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This paper explores solutions to linearized powerflow equations withbus-voltage phasors represented in rectangular coordinates. The key idea is tosolve for complex-valued perturbations around a nominal voltage profile from aset of linear equations that are obtained by neglecting quadratic terms in theoriginal nonlinear power-flow equations. We prove that for lossless networks,the voltage profile where the real part of the perturbation is suppressedsatisfies active-power balance in the original nonlinear system of equations.This result motivates the development of approximate solutions that improveover conventional DC power-flow approximations, since the model includes ZIPloads. For distribution networks that only contain ZIP loads in addition to aslack bus, we recover a linear relationship between the approximate voltageprofile and the constant-current component of the loads and the nodal activeand reactive-power injections.

Linear Approximations to AC Power Flow in Rectangular Coordinates