Cramer-Rao Bound for Admittance Matrix Estimation under Laplacian Constraints

In this paper, we consider the problem of estimating the admittance matrix in power systems, accounting for Laplacian and physical constraints. We assume the nonlinear alternating current (AC) model, which accurately represents the power flow model. We develop a closed-form expression for the oracle Cramér-Rao bound (CRB) on the mean-squared-error (MSE) of any unbiased estimator of the admittance matrix. The proposed oracle CRB takes into account the Laplacian parametric equality constraints, including symmetry and the null space property, through a reparametrization of the estimation problem as an unconstrained optimization. The oracle CRB assumes knowledge of the locations of the nonzero entries of the Laplacian matrix, and, thus, provides a valid lower bound. We evaluate and compare the oracle CRB with the MSE of: 1) the constrained maximum likelihood estimator (CMLE), which integrates the equality, inequality, and jointsparsity Laplacian constraints; and 2) the oracle CML estimator, which knows the location of the nonzero entries of the Laplacian matrix. It is shown that for data from the IEEE 33-bus power system, the MSEs of the estimators converge to the oracle CRB for a sufficient number of measurements.