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Abstract

In this paper, a discrete wavelet transform (DWT) based approach is proposed for power system frequency estimation. Unlike the existing frequency estimators mainly used for power system monitoring and control, the proposed approach is developed for fundamental frequency estimation in the field of energy metering of nonlinear loads. The characteristics of a nonlinear load is that the power signal is heavily distorted, composed of harmonics, inter-harmonics and corrupted by noise. The main idea is to predetermine a series of frequency points, and the mean value of two frequency points nearest to the power system frequency is accepted as the approximate solution. Firstly the input signal is modulated with a series of modulating signals, whose frequencies are those frequency points. Then the modulated signals are decomposed into individual frequency bands using DWT, and differences between the maximum and minimum wavelet coefficients in the lowest frequency band are calculated. Similarities among power system frequency and those frequency points are judged by the differences. Simulation results have proven high immunity to noise, harmonic and inter-harmonic interferences. The proposed method is applicable for real-time power system frequency estimation for electric energy measurement of nonlinear loads.

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Authors and Affiliations

Zhang Peng
Hong-Bin Li
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Abstract

In this paper, two new sinusoidal signal frequency estimators calculated on the basis of four equally spaced signal samples are presented. These estimators are called four-point estimators. Simulation and experimental research consisting in signal frequency estimation using the invented estimators have been carried out. Simulation has also been performed for frequency tracking. The simulation research was carried out applying the MathCAD computer program that determined samples of a sinusoidal signal disturbed by Gaussian noise. In the experimental research, sinusoidal signal samples were obtained by means of a National Instruments PCI-6024E data acquisition card and an Agilent 33220A function generator. On the basis of the collected samples, the values of four-point estimators invented by the authors and, for comparison, the values of three- and four-point estimators proposed by Vizireanu were determined. Next, estimation errors of the signal frequency were determined. It has been shown that the invented estimators can estimate a signal frequency with greater accuracy.
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Authors and Affiliations

Sergiusz Sienkowski
Mariusz Krajewski
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Abstract

A new solution to the problem of frequency estimation of a single sinusoid embedded in the white Gaussian noise is presented. It exploits, approximately, only one signal cycle, and is based on the well-known 2nd order autoregressive difference equation into which a downsampling is introduced. The proposed method is a generalization of the linear prediction based Prony method for the case of a single undamped sinusoid. It is shown that, thanks to the proposed downsampling in the linear prediction signal model, the overall variance of the least squares solution of frequency estimation is decreased, when compared to the Prony method, and locally it is even close to the Cramér–Rao Lower Bound, which is a significant improvement. The frequency estimation variance of the proposed solution is comparable with, computationally more complex, the Matrix Pencil and the Steiglitz–McBride methods. It is shown that application of the proposed downsampling to the popular smart DFT frequency estimation method also significantly reduces the method variance and makes it even better than the least squares smart DFT. The noise immunity of the proposed solution is achieved simultaneously with the reduction of computational complexity at the cost of narrowing the range of measured frequencies, i.e. a sinusoidal signal must be sufficiently oversampled to apply the proposed downsampling in the autoregressive model. The case of 64 samples per period with downsampling up to 16, i.e. 1/4th of the cycle, is presented in detail, but other sampling scenarios, from 16 to 512 samples per period, are considered as well.
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Bibliography

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[4] Zielinski, T. P., & Duda, K. (2011). Frequency and damping estimation methods - an overview. Metrology and Measurement Systems, 18(3), 505–528. https://doi.org/10.2478/v10178-011-0051-y
[5] Duda, K., & Zielinski, T. P. (2013). Efficacy of the frequency and damping estimation of a real-value sinusoid. IEEE Instrumentation & Measurement Magazine, 16(1), 48–58. https://doi.org/10.1109/ MIM.2013.6495682
[6] Borkowski, J., Kania, D., & Mroczka, J. (2018). Comparison of sine-wave frequency estimation methods in respect of speed and accuracy for a few observed cycles distorted by noise and harmonics. Metrology and Measurement Systems, 25(1), 283–302. https://doi.org/10.24425/119567
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Authors and Affiliations

