A New Approach of Exploiting Self-Adjoint Matrix Polynomials of Large Random Matrices for Anomaly Detection and Fault Location

Abstract

Synchronized measurements of a large power grid enable an unprecedented opportunity to study the spatialtemporal correlations. Statistical analytics for those massive datasets start with high-dimensional data matrices. Uncertainty is ubiquitous in a future's power grid. These data matrices are recognized as random matrices. This new point of view is fundamental in our theoretical analysis since true covariance matrices cannot be estimated accurately in a high-dimensional regime. As an alternative, we consider large-dimensional sample covariance matrices in the asymptotic regime to replace the true covariance matrices. The self-adjoint polynomials of largedimensional random matrices are studied as statistics for big data analytics. The calculation of the asymptotic spectrum distribution (ASD) for such a matrix polynomial is understandably challenging. This task is made possible by a recent breakthrough in free probability, an active research branch in random matrix theory. This is the very reason why the work of this paper is inspired initially. The new approach is interesting in many aspects. The mathematical reason may be most critical. The realworld problems can be solved using this approach, however.

Operation State Evaluation Method of Smart Distribution Network Based on Free Probability Theory

“…In statistical analysis of high-dimensional data, when the amount of data is large enough, the data as a whole will show certain random statistical characteristics after corresponding processing, such as the single ring theorem and the M-P law (Ling et al, 2018;Jain et al, 2019;Deepa et al, 2020;Xiong et al, 2020;Yang et al, 2020b;Li et al, 2021b;Li et al, 2021c;Yang et al, 2021c;Li et al, 2021d;Ye et al, 2021;Dong and Li, 2021;Liu et al, 2021;Mousavizadeh et al, 2021;Ouyang and Xu, 2021;Zhu et al, 2021). In the statistical analysis of highdimensional power data, the corresponding linear eigenvalue statistics (LES) are constructed, such as MSR, high-order moment, etc., which can effectively represent the state of the distribution network.…”

Research on a Location Method for Complex Voltage Sag Sources Based on Random Matrix Theory

“…Assume that X � x i,j 􏽮 􏽯 is a p × n random matrix with non-Hermitian features, and it consists of independent random variables, where the mean E(x i,j ) � 0 and the variance δ 2 (x i,j ) � 1. e matrix Z � z i,j 􏽮 􏽯 is obtained by quadrature of L the random matrix X, the matrix Z is subjected to standardization processing to obtain a standard matrix Z � z i,j 􏽮 􏽯, and each element satisfies with the mean E(z i,j ) � 0 and the variance δ 2 (z i,j ) � 1/N. When p, n ⟶ ∞ and the ratio y � p/n ∈ (0, 1], the ESD function of matrix Z satisfies the single ring theorem, and its probability density function is be expressed as follows [16]:…”

A Review of Smart Grid Anomaly Detection Approaches Pertaining to Artificial Intelligence

“…There are multiple approaches on the analysis of data coming from PMUs to detect anomalies in some of the measurements, for example: score-based detector arrays [43], hierarchical temporary memory (HTM) capable of unsupervised learning [54], application of big data tools (MapReduce) [45], random matrix theory, and new statistical models using massive data sets in the power grid [44]. In the case of HTM, a limitation is observed in not being able to classify anomalies based on the cause of the anomaly.…”