Using Wavelet Decomposition for Visualization and Understanding of Time-Varying Waveform Distortion in Power Systems

Transcription

1 1 Usin Wavelet Decomposition for Visualization and Understandin of Time-Varyin Waveform Distortion in Power Systems Paulo M. Silveira, Member, IEEE, Micael Steurer, Senior Member, IEEE and Paulo F. Ribeiro, Fellow, IEEE Abstract Harmonic distortion assumes steady state condition and is consequently inadequate to deal wit time-varyin waveforms. Even te Fourier Transform is limited in conveyin information about te nature of time-varyin sinals. Te objective of tis paper is to demonstrate and encourae te use of wavelets as an alternative for te inadequate traditional armonic analysis and still maintain some of te pysical interpretation of armonic distortion viewed from a time-varyin perspective. Te paper sows ow wavelet multi-resolution analysis can be used to elp in te visualization and pysical understandin of time-varyin waveform distortions. Te approac is ten applied to waveforms enerated by i fidelity simulation usin RTDS of a sipboard power system. Key Words. Time-Varyin Waveform Distortion, Harmonic Distortion, Wavelets, Multi-Resolution Analysis. I I. INTRODUCTION n recent years, utilities and industries ave focused muc attention in metods of analysis to determine te state of ealt of electrical systems. Te ability to et a pronosis of a system is very useful, because attention can be brout to any problems a system may exibit before tey cause te system to fail. Besides, considerin te increased use of te power electronic devices, utilities ave experienced, in some cases, a ier level of voltae and current armonic distortions. A i level of armonic distortions may lead to failures in equipment and systems, wic can be inconvenient and expensive. Traditionally, armonic analyses of time-varyin armonics ave been done usin a probabilistic approac and assumin tat armonics components vary too slowly to affect te accuracy of te analytical process [1-3]. Anoter paper as suested a combination of probabilistic and spectral metods also referred as evolutionary spectrum [4]. Te tecniques applied rely on Fourier Transform metods tat implicitly assume stationarity and linearity of te sinal components. Tis work was supported by CAPES, (Brazilian Researc Council)l and U.S. Office of Naval Researc. Paulo M. Silveira is wit te Power Quality Study Group (GQEE) of te Itajubá Federal University (UNIFEI) and Center for Advanced Power Systems (CAPS) / Florida State University (FSU) ( s: dasilveira@caps.fsu.edu). Micael Steurer is wit Center for Advanced Power Systems (CAPS) / Florida State University (FSU) ( steurer@caps.fsu.edu) and Paulo F. Ribeiro is wit Center for Advanced Power Systems (CAPS) / Florida State University (FSU)and Calvin Collee ( pfribeiro@ieee.or) In reality, owever, distorted waveforms are varyin continuously and in some case (durin transients, notces, etc) quite fast for te traditional probabilistic approac. Te ability to ive a correct assessment of time-varyin waveform / armonic distortions becomes crucial for control and proper dianose of possible problems. Te issue as been analyzed before and a number time-frequency tecniques ave been used [5]. Also, te use of te wavelet transform as a armonic analysis tool in eneral as been discussed [6]. But tey tend to concentrate on determinin equivalent coefficients and do not seem to quite satisfy te enineer s pysical understandin iven by te concept of armonic distortion. In eneral, armonic analysis can be considered a trivial problem wen te sinals are in stead state. However, it is not simple wen te waveforms are non-stationary sinals wose caracteristics make Fourier metods unsuitable for analysis. To address tis concern, tis paper reviews te concept of time-varyin waveform distortions, wic are caused by different operatin conditions of te loads, sources, and oter system events, and relates it to te concept of armonics (tat implicitly imply stationary nature of sinal for te duration of te appropriate time period). Also, te paper presents ow Multi-Resolution Analysis wit Wavelet transform can be useful to analyze and visualize voltae and current waveforms and unambiuously sow rapically te armonic components varyin wit time. Finally, te autors empasize te need of additional investiations and applications to furter demonstrate te usefulness of te tecnique. II. STEADY STATE, AND TIME-VARYING WAVEFORM DISTORTIONS AND FOURIER ANALYSIS To illustrate te concept of time-varyin waveform Fiure 1 sows two sinals. Te first is a steady state distorted waveform, wose armonic content (in tis case 3, 5 and 7 t ) is constant alon te time or, in oter words, te sinal is a periodic one. Te second sinal represents a time varyin waveform distortion in wic manitude and pase of eac armonic vary durin te observed period of time. In power systems, independently of te nature of te sinal (stationary or not), tey need to be constantly measured and

