Power Systems Topology and State Estimation by Graph Blind Source Separation

In this paper, we consider the problem of blind estimation of states and topology (BEST) in power systems. We use the linearized dc model of real power measurements with unknown voltage phases (i.e., states) and an unknown admittance matrix (i.e., topology) and show that the BEST problem can be formulated as a blind source separation (BSS) problem with a weighted Laplacian mixing matrix. We develop the constrained maximum likelihood (ML) estimator of the Laplacian matrix for this graph BSS problem with Gaussian-distributed states. The ML-BEST is shown to be only a function of the states' second-order statistics. Since the topology recovery stage of the ML-BEST approach results in a high-complexity optimization problem, we propose two low-complexity methods to implement it: First, two-phase topology recovery, which is based on solving the relaxed convex optimization and then finding the closest Laplacian matrix, and second, augmented Lagrangian topology recovery. We derive a closed-form expression for the associated Cram\acute{\text{e}}r-Rao bound (CRB) on the topology matrix estimation. The performance of the proposed methods is evaluated for three case studies: the IEEE-14 bus system, the IEEE 118-bus system, and a random network, and compared with the oracle minimum mean-squared-error state estimator and with the proposed CRB.