  The stable operation of the electric power grid relies on a precisely
synchronized state of all generators and machines. All machines rotate at
exactly the same frequency with fixed phase differences, leading to steady
power flows throughout the grid. Whether such a steady state exists for a given
network is of eminent practical importance. The loss of a steady state
typically leads to power outages up to a complete blackout. But also the
existence of multiple steady states is undesirable, as it can lead to sudden
transitions, circulating flows and eventually also to power outages. Steady
states are typically calculated numerically, but this approach gives only
limited insight into the existence and (non-)uniqueness of steady states.
Analytic results are available only for special network configuration, in
particular for grids with negligible Ohmic losses or radial networks without
any loops. In this article, we introduce a method to systematically construct
the solutions of the real power load-flow equations in the presence of Ohmic
losses. We calculate the steady states explicitly for elementary networks
demonstrating different mechanisms leading to multistability. Our results also
apply to models of coupled oscillators which are widely used in theoretical
physics and mathematical biology.
