Probability density function of three-phase ellipse parameters for the characterization of noisy voltage sags

This work deals with the statistical analysis of additive noise impact on the space-vector ellipse parameters used to detect and classify three-phase voltage sags. In fact, since voltage waveforms are always corrupted by additive noise and harmonics, the space vector is pre-processed through the Discrete Fourier Transform to extract the power frequency components. Thus, harmonics can be readily discarded, but additive noise can still have significant impact on the elliptical trajectory of the space vector on the complex plane. Therefore, by modeling the ellipse parameters (i.e., the shape index and the inclination angle) as random variables, the related statistical characterization is derived in the paper. In particular, the main results and the novelty of the paper are given by the analytical derivation in closed-form of the probability density function, cumulative distribution function, mean value, and variance of the ellipse parameters as functions of the additive noise variance. Since the ellipse shape index and inclination angle are commonly used to detect and classify voltage sags, the results derived in the paper are useful for both uncertainty propagation analysis, and assessment of detection capability in case of voltage dips close to the minimum value defined in the IEEE Standard 1159. Analytical results are validated through numerical simulation of noisy voltage sags.