  We present a detailed study on application of factor graphs and the belief
propagation (BP) algorithm to the power system state estimation (SE) problem.
We start from the BP solution for the linear DC model, for which we provide a
detailed convergence analysis. Using BP-based DC model we propose a fast
real-time state estimator for the power system SE. The proposed estimator is
easy to distribute and parallelize, thus alleviating computational limitations
and allowing for processing measurements in real time. The presented algorithm
may run as a continuous process.
  Using insights from the DC model, we use two different approaches to derive
the BP algorithm for the non-linear model. The first method directly applies BP
methodology, however, providing only approximate BP solution for the non-linear
model. In the second approach, we make a key further step by providing the
solution in which the BP is applied sequentially over the non-linear model,
akin to what is done by the Gauss-Newton method. The resulting iterative
Gauss-Newton belief propagation (GN-BP) algorithm can be interpreted as a
distributed Gauss-Newton method with the same accuracy as the centralized SE.
