valuable assistance in academic research and many other aspects during my five years of study here. Special thanks are also owed to Dr. Lianyu Chu for his technical support which inevitably made this work a success. My fruitful learning and achievement are based on their thoughtful and patient guidance, encouragement and support.

The success of my Ph.D. program would not have been possible without the continued consideration, encouragement and support of my family. I hereby would like to devote this dissertation to my parents and my wife. Their valuable helps are gratefully thanked.

Finally, thanks also go to all ITS friends and entities for their support during my five years of research and study here. Especially I need to thank Xu Yang, Yu Zhang, Lei Wang, Junping Duan, Shin-Ting Jeng and Chih-Lin Chung for their help.

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CURRICULUM VITAE

X ing Zheng

Education

2007 – 2010 Ph.D. in Civil Engineering (Traffic Engineering) University of California – Irvine, Irvine, California, U.S.A. 2005 – 2006 M.S. in Civil Engineering (Traffic Engineering) University of California – Irvine, Irvine, California, U.S.A. 1999 – 2003 B.S. in Civil Engineering (Highway and Urban Road Engineering) Chongqing Jiaotong University, Chongqing, China

Experience

2005 – 2010 Graduate Student Researcher Institute of Transportation Studies, University of California – Irvine, Irvine, California, U.S.A. 2008 – 2009 Traffic Simulation Analyst (Volunteer) CLR Analytics, Irvine, California, U.S.A. 2005 – 2005 Grader Department of Civil and Environmental Engineering, University of California – Irvine, Irvine, California, U.S.A.

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2004 – 2005 Designer H uaxi Jiaotong Design Company, Chengdu, Sichuan, China 2003 – 2004 Research Assistant Chongqing Jiaotong University, Chongqing, China

Training and Skill

Simulation: Visum/Vissim, Paramics, TransCAD/TransModeler Programming: C/C++, Visual Basic Software: Synchro, AutoCAD, Microsoft Office

Publication

1. Zheng, X. , Recker, W. and Chu, L. (2010). Optimization of Control Parameters for Adaptive Traffic-Actuated Signal Control, Journal of Intelligent Transportation Systems, 14 (2), 95 – 108.

2. Zheng, X. and Chu, L. (2008). Optimal Parameter Settings for Adaptive Traffic- Actuated Signal Control. In: Proceedings of the 11 th International IEEE Conference on Intelligent Transportation Systems, Beijing, China.

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ABSTRACT

A N ADAPTIVE CONTROL ALGORITHM FOR TRAFFIC-ACTUATED SIGNALIZED NETWORKS

By

Xing Zheng

Doctor of Philosophy in Civil Engineering University of California, Irvine, 2010 Professor Wilfred W. Recker, Chair

With advances in computation and sensing, real-time adaptive control has become an increasingly attractive option for improving the operational efficiency at signalized intersections. The great advantage of adaptive signal controllers is that the cycle length, phase splits and even phase sequence can be changed to satisfy current traffic demand patterns to a maximum degree, not confined by preset limits. To some extent, traffic- actuated controllers are themselves “adaptive” in view of their ability to vary control outcomes in response to real-time vehicle registrations at loop detectors, but this adaptability is restricted by a set of predefined, fixed control parameters that are not adaptive to current conditions. To achieve the functionality of truly adaptive controllers, a set of online optimized phasing and timing parameters are needed.

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This dissertation proposes a real-time, on-line control algorithm that aims to maintain the adaptive functionality of actuated controllers while improving the performance of signalized networks under traffic-actuated control. To facilitate deployment of the control, this algorithm is developed based on the timing protocol of the standard NEMA eight- phase full-actuated dual-ring controller. In formulating the optimal control problem, a flow prediction model is developed to estimate future vehicle arrivals at the target intersection, the traffic condition at the target intersection is described as “over-saturated” throughout the timing process, i.e., in the sense that a multi-server queuing system is continually occupied, and the optimization objective is specified as the minimization of total cumulative vehicle queue as an equivalent to minimizing total intersection control delay. According to the implicit timing features of actuated control, a modified rolling horizon scheme is devised to optimize four basic control parameters—phase sequence, minimum green, unit extension and maximum green—based on the future flow estimations, and these optimized parameters serve as available signal timing data for further optimizations. This dynamically recursive optimization procedure properly reflects the functionality of truly adaptive controllers. Microscopic simulation is used to test and evaluate the proposed control algorithm in a calibrated network consisting of thirty-eight actuated signals. Simulation results indicate that the proposed algorithm has the potential to improve the performance of the signalized network under the condition of different traffic demand levels.

