Clarke-Park Coordinates Transformation in the Quaternion Domain

Modern control techniques of three-phase power systems with electronics converters for domestic and utility applications, such as uninterruptible power supplies, variable speed or position AC drive, four-wire distribution generation system for renewable energy sources, controllers for power conditioning, etc., are employed synchronously rotating reference frame aligned with the reference vector. These control schemes involve coordinates transformation of AC quantities to constant DC quantities and provides a zero steady-state error while using integral action as an input. The transformation is ensured by rotation the plane of 3D space for appropriative angle described by square matrix with 9 elements. In order to reduce computational efforts and the complexity of control architecture, the quaternion algebra is applied to d,q,o transformation that allows to use only four variables and therefore it simplify the control implementation for digitally represented signals. The paper considers Clarke-Park transformation in quaternion domain as well as invariance condition of the scalar part after d,q,o transformation.