Complex-Vector Power and Cross-Phase Unbalance in Three-Phase Systems
Unbalanced three-phase systems still lack a compact phasor-domain representation of power that makes the joint cross-phase voltage-current structure explicit while remaining consistent with established apparent-power definitions. This paper addresses that point through a complex-vector power (CVP) formulation for sinusoidal steady-state operation. The proposed CVP is an element of a direct-sum space whose constituents are the conventional complex-power scalar and a cross-phase unbalance (CPU) vector obtained from the cross product of the voltage and current phasor vectors. The scalar constituent retains the usual active and reactive powers, whereas the CPU vector characterizes their antisymmetric cross-phase relation. Apparent power is thereby separated into intraphase and cross-phase contributions, and its norm is preserved under the power-invariant Fortescue transformation. The formulation is extended to three-phase four-wire systems by introducing equivalent coordinates that preserve the effective apparent-power norm for the chosen voltage reference. Only standard complex numbers and matrices are required. Numerical examples identify operating conditions in which a non-negligible part of the apparent-power structure is associated with cross-phase unbalance and is not displayed by the active- and reactive-power pair alone. A proof-of-concept shunt-compensation study further illustrates how the CPU descriptor can be incorporated into reproducible current references. The proposed formulation therefore provides a compact phasor-based description of unbalance that complements established scalar apparent-power representations.