POWER SYSTEM TRANSIENTS Theory and Applications AKIHIRO AMETANI • NAOTO NAGAOKA YOSHIHIRO BABA • TERUO OHNO www.TechnicalBooksPdf.com www.TechnicalBooksPdf.com Power SyStem tranSientS Theor y and Applications www.TechnicalBooksPdf.com www.TechnicalBooksPdf.com Power SyStem tranSientS theor y and Applications Akihiro AmetAni nAoto nAgAokA Yoshihiro BABA teruo ohno Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Group, an informa business www.TechnicalBooksPdf.com MATLAB® is a trademark of The MathWorks, Inc and is used with permission The MathWorks does not warrant the accuracy of the text or exercises in this book This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2013 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S Government works Version Date: 20140428 International Standard Book Number-13: 978-1-4665-7786-2 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under U.S Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers For permission to photocopy or use material electronically from this work, please access www.copyright com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400 CCC is a not-for-profit organization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com www.TechnicalBooksPdf.com Contents Introduction xv List of Symbols xix List of Acronyms xxi International Standards xxiii Theory of Distributed-Parameter Circuits and the Impedance/Admittance Formulas 1.1 Introduction 1.2 Impedance and Admittance Formula .2 1.2.1 Conductor Internal Impedance Zi 1.2.1.1 Derivation of an Approximate Formula 1.2.1.2 Accurate Formula by Schelkunoff 1.2.2 Outer-Media Impedance Z0 1.2.2.1 Outer-Media Impedance 1.2.2.2 Overhead Conductor 1.2.2.3 Pollaczek’s General Formula for Overhead, Underground, and Overhead/Underground Conductor Systems 14 1.2.3 Problems 16 1.3 Basic Theory of Distributed-Parameter Circuit 17 1.3.1 Partial Differential Equations of Voltages and Currents 17 1.3.2 General Solutions of Voltages and Currents 18 1.3.2.1 Sinusoidal Excitation 18 1.3.2.2 Lossless Line 21 1.3.3 Voltages and Currents on a Semi-Infinite Line 23 1.3.3.1 Solutions of Voltages and Currents 23 1.3.3.2 Waveforms of Voltages and Currents 24 1.3.3.3 Phase Velocity 25 1.3.3.4 Traveling Wave 27 1.3.3.5 Wave Length 28 1.3.4 Propagation Constants and Characteristic Impedance 28 1.3.4.1 Propagation Constants 28 1.3.4.2 Characteristic Impedance 31 1.3.5 Voltages and Currents on a Finite Line 32 1.3.5.1 Short-Circuited Line 32 1.3.5.2 Open-Circuited Line 35 1.3.6 Problems 38 v www.TechnicalBooksPdf.com vi Contents 1.4 1.5 1.6 Multiconductor System 38 1.4.1 Steady-State Solutions 38 1.4.2 Modal Theory 41 1.4.2.1 Eigenvalue Theory 41 1.4.2.2 Modal Theory 44 1.4.2.3 Current Mode 45 1.4.2.4 Parameters in Modal Domain 46 1.4.3 Two-Port Circuit Theory and Boundary Conditions 48 1.4.3.1 Four-Terminal Parameter 48 1.4.3.2 Impedance/Admittance Parameters 50 1.4.4 Modal Distribution of Multiphase Voltages and Currents 52 1.4.4.1 Transformation Matrix 52 1.4.4.2 Modal Distribution 53 1.4.5 Problems 55 Frequency-Dependent Effect 56 1.5.1 Frequency Dependence of Impedance 56 1.5.2 Frequency-Dependent Parameters 58 1.5.2.1 Frequency-Dependent Effect 58 1.5.2.2 Propagation Constant 59 1.5.2.3 Characteristic Impedance 61 1.5.2.4 Transformation Matrix 63 1.5.2.5 Line Parameters in the Extreme Case 68 1.5.3 Time Response 70 1.5.3.1 Time-Dependent Responses 70 1.5.3.2 Propagation Constant: Step Response 71 1.5.3.3 Characteristic Impedance 72 1.5.3.4 Transformation Matrix 74 1.5.4 Problems 77 Traveling Wave 77 1.6.1 Reflection and Refraction Coefficients .77 1.6.2 Thevenin’s Theorem 79 1.6.2.1 Equivalent Circuit of a Semi-Infinite Line 79 1.6.2.2 Voltage and Current Sources at the Sending End 79 1.6.2.3 Boundary Condition at the Receiving End 79 1.6.2.4 Thevenin’s Theorem 82 1.6.3 Multiple Reflection 84 1.6.4 Multiconductors 88 1.6.4.1 Reflection and Refraction Coefficients 88 1.6.4.2 Lossless Two Conductors 88 1.6.4.3 Consideration of Modal Propagation Velocities 91 1.6.4.4 Consideration of Losses in a Two-Conductor System 96 1.6.4.5 Three-Conductor System 99 www.TechnicalBooksPdf.com vii Contents 1.6.4.6 Cascaded System Composed of the Different Numbers of Conductors 102 1.6.5 Problems 103 1.7 Nonuniform Conductors 104 1.7.1 Characteristic of Nonuniform Conductors 105 1.7.1.1 Nonuniform Conductor 105 1.7.1.2 Difference from Uniform Conductors 108 1.7.2 Impedance and Admittance Formulas 109 1.7.2.1 Finite-Length Horizontal Conductor 109 1.7.2.2 Vertical Conductor 112 1.7.3 Line Parameters 114 1.7.3.1 Finite Horizontal Conductor 114 1.7.3.2 Vertical Conductor 117 1.7.3.3 Nonparallel Conductor 119 1.7.4 Problems 119 1.8 Introduction of EMTP 122 1.8.1 Introduction 122 1.8.1.1 History of a Transient Analysis 122 1.8.1.2 Background of EMTP 123 1.8.1.3 EMTP Development 124 1.8.2 Basic Theory of EMTP 124 1.8.2.1 Representation of a Circuit Element by a Current Source and a Resistance 126 1.8.2.2 Composition of Nodal Conductance 128 1.8.3 Other Circuit Elements 129 1.8.4 Solutions of the Problems 131 References 136 Transients on Overhead Lines 141 2.1 Introduction 141 2.2 Switching Surge on Overhead Line 142 2.2.1 Basic Mechanism of Switching Surge 142 2.2.2 Basic Parameters Influencing Switching Surge 143 2.2.2.1 Source Circuit 143 2.2.2.2 Switch 146 2.2.2.3 Transformer 147 2.2.2.4 Transmission Line 147 2.2.3 Switching