Bivariate holomorphic embedding applied to the power flow problem

Iterative methods for solving the power flow problem, including the Newton Raphson method and fast decoupled methods require good starting points, otherwise they may not converge or may converge to the wrong (low voltage) solution. The holomorphic embedding method (HEM) is a recursive, not iterative, method which uses the no-load condition as its starting points and is theoretically guaranteed to converge to the operable, high-voltage (HV) solution if one exists. The HEM method uses Padé approximant as a means of analytic continuation and is capable of finding the HV solution up to the saddle node bifurcation point (SNBP). The univariate HEM has been proven to be an efficient tool in experiments. However, a key drawback of the univariate HEM is that it lacks flexibility: the method can calculate the solutions only when the load/generation profile is scaled as a whole. A straightforward improvement is to use a multi-variate HEM combined with multi-variate Padé approximants. This paper presents a bivariate HEM formulation, which uses a corresponding bivariate Padé (Chisholm) approximant. Simulations on a three-bus and a modified IEEE 14-bus system show that the method can yield accurate voltage solutions and accurate values of the SNBPs.