Symmetry in the Narrow Sense: On the Linearity and Time-Invariance of DQ0 Models

This paper presents necessary and sufficient conditions for a linear three-phase circuit to have a linear and time-invariant $dq0$ model. More specifically, we show that a circuit with state and input matrices that can be partitioned into three-by-three circulant matrices at the $abc$ frame of reference, can be transformed into a linear and time-invariant model in the $dq0$ frame of reference. We also show that under this condition the circuit is symmetrically configured, in the sense that every three-phase balanced input results in a balanced state vector at steady-state. We explain that due to these properties, such circuits may be defined as 'symmetrically configured in the narrow sense'.