TIME-FREQUENCY ANALYSIS OF NON-STATIONARY THREE PHASE SIGNALS. Z. Leonowicz T. Lobos

Transcription

1 Copyright IFAC 15th Triennial World Congress, Barcelona, Spain TIME-FREQUENCY ANALYSIS OF NON-STATIONARY THREE PHASE SIGNALS Z. Leonowicz T. Lobos Wroclaw University o Technology Pl. Grunwaldzki 13, 537 Wroclaw, Poland Tel: (+48) , Fax: (+48) lobos@elektryk.ie.pwr.wroc.pl Abstract: New method o observation and diagnosis o inverter-ed induction motor drives is developed and tested. Unsymmetrical conditions concerning the machine impedances or valves operation are relected in the spectrum o the current spacephasor. We estimate the spectrum o the space-phasor with the help o the Wigner-Ville distribution (WVD) and we obtain its time-requency representation with excellent time and requency resolution. The proposed method is tested with nonstationary multiplecomponent signals occurring during the ault operation o inverter-ed drives and transmission lines. Copyright IFAC Keywords: spectrum estimation, requency conversion, power systems, signal analysis, time-requency representation. 1. INTRODUCTION Reliability o power electronic drive systems is important in many industrial applications. The analysis o ault mode behaviour can be utilised or development o monitoring and diagnostic systems. In this paper we present some results o simulation investigations o a converter-ed induction motor drive. PWM converters supplying asynchronous motor were simulated using the EMTP-ATP (Electromagnetic Transients Program - Alternative Transients Program). The EMTP-program allows the complete simulation o a converter-motor system and provides many tools to construct real-like situation (Lobos, et al., 1999). Three-phase systems can be described with complex phasors (Hosemann, 1984; Lobos, 199). On the plane o complex numbers we obtain a rotating phasor. Unsymmetrical conditions o the machine operation are relected in the spectrum o the voltage or current space-phasor as positive or negative sequence systems. We estimate the spectrum o the space-phasor with the help o the Wigner-Ville distribution (WVD). We also estimate the Wigner spectrum o the signals occurring during the line to ground ault o a transmission line. A huge amount o work was already done in analysis o stationary signals, but the real signals in vast majority are non-stationary. This has lead to the development o tools that are designed speciically to deal with non-stationary situations. Time-requency methods explicitly consider the time dependence o the requency contents o the signal. The WVD is especially appropriate or the analysis o nonstationary multicomponent signals due to its good temporal resolution, excellent perormance in the presence o noise, better requency concentration and less phase dependence than Fourier spectra (Velez and Absher, 199, Andria, et al., 1994 ).

