The no-gain theorem and localized response for the decoupled P -hetapower network with active power losses included

In this paper a no-gain theorem for electric power systems is presented using the
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model based decoupled active power-phase angle
P - \theta
network, in which each transmission element consists of two branches, representing transmitted power and loss power. The theorem gives necessary and sufficient conditions such that if a power system is perturbed from its normal operating point then the magnitude of the change in active power through either branch of a given transmission line
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graph is not greater than the sum of the magnitudes of the changes in active power through the sources. These conditions are given as a region of allowable perturbations, which are derived as a set of inequalities. Based on this result we present a proof, valid for systems with losses, of localized steady state response. Specifically, we prove that the effects of a single fault in a given power system diminish as one moves away from the fault, subject only to the constraint that the operating point of the system remains within the allowable regions given by the no-gain theorem. This result is often assumed without proof to be valid in the entire operating region.