# Traffic models in large-scale urban network

Gipsa-lab is a CNRS research unit joint with Grenoble-INP (Grenoble Institute of Technology), and Université Grenoble Alpes (Grenoble Alpes University). It has agreements with INRIA, Observatoire des Sciences de l'Univers de Grenoble. With 350 people, including about 150 doctoral students, Gipsa-lab is a multidisciplinary research unit developing both basic and applied researches on complex signals and systems. Gipsa-lab is internationally recognized for the research achieved in Automatic Control, Signal and Images processing, Speech and Cognition. The research unit develops projects in the strategic areas of energy, environment, communication, intelligent systems, Life and Health and language engineering. Thanks to the research activities, Gipsa-lab maintains a constant link with the economic environment through a strong partnership with companies. Gipsa-lab staff is involved in teaching and training in the various universities and engineering schools of the Grenoble academic area (Grenoble Alpes University).

Research is achieved in Gipsa-lab thanks to 12 research teams organized in 3 departments : Automatic control, Images-signal, Speech-cognition.

Gipsa-lab regroups 150 permanent staff and around 250 no-permanent staff (Phd, post-dotoral students, visiting scholars, trainees in master…)

Scale-FreeBack is an ERC hosted by the CNRS. The project will be conducted within the NeCS group ( a joint CNRS (GIPSA-Lab)-INRIA team), at Grenoble France. Scale-FreeBack is a project with the overall aim of developing holistic scale-free control methods of controlling complex network systems in the widest sense, and to set the foundations for a new control theory dealing with complex physical networks with an arbitrary size, see scale-freeback.eu

This research proposal deals with the problem of modeling and validating urban traffic network at an aggregated level. In this framework a field of research concentrates on two dimensional PDE models while another group of works concentrates on the notion of Macroscopic Fundamental Diagram (MFD). Starting with some empirical observation of traffic in a city, "Existence of urban-scale macroscopic fundamental diagram" show that it is possible to exhibit a relation between the average density and the average flow over a whole network. This result enables the introduction of accumulation models —also called reservoir models — which consist of representing the traffic state of a network with a single scalar field variable representing the total number of vehicles in the network. These models are practical because they are understandable, with few parameters to tune and a low computational cost. However, they contain little information about the traffic states. For example, they are not able to describe precisely where vehicles are located over the reservoir. This problem was later on addressed in some papers in which the authors separated different areas of the city with different reservoirs. Other models show that traffic in urban areas can be modeled with two-dimensional continuous and dynamic models. These models represent the traffic density as a variable over a 2D-plane. Such models are based on a two-dimensional conservation law and a review of some of these model have been done by "Dynamic traffic assignment using the macroscopic fundamental diagram: a review of vehicular and pedestrian flow models" . As 2D models are recent, there is little validation or calibration of these models. A first challenge in testing 2D models is to obtain a two-dimensional density function from real traffic data. In particular, the reconstruction of a density in the 2D-plane from vehicle data on the road network needs to be defined properly.

In the project we developed 2D-LWR model (which includes 2D wave propagations. This model can be seen as a natural extension to 2D of the well-known CTM. This was the first 2D model with a geometry-dependent flux where the magnitude depends on the density and the direction depends on space. However, at the current stage, the model is able to represent only monodirectional flow. The main goal of this post-doc is to extend this model to multidirectional flow (probably using a multi-layer approach) and to validate the model using synthetic and real data.

Several specific task will be expected:

- Extend our previous model to a 2-D multilayer PDE model for a large-scale urban traffic systems based on the 2D-LWR model ideas

- Starting from real data, recover the function in the PDE that models the flux function and the interaction between cars using inverse problems.

- Validate the model using a microscopic simulator

- Perform experiments in our micro-simulator to verify the aggregation process, and the validity of the detailed model.

Field tests and other realistic simulations to validate the theory will be performed using the equipment available at the Grenoble Traffic Lab center (see GTL), that is currently being extended at the level of city-center of Grenoble (GTL-Ville project) where we are collecting traffic related data and constructing a real-time data-collection systems. The algorithms developed in this work, will be integrated into the GTL-Ville project. Experiments that cannot be realized in vivo, will be tested on a microscopic traffic simulator replicating the full complexity of the Grenoble urban network.