Impulse Fault Classification in Transformers by Fractal Analysis

Transcription

1 IEEE Transactions on Dielectrics and Electrical Insulation Vol. 10, No. 1; February 2003 I09 Impulse Fault Classification in Transformers by Fractal Analysis P. Purkait Instrumentation Engineering Departmedt Haldia Institute of Technology Haldia, Midnapore, , India and s. Chakravorti Electrical Engineering Departmcnt ladavpur University Kolkata, , India ABSTRACT Transformers are usually subjected to lightning impulse tests after assembly for assessment of their insulation strength. In the case of a fault the resulting winding current gets changed to a certain extent. The pattern of the fault currents depends on the type of fault and its location along the length of the winding. This paper describes the application of the concept of fractal geometry to analyze the properties of fault currents. Fractal features such as fractal dimension, lacunarity used for image surface recognition and the sliding window algorithm used for fractal aualysis of waveform have been employed for classification of transformer impulse faults. Experimental results obtained for a 3 MVA transformer and simulation results ohtained for 3 MVA, 5 MVA and 7 MVA transformers are presented to illustrate the ability of this approach to classify insulation failures. The results indicate that this new approach possesses reasonable abilities for waveform pattern discrimination. Index Terms - Transformers, insulation, impulse fault, fault classification, fault simulation, fractal dimensions, lacunarity, sliding window algorithm. 1 INTRODUCTION MPULSE testing of transformers after assembly is an I accepted procedure for the assessment of their winding insulation strength to surge over-voltages. In such tests impulse voltage sequences are generated in the laboratory and applied to the transformers as per standards [l]. Sometimes, the windings fail to withstand the excessive voltage stress. The detection of the fault and the location of the damaged portion, which might take a long time, have to be found out for repair work. For many years the resulting current and voltage waveforms were analyzed manually by studying oscillographic records [2-61. Such manual interpretation of the waveform patterns for fault identification and classification was strongly dependent on the knowledge and experience of the experts performing the analysis. With the advent of digital recorders and analyzers, there has been an increasing trend to use the frequency domain analysis, particularly the transfer function approach [7-91 for fault classification. Manuscript receiurd on 9 Augun 2001, in final form 27 lune Fractals have been found to be successful to provide a description of naturally occurring phenomena and shapes, wherein conventional and existing mathematical models were found to be inadequate. In recent years, this technique has attracted increased attention for classification of textures and objects present in images and scenes [lo-141. The application of fractals in the field of electrical insulation commcnced in the early 1980 s. Researchers like Niemeyer et al. [El and Fuji et al. [161 presented results to show that the electrical tree shapes of the discharge process within the insulation material has fractal properties and also pointed out that the fractal dimension is correlated to the shape of the tree. Use of fractals for feature extraction and recognition of partial discharge (PD) patterns have also been reported by several researchers [ Fractal features like fractal dimension and lacunarity were used for the classification of PD patterns. Suitable and effective uses of fractal geometry for pattern classification of waveforms and profile-like cumes have also been reported by several researchers [19-22] /1/$ IEEE

2 I10 Purkait and Chakravorti: Impulse Fault Classification in Transformers by Fractal Analysis Insulation failure in transformers during impulse test is also a natural phenomenon and thc resulting winding currents have complex waveforms. The type and location of faults inside the transformer winding is related to the nature of the faulty current waveforms This complex nature of the current waveforms and the ability of the fractal geometry to discriminate complex patterns encouraged the authors to study its application for impulse fault classification in transformers. In this paper results are reported on the use of fractal geometry for the feature extraction and pattern recognition of winding current waveforms and classification of impulse faults therein. The study has been based on impulse faults simulated using the Electromagnetic Transient Program (EMTP) models of a range of power transformers commonly used up to a voltage rating of 33 kv. The results were found to he in good agreement with an experiment performed on the analog model of a 3 MVA, 33/11 kv transformer. 