  This paper proposes a convex optimization based method that either locates
all real roots of a set of power flow equations or declares no real solution
exists in the given area. In the proposed method, solving the power flow
equations is reformulated as a global optimization problem (GPF for short) that
minimizes the sum of slack variables. All the global minima of GPF with a zero
objective value have a one-to-one correspondence to the real roots of power
flow equations. By solving a relaxed version of GPF over a hypercube, if the
optimal value is strictly positive, there is no solution in this area and the
hypercube is discarded. Otherwise the hypercube is further divided into smaller
ones. This procedure repeats recursively until all the real roots are located
in small enough hypercubes through the successive refinement of the feasible
region embedded in a bisection paradigm. This method is desired in a number of
power system security assessment applications, for instance, the transient
stability analysis as well as voltage stability analysis, where the closest
unstable equilibrium and all Type-I unstable equilibrium is required,
respectively. The effectiveness of the proposed method is verified by analyzing
several test systems.
