Application of Wavelet Packet and S Transforms for Differential Protection of Power Transformer

: The differential protection of power transformers appears to be more difficult than any type of protection for any other part or element in a power system. Such difficulties arise from the existence of the magnetizing inrush phenomenon. Therefore, it is necessary to recognize between inrush current and the current arise from internal faults. In this paper, two approaches based on wavelet packet transform (WPT) and S-transform (ST) are applied to recognize different types of currents following in the transformer. In WPT approach, the selection of optimal mother wavelet and the optimal number of resolution is carried out using minimum description length (MDL) criteria before taking the decision for the extraction features from the WPT tree. In ST approach, the spectral energy index and the standard deviation (STD) are calculated from the S-matrix obtained by discrete S-transform. The two approaches are tested for generating a trip signal and disconnecting the transformer supply experimentally using 1KVA,


‫اﻟﻤﻔﺘﺎﺣﻴﺔ‬ ‫اﻟﻜﻠﻤﺎت‬
Frequencies and compressed at high frequencies so that large windows are used to obtain the low frequency components of the signal, while small windows reflect discontinuities.
The main objective of this paper is to develop and implement two approaches to recognize inrush current from faults based on wavelet packet transform (WPT) and Stransform.The MDL criterion is used to select both the mother wavelet and the level of decomposition for the WPT algorithm used.A signatures which represent the values and the time locations of the coefficients depend on the second level details 2 ad and 2 dd are considered in this algorithm.In the second approach, the spectral energy index and the STD are calculated from the S-matrix and used as an extraction features for signal recognition.The two approaches are examined experimentally on 1KVA, 220/110V, 50Hz, Y / ∆ three-phase transformer and the results revel fast, accurate and reliable methods to recognize different current following in the transformer.
The application of signal processing in power system transient has become increasingly popular in recent years due to its effectiveness in capturing short term transients.Basically, STFT transform, wavelet transform and S-transform are a means of obtaining a representation of both time and frequency contents of a differential current signal used in differential protection of power transformer.In this work, a short brief for WPT and ST are given.

Wavelet Packet Transform
Wavelet packet transform is a type of wavelet-based signal processing in a way that each level of resolution (octave) j consists of j 2 boxes generated by a tree of low pass filter (LPF) and high pass filter (HPF) operations.The frequency bandwidth of a box decreases with growing octave number i.e. with increasing octave number, the frequency resolution becomes higher while the time resolution is reduced.Starting with a discrete signal f[n] with length N , the first level, j=1 as follows [G.Stang , et., al., 1996 andM. A. S. K. Khan, et., al., 2007] are the first level approximations and details respectively, k is an integer and g(n), h(n) are the LPF and HPF associated with a selected mother wavelet function respectively.The output of both LPF and HPF are downsampled by two at the end of each filtering stage in order to increase the A higher level of decomposition can be produced in the same procedure above.
The main advantage of WPT over continuous and discrete wavelet transform is better, more accurate and more detailed representation of the decomposed signals [G.Stang, et., al., 1996] The implementation procedure of the WPT for two levels is shown in the decomposition tree of Fig. 1.

The Extended Wavelet Transform:
S-Transform The S-transform, is an extension to the ideas of wavelet transform, and is based on a moving and scalable localizing Gaussian window.From the S-matrix, an important information in terms of magnitude, phase and frequency can be extracted.Further, feature extraction is done by applying standard statistical techniques onto the S-matrix.
Many features such as amplitude, variance, mean, standard deviation and energy of the transformed signal are widely used for proper classification.The S-transform is fully convertible from the time domain to twodimensional (2-D) frequency translation domain and to then familiar Fourier frequency domain.The amplitude frequency-time spectrum and the phase-frequency-time spectrum are both useful in defining local spectral characteristics.
The S-transform produces a time frequency representation of a time varying signal by uniquely combining the frequency depended resolution with simultaneously localizing the real and imaginary spectra.The S-transform is similar to the wavelet transform but with a phase correction and here both the amplitude and phase spectrum of the signal are obtained.Since the S-transform provide the local spectrum of a signal, the time averaging of the local spectrum gives the Fourier transform [P.K. Dash, et., al., 2003 andM. V. Chilukuri, et., al., 2004].

CONTINUOUS S-TRANSFORM
It is well known that information is contained in the phase of the spectrum, as well as in the amplitude.In order to utilize the information contained in the phase of the continuous wavelet transform (CWT), it is necessary to modify of the mother wavelet.
where ) a , ( W τ is a scaled replica of the fundamental mother wavelet, the dilation a determines the width of the wavelet and this controls the resolution.The S-transform is obtained by multiplying the CWT with a phase factor as [S.Sendilkumar, et., al., 2009, P. K. Dash, et., al., 2003and M. V. Chilukuri, et., al., 2004].
where f is the frequency, and the quantity τ is a parameter which controls the position of Gaussian window on the t-axis and a is the dilation factor.The mother wavelet for this particular case is defined as In the equation above, the dilation factor "a" is the inverse of the frequency f.Thus, the final form of the continuous S-transform is obtained as

