The 6th International Conference on Renewable Power Generation (RPG) 19 20 October 2017 Sub/super-synchronous harmonics measurement method based on PMUs Hao Liu, Sudi Xu, Tianshu Bi, Chuang Cao State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources, North China Electric Power University, Beijing, People s Republic of China E-mail: hliu@ncepu.edu.cn Published in The Journal of Engineering; Received on 9th October 2017; Accepted on 2nd November 2017 Abstract: In recent years, there are more and more inter-harmonics in power system, with the rapid development of new energy and the application of the quantity power electronic equipment in power grid. These lead to the sub-synchronous oscillation having a negative effect on the stable of power grid. Nowadays, phasor measurement units (PMUs) are deployed in the power system in large quantities, which provides a possibility to monitor the inter-harmonics in grid. In this paper, a reduction method of sub/super-synchronous harmonics based on PMU phasor data is proposed. To decompose the sub/super-synchronous harmonics, first spectrum analysis on complex phasor sequences should be done. Second, an idea is proposed that calculates the frequency and amplitude of inter-harmonics separately. That is to say, the frequency can be got by spectrum analysis while zero-padding in short time window, and the amplitude can be calculated by solving equations. This can overcome the shortcoming of spectrum analysis which has larger amplitude error. Simulation and experimental tests show that the frequency and amplitude of inter-harmonics can be calculated using the proposed algorithm. 1 Introduction In recent years, renewable energy such as wind and solar energy has developed rapidly around the world, while power electronic equipment has been deployed in large quantities [1]. However, they lead to many inter-harmonics in power grid, which takes disadvantage of the stability of power system. Since July 2015, in some areas of western China, there are many sub-synchronous oscillations happening in the new energy pool area, which affected severely the production and life of local people. Therefore, it is a must to develop the technology to monitor the sub-synchronous oscillation. Up to now, phasor measurement units (PMUs) are deployed in the 220 kv and above substation in China, which is a possibility to develop the technology of monitoring the sub-synchronous oscillation based on PMUs [National Key R&D Program (2017YFB)]. The inter-harmonics information can be got through phasor data because of high reporting rate of PMUs. However, due to the synchronisation feature of PMUs, a comparative analysis of interharmonics information can be done in the same time. Meanwhile, the analysis process is only done in the master station when using the phasor data, so it is very convenient and easy to complete it. With the popularity of PMUs in the power system, many applications are arisen based on the phasor data [2 5], such as monitoring low-frequency oscillation [6], parameters identification [7] and model validation. However, the applications of sub-synchronous oscillation are far less. To increase the measurement accuracy of PMUs, many relevant international and national standards have been released, and some researches have been also done [8, 9]. Especially many new algorithms have been proposed to improve the dynamic characteristics of PMUs [10 12]. A dynamic phasor model is set in from [13, 14], and a method based on discrete Fourier transform (DFT) is proposed to measure and suppress harmonics. However, these methods only target harmonics and interharmonics above 100. In [15], it is analysed how the interharmonics in grid affect the phasor data, and spectrum characteristics of the phasor data are indicated, which provides the basis for reduction method based on the phasor data. However, the researches on reducing inter-harmonics information in grid based on the phasor data are few. A reduction method of sub/super-synchronous harmonics based on the PMU phasor data is proposed in this paper. First of all, do spectrum analysis on complex phasor sequences. Second, get frequencies of inter-harmonics by zero-padding in short time window. Then, calculate the amplitudes of inter-harmonics by solving the equations. Finally, valid the availability of algorithm by using the simulation and experimental tests. 