A Fractional Order Derivative Newton-Raphson Method for the Computation of the Power Flow Problem Solution in Energy Systems

Some nonlinear real-valued equations have no solution in the real number field, and only roots of this nature are of practical interest. However, complex roots associated with the solution may introduce an interpretation of the physical problem analysis. This paper investigates the solution of nonlinear equations exploiting fractional order derivative (FOD) calculus resources. The theory addresses a solver that considers the FOD and Newton-Raphson method. The problem is extended to a multivariate fractional order derivative (MFOD) method so that a set of nonlinear equations can be resolved iteratively. Applications to the computation of the power flow problem in energy systems are used to illustrate some types of equations and solutions. The MFOD uses a numerical limits technique to determine a Jacobian required for the fractional application. This work demonstrates how real-valued nonlinear equations with complex roots can be solved by the MFOD Newton-Raphson approach. The results indicate the potentiality of the method reaches complex-valued solutions despite the iterative process starting with a real-valued guess. The complex-valued results are interpreted considering the connection between the imaginary part of a root and the divergence of the classical Newton-Raphson (CNR) method.