2 TRANSFORMER MODELS EMTP based high frequency models [ of a range of power transformers commonly used up to 33 kv have been developed for the present study. In all the EMTP models, the delta-connected discs winding of the HV side of the transformers have been represented by a network with lumped parameters. The validity of the EMTP models can he observed from Figure 1, showing fault current waveforms for the analog as well as EMTP model of the 3MVA, 33/11 kv transformer. These parameters have been calculated from the practical design data of a number of transformers as summarized in Table la,lb. 2.1 TYPES OF FAULT Insulation failures may result in two classes of winding faults in a transformer during impulse tests, namely series and shunt faults. Series fault implies insulation failure between the discs or between the tums, while shunt fault represents insulation failure between the winding and grounded components like the tank, core, etc. Both of Table la. Transformer design data for digital model. P V D OD ID T /11 a / / Table 1 b. Symbols for Table la. Symbol Quantity Unit P Power rating kva V Voltage Rating kv D Number of discs on the HV winding OD Outer diameter of the HV winding mm ID Inner diameter of the HV winding mm 7 Total number of turns in the HV winding these classes of faults may occur anywhere along the entire length of the winding. In the present study, the entire winding has been divided into three sections: namely the line-end, the mid-winding and the ground-end sections, each involving 33.33% of the total length of the winding. Both of the classes of faults have been simulated separately in the three different sections to represent as many faulty conditions as possible. Each fault has been made to involve 5% to 10% of the winding length. All the current waves considered for analysis have been converted into a per-unit scale to effectively take into account the possible variations in current wave due to applied voltage wave magnitude variations. The acronyms used for different types of faults considered in this study are given in Table 2. Figure 2 shows tank-current 'waveforms for different fault conditions in the 5 MVA, 33/11 kv transformer. 2.2 TEST SET-UP FOR ANALOG MODEL To demonstrate the acceptability of the present study, experiments have becn performed on the analog model of a 3 MVA, 33/11 kv transformer. Performance of the 3 MVA transformer is studied with its whole winding being subjected to impulses from a recurrent surge generator (RSG). The winding response, captured by the tank-current method, is acquired by a personal computer (PC) via RS-232 interface circuit through a Tektronix TDS 320, 100 MHz, 500 M Samples/s digital storage oscilloscope. The test set-up is shown in Figure 3. I Mo 250 Time in micro-s Figure 1. Fault currents for 3MVA transformer. 1, analog model; 2, EMTP model. Table 2. Types of different faults simulated. Acronvms Faults NF No-Fault SE Series Fault at any section of the winding SH Shunt Fault at any section of the winding SEL Series Fault at Line-End SEM Series Fault at Mid-Winding SEE Series Fault at Earth-End SHL Shunt Fault at Line-End SHM Shunt Fdt at Mid-Winding SHE Shunt Fault at Earth-End

3 IEEE Transactions on Dielectrics and Electrical Insuhtion Vol. IO, No. 1; February (b) Figure 2. Currents in 5 MVA transformer. a, series faults; b, shunt faults; 1, NF; 2, SEL; 3, SEM; 4, SEE 5, SHL; 6, SHM; 7, SHE. 3 FRACTAL TECHNIQUES FOR IMPULSE FAULT CLASSIFICATION 3.1 FRACTAL DIMENSION The term fractal dimension was introduced by Mandelbrot [lo]. According to him, a set A in an Euclidean n- space, is said to be self-similar when A is the union of N distinct (non-overlapping) copies of itself, each of which has been scaled down by a ratio r in all co-ordinates. The fractal similarity dimension D of the set A is given by the relation Or NrD=I Natural fractal surfaces do not, in general, possess this deterministic self-similarity. Instead, they exhibit statistical self-similarity; that is they are composed of N distinct subsets each of which is scaled down by a ratio r from the original and is identical in all statistical respects to the scaled original. The fractal (similarity) dimension for these surfaces is also given by equation (1). While the definition of fractal dimension by self-similarity is straightforward, it is often difficult to estimate directly from the image data. However, a related measure of the fractal dimension, the box-dimension, can be more easily computed from a fractal set A in an n-dimensional Figure 3. Experimental sc1-m for 3MVA. 33/11 kv trsnsfonncr analog model. space [11-12]. Let, the entire set A is covered by a single n-dimensional box of size L,=. If the set A is now scaled down by the ratio r, then there are N = FD subsets, and so the number of boxes of size L = rl,, needed to cover the whole set is given by The simplest way to estimate D from equation (2) is to divide the n-dimensional space into a grid of boxes with side length L (i.e. square of side L for the present case of 2-dimensional space) and to count the number of nonempty boxes. Fractal dimension D is then obtained from the slope of a least square linear fit on the plot [log(l), -log(n(l))i. 3.2 LACUNARITY Among the various fractal features that can be computed from a complex pattern or texture, the fractal,dimension is the primaly one. However, it has been obselved that fractal dimension alone is insufficient for the purpose of discrimination, since two differently appearing surfaces can have the same value of D. To overcome this, Mandelbrot [lo] introduced the term called lacunarity A, which quantifies the denseness of an image surface. Many definitions of this term have been proposed and the basic

