Voltage-Collapse Point Estimation, Holomorphic Embedding Applied to the DCOPF Problem and the Padé Matrix Pencil Method

The power-flow problem has been solved using the Newton-Raphson and Gauss-Seidel methods. Recently the holomorphic embedding method (HEM), a recursive (non-iterative) method applied to solving nonlinear algebraic systems, was applied to the power-flow problem. HEM has been claimed to have superior properties when compared to the Newton-Raphson and other iterative methods in the sense that if the power-flow solution exists, it is guaranteed that a properly configured HEM can find the high voltage solution and, if no solution exists, HEM will signal that unequivocally. Provided a solution exists, convergence of HEM in the extremal domain is claimed to be theoretically guaranteed by Stahl’s convergence-in-capacity theorem, another advantage over other iterative nonlinear solver.In this work it is shown that the poles and zeros of the rational function from fitting the local PMU measurements can be used theoretically to predict the voltage-collapse point. Different numerical methods were applied to improve prediction accuracy when measurement noise is present. It is also shown in this work that the dc optimal power flow (DCOPF) problem can be formulated as a properly embedded set of algebraic equations. Consequently, HEM may also be used to advantage on the DCOPF problem. For the systems examined, the HEM-based interior-point approach can be used to solve the DCOPF problem. While the ultimate goal of this line of research is to solve the ac OPF; tackled in this work, is a precursor and well-known problem with Padé approximants: spurious poles that are generated when calculating the Padé approximant may, at times, prevent convergence within the functions domain. A new method for calculating the Padé approximant, called the Padé Matrix Pencil Method was developed to solve the spurious pole problem. The Padé Matrix Pencil Method can achieve accuracy equal to that of the so-called direct method for calculating Padé approximants of the voltage-functions tested while both using a reduced order approximant and eliminating any spurious poles within the portion of the function’s domain of interest: the real axis of the complex plane up to the saddle-node bifurcation point.