An Equivalent Second Order Model Based on Flow and Speed X. Y. Lu*, P. Varaiya**, and R. Horowitz*** *PATH, U. C., Berkeley, Richmond Field Station, Bldg 4521357 S. 46th Street, Richmond, CA 94804-4648, USA (Tel:510-665 3644; e-mail: xylu@path.berkeley.edu). ** EECS, U. C. Berkeley, 271M Cory Hall, Berkeley CA 94720-1770 USA (e-mail: variya@eecs.berkeley.edu) *** ME, U. C. Berkeley, 5138 Etcheverry Hall, Berkeley CA 94720-1770 USA (e-mail: horowitz@me.berkeley.edu) Abstract: Traffic models developed for control include first order LWR model and second order models. The first order model is essentially density dynamics which states the conservation of the number of vehicles in a section of highway. Most second order models are essentially a speed dynamics coupled with the first order density dynamics. There are several second order models but most are for explaining traffic flow. Only a couple of second order models were developed for traffic control purpose. The most popular one is the Payne-Papageorgiou model. Due to point sensor detection limit in highways such as inductive loops, flow estimation is easy and reasonably accurate while density estimation is very difficult. This is the motivation for us to have a look at flow dynamics in place of density. It turns out that flow dynamics of first order cannot be well-defined contrasted to the Cell Transmission Model. However, a flow-speed dynamics can be obtained which is equivalent to the density-speed dynamics of the Payne- Papageorgiou model. Keywords: traffic flow, density and speed; traffic flow model; first order Cell Transmission Model; second order Payne-Papageorgiou model, 1. INTRODUCTION In recent years, model-based traffic control design has been becoming more and more popular. The analysis and control design of ramp metering based on the first order Cell Transmission Model (CTM) is one example (Muñoz et al, 2004; Gomes and Horowitz, 2006). Another example is to use a second order model for combined Variable Speed Limit and Coordinated Ramp Meter control design in (Papageorgiou, 1983, 1990), Papamichail et al (2008), and Hegyi (2005). This paper is to addresses some modelling issue based on practical traffic control design considerations. The CTM is essentially a density dynamics. Thus corresponding control using ramp metering is to control the average density in each cell. For the control design using second order model, the density ρ and distance mean speed v are the state variables. By definition, density is a distance concept which, in principle, can be estimated by the vehicle count instantly captured by video camera for the stretch of highway involved. However, in practice, video camera is not easy to install and maintain. Most traffic sensors in the road are dominantly the inductive loops which provide point measurement. High density loop detector would give a better estimation of density but it is cost prohibitive. It is well- known that loop, particularly a dual loop station, has a good measurement of vehicle count in unit time, or flow. A question naturally arises: is it possible to use flow instead of density as state variable in traffic dynamics for control design? This paper provides an answer to this question. It turns out that the flow cannot be the state in first order model, but can be coupled with speed to form a second order model. The paper is organized as follows: section 2 is literature review; section 3 deduce the flow dynamics; section 4 discuss the possibility for flow to be a state in first order model, and coupled with speed dynamics to for a second order model; and section 5 is conclusions. 2. LITERATURE REVIEW There are at least three traffic modelling approaches in the literature for macroscopic traffic modelling. (1) Based on the physics of fluid flow – traffic is considered as compressible flow. Representative work in this approach is the well-known LWR model Lighthill and Whitham (1955a, b) and Richards (1956), and later development by Newell (1993a, b, c), Daganzo (1995a, b, c) and Zhang (1998, 1999a, 1999b, 2000). A good collection and review of the kinematic wave models and its development history can be found in Gartner et al (2001). (2) Based on driver behaviour and intuitive understanding of traffic behaviour such as prediction and delay. Representative work in this direction is the model by Payne (1971) and the improvement by Papageorgiou (1983), which is called Payne- Papageorgiou model here. It is essentially a second model with coupled density and speed dynamics. Since then, several second order models have been obtained by some modification/improvement on the model. The model has been further improved in (Papageorgiou, 1990) by introducing the