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CHAPTER 1 INTRODUCTION

1.1 O verview

Traditionally, traffic signals at a signalized intersection operate in one of two different control modes: either pre-timed or traffic-actuated (both semi-actuated and full-actuated). In pre-timed control, all of the control parameters, including cycle length, phase splits and phase sequence, are preset offline based on an assumed deterministic demand level at different time periods of day. This control mode has only a limited ability to accommodate the traffic fluctuations that are commonly found in reality. In traffic- actuated control, cycle length, phase splits and even phase sequence can be changed in response to the real-time vehicle actuations registered at loop detectors or other traffic sensors, but these changes are still subject to a set of predefined, fixed control parameters (e.g., minimum green, unit extension and maximum green, etc.) that are not accordingly responsive to the varying traffic condition.

Alternatively, an on-line control algorithm that decides real-time signal operation parameters offers a potential improvement to actuated control. Existing on-line control algorithms that have been deployed are typified by the so-called third-generation urban traffic control systems (UTCS), which can be further categorized into centralized traffic- responsive systems and distributed traffic-adaptive systems. In the centralized system, a master computer is used to adjust the cycle lengths, offsets and splits of all signals on a

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cycle-by-cycle basis, such that a “best” timing plan can be found for the operation of the entire network. In the distributed system, separate calculations are taken to determine the phase sequences and durations of each signal, with an attempt to optimize the performance of each local intersection. To some extent, actuated controllers are themselves “adaptive” in view of their ability to vary the same set of outcomes as in adaptive control, but as mentioned previously, this adaptability is restricted by a set of predefined, fixed control parameters that are not adaptive to current conditions. To achieve the functionality of truly adaptive controllers, a set of online optimized phasing and timing parameters are needed.

In general, three issues must be addressed in formulating an on-line optimal control problem: (1) development of a mathematical model that represents the current, or expected, traffic condition of the controlled system; (2) specification of the real-time control objective that can be expressed as a certain performance index; (3) design of an appropriate optimization technique such that the controlled system meets the specified criteria. In addition, the development and adoption of on-line control procedures have been hampered by two fundamental impediments to their successful implementation: (1) the theoretically-sound algorithms generally are specified in terms of those parameters and control options that are not simply within the lexicon of control devices and typically involve complex programming formulations (e.g., mixed-integer and/or piece-wise functions) that do not lend themselves to real-time solution, and (2) the practically feasible algorithms that do manipulate those parameters employed in modern control devices almost universally are formulated based on highly simplified approximations and

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assumptions (e.g., steady-state condition) to both control response and traffic measurement.

In consideration of these aspects, this dissertation introduces an on-line control algorithm that aims to maintain the adaptive functionality of actuated controllers while improving the performance of the signalized network under traffic-actuated control. In the interest of facilitating deployment, this algorithm is developed based on the timing process of the standard NEMA eight-phase full-actuated dual-ring controller. In formulating the optimal control problem, a flow prediction model is developed to estimate the future vehicle arrivals at the target intersection, the traffic condition at the target intersection is described as “over-saturated” through the whole timing process, and the optimization objective is specified as to minimize total cumulative vehicle queue as an equivalent to minimizing total intersection control delay. According to the implicit timing features of actuated control, a modified rolling horizon scheme is devised to optimize the values of four basic control parameters—phase sequence, minimum green, unit extension and maximum green—based on the future flow estimations, and these optimized parameters serve as available signal timing data for further optimizations. This dynamically recursive optimization procedure properly reflects the functionality of truly adaptive controllers. Microscopic simulation is used to test and evaluate the proposed control algorithm in a calibrated network consisting of thirty-eight actuated signals. Simulation results indicate that the proposed control algorithm has the potential to improve the performance of the signalized network under conditions arising from different traffic demand levels.

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1.2 R esearch Objective and Contribution

The objective of this research is to develop, test and evaluate (via microscopic simulation) a real-time, online control algorithm that aims to maintain the adaptive functionality of actuated controllers while improving the performance of the signalized network under traffic-actuated control. To be consistent with the operation logic of existing signal control devices, this algorithm is developed to optimize the basic control parameters that can be found in modern actuated controllers. The implicit timing features of actuated control are fully exploited to infer a relatively rich body of information that can be used in adapting the operation of an actuated controller to the current, or expected, traffic condition. By doing this, we hope to ensure that the procedures developed herein can be implemented with minimal adaptation of existing field devices and the software that controls their operation.

