MODELING UNCERTAINTY IN LARGE-SCALE URBAN TRAFFIC NETWORKS

Abstract

Recent work has proposed using aggregate relationships between urban traffic variables—

i.e., Macroscopic Fundamental Diagrams (MFDs)—to describe aggregate traffic dynamics in urban networks. This approach is particularly useful to unveil and explore the effects of various networkwide control strategies. The majority of modeling work using MFDs hinges upon the existence of well-defined MFDs without consideration of uncertain behaviors. However, both empirical data and theoretical analysis have demonstrated that MFDs are expected to be uncertain due to inherent instabilities that exist in traffic networks. Fortunately, sufficient amounts of adaptive drivers who re-route to avoid congestion have been proven to help eliminate the instability of MFDs. Unfortunately, drivers cannot re-route themselves adaptively all the time as routing choices are controlled by multiple factors, and the presence of adaptive drivers is not something that traffic engineers can control. Since MFDs have shown promise in the design and control of urban networks, it is important to seek another strategy to mitigate or eliminate the instability of MFDs. Furthermore, it is necessary to develop a framework to account for the uncertain phenomena that emerges on the macroscopic, network-wide level to address these unavoidable stochastic behaviors.

This first half of this work investigates another strategy to eliminate inherent network instabilities and produce more reliable MFDs that is reliable and controllable from an engineering perspective—the use of adaptive traffic signals. A family of adaptive signal control strategies is examined on two abstractions of an idealized grid network using an interactive simulation and analytical model. The results suggest that adaptive traffic signals should provide a stabilizing influence that provides more well-defined MFDs. Adaptive signal control also both increases average flows and decreases the likelihood of gridlock when the network is moderately congested. The benefits achieved at these moderately congested states increase with the level of signal adaptivity. However, when the network is extremely congested, vehicle movements become more constrained by downstream congestion and queue spillbacks than by traffic signals, and adaptive traffic signals appear to have little to no effect on the network or MFD. When a network is extremely congested, other strategies should be used to mitigate the instability, like adaptively routing drivers. Therefore, without sufficient amounts of adaptive drivers, the instability of MFDs could be somewhat controlled, but it cannot be eliminated completely. This is results in more reliable MFDs until the network enters heavily congested states.

The second half of this work uses stochastic differential equations (SDEs) to depict the evolutionary dynamics of urban network while accounting for unavoidable uncertain phenomena. General analytical solutions of SDEs only exist for linear functions. Unfortunately, most MFDs observed from simulation and empirical data follow non-linear functions. Even the most simplified theoretical model is piecewise linear with breakpoints that cannot be readily accommodated by the linear SDE approach. To overcome this limitation, the SDE well-known solutions are used to develop an approximate solution method that relies on the discretization of the continuous state space. This process is memoryless and results in the development of a computationally efficient Markov Chain (MC) framework. The MC model is also supported by a well-developed theory which facilitates the estimation of future states or steady state equilibrium conditions in a network that explicitly accounts for MFD uncertainty. Due to the fact that current formalization of Markov Chains is restricted with a countable state space, some assumptions which redefine the traffic state and stochastic dynamic process need to be set for the MC model application in dynamic traffic analysis. These assumptions could be sabotaged by inappropriate parameter selections, producing excessive errors in analytical solutions. Therefore, a parametric study is performed here to illustrate how to select two key parameters, i.e. bin size and time interval to optimize the MC models and minimize errors.

The major advantage of MC models is its wide flexibility, which has been demonstrated by showing how this method could well handle a wide variety of variables. A family of numerical tests are designed to include instability of MFD model, stochastic traffic demand, different city layouts and different forms of MFDs in the scenarios under static metering strategies. The results suggest that analytical solutions derived from MC models could accurately predict the future traffic state at any moment. Furthermore, the theoretical analysis also illustrates that Markov chains could easily model dynamic traffic control based on traffic state and pre-determined time-varying strategies by adjusting the transition matrix. Overall, the developed MC models are promising in the dynamic analysis of complicated urban network control under uncertainty for which simpler algebraic solutions do not exist.

 

 

