Recursive estimation of common partially observed disturbances in MFG systems with application to large scale power markets

Abstract

Power markets are modelled as dynamic large population games where suppliers and consumers submit their bids in real-time. The agents are coupled in their dynamics and cost functions through the price process. Here, a common unpredictable partially observed major agent is added to the system which represents common unpredictable disturbance factors (e.g. wind) and exogenous market factors (e.g. competing energy resource prices), etc. In previous work, the Mean Field Game (MFG) methodology was used to study the limit (Le., infinite population) behaviour of large population market systems without a major agent; this results in a decentralized algorithm where agents submit their bids solely using statistical information on the dynamics of the entire population. When a major agent is absent, the system exhibits the standard counter intuitive property of MFG solutions that agents need not observe the behaviour (i.e., inputs and state trajectories, market price evolution, etc.) of any other agent (individually or collectively) in order that simple decentralized control actions achieve a mass ϵ-Nash equilibrium (with ϵ vanishing as the population goes to infinity) and individual L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> stability. The contribution of this paper is the extension of the MFG theory to cover the addition of a major agent to the power market problem. In general, the addition of a major agent in the MFG framework makes the mean field stochastic in contrast to the situation with purely minor agents where the mean field is deterministic. In the general situation of sporadic noisy observations of the mean field and the state of the major agent, the extended MFG theory (with estimation of the mean field and the major agent state) yields simple decentralized control laws which achieve a mass ϵ-Nash equilibrium (with ϵ vanishing as the population goes to infinity) and individual L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> stability. In this paper, this is carried out for the MFG formulation of the power market problem in order to fit the situation where sporadic noisy observations of the state of the major agent and of the market price are available for recursive mean field state estimation.