Surges in Practice 148 2.2.3.1 Classification of Switching Surges 148 2.2.3.2 Basic Characteristic of Closing Surge: Field Test Results 149 2.2.3.3 Closing Surge on a Single-Phase Line 151 2.2.3.4 Closing Surges on a Multiphase Line 153 2.2.3.5 Effect of Various Parameters on Closing Surge 162 www.TechnicalBooksPdf.com viii Contents 2.3 2.4 2.5 2.6 Fault Surge 166 2.3.1 Fault Initiation Surge 166 2.3.2 Characteristic of a Fault Initiation Surge 169 2.3.2.1 Effect of Line Transposition 169 2.3.2.2 Overvoltage Distribution 169 2.3.3 Fault-Clearing Surge 172 Lightning Surge 175 2.4.1 Mechanism of Lightning Surge Generation 177 2.4.2 Modeling of Circuit Elements 179 2.4.2.1 Lightning Current 179 2.4.2.2 Tower and Gantry 180 2.4.2.3 Tower Footing Impedance 182 2.4.2.4 Arc Horn 184 2.4.2.5 Transmission Line 185 2.4.2.6 Substation 185 2.4.3 Lightning Surge Overvoltage 185 2.4.3.1 Model Circuit 185 2.4.3.2 Lightning Surge Overvoltage 187 2.4.3.3 Effect of Various Parameters 188 Theoretical Analysis of Transients: Hand Calculations 194 2.5.1 Switching Surge on an Overhead Line 195 2.5.1.1 Traveling Wave Theory 195 2.5.1.2 Lumped-Parameter Equivalent with Laplace Transform 202 2.5.2 Fault Surge 206 2.5.3 Lightning Surge 208 2.5.3.1 Tower Top Voltage 208 2.5.3.2 Two-Phase Model 208 2.5.3.3 No Back Flashover 210 2.5.3.4 Case of a Back Flashover 212 2.5.3.5 Consideration of Substation 212 Frequency-Domain Method of Transient Simulations 215 2.6.1 Introduction 215 2.6.2 Numerical Fourier/Laplace Transform 215 2.6.2.1 Finite Fourier Transform 215 2.6.2.2 Shift of Integral Path: Laplace Transform 217 2.6.2.3 Numerical Laplace Transform: Discrete Laplace Transform 218 2.6.2.4 Odd-Number Sampling: Accuracy Improvement 218 2.6.2.5 Application of FFT: Fast Laplace Transform (FLT) 221 2.6.3 Transient Simulation 228 2.6.3.1 Definition of Variables 228 www.TechnicalBooksPdf.com 468 Computation for Humanity: Information Technology to Advance Society Smith B and B Standler 1992 The effects of surge on electronic appliances IEEE Trans Power Deliv 7(3):1275 Imai, Y., N Fujiwara, H Yokoyama, T Shimomura, K Yamaoka, and S Ishibe 1993 Analysis of lightning overvoltages on low voltage power distribution lines due to direct lightning hits to overhead ground wire IEE Japan Trans PE 113-B: 881–888 Kawahito, M 2001 Investigation of lightning overvoltages within a house by means of an artificial lightning experiment R&D News Kansai Electric Power: 32–33 Nagai, Y and H Sato 2005 Lightning surge propagation and lightning damage risk across electric power and communication system in residential house IEICE Japan Research Meeting Tokyo, Japan, EMC-05-18 Hosokawa, T., S Yokoyama, and T Yokota 2005 Study of damages on home electric appliances due to lightning IEE Japan Trans PE 125-B (2): 221–226 Hosokawa, T., S Yokoyama, and M Fukuda 2009 Trend of damages on home appliances due to lightning and future problems, IEEJ Trans PE 129-B (8): 1033–1038 10 Ametani, A., H Motoyama, K Ohkawara, H Yamakawa, and N Sugaoka 2009 Electromagnetic disturbances of control circuits in power stations and substations experienced in Japan IET GTD (9): 801–815 11 Sonoda, T., Y Takeuchi, S Sekioka, N Nagaoka, and A Ametani 2003 Induced surge characteristics from a counterpoise to an overhead loop circuit IEEJ Trans PE 123 (11):1340–1349 12 Ametani, A 2006 EMTP study on electro-magnetic interference in low-voltage control circuits of power systems EEUG 2006, Dresden, Germany Paper D-3 (EEUG-Proc.), pp 24–27 13 Ametani, A., T Goto, S Yoshizaki, and H Motoyama 2006 Switching surge characteristics in gas-insulated substation UPEC 2006, Newcastle, U.K Paper 12–19 14 Ametani, A., T Goto, N Nagaoka, and H Omura 2007 Induced surge characteristics on a control cable in a gas-insulated substation due because of switching operation IEEJ Trans PE 127(12):1306–1312 15 International Electrotechnical Commission 2001 Standard for Electromagnetic Compatibility, IEC-61000-4 16 Japanese Electrotechnical Commission Test voltage for low-voltage control circuits in power stations and substations JEC-0103-2004 IEE Japan (in Japanese) 17 IEEJ WG 2002 The fact of lightning disturbances in a highly advanced ICT society and the subject to be investigated IEEJ Tech Report 902 18 IEEE 1980 Guide for Surge Voltages in Low-Voltage AC Power Circuits ANSI/ IEEE C62.41 19 Murota, N 1993 Characteristic of the lightning surge suppressors at low voltage responses IEICE Japan Research Meeting, Tokyo, Japan, EMCJ93-72 20 Ideguchi, T and M Hatori 1993 Measures for lightning protection of telecommunication equipment in the premises IEICE Japan Trans 13(1):16 21 Study report on insulation management of low voltage circuits 1993 J Elect InstIn Eng Japan 13 (11):1165 22 Ametani, A., K Matsuoka, H Ohmura and Y Nagai 2009 Surge voltages and currents into a customer due to nearby lightning Electric Power Syst Res 79:428–435 www.TechnicalBooksPdf.com Electromagnetic Disturbances in Power Systems and Customers 469 23 Yokoyama, S and H Taniguchi 1997 The third cause of lightning faults on distribution lines IEE Japan Trans PE 117-B (10):1332–1335 24 Ametani, A., K Hashimoto, N Nagaoka, H Omura, and Y Nagai 2005 Modeling of incoming lightning surges into a house in a low-voltage distribution system EEUG 2005 Warsaw, Poland 25 Nagai, Y and N Fukusono 2004 Lightning surge propagation on an electric