Using Differential Geometry to Revisit the Paradoxes of the Instantaneous Frequency

TL;DR

This paper introduces a geometric framework to interpret the Instantaneous Frequency in three-phase systems, clarifying longstanding paradoxes and providing new insights into power system analysis.

Contribution

It presents a novel geometric approach to understanding IF, revisiting paradoxes and linking analytic signals with the Clarke transform in power systems.

Findings

The geometric interpretation explains the five IF paradoxes.

The framework is validated on IEEE 39-bus system.

Provides new insights into IF analysis in power systems.

Abstract

This paper proposes a general framework to interpret the concept of Instantaneous Frequency (IF) in three-phase systems. The paper first recalls the conventional frequency-domain analysis based on the Fourier transform as well as the definition of IF which is based on the concept of analytic signals. The link between analytic signals and Clarke transform of three-phase voltages of an ac power system is also shown. Then the well-known five paradoxes of the IF are stated. In the second part of the paper, an approach based on a geometric interpretation of the frequency is proposed. This approach serves to revisit the five IF paradoxes and explain them through a common framework. The case study illustrates the features of the proposed framework based on a variety of examples and on a detailed model of the IEEE 39-bus system.