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In this paper, we investigate the problem of estimating a complex-valuedLaplacian matrix from a linear Gaussian model, with a focus on its applicationin the estimation of admittance matrices in power systems. The proposedapproach is based on a constrained maximum likelihood estimator (CMLE) of thecomplex-valued Laplacian, which is formulated as an optimization problem withLaplacian and sparsity constraints. The complex-valued Laplacian is asymmetric, non-Hermitian matrix that exhibits a joint sparsity pattern betweenits real and imaginary parts. Leveraging the l1 relaxation and the jointsparsity, we develop two estimation algorithms for the implementation of theCMLE. The first algorithm is based on casting the optimization problem as asemi-definite programming (SDP) problem, while the second algorithm is based ondeveloping an efficient augmented Lagrangian method (ALM) solution. Next, weapply the proposed SDP and ALM algorithms for the problem of estimating theadmittance matrix under three commonly-used measurement models, that stem fromKirchhoff's and Ohm's laws, each with different assumptions andsimplifications: 1) the nonlinear alternating current (AC) model; 2) thedecoupled linear power flow (DLPF) model; and 3) the direct current (DC) model.The performance of the SDP and the ALM algorithms is evaluated using data fromthe IEEE 33-bus power system data under different settings. The numericalexperiments demonstrate that the proposed algorithms outperform existingmethods in terms of mean-squared-error (MSE) and F-score, thus, providing amore accurate recovery of the admittance matrix.

Estimation of Complex-Valued Laplacian Matrices for Topology Identification in Power Systems