2 Characteristic analysis on phasor containing inter-harmonics signals 2.1 Phasor representation of inter-harmonics signals In general, a power signal is given by x(t) = ( ) 2Xm cos 2pf 0 t + w 0 where X m, f 0 and w 0 are the amplitude, frequency and initial phase angle of sinusoidal signal separately. Then, the corresponding phasor is (1) = X m e j(2p( f 0 50)t+w 0 ) (2) Therefore, if the signal is composed of fundamental wave and multiple inter-harmonics, like x(t) = 2Xm0 cos (2pf 0 t + w 0 ) 2 Xmi cos (2pf i t + w i ) where X mi, f i and w i (1 i n) are the amplitude, frequency and initial phase angle of the ith inter-harmonic signal separately. According to the concept of phasor in [8], the corresponding (3)
phasor of (3) is = X m0 e j(2p( f 0 50)+w 0 ) X mi e j(2p( f i 50)+w i ) (4) So, the phasor data can be decomposed in the form of (4) when the inter-harmonics exist in the power system. 2.2 Spectrum characteristic analysis on phasor containing inter-harmonics signals Assumed that the power signal contains two inter-harmonics whose frequencies are 20 and 80 separately, like x(t) = 10 2 cos (2p 50 t + p/3) + 2 cos (2p 20 t + p/4) (5) + 2 2 cos (2p 80 t + p/5) According to (4), the corresponding phasor is = 10e jp/3 + e j(2p( 30)t+p/4) + 2e j(2p30t+p/5) (6) As we can see from (6), there are 30 and 30 frequencies in the phasor data. Suppose reporting rate to be 100 and do spectrum analysis on amplitude and phase angle. The results are shown in Figs. 1 and 2. From Figs. 1 and 2, we can draw that the spectrum characteristics of amplitude and phase angle are symmetrical about 0. Thus the efficient frequency range is 0 50 or 50 to 0. If the frequency band is 0 50, 30 is the efficient frequency. However, we can know that the 30 is as a joint result of 20 sub-synchronous component and 80 super-synchronous component from (5) and (6), which causes to decompose the sub/ super-synchronous frequency components. Fig. 3 Amplitude frequency characteristics of phasor Therefore, it is an idea that do spectrum analysis on complex phasor sequences to solve the above problem. Given a signal as x(t) = Ae jv 1 t + Be jv 2 t (7) where A and B are the corresponding amplitudes of angular frequencies ω 1 and ω 2, while ω 1 is equal to minus ω 2. Equation (7) can be transformed into (8) after Fourier transform X (v) = 2pAd(v v 1 ) + 2pBd(v + v 2 ) (8) From (8), the different frequency components can be divided and their amplitudes cannot be affected each other when ω is set 0. Thus Fig. 3 can be got after doing fast Fourier transform (FFT) analysis on complex phasor sequences. It is a find that the efficient frequency range turns into 50 to 50, which indicates that sub/ super-synchronous inter-harmonics can be decomposed efficiently. That is to say, 30 represents the 20 sub-synchronous component and 30 represents the 80- super-synchronous component because of (5) and (6). To conclude, it is a must that do spectrum analysis on complex phasor sequences after getting the phasor data, which will lead to decomposing sub/super-synchronous inter-harmonics in power grid. 3 Fast extraction method of inter-harmonic frequencies From Fig. 3, if the frequency components are integer, the amplitudes can be calculated accurately within the 1 s time window. In general, the frequencies of inter-harmonic are not integer in the power system. As we can know from (9), to improve the precision of amplitude by increasing the frequency resolution, it is a must to add the number of valid data through expanding the time window of FFT Fig. 1 Amplitude frequency characteristics of amplitude DF = f s N (9) Fig. 2 Amplitude frequency characteristics of phase angle where ΔF represents frequency resolution, f s represents sample frequency and N represents the number of data. In (5), the frequencies are changed into 20.2 and 80.3, while the amplitudes are unchanged. Under such circumstances, do FFT analysis on the complex phasor sequences in the 1 s time window, and the amplitude characteristic is shown in Fig. 4. In Fig. 4, itisafind that the amplitude error of 20.2 is up to 7.3% and the amplitude error of 80.3 is up to 14%. Thus when we do spectrum analysis in short time window, the amplitude errors of inter-harmonics are much larger. The problem can be solved by increasing the time window of FFT. However, a new problem will be introduced that the inter-harmonic frequency can be changed dynamically in the time window, which will result in increasing the amplitude measurement error. Therefore, an idea is proposed that
