Three-phase Quaternion Power in Three-wire Systems

Instantaneous power theory has a central role in power systems analysis. Among mathematical settings used for the development of this theory, quaternion algebra has been used for describing electrical variables in recent works. In this context, this paper aims to describe three-phase power in a quaternion framework. We analyze quaternion power for balanced and unbalanced delta loads, comparing the expressions obtained to the usual expressions of complex power. The quaternion power expression obtained also makes it natural to introduce a decomposition of the unbalanced load in terms of a balanced component and an unbalanced load with null average power. It is also shown that delta unbalanced loads are equivalent to time-varying balanced loads. The results obtained extend the power systems theory in the quaternion domain and emphasize the advantages of using this framework.