A Noniterative Method for Solving Nonlinear Equations for Steady-State Regimes of Electrical Networks

Abstract

Methods for solving linear and nonlinear equations for steady-state operation modes are important for the development and operation of electrical networks. Linear equations are solved analytically or with the use of iterative methods, while nonlinear equations are solved only with use of iterative methods, usually the Newton–Raphson method. Iterative methods have two significant drawbacks. First, the iterative process can diverge, with the divergence being dependent on both the form of the nonlinear equation and the choice of the initial approximation. Second, in the case of convergence, the iterative method allows finding only one solution for each initial approximation, whereas a nonlinear equation can have several solutions corresponding to different operation modes of the electric network. Development of methods that are free of these drawbacks is an important problem. This paper proposes a noniterative method for solving nonlinear equations describing the steady-state operation modes of electric networks; the equations are written in the complex form. The method uses the resultant to transform set of nonlinear equations to a set of polynomials. The polynomials are composed in such a way that their zeros determine one of the coordinates of the solution vector of the nonlinear equation system. Thus, the problem of solving of the nonlinear equations system is reduced to the well-studied problem of finding the zeros of one-variable polynomials. There are numerous methods of solving this problem that do not require specifying the initial approximation and allow finding all zeros. The effectiveness of the method is illustrated by solving the equations of node network voltages in the form of power balance. The equations were solved by the traditional iterative method and the proposed method using the MathCAD-15 program. The drawbacks of the iterative method are illustrated by its divergence at certain values of the initial approximation and by the failure of finding all the solutions. In contrast, the use of the proposed noniterative method allows one to find two solutions and to establish that there is no other solutions. Further development of this method will be connected with taking into account the sparseness of the matrix of conductivities that would allow reducing the degree used polynomials, as well as with the extension of this method to the domain of real values in which the equations can be written down in the algebraic and trigonometric form.