the m frequency components, a composite phasor model is constructed as X = X m0 e jw 0 A i e j(2pdf it+w i ) B i e j( 2pDf i t+u i ) (10) Fig. 4 Amplitude frequency characteristics of phasor frequencies changed Fig. 5 Amplitude frequency characteristics and its envelopes of phasor after zero-padded the frequency and amplitude of inter-harmonics should be calculated separately. First, get the frequencies of inter-harmonics by doing FFT analysis in short time window. Second, calculate the amplitude based on the known frequencies. In this paper, a method that doing zero-padding FFT analysis on complex phasor sequence is adopted to improve the precision of frequency. Given the inter-harmonic frequencies are 20.2 and 80.3, the time window is 1 s and the number of zero-padding is 900, the corresponding amplitude characteristic is shown in Fig. 5. Compared with Fig. 4, we can find that the frequency can be calculated accurately. To be specific, the 29.8 frequency component represents the 20.2 sub-synchronous signal, and the 30.3 frequency component represents the 80.3 super-synchronous signal. Meanwhile, from Fig. 5, we can know that the amplitude errors are less, separately 2 and 4.5%. However, multiple inter-harmonics exist in the actual power system, the amplitude errors will increase. Simulation results show that the amplitude error will be up to 15% or even more when two pair inter-harmonics exist in the power system. Furthermore, if the proportion of inter-harmonic in power signal is different, the amplitude error will happen to high volatility which could be up to 5% from time to time. As a result, this method is suitable to calculate the inter-harmonic frequencies rather than amplitudes. From Fig. 5, it is a fact that there are plenty of side lobes nearby the main frequency component if adopting the zero-padding method, which will result in extracting the inter-harmonic frequencies. To extract the frequencies of inter-harmonics automatically, some actions should be taken. First of all, the envelope curve of phasor spectrum should be obtained. In the next place, acquire the maximum value of envelope curve and regard them as interharmonic frequencies. It is shown in Fig. 5 clearly. In addition, it is a valid step to set the threshold value to avoid the effect of noise. 4 Correction method of inter-harmonics amplitude Assume that m frequencies are got after doing FFT analysis, named Δf i (1 i m). From (4), Δf i is equal to f i 50. Based on where A i (1 i n) is the amplitude of super-synchronous signal, B i (1 i n) is the amplitude of sub-synchronous signal, w 0 is the initial phase angle of fundamental frequency signal, w i (1 i n) is the initial phase angle of super-synchronous signal, θ i (1 i n) is the initial phase angle of sub-synchronous signal and Δf i (1 i n) is the frequency difference between supersynchronous signal and fundamental frequency signal. It is an assume that sub-synchronous signal and its symmetrical super-synchronous signal are concurrent in (10). Here, the definition of symmetric is as follows. The f 1 and f 2 frequency components are an occurrence. If ( f 1 + f 2 )/2 is equal to 50 and f 1 50 equals f 2 50, the f 1 and f 2 frequency components will be referred to as symmetric. Consequently, it is necessary to judge the m frequencies whether there are symmetrical frequencies existed. Let n be the equal of m k on condition that k pairs of symmetrical frequencies exist in m frequency components. Although remaining m 2k frequencies do not have symmetrical frequencies, they are treated as symmetrical frequency components in the above model. If they do not exist in power signal, their amplitudes will be zero or be very close to zero after calculated. A method is taken over that solve equations to calculate the interharmonic amplitudes based on (10). In the first place, the fundamental frequency is not excursion to 50, and split phasor into the real and imaginary parts, as shown in (11) and (12) R(t) = X 0 cos f 0 I(t) = X 0 sin f 0 A i cos (2pDf i t + w i ) B i cos ( 2pDf i t + u i ) A i sin (2pDf i t + w i ) B i sin ( 2pDf i t + u i ) (11) (12) where R(t) is the real part of phasor and I(t) is the imaginary part of phasor. The meanings of other variables are as before. Let (11) and (12) expand trigonometrically, and adjust them to get (13) and (14) as follows: R(t) = X 0 cos w 0 I(t) = X 0 sin w 0 (A i cos w i + B i cos u i ) cos (2pDf i t) ( A i sin w i + B i sin u i ) sin (2pDf i t) (A i cos w i B i cos u i ) cos (2pDf i t) (A i sin w i + B i sin u i ) sin (2pDf i t) (13) (14)
