Power System
Transients
Parameter
CRC Press is an imprint of the
Taylor & Francis Group, an informa business Boca Raton London New York
Power System
Transients
Parameter
Determination
Edited by
Juan A. Martinez-Velasco
CRC Press
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Library of Congress Cataloging-in-Publication Data Power system transients : parameter determination / editor, Juan A. Martinez-Velasco.
p. cm. “A CRC title.”
Includes bibliographical references and index. ISBN 978-1-4200-6529-9 (hardcover : alk. paper)
1. Electric power system stability. 2. Transients (Electricity) Mathematical models. 3. Arbitrary constants. 4. Electric power systems--Testing. I. Martinez-Velasco, Juan A.
TK1010.P687 2010
621.319’21--dc22 2009031432
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v Preface ... vii Editor...ix Contributors ...xi
1. Parameter Determination for Electromagnetic Transient Analysis in
Power Systems ... 1
Juan A. Martinez-Velasco
2. Overhead Lines ... 17
Juan A. Martinez-Velasco, Abner I. Ramirez, and Marisol Dávila
3. Insulated Cables ... 137
Bjørn Gustavsen, Taku Noda, José L. Naredo, Felipe A. Uribe, and Juan A. Martinez-Velasco
4. Transformers ... 177
Francisco de León, Pablo Gómez, Juan A. Martinez-Velasco, and Michel Rioual
5. Synchronous Machines ... 251
Ulas Karaagac, Jean Mahseredjian, and Juan A. Martinez-Velasco
6. Surge Arresters ... 351
Juan A. Martinez-Velasco and Ferley Castro-Aranda
7. Circuit Breakers ... 447
Juan A. Martinez-Velasco and Marjan Popov
Appendix A: Techniques for the Identifi cation of a Linear System from Its
Frequency Response Data ... 557
Bjørn Gustavsen and Taku Noda
Appendix B: Simulation Tools for Electromagnetic Transients
in Power Systems ... 591
Jean Mahseredjian, Venkata Dinavahi, Juan A. Martinez-Velasco, and Luis D. Bellomo
vii The story of this book may be traced back to the Winter Meeting that the IEEE Power Engineering Society (now the Power and Energy Society) held in January 1999, when the Task Force (TF) on Data for Modeling Transient Analysis was created. The mandate of this TF was to produce a series of papers that would help users of transient simulation tools (e.g., EMTP-like tools) select an adequate technique or procedure for determining the parameters to be specifi ed in transient models. The TF wrote seven papers that were published in the July 2005 issue of the IEEE Transactions on Power Delivery.
The determination or estimation of transient parameters is probably the most diffi cult and time-consuming task of many transient studies. Engineers and researchers spend only a small percentage of the time running the simulations; most of their time is dedi-cated to collecting the information from where the parameters of the transient models will be derived and to testing the complete system model. Even today, with the availability of powerful numerical techniques, simulation tools, and graphical user interfaces, the selec-tion of the most adequate model and the determinaselec-tion of parameters are very often the weakest point of the whole task.
Signifi cant efforts have been made in the last two decades on the development of new transient models and the proposal of modeling guidelines. Currently, users of transient tools can take advantage of several sources to select the study zone and choose the most adequate model for each component involved in the transient process. However, the fol-lowing drawbacks must still be resolved: (1) the information required for the determination of some parameters is not always available; (2) some testing setups and measurements to be performed are not recognized in international standards; (3) more studies are required to validate models, mainly those that are to be used in high or very high-frequency tran-sients; and (4) built-in models currently available in simulation tools do not cover all mod-eling requirements, although most of them are capable of creating custom-made models.
Although procedures and studies of parameter determination are presented for only seven power components, it is obvious that much more information is required to cover all important aspects in creating an adequate and reliable transient model. In some cases, procedures of parameter determination are presented only for low-frequency models; in other cases, all the procedures required for creating the whole model of a component are not analyzed. In addition, there is a lack of examples to illustrate how parameters can be determined for real-world applications.
The core of this book is dedicated to current procedures and techniques for the deter-mination of transient parameters for six basic power components: overhead line, insulated cable, transformer, synchronous machine, surge arrester, and circuit breaker. Therefore, this book can be seen as a setup that has joined an expanded version of the transaction papers. It will help users of transient tools solve part of the main problems they face when creating a model adequate for electromagnetic transient simulations.
This book includes two appendices. The fi rst provides updated techniques for the identifi cation of linear systems from frequency responses; these fi tting techniques can be useful both for determining parameters and for fi tting frequency-dependent models of some of the components analyzed in this book (e.g., lines, cables, or transformers). The second reviews the capabilities and limitations of the most common simulation tools, taking into consideration both off-line and online (real-time) tools.
