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

TL;DR

This paper introduces a novel ADMM-based method for estimating complex-valued Laplacian matrices in power systems, improving accuracy in admittance matrix recovery under various measurement models.

Contribution

It proposes a group-sparse penalized likelihood approach with an ADMM algorithm for complex Laplacian estimation, applied to power system admittance matrices.

Findings

Outperforms existing methods in MSE and F-score

Effective across different power system models

Demonstrated on IEEE 33-bus system data

Abstract

In this paper, we investigate the problem of estimating a complex-valued Laplacian matrix 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. Thus, we present a group-sparse-based penalized log-likelihood approach for the Laplacian estimation. Leveraging the mixed \ell 2,1 norm relaxation of the joint sparsity constraint, we develop a new alternating direction method of multipliers (ADMM) estimation algorithm for the implementation of the CMLE of the Laplacian matrix under a linear Gaussian model. Next, we apply the proposed ADMM algorithms for the problem of estimating the admittance matrix under three commonly used measurement models that stem from Kirchhoff and Ohm 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 ADMM algorithm is evaluated using data from the IEEE 33-bus power system data under different settings. The numerical experiments demonstrate that the proposed algorithm outperforms existing methods in terms of mean-squared-error (MSE) and F-score, thus providing a more accurate recovery of the admittance matrix.