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

In this paper, we investigate the problem of estimating a complex-valued Laplacian matrix from a linear Gaussian model, with a focus on its application in the estimation of admittance matrices in power systems. The proposed approach is based on a constrained maximum likelihood estimator (CMLE) of the complex-valued Laplacian, which is formulated as an optimization problem with Laplacian and sparsity constraints. The complex-valued Laplacian is a symmetric, non-Hermitian matrix that exhibits a joint sparsity pattern between its real and imaginary parts. Leveraging the l1 relaxation and the joint sparsity, we develop two estimation algorithms for the implementation of the CMLE. The first algorithm is based on casting the optimization problem as a semi-definite programming (SDP) problem, while the second algorithm is based on developing an efficient augmented Lagrangian method (ALM) solution. Next, we apply the proposed SDP and ALM algorithms for the problem of estimating the admittance matrix under three commonly-used measurement models, that stem from Kirchhoff's and Ohm's laws, each with different assumptions and simplifications: 1) the nonlinear alternating current (AC) model; 2) the decoupled 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 from the IEEE 33-bus power system data under different settings. The numerical experiments demonstrate that the proposed algorithms outperform existing methods in terms of mean-squared-error (MSE) and F-score, thus, providing a more accurate recovery of the admittance matrix.