IDIA ISTITUTE O TECHOLOGY, KHARAGPUR 7232, DECEMBER 27-29, 22 59 umerical Differential Protection of Power Transformer using Algorithm based on ast Haar Wavelet Transform K. K. Gupta and D.. Vishwakarma Abstract-- A fast and simple numerical filtering algorithm is an important requirement for the efficient power system relaying. This paper presents a wavelet-based algorithm for numerical differential protection of power transformer, using harmonic restraint approach. This algorithm provides an accurate and computationally efficient tool for distinguishing internal faults from magnetizing inrush and over excitation inrush. This algorithm is essentially a numerical filter, which extract the fundamental frequency and second and fifth harmonic s of differential current, to provide operating and restraining signal respectively. This algorithm generates it s coefficient by additions and subtractions routines only and does not involve time consuming multiplication and division calculation. The computation complexity of the algorithm presented here is O() additions and subtractions as compared to DT based algorithm, whose computation complexity is O(log2) operations. Index Terms Differential Protection, Haar Transform, Wavelet Transform, ourier Transform. I. ITRODUCTIO A power transformer belongs to a class of very expensive and vital of electric power system and its protection is one of the most challenging problems in the area of power system relying. The frequency of occurrence faults in power transformer is less than on lines. But if a power transformer experiences a fault, it is necessary to take the transformer out of service as soon as possible so that damage is minimized. The cost associated with repairing a damaged transformer may be very high. An unplanned outage of a power transformer can cost electric utilities crores of rupees. Consequently, it is of great importance to minimize the frequency and duration of unwanted outages. Accordingly, high demands are imposed on power transformer protective relays. Requirements include Dependability (no missing operation) Security (no false tripping) Speed of operation (short fault clearing time) Differential protection scheme based on circulating current principle is widely used to protect the power transformer against internal faults [4], [9]. The authors are with the Institute of Technology, Banaras Hindu University, Varanasi - 225, India (telephone: 9-542-57733, e-mail: gupta_k_k@rediffmail.com, dnv@banaras.ernet.in). The differential protection scheme shows certain limitations. Detection of a differential current does not provide a clear distinction between internal faults and other conditions, e.g. Magnetizing inrush, over-excitation of the transformer core, external faults. The conventional approach to mitigate these problems is to apply percentage (biased) differential characteristic along with second and fifth harmonic restraints for inrush and over-excitation conditions, respectively. The conventional percentage differential relays used for the protection of power transformer against internal faults are either of electromagnetic or static type. Electromagnetic and static relays have several drawbacks. The concept of numerical protection, which evolved during the last two decades, shows much promise in providing improved performance. The main features of the umerical relays are their economy, reliability, compactness, flexibility and the possibility of integrating a umerical relay into the hierarchical computer system within the substation. II. AST HAAR WAVELET TRAORM (HWT) TECHIQUE The proposed algorithm is derived from the well-known Haar Transform based algorithm. In Haar Transform the Haar coefficients are calculated first and from Haar coefficients the sine and cosine ourier coefficients are calculated, using established relationship between Haar and ourier coefficients. The Haar transform can also be termed as Discrete Wavelet Transform (DWT), using Haar wavelet. The Discrete Wavelet (Haar) Transform requires log 2 operations to transform a sample vector. The DWT matrix is not sparse in general, so it has the same complexity issues as the discrete ourier transform. So it is solved in same way as for the T, by factoring the DWT into a product of a few sparse matrices using selfsimilarity properties. This result in an algorithm that requires only order operations to transform an -sample vector. This fast version of the DWT can be called as ast Wavelet Transform (WT). The WT using Haar Wavelet calculates Haar coefficients by 4(-) multiplications. Hence its computation complexity is O() operations. In the algorithm presented here these 4(-) multiplications are converted in 2(-) additions. Hence the computation complexity is reduced to O() additions. But the coefficients calculated by this modified algorithm will not be
