General theory of instantaneous power for multi-phase systems with distortion, unbalance and direct current components

Active instantaneous currents are generally defined as those compensated supply-wire currents that deliver a given instantaneous power with minimum line losses, without a change in voltage. Since the concept was introduced 60 years ago, many theories have been proposed to enable the calculation of those optimum supply currents, for various conditions of the supply system. This paper shows how these optimal wire currents can be obtained with a single general formula applicable to all supply systems. The solution depends on the number of wires considered, their resistances, which need not be equal, and their respective voltages measured from a common reference. The formula is derived through the properties of linear algebra in vector space, and is a direct consequence of Kirchhoff's current law and the law of conservation of energy. All the existing theories can be identified as particular cases of the general formula and most can be grouped into three common categories.