Abstract—Freeway traffic state estimation is one of the central components in real-time traffic management and information applications. Recent studies show that the classic kinematic wave model can be formulated and solved more efficiently and accurately in Lagrangian (vehicle number-time) coordinates. This paper investigates the opportunities of the Lagrangian form for state estimation. The main advantage for state estimation is that in Lagrangian coordinates, the numerical solution scheme is reduced to an upwind scheme. We propose a new model-based extended Kalman filter (EKF) state estimator where the discretized Lagrangian model is used as the model equation. This state estimator is applied to freeway traffic state estimation and validated using synthetic data. Different filter design specifications with respect to measurement aspects are considered. The achieved results are very promising for subsequent studies. I. INTRODUCTION HE kinematic wave model (referred to as first-order traffic flow model or LWR model) is generally formulated in Eulerian (space x – time t) coordinates [1, 2]. In the LWR model traffic flow operations are fully described by the dynamics of vehicle density k. The speed v and the flow q are (instantaneously) related to k by means of the so-called fundamental diagram q = Q e (k) = kV e (k). This first-order model is appealing because of its simplicity, parsimony, and its robustness in replicating basic traffic features [3, 4]. In Eulerian coordinates perturbations in a traffic stream may travel in both the upstream and downstream directions, depending on the prevailing traffic conditions. As a result, numerical solution approaches (e.g. the Godunov scheme [5]) need to switch between upwind and downwind schemes. Different authors have proposed traffic state estimators based on extended Kalman filters (EKF), using the discretized (Eulerian) kinematic wave model as the process model [6, 7]. The fundamental diagram, relating the system state to observable quantities as spot speed (v) and flow (q), is used as the observation equation. However, due to the upwind/downwind numerical scheme, the Eulerian process model is highly non-linear. This makes the justification (that the process model can be locally linearized) of the EKF approach questionable, and its implementation cumbersome, particularly when traffic Manuscript received Oct. 26, 2010. The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement No. INFSO-ICT- 223844, and supported by the Dutch Technology Foundation STW, applied science division of NWO and the Technology Program of the Ministry of Economic Affairs under project DCB.7814 "Reversing the Traffic Jam". Yufei Yuan, J.W.C. van Lint, T. Schreiter and S.P. Hoogendoorn are with the Delft University of Technology, Fac. of Civil Engineering. (e-mail: y.yuan@tudelft.nl , j.w.c.vanLint@tudelft.nl ) J.L.M. Vrancken is with the Delft University of Technology, Fac. of Technology, Policy and Management. (e-mail: j.l.m.vrancken@tudelft.nl ) operates near capacity. Recent studies show that the kinematic wave model can be formulated and solved more efficiently and accurately in Lagrangian coordinates. Leclercq et al. [3, 4] state that the main advantage of the Lagrangian approach is its exactness when the fundamental diagram is triangular. In [8], it is also shown there is less numerical diffusion when solving the LWR model in Lagrangian coordinates. In the Lagrangian coordinate system, traffic characteristics only move in the downstream direction (in the direction of increasing vehicle number instead of space), regardless of prevailing traffic conditions. The numerical solution method then simplifies to the upwind scheme. As a result, in the Lagrangian case the assumption that the process model can be locally linearized is more justifiable than in the Eulerian case. The linearization yields a more accurate approximation. In this paper, we propose a new model-based EKF state estimator where the discretized Lagrangian traffic flow model is used as the process equation. This state estimator is applied to freeway traffic state estimation and validated using synthetic data from a microscopic freeway simulation tool. Different filter design specifications with respect to measurement aspects, such as observation model, detection type and noise parameters, are considered. The paper is organized as follows. Section II first introduces the modeling of freeway traffic in Lagrangian coordinates. Section III addresses the design of the freeway traffic state estimator based on the EKF technique and the Lagrangian traffic flow model, where one of the applications of the state estimator to a (simplified) freeway stretch is discussed. In Section IV, the setup of a simulated experiment is presented, on the basis of which the proposed state estimator will be validated and evaluated in Section V. Finally, the conclusions and recommendations for further research are outlined in Section VI. II. TRAFFIC MODELING OF A FREEWAY IN LAGRANGIAN COORDINATES In this section, the Lagrangian kinematic wave model is introduced, followed by its numerical method of solution and the state space model of the Lagrangian traffic flow model. Next the modeling of traffic measurement in the Lagrangian system is discussed. A. The Continuous and Discretized Lagrangian Kinematic Wave Model The Lagrangian kinematic wave model consists of two equations [3, 4 and 8]: 0 = ∂ ∂ + ∂ ∂ n v t s , (Lagrangian Conservation Law) (1) Yufei Yuan, J.W.C. van Lint, S.P. Hoogendoorn, J.L.M. Vrancken, T. Schreiter Freeway Traffic State Estimation using Extended Kalman Filter for First-order Traffic Model in Lagrangian Coordinates T 2011 International Conference on Networking, Sensing and Control Delft, the Netherlands, 11-13 April 2011 978-1-4244-9573-3/11/$26.00 ©2011 IEEE 121