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2013 International Conference on Power Energy and Control ICPEC 978 1 4673 6030 2 13 31 00 2013 IEEE 520 Harmonic Analysis of Inrush Current using Fast Fourier Transform T Sridevi MIEEE PhD Scholar Jawaharlal Nehru Technological University Hyderabad Andhra Pradesh INDIA sridi t K Ramesh Reddy MIEEE Dean HOD EEE G Narayanamma Institute of Technology and Sciences Hyderabad Andhra Pradesh INDIA kollirameshreddy N Leela Jaya Syamala M Tech Student EPS Annamacharya Institute of Technology and Sciences Hyderabad Andhra Pradesh INDIA nljsnemali Abstract This paper presents analysis of magnetizing inrush current for different voltage angles and harmonic analysis of that inrush current using different FFT windowing techniques Time domain simulations were used to solve the inrush phenomenon Square Welch Hanning and Parzen windows were used to truncate the obtained inrush current harmonic spectrum for investigation and with parzen windowing the harmonic magnitude is getting reduced due to its reduced side lobes Inrush current harmonic spectrum obtained by using FFT algorithm in turn discrete fourier transform Out of different solvers to solve state equation ode23t solver is used For an unloaded single phase transformer inrush current in time domain and harmonic content of inrush current were presented The results are helpful to estimate harmonics effects in the transformer inrush currents The results establish a guide line for harmonic elimination due to inrush current and offer an important reference for system design Index Terms Inrush current harmonic analysis windowing techniques I INTRODUCTION Transformer is the most sensitive component in response to power system harmonics Inrush current is an important issue for transformer Due to temporary over fluxing in the transformer core at the energization of the transformer it results magnetizing inrush current The steady state magnetizing current of a transformer may be only 1 or 2 of the rated current but it may reach 10 20 times rated current When transformer is switched on to a source inrush current exists several seconds before the steady state condition is reached and cause unnecessary tripping of protective relays Inrush current magnitude mainly depends on switching parameters like resistance of primary winding voltage angles etc In this paper inrush current at different voltage angles were obtained initially Magnitude of the inrush current depends upon the magnitude and polarity of the remanent flux density with respect to those of the steady state flux density at the instant of energization Inrush phenomenon is of nonlinear nature and can only be reproduced by actual tests computer simulations Due to the non linear nature of inrush phenomenon it must be solved by iteration 1 In most cases the magnetizing current of single phase iron cores will be symmetric and the harmonic current spectrum just consists of the fundamental frequency component and odd harmonics no even harmonic and DC terms Mostly harmonic terms in the magnetizing current above 15th harmonic are negligibly small and harmless But practically if DC component results from any power electronic circuit fed directly from transformer find its way into the magnetic core it will cause the transformer to saturate asymmetrically Due to this the harmonic spectrum of the magnetizing current increases in magnitude and contains even harmonics and DC term in addition to fundamental frequency component and odd harmonics According to strategies of development diffusion of energy efficient distribution transformers SEEDT the losses caused by harmonics and reactive power in European Union Distribution Transformers are estimated at about 5000 GWh year 2 Therefore harmonic analysis plays an important role in transformers to reduce harmonics affect In the present work we developed a MATLAB code to get the time domain waveforms of the transformer excitation current and the magnitude and phase angles of the harmonic component of the inrush currents at different voltage angles with four different FFT windowing techniques II INRUSH CURRENT FOR DIFFERENT VOLTAGE ANGLES The circuit model for calculating magnetizing inrush current of an unloaded single phase transformer is shown in the Fig 1 Fig 1 Single phase transformer equivalent circuit 2013 International Conference on Power Energy and Control ICPEC 521 The nonlinear characteristic equation in this case can be represented by a polynomial equation as 3 0 7576 1 03 10 KCL loop equations of primary side of the transformer are The relationship between the applied voltage and the flux in the transformer s magnetic core is Therefore voltage drop across the iron core resistance is written as From the above equations 1 and 2 state equation obtained is 0 0 1 0 Where r1 0 192 r0 612 86 l1 0 9mH are the transformer parameters 3 The energization of a winding at a point different from the voltage peak will also result in a flux which is equivalent to the flux due to remanence Using this basic premise the inrush current is calculated The attenuation of inrush current is not significantly affected by eddy current and hysteresis losses in the iron core So no discretization of the laminations has been considered in this work 3 In the present work 110 t V 50 is selected and 200 cycles of simulation is done Out of different solvers available like ode23s ode15s ode113 etc ode23t solver is used to