power facility connected with feeder lines from a pole transformer KEPCO Research Committee of Insulation Condition Technologies, Osaka, Japan 26 Nagai, Y 2005 Lightning surge propagation into a model house from various places, in KEPCO Research Committee of Insulation Condition Technologies Osaka,Japan 27 Ametani, A., K Kasai, J Sawada, A Mochizuki, and T Yamada 1994 Frequencydependent impedance of vertical conductors and a multiconductor tower model IEE Proc GTD 141 (4):339–345 28 Ametani, A., K Shimizu, Y Kasai, and N Mori 1994 A frequency characteristic of the impedance of a home appliance and its equivalent circuit IEE Japan Annual Conference 1405, Tokyo, Japan 29 Soyama, D., Y Ishibashi, N Nagaoka, and A Ametani 2005 Modeling of a buried conductor for an electromagnetic transient simulation ICEE 2005 Kunming, China SM1-04 30 Nayel, M 2003 A study on transient characteristics of electric grounding systems PhD thesis, Doshisha University, Kyoto, Japan 31 Scott Meyer, W 1982 EMTP Rule Book B P A Portland, Ore 32 Mozumi, T., T Ikeuchi, N Fukuda, A Ametani, and S Sekioka 2002 Experimental formulas of surge impedance for grounding lead conductors in distribution lines IEEJ Trans PE 122-B (2): 223–231 33 Asakawa, S et al 2008 Experimental study of lightning surge aspect for the circuit mounted distribution and telecommunication and customer systems CRIEPI Research Report H07011 34 IEEJ WG (Covenor Ametani, A.) 2008 Numerical transient electromagnetic analysis method IEEJ ISBN 978-4-88686-263-1 35 Carson, J R 1926 Wave propagation in overhead wires with ground return Bell System Tech J 5: 539–554 36 Sunde, E D 1951 Earth Conduction Effect in Transmission System New York: Wiley 37 Taflov, A and J Dabkowski 1979 Prediction method for buried pipeline voltages due to 60 Hz AC inductive coupling, Part I: analysis’ IEEE Trans Power App Syst 98(3):780–787 38 Wedepohl, L.M 1963 Application of matrix methods to the solution of traveling wave phenomenon in polyphase systems Proc IEE 110 (12):2200–2212 39 CIGRE WG.36.02 1995 Guide on the Influence of High Voltage AC Power System on Metallic Pipeline Paris, France: CIGRE Publication 95 40 Sakai, H 1971 Induction Interference and Shielding Tokyo, Japan: Nikkan Kogyo Pub 41 Rickets, L W., S E Bridges, and S Mileta 1976 EMP Radiation and Protective Techniques New York: Wiley 42 Degauque, P and J Homelin 1993 Electromagnetic Compatibility Oxford: Oxford Univ Press 43 Paul, C R 1994 Analysis of Multiconductor Transmission Lines New York: Wiley www.TechnicalBooksPdf.com 470 Computation for Humanity: Information Technology to Advance Society 44 Koike, T 1995 Transmission and Distribution Engineering Tokyo, Japan: Yokohama Publication Company (in Japanese) 45 Ametani, A 1990 Distributed-Parameter Circuit Theory Tokyo, Japan: Corona Pub Co 46 Dommel, H W 1986 EMTP Theory Book B.P.A Portland, Ore 47 Tesche, F M., M V Ianoz, and T Karlsson, 1997 EMC Analysis Methods and Computational Models New York: Wiley 48 IEE Japan WG (Covenor Ametani, A.) 2002 Power system transients and EMTP analysis IEE Japan Technical Report 872 Tokyo (in Japanese) 49 Harrington, R F 1968 Field Computation by Moment Methods New York: Macmillan Company 50 Uno T 1998 Finite Difference Time Domain Method for Electromagnetic Field and Antenna Tokyo, Japan: Corona Pub Co (in Japanese) 51 IEE Japan WG (Covenor Ametani, A.) 2006 Recent trends of power system transient analysis—a numerical electromagnetic analysis J IEE Jpn 126(10):654–673 (in Japanese) 52 Frazier M.J 1984 Power line induced AC potential on natural gas pipelines for complex rights-off-way configurations EPRI Report EL-3106 AGA Cat L51418 53 Dawalibi, F P and R D Southey Analysis of electrical interference from power lines to gas pipelines Part Computation methods, IEEE Trans Power Deliv 1989, 4(3):1840–1846, Part 2, 1990 (1):415–421 54 Southey, R D., F P Dawalibi and W Vukonichi 1994 Recent advances in the mitigation of AC voltages occurring in pipelines located close to electric transmission lines IEEE Trans Power Deliv (2):1090–1097 55 Dawalibi, F P and F Dosono 1993 Integrated analysis software for grounding EMF and EMI IEEE Comput Appl Power (2):19–24 56 Haubrich, H J., B A Flechner and W A Machsynski 1994 A universal model for the computation of the electromagnetic interference on earth return circuits IEEE Trans Power Deliv (3):1593–1599 57 Safe Eng Service & Tec HIFREQ User Manual 2002 Montreal, CA 58 Chirstoforidis, G C., D P Labridis, and P S Dokopoulos 2005 Hybrid method for calculating the inductive interference caused by faulted power lines to nearby pipelines IEEE Trans Power Deliv 20 (2):1465–1473 59 Ametani, A., J Kamba, and Y Hosokawa 2003 A simulation method of voltages and currents on a gas pipeline and its fault location IEE Jpn Trans PE 123 (10):1194–1200 (in Japanese) 60 Isogai, H., A Ametani, and Y Hosakawa 2006 An investigation of induced voltages to an underground gas pipeline from an overhead transmission line IEE Jpn Trans PE 126 (1):43–50 (in Japanese) 61 Boker, H and D Oeding 1996 Induced voltage in pipelines on right-of-way to high voltage lines Elektrizitatswirtschaft 65:157–170 62 Ametani, A 2008 Four-terminal parameter formulation of solving induced voltages and currents on a pipeline system IET Sci Meas Technol 2(2):76–87 63 Ametani, A 1994 EMTP Cable Parameters Rule Book B P A Portland, Ore 64 