On the problem of optimal estimation of balanced and symmetric three-phase signals

This paper revisits the fundamental problem of optimal estimation of the magnitude and phase of balanced and symmetric three-phase voltage or current signals. We analyze and compare various setups for the corresponding optimal Kalman filter, including the direct use of three-phase measurements, as well as measurements subjected to the Clarke transform in real or complex form. One contribution is to show that the standard practice of disregarding the transformed zero-component of the Clarke transformed three-phase signal almost always leads to a sub-optimal performance of the Kalman estimator. Our analysis extends to show that the closely related complex Kalman estimator is also sub-optimal and that optimal performance can be recovered if the zero-component is made available to the filter provided that the noises are properly characterized. These results are illustrated by means of simple numerical examples, which also highlight the importance of correctly modeling the noise characteristics if a real or complex form of the Clarke transformation is to be used. We conclude the paper with a unified set of guidelines or best practices regarding the use of optimal Kalman estimators for balanced and symmetric three-phase signals.