An algebraic-geometric and topological analysis of the solution to the load-flow equations for a power system

The load flow equations for a lossless electric power network are transformed by trigonometric substitutions into algebraic equations. This makes it possible to apply some deep and powerful results from algebraic geometry and intersection theory to study these equations. An obvious tool for determining the number of solutions is provided by the classical theorem of Bezout, but it is shown that for systems describing an n-machine network with n ?? 4, this result cannot be directly applied because the solutions contain solution components of positive dimension "at infinity." A major result in this paper is a modified Bezout technique which allows us to compute the number of complex (and a fortiori an upper bound on the number of real) solutions to the load flow equations. Combining this with the classical Morse inequalities we obtain very explicit results regarding the number of stable load flows for a given network topology and set of power injections. The cases of three and four machine networks are considered in detail.