Krzysztof Duda
1
Tomasz P. Zieliński
2

  1. AGH University of Science and Technology, Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, Department of Measurement and Electronics, al. Mickiewicza 30, 30-059 Kraków, Poland
  2. AGH University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Institute of Telecommunications, al. Mickiewicza 30, 30-059 Kraków, Poland
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Abstract

This paper presents a new simple and accurate frequency estimator of a sinusoidal signal based on the signal autocorrelation function (ACF). Such an estimator was termed as the reformed covariance for half-length autocorrelation (RC-HLA). The designed estimator was compared with frequency estimators well-known from the literature, such as the modified covariance for half-length autocorrelation (MC-HLA), reformed Pisarenko harmonic decomposition for half-length autocorrelation(RPHD-HLA), modified Pisarenko harmonic decomposition for half-length autocorrelation (MPHD-HLA), zero-crossing (ZC), and iterative interpolated DFT (IpDFT-IR) estimators. We determined the samples of the ACF of a sinusoidal signal disturbed by Gaussian noise (simulations studies) and the samples of the ACF of a sinusoidal voltage(experimental studies), calculated estimators based on the obtained samples, and computed the mean squared error(MSE) to compare the estimators. The errorswere juxtaposed with the Cramér–Rao lower bound (CRLB). The research results have shown that the proposed estimator is one of the most accurate, especially for SNR > 25 dB. Then the RC-HLA estimator errors are comparable to the MPHD-HLA estimator errors. However, the biggest advantage of the developed estimator is the ability to quickly and accurately determine the frequency based on samples collected from no more than five signal periods. In this case, the RC-HLA estimator is the most accurate of the estimators tested.

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Authors and Affiliations

Sergiusz Sienkowski
Mariusz Krajewski
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Abstract

This paper presents a simple DFT-based golden section searching algorithm (DGSSA) for the single tone frequency estimation. Because of truncation and discreteness in signal samples, Fast Fourier Transform (FFT) and Discrete Fourier Transform (DFT) are inevitable to cause the spectrum leakage and fence effect which lead to a low estimation accuracy. This method can improve the estimation accuracy under conditions of a low signal-to-noise ratio (SNR) and a low resolution. This method firstly uses three FFT samples to determine the frequency searching scope, then – besides the frequency – the estimated values of amplitude, phase and dc component are obtained by minimizing the least square (LS) fitting error of three-parameter sine fitting. By setting reasonable stop conditions or the number of iterations, the accurate frequency estimation can be realized. The accuracy of this method, when applied to observed single-tone sinusoid samples corrupted by white Gaussian noise, is investigated by different methods with respect to the unbiased Cramer-Rao Low Bound (CRLB). The simulation results show that the root mean square error (RMSE) of the frequency estimation curve is consistent with the tendency of CRLB as SNR increases, even in the case of a small number of samples. The average RMSE of the frequency estimation is less than 1.5 times the CRLB with SNR = 20 dB and N = 512.
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Authors and Affiliations

Xin Liu
Yongfeng Ren
Chengqun Chu
Wei Fang
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Abstract

Based on real-time multi-domain communication signal analysis architecture, a high-efficiency blind carrier frequency estimation algorithm using the power spectrum symmetry of the measured modulated signal is presented. The proposed algorithm, which utilizes the moving averaged power spectrum achieved by the realtime spectrum analysis, iteratively identifies the carrier frequency in according to the power difference between the upper sideband and lower sideband, which is defined and revised by the estimated carrier frequency in each iteration. When the power difference of the two sidebands converges to the preset threshold, the carrier frequency can be obtained. For the modulation analysis, the measured signal can be coarsely compensated by the estimated result, and the residual carrier frequency error is eliminated by a following carrier synchronization loop. Compared with previous works, owing to the moving averaged power spectrum normalization and the smart iterative step variation mechanism for the two sidebands definition, the carrier frequency estimation accuracy and speed can be significantly improved without increasing the computational effort. Experimental results are included to demonstrate the outstanding performance of the proposed algorithm.