2 . WIGNER-VILLE REPRESENTATION The Wigner-Ville distribution (WVD) (Qian and Chen, 1996) is a time-requency representation given by: τ (1) τ τ ( ) ( ) ( ) ( jωτ ω = + ) W t, x t x t e d x where t is a time variable, ω is a requency variable and * denotes complex conjugate. The Wigner distribution is a two-dimensional unction describing the requency content o a signal as a unction o time (Lin, 1997) and possesses many advantageous properties, among them: 1. Instantaneous requency property which says that, at the time instant t, the mean instantaneous requency o the Wx ( t, ω ) is equal to the mean instantaneous requency o the signal (Lin, 1997).. Because the Wigner-Ville distribution satisies both time and requency marginal conditions, it can be shown (Lin, 1997), that energy contained in the Wx ( t, ω ) is equal to the energy possessed by the original signal. Evaluation o Wigner-Ville distribution combines the non-linear operation (quadratic operation applied to the signal) with linear Fourier transorm. This constitutes an essential dierence between Wigner- Ville distribution and spectrogram which combines linear Fourier transorm with quadratic operation in the opposite order. Spectrogram originates rom a short-time Fourier transorm with an external window unction, but the Wigner-Ville distribution can be regarded as a similar way o analysis with window matched to the signal. This window is the mirror o the signal itsel. Moreover, the Wigner- Ville distribution, in its original orm, does not require the introduction o a window unction, which remains external to the signal. For a discrete-time signal x( n ) the discrete pseudo- Wigner-Ville distribution (PWD) is evaluated using a sliding symmetrical inite-length analysis window h( τ ) (Qian and Chen, 1996). xh L 4 πτ/ (, ) = ( + τ) ( τ) ( τ) ( τ) W n k x n x n h h e τ = L j k N () where h( τ ) is a windowing unction that satisies the condition: h( τ) = ; τ > L. Variables n and k correspond respectively to the discrete time and requency variables. The short-time Fourier transorm (STFT) was the irst tool or analysing the signal in joint timerequency domain. The crucial drawback inherent to the STFT is a trade-o between time and requency resolution. WVD does not suer rom interaction between time and requency resolutions, but presents some other undesired properties. One main deiciency o the WVD is the cross-term intererence. WVD o the sum o multicomponents is a linear combination o auto- and cross-terms. Each pair o the signal components creates one additional crossterm in the spectrum, thus the desired time-requency representation may be conusing. Traditionally, the cross-terms are considered as something undesired in the WVD (Qian and Chen, 1996) and should be removed. It is not entirely correct. Some types o cross-terms (e.g. caused by two Gaussian unctions whose time and requency centres are ar apart when the cross-term is highly oscillating) have near-zero average and limited inluence on the time-marginal conditions as well as other useul properties. In other situations when a cross-term is discarded, the resulting representation will leave signiicant energy out. One way o lowering cross-term intererence is to apply a low-pass ilter to the WVD. The smoothing, however, will reduce the requency resolution o the WVD and cause the loss o many useul properties o the transormation (Qian and Chen, 1996). Another way o lowering cross-term intererence is to use the WVD o analytical signal. Analytical unction o the signal is a hal hand unction and thereore the resulting WVDa (WVD o analytical signal) avoids all cross-terms associated with negative requency components. The analytical unction, however, diers rom the original signal in dierent ways; e.g. its instantaneous properties may substantially dier rom that o the original signal (Martin and Flandrin, 1985). Also the WVDa reduces the cross-term intererence at the cost o loosing some useul properties. 3. SIMULATION OF THE FAULT OPERATION In the recent years, simulation programs or complex electrical circuits and control systems have been improved essentially. The EMTP-ATP (Electromagnetic Transients Program - Alternative Transients Program) as a FORTRAN based and to MS-DOS/WINDOWS adapted program serves or modelling complex 1- or 3-phase networks occurring in drive, control and energy systems. In the paper we show investigation results o a 3 kva-pwm-converter with a modulation requency o 1 khz supplying a -pole, 1 kw asynchronous motor (U=38 V, I=.8 A). To design the intermediate circuit, the L, C values o a typical 3- kva converter are chosen (Lobos, et al, 1999). Figure 1 shows voltage waveorm at the converter output (phase R) or the requency 4 Hz during a short circuit between two phases with ault resistance

3 o 1 Ω. The motor is provided with a positivesequence 3-phase voltage system R, S, T Complex space-phasor p = + j o a threephase system R, S, T is given by (Hosemann, 1984) R α S = (3) β T It describes, in addition to the positive-sequence component, an existing negative-sequence component, harmonic and non-harmonic requency components o the signal. The complex space-phasor o the converter output voltages is investigated using WVD. We also show the investigation results o the simulated transmission line ault (Lobos, 199). The voltage waveorm shown in Fig. 7 in the aulted phase ollowing a single line to ground ault is composed o the exponentially damped oscillating component with the requency R = 6 Hz. In this case only a real-valued one-phase signal is investigated. 4. INVESTIGATIONS Figures and 3 show the estimated requency representation o the space-phasor. In this case the investigated signal was sampled with the requency o 5 khz and samples were taken into calculation. Beore the ault the undamental component requency 4 Hz and the 1th harmonic are present, both positive-sequence components (situated in the right hal plane). 6 4 α β Fig.. Estimated WVD o the space-phasor (signal in Figure 1), at the time point t=.15 s. (beore the ault). Under ault conditions the negative-sequence component (situated in the let hal plane) o the undamental requency appears and also irregular requencies 44 and 56 Hz (mainly negativesequence components) have also been detected. Detection o the negative-sequence components (with negative requencies) can be applied as a ault indicator. In Figure 4 is shown the power spectrum o the signal or comparison. Despite the dramatic change in the signal shape ater the ault its power spectrum remains nearly identical beore and ater the ault. It proves the superiority o the proposed approach over the conventional FFT-based tool. 1 1 amplitude [V] Fig. 1. Voltage waveorm (phase R) at the motor input during a short circuit. Fault occurs ater the 1 th sample Fig. 3. Estimated WVD o the space-phasor (signal in Figure 1), at the time point t=.6 s. (ater the ault).

