IEEE Proof IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS 1 Interval Macroscopic Models for Traffic Networks 1 Amadou Gning, Lyudmila Mihaylova, Senior Member, IEEE, and René K. Boel, Senior Member, IEEE 2 Abstract—The development of real-time traffic models is of 3 paramount importance for the purposes of optimizing traffic flow. 4 Inspired by the compositional model (CM) and the METANET AQ1 5 model, this paper proposes an interval approach for macroscopic 6 traffic modeling. We develop an interval CM (ICM) and an in- 7 terval implementation of the METANET model (IMETANET) 8 that provide a natural way of predicting traffic flows without 9 the assumption of uniform distribution of vehicles in a cell. The 10 interval macroscopic models are suitable for real-time applications 11 in road networks and can be part of road traffic surveillance and 12 control systems. The performances of the interval approaches are 13 investigated for both the ICM and the IMETANET models. The 14 efficiency of the interval models is demonstrated over simulated 15 data, and as well as over real traffic data from MIDAS data sets AQ2 16 from the United Kingdom. 17 Index Terms—Compositional model (CM), interval methods, 18 macroscopic models, METANET model, traffic modeling. 19 I. I NTRODUCTION 20 M ODELING traffic flow is important for both motorway 21 networks and urban traffic systems. For instance, traffic 22 models are necessary for planning, traffic light settings, the 23 design of online control, and traffic guidance systems. The 24 traffic models available in the literature can be classified into 25 three groups: 1) macroscopic; 2) microscopic; and 3) meso- 26 scopic [4], [5]. Macroscopic models require less computation 27 than microscopic models and are particularly suitable for real- 28 time applications like online traffic state estimation and control. 29 This is a strong motivation for considering macroscopic traffic 30 models. 31 The simplest macroscopic traffic models, as first proposed 32 by Lighthill and Whitham, express the conservation equations 33 of the vehicles traveling through the network in the form of 34 a first-order partial differential equation describing the time 35 evolution of the traffic density. Speed and traffic flow are 36 defined in these first-order models by an algebraic relationship 37 called the fundamental diagram. These first-order models were 38 extended by dynamical equations expressing the inertia of 39 platoons of vehicles, leading to the second-order partial differ- 40 ential equations also expressing the time evolution of the speed 41 Manuscript received December 10, 2008; revised November 14, 2009 and September 30, 2010; accepted January 6, 2011. This work was supported by the Engineering and Physical Sciences Research Council under Project EP/E027253/1, by the EU FP7 projects DISC and CON4COORD, and by the AQ3 IUAP DYSCO project. The Associate Editor for this paper was D.-H. Lee. A. Gning and L. Mihaylova are with the School of Computing and Communications, Lancaster University, LA1 4WA Lancaster, U.K. (e-mail: e.gning@lancaster.ac.uk; mila.mihaylova@lancaster.ac.uk). R. K. Boel is with the SYSTeMS group, University of Ghent, 9000 Ghent, Belgium (e-mail: rene.boel@ugent.be). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TITS.2011.2107900 (or equivalently of the flow) [6]. Many further refinements 42 to the model and an efficient numerical approximation of the 43 second-order partial differential equations have been imple- 44 mented in the METANET model [2], [3], which has become 45 a de facto standard for freeway traffic modeling. Nevertheless, 46 Daganzo [7], [8] has pointed out that there are inherent limi- 47 tations to the numerical approximations of second-order traffic 48 models (such as the inevitability that some conditions lead to 49 negative speed predictions). Therefore, Daganzo has proposed 50 in [7] a cell transmission model (CTM) that dynamically up- 51 dates only the evolution of the number of vehicles per cell along 52 a freeway that is decomposed in successive cells. 53 This also leads to reliable and relatively simple models of 54 freeway traffic. The fact that many traffic measurement systems 55 provide data, often even in real time, including the speed, led 56 Boel and Mihaylova [1] to propose a simple extension of the 57 CTM model called the compositional model (CM). In this 58 CM model, the speed inertia of vehicles moving from one 59 cell to the next cell is combined with the sending and re- 60 ceiving function of the CTM model, leading to a discrete- 61 time discrete-space second-order model that does not suffer 62 from the drawbacks that Daganzo pointed out for second-order 63 partial differential equations [8]. Moreover, the CM model 64 explicitly represents the randomness in the evolution of the 65 density and speed of the traffic, making this CM model suitable 66 for recursive Bayesian estimation methods [9]. This noise in the 67 sending and receiving functions describing the number of vehi- 68 cles crossing a cell boundary partially deals with the difficulty 69 that the sending function assumes implicitly that the vehicles 70 are uniformly distributed in the upstream cell. This uniform 71 distribution is also assumed in METANET [2], [3]. In reality, 72 vehicles cross cell boundaries in platoons, corresponding to a 73 possibly significant deviation of this uniform distribution. To 74 provide a better description of this uncertainty in the evolution 75 of traffic states, we propose in this paper an alternative to the 76 CM and METANET models using interval analysis [10], [11] 77 to represent uncertainty. 78 Interval methods and interval models have appealing proper- 79 ties that can be beneficial when applied to traffic phenomena. 80 Interval methods do not provide one single value for the traffic 81 variables, e.g., for the speed, flow, and density. Rather, interval 82 methods provide lower and upper bounds of possible values for 83 these variables in speed, flow, and density. Interval methods do 84 not rely on any assumption about the distribution of location 85 or speed of vehicles within traffic segments, but rather, they 86 express the range of values that is compatible with the model. 87 The main contribution of this paper is in the introduced 88 new class of traffic models using interval methods to represent 89 uncertainty in the distribution of vehicles in a cell. This new 90 approach is applied in this paper to both the CM model [1] 91 1524-9050/$26.00 © 2011 IEEE