A crucial aspect that needs to be emphasized is the importance of standards. As pointed out in several chapters, standards are a very valuable source of information for the deter-mination of parameters, for the characterization of components, and even for their selec-tion (e.g., surge arresters). Whenever possible, recommendaselec-tions presented in standards have been summarized and included in the appropriate chapter.
Although this is a book on electromagnetic transients, many of the most important top-ics related to this fi eld have not been well covered or have not been covered at all. For instance, Appendix B on simulation tools contains only a short summary of computational methods for transient analysis.
Most of the topics covered require one to have some basic knowledge of electromagnetic transient analysis. This book is addressed mainly to graduate students and professionals involved in transient studies.
I want to conclude this preface by thanking the members of the IEEE TF for their pio-neering work, and by expressing my gratitude to all our colleagues, friends, and relatives for their help, and, in many circumstances, for their patience.
Juan A. Martinez-Velasco
Barcelona, Spain
MATLAB® is a registered trademark of The MathWorks, Inc. For product information, please contact:
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ix
Juan A. Martinez-Velasco was born in Barcelona, Spain. He received his Ingeniero Industrial
and Doctor Ingeniero Industrial degrees from the Universitat Politècnica de Catalunya (UPC), Spain. He is currently with the Departament d’Enginyeria Elèctrica of the UPC.
Martinez-Velasco has authored and coauthored more than 200 journal and conference papers, most of them on the transient analysis of power systems. He has been involved in several ElectroMagnetic Transients Program (EMTP) courses and has worked as a consultant for Spanish companies. His teaching and research areas cover power systems analysis, transmission and distribution, power quality, and electromagnetic transients. He is an active member of several IEEE and CIGRE working groups (WGs). Currently, he is the chair of the IEEE WG on Modeling and Analysis of System Transients Using Digital Programs.
Dr. Martinez-Velasco has been involved as an editor or a coauthor of eight books. He is also the coeditor of the IEEE publication Modeling and Analysis of System Transients Using Digital Programs (1999). In 1999, he received the 1999 PES Working Group Award for Technical Report, for his participation in the tasks performed by the IEEE Task Force on Modeling and Analysis of Slow Transients. In 2000, he received the 2000 PES Working Group Award for Technical Report, for his participation in the publication of the special edition of Modeling and Analysis of System Transients Using Digital Programs.
xi
Luis D. Bellomo
École Polytechnique de Montréal Montreal, Quebec, Canada
Ferley Castro-Aranda
Universidad del Valle
Escuela de Ingeniería Eléctrica y ElectrÓnica
Cali, Colombia
Marisol Dávila
Universidad de los Andes Escuela de Ingeniería Eléctrica Mérida, Venezuela
Francisco de León
Department of Electrical and Computer Engineering
Polytechnic Institute of NYU Brooklyn, New York
Venkata Dinavahi
University of Alberta
Department of Electrical and Computer Engineering
Edmonton, Alberta, Canada
Pablo Gómez
Instituto Politécnico Nacional Departmento de Ingeniería Eléctrica México, Mexico
Bjørn Gustavsen
SINTEF Energy Research Trondheim, Norway
Ulas Karaagac
École Polytechnique de Montréal Montreal, Quebec, Canada
Jean Mahseredjian
École Polytechnique de Montréal Montreal, Quebec, Canada
Juan A. Martinez-Velasco
Universitat Politècnica de Catalunya Department d’Enginyeria Elèctrica Barcelona, Spain
José L. Naredo
Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional
Guadalajara, Mexico
Taku Noda
Central Research Institute of Electric Power Industry
Yokosuka, Japan
Marjan Popov
Delft University of Technology Faculty of Electrical Engineering,
Mathematics and Computer Science Delft, the Netherlands
Abner I. Ramirez
Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional
Guadalajara, Mexico
Michel Rioual
Électricité de France R & D Clamart, France
Felipe A. Uribe
Universidad de Guadalajara
Departmento de Ingeniería Mecánica Eléctrica
1
Parameter Determination for Electromagnetic
Transient Analysis in Power Systems
Juan A. Martinez-Velasco
CONTENTS
1.1 Introduction ... 1
1.2 Modeling Guidelines ... 2
1.3 Parameter Determination ... 6
1.4 Scope of the Book ... 11
References ... 15
1.1 Introduction
Power system transient analysis is usually performed using computer simulation tools like the Electromagnetic Transients Program (EMTP), although modeling using Transient Network Analyzers (TNAs) is still done, but decreasingly. There is also a family of tools based on computerized real-time simulations, which are normally used for testing real control system components or devices such as relays. Although there are several common links, this chapter targets only off-line nonreal-time simulations.
