The Frenet Frame as a Generalization of the Park Transform

TL;DR

This paper introduces a geometric generalization of the Park transform using the Frenet frame, extending its application to multi-phase circuits and providing a differential geometric perspective.

Contribution

It presents a novel Frenet frame-based transform for multi-phase systems, extending the classical Park transform with differential geometry tools and demonstrating its applicability.

Findings

Effective extension of Park transform to multi-phase systems.

Illustrated with IEEE 39-bus system case study.

Provides differential geometric insights into circuit analysis.

Abstract

The paper proposes a generalization of the Park transform based on the Frenet frame, which is a special set of coordinates defined in differential geometry for space curves. The proposed geometric transform is first discussed for three dimensions, which correspond to the common three-phase circuits. Then, the expression of the time derivative of the proposed transform is discussed and the Frenet-Serret formulas and the Darboux vector are introduced. The change of reference frame and its differentiation based on Cartan's moving frames and attitude matrices are also described. Finally, the extension to circuits with more than three phases is presented. The features of the Frenet frame are illustrated through a variety of examples, including a case study based on the IEEE 39-bus system.