# Using Wavelet for Visualization and Understanding

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Section 6 Chapter 1 Chapter 15 Using Wavelet for Visualization and Understanding of Time Varying Waveform Distortions in Power Systems Paulo M Silveira Michael Steurer and Paulo F Ribeiro 1 1 Introduction Harmonic distortion assumes steady state condition and is consequently inadequate to deal with time varying waveforms Even the Fourier Transform is limited in conveying information about the nature of time varying signals The objective of this chapter is to demonstrate and encourage the use of wavelets as an alternative for the inadequate traditional harmonic analysis and still maintain some of the physical interpretation of harmonic distortion viewed from a time varying perspective The chapter shows how wavelet multi resolution analysis can be used to help in the visualization and physical understanding of time varying waveform distortions The approach is then applied to waveforms generated by high fidelity simulation using RTDS of a shipboard power system In recent years utilities and industries have focused much attention in methods of analysis to determine the state of health of electrical systems The ability to get a prognosis of a system is very useful because attention can be brought to any problems a system may exhibit before they cause the system to fail Besides considering the increased use of the power electronic devices utilities have experienced in some cases a higher level of voltage and current harmonic distortions A high level of harmonic distortions may lead to failures in equipment and systems which can be inconvenient and expensive Traditionally harmonic analyses of time varying harmonics have been done using a probabilistic approach and assuming that harmonics components vary too slowly to affect the accuracy of the analytical process 1 2 3 Another paper has suggested a combination of probabilistic and spectral methods also referred as evolutionary spectrum 4 The techniques applied rely on Fourier Transform methods that implicitly assume stationarity and linearity of the signal components In reality however distorted waveforms are varying continuously and in some case during transients notches etc quite fast for the traditional probabilistic approach The ability to give a correct assessment of time varying waveform harmonic distortions becomes crucial for control and proper diagnose of possible problems The issue has been analyzed before and a number time frequency techniques have been used 5 Also the use of the wavelet transform as a harmonic analysis tool in general has been discussed 6 But they tend to concentrate on determining equivalent coefficients and do not seem to quite satisfy the engineer s physical understanding given by the concept of harmonic distortion In general harmonic analysis can be considered a trivial problem when the signals are in stead state However it is not simple when the waveforms are non stationary signals whose characteristics make Fourier methods unsuitable for analysis To address this concern this chapter reviews the concept of time varying waveform distortions which are caused by different operating conditions of the loads sources and other system events and relates it to the concept of harmonics that implicitly imply stationary nature of signal for the duration of the appropriate time period Also the chapter presents how MultiResolution Analysis with Wavelet transform can be useful to analyze and visualize voltage and current waveforms and unambiguously show graphically the harmonic components varying with time Finally the authors emphasize the need of additional investigations and applications to further demonstrate the usefulness of the technique 1 2 State and time varying Waveform Distortions and Fourier Analysis To illustrate the concept of time varying waveform Figure 1 shows two signals The first is a steady state distorted waveform whose harmonic content in this case 3 5 and 7th is constant along the time or in other words the signal is a periodic one The second signal represents a time varying waveform distortion in which magnitude and phase of each harmonic vary during the observed period of time In power systems independently of the nature of the signal stationary or not they need to be constantly measured and analyzed by reasons of control protection and supervision Many of these tasks need specialized tools to extract information in time in frequency or both a b Figure 1 a Steady state distorted waveform b time varying waveform distortion The most well known signal analysis tool used to obtain the frequency representation is the Fourier analysis which breaks down a signal into constituent sinusoids of different frequencies Traditionally it is very popular mainly because of its ability in translating a signal in the time domain for its frequency content As a consequence of periodicity these sinusoids are very well localized in the frequency but not in time since their support has an infinite length In other words the frequency spectrum essentially shows which frequencies are contained in the signal as well as their corresponding amplitudes and phases but does not show at which times these frequencies occur Using the Fourier transform one can perform a global representation of a time varying signal but it is not possible to analyze the time localization of frequency contents In other words when non stationary information is transformed into the frequency domain most of the information about the non periodic events contained on the signal is lost In order to demonstrate the FFT lack of ability with dealing with time varying signals let us consider the hypothetical signal represented by 1 in which during some time interval the harmonic content assumes variable amplitude sin 2p60t 0 2sin 10p60t 0 t 0 2 s f sin 2p60t 0 2 t sin 10p60t 0 2 t 2 s sin 2p60t 0 2sin 10p60t 2 t 5 s 1 Fourier transform has been used to analyze this signal and the result is presented in Figure 2 Unfortunately as it can be seen this classical tool is not enough to extract features from this kind of signal firstly because the information in time is lost and secondly the harmonic magnitude and phase will be incorrect when the entire data window is analyzed In this example the magnitude of the 5th harmonic has been indicated as 0 152 pu Considering the simplicity of the case the result may be adequate for some simple application however large errors will result when detailed information of each frequency is required Signal Magnitude 2 1 0 1 2 0