The major contributions of this research can be summarized as follows:

1. A traffic flow prediction model that fully utilizes the available signal timing data and phase state is developed to estimate the future vehicle arrival flow rate at the target intersection. Specifically, this model first calculates the average approaching flow toward the downstream intersection based on the vehicle departure numbers from upstream signal phases, and then estimates the turning fraction associated with each downstream signal phase based on the vehicle arrival flow rates in previous cycles at the downstream intersection. The future vehicle arrival flow rate for each signal

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phase at the downstream (i.e., the target) intersection is determi ned by multiplying the approaching flow and the corresponding turning fraction.

2. An intersection control delay model is developed to estimate the real-time total intersection control delay. A major difference between this model and other well- known intersection delay models is that, in this model, all signal control phases are considered simultaneously, and the intersection is considered to be “over-saturated” throughout the timing process. Therefore, the total intersection control delay for a certain period is approximately equal to the integral of the difference between the total arrival and departure flow rates (which is equivalent to the cumulative vehicle queue) over this period.

3. Based on the implicit timing features of actuated control, a modified rolling horizon scheme is devised to minimize the total intersection control delay, which serves as the optimal control objective of the proposed algorithm. A major advantage of this modified rolling horizon scheme is that the project horizon, head portion, tail portion and roll period are all time-variant since they are actually determined in response to real-time traffic conditions. This advantage mitigates those shortcomings underlying the traditional rolling horizon scheme that adopts pre-set and fixed projection and rolling intervals. Additionally, in the modified rolling horizon scheme, the optimized control parameters (especially the unit extension) can serve as a feed-back mechanism that has the ability to adjust signal timing in response to real-time vehicle

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actuations, and, according to the setting of roll period, the optimiz ation frequency is increased to be as twice that employed in the traditional cycle-by-cycle procedure.

4. A non-linear optimal control problem is formulated with the solution being the optimal phase sequence and a set of optimal phase splits for all signal control phases. These optimal phase splits are not parameter settings but serve as a reference for the decision of the optimal minimum green, unit extension and maximum green for each signal phase. In this research, in order that the adaptive signal control algorithm developed herein can be implemented with minimal adaptation of existing field devices and the software that controls their operation, only those four basic control parameters that can be found in modern actuated controllers are considered as decision variables.

5. The proposed adaptive control algorithm is tested and evaluated within a scalable, high-performance microscopic simulation package, PARAMICS. Users can access the core functions provided in PARAMICS through API (Application Programming Interfaces) to customize and extend many features of the underlying simulation model without having to deal with the proprietary source codes. In this research, the proposed adaptive control algorithm is successfully programmed into a plug-in through API programming and applied to a calibrated signalized network, which also serves as an example of realizing a user-developed optimal control strategy in PARAMICS.

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1.3 D issertation Outline

This dissertation consists of five chapters and is organized as follows. Chapter 2 states the background of this research, which includes the expatiation of the general issues regarding on-line optimal control problems, the literature review of existing state of the art on-line signal control algorithms, and the description of the general traffic-actuated signal control strategy. Chapter 3 describes in detail the methodology of the proposed adaptive control algorithm for traffic-actuated signalized networks. This methodology addresses the general issues regarding on-line optimal control problems, and it can be illustrated as a dynamically recursive optimization procedure. In Chapter 4, the proposed adaptive control algorithm is tested and evaluated through microscopic simulation, and the performance of a calibrated network under this adaptive control is compared with that under the free-mode traffic-actuated control. Chapter 5 concludes this dissertation by summarizing the research and indicating the advantages and limitations of the proposed algorithm. Recommendations for future work are also provided.

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CHAPTER 2 BACKGROUND

2.1 G eneral Issues regarding Online Optimal Control

In general, three issues must be addressed in formulating an online optimal control problem: (1) development of a mathematical model that represents the current, or expected, traffic condition of the controlled system; (2) specification of the real-time control objective that can be expressed as a certain performance index; (3) design of an appropriate optimization technique such that the controlled system meets the specified criteria.

