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We employ an adjoint network concept based on an augmented form of Tellegen's theorem to describe a novel method for solving the load flow problem. The method incorporates successive adjoint network simulations with a sparse, mostly constant matrix of coefficients, the majority of its elements representing basic data of the problem already stored in computer memory. Nevertheless, the exact version of the method enjoys the same rate of convergence as the Newton-Raphson method. Moreover, it automatically supplies the sensitivities of all system states with respect to adjustable variables at the load flow solution without any additional adjoint simulation. An approximate version of the method is also presented. It partly employs very fast repeat forward and backward substitutions with constant LU factors of a reduced matrix of coefficients and is applicable to both the rectangular and the polar formulations of the power flow equations. Numerical examples are presented for illustration and comparison.