Accurate Estimation of Amplitude, Phase, and Frequency of a Sinusoidal Signal Contaminated with Harmonics and DC Decaying Components

Abstract

This paper proposes the modified Newton–Raphson method to estimate a sinusoidal signal's amplitude, phase, and frequency polluted with harmonics and decaying DC components. Discrete Fourier Transform (DFT) can only determine harmonics at a specific frequency. A simple method is proposed to filter DC offset (DCO) from the main signal and prepare pure sinusoidal input. This pre-processing method prepares a signal without DCO for accurate phasor estimation in DFT, widely used for frequency estimation in protection systems. The effectiveness of the proposed method for static and dynamic signals in the presence of noise was studied. The simulation studies performed in this work found that the performance of the proposed method is fully satisfactory.

Appendix

Due to space limitation derivation only some of the Jacobian coefficients of Eq. (11) are presented here:

$$ \begin{gathered} \frac{{\partial^{2} {\text{F}}}}{{\partial \omega^{2} }} = \sum\limits_{k = 1}^{N} {\left\{ {2t_{{\text{k}}}^{2} \left[ {\sum\limits_{i = 1}^{7} {{\text{i}}^{2} {\text{A}}_{{\text{i}}} \sin \left( {{\text{i}}\omega {\text{t}}_{{\text{k}}} + \phi_{{\text{i}}} } \right)} } \right]} \right.} \hfill \\ \left. {\left[ {{\text{Y}}_{{\text{k}}} - {\text{A}}_{{{\text{DC}}}}^{{{\text{e}} - \left( {\frac{{{\text{t}}_{{\text{k}}} }}{\tau }} \right)}} - \sum\limits_{i = 1}^{7} {{\text{A}}_{{\text{i}}} \sin \left( {{\text{i}}\omega {\text{t}}_{k} + \phi_{{\text{i}}} } \right)} } \right] + \sum\limits_{i = 1}^{7} {2\left( {{\text{iA}}_{{\text{i}}} {\text{t}}_{{\text{k}}} } \right)^{2} \left[ {\cos \left( {{\text{j}}\omega {\text{t}}_{{\text{k}}} + \phi_{{\text{j}}} } \right)} \right]^{2} } } \right\} \hfill \\ \end{gathered} $$

(27)

$$ \begin{gathered} \left[ {\frac{{\partial ^{2} F}}{{\partial A_{l} \partial \omega }}} \right]_{{1 \times 7}} = \sum\limits_{{k = 1}}^{N} 2 \left[ {\sum\limits_{{i = 1}}^{7} {ji} t_{k} \cos \left( {i\omega t_{k} + \phi _{i} } \right)\cos \left( {j\omega t_{k} + \phi _{j} } \right)} \right] \hfill \\ - 2t_{k} j\cos \left( {j\omega t_{k} + \phi _{j} } \right)\left[ {Y - A_{{DC}} e^{{ - \left( {\frac{{t_{k} }}{\tau }} \right)}} - \sum\limits_{{i = 1}}^{7} {A_{i} } \sin \left( {i\omega t_{k} + \phi _{i} } \right)} \right];j = 1,2,\ldots 7 \hfill \\ \end{gathered} $$

(28)

