Since its invention, the phasor has essentially been considered as a steady-state concept. Up to now, this assumption has shaped most of the algorithms for phasor estimation. This paper breaks that old paradigm by relaxing the static phasor concept to a dynamic one, i.e., the dynamic phasor, which is one complex time function with movement freedom. This paper presents the algorithm to approximate the dynamic phasor by a second-order Taylor polynomial and compares its phasor estimate to the traditional one. This approximation leads to the definition of the phasor state vector, which contains not only the estimate of the dynamic phasor but the estimates of its derivatives as well. These new estimates improve the accuracy of oscillation estimation by including new Taylor details into the interpolation process. Errors on the order of 10^-4 are achieved with this approximation in bandpass signals over observation intervals of two cycles.

Estimates of the dynamic phasor and its derivatives are obtained through the weighted least squares solution of a Taylor approximation using classical windows as weighting factors. This solution leads to differentiators with ideal frequency response around the fundamental frequency and to very low sidelobe level over the stopband, which implies low noise sensitivity. The differentiators are maximally flat in the interval centered at the fundamental frequency and have a linear phase response. Therefore, their estimates are free of amplitude and phase distortion and are obtained at once. No further patch is needed to improve their accuracy. Examples of dynamic phasor estimates are illustrated under transient conditions. Special emphasis is put on frequency measurements.

A new dynamic harmonic estimator is presented as an extension of the fast Fourier transform (FFT), which assumes a fluctuating complex envelope at each harmonic. This estimator is able to estimate harmonics that are time varying inside the observation window. The extension receives the name “Taylor-Fourier transform (TFT)” since it is based on the McLaurin series expansion of each complex envelope. Better estimates of the dynamic harmonics are obtained due to the fact that the Fourier subspace is contained in the subspace generated by the Taylor-Fourier basis. The coefficients of the TFT have a physical meaning: they represent instantaneous samples of the first derivatives of the complex envelope, with all of them calculated at once through a linear transform. The Taylor-Fourier estimator can be seen as a bank of maximally flat finite-impulse-response filters, with the frequency response of ideal differentiators about each harmonic frequency. In addition to cleaner harmonic phasor estimates under dynamic conditions, among the new estimates are the instantaneous frequency and first derivatives of each harmonic. Two examples are presented to evaluate the performance of the proposed estimator.

Recently, the TaylorK 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 TaylorK 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 K = 0 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 K = 2 is similar to that of the Taylor² 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.

Prony's method can be used as a dynamic phasor estimator. It can be regarded as the adaptive approximation of its complex exponential signal model to the dynamic phasor of an oscillation over a finite time interval. Equipped with a closed signal model, it is possible to implement it in one cycle. In its first adaptive stage, it estimates the best damping and frequency for its signal model; and, in the second one, the best phasor over the considered time window. This paper compares the performance of Prony's method with that of the very wellknown one-cycle Fourier filter. With a higher flexibility, due to its adaptive nature, Prony estimates improve the Fourier ones under oscillation conditions. The Fourier filter can be considered as a static subclass of the Prony filters. With its static signal model, it is unable to accurately follow oscillations when the frequency fluctuates. Additionally, the Prony filter, together with its phasor estimates, provides instantaneous estimates of damping and frequency, corresponding to the first derivative of amplitude and phase, which are very useful to assess the power system stability. Finally, by being implemented in one-cycle windows, and its good rejection of the dc or exponentially attenuated components, it can also be used in protection applications.

The state-transition matrix of the Kth order Taylor approximation to the dynamic phasor and its first derivatives leads to a plurality of state-space representations to approach the bandpass signal model of a power oscillation. With these truncated signal models, the Kalman filter algorithm can be applied to their state vectors in order to find observers able to estimate the dynamic phasor and its first derivatives. The estimates obtained through this technique, from oscillatory signals, are not only instantaneous (no delay) but also synchronous, an important attribute for control applications. They also provide frequency estimates. The new filters reduce the total vector error achieved with the traditional Kalman filter; are much more stable, with settling times five times lower; and improve the phasor estimates of oscillations with frequency offset.