Introducing Modeling, Analysis, and Control of Three-Phase Electrical Systems Using Geometric Algebra

This article introduces a novel framework based on geometric algebra (GA) for the modeling, analysis, and control of three-phase electrical systems, with a focus on unbalanced scenarios. The proposed approach utilizes GA to represent these systems using simplified single-input/single-output (SISO) models, significantly reducing their mathematical complexity while maintaining accuracy. The framework introduces GA-valued transfer functions (GA-TFs), enabling the extension of well-established multivariable (MIMO) techniques, such as Youla-Kučera parameterization, into the GA domain as an illustrative example. This demonstrates the adaptability of classical tools within the new algebra. The validity and potential of the approach are showcased through numerical examples of balanced and unbalanced systems, as well as physical implementations in a laboratory setup. The results confirm that GA provides a robust and scalable foundation for simplifying the analysis and control of three-phase systems, paving the way for future theoretical and practical advancements.