$$ \begin{gathered} \frac{{\partial ^{2} {\text{F}}}}{{\partial \omega \partial \phi _{{\text{l}}} }} = \sum\limits_{{{\text{k}} = 1}}^{{\text{N}}} {\left\{ {2\left[ {\sum\limits_{{{\text{i}} = 1}}^{7} {\text{i}} {\text{t}}_{{\text{k}}} {\text{A}}_{{\text{i}}} {\text{cos}}\left( {{\text{i}}\omega {\text{t}}_{{\text{k}}} + \phi _{{\text{i}}} } \right){\text{A}}_{{\text{l}}} {\text{cos}}\left( {{\text{l}}\omega {\text{t}}_{{\text{k}}} + \phi _{{\text{l}}} } \right)} \right]} \right.} \hfill \\ \left. { + 2{\text{lA}}_{{\text{l}}} {\text{t}}_{{\text{k}}} {\text{sin}}\left( {{\text{l}}\omega {\text{t}}_{{\text{k}}} + \phi _{{\text{l}}} } \right)\left[ {{\text{Y}} - {\text{A}}_{{{\text{DC}}}} {\text{e}}^{{ - \left( {\frac{{{\text{t}}_{{\text{k}}} }}{\tau }} \right)}} - \sum\limits_{{{\text{i}} = 1}}^{7} {{\text{A}}_{{\text{i}}} } {\text{sin}}\left( {{\text{i}}\omega {\text{t}}_{{\text{k}}} + \phi _{{\text{i}}} } \right)} \right]} \right\};{\text{i}} = 1,2,\ldots 7 \hfill \\ \end{gathered} $$

(29)

$$ \left[ {\frac{{\partial ^{2} {\text{F}}}}{{\partial \phi _{{\text{i}}} \partial \phi _{{\text{j}}} }}} \right]_{{7 \times 7}} = \sum\limits_{{{\text{k}} = 1}}^{{\text{N}}} 2 {\text{A}}_{{\text{i}}} {\text{A}}_{{\text{j}}} {\text{cos}}\left( {{\text{i}}\omega {\text{t}}_{{\text{k}}} + \phi _{{\text{i}}} } \right){\text{cos}}\left( {{\text{j}}\omega {\text{t}}_{{\text{k}}} + \phi _{{\text{j}}} } \right){\text{i}}\,{\text{and}}\,j = 1,2, \ldots , 7 $$

(30)

$$\frac{{\partial }^{2}{\text{F}}}{\partial {\upomega \partial {\text{A}}}_{{\text{DC}}}}= \sum_{{\text{k}}=1}^{{\text{N}}}\left\{2 {{\text{t}}}_{{\text{k}}} {{\text{e}}}^{- \left(\frac{{{\text{t}}}_{{\text{k}}}}{\uptau }\right)}\left[\sum_{{\text{i}}=1}^{7}{{\text{iA}}}_{{\text{i}}}\mathrm{cos }\left(\mathrm{i\omega }{{\text{t}}}_{{\text{k}}}+{\upphi }_{{\text{i}}}\right)\right] \right\}$$

(31)

$$\frac{{\partial }^{2}{\text{F}}}{\partial\upomega \partial\uptau }= \sum_{{\text{k}}=1}^{{\text{N}}}\left\{2 {{\text{t}}}_{{\text{k}}} {{\text{e}}}^{- \left(\frac{{{\text{t}}}_{{\text{k}}}}{\uptau }\right)}\left(\frac{{{\text{t}}}_{{\text{k}}}}{{\uptau }^{2}}\right)\left[\sum_{{\text{i}}=1}^{7}{{\text{iA}}}_{{\text{i}}}\mathrm{cos }\left(\mathrm{i\omega }{{\text{t}}}_{{\text{k}}}+{\upphi }_{{\text{i}}}\right)\right]\right\}$$

(32)

$$\frac{{\partial }^{2}{\text{F}}}{{\partial {\text{A}}}_{{\text{i}}}\partial\uptau }=\sum_{{\text{k}}=1}^{{\text{N}}}2 {{\text{A}}}_{{\text{DC}}} {{\text{e}}}^{- \left(\frac{{{\text{t}}}_{{\text{k}}}}{\uptau }\right)}\left(\frac{{{\text{t}}}_{{\text{k}}}}{{\uptau }^{2}}\right)\mathrm{ sin}\left(\mathrm{i\omega }{{\text{t}}}_{{\text{k}}}+{\upphi }_{{\text{i}}}\right);\mathrm{i }=1, 2, \dots ..7$$

(33)