2 2 analyzed by reasons of control, protection, and supervision. Many of tese tasks need specialized tools to extract information in time, in frequency or bot. Considerin te simplicity of te case, te result may be adequate for some simple application, owever lare errors will result wen detailed information of eac frequency is required. 2 Sinal (a) (b) Fiure 1 (a) Steady state distorted waveform; (b) time-varyin waveform distortion. Te most well-known sinal analysis tool used to obtain te frequency representation is te Fourier analysis wic breaks down a sinal into constituent sinusoids of different frequencies. Traditionally it is very popular, mainly because of its ability in translatin a sinal in te time domain for its frequency content. As a consequence of periodicity tese sinusoids are very well localized in te frequency, but not in time, since teir support as an infinite lent. In oter words, te frequency spectrum essentially sows wic frequencies are contained in te sinal, as well as teir correspondin amplitudes and pases, but does not sow at wic times tese frequencies occur. Usin te Fourier transform one can perform a lobal representation of a time-varyin sinal but it is not possible to analyze te time localization of frequency contents. In oter words, wen non-stationary information is transformed into te frequency domain, most of te information about te nonperiodic events contained on te sinal is lost. In order to demonstrate te FFT lack of ability wit dealin wit time-varyin sinals, let us consider te ypotetical sinal represented by equation (1), in wic, durin some time interval, te armonic content assumes variable amplitude. sin(2π 60) t + 0.2sin(10π60) t => 0< t 0.2s f = sin(2π 60 t) t.sin(10π60 t) => 0.2 < t 2 s sin(2π 60 t) + 0.2sin(10π60 t) => 2 < t 5 s Fourier transform as been used to analyze tis sinal and te result is presented in Fiure 2. Unfortunately, as it can be seen, tis classical tool is not enou to extract features from tis kind of sinal, firstly because te information in time is lost, and secondly, te armonic manitude and pase will be incorrect wen te entire data window is analyzed. In tis example te manitude of te 5 t armonic as been indicated as pu. (1) Manitude Manitude Time [s] Discrete Fourier Transform pu Frequency [Hz] Fiure 2 Time-Varyin Harmonic and its Fourier (FFT) analysis. III. DEALING WITH TIME-VARYING WAVEFORM DISTORTIONS Frequently a particular spectral component occurrin at a certain instant can be of particular interest. In tese cases it may be very beneficial to know te beavior of tose components durin a iven interval of time. Time-frequency analysis, tus, plays a central role in sinal processin sinal analysis. Harmonics or i frequency bursts for instance cannot be identified. Transient sinals, wic are evolvin in time in an unpredictable way (like time-varyin armonics) needs te notion of frequency analysis tat is local in time. Over te last 40 years, a lare effort as been made to efficiently deal te drawbacks previously cited and represent a sinal jointly in time and frequency. As result a wide variety of possible time-frequency representations can be found in specialized literature, as for example in [5]. Te most traditional approaces are te sort time Fourier transform (STFT) and Winer-Ville Distribution [7]. For te first case, (STFT), wose computational effort is smaller, te sinal is divided into sort pseudo-stationary sements by means of a window function and, for eac portion of te sinal, te Fourier transform is found. However, even tese tecniques are not suitable for te analysis of sinals wit complex time-frequency caracteristics. For te STFT, te main reason is te widt of fixed data window. If te time-domain analysis window is made too sort, frequency resolution will suffer, and lentenin it could invalidate te assumption of stationary sinal witin te window. IV. WAVELET MULTI RESOLUTION DECOMPOSITION

