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A new measurement algorithm for PMU in power system based on allphase Fourier transform
EURASIP Journal on Wireless Communications and Networkingvolume 2019, Article number: 165 (2019)
Abstract
One new vector estimation approach for phasor measurement units (PMUs) within power system is put forward, which is built upon the basis of apFFT, namely allphase fast Fourier transform. On account of the remarkable accomplishment of apFFT when subduing property of “phase invariant” and spectral leakage, the estimation approach is capable of achieving quick evaluation of PMU phase angles, and subsequently, the spectrum correction of timeshift phase discrepancy is used so as to evaluate the amplitude and frequency parameters. The calculation flow of the algorithm is given, and the hardware design of the new PMU measuring system is completed by using ARM9 microprocessor. Simulation result indicates that the method in this paper is roughly the same to an estimator without bias for noisefree conditions, and this calculation precision is also better than the existing approach.
Introduction
Against the background of increasing technical growth and usage of smart grid, stability and security of electric power system is of particular significance [1, 2]. How to construct the dynamic stability monitoring and control system for whole power grid is a problem urgent to be solved. Phasor measurement unit is built on the basis of the GPS, namely the global position system, and great accuracy timing signals, which enjoys widespread applications within widearea measurement system so as to achieve simultaneous sampling of various nodes in the system mentioned above [3]. Various kinds of phasor measurement units (PMUs) get used in areas containing online parameter estimation, widearea protection, fault location, and power grid dynamic calculation [4,5,6], and the PMU algorithm’s performance is prone to exert direct influence on the capability of this system.
Phasor measurement algorithms chiefly get created so as to evaluate the phase angle, amplitude, and frequency parameters of electric power system, mainly including discrete Fourier transform method, zerocrossing detection method, and Kalman filtering method [7, 8]. Because of the wonderful harmonic suppression of discrete Fourier transform (DFT) for stationary signal, this method is widely applied at present. However, when the operating frequency of the system deviates from 50 Hz, the receiving data of PMUs cannot satisfy the condition of integral period sampling, and the frequency aliasing and spectral leakage of DFT method will cause big errors in parameter estimation. In paper [9], length of the data window gets designed to function as a variable so as to promote frequency estimation accuracy, while the computational complexity gets great increase. In paper [10], sampling frequency gets regulated flexibly by system operating frequency so as to meet the restriction of integerperiod sampling. Nevertheless, bay level IEDs cannot get direct command of the sampling rate of process level in typical substations, making a problem of practical applications. Paper [11] presents one vector measurement approach on the basis of recursive DFT, which reduces the estimation error of frequency significantly. But this method cannot solve the problem of spectral leakage or frequency aliasing also, and the amplitude of signal is assumed to be fixed, which is not suitable for dynamic condition.
In this paper, a new power system vector measuring algorithm on the basis of allphase fast Fourier transform (apFFT) [12,13,14] is proposed, in which phase, frequency, and amplitude parameters of received PMU signal can be estimated synchronously, and the input data need not be sampled by strictly integrated periods.
The remaining part of this paper is designed as shown below: Section 2 will give a brief description about the relevant concepts of allphase spectral analysis, and the evaluation performance of phase parameters for deterministic condition will come up in Section 3. In Section 4, timeshift phase difference correcting method is put forward so as to evaluate the amplitude and frequency of observed signals. Finally, simulation results and conclusion are presented respectively.
Description of current analysis method
The apFFT’s data process flow is shown in Fig. 1.
