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 insights from the DC model, we use two different approaches to derive the BP algorithm for the non-linear AC model. The first met...

Hessian information speeds convergence substantially in motion optimization. The better the Hessian approximation the better the convergence. But how good is a given approximation theoretically? How much are we losing? This paper addresses that question and proves that for a particularly popular and empirically strong approximation known as the Gauss-Newton approximation, we actually lose very ...

Inverse problems are typically ill-posed or ill-conditioned and require regularization. Tikhonov regularization is a popular approach and it requires an additional parameter called the regularization parameter that has to be estimated. The χ method introduced by Mead in [8] uses the χ distribution of the Tikhonov functional for linear inverse problems to estimate the regularization parameter. H...

The curvature matrix depends on the specific optimisation method and will often be only an estimate. For notational simplicity, the dependence of f̂ on θ is omitted. Setting C to the true Hessian matrix of f would make f̂ the exact secondorder Taylor expansion of the function around θ. However, when f is a nonlinear function, the Hessian can be indefinite, which leads to an ill-conditioned quadra...