Taylor–Kalman–Fourier Filters for Instantaneous Oscillating Phasor and Harmonic Estimates

Abstract

Recently, the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\hbox{Taylor}^{K}$</tex></formula> Kalman filter was proposed for estimating instantaneous oscillating phasors. Its performance was examined through time-domain simulations using the benchmark test signals specified in the IEEE Standard for Synchrophasors for Power Systems. It was discovered that the estimation error level was abruptly reduced by a factor of ten from the second order, mainly because those filters were able to provide instantaneous phasor estimates. In this paper, the frequency response of the zeroth- and second-order filters is established and illustrated. They demonstrate that, for orders greater than or equal to two, the filters are able to form zero flat phase response about the operation frequency and then able to provide instantaneous estimates. By assessing the behavior of the estimates before signals with harmonics, or noise, not contemplated in the signal model, the frequency response method leads us to design more robust filters, referred to as <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\hbox{Taylor}^{K}$</tex></formula> Kalman–Fourier, because they incorporate the whole set of harmonics in their multiharmonic signal model. It turns out that the bank of comb filters achieved with <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$K = 0$</tex></formula> is equivalent to that of the discrete Fourier transform, with a computational cost of one and a half products per harmonic estimate, which is lower than the FFT cost for more than eight components, and the bank of fence filters obtained with <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$K = 2$</tex></formula> is similar to that of the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$\hbox{Taylor}^{2}$</tex></formula> Fourier transform but with the advantage of providing estimates devoid of delay and needing only four products per harmonic set of estimates. Due to their instantaneous character, and computational simplicity, those estimates are certainly very useful for real-time harmonic analysis and power system control applications.