Vector Definitions for Control of Single-Phase and Unbalanced Three-Phase Systems

The d- q transformation applied to three-phase balanced systems allows for convenient definitions of vectors, which are consistent with the usual notion of phasors in sinusoidal steady state. If harmonics are neglected, then the dynamic models of balanced power-electronic systems in the d- q variables are also time-invariant, which facilitates vector control design. Alternative definitions are required for single-phase systems and unbalanced three-phase systems, around which similar controllers can then be built. This paper presents a unified theoretical analysis of these definitions. Generalizations of the time-delay-based definition are presented, which are also shown to be amenable to vector control. The dynamic equations for zero-sequence components reveal that the strategy to avoid cross-coupling between in-phase and quadrature terms in typical vector-control schemes is applicable to the control of zero-sequence vector components as well.