Robust complex-valued Levenberg-Marquardt algorithm as applied to power flow analysis

Abstract

This paper deals with a robust complex-valued Levenberg-Marquardt algorithm specially developed for solving ill-conditioned power flow problems. Moreover, it can also be a useful tool for voltage instability and voltage collapse studies. Because power flow models are nonlinear, the Wirtinger calculus is applied to develop iterative algorithms based on Taylor series expansions of nonlinear functions of complex variables and their complex conjugates. Our proposal in complex plane is straightforward derived in rectangular coordinates. Consequently, its performance is compared to the well-known optimized multiplier based load flow method. Aiming this purpose, we show that few changes in the codes are required to transform the complex-valued Newton-Raphson power flow algorithm into the complex-valued Levenberg-Marquardt power flow one. Furthermore, we show that the latter lends itself well to modeling new smart grid technologies while exhibiting a bi-quadratic convergence rate and superior performance as compared to the former procedure. The performance of our proposal is demonstrated and analyzed on well-conditioned IEEE-14, −30, −57 and −118 bus systems and the Brazilian Southern-equivalent system termed SIN-1916 bus. Furthermore, its performance is also demonstrated on the ill-conditioned IEEE-11, −43 bus systems besides the SIN-1916 under stressed operating conditions or higher R/X ratios of transmission lines.