INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 721302, DECEMBER 27-29, 2002 509 Numerical Differential Protection of Power Transformer using Algorithm based on Fast Haar Wavelet Transform K. K. Gupta and D. N. Vishwakarma Abstract-- A fast and simple numerical filtering algorithm is The differential protection scheme shows certain limitations. an important requirement for the efficient power system Detection of a differential current does not provide a clear relaying. This paper presents a wavelet-based algorithm for distinction between internal faults and other conditions, e.g. numerical differential protection of power transformer, using Magnetizing inrush, over-excitation of the transformer core, harmonic restraint approach. This algorithm provides an external faults. The conventional approach to mitigate these accurate and computationally efficient tool for distinguishing internal faults from magnetizing inrush and over excitation problems is to apply percentage (biased) differential inrush. This algorithm is essentially a numerical filter, which characteristic along with second and fifth harmonic restraints extract the fundamental frequency component and second and for inrush and over-excitation conditions, respectively. fifth harmonic components of differential current, to provide operating and restraining signal respectively. This algorithm The conventional percentage differential relays used for generates it’s coefficient by additions and subtractions routines the protection of power transformer against internal faults are only and does not involve time consuming multiplication and either of electromagnetic or static type. Electromagnetic and division calculation. The computation complexity of the static relays have several drawbacks. The concept of algorithm presented here is O(N) additions and subtractions as numerical protection, which evolved during the last two compared to DFT based algorithm, whose computation complexity is O(Nlog2N) operations. decades, shows much promise in providing improved performance. The main features of the Numerical relays are Index Terms—Differential Protection, Haar Transform, their economy, reliability, compactness, flexibility and the Wavelet Transform, Fourier Transform. possibility of integrating a Numerical relay into the hierarchical computer system within the substation. I. INTRODUCTION II. FAST HAAR WAVELET TRANFORM (FHWT) A power transformer belongs to a class of very TECHNIQUE expensive and vital component of electric power system and its protection is one of the most challenging problems in the The proposed algorithm is derived from the well-known area of power system relying. The frequency of occurrence Haar Transform based algorithm. In Haar Transform the Haar faults in power transformer is less than on lines. But if a coefficients are calculated first and from Haar coefficients the power transformer experiences a fault, it is necessary to take sine and cosine Fourier coefficients are calculated, using the transformer out of service as soon as possible so that established relationship between Haar and Fourier damage is minimized. The cost associated with repairing a coefficients. The Haar transform can also be termed as damaged transformer may be very high. An unplanned outage Discrete Wavelet Transform (DWT), using Haar wavelet. of a power transformer can cost electric utilities crores of The Discrete Wavelet (Haar) Transform requires Nlog2N rupees. Consequently, it is of great importance to minimize operations to transform a N sample vector. the frequency and duration of unwanted outages. Accordingly, high demands are imposed on power The DWT matrix is not sparse in general, so it has the transformer protective relays. Requirements include same complexity issues as the discrete Fourier transform. So it is solved in same way as for the FFT, by factoring the DWT into a product of a few sparse matrices using self- · Dependability (no missing operation) similarity properties. This result in an algorithm that requires · Security (no false tripping) only order N operations to transform an N-sample vector. · Speed of operation (short fault clearing This fast version of the DWT can be called as Fast Wavelet time) Transform (FWT). The FWT using Haar Wavelet calculates Haar coefficients by 4(N-1) multiplications. Hence its Differential protection scheme based on circulating current computation complexity is O(N) operations. principle is widely used to protect the power transformer against internal faults [4], [9]. In the algorithm presented here these 4(N-1) multiplications are converted in 2(N-1) additions. Hence the The authors are with the Institute of Technology, Banaras Hindu computation complexity is reduced to O(N) additions. But the University, Varanasi - 221005, India (telephone: 91-5412-57733, e-mail: coefficients calculated by this modified algorithm will not be [email protected], [email protected]). 510 NATIONAL POWER SYSTEMS CONFERENCE, NPSC 2002 same as Haar coefficients. We can call them as Modified é0.5 0.5 ù Haar coefficients. A relationship between the Modified Haar ê0.5 -0.5 ú [H ] =ê ú (4.3) coefficient and Fourier coefficients can be established and 3 ê 0.5 0.5 ú from that the sine and cosine Fourier coefficients of the input ê ú ë 0.5 -0.5û signal can be calculated. é0.5 0.5 ù [H ] =ê ú (4.4) A signal vector [ X ] , whose length is an integer power of 4 0.5 -0.5 two, can expressed as ë û Here blank entries signify zeros. [ X ] = [ H ] [ B ] (1) 16 Like the Fast Fourier Transform (FFT), the Fast wavelet Where H is the Haar matrix and B is the Haar coefficient 16 Transform (FWT) is a fast, linear operation that operates on a vector, which can be calculated as data vector whose length is an integer power of two, [ B ]= [ H ]-1. [ X ] (2) 16 transforming it into a numerically different vector of the same length. The Haar matrix H16 The FWT consists of applying above matrix [ H ] é1 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0 ù ê ú hierarchically, first to full data vector [ X ] of length N (i.e. ê1 1 1 0 1 0 0 0 -1 0 0 0 0 0 0 0 ú ê1 1 1 0 -1 0 0 0 0 1 0 0 0 0 0 0 ú multiplying [H1] to data vector [ X ]), next all the high ê ú ê1 1 1 0 -1 0 0 0 0 -1 0 0 0 0 0 0 ú frequency information is shifted to bottom of the vector by ê1 1 -1 0 0 1 0 0 0 0 1 0 0 0 0 0 ú ê ú using a permutation matrix. This process is applied again to ê1 1 -1 0 0 1 0 0 0 0 -1 0 0 0 0 0 ú the vector of low frequency information of length N/2 (i.e. ê1 1 -1 0 0 -1 0 0 0 0 0 1 0 0 0 0 ú ê ú multiplying [H 2] to data vector of low frequency H = ê1 1 -1 0 0 -1 0 0 0 0 0 -1 0 0 0 0 ú (3) [ 16] ê ú information), then to the vector of length N/4 and so on, until ê1 -1 0 1 0 0 1 0 0 0 0 0 1 0 0 0 ú ê1 -1 0 1 0 0 1 0 0 0 0 0 -1 0 0 0 ú only two components remains. The procedure is sometimes ê ú ê1 -1 0 1 0 0 -1 0 0 0 0 0 0 1 0 0 ú called a pyramidal algorithm [5], for obvious reason. The ê1 -1 0 1 0 0 -1 0 0 0 0 0 0 -1 0 0 ú ê ú equation below makes the procedure clear: ê1 -1 0 -1 0 0 0 1 0 0 0 0 0 0 1 0 ú ê1 -1 0 -1 0 0 0 1 0 0 0 0 0 0 -1 0 ú ê ú ê1 -1 0 -1 0 0 0 -1 0 0 0 0 0 0 0 1 ú ê ú ë1 -1 0 -1 0 0 0 -1 0 0 0 0 0 0 0 -1û Hence by using above matrix and equation (2), the Haar coefficients vector [ B ] can be calculated and by relating these coefficients with Fourier coefficients, the sine and cosine Fourier coefficients can be determined. This algorithm requires order Nlog2N operations. The same Haar coefficients can be computed with less number of mathematical operations by factorizing above matrix as follows [11] é0.5 0.5 ù ê0.5 -0.5 ú ê ú ê 0.5 0.5 ú ê ú ê 0.5 -0.5 ú ê 0.5 0.5 ú ê ú 0.5 -0.5 ê ú ê 0.5 0.5 ú ê ú In the above procedure the last vector is the vector of Haar [H ]=ê 0.5 -0.5 ú (4.1) 1 ê 0.5 0.5 ú coefficients [B] we got before. This procedure requires only ê ú ê 0.5 -0.5 ú 4(N-1) multiplications and 2(N-1) additions. Now if above ê ú ê 0.5 0.5 ú procedure is followed in reverse direction without Haar ê 0.5 -0.5 ú -1 ê ú coefficient vector [B], we get a matrix [H] , which is same as ê 0.5 0.5 ú -1 ê 0.5 -0.5 ú the matrix [H16] mentioned before. ê ú ê 0.5 0.5 ú ê ú ë 0.5 -0.5û Now if all the non-zero coefficients (0.5 & -0.5) of matrix [H] are replaced by ones (1 & -1 respectively) and above é0.5 0.5 ù ê ú procedure is followed, we will get a vector of coefficients, ê0.5 -0.5 ú like [ B ]. But the coefficients of this vector will be different ê 0.5 0.5 ú ê ú then coefficients of vector [ B ].