5 ATIOAL POWER SYSTEMS COERECE, PSC 22 same as Haar coefficients. We can call them as Modified Haar coefficients. A relationship between the Modified Haar coefficient and ourier coefficients can be established and from that the sine and cosine ourier coefficients of the input signal can be calculated. A signal vector [ X ], whose length is an integer power of two, can expressed as [ X ] [ H 6 ] [ B ] () Where H 6 is the Haar matrix and B is the Haar coefficient vector, which can be calculated as [ B ] [ H 6 ]. [ X ] (2) The Haar matrix H 6 (3) 6.5.5 3 3.5.5.5.5.5.5 (4. ).5.5 (4.4) 4.5.5 Here blank entries signify zeros. Like the ast ourier Transform (T), the ast wavelet Transform (WT) is a fast, linear operation that operates on a data vector whose length is an integer power of two, transforming it into a numerically different vector of the same length. The WT consists of applying above matrix [ H ] hierarchically, first to full data vector [ X ] of length (i.e. multiplying [H ] to data vector [ X ]), next all the high frequency information is shifted to bottom of the vector by using a permutation matrix. This process is applied again to the vector of low frequency information of length /2 (i.e. multiplying [H 2 ] to data vector of low frequency information), then to the vector of length /4 and so on, until only two s remains. The procedure is sometimes called a pyramidal algorithm [5], for obvious reason. The equation below makes the procedure clear: Hence by using above matrix and equation (2), the Haar coefficients vector [ B ] can be calculated and by relating these coefficients with ourier coefficients, the sine and cosine ourier coefficients can be determined. This algorithm requires order log 2 operations. The same Haar coefficients can be computed with less number of mathematical operations by factorizing above matrix as follows [].5.5.5.5.5.5.5.5.5.5.5.5 (4.).5.5.5.5.5.5.5.5.5 2.5.5.5.5.5 5..5 5..5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5.5 (4.2) In the above procedure the last vector is the vector of Haar coefficients [B] we got before. This procedure requires only 4(-) multiplications and 2(-) additions. ow if above procedure is followed in reverse direction without Haar coefficient vector [B], we get a matrix [H] -, which is same as the matrix [H 6 ] - mentioned before. ow if all the non-zero coefficients (.5 & -.5) of matrix [H] are replaced by ones ( & - respectively) and above procedure is followed, we will get a vector of coefficients, like [ B ]. But the coefficients of this vector will be different then coefficients of vector [ B ]. Let call this vector as modified Haar coefficient vector [B m ]. ow it will take only 2(-) additions to get these coefficients.
IDIA ISTITUTE O TECHOLOGY, KHARAGPUR 7232, DECEMBER 27-29, 22 5 [ ] (6.) Η a [ ] (6.2) Η a 2 [ ] (6.3 ) Η a 2 [ ] (6.4 ) Η a 4 By following the reverse procedure without involving modified Haar coefficient vector [ B m ], we will get a x matrix [ H a ] -. Hence the input vector X(m) in terms of this matrix can be expressed as [ X ] [ H a ] [ B m ] (7) Using matrix [ Ha ], a relationship between the modified Haar coefficients and ourier coefficients can be established as follows [8] ourier expansion of the periodic signal x(t) in the interval (, T) is defined as X(t) + 2 sin wt + 2 2 cos wt + 2 3 sin 2wt + 2 4 cos 2wt +..