Higher-Order Statistics Extension to Three-Phase Systems in the Power Quality Framework

In this paper, Higher-Order Statistics has been proposed to characterize the voltage quality of a three-phase system by evaluating the moment vectors of order two, three and four of a multivariate probability distribution function. This approach represents the natural extension of the well know single-phase HOS theory applied to the PQ framework which has been widely adopted for both power quality and reliability applications. The motivations to move towards a three-phase approach to higher-order statistics are twofold: i) the extension to three-phase allows to uniquely characterize the whole system ii) the adoption of a multivariate statistics avoids misleading considerations that can arise by considering as independent random variables the three-phase quantities. In particular, the reference moments vectors were calculated for a multivariate probability distribution function defined by a symmetric three-phase systems. Then, a method to measure the variance, the skewness and the kurtosis based on the $L$2-norm considering only the independent moment vectors elements has been presented so that a global PQ index has been defined for the entire three-phase system. Numerical simulations showed the applicability of the three-phase HOS approach proposing a more correct and comprehensive formulation in three-phase system under distorted and asymmetry conditions with respect to the single-phase approach.