# Power Grid Frequency Estimation Based on Zero Crossing Technique Using Least Squares Method to Approximate Sampled Voltage Signal Around Zero Level

## Abstract

The article presents the application of the least squares method to the estimation of voltage transition points through the zero level in order to determine the frequency of the power grid system. A linear approximation of the quantized voltage signal samples near zero has been applied, which ensures effective noise suppression and reduces measurement errors. An even sampling of the sinusoidal voltage disturbed by noise was assumed, and on this basis the corresponding mathematical relations were derived. The presented algorithm for processing signal samples is computationally simple and can be easily implemented in a microprocessor system. The dependence of the accuracy of the signal frequency measurement on: SNR signal ratio, ADC converter resolution, signal sampling rate and the number of samples used for approximation were investigated. The simulation tests of the presented method were carried out and the results were presented, which enable proper design of the measurement system (ADC converter resolution, sampling rate, number of points to approximate) depending on the expected signal SNR ratio and the expected accuracy of measurements. In order to verify the presented method in practice, a measuring system was implemented using the National Instruments NI USB 6009 Data Acquisition Card and a personal computer. The measurement algorithm was implemented in the LabVIEW environment. The structure of the program was presented and the method of implementation of the most important parts of the algorithm was discussed, as well as examples of measurement results. The developed method can be used to build an independent measuring instrument or it can be an element of a larger measuring system.

## Keywords

Frequency measurement Power grid frequency Zero crossing technique Least squares method Linear approximation Sampled signal## References

- 1.IEC 61000-4-30:2015, Electromagnetic compatibility (EMC)-Part 4-30: Testing and measurement techniques - Power quality measurement methodsGoogle Scholar
- 2.Liu, Y., Yuan, Z., Markham, P.N., Conners, R.W., Liu, Y.: Application of power system frequency for digital audio authentication. IEEE Trans. Power Deliv.
**27**(4), 1820–1828 (2012)CrossRefGoogle Scholar - 3.Fundamentals of the Electronic Counters, Application Note 200, Electronic Counter Series, Hewlett Packard Co. (1997)Google Scholar
- 4.Lobos, T.: Nonrecursive methods for real-time determination of basic waveforms of voltages and currents. IEE Proc. C Gener. Transm. Distrib.
**136**(6), 347–352 (1989)CrossRefGoogle Scholar - 5.Phadke, G., Thorp, J.S., Adamiak, M.G.: A new measurement technique for tracking voltage phasors, local system frequency, and rate of change of frequency. IEEE Trans. Power Appar. Syst.
**102**(5), 1025–1038 (1983)CrossRefGoogle Scholar - 6.Moore, P.J., Allmeling, J.H., Johns, A.T.: Frequency relaying based on instantaneous frequency measurement. IEEE Trans. Power Deliv.
**11**(4), 1737–1742 (1996)CrossRefGoogle Scholar - 7.Akke, M.: Frequency estimation by demodulation of two complex signals. IEEE Trans. Power Deliv.
**12**(1), 157–163 (1997)CrossRefGoogle Scholar - 8.Lobos, T., Rezmer, J.: Real-time determination of power system frequency. IEEE Trans. Instrum. Meas.
**46**(4), 877–881 (1997)CrossRefGoogle Scholar - 9.Zhang, C., Tan, J., Kirby, B., Bo, Z.: A derivative based instantaneous frequency tracking algorithm. In: 2008 43rd International Universities Power Engineering Conference, Padova, pp. 1–3 (2008)Google Scholar
- 10.Girgis, A.A., Ham F.M.: A new FFT-based digital frequency relay for load shedding. IEEE Trans. Power Appar. Syst.
**101**(2), 433–439 (1982)CrossRefGoogle Scholar - 11.Sachdev, M.S., Giray, M.M.: A least error squares technique for determining power system frequency. IEEE Trans. Power Appar. Syst.
**104**(2), 437–444 (1985)CrossRefGoogle Scholar - 12.Hwang, J.K., Markham, P.N.: Power system frequency estimation by reduction of noise using three digital filters. IEEE Trans. Instrum. Meas.
**63**(2), 402–409 (2014)CrossRefGoogle Scholar - 13.Cai, H.: Fast frequency measurement algorithm based on zero crossing method. In: 2010 2nd International Conference on Computer Engineering and Technology, Chengdu, pp. V4-606-V4-608 (2010)Google Scholar
- 14.Vizireanu, D.N.: A simple and precise real-time four point single sinusoid signals instantaneous frequency estimation method for portable DSP based instrumentation. Measurement
**44**(2), 500–502 (2011)CrossRefGoogle Scholar - 15.Alegria, F.C., Molino-Minero-Re, E., Shariat-Panahi, S.: Evaluation of a four-point sine wave frequency estimator for portable DSP based instrumentation. Measurement
**45**(7), 1866–1871 (2012)CrossRefGoogle Scholar - 16.Mendonça, T.R.F., Pinto, M.F., Duque, C.A.: Least squares optimization of zero crossing technique for frequency estimation of power system grid distorted sinusoidal signals. In: 2014 11th IEEE/IAS International Conference on Industry Applications, Juiz de Fora, pp. 1–6 (2014)Google Scholar
- 17.Maru, K., Fujii, Y., Hessling, J.P., Shimada, K.: Frequency estimation method for measuring time-varying single frequency from digitized waveform. In: 2009 4th IEEE Conference on Industrial Electronics and Applications, Xi’an, pp. 3687–3689 (2009)Google Scholar
- 18.Wall, R.W.: Simple methods for detecting zero crossing. Industrial Electronics Society, 2003. In: IECON’03, The 29th Annual Conference of the IEEE, Vol.3, pp. 2477–2481 (2003)Google Scholar
- 19.Pawłowski, E., Warda, P.: Power grid frequency measurement in LabVIEW environment using the least mean squares method to signal phase approximation in the presence of noise. In: 2017 International Conference on Electromagnetic Devices and Processes in Environment Protection (ELMECO), Lublin, pp. 1–4 (2017)Google Scholar
- 20.Świsulski, D., Hanus, R., Zych, M., Petryka, L.: Methods of measurement signal acquisition from the rotational flow meter for frequency analysis. Eur. Phys. J. Web Conf.
**143**, 02124 (2017)CrossRefGoogle Scholar - 21.Taylor, J.R.: An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. 2nd edn. University Science Books, Sausalito, California (1997)Google Scholar