Specifications of Fundamental Diagrams for Dynamic Traffic Modeling Andy H. F. Chow 1 ; Ying Li 2 ; and Konstantinos Gkiotsalitis 3 Abstract: This paper studies different specifications of fundamental diagrams for first-order dynamic traffic modeling. The fundamental diagram, which refers to the relationship between traffic flow and density at a given location, plays an important role in modeling the dy- namics of traffic propagation. Many forms of fundamental diagrams have been proposed since the seminal work of Greenshields 80 years ago. Nevertheless, empirical validation of fundamental diagrams, in particular, for dynamic modeling purposes, is limited. The study is conducted with an extensive dataset collected from the U.K. M25 motorway. This paper highlights the difficulty in modeling congested traffic with single-valued fundamental diagrams and fine-grained loop detector data. Consequently, this paper first analyzes and discusses the effect of data granularity on model calibration and application. Following this, a set-valued fundamental diagram modeling approach with fine-grained data is also presented. This paper contributes to the development and validation of dynamic traffic simulation tools. DOI: 10.1061/(ASCE) TE.1943-5436.0000781. © 2015 American Society of Civil Engineers. Author keywords: Motorway Incident Detection and Automatic Signalling (MIDAS); Quantile regression; Data granularity; Set-valued fundamental diagrams; Robust control. Introduction This paper investigates the use of fundamental diagrams for macro- scopic dynamic traffic modeling. The importance and value of ef- fective traffic management have recently been revealed empirically in Chow et al. (2014). Having an efficient and reliable model of traffic flow is the prerequisite for the development of an effective transport management system (Kotsialos and Papageorgiou 2001). From a macroscopic perspective, traffic models can be distin- guished into two approaches: first order and second order. Lighthill and Whitham (1955) and Richards (1956) proposed a pioneering first-order macroscopic model of traffic flow, which is now known as the kinematic wave model or Lighthill-Whitham-Richards (LWR) model. The kinematic wave assumes the traffic flow fðx; tÞ at each time t and location x adjusts instantaneously to the asso- ciated traffic density ρðx; tÞ through a predefined fundamental diagram. Because of the assumption of this instantaneous adjust- ment, the LWR model is regarded as a first-order model that does not consider the transient process in traffic state transition. The LWR model has been one of the most widely accepted models of traffic due to its plausibility and simplicity. The principle of the LWR model led to the development of numerous modeling and optimization tools, including Daganzo (1994), Lo (1999), Gomes and Horowitz (2006), Kurzhanskiy and Varaiya (2010), and many others. A major criticism of the LWR model is its implicit assumption of unrealistically high acceleration and deceleration of vehicles and its incapability of capturing complex phenomenon such as capacity drop and stop-and-go wave. In an attempt to rem- edy this deficiency of first-order models, Payne (1971) developed a second-order modeling approach that considers the transient dy- namics of acceleration and deceleration, and drivers’ reaction time during the state transition. The principle of Payne’ s second-order model has also led to the development of a number of modeling and optimization tools, most notably the METANET model by Papageorgiou et al. (1990). Nevertheless, Daganzo (1995) identi- fies several deficiencies of second-order traffic models, including the possibility of violation of causality and negative flows. More- over, calibration of a second-order model will involve the determi- nation of additional parameters such as acceleration/deceleration rate and drivers’ reaction time. The settings of these additional var- iables may mask the significance of the core fundamental diagram. Consequently, this study will adopt the first-order model and the analysis of the additional model parameters and implications for second-order modeling are left for future study. No matter for first- or second-order model, the underlying fun- damental diagram of traffic, i.e., the flow-density relationship in equilibrium, plays a vital role in representing the resulting flow dy- namics. Many different forms of fundamental diagrams have been proposed over the years since the seminal work of Greenshields (1935). Kerner and Rehborn (1996) show the three distinct features (free-flow, synchronized flow, and wide-moving jam) of traffic that a fundamental diagram should possess. Gomes and Horowitz (2006) discuss the importance of incorporating the capacity drop phenome- non in the fundamental diagram for control design purposes. Chiabaut et al. (2009) present an estimation of fundamental dia- grams with traffic data collected from measurements along moving observer paths. Heydecker and Addison (2011) proposed that a fundamental diagram should contain a convex region at high traffic densities to capture the stop-and-go dynamics in congestion. Re- cently, Carey and Bowers (2012) reviewed and summarized a set of desirable properties that fundamental diagrams should possess. Empirical validation of a fundamental diagram, in particular, for dynamic traffic modeling, is very limited in the literature. This study fills the gap by presenting an empirical study on different specifications of a fundamental diagram for dynamic traffic 1 Lecturer, Centre for Transport Studies, Univ. College London, London WC1E 6BT, U.K. (corresponding author). E-mail: andy.chow@ucl.ac.uk 2 Ph.D. Student, Centre for Transport Studies, Univ. College London, London WC1E 6BT, U.K. E-mail: ying.lixin.11@ucl.ac.uk 3 Research Associate, NEC Laboratories Europe, 69115 Heidelberg, Germany. E-mail: konstantinos.gkiotsalitis@neclab.eu Note. This manuscript was submitted on January 15, 2014; approved on March 4, 2015; published online on May 4, 2015. Discussion period open until October 4, 2015; separate discussions must be submitted for indivi- dual papers. This paper is part of the Journal of Transportation Engineer- ing, © ASCE, ISSN 0733-947X/04015015(11)/$25.00. © ASCE 04015015-1 J. Transp. Eng. J. Transp. Eng. Downloaded from ascelibrary.org by University College London on 05/12/15. Copyright ASCE. 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