1.1 Introduction1.2 State, and time-varying Waveform Distortions and Fourier Analysis1.3 Dealing with Time-Varying Waveform Distortions1.4 Wavelet Multi—Resolution Decomposition1.5 The Selection of the Mother Wavelet1.6 Impact of Sampling Rate and Filter Characteristic1.7 Time-Varying Waveform Distortions with Wavelets1.8 Application to Shipboard Power Systems Time-Varying Distortions1.9 Conclusions1.10 ReferencesSection 6 - Chapter 1 – (Chapter 15)Using Wavelet for Visualization and Understanding of Time-Varying Waveform Distortions in Power SystemsPaulo M. Silveira, Michael Steurer and Paulo F. Ribeiro1.1 IntroductionHarmonic distortion assumes steady state condition and is consequently inadequate todeal with time-varying waveforms. Even the Fourier Transform is limited in conveyinginformation about the nature of time-varying signals. The objective of this chapter is todemonstrate and encourage the use of wavelets as an alternative for the inadequate traditionalharmonic analysis and still maintain some of the physical interpretation of harmonic distortionviewed from a time-varying perspective. The chapter shows how wavelet multi-resolutionanalysis can be used to help in the visualization and physical understanding of time-varyingwaveform distortions. The approach is then applied to waveforms generated by high fidelitysimulation using RTDS of a shipboard power system. In recent years, utilities and industries havefocused much attention in methods of analysis to determine the state of health of electricalsystems. The ability to get a prognosis of a system is very useful, because attention can bebrought 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, insome cases, a higher level of voltage and current harmonic distortions. A high level of harmonicdistortions may lead to failures in equipment and systems, which can be inconvenient andexpensive. Traditionally, harmonic analyses of time-varying harmonics have been done using aprobabilistic approach and assuming that harmonics components vary too slowly to affect theaccuracy of the analytical process [1][2][3]. Another paper has suggested a combination ofprobabilistic and spectral methods also referred as evolutionary spectrum [4]. The techniquesapplied rely on Fourier Transform methods that implicitly assume stationarity and linearity ofthe signal components.In reality, however, distorted waveforms are varying continuously and in some case (duringtransients, notches, etc) quite fast for the traditional probabilistic approach.The ability to give a correct assessment of time-varying waveform / harmonic distortionsbecomes crucial for control and proper diagnose of possible problems. The issue has beenanalyzed before and a number time-frequency techniques have been used [5]. Also, the use ofthe wavelet transform as a harmonic analysis tool in general has been discussed [6]. But theytend to concentrate on determining equivalent coefficients and do not seem to quite satisfy theengineer’s physical understanding given by the concept of harmonic distortion. In general, harmonic analysis can be considered a trivial problem when the signals are instead state. However, it is not simple when the waveforms are non-stationary signals whosecharacteristics make Fourier methods unsuitable for analysis.To address this concern, this chapter reviews the concept of time-varying waveformdistortions, which are caused by different operating conditions of the loads, sources, and othersystem events, and relates it to the concept of harmonics (that implicitly imply stationary natureof signal for the duration of the appropriate time period). Also, the chapter presents how Multi-Resolution Analysis with Wavelet transform can be useful to analyze and visualize voltage andcurrent waveforms and unambiguously show graphically the harmonic components varying withtime. Finally, the authors emphasize the need of additional investigations and applications tofurther demonstrate the usefulness of the technique. 1.2 State, and time-varying Waveform Distortions and Fourier AnalysisTo illustrate the concept of time-varying waveform Figure 1 shows two signals. The firstis a steady state distorted waveform, whose harmonic content (in this case 3, 5 and 7th) isconstant along the time or, in other words, the signal is a periodic one. The second signalrepresents a time varying waveform distortion in which magnitude and phase of each harmonicvary during the observed period of time. In power systems, independently of the nature of the signal (stationary or not), theyneed to be constantly measured and analyzed by reasons of control, protection, andsupervision. Many of these tasks need specialized tools to extract information in time, infrequency 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 isthe Fourier analysis which breaks down a signal into constituent sinusoids of differentfrequencies. Traditionally it is very popular, mainly because of its ability in translating a signal inthe time domain for its frequency content. As a consequence of periodicity these sinusoids arevery well localized in the frequency, but not in time, since their support has an infinite length. Inother words, the frequency spectrum essentially shows which frequencies are contained in thesignal, as well as their corresponding amplitudes and phases, but does not show at which timesthese frequencies occur.Using the Fourier transform one can perform a global representation of a time-varyingsignal but it is not possible to analyze the time localization of frequency contents. In otherwords, when non-stationary information is transformed into the frequency domain, most of theinformation 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, letus consider the hypothetical signal represented by (1), in which, during some time interval, theharmonic content assumes variable amplitude.sin(2 60 ) 0.2sin(10