Engineers and researchers who perform transient simulations typically spend only a small amount of their total project time running the simulations. The bulk of their time is spent obtaining parameters for component models, testing the component models to confi rm proper behaviors, constructing the overall system model, and benchmarking the overall system model to verify overall behavior. Only after the component models and the overall system representation have been verifi ed, one can confi dently proceed to run meaningful simulations. This is an iterative process. If there are some transient event records to compare against, more model benchmarking and adjustment may be required.
This book deals with parameter determination and is aimed at reviewing the proce-dures to be performed for deriving the mathematical representation data of the most important power components in electromagnetic transient simulations. This chapter presents a summary on the current status and practice in this fi eld and emphasizes needed improvements for increasing the accuracy of modeling tasks in detailed tran-sient analysis.
Figure 1.1 shows a fl ow chart of the procedure suggested to obtain the complete repre-sentation of a power component [1]:
First, choose the mathematical model. •
Second, collect the information that could be useful to determine the values of •
parameters to be specifi ed.
Third, decide whether the available data are enough or not to derive all •
parameters.
The procedure depicted in Figure 1.1 assumes that the values of the parameters to be speci-fi ed in some mathematical descriptions are not necessarily readily available and they must be deduced from other information using, in general, a data conversion procedure.
1.2 Modeling Guidelines
An accurate representation of a power component is essential for reliable transient analy-sis. The simulation of transient phenomena may require a representation of network com-ponents valid for a frequency range that varies from DC to several MHz. Although the ultimate objective in research is to provide wideband models, an acceptable representation of each component throughout this frequency range is very diffi cult, and for most compo-nents is not practically possible. In some cases, even if the wideband version is available, it may exhibit computational ineffi ciency or require very complex data.
Modeling of power components that take into account the frequency-dependence of parameters can be currently achieved through mathematical models which are accurate FIGURE 1.1
Procedure to obtain a complete representation of a power component. (From Martinez, J.A. et al., IEEE Power Energy Mag., 3, 16, 2005. With permission.)
Choose the mathematical representation of the
power component
Collect the information needed to derive the mathematical model
Is this information
enough?
No
Estimate the value of some parameters Perform the conversion
procedure Yes
enough for a specifi c range of frequencies. Each range of frequencies usually corresponds to some particular transient phenomena. One of the most accepted classifi cations is that proposed by the International Electrotechnical Commission (IEC) and CIGRE, in which frequency ranges are classifi ed into four groups [2,3]: low-frequency oscillations, from 0.1 Hz to 3 kHz; slow-front surges, from 50/60 Hz to 20 kHz; fast-front surges, from 10 kHz to 3 MHz; and very fast-front surges, from 100 kHz to 50 MHz. One can note that there is overlap between frequency ranges.
If a representation is already available for each frequency, the selection of the model may suppose an iterative procedure: the model must be selected based on the frequency range of the transients to be simulated; however, the frequency ranges of the test case are not usually known before performing the simulation. This task can be alleviated by looking into widely accepted classifi cation tables. Table 1.1 shows a short list of common transient phenomena.
An important effort has been dedicated to clarify the main aspects to be consid-ered when representing power components in transient simulations. Users of electro-magnetic transient tools can nowadays obtain information on this fi eld from several sources:
1. The document written by the CIGRE WG 33-02 covers the most important power components and proposes the representation of each component taking into account the frequency range of the transient phenomena to be simulated [2]. 2. The fourth part of the IEC standard 60071 (TR 60071-4) provides modeling
guide-lines for insulation coordination studies when using numerical simulation, e.g., EMTP-like tools [3].
3. The documents produced by the Institute of Electrical and Electronics Engineers (IEEE) WG on modeling and analysis of system transients using digital programs and its task forces present modeling guidelines for several particular types of studies [4].
Table 1.2 provides a summary of modeling guidelines for the representation of the most important power components in transient simulations taking into account the frequency range.
The simulation of a transient phenomenon implies not only the selection of models but the selection of the system area that must be represented. Some rules to be considered in
TABLE 1.1
Origin and Frequency Ranges of Transients in Power Systems
Origin Frequency Range
Ferroresonance 0.1 Hz to 1 kHz
Load rejection 0.1 Hz to 3 kHz
Fault clearing 50 Hz to 3 kHz
Line switching 50 Hz to 20 kHz
Transient recovery voltages 50 Hz to 100 kHz Lightning overvoltages 10 kHz to 3 MHz Disconnector switching in GIS 100 kHz to 50 MHz
T ABLE 1.2 Mod el ing o f P o w er C o m p one n ts for T ra n sie n t Si m u la ti on s Component Low-Frequency T ransients 0.1 HZ to 3 kHz Slow-Front T ransients 50 Hz to 20 kHz Fast-Front T ransients 10 kHz to 3 MHz V ery Fast-Front T ransients 100 kHz to 50 MHz Over head line
Multiphase model with lumped and constant parameters, including conductor asymmetry
. Fr
equency-dependence of parameters can be important for the gr
ound pr
opagation
mode. Cor
ona ef
fect can be also
important if phase conductor voltages exceed the cor
ona inception voltage.
