Resolution of line-transferred power in grids yielded by circuit-laws´ symmetry under deductive reasoning of shapley theorem

This study presents a strict method for resolving line-transferred power (or decomposing line-power flow) into source-driven component powers over lines. First, three properties of conservation and symmetry and additivity inherent in circuit laws are derived. Incorporating them with circuit-laws´ equation, a model for resolving line-transferred power is built. The three properties make all conditions or hypothesis in Shapley theorem satisfied. Then the deductive reasoning of Shapley theorem is used to solve the model, which immediately gives a discrete and non-analytical resolution formula in terms of line-transferred powers caused by excitations of possible combination of sources. Representing the power by source currents (or electromotive forces), a continuous and analytical resolution formula in terms of source currents (or electromotive forces) is then proved mathematically. The resolution formula is invariantly the same for all sources including the slack source. It is also applicable to find the source-driven component powers flowing into loads and out of sources in arbitrarily complicated grids. Simulation results show the features of the proposed method.