Table of Contents

List of Figures ………………………………………………………………………………………………………….. viii

List of Tables …………………………………………………………………………………………………………… xi

Acknowledgement ……………………………………………………………………………………………………. xii

Introduction ………………………………………………………………………………………………. 1

Literature Review ………………………………………………………………………………………. 5

2.1 Aggregated Modeling of Urban Traffic Networks …………………………………………….. 5

2.1.1 Older Models ……………………………………………………………………………………… 5

2.1.2 MFD …………………………………………………………………………………………………. 9

Concept of MFD ………………………………………………………………………………….. 9

Well-defined Urban MFD …………………………………………………………………….. 12

Freeway MFD ……………………………………………………………………………………… 14

Adaptive Signals ………………………………………………………………………………….. 16

2.2 Traffic Control with MFD ……………………………………………………………………………… 18

2.2.1 Perimeter Flow Control (Metering/Gating) …………………………………………….. 18

2.2.2 Road Congestion Pricing ……………………………………………………………………… 20

2.2.3 Network Design & Routing ………………………………………………………………….. 22

Research Objectives …………………………………………………………………………………… 24

MFD Model with Adaptive Signals Control ………………………………………………….. 27

4.1 Idealized Urban Network Description……………………………………………………………… 27

4.1.1 Idealized Grid Network ……………………………………………………………………….. 27

4.1.2 Two-Ring Abstraction …………………………………………………………………………. 31

4.1.3 Two-Bin Abstraction …………………………………………………………………………… 31

4.2 Two-Bin Analysis ………………………………………………………………………………………… 32

4.2.1 Analytical Description of the Two-Bin Model ………………………………………… 32

4.2.2 Stability of Adaptive Signal Control Only ……………………………………………… 36

4.2.3 Stability of Adaptive Drivers Routing Only ……………………………………………. 40

4.2.4 Stability with Adaptive Signals and Drivers …………………………………………… 42

4.3 Two-Ring Simulation ……………………………………………………………………………………. 43

4.3.1 Two-Ring Simulation Model………………………………………………………………… 43

4.3.2 Stability without Adaptive Signals Control …………………………………………….. 45

4.3.3 Stability with Adaptive Signals Control Only …………………………………………. 464.3.4 Stability with Adaptive Drivers Routing Only ………………………………………… 48

4.3.5 Stability Test with Adaptive Signals and Drivers ……………………………………. 49

4.4 Grid Network Simulation ………………………………………………………………………………. 50

4.4.1 Idealized Grid Network Description ……………………………………………………… 50

4.4.2 Stability in Idealized Network ………………………………………………………………. 51

4.4.3 Stability with a More Realistic Network ………………………………………………… 53

Illustration of New Traffic Dynamics with Uncertainty ………………………………….. 55

5.1 Control Strategy Selection …………………………………………………………………………….. 55

5.2 Problem Scenario of Single-Region Network …………………………………………………… 57

5.3 Analysis under No Uncertainty (G0 = 0; Gf = 0) ……………………………………………. 61

5.4 Analysis with Uncertainty (G0 ≠ 0; Gf = 0 ) ………………………………………………….. 63

5.4.1 Analytical Solution using Stochastic Differential Equation for Linear NEFs

………………………………………………………………………………………………………….. 64

5.4.2 Numerical Simulation Tests …………………………………………………………………. 68

5.4.3 Setting Optimal Metering Rate …………………………………………………………….. 72

Markov Chain (MC) Model for a Single Region ……………………………………………. 76

6.1 MC Model Application …………………………………………………………………………………. 766.2 Insights from MC Model and Extensions ………………………………………………………… 80

6.3 Numerical Simulation Tests …………………………………………………………………………… 83

6.3.1 Triangular NEF Model ………………………………………………………………………… 84

Type I Uncertainty ……………………………………………………………………………….. 84

Type I & II Uncertainty ………………………………………………………………………… 88

6.3.2 Nonlinear NEF Model …………………………………………………………………………. 92

Type I Uncertainty ……………………………………………………………………………….. 92

Type I & II Uncertainty ………………………………………………………………………… 96

Parametric Study ……………………………………………………………………………………….. 100

7.1 Parametric Analysis ……………………………………………………………………………………… 100

7.2 Numerical Simulation Tests …………………………………………………………………………… 106

7.2.1 Triangular NEF Model ………………………………………………………………………… 1077.2.2 Nonlinear NEF Model …………………………………………………………………………. 113

Conclusions ………………………………………………………………………………………………. 119

8.1 Summary of Major Findings ………………………………………………………………………….. 119

8.2 Future Work ………………………………………………………………………………………………… 121

REFERENCES ………………………………………………………………………………………………………… 122

 

 

Introduction

Urban traffic congestion is a significant problem in the United States, particularly densely populated areas with over 1 million inhabitants. The costs of congestion in the US are severe: in 2011, congestion accounted for more than 5.5 billion hours of wasted productivity and 2.9 billion gallons of wasted fuel with a total estimated cost to society of $121 billion (Schrank et al., 2012). Furthermore, the worsening congestion has been a real threat to quality of life in terms of how it affects people’s daily schedules. In cities with more than 3 million inhabitants, for instance, the average commuter suffers about 6 hours of congested road conditions during an average weekday (Schrank et al., 2012).

Building additional transportation infrastructure is not an optimal way to mitigate this congestion problem not only due to the high monetary costs, but because additional infrastructure construction also generates new travel demand (the induced demand phenomenon). This means that adding new roadway capacity will cause more people to travel, and this new travel demand might still exceed the expanded capacities of the renewed traffic network. Furthermore, building sufficient traffic infrastructure for peak-hour travel demands is also a waste of resources during the off-peak hours when it goes unused. Instead, an alternative to capacity expansion is the implementation of various traffic control strategies to alleviate the imbalance between increasing traffic demands and inability of the transportation infrastructure to support drivers. Such strategies, combined with advanced traveler information systems, communications and sensors, enable drivers to make better-informed decisions and allow existing transportation networks to be used more

efficiently.