Multiphase model with distributed parameters, including conductor asymmetry
. Fr
equency-dependence
of parameters is important for the ground pr
opagation mode.
Multiphase model with distributed parameters, including conductor asymmetry and cor
ona ef
fect. Fr
equency-dependence of parameters is important for the gr
ound
pr
opagation mode.
Single-phase model with distributed parameters. Fr
equency-dependence of
parameters is important for the gr
ound pr
opagation mode.
Insulated cable Multiphase model with lumped and constant parameters, including conductor asymmetry
. Fr
equency-dependence
of parameters can be important for the ground pr
opagation mode.
Multiphase model with distributed parameters, including conductor asymmetry
. Fr
equency-dependence
of parameters is important for the ground pr
opagation mode
Multiphase model with distributed parameters. Fr
equency-dependence of
parameters is important for the ground pr
opagation mode.
Single-phase model with distributed parameters. Fr
equency-dependence of
parameters is important for the gr
ound pr
opagation mode.
T
ransformer
Models must incorporate saturation ef
fects, as well as cor
e and winding
losses. Models for single- and thr
ee-phase cor
e can show signifi
cant
dif
fer
ences.
Models must incorporate saturation ef
fects, as well as cor
e and winding
losses. Models for single- and thr
ee-phase cor
e can show signifi
cant
dif
fer
ences.
Cor
e losses and saturation can be
neglected. Coupling between phases is mostly capacitive. The infl
uence of the short cir
cuit
impedance can be signifi
cant.
Cor
e losses and saturation can
be neglected. Coupling between phases is mostly capacitive. The model should incorporate the sur
ge impedance of windings. Synchr onous generator Detailed r epr
esentation of the electrical
and mechanical parts, including saturation ef
fects and contr
ol of excitation. The machine is r epr esented as a sour
ce in series with its subtransient
impedance. Saturation ef
fects can be
neglected. The contr
ol excitation and
the mechanical part ar
e not included. The r epr esentation is based on a linear cir cuit whose fr equency
response matches that of the machine seen fr
om its terminals.
The r
epr
esentation may be
based on a linear lossless capacitive cir
cuit.
Metal oxide sur
ge arr ester Nonlinear r esistive cir cuit, characterized by its r
esidual voltage to switching
impulses. Nonlinear r esistive cir cuit, characterized by its r esidual voltage to switching impulses. Nonlinear r esistive cir cuit, characterized by its r esidual
voltage to switching impulses, including the ef
fect of the peak
curr
ent and its fr
ont. Nonlinear r esistive cir cuit, characterized by its r esidual
voltage to switching impulses, including the ef
fect of the peak
curr
ent and its fr
ont.
Cir
cuit
br
eaker
The model has to incorporate mechanical pole spr
ead, and ar
c equations for
interr
uption of high curr
ents.
The model has to incorporate mechanical pole spr
ead, the
sparkover characteristic vs. time, ar
c instability , and interr uption of high-fr equency curr ents.
The model has to incorporate the sparkover characteristic vs. time, ar
c instability , and interr uption of high-fr equency curr ents.
The model has to incorporate the sparkover characteristic vs. time, and interr
uption of high-fr equency curr ents. Sour ce: Martinez, J.A.,
IEEE PES General Meeting
, T ampa, 2007. W ith permission. Note: The r epr
the simulation of electromagnetic transients when selecting models and the system area can be summarized as follows [5]:
1. Select the system zone taking into account the frequency range of the transients; the higher the frequencies, the smaller the zone modeled.
2. Minimize the part of the system to be represented. An increased number of com-ponents does not necessarily mean increased accuracy, since there could be a higher probability of insuffi cient or wrong modeling. In addition, a very detailed representation of a system will usually require longer simulation time.
3. Implement an adequate representation of losses. Since their effect on maximum voltages and oscillation frequencies is limited, they do not play a critical role in many cases. There are, however, some cases (e.g., ferroresonance or capacitor bank switching) for which losses are critical to defi ning the magnitude of overvoltages. 4. Consider an idealized representation of some components if the system to be sim-ulated is too complex. Such representation will facilitate the edition of the data fi le and simplify the analysis of simulation results.
