#### MPEL Seminar

# Mean field games and their application to energy systems

Add to Google Calendar

Game theory has since long been the mathematical language used to analyze interacting multi-agent systems driven by often mutually conflicting objectives. Games quickly become computationally intractable when the number of interacting agents is large. This is because any change in the position of a single agent induces, in general, reactions from all the other agents. However, if as the number of agents grows, their individual influence on other agents becomes vanishingly small, then in the limit, a decoupling occurs whereby the group"s behavior gains inertia thus becoming insensitive to the isolated actions of an individual. At that point, each individual optimally reacts to some assumed immutable behavior of the group, and the infinite regress of mutual influences is broken. This is the situation analyzed in so-called mean field games, a recent theory independently and about the same time, developed in Canada and France. Besides optimal control theory, it relies on the ideas of statistical mechanics to help anticipate the aggregate group behavior under the optimal control laws of individuals.

The talk will be divided in two parts. In the first part, we draw inspiration from the motion of fish schools to develop a simplified engineering view of how collective motion can be achieved through decentralized control actions. In the process, the intuitive ideas underlying mean field game theory are brought forth. In the second part, we illustrate the application of this body of ideas to energy systems: An approach for aggregator coordinated decentralized control of large numbers of heating-cooling electric loads is presented. It is meant as an environmentally friendly way of helping to mitigate the increased generation volatility brought about by a higher penetration of wind and solar energy sources in today grids' generation mixes.

This is joint work with Arman Kizilkale and Rabih Salhab.

Roland Malhamé received the Bachelor's, Master's and Ph.D. degrees in Electrical Engineering from the American University of Beirut, the University of Houston, and the Georgia Institute of Technology in 1976, 1978 and 1983 respectively.

After single year stays at University of Quebec , and CAE Electronics Ltd (Montreal), he joined in 1985 ‰cole Polytechnique de Montréal, where he is Professor of Electrical Engineering. In 1994, 2004, and 2012 he was on sabbatical leave respectively with LSS CNRS (France), ‰cole Centrale de Paris, , and University of Rome Tor Vergata.

His interest in statistical mechanics inspired approaches to the analysis and control of large scale systems has led him to contributions in the area of aggregate electric load modeling, and to the early developments of the theory of mean field games. His current research interests are in collective decentralized decision making schemes, and the development of mean field based control algorithms in the area of smart grids and communication systems. From june 2005 to june 2011, he headed GERAD, the Group for Research on Decision Analysis. He is an Associate Editor of International Transactions on Operations Research, and IEEE Transactions on Automatic Control.