Traffic control strategies operate on a variety of scales within urban transportation networks. At the local level are strategies designed to improve traffic conditions at specific locations—e.g., a specific intersection or an arterial road. These strategies require a tremendous amount of detailed data to facilitate their design and to estimate their performance. Such data are possible to collect at individual locations through manual or automatic sources (such as inpavement sensors and detectors). Reliable traffic models to describe traffic dynamics at the local scale have also been developed and refined for quite some time, including detailed car following models, the LWR theory of kinematic waves, and cell transmission models (Gipps, 1981; Lighthill and Whitham, 1955; Richards, 1956; Daganzo, 1994).

These aforementioned models can also be used to describe detailed traffic dynamics throughout an entire network to test large-scale traffic control strategies (e.g., a comprehensive coordinated signal timing plan that impacts multiple intersections). The network model is typically constructed by modeling each of its individual components. However, the process of collecting enough data to use each detailed model to assess the network-wide impacts of large-scale control strategies generally makes these approaches infeasible. Those approaches are also very time- and resource-intensive since very large urban networks are often made up of thousands or millions of individual agents, links and nodes, and each of these must be modeled individually. Furthermore, those approaches make it more difficult to see big picture trends and insights that are often lost among the complex analysis of detailed data.

Further complicating this is the fact that congestion is not a local phenomenon: queue spillovers from nearby junctions means that traffic in one region is impacted by traffic conditions in neighboring regions. In other words, congestion is actually a large-scale phenomenon, especially recurring congestion that occurs during busy peak hours. Thus, a network-level perspective is required when designing and implementing large-scale traffic control strategies. Such a perspective of studying large-scale traffic control strategies is particularly useful and has shown promise in several urban settings across the world. Examples of large-scale traffic control include pricing entire busy downtown regions (known more commonly as congestion pricing), which has recently been implemented in London, Stockholm and Milan (de Palma and Lindensy, 2011), car entry restrictions in congested areas, for instance, in Singapore and Beijing (Zhang et al., 2003), and zonal-based adaptive signal control, such has been applied in Zurich (Benesty and Huang, 2003).

To alleviate this concern, recent work has developed aggregated traffic models that can be used to directly study and model large-scale network dynamics, and these models are useful to both develop and test various network-wide traffic control strategies. This model is commonly known as the Macroscopic Fundamental Diagram (MFD), and provides a relationship between average network density and flow on an urban traffic network. The former metric describes the accumulation of vehicles in the network and is analogous to how busy the network currently is, while the latter describes the productivity of the network. The model is able to describe in a physically realistic fashion all traffic states that may arise in a urban traffic network, from nearly empty networks that are not productive due to lack of demand to highly congested networks that are not productive because they are too busy. As will be described in the next chapter, recent work has shown that when a reliable and reproducible MFD model of an urban network exists, it can be used to feasibly describe traffic dynamics in a way that is sufficient to analyze various traffic management schemes, such as pricing strategies or perimeter flow control between distinct neighborhoods or regions. However, the existence of a reproducible and well-defined MFD is not universally expected. According to field data, these relationships are highly stochastic such that multiple levels of network productivity (or average flow) are observed for a given accumulation (or average density). Furthermore, simulated and empirical MFDs exhibit more complex phenomena such as multivaluedness and hysteresis behavior. Overall, these phenomena serve to decrease the applicability of MFDs for the development and analysis of large-scale traffic control strategies.

Besides the endogenous instability that exists within MFDs, other types of uncertainty also arise during the application of traffic control schemes that further limit the applicability of using MFDs to analyze large-scale control policies. Such random behavior might be caused by limitations of our ITS technologies and the randomness of individual driver behavior. Take the traffic signal control on the perimeter of urban network as an example. Existing research has used the existence of a well-defined MFD to develop perimeter flow control plans that limit the rate at which vehicles enter congested urban centers to ensure that overall network productivity is always maximized. In theory, it should then be fairly easy to carefully control traffic signals on the perimeter of the network in an attempt to achieve the desired vehicle entry rates. However, the time-varying nature of travel demand and the presence of aggressive and timid drivers would cause actual entries into the network to randomly fluctuate around the desired values. These factors could result in unexpected overall behavior and less efficient control strategies if this stochastic behavior is not explicitly accounted for in any modeling representation.

Thus, the purpose of this dissertation work is 1) to examine in more detail some of the features that might impact the uncertainty that is endogenous to the MFD and 2) to develop a probabilistic modeling framework that can explicitly incorporate this uncertainty (and other types of uncertainty) into large-scale network representations of urban traffic networks. The former will help unveil conditions under which well-defined MFDs might be expected in practice, which would allow transportation engineers the ability to model networks using this novel approach. The latter will then provide engineers with the tools necessary to adequately develop traffic control strategies that account for inherent uncertainties that arise on a macroscopic scale, which will hopefully lead to more efficient and effective large-scale traffic control strategies.

MODELING UNCERTAINTY IN LARGE-SCALE URBAN TRAFFIC NETWORKS

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