An introduction to energy markets: a mathematical viewpoint

The electricity market is undergoing a profound transformation, driven by the dual imperatives of decarbonization and energy security. Traditional architectures–long reliant on fossil fuel generation and centralized grid management–are progressively giving way to systems built around renewable integration and decentralized energy production, made feasible and economically sustainable through the widespread adoption of digital technologies. This paper explores, from a mathematical standpoint, the fundamental challenges underlying this paradigm shift. In the form of a review, it offers a mathematical analysis of contemporary electricity markets, emphasizing the roles of system actors: producers, consumers, system operators, and regulatory authorities. Particular attention is devoted to the European framework, where regulation plays a crucial role in safeguarding transparency and integrity in electricity exchange. We discuss the structural issues associated with renewable penetration, as well as the growing need for grid stability, and examine how optimization theory, financial mathematics, and advanced statistical and machine learning techniques contribute to improved forecasting and enhanced market efficiency. The paper first offers a systematic overview of the structure and functioning of the electricity market; it then identifies the mathematical methodologies and modeling tools essential for understanding and managing such a complex and regulation-intensive energy ecosystem. A reproducible case study, calibrated on real Italian day-ahead load data, illustrates the application of these concepts. Finally, we outline future perspectives, focusing on the potential impact of emerging technologies–particularly artificial intelligence and quantum computing–on electricity markets.