An integral control formulation of Mean-field game based large scale coordination of loads in smart grids

Pressure on ancillary reserves, i.e.frequency preserving, in power systems has significantly mounted due to the recent generalized increase of the fraction of (highly fluctuating) wind and solar energy sources in grid generation mixes. The energy storage associated with millions of individual customer electric thermal (heating-cooling) loads is considered as a tool for smoothing power demand/generation imbalances. The piecewise constant level tracking problem of their collective energy content is formulated as a linear quadratic mean field game problem with integral control in the cost coefficients. The introduction of integral control brings with it a robustness potential to mismodeling, but also the potential of cost coefficient unboundedness. A suitable Banach space is introduced to establish the existence of Nash equilibria for the corresponding infinite population game, and algorithms are proposed for reliably computing a class of desirable near Nash equilibria. Numerical simulations illustrate the flexibility and robustness of the approach

Schr\"odinger approach to Mean Field Games with negative coordination

Mean Field Games provide a powerful framework to analyze the dynamics of a large number of controlled agents in interaction. Here we consider such systems when the interactions between agents result in a negative coordination and analyze the behavior of the associated system of coupled PDEs using the now well established correspondence with the non linear Schr\"odinger equation. We focus on the long optimization time limit and on configurations such that the game we consider goes through different regimes in which the relative importance of disorder, interactions between agents and external potential varies, which makes possible to get insights on the role of the forward-backward structure of the Mean Field Game equations in relation with the way these various regimes are connected

Algorithms for Optimal Paths of One, Many, and an Infinite Number of Agents

In this dissertation, we provide efficient algorithms for modeling the behavior of a single agent, multiple agents, and a continuum of agents. For a single agent, we combine the modeling framework of optimal control with advances in optimization splitting in order to efficiently find optimal paths for problems in very high-dimensions, thus providing alleviation from the curse of dimensionality. For a multiple, but finite, number of agents, we take the framework of multi-agent reinforcement learning and utilize imitation learning in order to decentralize a centralized expert, thus obtaining optimal multi-agents that act in a decentralized fashion. For a continuum of agents, we take the framework of mean-field games and use two neural networks, which we train in an alternating scheme, in order to efficiently find optimal paths for high-dimensional and stochastic problems. These tools cover a wide variety of use-cases that can be immediately deployed for practical applications