The Modeling and Control of (Renewable) Energy Systems by Partial Differential Equations—An Overview

Mathematical models of energy systems have been mostly represented by either linear or nonlinear ordinary differential equations. This is consistent with lumped-parameter dynamic system modeling, where dynamics of system state variables can be fully described only in the time domain. However, when dynamic processes of energy systems display both temporal and spatial evolutions (as is the case of distributed-parameter systems), the use of partial differential equations is necessary. Distributed-parameter systems, being described by partial differential equations, are mathematically (and computationally) much more difficult for modeling, analysis, simulation, and control. Despite these difficulties in recent years, quite a significant number of papers that use partial differential equations to model and control energy processes and systems have appeared in journal and conference publications and in some books. As a matter of fact, distributed-parameter systems are a modern trend in the areas of control systems engineering and some energy systems. In this overview, we will limit our attention mostly to renewable energy systems, particularly to partial differential equation modeling, simulation, analysis, and control papers published on fuel cells, wind turbines, solar energy, batteries, and wave energy. In addition, we will indicate the state of some papers published on tidal energy systems that can be modelled, analyzed, simulated, and controlled using either lumped or distributed-parameter models. This paper will first of all provide a review of several important research topics and results obtained for several classes of renewable energy systems using partial differential equations. Due to a substantial number of papers published on these topics in the past decade, the time has come for an overview paper that will help researchers in these areas to develop a systematic approach to modeling, analysis, simulation, and control of energy processes and systems whose time–space evolutions are described by partial differential equations. The presented overview was written after the authors surveyed more than five hundred publications available in well-known databases such as IEEE, ASME, Wiley, Google, Scopus, and Web of Science. To the authors’ best knowledge, no such overview on PDEs for energy systems is available in the scientific and engineering literature. Throughout the paper, the authors emphasize novelties, originalities, and new ideas, and identify open problems for future research. To achieve this goal, the authors reviewed more than five hundred journal articles and conference papers.