3 3 Wavelet transforms provide a way to overcome te problems cited previously by means of sort widt windows at i frequencies and lon widt windows at low frequencies. In bein so, te use of wavelet transform is particularly appropriate since it ives information about te sinal bot in frequency and time domains. Te continuous wavelet transform of a sinal f(t) is ten defined as ψ CWTf ( a, b) = f ( t) ψab ( t) dt (2) were 1 t b ψ ab () t = ψ a, b R; a 0 a a bein ψ te moter wavelet wit two caracteristic parameters, namely, dilation (a) and translation (b), wic vary continuously. Te results of (2) are many wavelet coefficients, wic are a function of a and b. Multiplyin eac coefficient by te appropriately scaled and sifted wavelet yields te constituent wavelets of te oriinal sinal. Just as a discrete Fourier transform can be derived from Fourier transform, so can a discrete wavelet transform be derived from a continuous wavelet transform. Te scales and positions are discretized based on powers of two wile te sinal is also discretized. Te resultin expression is sown in (4). ψ 1 DWT ( jk, ) ( ) f j 0 0 = f nψ j j a n= 0 a0 were j, k, n Z and a 0 > 1. (3) n a kb (4) Te simpler coice is to make a = 2 and b = 1. In tis case te Wavelet transform is called dyadic-ortonormal. Wit tis approac te DWT can be easily and quickly implemented by filter bank tecniques normally known as Multi-Resolution Analysis (MRA) [8]. Te multi-resolution property of wavelet analysis allows for bot ood time resolution at i frequencies and ood frequency resolution at low frequencies. Te Fiure 3 sows a MRA diaram, wic is built and performed by means of two filters: a i-pass filter wit impulse response (n) and its low-pass mirror version wit impulse response (n). Tese filters are related to te type of moter wavelet (ψ) and can be cosen accordin to te application. Te relation between (n) and (n) is iven by: n n ( ) = ( 1) K ( 1 n) (5) were k is te filter lent and n Z. Eac i-pass filter produces a detailed version of te oriinal sinal and te low-pass a smooted version. Te same Fiure 3 summarizes several kinds of power systems application usin MRA, wic as been publised in te last decade [9-13]. Te samplin rate (fs) sowed in tis fiure represents just only a typical value and can be modified accordin to te application wit te faster time-varyin events requirin ier samplin rates. It is important to notice tat several of tese applications ave not been compreensively explored yet. Tis is te case of armonic analysis, includin sub-armonic, inter-armonic and time-varyin armonic. And te reason for tat is te difficulty to pysically understand and analytically express te nature of time-varyin armonic distortions from a Fourier perspective. Te oter aspect, from te wavelet perspective, is tat not all wavelet moters enerate pysically meaninful decomposition. Anoter important consideration, mainly for protection applications is te computational speed. Te time of te aloritm is essentially a function of te penomena (te sort of information needed to be extracted), te samplin rate and processin time. Applications wic require detection of fast transients, like travelin waves [10,11], normally ave a very sort time of processin. Anoter aspect to be considered is samplin rate and te frequency response of te conventional CTs and VTs. fs = 200 khz fs = 100 khz 0 50 khz Sure capture khz Fault location fs = 50 khz 0 25 khz khz fs = 25 khz 0 12,5 khz Fault identification 12,5 25 khz... 6,25 12,5 khz Detection of insipient failures 0 fs/(2 j+1 ) Hz fs/(2 j+1 ) fs/(2 j ) Hz Harmonic Analyses Intellient events records for power quality Fiure 3 Multi-Resolution Analysis and Applications in Power Systems. V. THE SELECTION OF THE MOTHER WAVELET Unlike te case of Fourier transform, tere exists a lare selection of wavelet families dependin on te coice of te moter wavelet. However, not all wavelet moters are suitable for assistin wit te visualization of time-varyin (armonic) frequency components. For example, te celebrated Daubecies wavelets (Fiure 4a) are ortoonal and ave compact support, but tey do not ave a closed analytic form and te lowest-order families do

4 4 not ave continuous derivatives everywere. On te oter and, wavelets like modulated Gaussian function or armonic waveform are particularly useful for armonic analysis due to its smootness. Tis is te case of Morlet and Meyer (fiure 4b) wic are able to sow amplitude information [14]. Fiure 4 (a) Daubecies-5 and (b) Meyer wavelets. Te "optimal" coice of te wavelet basis will depend on te application. For discrete computational te Meyer wavelet is a ood option to visualization of time-varyin frequency components because te MRA can clearly indicate te oscillatory nature of time-varyin frequency components or armonics in te Fourier sense of te word. In order to exemplify suc an application te MRA as been performed to decompose and visualize a sinal composed by 1 pu of 60 Hz, 0.3 pu of 7 t armonic, 0.12 pu of 13 t armonic and some noise. Te oriinal sinal as a samplin rate of 10 khz and te armonic content as not been present all te time. Fiure 5 sows te results of a MRA in six level of