+ 2 9 sin 5wt + 2 cos 5wt +.. (8) Where w is the fundamental angular frequency which is given by w 2πf 2π/T, and,, 2, 9,,. are the ourier coefficients. A finite series expansion is obtained by truncating the above series of equation (8) and reducing the number of terms from infinite to, where is of the form 2 n. The ourier coefficients are defined in the discrete form as follows. X ( m) m 2 m 2πm X ( m) sin 2 2πm 2 X ( m) cos m 2 X ( m) sin πm m The above equations can be written in matrix form as follows. [ ] [ S ] [ X ] (9) Where [ ] [,, 2,.. - ] T is a vector whose s are the ourier coefficients, [ X ] is the data vector, and [ S ] is a x matrix forms by values of sine and cosine functions at different sampling points, along with the factor, 2/. Considering to be even and of the form 2 n, the elements of the matrix S can be obtained as follows. S km 2 ( k + ) mπ sin 2 k mπ cos for k foroddk, ie.. k,3, KK( ) forevenk, ie.. k 2,4, KK ( 2) () Where, m,,2 (-) and S km is the element of matrix S in row k and column m. Thus all the elements of [ S ] for 6 can be obtained easily. The ourier coefficients are the s of vector [ ] representing x(t) over the interval (,T). The square of the length of this vector, [ ] T [ ] 2 k, is equal to the k mean square value of x(t) over this interval. Similarly the modified Haar coefficients are the s of another vector [ B m ] that has the same length and represents x(t) in terms of a different basis. By substituting equation (7) into equation (9) [ ] [ S ] [ H a ] [ B m ] () Hence, by calculating the matrix [ S ] from Equ. () and matrix [ H a ], the ourier coefficient vector [ ] can be determined directly from the modified Haar coefficient vector [ B m ].
52 ATIOAL POWER SYSTEMS COERECE, PSC 22 III. ALGORITHM OR DIERETIAL PROTECTIO O POWER TRASORER The algorithm for extracting the fundamental frequency and the harmonic s from differential current signal is based on the ast Haar Wavelet Transform (HWT) technique. Based on the relative magnitude of these fundamental and harmonic s the operating condition of the relay is decided. This algorithm for the umerical Differential protection of power transformer can be applied in the following way: The differential transformer current, whether due to fault or inrush, is represented as [6] I ατ βτ m ( τ) IDCe + IkSin( kwτ + Φk ) e k (2) Where the time constant /α and /β are associated with decaying DC and harmonic s, respectively. The current signal is sampled at regular intervals. In order to prevent aliasing effect due to sampling, the sampling rate (samples per cycle) should be greater than (2n+). Where n is the order of harmonic to be detected. Therefore, to satisfy the computational requirement the sampling rate is chosen as 6 samples per cycle, i.e. a sampling frequency of 8Hz for the 5Hz power frequency. The selection of data window is an important factor in the computation of the threshold quantities. The full cycle data window is used for an accurate estimation of the fundamental and harmonic s and the rejection of the DC s. The modified Haar coefficients of sampled current vector are obtained by using equation (5) and equation (6). These coefficients are obtained mere addition and subtraction of the current samples. The sine and cosine ourier coefficients corresponding to modified Haar coefficients are obtained by the relationship between them, which is represented by the equation (). rom these ourier coefficients the fundamental and different harmonic s of input differential current are obtained. IV. RESULTS The algorithm based on ast Haar Wavelet Transform for the numerical differential protection of power transformer, to extract fundamental and harmonic s from the differential current signal, has been programmed in the MATLAB 5.3 and tested successfully. The flow chart to find the modified Haar coefficients and relating them with ourier coefficients, to find different ourier coefficients is given in the ig.. Software on MATLAB 5.3, based on this flow chart is given in Appendix I. This software is tested for the following type of post fault test signal: i(t) I m Sin (wt - φ ) + I m2 Sin (2wt - φ 2 ) + I m5 Sin (5wt - φ 5 ) + I d e (- t / τ) (3) Where the terms used are as follows: I m maximum value of fundamental current I m2 maximum value of second harmonic current I m5 maximum value of fifth harmonic current φ angle lag from reference for fundamental φ 2 angle lag from reference for second