  The solution space of any set of power flow equations may contain different
number of real-valued solutions. The boundaries that separate these regions are
referred to as power flow solution space boundaries. Knowledge of these
boundaries is important as they provide a measure for voltage stability.
Traditionally, continuation based methods have been employed to compute these
boundaries on the basis of initial guesses for the solution. However, with
rapid growth of renewable energy sources these boundaries will be increasingly
affected by variable parameters such as penetration levels, locations of the
renewable sources, and voltage set-points, making it difficult to generate an
initial guess that can guarantee all feasible solutions for the power flow
problem. In this paper we solve this problem by applying a numerical polynomial
homotopy based continuation method. The proposed method guarantees to find all
solution boundaries within a given parameter space up to a chosen level of
discretization, independent of any initial guess. Power system operators can
use this computational tool conveniently to plan the penetration levels of
renewable sources at different buses. We illustrate the proposed method through
simulations on 3-bus and 10-bus power system examples with renewable
generation.