4 ~ 112 Purkait and Chakrauorti: Impulse Fault Classification in Transformers by Fractal Analysis idea is to quantify the gaps or lacunae present in the given pattern. One of the useful definitions of this term as suggested by Mandelbrot is (3) where M is the mass of the fractal set, and E(M) is the expected mass. The mass M of a fractal set is dependent on the length L of the measuring device-governed by the power law M( L) = K LD (4) where K is a constant. The lacunarity, thus, is a function of L. A procedure for calculating Lacunarity is given in [11-12]. Let P(m,L) be the possibility that there are m points within a box of side L Le. square of side L), which is centered about an arbitraly point on the waveform. Then P(m,L) is normalized, as below for all L N m=1 P(m,L)=i (5) where N is the number of possible points within the box. Let, the total number of points in the image is M. If the entire waveform is overlayed with boxes of size L, then the number of boxes with m points inside the box is (M/m)P(m,L). Defining M(L) and Mz(L), Voss [ll] defined lacunarity as N M(L)= mp(m, L) (6) rn = 1 N W(L) = m2p(m, L) (7) m=l 3.3 APPLICATION OF BOX-DIMENSION AND LACUNARITY FOR IMPULSE FAULT CLASSIFICATION In order to calculate an estimate of P(m,L), each square of side L must be centered around a point on the current waveform and count the number of neighboring points that fall within the square. Accumulating the occurrences of each number of neighboring points over the waveform gives the frequency of occurrence of m. This is normalized to obtain P(m,L). Values of L are chosen to be odd to simplify the centering process. Also, the centering and counting activity is restricted to pixels having all their neighbors inside the waveform. This obviously will leave out image portions of width = (L - 1)/2 at both ends. As is seen, large values of L result in increased areas being excluded during the counting process, thereby Wr-I Figure 4. Sampld plot of the set (lag(l),-log(n(l))) for different box sizes and different faults for 3 MVA transformer EMTP model. 1, NF 2, SEL; 3, SEM; 4, SEE 5, SHL; 6, SHM; 7, SHE. increasing the uncertainty about counts near the two ends of the waveform. Additionally, the computation time grows with the value of L. Hence L = 3,5,7,9,11 and 13 were chosen for this work. Figure 4 is a sample plot of the set (log(l),-log(n(l))l for the six chosen values of L computed for all the different faults simulated on the EMTP model of the 3 MVA transformer. It is observed from the Figure 4, the plots for SEL and SEM are too close to each other to be distinguished and so are the case for SEE and SHE. A least.square fit to these data sets are performed to obtain the fractal dimensions corresponding to each faulty current waveform. The fractal dimension has been calculated by box-counting method by treating the waveforms as a curved line within a fixed boundaly. Since the inherent oscillations in the waveforms are not very high, the fractal dimensions are found to have values close to 1.0, as expected. Table 3 summarizes ihe fractal dimensions corresponding to different faults for all the three transformer ratings. It is observed from the values of the fractal dimensions in Table 3 and also from the nature of the plots in Figure 4 that fractal dimension alone is not sufficient to provide adequate discrimination between different patterns. However, the incorporation of the second fractal feature, namely the lacunarity, improves the situation. Lacunarity essentially is a term that describes the feature of a 2-dimensional image. Lacunarity of the current waveforms has been calculated considering the area under the current wave over a fixed current-time range. In this way, lacunar- Table 3. Fractal dimension as obtained by box-counting method for different faults. Fault Fractal Dimension 3 MVA 5 MVA 7MVA NF , SEL , SEM 1, SEE SHI ~.nn9m ~~~~~ 1.nn2xx4 ~ ~~~~~ SHM SHE

5 IEEE Transactions on Dielectrics and Electrical Insulation Vol. 10, No. I; February Bo*sl2e Figure 5. Sample plot for box-size vs lacunarity for different faults in 5 MVA transformer. 