Determining the solutions of the load flow of power systems: Theoretical results and computer implementation

Powerful analytical tools and modern techniques from algebraic geometry are used to determine the number of solutions of the full-fledged load flow of power systems. Sufficient conditions are provided which guarantee the precise number of solutions to the load flow. The sufficient conditions are cast in terms of properties of the physical admittance matrix of the power grid. When the sufficient conditions are not satisfied, the cluster method is used to provide a `tighter´ upper bound on the load flow solutions for special power grid structures. The authors then develop an imbedding-based continuation method to reduce the computational complexity in finding all (or some of) the solutions of the so-called class of deficient systems. The authors specialize some of the results to example models and illustrate the computational efficiency of the proposed procedures by numerically finding all of the load flow solutions of the examples