Remarks on the number of solutions to the load flow equations for a power system with electrical losses

It is shown that for a wide class of electrical power networks whose nodes are PV-buses and in which transfer conductances are nonzero, solutions to the load flow equations are almost always unique, and even when multiple solutions occur, there are fewer of them than would be expected in the case of zero transfer conductances. The reason for this difference is that the load flow equations for the nonzero conductance case constitute a system which is in a sense "overdetermined", and thus our previous work on intersection theory does not completely solve the problem of determining the number of solutions. In fact, the algebraic geometry for the nonzero conductance case is so strikingly different from the zero conductance case that many results which are based on the a priori assumption that transmission lines are lossless are suspect for networks with losses of any significant size. Nevertheless, it will be argued that previous work by the authors and others on models for lossless power networks provides valuable insight and understanding for systems with small transfer conductances.