1, NF; 2, SEL; 3, SEM; 4, SEE; 5 SHL; 6, SHM; 7, SHE. ity, which is a fractal property of a 2-dimensional data, has been calculated from the 1-dimensional waveforms. Lacunarity is computed, as explained in the previous section, for each value of L and Figure 5 shows its variation with respect to the box size L for different faults simulated for the EMTP model of the 5 MVA transformer. Similar nature of variation of lacunarity with box-size L have been reported for texture data in [12] and for PD pattern data in [171. Figure 6 is a plot for lacunarity with respect to fractal dimension for all the different faults simulated in the EMTP models of (a) 5 MVA and (b) 7 MVA transformers. Lacunarity was found to be maximum for all the waveform patterns considered, at L = 3 and hence, this L value was chosen in Figure 6 for convenience. Table 4 summarizes the values lacunarity and fractal dimension for different faults simulated in all three transformer EMTP models. Individual classes of series and shunt faults could be separately identified and are circled (SH and SE) on Figure 6. As can be seen from Figure 6, patterns belonging to a particular class of fault (SH or SE) are found to lie close to each other forming two different clusters. It also depicts that further sub-clusters can be formed to classify SHE, SHM and SHE. However, the exact location of the SE namely SEL, SEM or SEE still cannot he discriminated by this method. This proves, though to a limited extent, the fairly reasonable discrimination abilities possessed by these two fractal features for impulse fault recognition. In an effort to improve further, investigations have been carried out using the sliding window algorithm, as described by Sevcik [19], for further classification of transformer impulse faults. 3.4 SLIDING WINDOW ALGORITHM The term waveform applies to the shape of a wave containing instantaneous values of a periodic quantity versus time. Any waveform is ideally an infinite series of points. Apart from classical methods such as moment statistics and regression analysis, properties such as the Kolmogorov-Sinai entropy [271, the apparent entropy and the fractal dimension have also been proposed to tackle the problem of pattern analysis of waveforms. As reported by Fractal Dimension I Fractal Dkqenslon I 6 NP ASEL* SEM B SEE I (b) Figure 6. Sample plot for lacunarily vs fractal dimension for all the different faults. a, 5 MVA, b, 7 MVA transformer. Katz [20], for fractal analysis of waveforms the fractal dimension might be measured empirically by sampling the waveform at Ns points evenly spaced on the abscissa. This procedure discretizes the waveform into N' = Ns - 1 segments and then, according to Katz's equation where d is the planar extent of the curve and L, is the length of the curve, both of them defined as d = max[dist(l, j)] N Ls= dist(i,i+l) (10) L=l dist(i,j) is the distance between points i and j of the curve and d = max(dist(1,i)). Refinements of the above Katz's equation (9) were proposed by Sevcik [19]. He derived a method for calculating the approximate fractal dimension (D) from a set of N, values of y, sampled from a waveform between time zero and, f with a sampling interval At. Sevcik obtained an expression to calculate the fractal dimension of a waveform starting from the definition of Hausdorff dimension D,. The Hausdorff dimension [lo] of a set in a metric space may be expressed as

6 114 Purkait and Chakrauorti: Impulse Fault Classification in Transformers by Fractal Analysis Table 4. Lacunarity and fractal dimensions for different classes of faults in 3 MVA, 5 MVA and 7 MVA transformers. FBUItS 3 MVA 5 MVA 7 MVA Dimension Lacunarity Dimension Lacunarity Dimension Lacunarity NF SEL SEM SEE SH1 1 nns nn ~ ~ ~~." SHM SHE where NS(e) is the number squares of side 2t needed to cover the set. In a metric space, given any point P, a square centered at P and side 2~ is a set of all points x for which dist(p,x)< E. A line of length L,? may be divided into NS(6)= L.42~) segments of length ZE, and may be covered by NS(t) squares of side 2 ~ Thus,. (11) may be rewritten as 121 I in 0 'CI B Y c 101) Waveforms are planar cumes in a space with coordinates usually having different units. Since the topology of. a metric space does not change under linear transformation, it is convenient linearly to transform a waveform into another in a normalized space, where all axes are equal. Two linear transformations are used that map the original waveform into another embedded in an equivalent metric space. The first transformation normalizes every point in the abscissa as x'=- xi xm, (13) where xi are the original values of the abscissa, and nmai is the maximum xi. The second transformation normalizes the ordinate as follows 1.25 g 1.2 'I B a e sampling lntaval (us) (b) Figure 7. Fractal dimcnsion vs sampling intend for different transformer faults. a, EMTP model of 7 MVA; b, analog model of 3 MVA. 1, NF; 2, SEL; 3, SEM, 4, SEE 5, SHL; 6, SHM; 7, SHE. Yi - Ymi, y' = Ym, - Ymin (14) comes where y, are the original values of the ordinate, and ymj, and ymax are the minimum and maximum yi, respectively. These two linear transformations map the Ns points of the waveform into another that belongs to a unit square. This unit square may be visualized as covered by a grid of N? x Ns cells. Ns of them containing one point of the transformed waveform. Calculating L, of the transformed waveform and taking E = 1/(2 X N'), equation (12) be- (15) Figure 7 represents the variation of fractal dimension with the sampling intend At of the current waveforms corresponding to different faults for (a) the EMTP model of the 7 MVA transformer and (b) the analog model of the 3 MVA, 33/11 kv transformer.