Wedepohl, L M and D J Wilcox 1973 Transient analysis of underground power transmission systems Proc IEE 120123:253–260 65 Ametani, A., T Yoneda, Y Baba, and N Nagaoka 2009 An investigation of earth-return impedance between overhead and underground conductors and its approximation IEEE Trans EMC 51(3):860–867 www.TechnicalBooksPdf.com Problems and Application Limits of Numerical Simulations Because the electromagnetic transients program (EMTP) is based on a circuit theory assuming transverse electro-magnetic (TEM) mode propagation, it cannot give an accurate solution for a high-frequency transient which involves non-TEM mode propagation Also, the EMTP cannot deal with a circuit of unknown parameters On the contrary, a numerical electromagnetic analysis (NEA) method can deal with a transient associated with both TEM and non-TEM mode propagation Furthermore, it requires no circuit parameter However, it results in numerical instability if the analytical space, the boundary conditions, the cell size, etc., are not appropriate Also, it requires a large amount of computer resources, and existing codes are not general enough to deal with various types of transients, especially in a large network 7.1 Problems of Existing Impedance Formulas Used in Circuit Theory–Based Approaches 7.1.1 Earth-Return Impedance 7.1.1.1 Carson’s Impedance [1] The reason why Carson’s impedance is very popular and has been widely used is simply due to its asymptotic expression During the early days of computing, the calculation of Pollaczek’s infinite integral [2] was very hard because of the limited computation capability Therefore, Carson’s asymptotic formula was the only possible way to evaluate the earth-return impedance [3] However, the asymptotic expression inherently necessitates formulas for a small variable, that is, a low frequency, and for a large variable, and this results in a discontinuity of the calculated impedance as a function of frequency Also, the accuracy in the boundary region is not high enough The same is true with Schelkunoff’s formula for internal impedance of a conductor [4] Nowadays, the advancement in computing capabilities makes it possible to calculate an infinite integral, and various methods of evaluating Pollaczek’s 471 www.TechnicalBooksPdf.com 472 Computation for Humanity: Information Technology to Advance Society impedance have been proposed A typical example is the work by Noda [5] This author, however, doubts Pollaczek’s formula itself 7.1.1.2 Basic Assumption of the Impedance Pollaczek’s and Carson’s formulas were derived under the assumption that length x height h radius r (7.1) It should be noted that most formulas of capacitance and inductance of conductors given in textbooks are based on the aforesaid condition It is easily confirmed that any capacitance formula gives an erroneously large value when the radius reaches the height Correspondingly, the inductance of an infinite conductor becomes larger than that of a real finite conductor [6,7] Furthermore, the formulas neglect displacement currents, that is, ρe ωε e or f (7.2) 2πε eρe where ρe is the earth resistivity εe is the permittivity ω = 2πf For example, the applicable range of a frequency in the case of ρe = 1000 Ω-m and εe = ε0 is given by f 18 MHz or t 50 ns Even in the case of ρe = 100 Ω-m, a transient of 10 ns time region cannot be simulated by Pollaczek’s and Carson’s impedance [8–11] It should be noted that most of the frequency-dependent line models are not applicable to the aforesaid cases, because those models are based on Pollaczek’s and Carson’s impedances Under conditions where Equations 7.1 and 7.2 are not satisfied, only Kikuchi’s and Wedepohl’s impedance formulas are applicable at present [8–10] This requires a far advanced numerical integration than that applied to Pollaczek’s formula 7.1.1.3 Nonparallel Conductor Pollaczek’s and Carson’s impedances are for a horizontal conductor In reality, there are a number of nonhorizontal conductors such as vertical and inclined Although many papers have been published on the impedance of the vertical conductor such as a transmission tower, it is still not clear if the www.TechnicalBooksPdf.com Problems and Application Limits of Numerical Simulations 473 proposed formulas are correct An empirical formula in Ref [12] is almost identical to an analytical formula [13], which agrees quite well with measured results However, the analytical formula requires further investigation if the derivation is correct Impedance formulas for inclined and nonparallel conductors have been proposed in Refs [6,7,14] Since the formulas have been derived by the idea of complex penetration depth [15] using Neumann’s inductance formula, those require a further theoretical analysis 7.1.1.4 Stratified Earth Earth is stratified, as is well known, and its resistivity varies significantly at the top layer depending on the weather and climate The earth-return impedance of an overhead conductor above the stratified earth was derived in Ref [16], and the stratified-earth effect was investigated in Ref [17] The stratified-earth effect might be far more significant than the accurate evaluation of the homogenous earth-return impedance of Pollaczek and Carson, and this requires a further investigation 7.1.1.5 Earth Resistivity and Permittivity Earth resistivity, as mentioned earlier, is quite weather/climate