From this figure, convolution window w_{c} = [w_{c}(− N + 1),…, w_{c}(− 1), w_{c}(0), w_{c}(1),…, w_{c}(N − 1)]^{T} gets framed by front window f convolved with reversal back window b, which is the result of the crosscorrelation operation of two window sequences with length N, counted by:
Here, the front window f and the back window b are usually required to be the same symmetrical window, and none of them are rectangular windows. Due to the symmetry of f and b, the convolution window w_{c} is also a symmetrical window as w_{c}(n) = w_{c}(− n).
The FT of w_{c} is as follows:
It is found that the Fourier transform of the whole N types of data units containing point x(0) are taken into full consideration in the apFFT result, which brings many excellent performance such as flat phase distribution in the vicinity of the spectral peak and suppression of amplitude spectral leakage. Besides, the FFTs of various data units are taken into account, and the calculation of apFFT is realized by simply one FFT, significantly improving the computational efficiency.
For the applications of phasor measurement in power system, the measured current and voltage signals often have typical stationary signal characteristics and are mostly singlefrequency signals. For this type of input data, the window function used for data preprocessing is a key factor to improve the performance of the algorithm.
Take the most common cosine voltage amplitude signal as an example, the sampling point N = 64, frequency sampling interval Δω = 2π/64 rad/s, and three single frequency cosine signals have digital angular frequencies of 3.25Δω, 3.375Δω, and 3.5Δω, respectively. The three groups of data were preprocessed with nowindow, singlewindow, and doublewindow functions, then DFT operations were performed and the resulting apFFT discrete spectrum is shown in Fig. 2.
From the comparison results, for the singlefrequency sampling data of power system, the preprocessing of window function can effectively suppress the energy leakage caused by the offset of the sampling frequency. In particular, when the frequency offset reaches 0.5Δω, the results of the doublewindow apFFT spectrum are basically only focused on the two main spectral lines. In the absence of nowindow and singlewindow, besides the main spectral lines, there are also some side spectral leaks. Based on this situation, this paper chooses the doublewindow allphase data preprocessing scheme and selects the Hanning window function with better performance against highfrequency interference.
Phase estimation of deterministic signal
Think about one compound test signal including 3 elements with different signal parameters:
where ω_{1} = 20.0, ω_{2} = 60.2, ω_{3} = 100.4, φ_{1} = 10^{°}, φ_{2} = 50^{°}, φ_{3} = 90^{°}, and N = 256. Figures 3 and 4 have shown the windowed FFT of x(n) and the doublewindow apFFT of x(n), respectively, and risingcosine window w_{h} is used as a window function with an expression as follows:
From the figures listed above, apFFT enjoys a greater feature of spectrum decrease of leakage in comparison with FFT, the leakage of energy because of noninteger process sampling, and cross talks among various frequency elements get significantly inhibited within 2 spectral lines.
When it comes to the phase spectrum, conventional FFT spectrum is irregular, and the spectral value approaches theoretical value 10° at position k = 20 (integral period sampling point), but there is a big deviation between measured phase and real phase at the points k = 60 and k = 100. The apFFT spectrum shows one organized allocation nearby object frequency, that is, the feature of “phase invariant.” Using this advantage, the initial phases of all signal components can be estimated without any error correction. The phase measurement of these two approaches gets listed in Table 1.
Frequency and amplitude estimation
Similar to discrete Fourier transform, the barrier effect of apFFT spectral analysis also exists. If the length of calculation samples is N, then the minimum resolution of digital angular frequency Δω = 2π/N, and the real frequency position of signal lies between the two adjacent spectrum lines with the interval of Δω.
In the paper, data intercepting approach gets improvement, and brandnew timeshift phase discrepancy correcting method is put forward so as to evaluate PMU vectors’ frequency and amplitude. Detailed procedures get shown below:

Received signal is taken as a sample (perhaps not be integer process sampled) and falls into 2 sets, including x_{1}(n) and x_{2}(n) having set length 2N + 1. Time delay of x_{1}(n) and x_{2}(n) is n_{0}:

Perform doublewindowed apFFT concerning x_{1}(n) and x_{2}(n), and phase difference between them given by Eq. (7) is obtained from phase spectrum in the major spectral line k^{*}, in which 2n_{0}k^{*}π/N is the compensation value of digital frequency 2 k^{*}π/N of spectrum k^{*} with time delay n_{0}:

On the basis of Eq. (7), the evaluation of frequency with phase compensation and frequency deviation at spectral line k^{*} gets generated respectively as Eqs. (8) and (9):

Calculate the signal amplitude by deviation of frequency dω; in the case of doublewindow apFFT, the evaluation equation is:

In the paper, the Hanning window is used for window function, then the amplitude parameter is adjusted to Eq. (11):
All in all, the processing flow of proposed PUM vector measuring algorithm is offered (Fig. 5).
System hardware design
In order to implement the algorithm, the optimal implementation of PMU measurement system based on ARM9 processor is shown in Fig. 6.
The system consists of frontend analog access circuit, signal conditioning circuit, A/D converter, ARM9 LPC3250 microprocessor, and external system. The frontend analog access circuit is responsible for isolating the measured AC signal before accessing the system and improving the security of the entire measurement system. The analog signal conditioning circuit consists of an integrated operational amplifier and a digital potentiometer, the voltage and current signals are amplified, and the digital potentiometer performs feedback resistance adjustment under the control of the ARM processor to achieve automatic variable range measurement. The ADS8568 type 8 channel synchronous sampling converter is used in the A/D conversion circuit. An industrial class ARM9 microcontroller LPC3250 is used as the control and operation core of the PMU measurement system. The external SPItype flash memory is used to store measurement information and system parameters. The Ethernet port is used to achieve communication with the scheduling terminal. SDRAM is used to store and execute the test master program code.
Simulation results and discussion
Experiment 1: Performance of algorithm under harmonics condition
A highprecision threephase synchronized vector measuring device is used as signal generator in this experiment. The essential frequency phase parameters of test signal get fixed with 50.5 Hz and 40^{°}, and two harmonic signal elements with frequency 200 Hz and 300 Hz also get included in the test signal, as Eq. (12).
x(t)’s doublewindow apFFT is given in Fig. 7.
Evaluation approaches designed in this paper and paper [9] get applied so as to evaluate the phase angle, frequency, and amplitude of this signal, and estimation results are given in Table 2.
Experiment 2: Performance of algorithm under noise condition
White Gaussian noise is injected into signal x(t) with signaltonoise (SNR) 0~30 dB, and the testing flow is repeated. The root mean square error (RMSE) of above two parameter estimation methods is shown in Fig. 8 versus SNR, where the estimation RMSE of frequency, phase angle, and amplitude parameters is defined as follows:
From the simulation outcomes mentioned above and comparison between the DFT and FFT method, the apFFT approach of the paper gets a greater performance in terms of accuracy and noise suppression. The algorithm of this paper can gain 7 dB signaltonoise ratio gain in comparison with the traditional approach.
Conclusions
The accomplishment of PMU vector estimation algorithm tends to exert direct influence on the dependability of power system applications, like controlling, measuring, and relay protection. In the paper, one brandnew vector measuring approach for power system is put forward on the basis of allphase spectrum analysis. In this approach, noticed data do not need to be taken as samples by thoroughly integrated processes, and the spectral leakage and barrier influence get inhibited in a significant manner. Simulation outcomes indicate that the apFFT approach enjoys a greater property in calculation accuracy and noise controlling than existing approach. There is a possibility that this approach will get widespread applications in ranges of harmonic and vector measurement explanation within power system automation, enjoying huge value of researching.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
Abbreviations
 apFFT:

Allphase fast Fourier transform
 DFT:

Discrete Fourier transform
 IED:

Intelligent Electronic Device
 PMU:

Phasor measurement unit
 RMSE:

Root mean square error
 SNR:

Signaltonoise ratio
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Acknowledgements
There are no other participants in this work except those in the author’s list.
Funding
The research was supported by the technology program of State Grid Hebei Electric Power Research Institute (No. KJ2017016).
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PL conceived and created the research, then performed the simulations, and wrote the draft. HF and SZ reviewed and corrected the manuscript. All writers read and approved the final manuscript.
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Correspondence to Peng Luo.
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Keywords
 Phasor measurement unit (PMU)
 Vector estimation
 Allphase fast Fourier transform (apFFT)
 Phase invariant
 ARM9 microprocessor