In the case of (13), make X 0 cos w 0, (A i cos w i +B i cos θ i ) and ( A i sin w i +B i sin θ i ) as a whole and regard them as unknowns, which will result in 2n + 1 unknowns. As Δf i (1 i n) is known after doing spectrum, the values of cos(2πδf i t) and sin(2πδf i t) can be calculated. While R(t) is also known, hence 2n + 1 data points are involved at least to calculate the unknowns. Let cos(2πδf i t) equals C i (t), sin(2πδf i t) equals S i (t), the linear equations unresolved will be Thus (11) and (12) will turn into (21) and (22) R(t) = X 0 cos (2pDf 0 t + f 0 ) A i cos (2pDf i t + w i ) B i cos ( 2pDf i t + u i ) (21) R(t 1 ) K(t 1 ) R(t 2 ) K(t.. = 2 ).. Y (15) R(t 2n+1 ) K(t 2n+1 ) where K(t i ) is equal to [1, C 1 (t i ),, C 2n+1 (t i ), S 1 (t i ),, S 2n+1 (t i )] (1 i n); Y is equal to [X 0 cos w 0, (A 1 cos w 1 +B 1 cos θ 1 ),, (A n cos w n +B n cos θ n ), ( A 1 sin w 1 +B 1 sin θ 1 ),, ( A n sin w n +B n sin θ n )] (1 i n). Equation (14) is dealt with in the same way, and the values of X 0 sin w 0,(A i cos w i B i cos θ i ) and (A i sin w i +B i sin θ i ) can be calculated. For instance, the process is introduced how to calculate the inter-harmonic amplitudes in the case of i = 1: A 1 cos w 1 B 1 cos u 1 = a (16) A 1 sin w 1 + B 1 sin u 1 = b (17) A 1 cos w 1 + B 1 cos u 1 = c (18) A 1 sin w 1 + B 1 sin u 1 = d (19) Using simultaneous equations (16) and (18) we get the values of A 1 cos w 1, and using simultaneous equations (17) and (19) we get the values of A 1 sin w 1. Based on A 1 cos w 1 and A 1 sin w 1, the amplitude A 1 can be got. The amplitude B 1 can also be calculated in the same way. In the second place, the fundamental frequency deviating from 50 takes into account, which will make (10) changed into (20) X = X m0 e j(2pdf 0 t+w 0 ) B i e j( 2pDf it+u i ) A i e j(2pdf i t+w i ) Fig. 6 Flowchart of reduction method of inter-harmonics (20) I(t) = X 0 sin (2pDf 0 t + f 0 ) B i sin ( 2pDf i t + u i ) A i sin (2pDf i t + w i ) (22) where Δf 0 represents the offset of the fundamental frequency deviating to 50. The meanings of other variables are as before. After expanding (21) and (22) trigonometrically, it is a discovery that an unknown is increase compared with (13) and (14), which lead to the number of equations up to 2n + 2. The other processes are unchanged. To conclude, the procedure of inter-harmonic reduction method based on the PMU phasor data is shown in Fig. 6. 5 Simulation and experimental verification The simulation and experimental data are calculated to verify the validity of the above algorithm. In the process of the reduction of the inter-harmonics, the time window of FFT analysis is 1 s, the sample frequency is 100 and the number of zero-padding during the spectrum analysis is 900. In the actual power system, multiple inter-harmonics are possible to exist, in general, which are symmetrical. Meanwhile, the fundamental frequency could be excursion to 50. As a result, the origin signal is the superposition of the fundamental wave and multiple inter-harmonic signals, while the fundamental frequency is 50.2. After phasor measurement, the calculation results are shown in Table 1 by using the above reduction algorithm. To conclude from Table 1, though multiple inter-harmonics existing in the power system, the precision of frequencies is very high and the amplitudes can be enough to be calculated accurately. Moreover, 12.35 and 87.40 are almost symmetrical about 50, but they are treated as single frequency components in (10) while 24.66 and 75.20 being symmetrical, which results in small error. Therefore, it indicates that the reduction algorithm can be worked in the case that the single inter-harmonic or asymmetrical inter-harmonics exist in the actual power system. The experimental verification is on the basis of Fig. 7. In the