5. Perform a sensitivity study if one or several parameters cannot be accurately deter-mined. Results derived from such a sensitivity study will show what parameters are of concern.
Figure 1.2 shows a test case used for illustrating the differences between simulation results from two different line models with distributed parameters: the lossless constant-parameter (CP) line and the lossy frequency-dependent (FD) line. Wave propagation along an overhead line is damped and the waveform is distorted when the FD-line model is used, while the wave propagates without any change if the line is assumed ideal (i.e., lossless), as expected. In addition the propagation velocity is higher with the FD-line model.
FIGURE 1.2
Comparative modeling of lines: FD and CP models. (a) Scheme of the test case; (b) wave propogation along the line.
Single-phase overhead line with distributed parameters 1 e(t) + – 2 (a) CP line (b) FD line CP line FD line Time Voltage Voltage
Frequency-dependent modeling can be crucial in some overvoltage studies; e.g., those related to lightning and switching transients. In breaker performance studies, since the breaker arc voltage excites all circuit frequencies, it might be important to apply fre-quency-dependent models even for substation bus-bar sections where the breaker effect is analyzed.
Figure 1.3 shows the fi eld current in a synchronous generator during a three-phase short circuit at the armature terminals obtained with two different models. The fi rst model assumes a coupling between the two rotor circuits located on the d-axis (fi eld and damp-ing winddamp-ings), while this coupldamp-ing has been neglected in the simulation of the second case. Although the differences are important, both modeling approaches can be acceptable if the main goal is to obtain the short circuit current in the armature windings or the deviation of the rotor angular velocity with respect to the synchronous velocity.
1.3 Parameter Determination
Given the different design and operation principles of the most important power compo-nents, various techniques can be used to analyze their behavior. The following paragraphs discuss the effects that have to be represented in mathematical models and what are the approaches that can be used to determine the parameters, without covering the determi-nation of parameters needed to represent mechanical systems, control systems, or semi-conductor models.
Basically, the mathematical model of a power component (e.g., line, cable, transformer, rotating machine) for electromagnetic transient analysis must represent the effects of elec-tromagnetic fi elds and losses [1]:
1. Electromagnetic fi eld effects are, in general, represented using a circuit approach: magnetic fi eld effects are represented by means of inductors and coupling between them, while electric fi elds effects are replaced by capacitors. In increased preci-sion models, such as distributed-parameter transmispreci-sion lines, parameters cannot be lumped, and mathematical models are based on solving differential equations with matrix coupling.
FIGURE 1.3
Field winding current in a synchronous generator during a three-phase short circuit. (a) Field current when coupling between the rotor d-axis circuits is assumed; (b) fi eld current when coupling between the rotor d-axis circuits is neglected. (From Martinez, J.A. et al., IEEE Power Energy Mag., 3, 16, 2005. With permission.)
0 0 4 8 Current (kA) 12 200 400 Time (ms) 600 800 1000 (a) 0 0 4 8 Current (kA) 12 200 400 Time (ms) 600 800 1000 (b)
2. Losses can be caused in windings, cores, or insulations. Other sources of losses are corona in overhead lines or screens and sheaths in insulated cables. As for electromagnetic field effects, they are represented using a circuit approach. In many situations, losses cannot be separated from electromagnetic fields: skin effect is caused by the magnetic field constrained in windings, and pro-duces frequency-dependent winding losses; core losses depend on the peak magnetic flux and the frequency of this field; corona losses are caused when the electric field exceeds the inception corona voltage; insulation losses are caused by the electric field and show an almost linear behavior. Approaches that can be used to represent losses would include a resistor, with either linear or nonlinear behavior, a hysteresis cycle, or a combination of various types of circuit elements. More sophisticated loss models must include frequency dependence.
The parameters used to represent electromagnetic fi eld effects and losses can be deduced using the following techniques [1]:
Techniques based on geometry, for instance a numerical solution aimed at solving •
the partial-differential equations of the electromagnetic fi elds developed within the component and based on the fi nite element method (FEM), a technique that can be used with most components. However, more simple techniques have also been developed; for instance, an analytical solution based on a simplifi ed geom-etry and the separation of the electric and magnetic fi eld are used with lines and cables. Factory measurements can be needed to obtain material properties (i.e., resistivity, permeability, and permittivity), although very often these values can be also obtained from standards or manufacturer catalogues. If the behavior of the component is assumed linear, permeabilities are approximated by that of the vacuum. If the behavior of the component is nonlinear (i.e., made of ferromagnetic material), factory tests may be needed to obtain saturation curves and/or hyster-esis cycles.
