Robust Mahalanobis distance in power systems state estimation

Abstract

The dissertation presents a new robust method for estimating the standardized distances of the data points associated with the weighted Jacobian matrix in power system state estimation. These distances, called robust Mahalanobis distances, can be used as weight functions to robustify the residuals of both the M-estimators and the least median of squares estimators for outlier diagnostics. They can also be used for leverage diagnostics and for alleviating the ill-conditioning problem of the Jacobian matrix. The robust Mahalanobis distances are calculated in three steps. First, projection distances are calculated and statistical tests applied to them to identify leverage points. Then, the sample covariance matrix is estimated from the data set without the identified leverage points. Finally robust Mahalanobis distances are calculated from the estimated covariance matrix.

The projection distances are provided by a new version of the projection algorithm proposed by Donoho and Stahel, which has been specially adapted for power systems. The new projection algorithm consists of selecting relevant directions for each measurement in the factor space and projecting on these directions only the subset of data points that have non-zero projections. It is shown that this subset is the union of the fundamental sets containing the selected measurement. The fundamental set of a state variable consists of all those measurements that observe this state variable. The probability distributions of the projection distances and the statistical cutoff values for leverage point identification have been determined through Monte Carlo simulations and Q-Q plots. It is found that the projection distances follow x²-distributions with degrees of freedom much smaller than the dimension of the factor space.

Simulation results performed on various test systems have revealed that the projection algorithm can handle a large fraction of leverage points, whatever their positions in the factor space. In addition, it is very fast and compatible with real-time environment, even for very large systems. Its computing times grow linearly with system size.