Interpreting Frame Transformations in AC Systems as Diagonalization of Harmonic Transfer Functions
Yitong Li, Yunjie Gu, T.C. Green
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 frame transformation on the input-output signals creates a direct equivalence to a similarity transformation to the HTF matrix of the system. 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. Therefore, the paper not only establishes a unified view in theory but also offers an effective practical tool for stability assessment.