Newton-Raphson Load-Flow Method based on an Extended Complex-Number Plane

Abstract

This paper presents a load-flow algorithm using the Newton-Raphson method on an extended complex-number plane, in which the real and imaginary parts of each bus volatage are treated as complex numbers respectively. As the results of solving double-dimensional power-flow equations, the solutions converge on real numbers for feasible conditions of power systems, that is, usual system states. If the solutions converge on complex numbers for ill-conditioned system, e. g., heavy-loaded conditions, it can be confirmed that the system state does not have feasible solutions. The authors also develop a method for obtaining feasible system conditions from complex-number solutions calculated in the above-mentioned approach. This algorithm consists of two steps; selecting the specified variable which is the most effective for updating the system state, and estimating a boundary value of the specified variable selected. While the former is based on the so-called sensitivity analysis, the estimation is carried out by approximating the characteristic curve of the relation between imaginary parts of the solutions and the specified values. Numerical examples for various model systems show that the proposed technique works excellently.