7 IEEE Transactions on Dielectrics and Electrical Insulation Val. 10, No. I; February The procedure to estimate fractal dimension of a waveform expressed by equation (15) is very fast and can be applied to small amounts of data. The sliding window algorithm [19] is applicable in this case for segmentation of the data set of the entire waveform into smaller window sizes. The fractal dimensions for all the different classes of faults for all the transformers were estimated using equation (15) for each individual window. After each estimate, the window was moved one sampling intelval forward, the calculation repeated, and so on. The clarity of the results however, was found to depend on the window size chosen. A smaller window size, though saves computation time, often loses distinctive features of the dimensional analysis. 3.5 APPLICATION OF THE SLIDING WINDOW ALGORITHM TO IMPULSE FAULT CLASSIFICATION Figure 8 gives the sample plot of fractal dimension for a sliding window size of 75 sampling points for all the fault current waveforms for the 3 MVA transformer EMTP model. It is found that though there is a noticeable demarcation between the plots for each fault class, the clarity needs to be improved. Moreovcr the primary distinguishing features of the fractal-dimension plots for different faults are present well within the first 300 to 400 sampling points, it is thus irrelevant to plot the entire range for necessary feature extraction. Figure 9 represents the dimensions calculated for a higher sliding window size of 151 sampling points for all classes of faults for all the three transformer models, and plotted on a zoomed scale for better clarity. The plot of Figure 9 effectively depicts the applicability of the proposed algorithm for classification of all the different kinds of faults studied in the present work. The proposed method thus has reasonable differentiation capabilities between different fault currents as can he observed by a comparative study of the timedomain plots of Figure 2 and the fractal dimension plots of Figure 9. The acceptability of the present work is aug- 1.1 I 1.1, 1 1 1oD 150 zoo Sampling pomis (a) I 4 I (C) Fiaure "~ 9. Fractal dimension dots for a slidine window size of 151 Y sampling points for all classes of faults in the EMTP models of transformers. a, 7 MVA, b, 5 MVA; c, 3 MVA. 1, NF; 2, SEL; 3, SEM; 4, SEE; 5, SHL; 6, SHM; 7, SHE. a 1 tr ID5 1 la) 2a) 3w 400 sw) DXl smpirg Pcilpr Figure 8. Sample plot of fractal dimension for a sliding window size of 75 sampling points for all the fault current waveforms for the EMTP model of 3 MVA transformer. 1, NF; 2, SEL; 3, SEM; 4, SEE; 5, SHL; 6, SHM; I, SHE. mented by fault classification study on the analog model of a 3 MVA, 33/11 kv transformer. One representative plot of the fractal dimension of results obtained from the analog model is provided in Figure 7b. The results obtained were found to be acceptable. W 4 CONCLUSIONS INDING currents in transformers vary to different extent depending upon the type of impulse fault. Fractal analyses of such complex current waveforms have been reported in this paper for classification of impulse faults in transformers. Studies have been carried out using