dependent The resistivity after rain is lower than that measured during dry days Also, it may be frequency dependent The frequency dependence of the earth permittivity might be far more significant than that of the earth resistivity Furthermore, water (H2O), which is a dominant factor of the earth permittivity, is extremely temperature dependent [18] As a result, an error due to uncertainty of the earth resistivity and permittivity might be far greater than that due to the incompleteness of the earth-return impedance derived by Carson and Pollaczek This fact should be reminded as a physical reality and engineering practice 7.1.2 Internal Impedance 7.1.2.1 Schelkunoff’s Impedance Schelkunoff’s impedance was derived under the condition that a conductor was in a free space corresponding to Equation 7.1 Therefore, the impedance is not applicable to finite-length conductors with proximity This fact suggests that an internal impedance of the finite length with the proximity effect is to be developed 7.1.2.2 Arbitrary Cross-Section Conductor Schelkunoff’s impedance assumes that a conductor is circular or cylindrical In reality, there exist many conductors of which the cross-section is not www.TechnicalBooksPdf.com 474 Computation for Humanity: Information Technology to Advance Society circular or cylindrical Reference [19] derived the internal impedance of a conductor with an arbitrary cross-section, which has been implemented into the EMTP Cable Parameters program [20] Reference [21] shows an approximation of a conductor with T or hollow rectangular shape by a cylindrical shape conductor Although the internal impedance of a conductor with an arbitrary cross-section can be accurately evaluated by a finite-element method of numerical calculation, it requires a lot of time and memory Either an analytical formula or an efficient numerical method needs to be developed 7.1.2.3 Semiconducting Layer of Cable It is well known that there exists a semiconducting layer on the surface of a cable conductor, which occasionally shows a significant effect on a cable transient The impedance of the semiconducting layer was derived in Ref [22] and may be implemented into a cable-impedance calculation It should be noted that the admittance of the semiconducting layer is far more important than the impedance, from the viewpoint of a transient analysis 7.1.2.4 Proximity Effect The significance of the proximity effect on conductor impedance is well known, and there are a number of papers that derive a theoretical formula of the impedance and admittance [23–29] and discuss the impedance variation due to the proximity based on numerical simulations [30–33] The proximity effect might be very important in a steady-state power system performance from the viewpoint of power loss, and some measured results in a power frequency were published [34–37] It has been pointed out that the proximity effect is also significant in a transient state for a surge waveform is noticeably distorted by an increase of a conductor resistance due to the proximity effect Unfortunately, there exists almost no measured data investigating the proximity effect on a transient [33] There exists a formula that considers the proximity or the eccentricity of a conductor enclosed within a conducting-pipe enclosure [24], that is, a pipetype cable [20,38] However, there is no formula that considers the proximity between two conductors above the earth 7.1.3 Earth-Return Admittance The earth-return impedance has been well discussed, and its effect on the wave-propagation characteristic and the transient waveform is well known, as is clear from a number of publications The earth-return admittance [8,9,39,40], however, is neglected in most studies on wave propagation and surge characteristics, and its significant effect is not well realized [8,9,40–43] www.TechnicalBooksPdf.com Problems and Application Limits of Numerical Simulations 475 It has been pointed out in Refs [8,9,40,43] that the attenuation starts to decrease at a critical frequency, which is inversely proportional to the earth resistivity and the conductor height This phenomenon is caused by a negative conductance, and corresponds to transition between TEM mode propagation called “earth-return wave” and transverse magnetic (TM) mode propagation called “surface wave,” as discussed by Kikuchi in 1957 When the earth-return admittance is neglected as usual, the attenuation increases monotonously as frequency increases The wave-propagation velocity and the characteristic impedance become greater when the earth-return admittance is considered A study on the earth-return admittance might be another challenging and fruitful field of a transient analysis including transition between TEM, TM, and TE modes of propagation [44] 7.2 Existing Problems in Circuit Theory–Based Numerical Analysis 7.2.1 Reliability of a Simulation Tool Quite often a problem appears unexpectedly from a user, but not from developers of a simulation tool, and it is hard for the developers to predict the problem at the development stage The problem is dependent quite often on the user’s misuse of the tool Therefore, reliability and severity tests of a simulation tool become very significant For example, the EMTP Cable Constants had taken nearly 10 years to carry out the reliability and severity tests with more than tens of thousands of