1. Time-Frequency Analysis of Non-stationary Phenomena in Electrical Engineering Antonio Bracale, Guido Carpinelli Universita degli Studi di Napoli “Federico II” Zbigniew Leonowicz, Tomasz Sikorski, Krzysztof Wozniak Wroclaw University of Technology, Poland Modern Electrical Power Systems - MEPS 06 September 06-08, 2006, Wroclaw, Poland

2. Contents of presentation Motivations for applying time-frequency analysis in electrical engineering How to obtain time-frequency representations? Mathematical backgrounds of applied tools Investigated non-stationary phenomena Results of investigations Conclusions

3. Motivations Increasing level of non-stationary phenomena in contemporary power systems and its influence on power quality Converter systems generate a wide range of characteristic harmonics typical for the ideal converter operations, but also in some cases they become a source of non-characteristic harmonics. The duration time of some transient states can reach values up to 5-10 periods of basic component. Limitation of one-dimensional Fourier spectrum and new trends in signal processing for designing comprehensive and adaptive algorithms Classical Fourier spectrum loses the information about transient character of investigated phenomena. Non-stationarity is spread out over the whole frequency domain. Fourier algorithm with sliding window has a inseparable tradeoff between window width and time-frequency resolution.

4. To know how to merge two dimensions in one How to obtain joint time-frequency representations? Sliding window and different spectrum estimation methods Two-dimensional non- parametric equations Linear Non-Linear

5. Mathematical backgrounds – STFT STFT is classical method of time frequency analysis Involves both time and frequency and allows a time-frequency analysis or in other words a signal representation in the time-frequency plane The width of analysis window is fixed = constant time-frequency resolution for all frequency components Time-frequency resolution is dependent of analysis window width Wide window good frequency resolution, poor time resolution Narrow window good time resolution, poor frequency resolution STFT cannot be used successfully to analyze transient signals which contain high and low frequency components simulatneously

6. Mathematical backgrounds – S-Transform S transform is conceptually a hybrid of STFT and wavelet analysis, containing elements of both but falling entirely into neither category S transform uses a moving analyzing window but unlike STFT the width of the window is scaled with frequency as in wavelets The width of analysis window is the inverse of the frequency = frequency- dependent resolution S transform performs multi-resolution analysis on the signal, gives high time resolution at high frequencies and high frequency resolution at low frequencies

7. Mathematical backgrounds – Cohen’s Class General comments: The equation leads to two-dimensional time-varying spectrum which represents the energy changes of frequency components, here called auto-terms (a-t). Unfortunately, bilinear nature of discussed transformations manifests itself in existing of undesirable oscilating components, called cross-terms (c-t).

8. Mathematical backgrounds – Cohen’s Class Basic level of adaptation for signal analysis –selection of the kernel function: Additional level of adaptation for signal analysis – applying smoothing function:

9. Investigated phenomena – switching of capacitor banks T – transformer HV/MV, Δ-Y connected,25 MVA, 110kV/15kV First capacitor: 900kVar, 0.2km from the station, switching on at 0.03s Second capacitor: 1200kVar, 1.2km from the station, switching on at 0.09s One-phase diagram of simulated distribution system Fragment of current waveform at MV busbar (a) and its spectrum (b)

10. Fig. 2a. Time-varying spectrum of switching on the capacitor banks phenomena obtained using S- Transform. Fig. 2b. Comparison of STFT with S-Transform for tracking capability. Time-frequency analysis of investigated phenomena – switching on the capacitor banks Fig. 1. Time-varying spectrum of switching on the capacitor banks phenomena obtained using STFT. Short-Time Fourier Transform vs S-Transform

11. Fig. 4a. Time-varying spectrum of switching on the capacitor banks phenomena obtained using Wigner-Ville Distribution. Fig. 4b. Time-varying spectrum of switching on the capacitor banks phenomena obtained using transformation with Gaussian kernel. Fig. 4c. Time-varying spectrum of switching of the capacitor banks phenomena obtained using transformation with cone-shaped kernel. Time-frequency analysis of investigated phenomena – switching on the capacitor banks Fig. 3. Time-varying spectrum of switching on the capacitor banks phenomena obtained using STFT. Short-Time Fourier Transform vs Cohen’s class of transformation

12. Time-frequency analysis of investigated phenomena – switching on the capacitor banks Fig. 5. Comparison of STFT with Cohen’s class on account of tracking ability, when transient 475Hz component (a) and basic 50Hz component (b) are detected. Short-Time Fourier Transform vs Cohen’s class of transformation General comment: Selection of appropriate kernel function allows to achieve sharp detection of the beginning of transient states.

13. Time-frequency analysis of investigated phenomena – switching on the capacitor banks Fig. 6. Comparison of STFT (a) with smoothed version of Cohen’s class of transformation (b), on account of tracking tracking ability, when different width of smoothing window are applied. Short-Time Fourier Transform vs Smoothed version of Cohen’s class of transformation General comment: Applying additional smoothing windows in Cohen’s equation allows to control time and frequency resolution separately. In STFT the relationship between window width and time-frequency resolution is inseparable

14. Conclusions Time-frequency representations can be treated as a comprehensive tool for analysis of transient states. Simultaneous representation of two dimensions delivers much more information about character of investigated phenomena than known tools. Proposed S-Transform and transformation from Cohen’s family have adaptability to investigated phenomena: in case of S-Transform, moving and scalable Gaussian window uniquely combines frequency dependent resolution with desirable information about time-varying spectrum, referring to Cohen’s generalization, selection of appropriate kernel function enables adaptation of the representation method to investigated non-stationarity. Undesirable cross-term components, characteristic for bilinear nature of Cohen’s family, can be successfully reduced through the selection of appropriate kernel function as well as application of additional smoothing functions.

15. Conclusions In comparison with classical spectrogram, proposed methods break of the inherent relation between time-frequency resolution and width of the window. Smoothed version of Cohen’s transformation allows to control the time and frequency resolution independently. In case of S-Transform, achieved time-frequency resolution comes from combination of scalable localizing Gaussian function and width of spectrum. Allows detection of the begging of distortion with better precision than classical spectrogram.