harmonic φ 5 angle lag from reference for fifth harmonic I d maximum value of dc decaying τ time constant associated with decaying dc The post fault current signal based on which results are calculated is i(t) 2 Sin (wt - 8 )+.5 Sin (2wt - 27 )+.3 Sin (5wt - 8 ) +.7 e (- t /.) (4) The value of fundamental current obtained using the algorithm based on ast Haar Wavelet transform. I.9938 The value of second harmonic I 2.556 The value of fifth harmonic I 5.32 Calculated errors and frequency response The error in the fundamental frequency, second harmonic and fifth harmonic can be calculated from the formula given below. % Error ((Actual value Calculated value) / Actual value) * (5) or the fundamental frequency : % error ((2.9938) / 2)*.3 or second harmonic : % error ((.5.556) /.5)* -.2 or second harmonic : % error ((.3.32) /.3)* -.4 The frequency response is a measure of the filtering characteristics of algorithm. The frequency response for the umerical filter using proposed algorithm, for the test signal is shown in ig. 2. requency response is curve between normalized frequency and gain, where normalized frequency is assumed frequency divided by power frequency and gain is the calculated value of current at assumed frequency divided by actual current. The response to all undesired frequencies is
IDIA ISTITUTE O TECHOLOGY, KHARAGPUR 7232, DECEMBER 27-29, 22 53 ig.. Program flow chart for computation of I m, I m2 and I m5 nonzero, but small. The efficacy of the filter is clear from the response shown in ig. 2., that for particular frequency, the corresponding frequency is extracted and other are almost rejected. V. COCLUSIO The algorithm based on fast Haar wavelet transform, generates its coefficients using additions and subtractions of the data samples more accurately than earlier schemes. The equations for extraction of fundamental frequency and harmonic s using ast Haar Wavelet transform are derived. Software has been developed to calculate modified Haar coefficients and ourier coefficients and hence magnitude of fundamental and second and fifth harmonic of differential current. The calculated errors for different frequency of differential current are very small and do not affect the accuracy of algorithm. The frequency responses of the fundamental frequency filter, second harmonic filter and fifth filer shows good filtering characteristics of the algorithm. During magnetizing inrush and over excitation, the algorithm is flexible enough to generate the necessary restrain signal. ig. 2. requency Response of HWT Based ilter VI. REERECES [] Z. Moravej, D.. Vishwakarma, and S. P. Singh, Digital filtering algorithms for the differential relaying of power transformer: An Overview, Electrical Machines and Power Systems, Vol. 28, pp. 485 5, 2. [2] M. Gomez-Morgante and D. W. icollete, A wavelet based differential transformer protection, IEEE Transactions on Power Delivery, vol. 4, o. 4, pp. 35-358, 999. [3] B. Kasztenny, and M. Kezunovic, Digital relays improve protection of large transformers, IEEE Computer Application in Power, pp. 39-45, Oct. 998. [4] A. D. Guzman, S. Zocholl, G. Benmouyal, and H. J. Altuve, A current based solution for transformer differential protection Part-I; Problem statement, IEEE Transactions on Power Delivery, vol. 6, o. 4, pp. 485-49, Oct. 2. [5] O. Rioul and M. Vetterli, Wavelets and Signal Processing, IEEE Signal Processing Magazine, pp. 4-38, Oct. 99. [6] H. K. Verma, and A. M. Basha, Transformer differential relay based on waveform approach, The institute of Engineers (India) Journal (EL), Vol. 67, pp. 32-37, Dec. 986. [7] M. A. Rahman, B. so, M. R. Zaman, and M. A. Hoque, Testing of algorithm for a stand-alone digital relaying for power transformers, IEEE Transactions on Power Delivery, vol. 3, o. 2, pp. 374-385, April 997. [8] D.. Vishwakarma and B. Ram, Power System Protection and Switch Gear, Tata McGraw-Hill Publishing Company Limited, ew Delhi, 995. [9] L. P. Singh, Digital Protection Protective Relaying rom Electro - Mechanical to Microprocessor, Willey Eastern Limited, 994. [] A. G. Phadke and J. S. Thorpe, Computer Relaying for Power Systems, Research Studies Press Ltd., England 988. [] http://www.math.hmc.edu/math85/pdfs