cases It should be noted that the reliability of a tool, that is, the probability of trouble occurrence, is proportional to the number of elements, that is, the number of subroutines and options, even though each element keeps very high reliability Also, the input data often cause numerical instability when the data physically not exist This trouble relates to the assumption of formulas adopted in the simulation tool as explained in the previous section To avoid such trouble, a “KILL CODE” is prepared in the EMTP The kill code is to judge whether the input data are beyond the assumption and the limit It might be noteworthy that nearly half of the EMTP codes are kill codes This might be considered in another simulation tool 7.2.2 Assumption and Limit of a Simulation Tool It should be noted that most of the existing or well-known formulas of conductor impedances and admittances are derived based on the assumption of an infinity long conductor The frequency of an interested electrical phenomenon is increasing year by year corresponding to the advancement in measuring equipment For example, the sampling frequency of an oscilloscope, which is GHz today, was approximately 10 MHz 10 years ago The length is inversely www.TechnicalBooksPdf.com 476 Computation for Humanity: Information Technology to Advance Society proportional to the frequency, and therefore it becomes necessary to deal with a transient on a m conductor, of which the natural resonant frequency is in the order of 100 MHz Then, Schelkunoff’s, Pollaczek’s, and Carson’s impedances adopted in any circuit theory–based simulation tool such as the EMTP may not be applied [3] The earlier-explained assumption and the limit should be clearly explained in a rule book of a simulation tool, and the kill codes corresponding to the limit of the impedance and the admittance should be prepared in the tool The aforementioned problem often appears when the user adopts a commercial software, unless a developer or a user group gives a guide for the usage Even in the case of a publicized simulation tool such as the EMTP, it happened many times To avoid the problem, we have to realize that we are electrical engineers The best solution that overcomes the aforementioned problem is the physical understanding of the phenomenon to be simulated That is engineering We are not computer engineers, nor IT engineers 7.2.3 Input Data Corresponding to what is mentioned in the earlier section, a user of a simulation tool should be careful of input data Quite often, input data beyond the assumption and the limit of the tool are used, and the user complains that the tool gives erroneous output—this is the author’s experience as a developer of the original EMTP since 1976 At the same time, both the user and the developer should recognize that there are a number of uncertain physical, typically the earth resistivity parameters, which vary along a transmission line and also along the depth of the earth [16,17] The stratified-earth effect on a transient may be far more influential than the accuracy of numerical calculations of Pollaczek’s and Carson’s earth-return impedance, assuming a homogenous earth It is interesting to state the fact that the stratified-earth option of the EMTP Cable Constants has never been used since 1978 Also, stray capacitances and residual inductances of a power apparatus are, in general, not available from a manufacturer The same is the case with the nonlinear characteristic of the apparatus, and the resistivity and permittivity of a cable insulator and a semiconducting layer [18] 7.3 Numerical Electromagnetic Analysis for Power System Transients The numerical electromagnetic analysis (NEA) method [45–50] is becoming one of the most promising approaches to solve transient phenomena that are very hard to be solved by existing circuit theory–based simulation tools such as the EMTP Existing circuit theory–based approaches cannot solve a three-dimensional (3D) transient and a transient involving sphere-wave www.TechnicalBooksPdf.com Problems and Application Limits of Numerical Simulations 477 propagation and a scattered field, such as a transient across an archon, a wavefront transient at a transmission tower due to lightning, and the voltage and current at the corner or across a spacer of a gas-insulated bus due to a switching surge Also, the circuit theory–based approach has the difficulty of solving a transient in a complex medium, such as the transient on a grounding electrode and that on a semiconducting layer of a cable Furthermore, the circuit theory approach cannot be applied if circuit parameters are not known NEA can solve such problems, because it calculates Maxwell’s equation directly A working group of the IEE Japan was founded in April 2004, and was carrying out an investigation on NEA and its application examples The results derived by the working group were published as a book from the IEE Japan [49] Also, CIGRE WG C4 501 was established [50] in 2009, and a CIGRE technical brochure (TB) has been completed and will be published soon The NEA method is powerful to deal with power system transients, for example, in the following subjects: • Surge characteristics of overhead transmission-line towers • Surge characteristics of vertical grounding electrodes and horizontally-placed square-shape grounding electrodes • Surge characteristics of