Comparison of Rationalised Haar Transform and Block Pulse Function based algorithms for Transformer Protection

This paper describes application of Rationalized Haar Transform and Block Pulse Function for digital protection of power transformers. Digital relay algorithms are developed to extract fundamental, second harmonic and fifth harmonic components. These components are then used for harmonic restraint differential protection of power transformers. The Block Pulse Function based method is computationally simple and flexible to use with any sampling frequency with respect to Rationalized Haar Transform. In Rationalized Haar Transform method one extra step of computation of Haar co-efficient is involved. Different graphs of Rationalized Haar Transform and Block Pulse Function based methods for Inrush, Over-excitation and Internal fault conditions have been plotted and compared. Off-line testing of the method with simulated inrush, over-excitation and internal fault current data clearly indicate that the Block Pulse Function method can provide fast and reliable trip decision. Key WordsRationalized Haar Transform, Block Pulse Functions, Power Transformer Protection And Digital Differential Relay. I.INTRODUCTION The differential relaying principle is commonly used for the protection of power transformers [1]. This is based on comparison of the fundamental, second and fifth harmonic currents. A differential protection scheme with harmonic restraint is the usual way of protecting a power transformeragainst internal faults and restraining the tripping operation during non fault conditions, such as magnetizing inrush currents and over-excitation currents [2-9]. Several algorithms have been proposed for digital protection of power transformers. Among these algorithms, Discrete Fourier Transform based algorithm has been used for a very long time and it is still being used but there have been developments which provided better algorithms such as HAAR function and Block Pulse Function based algorithms. Schemes using Rationalized Haar Transform and BPF have been compared for differential protection of power transformers. II.RATIONALISED HAAR TRANSFORM The Rationalized Haar Transform (RHT) is the rationalized version of Haar transform (HT).The RHT coefficients Crhk, k =0, 1,2,...N-1 are obtained by using the rationalized Haar transform on the incoming data samples, i.e. voltage and current samples acquired over a full cycle data window or a half-cycle data window at a sampling rate of 16 samples per cycle. These coefficients are obtained by mere addition and subtraction of data sampleThe RHT coefficients are calculated by the given formula:Crh0=(x0+x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11+x12 +x13+x14+x15) Crh1 =(x0+x1+x2+x3+x4+x5+x6+x7)(x8+x9+x10+x11+x12+x13+x14+x15) Crh2 =(x0+x1+x2+x3)-(x4+x5+x6+x7) Crh3 =(x8+x9+x10+x11)-(x12+x13+x14+x15) Crh4 =(x0+x1)-(x2+x3) Crh5=(x4+x5)-(x6+x7) Crh6=(x8+x9)-(x10+x11) Crh7=(x12+x13)-(x14+x15) Crh8=(x0+x1) Crh9=(x2-x3) Crh10=(x4-x5) Crh11=(x6-x7) Crh12=(x8-x9) Crh13=(x10-x11) Crh14=(x12-x13) Crh15=(x14-x15) Current ) (t i which is given by time function can be expressed in terms of Fourier coefficients as: ) 4 ( sin 2 ) 2 ( cos 2 ) 2 ( sin 2 ) ( 2 1 1 0 t A t B t A A t i        ...... ) 4 ( cos 2 2    t B ) 10 ( cos 2 ) 10 ( sin 2 ... 5 5 t B t A     (1) Saurabh Kumar Gautam, Dr.Ramesh Kumar / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue6, NovemberDecember 2012, pp.1276-1281 1277 | P a g e In terms of RHT coefficients: A1=0.0555Crh1-0.011Crh2+0.011Crh30.0276Crh4+0.0184Crh5+0.0276Crh6-0.0184Crh70.0169Crh80.0096Crh9+0.0033Crh10+0.0143Crh11+0.0169Crh12+ 0.0096Crh13-0.0033Crh14-0.0143Crh15 ............... (2) B1=0.011Crh1+0.0555Crh20.0555Crh3+0.0184Crh4+0.0276Crh5-0.0184Crh60.0276Crh7+0.0033Crh8+0.00143Crh9+0.0169Crh10+0 .0096Crh11-0.0033C12-0.0143Crh13-0.0169Crh140.0096Crh15 ................ (3) Similarly 2 harmonic is calculated as follows:A2=0.0533Crh2+0.0533Crh3-0.022Crh4+0.022Crh50.022Crh6+0.022Crh7-0.0312Crh80.0129Crh9+0.0312Crh10-0.0129Crh110.0312Crh12+0.0129Crh13-0.0312Crh14-0.0129Crh15 ............... (4) B2=0.022Crh2+.022Crh3+0.0533Crh40.0533Crh5+0.533Crh60.0533Crh7+0.0129Crh8+0.0312Crh9-0.0129Crh100.0312Crh11+0.0129C12+0.0312Crh13-0.0129Crh140.0312Crh15 .... (5) Likewise 5 harmonic is also calculated as:-A5=0.0074Crh1-0.011Crh2+0.011Crh30.044Crh4+0.0088Crh5-0.044Crh6-0.0088Crh70.040Crh8-0.014Crh9+0.061Crh100.072Crh11+0.040Crh12+0.014Crh130.061Crh14+0.072Crh15 .....(6) B5=0.011Crh1+0.0074Crh2-0.0074Crh3+0.0088Crh40.044Crh5-0.0088Crh6-0.044Crh7+0.061Crh80.072Crh9+0.040Crh10+0.014Crh11-0.061C120.072Crh13-0.040Crh14-0.014Crh15 ........... (7) III.BLOCK PULSE FUNCTIONS The algorithm based on BPF is computationally simple and flexible to use with any sampling frequency [11].The BPF coefficients are obtained by merely calculatingthe values of current samples.The current samples are acquired over a full cycle data window at the sampling rate of 12samples per cycle. Relationship between Fourier and BPF coefficients has been established. Current ) (t i which is given by time function can be expressed in terms of Fourier coefficients as: ) 4 ( sin 2 ) 2 ( cos 2 ) 2 ( sin 2 ) ( 2 1 1 0 t A t B t A A t i        ...... ) 4 ( cos 2 2    t B ) 10 ( cos 2 ) 10 ( sin 2 ... 5 5 t B t A     ........ (1) In terms of BPF coefficient an: ) ( 0824 . 0 ) ( 0302 . 0 11 8 5 2 12 7 6 1 1 a a a a a a a a A         ) ( 1125 . 0 10 9 4 3 a a a a    