Complex-Valued Least Squares Frequency Estimation for Unbalanced Power Systems

A class of fundamental frequency estimation algorithms for unbalanced three-phase power systems is proposed. This is achieved in the complex domain by establishing and exploiting three time-series relationships among equidistantly spaced Clarke's transformed system voltage samples. To enable meaningful frequency estimation with high immunity to noise and higher order harmonic pollution, complex-valued least squares (CLS) framework is employed in conjunction with those relationships. Theoretical bias and variance analysis are further conducted to yield the optimal distance (phase interval) between the voltage samples within the models considered. It is shown that although all the CLS approaches, with their own optimal phase intervals, ideally have identical minimum mean square estimation errors, the CLS approaches based on four-point relationships are favorable in practical applications, since the optimal phase interval attains statistically unbiased frequency estimation in the presence of both noise and harmonic pollution, which is impossible when using their three-point counterpart. Simulations over a range of unbalanced system conditions and in the presence of noise and higher order harmonics, as well as for real-world measurements, support the analysis.