2.1.1 Literature review of signal-controlled street models

Mathematical models that represent the traffic condition of the controlled system, which in this research corresponds to the traffic phenomena on signalized street networks, can be classified into the following three generalized categories (Pavlis and Recker, 2004): (1) store-and-forward model; (2) dispersion-and-store models; (3) kinematic wave models. Store-and-forward models are essentially adopted from the theory of communication networks. In this modeling approach, it is assumed that vehicles entering a link travel at a constant travel time, and are either stored at the end of this link in case of red signal, or further forwarded to downstream links at saturation flow rate during the time of green. This approach was first suggested to represent the traffic condition at oversaturated

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intersections (Gazis and Potts, 1963), and has since been used in various works notably for street network control problems (e.g., Hakimi, 1969; Singh and Tamura, 1974; D’Ans and Gazis, 1976, Aboudolas et al, 2009).

Dispersion-and-store models are based on empirical observations regarding the traffic flow behavior after the onset of green interval. This modeling approach describes the behavior of platoons of vehicles that enter a street link and are dispersed until they are uniformly distributed on the street stretch. The dispersed platoon is subsequently either stored at the end of the link when the signal turns to red, or further diffused on the downstream link when the signal stays in green. A number of dispersion models have been developed to describe the behavior of platoons between signalized intersections, typical of which being the normal distribution model (Pacey, 1956), as expressed in Eq. (2-1),

( ) ( ) ( ) ( ) ( ) 1 2 2 2

exp 2 2 j d u i q j q i g j i a T v a g T S T S π = = − − = − ∑ (2-1) where

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( ) ( ) upstream flow rate at time i downstream flow rate at time j travel time between upstream and downst ream points distance from the upstream to the downs tream point average vehicle speed u d q i q j T a v S = = = = = standard deviation of vehicle speed=

a nd the geometric distribution model (Robertson, 1969), as expressed in Eq. (2-2). This modeling approach has also been applied and extended in several street traffic control strategies (e.g., Cremer and Schoof, 1989; Chang et al., 1994).

( ) ( ) ( ) ( ) 1 1 1 1 d u d q j F q j T F q j F T β αβ = ⋅ − + − ⋅ − = + (2-2) where ( ) ( ) upstream flow rate at time interval j downstream flow rate at time interval j travel time between upstream and downst ream points dispersion factor travel time factor u d q j q j T α β = = = = =

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Kinematic wave models are derived through an analogy with hydrodyn amic theory and assumptions based on empirical evidence. This modeling approach was developed based on the well-known LWR model (Lighthill and Whitham, 1955; Richards, 1956), in which traffic flow behaviors are characterized using flow, speed and density, as shown in Eq. (2-3).

i i i j i ij j i v k u v v SW k k = ⋅ − = − (2-3) where traffic flow rate in road section i traffic density in road section i traffic speed in road section i speed of shock wave between road sectio n i and j i i i ij v k u SW = = = =

In their discretized forms, it is assumed that the street link is divided into a number of segments. The corresponding flow rate at the segment’s boundary is the minimum between the saturation flow rate, the desired departing flow rate at the upstream segment, and the flow capacity associated with the downstream segment. The application of kinematic wave models can also be found in several street signal control strategies (e.g., Stephanedes and Chang, 1993; Lo, 2001).

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Generally, when developing a signal-controlled stree t model, there is a trade-off between the model’s consistency with the real-life process, and the difficulty in solving the corresponding optimization problem. For example, kinematic wave models are capable of describing the traffic flow on signalized streets under different traffic conditions, but their major disadvantage in the formulation of optimal control problems is the creation of large dimensional state vectors due to the discretization of the street link into a number of segments. Consequently, these models lead to the impracticality of their solution in real- time and render their practical implementation virtually impossible. As another example, the optimal signal control strategies based on store-and-forward models usually assume that oversaturated conditions prevail. The control variables are the green per cycle ratios given a cycle of fixed duration, so that the outflow discharge is calculated as the product of the saturation flow rate and the green per cycle ratio. In their formulations, the traffic signal operation is not explicitly modeled, and the oversaturation assumption restricts the applicability of the control strategy only to a single-ring, two-phase, fixed cycle controller.

2.1.2 Literature review of intersection control delay models

The most common objective for real-time signal control strategies is to design a signal timing plan that minimizes the total (or, average) intersection control delay, which serves as the primary performance indicator of level of service at signalized intersections. Existing intersection control delay models are summarized into four categories (Dion et

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al., 2004): (1) deterministic queuing models, (2) sh ock wave models, (3) steady-state stochastic models, and (4) time-dependent stochastic models.

