  Belief propagation is an algorithm that is known from statistical physics and
computer science. It provides an efficient way of calculating marginals that
involve large sums of products which are efficiently rearranged into nested
products of sums to approximate the marginals. It allows a reliable estimation
of the state and its variance of power grids that is needed for the control and
forecast of power grid management. At prototypical examples of IEEE-grids we
show that belief propagation not only scales linearly with the grid size for
the state estimation itself, but also facilitates and accelerates the retrieval
of missing data and allows an optimized positioning of measurement units. Based
on belief propagation, we give a criterion for how to assess whether other
algorithms, using only local information, are adequate for state estimation for
a given grid. We also demonstrate how belief propagation can be utilized for
coarse-graining power grids towards representations that reduce the
computational effort when the coarse-grained version is integrated into a
larger grid. It provides a criterion for partitioning power grids into areas in
order to minimize the error of flow estimates between different areas.