process of the test, the voltage signal containing the 20 and 80 inter-harmonics is transmitted into the PMU firstly. In the next place, the phasor data should be extracted from the PMU as the input data of the reduction algorithm. Then, the test results are shown in Table 2, which can provide a conclusion that the reduction algorithm can be performed excellently in the actual power signal. Table 1 Simulation results Frequency theoretical value, Frequency actual value, Frequency error, Amplitude theoretical value, V Amplitude actual value, V Amplitude error, % 12.35 12.35 0.00 5.773 5.678 1.65 24.66 24.66 0.00 11.546 11.466 0.69 75.20 75.20 0.00 5.773 5.656 2.72 87.40 87.40 0.00 8.660 8.669 0.11
Table 2 Experiment results Frequency theoretical value, Frequency actual value, Frequency error, Amplitude theoretical value, V Amplitude actual value, V Amplitude error, % 20 20.00 0.00 8.806 8.606 2.27 80 80.00 0.00 18.520 18.416 0.56 Fig. 7 Flowchart of experimental tests Through the results of simulation and experimental tests, a conclusion can be drawn that the reduction method has a good performance in multiple cases. 6 Conclusion A reduction algorithm of inter-harmonics is proposed based on the phasor data in this paper. Not only the necessity of spectrum analysis on the complex phasor data is revealed, but also the influence of the short time window is analysed. Moreover, a method of zerospadding during FFT analysis is proposed to extract the interharmonic frequencies. Furthermore, another method of constructing the equations is proposed to calculate amplitudes through theoretical deviation. The simulation and experimental tests indicate that the reduction algorithm can calculate the frequencies and amplitudes of inter-harmonic signals accurately on the condition that the PMU phasor data being precision. 7 References [1] Wu M., Xie L., Cheng L., ET AL.: A study on the impact of wind farm spatial distribution on power system sub-synchronous oscillations, IEEE Trans. Power Sys., 2016, 31, (3), pp. 2154 2162 [2] Phadke A.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., 1983, 102, (5), pp. 1025 1038 [3] Terzija V., Valverde G., Cai D., ET AL.: Wide-area monitoring, protection, and control of future electric power networks, Proc. IEEE, 2011, 99, (1), pp. 80 93 [4] De La Ree J., Centeno V., Thorp J.S., ET AL.: Synchronized phasor measurement applications in power systems, IEEE Trans. Smart Grid, 2010, 1, (1), pp. 20 27 [5] Phadke A.G., de Moraes R.M.: The wide world of wide-area measurement, IEEE Power Energy Mag.., 2008, 6, (5), pp. 52 65 [6] Wu H.R., Wang Q., Li X.H.: PMU-based wide area damping control of power systems. Proc. Joint Int. Conf. Power Syst. Technol., IEEE Power India Conf., New Delhi, 12 15 October 2008, pp. 1 4 [7] Chen J., Shrestha P.: Use of synchronized phasor measurements for dynamic stability monitoring and model validation in ERCOT. Proc. 2012 IEEE Power Energy Society General Meeting, San Diego, CA, USA, 24 29 July 2012, pp. 1 7 [8] IEEE Std. C37.118.1-2011: IEEE standard for synchrophasor measurements for power systems, December 2011 [9] IEEE Std C37.118.1-2005: IEEE standard for synchrophasors for power systems, March 2006 [10] Liu H., Bi T., Yang Q.X.: The evaluation of phasor measurement units and their dynamic behavior analysis, IEEE Trans. Instrum. Meas., 2013, 62, (6), pp. 1479 1485 [11] Mai R.K., He Z.Y., Fu L., ET AL.: A dynamic synchrophasor estimation algorithm for online application, IEEE Trans. Power Delivery, 2010, 25, (1), pp. 570 578 [12] Bi T., Liu H., Feng Q., ET AL.: Dynamic phasor model-based synchrophasor estimation algorithm for M class PMU, IEEE Trans. Power Delivery, 2015, 30, (3), pp. 1162 1171 [13] Kang S.H., Lee D.G., Nam S.R.: Fourier transform-based modified phasor estimation method immune to the effect of the DC offsets, IEEE Trans. Power Delivery, 2009, 24, (2), pp. 1104 1111 [14] Frigo G., Narduzzi C., Colangelo D., ET AL.: Definition and assessment of reference values for PMU calibration in static and transient conditions. 2016 IEEE Int. Workshop Applied Measurements for Power Systems (AMPS), Aachen, 2016, pp. 1 6 [15] Liu H., Bi T., Chang X., ET AL.: Impacts of subsynchronous and supersynchronous frequency components on synchrophasor measurements, J. Mod. Power Syst. Clean Energy, 2016, 4, (3), pp. 263 269