decomposition, firstly usin Daubecies lent 5 (Db5) and next te Meyer wavelet (dmey). It is clear tat even a i order wavelet (Db5 10 coefficients) te output sinals will be distorted wen usin Daubecies wavelet and, oterwise, will be perfect sinusoids wit Meyer wavelet as it can be seen at level 3 (13 t ) and in level 4 (7 t ). Some of te detail levels are not te concern because tey result represent only noise and transitions state. VI. IMPACT OF SAMPLING RATE AND FILTER CHARACTERISTIC It is important to reconize tat te samplin rate and te caracteristic of filter in te frequency domain, will affect te ability of te MRA to separate te frequency components and avoid frequency crossin in two different detail levels as previously reconized [15]. Tis problem can be better clarified wit te aid of Fiure 6. Te pass-band filters location is defined by te samplin rate and, te frequency support of eac filter ([n] and [n]) by te moter wavelet. If certain frequency component of interest is positioned inside te crossin rane of te filters, tis component will be impacted by te adjacent filters. As a consequence, te frequency component will appear distorted in two different levels of decomposition. Fiure 5 MRA wit six level of decomposition usin (a) Daubecies 5 wavelet and (b) Meyer wavelet. In order to illustrate tis question let us consider a 60 Hz sinal in wic a 5 t armonic is present and wose samplin rate is 10 khz. A MRA is performed wit a dmey filter. Accordin to te Fiure 6 te 5 t armonic is located in te crossin between detail levels d4 and d5. Te result can be seen in Fiure 7 were te 5 t armonic appears as a beat frequency in levels d4 and d5. a d d d Fiure 6 MRA Filters: Frequency Support khz Tis problem previously cited may reduce te ability of te tecnique to track te beavior of a particular frequency in time. However, artificial tecniques can be used to minimize tis problem. For example, te simple alebraic sum of te two sinals (d4+d5) will result te 5 t armonic. Of course, if oter components are present in te same level, a more complex tecnique must be used. As a matter of practicality te problem can be sometimes easily overcome as sown in Fiure 5 were te 5 t and 7 t armonics (two of te most common components found in power systems distortions) can be easily separated if an adequate samplin rate and number of scales / detail coefficients are used.

5 5 Fiure 7 MRA wit five decomposition levels. Te 5 t armonic is revealed on level d4 and d5. VII. TIME-VARYING WAVEFORM DISTORTIONS WITH WAVELETS Let us consider te same sinal of te equation (2), wose variable amplitude of te 5 t armonic is not revealed by Fourier transform. By performin te MRA wit Mayer wavelet in six decomposition levels te 5 t armonic as been revealed in d5 (fift detailed level) durin all te time of analysis, includin te correct amplitude of te contents. Fiure 8 MRA of te sinal (equation (1)) sowin timevaryin 5 t armonic in level d5. As it can be seen from Fiure 8 tis decomposition can be very elpful to visualize time-varyin waveform distortions in wic bot frequency / manitude (armonics) and time information is clearly seen. Tis can be very elpful for understandin te beavior of distortions durin transient penomena as well as to be used for possible control and protection action. It is important to remark tat oter information suc as rms value and pase can be extract from te detailed and te approximation levels. For te previous example te rms value durin te interval from 0.2 to 2 s as been easily acieved. VIII. APPLICATION TO SHIPBOARD POWER SYSTEMS TIME- VARYING DISTORTIONS In order to apply te concept a time-varyin voltae waveform, resultin from a pulse in a sipboard power system [16], is decomposed by MRA usin a Meyer moter wavelet wit five levels (only detail levels 5 and 4 and te approximation coefficients are sown) is sown in Fiure 9. Te approximation coefficient A5 sows te beavior of te fundamental frequency wereas D5 and D4 sow te timevaryin beavior of te 5 t and 7 t armonics respectively. Te time-varyin beavior of te fundamental, 5 t and 7 t armonics can be easily followed. Fiure 9 Time-varyin voltae waveform caused by pulsed load in a sipboard power system : MRA Decomposition usin Meyer moter wavelet. IX. CONCLUSIONS Tis paper attempts to demonstrate te usefulness of wavelet MRA to visualize time-varyin waveform distortions and track independent frequency component variations. Tis application of MRA can be used to furter te understandin of time-varyin waveform distortions witout losin te pysical meanin of frequency components (armonics) variation wit time. It is also possible tat tis approac could be used in control and protection applications.

