On the Uniqueness of the Solution of Electric Power System Load Flow Nonlinear Equation Systems in the Form of a Power Balance

The article addresses the problem of finding the uniqueness of a solution to electric power system load flow nonlinear equation systems in the form of a power balance. A mathematical formulation of the problem is presented, and expressions for determining power losses in transmitting power along power lines are given. It is pointed out that the stated problem does not have a unique solution, and to obtain it, a special criterion has to be formulated that would reflect the physical essence of the power flow distribution process in electric power systems. Based on D.K. Maxwell’s theorem “about the distribution of currents in a DC electrical circuit”, a theorem “of determining the uniqueness of the solution of nonlinear equation systems for arbitrary (AC and DC) electric power systems” has been formulated along with the uniqueness criterion: the minimum modulus of the total power loss values. A discrete optimization problem is stated using the formulated uniqueness criterion. In the experimental part of the article, the multiplicity of solutions is considered using the example of a four-node electric power system. It has been revealed that, when directly solving a system of nonlinear steady-state equations, eighteen solutions are obtained. Based on the formulated uniqueness criterion, a physically realizable solution is determined.