Graphon Mean Field Games with Finite States and Forecasting Models for the Energy Market

This thesis is divided into two parts, combining theoretical advancements in the field of mean field games (MFGs) with applied contributions to the modelling and forecasting of renewable energy systems. The first part, carried out in collaboration with Prof. Francesco Giuseppe Cordoni, focuses on the mathematical analysis of graphon mean field games (GMFGs), a generalisation of classical MFGs that allows for heterogeneous and structured interactions among agents. The study considers finite state spaces and establishes results on the existence and uniqueness of equilibria under various conditions, including fixed measure flows, short time horizons, and monotonicity assumptions. It also presents two case studies in which uniqueness fails due to anti-monotonic terminal costs and non-separable Hamiltonians. The second part, developed within the framework of the PON “Research and Innovation 2014–2020” scholarship titled “Renewable Energy Forecasting for a Green and Sustainable Development”, addresses applied problems in energy systems engineering. Three data-driven applications are explored: short-term load forecasting in the Italian electricity market using machine learning; optimisation of financial transmission rights (FTRs) allocation via linear programming; and stochastic modelling of wind energy production through differential equations with jumps.