4 1 1 - beore ault ater ault Fig. 4. Power spectrum o the signal in Figure 1. Another set o investigations concerned the motor lead to ground ault. The Wigner-Ville representation o the space phasor computed rom the inverter output currents is presented in the Figures 5 and 6. In the case o lower short circuit current (Figure 5), the negative-sequence 5 Hz component is visible as a result o the asymmetry o the supply. In the second case (Figure 6) with lower short-circuit resistance, additional sub-harmonic components can be detected. WVD oers also the possibility to track the requency and amplitude changes o non-stationary signals. In this case the investigated signal was sampled with the requency o,5 khz and samples were taken into calculation. In Figure 8 the estimated instantaneous requency o the oscillating component o the signal in Figure 7 is shown. In Figure 9 the estimated instantaneous amplitude (power) o the main component is shown. Both characteristics show good correlation with the true changes o investigated signal Fig. 5. Estimated WVD o the space-phasor o the ault signal during the motor lead to ground shortcircuit in the converter drive (high short-circuit resistance) Fig. 6. Estimated WVD o the space-phasor o the ault signal during the motor lead to ground shortcircuit in the converter drive (low short-circuit resistance). amplitude [V] Fig. 7. Voltage waveorm (aulted phase) during single line to ground short circuit. Fault occurs ater the th sample Fig. 8. Estimated instantaneous requency o the signal in Figure 7.

5 18 amplitude Fig. 9. Estimated instantaneous amplitude o the signal in Figure CONCLUSIONS Visualisation o requency converter supplied drives by means o a static space-phasor is a very useul and compact observation and diagnosis method. Spectrums o the space-phasor and o the real-valued signal have been investigated under dierent operation conditions using the Wigner-Ville distribution. Superiority o the proposed approach over the conventional FFT-based tool was shown. Detection o irregular requencies may be useul or diagnosis o some drive aults. WVD oers also the possibility to track the requency and amplitude changes o non-stationary signals. ACKNOWLEDGEMENT The authors would like to thank the North Atlantic Treaty Organization (Advanced Science Fellowship or Dr. Z. Leonowicz) REFERENCES Andria G., M. Savino and A. Trotta (1994), Application o Wigner-Ville Distribution to Measurements on Transient Signals, IEEE Trans. on Instr. and Meas. 43, pp Hosemann G. (1984), Space-Phasors or Dynamic Minimum-Time Computing and Digital Real- Time Three-Phase Display. Proceedings o the Int. 8 th PSCC Conerence, pp Lobos T. (199), Fast Estimation o Symmetrical Components in Real-Time, IEE-Proc.-Gener. Trasm. Distrib.139 (1), pp Lobos T., D. Ruhm and Z. Waclawek (1999), Signal analysis in converter-ed drives using adaptive neural networks, in: R. Parenti & F. Masulli (ed.), Proceedings o the Int. Symposia "Intelligent Industrial Automation, pp Martin W. and P. Flandrin (1985), Wigner-Ville Spectral Analysis o Nonstationary Processes, IEEE Trans. on Acoustics, Speech and Signal Processing 33, pp Mingui Sun, Ching Chung Li, Laligam H. Sekhar and R.J. Sclabassi (1989), A Wigner Spectral Analyser or Nonstationary Signals, IEEE Trans. on Instr. and Meas., 38, pp Qian S. and D. Chen (1996), Joint Time-Frequency Analysis - Methods and Applications, Prentice- Hall, Upper Saddle River, NJ, p Velez E.F and R.G. Absher (199), Spectral estimation based on the Wigner-Ville representation, Signal Processing,, pp Zhiyue Lin (1997), An introduction to timerequency signal analysis, Sensor Review, 17, pp