### Mobile Users

## Traffic state estimation

Motorway traffic state estimation and prediction are central components in real-time traffic management/control and information applications (Figure 1 schematically outlines a model-based traffic decision-support system, which performs three closely intertwined tasks: traffic state estimation, traffic state prediction and the optimization of traffic control measures). In road networks, traffic states usually refer to traffic related quantities, such as travel time, traffic speed, traffic flow and density. These quantities reflect traffic conditions. Based on which, we might know how many vehicles there are on motorways, when traffic becomes busy, where and when a traffic congestion/accident occurs, and etc. However, in real traffic, it is not able to get the full picture of traffic states from the current sensor systems. Due to the cost and technical constraint, we can only obtain the spatial and temporal discretized traffic data. Therefore, the essence of traffic state estimation is to reproduce motorway traffic conditions based on limited amount of traffic measurements.

In general, model-based traffic state estimators consist of a dynamic model for the state variables which is used to describe the basic mechanism of traffic flow (e.g. a first or second-order macroscopic traffic flow model); a set of observation equations relating sensor observations to the system state (e.g. a fundamental diagram) and a data assimilation technique to combine the model predictions with the sensor observations (e.g. the extended Kalman filter).

## Lagrangian formulation

Traffic flow operations can be observed from fixed points (e.g. loop detectors, cameras). Alternatively, observers can travel along with the traffic flow (e.g. GPS, mobile phone). These two observer “modes” relate to two different coordinate systems: the Eulerian (space x – time t) coordinates and the Lagrangian (vehicle number n – time t). Commonly, both traffic dynamic and observation models are formulated in Eulerian (space-time) coordinates. Recent studies show that the dynamic model can be formulated and solved more efficiently and accurately in Lagrangian coordinates. Meanwhile, the Lagrangian formulation provides a natural set of observation equations to deal with GPS data (floating car data) which are becoming popular nowadays.

## Main contribution and outcomes

This research is to promote a new model-based state estimator based on the Extended Kalman Filtering (EKF) technique, in which the discretized Lagrangian first-order traffic model is used as the process equation, and in which observation models for both Eulerian and Lagrangian sensor data (from loop detectors and probe vehicle trajectories respectively) are incorporated. This Lagrangian state estimator is validated and compared to an Eulerian state estimator based on the same traffic flow model using an empirical microscopic traffic dataset from part of the managed motorway M42 in the UK.

Figures 2, 3 and 4 show some graphical results based on the Eulerian and the Lagrangian state estimation with the Eulerian sensing data. In both simulations, a shock (low speed) is observed by a local detector. In the Lagrangian estimation (see Figure 4), the shock propagation is clearly visible in the result, whereas the shock diffuses quickly further upstream in the Eulerian case (see Figure 3). Moreover, the edges of vehicle platoons clearly distinguish the shock boundary, whereas a step-wise boundary (in a resolution determined by the grid-size) is present in the Eulerian simulation. These figures illustrate that indeed the numerical method in Lagrangian formulation causes less numerical diffusion and more accurate results than the Eulerian method.

This research indicates that the Lagrangian traffic state estimator outperforms the Eulerian approach. First of all, the Lagrangian formulation enables more accurate simulation. Secondly, the Lagrangian formulation of the traffic model also leads to better data assimilation.