air-insulated substations • Lightning-induced surges on overhead distribution lines • Surge characteristics of a wind-turbine tower struck by lightning and its inside transient magnetic field • Very fast transients in gas-insulated switchgears • Three-dimensional electromagnetic-field analysis The details of NEA have been explained in Chapter In summary, NEA methods can provide better accuracy in comparison with simulation results obtained using circuit theory–based approaches However, as large computation resources are, in general, required, NEA methods can be considered useful tools to set reference cases and study specific problems Also, a perfect conductor assumption in a finite-difference time-domain (FDTD) method, for example, results in a difficulty to analyze TEM, TM, and TE transition of wave propagation along a lossy conductor above a lossy earth [8,9,43,44] References Carson, J R 1926 Wave propagation in overhead wires with ground return Bell Syst Tech J 5:539–554 www.TechnicalBooksPdf.com 478 Computation for Humanity: Information Technology to Advance Society Pollaczek, F 1926 Uber das Feld einer unendlich langen wechselstromdurchflossenen Einfachleitung ENT 9(3):339–359 Dommel, H W 1986 EMTP Theory Book Portland, OR: B.P.A Schelkunoff, S A 1934 The electromagnetic theory of coaxial transmission line and cylindrical shields Bell Syst Tech J 13:532–579 Noda, T 2006 Development of accurate algorithms for calculating groundreturn and conductor-internal impedances CRIEPI Report H05003 by Central Research Institute of Electric Power Industries in Japan (CRIPEI Tokyo) Report H05003 Ametani, A and A Ishihara 1993 Investigation of impedance and line parameters of a finite-length multiconductor system Trans IEE Jpn 113-B(8):905–913 Ametani, A and T Kawamura 2005 A method of a lightning surge analysis recommended in Japan using EMTP IEEE Trans Power Deliv 20(2):867–875 Kikuchi, H 1955 Wave propagation on the ground return circuit in high frequency regions J IEE Jpn 75(805):1176–1187 Kikuchi, H 1957 Electro-magnetic field on infinite wire at high frequencies above plane-earth J IEE Jpn 77:721–733 10 Wedepohl, L M and A E Efthymiais 1978 Wave propagation in transmission line over lossy ground-A new complete filed solution IEEE Proc 125(6):505–510 11 Ametani, A., T Yoneda, Y Baba, and N Nagaoka 2009 An investigation of earth-return impedance between overhead and underground conductors and its approximation IEEE Trans EMC 51(3): 860–867 12 Hara, T., O Yamamoto, M Hayashi, and C Uenosono Empirical formulas of surge impedance for single and multiple vertical conductors Trans IEE Jpn 110-B:129–136 13 Ametani, A., Y Kasai, J Sawada, A Mochizuki, and T Yamada 1994 Frequencydependent impedance of vertical conductors and a multiconductor tower model IEE Proc Generat Transm Distrib 141(4):339–345 14 Ametani, A 2002 Wave propagation on a nonuniform line and its impedance and admittance Sci Eng Rev Doshisha Univ 43(3):135–147 15 Deri, A et al 1981 The complex ground return plane: A simplified model for homogeneous and multi-layer earth return IEEE Trans Power App Syst 100(8): 3686 16 Nakagawa, M., A Ametani, and K Iwamoto 1973 Further studies on wave propagation in overhead lines with earth return—Impedance of stratified earth Proc IEE 120(2):1521–1528 17 Ametani, A 1974 Stratified effects on wave propagation—Frequency-dependent parameters IEEE Trans Power App Syst 93(5):1233–1239 18 Ametani, A 2000 Problems and countermeasures of cable transient simulations EMTP J 5:3–11 19 Ametani, A and I Fuse 1992 Approximate method for calculating the impedances of multi conductors with cross-section of arbitrary shapes Elect Eng Jpn 111(2):117–123 20 Ametani, A 1994 Cable Parameters Rule Book Portland, OR: B.P.A 21 Ametani, A., N Nagaoka, R Koide, and T Nakanishi 1999 Wave propagation characteristics of iron conductors in an intelligent building Trans IEE Jpn B-120(1): 271–277 22 Ametani, A., Y Miyamoto, and N Nagaoka 2004 Semiconducting layer impedance and its effect on cable wave-propagation and transient characteristics IEEE Trans Power Deliv 19(4):523–531 www.TechnicalBooksPdf.com Problems and Application Limits of Numerical Simulations 479 23 Tegopoulos, J A and E E Kriezis 1971 Eddy current distribution in cylindrical shells of infinite length due to axial currents, Part II—Shells of infinite thickness IEEE Trans Power App Syst 90:1287–1294 24 Brown, G W and R G Rocamora 1976 Surge propagation in three-phase pipe-type cables, Part I—Unsaturated pipe IEEE Trans Power App Syst 95:88–95 25 Dugan, R C et al 1977 Surge propagation in three-phase pipe-type cables, Part II—Duplication of filed test including the effects of neutral wires and pipe saturation IEEE Trans Power App Syst 96:826–833 26 Schinzinger, R and A Ametani 1978 Surge propagation characteristics of pipe enclosed underground cables IEEE Trans Power App Syst 97:1680–1687 27 Dokopoulos, P and D Tampakis 1984 Analysis of field and losses in threephase gas cable with thick walls: Part I Field analysis IEEE Trans Power App Syst 103(9):2728–2734 28 Dokopoulos, P and D Tampakis 1985 Part II Calculation of losses and results IEEE Trans Power App Syst 104(1):9–15 29 Poltz, J., E Kuffel, S Grzybowski, and M R Raghuveer 1982 Eddy-current losses in pipe-type cable systems IEEE Trans Power App Syst 101(4):825–832 30 Fortin, S., Y Yang, J Ma, and F P Dawalibi 2005 