A Holomorphic embedding approach for finding the Type-1 power-flow solutions

A holomorphic embedding method (HEM)-based algorithm for finding Type-1 power-flow solutions is introduced whose complexity is the same as that of the HEM power-flow algorithm for calculating the high voltage (operable) power-flow solution. The algorithm is tailored to finding Type-1 solutions by using a modified embedded system and a numerical mapping from a set of integer-based boundary conditions to a floating-point-number-based reference state. The modified system can also be viewed as a homotopy whose initial point (no-load reference state) is consistent with the Type-1 solution homotopy path of the modified system, with an embedding parameter that functions simultaneously as the homotopy parameter. By using analytic continuation, starting from the initial point/reference state, the solution obtained for the modified system matches the one obtained from the original system model at the load of interest. Numerical results for three-/five-/seven-/14- and 118-bus systems are presented to demonstrate the numerical robustness.