Implementation of Wavelet Transform

There are many types of mother wavelets, such as Harr, Daubichies, Coiflet and Symmlet wavelets. The choice of mother wavelet plays a significant role in detecting and localising different types of transients. In addition to this, the choice also depends on a particular application. The interest in this application is to detect and analyze low amplitude, short duration, fast decaying and oscillating type of high frequency currents. Daubichies's wavelet D4 is used in this report.

Figure 95 illustrates the implementation procedure of a Discrete Wavelet Transform (DWT), in which x[n] is the original signal, h[n] and g[n] are low-pass and high-pass filters, respectively. At the first stage, an original signal is divided into two halves of the frequency bandwidth by the low-pass and high-pass filters. The outputs of the filters are down-sampled by a factor of 2. The procedure is repeated until the signal is decomposed to a pre-defined level. The set of outputs thus attained represent the original waveform but all correspond to different frequency bands. The frequency band of each detail of the DWT is directly related to the sampling rate of the

original waveform. The sampling rate, Fs Hz, should be chosen so that the Nyquist‟s criterion is

satisfied. The highest frequency that can be extracted would be 0.5-Fs Hz. This frequency would

frequencies between 0.5-Fs and 0.25-Fs would be captured in detail 1; similarly, the band of

frequencies between 0.25-Fs and 0.125-Fs would be captured in detail 2, and so on.

Figure 95  Implementation procedure of a Discrete Wavelet Transform

6.7.2.1 Wavelet Transform of Magnetizing Inrush Current

A transient magnetising inrush current flows in the primary winding when a power transformer is energized from the primary side with the secondary winding open-circuited. This current may reach instantaneous peaks of 6-8 times full-load current because of the extreme saturation of the core in the power transformer. Figure 96 (a) typifies the magnetising inrush current waveforms of differential currents from the CT secondary sides of phases a, b and c of transformer system 1.

This figure shows that the current waveforms are distorted significantly; gaps appear in the inrush current waveforms. At the edges of the gaps, the current magnitude changes from near zero to a significant value or from a significant value to near zero; this is expected by virtue of the fact that sudden changes from one state to other states produce small ripples that are not often not visible due to the large fundamental frequency signals as apparent from Figure 96 (a). However, these phenomena can be discerned (in terms of the various signatures) and is clearly demonstrated by the wavelet transform. For brevity, only the DWT of the a-phase differential current is shown in this figure.

The original inrush current waveform has been sampled at 25 kHz and passed through a DWT based on the structure shown in Figure 95. Five detailed outputs that contain a frequency band of 12.5 kHz ~ 6.25 kHz at detail 1, 6.25 kHz ~ 3.125 kHz at detail 2, 3.125 kHz ~ 1.562 kHz at detail 3, 1.562 kHz ~ 781 Hz at detail 4, 781 Hz ~ 390 Hz at detail 5 and one output in the frequency band 390 Hz ~ DC level are shown in Figure 96 (b)-(g). These figures show that there are very useful features in the decomposed magnetising inrush signals. A certain high frequency component can be located better in time than a low frequency component. In contrast, a low frequency component can be located better in the frequency domain than the high frequency component. This means that all the features for a particular signal are obtained. In this study, the interest is in the components that are located better in time; details 1-3 are analyzed and their features are extracted.

Figure 96 (b)-(d) that correspond to details 1-3 show a number of sharp spikes during the period of the inrush-current transient. A number of the spikes arise at the edges of gaps at which the inrush current suddenly changes from one state to other different states; others are produced because the primary windings of the power transformer are connected in delta configuration and the differential current of phase-a is in fact the difference between the phase-a and phase-c magnetising inrush currents. This results in the non-smooth points in the current waveforms that in turn cause sharp spikes to appear in the DWT of the current waveforms.

6.7.2.2 Wavelet Transform of Internal Fault Current

Figure 97 shows waveforms of currents and DWT outputs for phase-a and phase-b to ground fault on the high-voltage side. Figure 97 (a) shows differential currents of phases a, b and c out the CT secondary windings for this fault internal fault. The figure shows that there is high frequency distortion in the current waveforms. This is due to the distributed nature of the inductance and capacitance of the transmission line.

For brevity, only the DWT of the phase-a differential current is presented in Figure 97 (b)-(g); details 1-3 of the DWT show several sharp spikes immediately following the inception of the fault. However, there is a marked contrast to the spikes in the inrush current case. The spikes in the fault case decay to near zero within one cycle, whereas the spikes associated with the inrush current suffer from little attenuation during the entire inrush transient that lasts from 0.20 s for small transformers to 1 minutes for large units [26]. It is apparent that this difference can be effectively used as the key feature to distinguish the internal fault from the inrush current.

6.7.2.3 Wavelet Transform of External Fault Current with CT Saturation

Differential current is used in transformer protection systems to restrain during normal load flows and during external faults. However, external short circuits can result in large differential currents if one or more CTs saturate. It is, therefore, crucial to check the impact of CT saturation on the measured currents during external faults. The severity of CT saturation is accentuated by the presence of remnant flux in the CT core. Figure 98 (a) shows the waveforms of simulated differential current out of secondary windings of CTs during an external three-phase short circuit

at Bus 2 in Figure 93. The initial remnant flux was 65% of rated flux in the core of the CT of phase-a on the low voltage side, and a zero remnant flux in phase-a CT core on the high voltage side. Figure 98 (a) shows that the differential in phase-a is substantial.

Figure 98 (b)-(d) show details 1-3 of the DWT; these figures shown that there are several spikes that last during the entire period of the transient that depends on the DC component and the remnant flux of CT core. However, unlike the case for inrush current, these bursts comprise of a number of spikes clustered very close to each other. Here again, it is the apparent unique features that can be used to distinguish between an internal fault and an external fault with CT saturation.

Figure 97  Details and approximation for a phase-a to phase-b to earth fault on high-voltage side