# FREQUENCY ESTIMATION OF DISTORTED POWER SYSTEM SIGNALS USING EXTENDED COMPLEX KALMAN FILTER

The paper proposes an extended complex Kalman filter and employs it for the estimation of power system frequency in the presence of random noise and distortions. From the discrete values of the 3-phase voltage signals of a power system, a complex voltage vector is formed using the well known @-transform. A nonlinear state space formulation is then obtained for this complex signal and an extended Kalman filtering approach is used to compute the true state of the model iteratively with significant noise and harmonic distortions. As the frequency is modeled as a state, the estimation of the state vector yields the unknown power system frequency. Several computer simulations test results are presented in the paper to highlight the usefulness of this approach in estimating near nominal and off-nominal power system frequencies.

Digital control and protection of power systems require the estimation of supply frequency and its variation in real-time. Variations in system frequency from its normal value indicate the occurrence of a corrective action for its restoration. A large number of numerical methods is available for frequency estimation from the digitized samples of the system voltage. Conventional methods assume that the power system voltage waveform is purely sinusoidal and therefore the time between two zero crossings is an indication of system frequency. Discrete Fourier transforms Kalman filtering, Recursive Newton-type algorithm , Adaptive notch filters etc. are known signal processing techniques used for frequency measurements of power system signals. A new numeric technique and its practical implementation are presented in reference 171. This approach suffers from inaccuracies due to the presence of noise and harmonics. An iterative technique for fast and accurate estimation of nominal and off-nominal power system frequency has been presented in [SI. This technique requires a correct guess of the system frequency for fast estimation and suffers from inaccuracies in the presence of noise (with an SNR value of 20 dB or less) and harmonics. Amongst the several numerical techniques described above both linear and extended Kalman filtering approaches have attracted widespread attention, as they accurately estimate the amplitude, phase and frequency of a signal buried with noise and harmonics.

In this paper, a variation of nonlinear Kalman filter in the complex form is presented which simplifies the modeling requirement for frequency and amplitude estimation of a signal. It has been recently shown in reference that the extended complex Kalman filter (ECKF) is more attractive than the real one from the point of view of modeling and stability considerations. The discrete values of the three phase voltage signals of a power system are transformed into a complex vector using the well known ap-transform used in power system analysis. This complex voltage vector is then modeled along with frequency in a nonlinear state-space form and the theory of extended Kalman filter is used to obtain the state vectors iteratively. The computation of Kalman gain and choice of initial covariance matrix is crucial in determining the speed of convergence of the new algorithm and its noise rejection property. A variety of simulated power system conditions is used €or the application of this new technique and frequency estimation error is close to .01 Hz to .02 Hz in most cases. The application of this algorithm for frequency relaying in power system is expected to be simple with very little computation for 2-state complex Kalman filter

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