  We propose solving the power flow equations using monodromy. We prove the
variety under consideration decomposes into trivial and nontrivial subvarieties
and that the nontrivial subvariety is irreducible. We also show various
symmetries in the solutions. We finish by giving numerical results comparing
monodromy against polyhedral and total degree homotopy methods and giving an
example of a network where we can find all solutions to the power flow equation
using monodromy where other homotopy techniques fail. This work gives hope that
finding all solutions to the power flow equations for networks of realistic
size is possible.