Factory tests, mainly used with transformers and rotating machines. Tests devel-•
oped with this purpose can be grouped as follows:
Steady-state tests, which can be classifi ed into two groups: fi xed frequency •
tests (no load and short circuit tests are frequently used) and variable fre-quency tests (frefre-quency response tests).
Transient tests; e.g., those performed to obtain parameters of the equivalent •
electric circuits of a synchronous machine.
When parameter determination is based on factory tests, a data conversion procedure can be required; that is, in many cases, parameters to be specifi ed in a given model are not directly provided by factory measurements.
Factory tests are usually performed according to standards. However, the tests defi ned by standards do not always provide all of the data needed for transient modeling, and there are some cases for which no standard has been proposed to date. This is applicable to both transformers and rotating machines, although the most signifi cant case is related to the representation of three-phase core transformers in low- and midfrequency transients [6]. The simulation of the asymmetrical behavior that can be caused by some transients
must be based on models for which no standard has been yet developed, although several tests have been proposed in the specialized literature.
Figure 1.4 shows a fl owchart on the parameter determination approaches presented above.
Some common approaches followed to obtain the mathematical model for lines and synchronous machines when using time-domain simulation tools will clarify the above discussion.
Parameters and/or mathematical models needed for representing an overhead line in simulation packages are obtained by means of a “supporting function” which is available in most EMTP-like tools and can be named, for the sake of generality, as line constants (LC) routine [7] (see Chapter 2). LC routine users must enter the physical parameters of the line and select the desired model. The following data must be inputted: (x, y) coordinates of each conductor and shield wire, bundle spacing, and orientations, sag of phase conduc-tors and shield wires, phase and circuit designation of each conductor, phase rotation at transposition structures, physical dimensions of each conductor, DC resistance of each conductor (including shield wires) and ground resistivity of the ground return path. Note that all the above information except conductor resistances and ground resistivity is from geometric line dimensions. The following models can be created: lumped-parameter model or nominal pi-circuit model, at the specifi ed frequency; constant distributed- parameter model, at the specifi ed frequency; frequency-dependent distributed-parameter model, fi tted for a given frequency range (see Figure 1.5). In addition, the following information
Parameter determination Geometry Factory tests Analitycal solution of electromagnetic fields (simplified geometry, field separation) Steady-state tests Fixed-frequency tests Variable-frequency tests Data conversion procedure Transient tests Numerical solution of a continuum problem (electromagnetic field PDEs) FIGURE 1.4
Classifi cation of methods for parameter determination. (From Martinez, J.A. et al., IEEE Power Energy Mag., 3, 16, 2005. With permission.)
usually becomes available: the capacitance or the susceptance matrix; the series imped-ance matrix; resistimped-ance, inductimped-ance, and capacitimped-ance per unit length for zero and positive sequences, at a given frequency or for the specifi ed frequency range; surge impedance, attenuation, propagation velocity, and wavelength for zero and positive sequences, at a given frequency or for a specifi ed frequency range. Line matrices can be provided for the system of physical conductors, the system of equivalent phase conductors, or with sym-metrical components of the equivalent phase conductors. Note that the model created by the LC routine is representing only phase conductors and, if required, shield wires, while in some simulations the representation of other parts is also needed; for instance, towers, insulators, grounding impedances, and corona models are needed when calculating light-ning overvoltages in overhead transmission lines.
The conversion procedures that have been proposed for the determination of the electri-cal parameters to be specifi ed in the equivalent circuits of a synchronous machine use data from several sources; e.g., short circuit tests or standstill frequency response (SSFR) tests [8]. The diagram shown in Figure 1.6 illustrates the determination of the electrical param-eters of a synchronous machine from SSFR tests. Low-voltage frequency response tests at standstill are becoming a widely used alternative to short circuit tests due to their advan-tages: they can be performed either in the factory or in site at a relatively low cost, equiva-lent circuits of high order can be derived, identifi cation of fi eld responses is possible. The equivalent circuits depicted in Figure 1.6 are valid to represent a synchronous machine in
A
Physical properties (wire and soil resistivity)
Line constants routine
Model for lightning transients
Re
zt, τt
zt, τt (Z) (Y) Conductors
Shield wires Model for switching transients
Z0 t0 R0
Z1 t1 R1
Z1 t1 R1
Modes
Steady-state and low-frequency model
Corona (v) Re B C Coupledphases 3 3 2 2 1 3 2 (T) (T) 1 1 3 2 1 FIGURE 1.5
Application of a line constants routine to obtain overhead line models. (From Martinez, J.A. et al., IEEE Power Energy Mag., 3, 16, 2005. With permission.)
low-frequency transients; e.g., transient stability studies. However, the information pro-vided by SSFR tests can be also used to obtain more complex equivalent circuits.
