Discrete solutions of nonlinear electric systems: A differentiation matrix and newton-based approach

A discrete-time approach based on the use of a finite-dimensional matrix representation of the differentiation operation is presented in this paper to compute periodic steady-state solutions of electric power networks. The discrete representation of the derivative operator reproduces the exact derivative of a trigonometric polynomial. The set of ordinary differential equations that represents the dynamic behavior of the electric system is transformed into a nonlinear algebraic formulation and solved using a Newton algorithm. The unknowns of the algebraic equations are the samples of the state variables at the periodic steady-state. Furthermore, a direct method for solving sparse systems of equations is incorporated in the Newton method, where the Jacobian matrix is highly sparse for large-scale power systems. The incorporation of sparse techniques to this discrete-time domain method improves its efficiency in terms of storage and computational effort. Two test cases based on a electric arc furnace and a transmission power network are presented to illustrate the applicability and effectiveness of this method to solve electric power systems. Results in terms of convergence properties and computational effort are reported for the proposal presented in this work and compared with the well-known finite-difference method.