Effects of eddy current on the impedance of pipe-type cables with arbitrary pipe thickness ICEE 2005, Gliwice, Poland Paper TD2-09 31 Chien, C H and R W G Bucknall 2009 Harmonic calculation of proximity effect on impedance characteristics in subsea power transmission cables IEEE Trans Power Deliv 24(2):2150–2158 32 Gustavsen, B., A Bruaset, J J Bremnes, and A Hassel 2009 A finite element approach for calculating electrical parameters of umbilical cables IEEE Trans Power Deliv 24(4):2375–2384 33 Ametani, A., K Kawamura et al 2013 Wave propagation characteristics on a pipe-type cable in particular reference to the proximity effect IEE J High Voltage Eng Conf Kyoto, Japan, Paper HV-13-005 34 Ishikawa, T., K Kawasaki, and I Okamoto 1976 Eddy current losses in cable sheath (1) Dainichi Nihon Cable J 61:34–42 35 Ishikawa, T., K Kawasaki, and O Okamoto 1977 Eddy current losses in cable sheath (2) Dainichi Nihon Cable J 62:21–64 36 Kawasaki, K., M Inami, and T Ishikawa 1981 Theoretical consideration on eddy current losses on non-magnetic and magnetic pipes for power transmission systems IEEE Trans Power App Syst 100(2):474–484 37 Mekjian, A and M Sosnowski 1983 Calculation of altering current losses in steel pipe containing power cables IEEE Trans Power App Syst 102(2): 382–388 38 Ametani, A 1980 A general formulation of impedance and admittance of cables IEEE Trans Power App Syst 99(3):902–910 39 Wise, W H 1948 Potential coefficients for ground return circuits Bell Syst Tech J 27:365–371 40 Nakagawa, M 1981 Further studies on wave propagation along overhead transmission lines: Effects of admittance correction IEEE Trans Power App Syst 100(7):3626–3633 41 Rachidi, F., C A Nucci, and M Ianoz 1999 Transient analysis of multiconductor lines above a lossy ground IEEE Trans Power Deliv 14(1):294–302 www.TechnicalBooksPdf.com 480 Computation for Humanity: Information Technology to Advance Society 42 Hashmi, G M., M Lehtonen, and A Ametani 2010 Modeling and experimental verification of covered conductors for PD detection in overhead distribution networks IEE J Trans PE 130(7):670–678 43 Ametani, A., M Ohe, Y Miyamoto, and K Tanabe 2012 The effect of the earthreturn admittance on wave propagation along an overhead conductor in a highfrequency region EEUG Proceedings, Zwickau, Germany 1:6–22 44 Sommerfeld, A 1964 Partial Differential Equation in Physics New York: Academic Press 45 Yee, K S 1966 Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media IEEE Trans Antennas Propagat 14(3):302–307 46 Uno, T 1998 FDTD Method for Electromagnetic Fields and Antennas Tokyo, Japan: Corona Pub Co 47 Taflove, A and S.C Hagness 2000 Computational Electromagnetics The FiniteDifference Time-Domain Method Norwood, MA: Artech House 48 CRIEPI by Central Research Institute of Electric Power Industry, Tokyo, Japan, 2007 Visual Test Lab (VSTL) http://cripei.denken.or.jp/jp/electric/ substance/09.pdf 49 IEE Japan WG Working Group of Numerical Transient Electromagnetic Analysis (Convenor: A Ametani) 2008 Numerical Transient Electromagnetic Analysis Methods IEE Japan, ISBN 978-4-88686-263-1 50 Ametani, A., T Hoshino, M Ishii, T Noda, S Okabe, and K Tanabe 2008 Numerical electromagnetic analysis method and its application to surge phenomena CIGRE 2008 General Meeting, Paris, France Paper C4-108 www.TechnicalBooksPdf.com Power Engineering POWER SYSTEM TRANSIENTS Theor y and Applications AKIHIRO AMETANI • NAOTO NAGAOKA YOSHIHIRO BABA • TERUO OHNO As a transient phenomenon can shut down a building or an entire city, transient analysis is crucial to managing and designing electrical systems Power System Transients: Theory and Applications discusses the basic theory of transient phenomena—including lumped- and distributed-parameter circuit theories— and provides a physical interpretation of the phenomena It covers novel and topical questions of power system transients and associated overvoltages Using formulas simple enough to be applied using a pocket calculator, the book presents analytical methods for transient analysis It examines the theory of numerical simulation methods such as the EMTP (circuit theory–based approach) and numerical electromagnetic analysis The book highlights transients in clean or sustainable energy systems such as smart grids and wind farms, since they require a different approach than overhead lines and cables Simulation examples provided include arcing horn flashover, a transient in a grounding electrode, and an induced voltage from a lightning channel an informa business www.taylorandfrancisgroup.com 6000 Broken Sound Parkway, NW Suite 300, Boca Raton, FL 33487 711 Third Avenue New York, NY 10017 Park Square, Milton Park Abingdon, Oxon OX14 4RN, UK www.TechnicalBooksPdf.com K16810 www.TechnicalBooksPdf.com ... Power SyStem tranSientS Theor y and Applications www.TechnicalBooksPdf.com www.TechnicalBooksPdf.com Power SyStem tranSientS theor y and Applications Akihiro AmetAni nAoto nAgAokA Yoshihiro BABA. .. voltage and power generation vary depending on the amount of sunshine the photo cells are exposed to, which is based on the time of the day and the weather A power conditioner and a storage system. .. electromagnetic transients program (EMTP), originally developed by the US Department of Energy, Bonneville Power Administration, which is useful in dealing with a real transient in a power system Chapter