There are two important exceptions which are worth mentioning: circuit breakers and arresters. The approaches usually followed for the determination of parameters to be specifi ed in the most common models proposed for these two components are discussed below.
The currently available approach for accurate analysis of a circuit breaker per-•
formance during fault clearing is based on using arc equations. Such black-box model equations are able to represent, with suffi cient precision, the dynamic con-ductance of the arc and predict thermal reignition [9]. Although some typical parameters are available in the literature, there are no general purpose methods for arc model parameter evaluation. In some case studies, it becomes compulsory to perform or require specifi c laboratory tests for establishing needed parameters. The detailed arc model is useful for predicting arc quenching and arc instability. In the case of transient recovery voltage studies, a simple ideal switch model can be suffi cient.
The metal oxide surge arrester is modeled using a set of nonlinear exponential •
equations. The original data is deduced from arrester geometry and manufac-turer’s data [10]. Techniques based on factory measurements for determination of parameters have also been developed. A combination of arrester equations with inductances and resistances can be used to derive an accurate arrester model for both switching and lightning surges.
Standstill frequency response tests (IEEE std. 115) Oscillator Frequency response analyzer Oscillator Frequency response analyzer Procedure for parameter identification
d-axis equivalent circuit
q-axis equivalent circuit Ra Lad ωλq ωλd L1d R1d Rf Ra Laq L1q L2q R2q R1q Vf + – + – + – Lfℓ Lf1d Lℓ Lℓ Power amplifier a b c Shunt Shunt Field Field Field Field c b a Power amplifier iarm var m iarm var m Oscillator Frequency response analyzer Shuntb c a Shunt c b a Power amplifier Power amplifier iarm ifd Oscillator Frequency response analyzeriarm efd FIGURE 1.6
Determination of the electrical parameters of a synchronous machine from SSFR tests. (From Martinez, J.A. et al., IEEE Power Energy Mag., 3, 16, 2005. With permission.)
1.4 Scope of the Book
This book presents techniques and methods for parameter determination of six basic power components: overhead line, insulated cable, transformer, synchronous machine, surge arrester, and circuit breaker.
The role that each of these components plays in a power system is different, so it is their design, functioning, and behavior, as well as the approaches to be considered when deter-mining parameters that are adequate for transient models. Synchronous machines are the most important components in generation plants; lines, cables, and transformers transmit and distribute the electrical energy; surge arresters and circuit breakers protect against overvoltages and overcurrents, respectively, although circuit breakers can make and break currents under both normal and abnormal (e.g., faulted) conditions, and are the origin of switching transient caused by opening and closing operations.
There are, however, some similarities between some of these components: lines and cables are both aimed at playing the same role in distribution and transmission networks; transformers and rotating machines are made of windings whose behavior during high-frequency transients (several hundreds of kHz and above) is very similar. Consequently, similar approach can be applied to determine their parameters for transient models: the most common approach for parameter determination of lines and cables at any frequency range is based on the geometry arrangement of conductors and insulation; transformer and rotating machine parameters for high-frequency models are also obtained from the geometry of windings, magnetic cores, and insulation.
A single chapter has been dedicated to any of the aforementioned components. Although there are some exceptions, the organization is similar in all the chapters: guidelines for rep-resentation of the component in transient simulations, procedures for parameter determi-nation, and some practical examples are common features of most chapters. It is important to keep in mind that although this is a book related to transient analysis, and some tran-sient cases are analyzed in all chapters, most examples are mainly addressed to emphasize the techniques and conversion procedures that can be used for parameter determination.
The various approaches that can and often must be used to represent a power component, when considering transients with different frequency ranges, do not require a parameter specifi cation with the same accuracy and detail. The literature on modeling guidelines can help users of transient tools to choose the most adequate representation and advice about the parameters that can be of paramount importance, see Section 1.2.
Another important aspect is the usefulness of international standards, basically those developed by IEEE and IEC. As discussed in the previous section, parameter to be speci-fi ed in transient models can be obtained from factory tests. Setups and measurements to be performed in these tests are specifi ed in standards, although present standards do not cover the testing setups and the requirements that are needed to obtain parameters for any frequency range. In addition, it is important to make a distinction between test procedures established in standards, the determination of characteristic values derived from those measurements and the calculation/estimation of parameters to be specifi ed in mathemati-cal models, and for which a data conversion procedure can be needed. All these aspects, when required, are covered in the book.
Table 1.3 lists some IEEE and IEC standards that can be useful to understand either fac-tory tests or the guidelines to be used for constructing electromagnetic transient models.