8 I16 Purkait and Chakrauorti: Impulse Fault Classification in Transformers by Fractal Analysis results obtained from EMTP model of the transformers. Results show that fractal dimension alone is not good enough for fault classification. But the use of lacunarity along with fractal dimension can discriminate between shunt and series faults in a better way. A sliding window algorithm for the determination of fractal dimension has bee,, found to be able to distinguish different faults effectively. Fractal analyses of experimental waveforms Obtained from the analog Inode Of a MVA transformer also corroborated the observation made from EMTP model results. REFERENCES, Power Transformer - Insulation Levels and Dielectric Tests, IEC Publication 76-3, [21 Guide to the Lightning and Switching Impulse Testing of Power Transformers and Reactors, IEC Standard, Publication 722, [31 F. Beldi, The Impulse Testing of Transformers, Brown Boveri Rev., Vol. 37, pp , [4] J. H. Hagenguth, and 1. R. Meador, Impulse Testing of power Transformers. AlEE Trans., Vol. 71, PP , [51 C. Aicher, Experience With Transformer Impulse Failure Detection Methods,AIEE Trans. Vol. 67, pp , [6l G. B. Harper, Detection and Diagnosis of Dcterioration and Faults in Power Transformers, CIGRE, paper 12.01, pp , [7] R. Malewski and B. Poulin, IApulse Testing of Power Transformers Using the Transfer Function Method, IEEE Trans. Power Delivery, Val. 3, pp , [SI J. Bak-Jensen, B. Bak-Jensen and S. D. Mikkelsen, Detection of Faults and Aging Phenomena in Transformers by Transfer Functions, IEEE Trans. Power Delivery, Val. 10, pp , [9] R. Vajana and K. Udayakumar, Fault Lacation in Power Transformers During Impulse Tests, in Proc leee PES Winter Meeting, paper No , [lo] B. B. Mandelbrot, ~ Frnctol Geomet~ofNoture, Freeman, New York, [Ill R. F. Voss, Random Fractals: Characterisation and Measurement,in Scaling Phenomena in Dimrderrd Systems, Eds. R. Pynn and A. Skjcltrop, Plenum Press, New York, pp. 1-11, J. M. Keller, S. Chin and R. M. Crownaver, Texture Description and Segmentation Through Fractal Geometry, Computer Vision, Graphics, and Image Processing, Val. 45, pp , S. Chen, J. M. Keller and R. M. Crownover, On the Calculation of Fractal Features from Images, IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 15, pp , [I41 s, Lovejoy and D, Schertrer, Scale Invariance, symmetries, Fractals and Stochastic Simulations of Atmospheric Phenomena, Bulletin of American Meteorological Society, Vol. 67, pp , L. Nicmeyer, L. Pietronero and H. J. Wisemann, Fractal Dimension of Dielectric Breakdown, Phys. Rev. Lett., Vol. 6, pp , [16] M, Fuji, M, Watanabe, 1. Kitani, K, Arii and K, Yoshino, Fractal Character of dc Trees in Palymethylmethacrylate, IEEE Trans. El, Vol. 26, pp , [17] L. Satish and W.S. Zaengl, Can Fractal Features be Used for Recognizing #-D Partial Discharge Patterns?, IEEE Trans. DEI. Vol. 2. OD [181 R, Candela, G, ~ i~~lli, of Statistical and Fractal Parameters and a Neural Neovork, IEEE Trans. DEI, Vol. 7, [19l C. Sevcik, A Procedure to Estimate the Fractal Dimension of Waveforms, Cmplexity Intem. J., Vol. 5, [Online]. Available: http~//l~~/~~~/~d~/d~/~/edu/du/complex/cl/uols/seucik. [201 M. J. Katz, Fractals and the Analysis of Waveforms,Comput. Bid. Med., Val. 18, pp. 145, [21l P. Vanouplines, Rescaled Range Analysis and the Fractal Dimension of pi.[onlinrl. Available: htrp://homepages. uub.ac.be/- poupli/pi/compfdim.htm R. Esteller, G. Vachtsevanos, J. Echauz and B. Litt, A Comparison of Waveform Fractal Dimension Algorithms, IEEE Trans. Circuits and Systems-I, Vol. 48, pp , I231 P. Purkait and S. Chakravorti, An Expert System for Fault Diagnosis in Transformers During Impulse Tests, in Proc IEEE PES Winter Meeting, paper no , [241 P. Purkait and S. Chakravorti, Time and Frequency Domain Analysis Based Expert System for Impulse Fault Diagnosis in Transformers, IEEE Trans. DEI, Vol. 9, pp , P. T. M. Vaessen, Transformer Model for High Frequencies, IEEE Trans. Power Delivery, Val. 3, pp , ~~ [26l F. de. Leon and A. Semlyen, Complete Transformer Model for Electromagnetic Transients, IEEE Trans. Power Delivery, Vol. 9, pp , [27l P. Grassberger and I. Procaccia, Estimation of the Kolmogoran Entropy from Chaotic Signal, Phys. Rev.Lett. A, Vol. 28, pp , R, schifani, -pd Recognition by Means