6 6 Te paper reconizes tat te samplin rate / location of te filters for te successful trackin of a particular frequency beavior, te sinificance of te wavelet moter type for te meaninful information provided by te different detail levels decomposition, and te number of detail levels. A sipboard system simulation voltae output durin a pulsed load application is ten used to verify te usefulness of te metod. Te MRA decomposition applyin Meyer moter wavelet is used and te transient beavior of te fundamental, 5 t and 7 t armonic clearly visualized and properly tracked from te correspondin MRA. X. REFERENCES [1] IEEE Task Force on Harmonics Modelin and Simulation: Modelin and Simulation of te Propaation of Harmonics in Electric Power Networks Part I: Concepts, Models and Simulation Tecniques, IEEE Trans. on Power Delivery, Vol. 11, No. 1, 1996, pp [2] Probabilistic Aspects Task Force of Harmonics Workin Group (Y. Bazouz Cair): Time-Varyin Harmonics: Part II Harmonic Summation and Propaation IEEE Trans. on Power Delivery, No. 1, January 2002, pp [3] R. E. Morrison, Probabilistic Representation of Harmonic Currents in AC Traction Systems, IEE Proceedins, Vol. 131, Part B, No. 5, September 1984, pp [4] P.F. Ribeiro; A novel way for dealin wit time-varyin armonic distortions: te concept of evolutionary spectra Power Enineerin Society General Meetin, 2003, IEEE, Volume: 2, July 2003, Vol. 2, pp [5] P. Flandrin, Time-Frequency/Time-Scale Analysis. London, U.K.: Academic, [6] Newland D. E., Harmonic Wavelet Analysis, Proc. R. Soc., London, A443, pp , [7] G. Matz and F. Hlawatsc, Winer Distributions (nearly) everywere: Time-frequency Analysis of Sinals, Systems, Random Processes, Sinal Spaces, and Frames, Sinal Process., vol. 83, no. 7, 2003, pp [8] C. S. Burrus; R. A. Gopinat; H. Guo; Introduction to Wavelets and Wavelet Transforms - A Primer. 10 Ed. New Jersey: Prentice-Hall Inc., [9] O. Caari, M. Meunier, F. Brouaye. : Wavelets: A New Tool for te Resonant Grounded Power Distribution Systems Relayin. IEEE Transaction on Power System Delivery, Vol. 11, No. 3, July pp [10] F. H. Manao; A. Abur; Fault Location Usin Wavelets. IEEE Transactions on Power Delivery, New York, v. 13, n. 4, October 1998, pp [11] P. M. Silveira; R. Seara; H. H. Zürn; An Approac Usin Wavelet Transform for Fault Type Identification in Diital Relayin. In: IEEE PES Summer Meetin, June 1999, Edmonton, Canada. Conference Proceedins. Edmonton, IEEE Press, pp [12] V. L. Pam and K. P. Won, Wavelet-transform-based Aloritm for Harmonic Analysis of Power System Waveforms, IEE Proc. Gener. Transm. Distrib., Vol. 146, No. 3, May [13] O. A. S. Youssef; A wavelet-based tecnique for discrimination between faults and manetizin inrus currents in transformers, IEEE Trans. Power Delivery, vol. 18, Jan. 2003, pp [14] Norman C. F. Tse, Practical Application of Wavelet to Power Quality Analysis; CEn, MIEE, MHKIE, City University of Hon Kon; /06/2006 IEEE. [15] Mat H.J. Bollen and Irene Y.H. Gu, Sinal Processin of Power Quality Disturbances, IEEE Press [16] M. Steurer, S. Woodruff, M. Andrus, J. Lanston, L. Qi, S. Suryanarayanan, and P.F. Ribeiro, Investiatin te Impact of Pulsed Power Carin Demands on Sipboard Power Quality, Te IEEE Electric Sip, Tecnoloies Symposium (ESTS 2007) will be eld from May 21 to May 23, 2007 at te Hyatt Reency Crystal City, Arlinton, Virinia, USA.