DISCRETE S-TRANSFORM
The signal h(t) can be expressed in a discrete from as h(kT), k=0,1,…......,N-1 and T is the sampling time interval.The discrete Fourier transform of h(kT) is obtained as [P.K. Dash, et., al., 2003 andM. V. Chilukuri, et., al., 2004] where n=0, 1,………., N-1.Using eq. ( 13), the S-Transform of a discrete time series h(kT) is obtained by making f = n/NT and τ = jT as where j, m =0, 1, 2,…...., N-1, and n= 1, 2,…...., N-1.For n=0 Equation 18 gives the constant average of the time series into zero frequency, so that the amplitude of the S-matrix over the time results in Fourier spectrum.The amplitude and phase of the S-transform are obtained as The multiresolution S-transform output is a complex matrix, the row of which is the frequency and the columns are the time values.Each column thus represents the "local spectrum" for that point in time.Also, frequency-time contours which have the same amplitude spectrum are obtained to detect, and localize signal disturbance events.A mesh three-dimensional (3-D) of the S-transform output yields frequency-time, amplitude-time, and frequency-amplitude plots.

EXPERIMENTAL SETUP AND DATA COLLECTION
The initial and important step in this work is to setup the protected transformer with all necessary equipments to accurately collect the needed data for analytical purpose and then to protect the transformer from internal faults.Figure 2 shows the schematic diagram for the implemented protection system given in Fig. 3 The collected data consists of the differential current of different magnetizing inrush current, different external fault currents and different internal current under load and noload conditions.For the case of collecting data, a conductor, timer and switch (switching circuit), as shown in Fig. 5, is used to isolate the transformer after ten cycles from the running instant to avoid damage in the transformer and the equipments used.
When the decision features are extracted and the algorithm of WPT and ST are ready to use, the tripping signal, if initiated, is used to isolate the supply by a relay circuit connected instead of the switching circuit.The schematic and the implemented relay circuits are shown in Figs. 6 and  Generated from the recognition algorithms is converted to analogue by the LabJack and sent to the relay circuit to use it for the complete system protection.
The data for different operating conditions is sampled at 6.4kHz, 128 samples per cycle and stored to be processed for constructing the features in the two mentioned algorithms.

DECISIONS FOR EXTRACTION FEATURES
In order to detect any fault following a disturbance, it is necessary to build an extraction features.A successful feature involves the identification of abnormal conditions by the mathematical modeling of the complete system or the analysis of the fault current signature resulting from different transient disturbances.A signature analysis method is used in the two approaches used in this work.

WPT Approach
Three various conditions from the collected data are used for the analysis of selecting the optimal mother wavelet filter and the number of level of decomposition.These various conditions are; unloaded magnetizing inrush current, primary double line to ground internal fault loaded and single line to ground external fault loaded.
The optimal mother wavelet filter can be selected by calculating the MDL index using the following formula [N.Saito, 1994 andS. A. Saleh, et. al., 2005

α
. It should be noted that the residual energy decreases as k increases.The number of coefficients k for which the MDL function reaches the minimum value, is considered as an optimal one.
The MDL index is calculated up to the second level of decomposition for five types of orthogonal and bi-orthogonal mother wavelet functions.The results for the three conditions are given in Tables 1, 2 and 3. From the these tables, it can be noted that Daubechies (db4) has the smallest value of MDL index in the first level.Therefore it can be considered the optimal mother wavelet function.These low values of MDL(1) may be due to the jumps of the data at the switching instant and some missing data.Thus, MDL(1) may not provide accurate indication and the higher levels of resolution will include more detailed representation of the signal.The optimal level of decomposition j is reached when MDL( j ) is less than levels of higher j .Table 4 presents four levels of MDL index using db4 mother wavelet.The second level has the lowest values of MDL index.Therefore, the second level is selected as the optimal level of decomposition.
To extract features for the WPT algorithm, the signals of the different differential currents are decomposed up to the second level of resolution by WPT using the selected mother wavelet function (db4).[S. A. Saleh, et. al., 2005] proposes an algorithm for the same features depends on the second level high frequency details (dd 2 ).This approach looks at the second level high and low frequency details (dd 2 and ad 2 ) for different current signals.These details are given in Fig. 8.
The values of these coefficients and their time locations represent the needed signatures to recognize the type of the investigated current.It is clear that the normal (inrush) current and the external fault current do not have any high frequency details components in the second level dd 2 and a threshold value for low frequency details in the second level.When an internal fault occurred, the details dd 2 and ad 2 show a threshold value frequency components, which can be used as a features extraction recognition.Therefore, the WPT algorithm is depended on the identification of dd 2 and ad 2 components or coefficients in the second level.

S-Transform Approach
The computation of the discrete Stransform can be outlined as follows.
1. Denote n/NT, m/NT, kT, and jT as n, m, k, and j respectively, for all of the computations.

Obtain discrete Fourier transform H[n] of the original time-varying signal h[k],
with N points and sample interval T, using DFT routine from eq. ( 15).

Compute the localizing Gaussian G[n,m]
for the required frequency n using eq.( 17).

