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A Theoretical Study of the Application of Hartley Transform for Power Quality Analyses

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A Theoretical Study of the Application of Hartley Transform for Power Quality Analyses A THEORETICAL STUDY OF THE APPLICATION OF HARTLEY TRANSFORM FOR POWER QUALITY ANALYSES By PARANJYOTlMAHANTA Bachelor of Science Gauhati University Gauhati, India 1978 Bachelor of Engineering lorhat Engineering College J orhat, India 1985 Submitted to the Faculty of the Graduate college of the Oklahoma State University in partial fulfillment of the requirements for the Degree of MASTER OF SCIENCE May, 1993 A THEORETICAL STUDY OF THE APPLICATION OF HARTLEY TRANSFORM FOR POWER QUALITY ANALYSES Thesis Approved : Dean of the Graduate College ii PREFACE The subject related of this thesis is Hartley transform and its usage in the study of power quality. The main purpose of this is to conduct a theoretical study of the application of Hartley Transform In power system analyses and explore its potential in solving power quality problems. An effort has been made to keep the level of this thesis as simple as possible. There is a wide scope for further studies. Hartley Transform, although mainly used in Communication Engineering, is seldom used to solve power related problems. Literature on this is just becoming available. I wish to express my sincere gratitude to Dr R. Yarlagadda of the School of Electrical and Computer Engineering of Oklahoma State University for his advice in literature search for Hartley Transform. I am also specially grateful to Dr. J . A. Allison for his help in allowing me to use the VMS mainframe computer system for computer programming. Special Thanks are due to my uncle Dr. N. C. Das of Wilksbarre, Pennsylvania and Mr D. P. Das of Philadelphia for their constant support and understanding. I offer my deepest appreciation to my parents for their moral encouragement. III ACKNOWlEDGEMENT I wish to express my sincere appreciation to Dr R. G. Ramakumar for his encouragement and advice throughout my graduate program. Many thanks also go to Dr B. L. Basore, Dr R. Rhoten, Dr R. L. Lowery for serving on my graduate Committee. Their suggestions and support were very helpful throughout the study. f IV TABLE OF CONTENfS Chapter Page I. INTRODUCTION Background................................................................................... 1 Scope ................................................................................................ 2 Purpose .......................................................................................... 4 II. POWER QUALITY Introduction........................................................................... ....... 6 Definitions of power quality ............................................... 1 1 Types of Power quality disturbances . ................. ......... 1 2 Planning a power Quality Survey....... ............................. 34 Perfonning the survey ......................................... .............. 4 3 Analyzing survey Data ......................................................... 5 1 Measure of Power quality .................................................... 59 Discussion of various Methods to improve Power Quality ... .................................. .................................. 6 1 III. HARTLEY TRANSFORM Introduction ................................ "............................................. 6 6 Definitions ................................................................................... 6 7 Continuous Hartley Transform ........................................... 69 Discrete Hartley Transfonn ................................................. 7 3 Two Dimensional Hartley Transform ............................... 80 General Discussion ................................................................... 8 3 v Chapter page IV. APPLICATION OF HARTLEY TRANSFORM FOR POWER QUALITY ANALYSES Introduction....... ................... ... .............. .......... .... ........................ 8 4 Analyses .......................................................................................... 8 8 Discussion................................................ ....................... ..... ............ 9 5 V. COMPUTER SOLUTION Introduction ............................................ '" ..................................... 9 6 Procedure ............................... '" ....................................................... 9 8 VI. SUMMARY AND CONCLUSIONS .................................................. 1 08 BmLIOORAPHY ..................................................................................................... 111 APPENDIX-(COIvlPlITER PROGRAM OF APPLICATION OF HARlLEY 1RANSFORM FOR POWER QUALITY ANALySES).............. 1 14 vi LIST OF TABLES Table Page I. Typical range of input power quality and load parameters of major computer manufacturers ............ ... ....... ........................... 1 3 II. Types of disturbances and other information .............................. 1 5 III. Typical sources of spikes, impulses & surges .............................. 2 0 IV. Common power monitoring features ............................................... .40 V. Electrical distribution system and grounding measurements and consideration ...................................................................................46 VI. Equipment specifications and tolerance ......................................... 5 4 VII. Values of discrete Hartley transform.. ....... ....... .. .............. .............. 78 VITI. Algorithm summary .............................................................................. l 07 -1 T vii o LIST OF FIGURES Figure Page 1. Voltage Sag and Swell.............................................. ................................ 18 2. Spikes, impulses and surges .................................................................. 2 2 3. Simplified Circuit illustrating magnification phenomena ................................................................................................. 2 4 4. Waveforms on primary and secondary illustrating magnification phenomena............................................................................... ............... 2 5 5 . Outage ............................................................................................................... 2 7 6 . Harmonic distortion .................................................................................... 2 8 7. Current waveform typical of most single phase electronic loads........... ........ .......... ........... ...... ......... .......... ..... ................. 3 0 8. Typical transformer magnetizing Current. ...................................... 3 1 9. Arc furnace voltage and current.. ......................................................... 3 2 10. Electrical noise .................................................................... .......................... 3 3 11. Undervoltage & overvolge ....................................................................... 3 5 12. Seven day event summary ..................................................................... 5 5 13. Waveform of SCR-controlled load ........................................................ 5 8 14. Power Quality Indices for non-periodic phenomena........... ..................................................................................... 60 viii Figure Page 15. Plot of continuous Hartley transform function H(f) ................................................................................................................ 72 16. A 16-point representation of the truncated exponential waveform (left) used for illustrating the continuous transform and it's DHT right. ............................................................ 79 17. Distribution network decomposed into SE and SH Networks ...................................................................................... 89 18. Diagram of n - port circuits .................................................................... 9 2 19. Decomposition of a power network into SH and SE network ...................................................................................................... 94 20. Non sinusoidal rectifier injected current waveform .................... 97 21. Three bus power system ......................................................................... 9 9 22. Equivalent starnetwork ......................................................................... 1 00 23. Harmonic voltageplot.. .............. ;.................................. ........................... 1 27 ... 1 ) ix LIST OF SYMBOLS Fourier coefficient Fourier coefficient. Hartley coefficient. Cas Sum of sine and cosine function (Argument). CSH Capacitance of the SH network. DIN Distortion index used mostly 10 Europe. e Exponential base (e=2.7183) E(f) Even part of a function. E(v) Even part of Hartley transformed function. f(t) An arbitrary function. F(v)f Discrete Fourier transform of function F(t). Imaginary part of a complex function F(f). Real part of a complex function F(f) fsymm(x, y) Symmetrical part of f(x, y). fantisymm(X, y) Antisymmetrical part of f(x, y). GSH Conductance of (SH network.) H(s) or H(f) Hartley transformed function. H(u, v) Two Dimensional Hartley transform of a function f(x, y). Hartley current. Positive sequence Hartley current. Negative sequence Hartley current. x I(v) Discrete Hartley transformed function ofcurrent function i(t). I(kO) Discrete Fourier transformed function I(kt). ISE(t) Current of SE network. i and j Complex operator which IS equal to ~-::I. k or K Any arbitrary integer. KVT Expression for Audio weight. LSE Inductance of the SE network. n or N Any Integer. Nl and N2 Two arbitrary Matrices. O(t)
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