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Authors and Affiliations

Qian Wang
Xiaomei Yang
Xiao Yan
Kaiyu Qin
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Abstract

International standards from IEC and IEEE regulate power grid parameters such as theRMSvalue, frequency, harmonic and interharmonic distortion, unbalance or the presence of transients, that are important to assure the quality of distributed power. Standard IEC 61000-4-30 suggests the zero crossing algorithm for the measurement of the power grid frequency, but also states that different algorithms can be used. This paper proposes a new algorithm, the Fractional Interpolated Discrete Fourier Transform, FracIpDFT, to estimate the power grid frequency, suitable for implementation in resource limited embedded measurement systems. It is based on the non-integer Goertzel algorithm followed by interpolation at non-integer multiples of the DFT frequency resolution. The proposed algorithm is validated and its performance compared with other algorithms through numerical simulations. Implementation details of the FracIpDFT in an ARM Cortex M4 processor are presented along with frequency measurement results performed with the proposed algorithm in the developed system.
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Authors and Affiliations

Nuno M. Rodrigues
1
Fernando M. Janeiro
2
Pedro M. Ramos
1

  1. Instituto de Telecomunicações, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal
  2. Instituto de Telecomunicações, Universidade de Évora, 7000-671 Évora, Portugal
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Abstract

This overview paper presents and compares different methods traditionally used for estimating damped sinusoid parameters. Firstly, direct nonlinear least squares fitting the signal model in the time and frequency domains are described. Next, possible applications of the Hilbert transform for signal demodulation are presented. Then, a wide range of autoregressive modelling methods, valid for damped sinusoids, are discussed, in which frequency and damping are estimated from calculated signal linear self-prediction coefficients. These methods aim at solving, directly or using least squares, a matrix linear equation in which signal or its autocorrelation function samples are used. The Prony, Steiglitz-McBride, Kumaresan-Tufts, Total Least Squares, Matrix Pencil, Yule-Walker and Pisarenko methods are taken into account. Finally, the interpolated discrete Fourier transform is presented with examples of Bertocco, Yoshida, and Agrež algorithms. The Matlab codes of all the discussed methods are given. The second part of the paper presents simulation results, compared with the Cramér-Rao lower bound and commented. All tested methods are compared with respect to their accuracy (systematic errors), noise robustness, required signal length, and computational complexity.

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Authors and Affiliations

Tomasz Zieliński
Krzysztof Duda
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Abstract

The paper deals with frequency estimation methods of sine-wave signals for a few signal cycles and consists of two parts. The first part contains a short overview where analytical error formulae for a signal distorted by noise and harmonics are presented. These formulae are compared with other accurate equations presented previously by the authors which are even more accurate below one cycle in the measurement window. The second part contains a comparison of eight estimation methods (ESPRIT, TLS, Prony LS, a newly developed IpDFT method and four other 3-point IpDFT methods) in respect of calculation time and accuracy for an ideal sine-wave signal, signal distorted by AWGN noise and a signal distorted by harmonics. The number of signal cycles is limited from 0.1 to 3 or 5. The results enable to select the most accurate/ fastest estimation method in various measurement conditions. Parametric methods are more accurate but also much slower than IpDFT methods (up to 3000 times for the number of samples equal to 5000). The presented method is more accurate than other IpDFT methods and much faster than parametric methods, which makes it possible to use it as an alternative, especially in real-time applications.
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Authors and Affiliations

Józef Borkowski
Dariusz Kania
Janusz Mroczka
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Abstract

Fast and accurate grid signal frequency estimation is a very important issue in the control of renewable energy systems. Important factors that influence the estimation accuracy include the A/D converter parameters in the inverter control system. This paper presents the influence of the number of A/D converter bits b, the phase shift of the grid signal relative to the time window, the width of the time window relative to the grid signal period (expressed as a cycle in range (CiR) parameter) and the number of N samples obtained in this window with the A/D converter on the developed estimation method results. An increase in the number b by 8 decreases the estimation error by approximately 256 times. The largest estimation error occurs when the signal module maximum is in the time window center (for small values of CiR) or when the signal value is zero in the time window center (for large values of CiR). In practical applications, the dominant component of the frequency estimation error is the error caused by the quantization noise, and its range is from approximately 8×10-10 to 6×10-4.

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Authors and Affiliations

Janusz Mroczka
Józef Borkowski
Dariusz Kania

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