solve the initial value problem for ordinary differential equation Figure 2 specifies the flowchart for obtaining magnetizing current of transformer at different voltage angles Fig 2 Flow chart for evaluating inrush current at specified voltage angles 1 2 No Yes Start Evaluate State Matrix Input Matrix Initialize flux specify Iteration time span Enter some voltage angle End P Voltage Angle Plot Inrush current versus Time with Red colour Solve State Equation for finding Inrush current Flux Plot Inrush current versus Time with Green colour Plot Inrush current versus Time with Blue colour Increment P 0 P n Yes Yes No No 2013 International Conference on Power Energy and Control ICPEC 522 Fig 3 Inrush Current waveform for 43 Fig 4 Inrush Current waveform for 90 Fig 5 Inrush Current waveform for 150 Figure 3 to Fig 5 shows the inrush current phenomenon at different voltage angles It was observed that the time taken to reach the steady state value of current varies for each voltage angle From zero to 180 degrees the slope of the current spectrum is decreasing and for higher angles the slope is increasing due to negative remanent flux density III HARMONIC CONTENT OF INRUSH CURRENT As the current spectrum of inrush current takes much larger time to reach steady state it has been exploited to conduct the simulation using harmonic domain techniques arguing that the inrush current may be considered like a non sinusoidal current in a quasi steady state condition if one only looks at a reduced number of cycles 4 For the present work FFT algorithm was chosen since direct evaluation of DFT using requires N2 complex multiplication and N N 1 complex addition s But for an N point DFT the number of complex multiplications required using FFT is log FFT is highly efficient procedure for computing the DFT of a finite series and requires less number of computation than that of direct evaluation of DFT FFT reduces the computation time Abrupt truncation of the series for analysis results in oscillations which are due to slow convergence of the series To reduce their oscillations fourier coefficients of the filter are modified by multiplying the infinite impulse response with a finite weighing sequence W n known as window Where 0 1 2 0 The sequences for Square Hanning Welch and Parzen Ws Wh Ww and Wp windows which are used in harmonic analysis for the present work are 1 1 2 1 2 0 1 2 1 cos 2 1 1 1 2 1 1 2 1 1 1 2 1 1 2 1 with number of points per cycles T as 50 and N varies from 1 to number of points per cycles T and total number of points are taken as 10000 for carrying out the work Figure 6 specifies the flow chart for finding harmonic content in inrush current using different FFT windowing techniques For different voltage angles how it is varying is shown from Fig 7 to Fig 9 The Curves represents fundamental second third DC fourth terms etc from top to down in the Fig s 2013 International Conference on Power Energy and Control ICPEC 523 Fig 6 Flow chart for determining harmonic content of inrush current Fig 7 Harmonic content of Inrush Current waveform for 30 Fig 8 Harmonic content of Inrush Current waveform for 43 Fig 9 Harmonic content of Inrush Current waveform for 90 End Start Read r1 r0 l f Evaluate State Matrix Input Matrix Initialize flux specify Iteration time span Enter some voltage angle n FFT of the Inrush current and shifting them about origin and store them in result Solve State Equation for finding Inrush current Flux Switch to FFT windowing technique for any specific window function K 1 K NP T 1 Store result in complex form of harmonics harmonic magnitude angle K K NP T 1 Yes No 2013 International Conference on Power Energy and Control ICPEC 524 IV RESULTS CONCLUSION TABLE I COMPARISON OF DIFFERENT WINDOWING TECHNIQUES Parzen Hanning Welch Square 30 Im 0 1723 0 3662 0 182 0 389 0 220 0 476 0 263 0 554 Ia 0 112 00 0 114 57 0 108 58 0 101 65 43 Im 0 9878 1 8747 1 077 2 054 1 183 2 246 1 271 2 379 Ia 0 132 56 0 132 89 0 130 49 0 128 15 60 Im 3 8164 7 1385 4 110 7 698 4 306 8 022 4 445 8 233 Ia 0 149 16 0 148 77 0 147 57 0 146 60 90 Im 7 4111 13 7083 7 927 14 63 8 166 14 99 8 320 15 22 Ia 0 174 97 0 175 24 0 175 46 0 175 54 Table I represents two sample values of harmonic magnitude Im and angle Ia for the four windowing techniques Elaboration of the inrush current waveform for 43 is exposed in Fig 10 Figure 11 displays bar representation of harmonic content The decay of the magnetizing inrush current has been observed It was found that switching at 90 of the voltage wave reduces the magnitude of the magnetizing inrush current Switching at lower switching angles produces a first cycle of relatively large magnitude whereas a low peak current appeared when the switching took place at 180 The reason for this can be explained in terms of the lagging nature of the flux density waveform and the value of remenant flux density Due to the asymmetrical nature of the magnetizing inrush current even harmonics are more dominant ones in the harmonic spectrum in particular the second harmonic Parzen window sequence is giving better evaluation of harmonic magnitudes as the central lobe of the frequency response of the window contains most of the energy and is narrow compared to other windows The side lobes of the frequency response is also in decreasing manner in the case of parzen window Due to which the harmonic magnitude is reduced The decay of the inrush currents and its DC component with t