The classic deterministic queuing models were the first theoretical approaches that estimate the control delay at signalized intersections. These models view the traffic on each intersection approach as a uniform stream of vehicles that arrive at a constant rate, accelerate and decelerate instantaneously, and queue vertically at the stop line. Refer to Figure 2-1 as an example showing how deterministic queuing models estimate control delay at an under-saturated intersection. The arrival and departure curves represent the corresponding cumulative number of vehicle arrivals and departures respectively for a certain traffic movement (i.e., signal phase). Since deterministic queuing models consider no randomness of traffic that may cause vehicle queue at the end of green interval in under-saturated cases, the vehicles arriving at the intersection during each control cycle can always be cleared before the return of red signal. Therefore, the total uniform delay per cycle associated with the target signal phase can be represented by the shaded area between the arrival and departure curves, and, through algebraic manipulations, it can be determined by Eq. (2-4). Further, the average uniform delay, which is determined by dividing the total delay by the number of vehicles processed during the cycle, is expressed in Eq. (2-5).

( ) 2 2 1 1 2 1 uniform e v D C g C v s = − − (2-4)

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( ) ( ) 2 1 2 1 e uniform C g C D d v C v s − = = ⋅ −

(2-5) where total uniform delay average uniform delay cycle length effective green time vehicle arrival flow rate saturation flow rate uniform uniform e D d C g v s = = = = = =

Figure 2-1 Uniform Delay Estimation

g e

Arrival Curve (slope = v) Departure Curve (slope = s, v or 0) Time

Cumulative Vehicles C r e

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F igure 2-2 Uniform and Overflow Delay Estimation

Under over-saturated conditions, the number of vehicles arriving at the intersection exceeds the number of vehicles that can be served by the traffic signal. This situation causes a residual vehicle queue to keep growing, as illustrated in Figure 2-2. The total control delay associated with this situation consists of two components: the uniform delay corresponding to the area between the departure curve and the capacity curve, and the overflow delay corresponding to the area between the capacity curve and the arrival curve, where the capacity curve represents the cumulative number of vehicle departures at capacity (denoted by c). Since all queuing vehicles depart at saturation flow rate under over-saturated conditions, the average uniform delay can be determined by substituting (g e /C)·(v/c) = v/s and inserting v/c = 1.0 into Eq. (2-5), which yields ( ) 1 2 2 e e uniform C g C C g d − − = = (2-6) g e

Capacity Curve (slope = c) Time Arrival Curve (slope = v) Departure Curve (slope = s or 0)

Cumulative Vehicles C

r e

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Suppose the analysis period of the over-saturated cond ition is equal to T, then the total overflow delay can be determined by

( ) ( ) 2 1 2 2 overflow T D T vT cT v c = − = − (2-7)

And, the average overflow delay per vehicle can be determined by dividing the total overflow delay by the number of vehicles discharged during time T, i.e. cT, then

( ) 1 2 overflow T d v c = − (2-8)

Shock wave models are developed based on an analogy with fluid dynamics, in which traffic flow behavior is characterized using flow, speed and density. A fundamental research contribution regarding this description is the LWR model (Lighthill and Whitham, 1955; Richards, 1956), and based on this work, many researchers have investigated and analyzed the dynamics of queue formation and dissipation at signalized intersections (e.g., Rorbech, 1968; Stephanopoulos and Michalopoulos, 1979; Michalopoulos et al., 1980). A number of real-time signal control strategies have been developed based on shock wave theory that minimize total intersection control delay subject to the constraints regarding maximum queue length (e.g., Michalopoulos et al. 1981; Michalopoulos and Pisharody, 1981).

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The major difference between shock wave and determin istic queuing models is that shock wave models consider horizontal vehicle queue (versus the vertical queue assumption used by deterministic queuing models), which enables the capturing of more realistic queuing behavior and the determination of maximum queue length. Figure 2-3 illustrates the shock wave analysis associated with the under-saturated condition, where vehicles are also assumed to arrive at a uniform rate and accelerate and decelerate instantaneously. The total travel time spent by all vehicles going through the intersection can be estimated using the density and flow rate associated with each area, which is represented by a certain style of vehicle trajectory. Since the signal control delay represents the added travel time caused by traffic signal operations, the total delay incurred within one signal control cycle can be estimated by calculating the difference between the total travel time with and without traffic signals.

Figure 2-3 Shock Wave Analysis under Under-Saturated Conditions r e