Shift the spectrum H[n] to H[m+n] for
the frequency n by using convolution theorem. 5. Obtain the product of H(m+n) and G(m+n) and take inverse Fourier transform of the product to get S transform as eq.( 16).
The multiresolution S-transform output is a complex matrix, the row of which is the frequency and the columns are the time values.Each column thus represents the "local spectrum" for that point in time.Also, frequency-time contours which have the same amplitude spectrum are obtained to detect, and localize signal disturbance events.A mesh three-dimensional (3-D) of the S-transform output yields frequency-time, amplitude-time, and frequency-amplitude plots.
To illustrate the use of multiresolution Stransform for nonstationary signal analysis, a sample of inrush current waveform in a data window of eight cycles (using a sampling rate of 128 samples/ cycle) is considered.The three-dimensional (3-D) mesh for the signal shown in Fig. 9(a) is shown in Fig. 9(b).From the 3-D plot, one can find the magnitude, frequency, and time information's to detect, localize, and visually classify the event.Also, it is observed that the increase or decrease of the signal magnitude can be deduced from the innermost (or the lowest level) contour.Similar plots are shown for a sample of internal fault current waveform in Fig. 10.It is mentioned here that the time axis is replaced by the number of samples.
After the signal is retrieved, S-transform is used to process the signal samples to provide the relevant features for identifying the type of fault.The energy of inrush current and fault current are computed through Parseval's theorem [M.O. Oliveira, et., Al., 2008] from S-matrix output.On the other hand, the standard deviation is directly applied to the S-matrix output to derive the Standard Deviation (STD) values for the corresponding phase, for example, the feature extraction of energy and standard deviation of S-transform contour are obtained from Matlab functions as [S.Sendilkumar, et., al., 2009] Energy a = (S-matrix a) ^ 2 (20) S-matrix a = S-matrix of phase a and STD a = std (abs (S-matrix a)) Table 5 shows the spectral energy and the STD for different inrush, internal and external fault current for different connections of the three-phase transformer.It can be seen from this table that the values of energy vector and STD are able to be differentiable effectively between them or as pattern recognition to discriminate between inrush, external and internal fault current.This pattern is used to

ALGORITHMS IMPLEMENTATION AND EXPERIMENTAL RESULTS
The two algorithms are implemented experimentally to examine the ability of recognition between the internal fault currents and the current of inrush and external faults.Figure 11 shows the flowcharts for the software implementation of the two algorithms.In WPT algorithm, the three-phase differential current is read through the CTs and root summed squared together into one value before being passed to low and high pass filters to construct the first level d 1 and a 1 .The output is downloaded by two and then passed through another low and high pass filter to construct dd 2 and ad 2 .If the absolute value of dd 2 and ad 2 are higher than threshold, then it generates a trip signal in order to disconnect the transformer from the supply.
In ST algorithm, the S-matrix is calculated for each phase current.The energy and the standard deviation are also calculated.If E and STD are greater than the threshold, then a trip signal is initiated.Otherwise, a restrain signal is appear.
Several magnetizing inrush currents, external and internal fault currents both in load and noload cases are used to test the two algorithm with the hardware of the complete protection system.Three cases differ from that used in MDL calculation.Figure 12 shows the three-phase differential current and the trip signal generated by WPT algorithm (Fig. 12 a) and by ST algorithm (Fig. 12 b) for the case of magnetizing inrush current with a balanced Yconnected full load unity power factor.The two algorithms used do not generate any trip signal.Figure13 shows the same results for the case of loaded external three-phase to ground fault current after energization.The trip signal status is also not changed.
Two cases of different internal faults are considered to show the ability of disconnecting the transformer from the supply.Figures 14  and 15 show the results unloaded secondary three-phase to ground fault and unloaded secondary internal two-phase to ground fault respectively.In the two cases, the trip signal is generated

CONCULSIONS
This paper presents and applies two approaches for differential protection of power transformer based on WPT and S transforms.The main function of these approaches is how to recognize between (internal faults) and (inrush and external faults).The two approaches do not need any kind of training or harmonic analysis.The WPT method must select the optimal for both the mother wavelet filter and the number level of resolution.This limitation is overcome using Stransform but WPT is fast for recognition of current type.The two methods are quite simple to implement and easy to be coded with a small amount memory for storage and compilation.The two methods are applied with hardware implementation of the overall system and the experimental results are accurate, prompt and can be used for differential protection of power transformer.

7-REFERENCES
. The system is applied to 1KVA, 220/110V, 50Hz, Y / ∆ three-phase transformer.Three identical current transformer (CTs) are connected in Y on the ∆ connected primary side while another three identical CTs are connected in ∆ on the Y connected secondary side.The ratios of the primary side CTs and the secondary side CTs are chosen in a manner that guarantees a negligible value of the differential current under steady state operation conditions.The differential current signal of the three phases passes through Ω 1 resistor combination.Theses signals are sent into personal computer (PC) through analogue to digital circuit (LabJack-U3-HV), shown in Fig.4.The LabJack can operate also as digital to analogue converter.
7. The digital trip signal continuous wavelet transform function N α : vector of the decomposition coefficients of f ) k ( N α : vector of the continuous nonzero elements

Fig. 8
Fig.1 Two levels of decomposition by WPT