Mean Field Games in Energy Systems Applications

Roland Malhame
École Polytechnique de Montréal

Solar and wind energy power sources, while ubiquitous in many areas around the world, are plagued with intermittency and uncontrollable variability. This has emerged as the single most formidable challenge to their massive penetration in the energy mix of power systems. Indeed, the operation of the latter requires a constant balancing between power generated and load. This is normally achieved via controllable generation sources be they fossil fuel based, or hydroelectric. While storage, in the form of large batteries can partly alleviate the problem, more and more hope is directed at controlling the other side of the equation, namely the load. One particularly suitable candidate for control is the class of thermal loads (e.g. electric space heating /cooling loads, or electric water heating loads), which because they are associated with energy storage can be partially deferred or anticipated as a device for respectively facing a renewable power deficit, or a not immediately usable renewable power surge.

The objective of using the customer aggregate load as an effective storage device for smoothing anticipated solar or wind energy fluctuations translates in mathematical terms into a large scale, small loads (occasionally in the millions) coordination problem. While the challenge, both in terms of computations and communications, is nearly insurmountable if one approaches it in a centralized fashion, it turns out that this control problem has a very natural formulation as a prescriptive mean field game.

We consider the class of heating-cooling loads and start, as is customary in MFG formulations, from a microscopic individual stochastic load dynamic description. It is assumed that the loads are to be coordinated by an aggregator who solves an optimization problem at the macroscopic aggregate load level, given a forecasted renewable energy profile. The result of this optimization is a sequence of either (i) target mean energy content levels (i.e. mean target temperatures) or (ii) target mean power consumption levels over a sequence of discrete time intervals. Individual users are prescribed a cost function designed to be a compromise between their individual comfort objectives and the need to fulfill aggregate objectives. Both problems are formulated as non standard linear quadratic Gaussian MFGs. (i) is solved using cost coefficients in the LQG which involve an integral of the mean temperature tracking error; (ii) is solved using for the first time we believe an inverse Nash cost formulation approach where the time varying cost coefficients in the LQG are computed as solutions of a systems of ordinary differential equations leading to a desired equilibrium trajectory. Existence results are given, and numerical results are presented.

This is joint work with Arman Kizilkale, Rabih Salhab and Quentin Lenet.

Presentation (PDF File)

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