Interpreting Frame Transformations as Diagonalization of Harmonic Transfer Functions

TL;DR

This paper links time-domain frame transformations with frequency-domain harmonic transfer functions, showing that similarity transformations can diagonalize HTFs, simplifying analysis and providing new stability insights for ac electrical systems.

Contribution

It introduces a novel theoretical connection between frame transformations and harmonic transfer functions, enabling diagonalization of HTFs to simplify stability analysis.

Findings

Similarity transformations create a direct equivalence to frame transformations.

Diagonalization reduces HTF matrices from infinite to finite dimensions.

The derived stability criterion aligns with traditional models but offers deeper insights.

Abstract

Analysis of ac electrical systems can be performed via frame transformations in the time-domain or via harmonic transfer functions (HTFs) in the frequency-domain. The two approaches each have unique advantages but are hard to reconcile because the coupling effect in the frequency-domain leads to infinite dimensional HTF matrices that need to be truncated. This paper explores the relation between the two representations and shows that applying a similarity transformation to an HTF matrix creates a direct equivalence to a frame transformation on the input-output signals. Under certain conditions, such similarity transformations have a diagonalizing effect which, essentially, reduces the HTF matrix order from infinity to two or one, making the matrix tractable mathematically without truncation or approximation. This theory is applied to a droop-controlled voltage source inverter as an illustrative example. A stability criterion is derived in the frequency-domain which agrees with the conventional state-space model but offers greater insights into the mechanism of instability in terms of the negative damping (non-passivity) under droop control. The paper not only establishes a unified view in theory but also offers an effective practical tool for stability assessment.