Readers are encouraged to consult standards related to the power components ana-lyzed in this book; they are a very valuable source of information, regularly revised and
T ABLE 1.3 Som e I n ter n at ion a l S ta n da rd s Component IEEE Standards IEC Standards Over head line
IEEE Std. 738-1993, IEEE Standar
d for Calculating the Curr
ent–T emperatur e Relationship of Bar e Over head Conductors.
IEC 61089, Round wir
e concentric lay over
head electrical
stranded conductors, 1991.
IEEE Std. 1243-1997, IEEE Guide for Impr
oving the Lightning Performance of
T
ransmission Lines.
IEC 61597, Over
head electrical conductors—Calculation
methods for stranded bar
e conductors, 1997.
IEEE Std. 1410–1997, IEEE Guide for Impr
oving the Lightning Performance of
Electric Power Over
head Distribution Lines.
Insulated cable
IEEE Std. 575-1988, IEEE Guide for the
Application of Sheath-Bonding Methods
for Single-Conductor Cables and the Calculation of Induced V
oltages and
Curr
ents in Cable Sheaths.
IEC 60141-X, T
ests on oil-fi
lled and gas-pr
essur
e cables and
their accessories, 1980.
IEEE Std. 635–1989, IEEE Guide for Selection and Design of
Aluminum Sheaths
for Power Cables.
IEC 60228, Conductors of insulated cables, 2004.
IEEE Std. 848–1996, IEEE Standar
d Pr
ocedur
e for the Determination of the
Ampacity Derating of Fir
e-Pr
otected Cables.
IEC 60287-X, Electric cables. Calculation of the curr
ent rating,
2001.
IEEE Std. 844-2000, IEEE Recommended Practice for Electrical Impedance, Induction, and Skin Ef
fect Heating of Pipelines and V
essels.
IEC 60840, Power cables with extr
uded insulation and their
accessories for rated voltages above 30
kV (Um = 36
kV) up to
150
kV (Um = 170
kV)—T
est methods and r
equir
ements, 2004.
T
ransformer
IEEE Std. C57.12.00-2000, IEEE Standar
d General Requir
ements for
Liquid-Immersed Distribution, Power
, and Regulating T
ransformers.
IEC 60076-X, Power T
ransformers, 2004.
IEEE Std. C57.12.01-1998, IEEE Standar
d General Requir
ements for Dry-T
ype
Distribution and Power T
ransformers Including Those with Solid-Cast and/or
Resin-Encapsulated W
indings.
IEC 60905, Loading guide for dry-type power transformers, 1987.
IEEE Std. C57.12.90-1999, IEEE Standar
d T
est Code for Liquid-Immersed
Distribution, Power
, and Regulating T
ransformers and IEEE Guide for
Short-Cir
cuit T
esting of Distribution and Power T
ransformers.
IEEE Std. C57.12.91-2001, IEEE Standar
d T
est Code for Dry-T
ype Distribution
and Power T
ransformers.
IEEE Std. C57.123-2002, IEEE Guide for T
ransformer Loss Measur
ement.
Synchr
onous
machine
IEEE Std. 1
110-1991, IEEE Guide for Synchr
onous Generator Modeling
Practices in Stability Studies.
IEC 60034-4, Rotating electrical machines—Part 4: Methods for determining synchr
onous machine quantities fr
om tests, 1985. IEEE Std. 1 15-1995, IEEE Guide: T est Pr ocedur es for Synchr onous Machines.
Metal oxide sur
ge arr
ester
IEEE Std. C62.1
1-1999, IEEE Standar
d for Metal-Oxide Sur
ge
Arr
esters for
Alternating Curr
ent Power Cir
cuits (>1
kV).
IEC 60099-4, Sur
ge arr
esters—Part 4: Metal-oxide sur
ge
arr
esters without gaps for a.c. systems, 2004.
IEEE Std. C62.22-1997, IEEE Guide for the
Application of Metal-Oxide Sur
ge Arr esters for Alternating-Curr ent Systems. Cir cuit br eaker
IEEE Std. C37.04-1999, IEEE Standar
d Rating Str uctur e for AC High-V oltage Cir cuit Br eakers.
IEC 60427, Synthetic testing of high-voltage alternating curr
ent
cir
cuit-br
eakers, 2000.
IEEE Std. C37.09-1999, IEEE Standar
d T est Pr ocedur e for AC High-V oltage Cir cuit Br
eakers Rated on a Symmetrical Curr
ent Basis.
IEC 60694, Common specifi
cations for high-voltage switchgear
and contr
olgear standar
ds, 2002.
ANSI/IEEE C37.081-1981, IEEE Guide for Synthetic Fault T
esting of
AC
High-V
oltage Cir
cuit Br
eakers Rated on a Symmetrical Curr
ent Basis.
IEC 61233, High-voltage alternating curr
ent cir
cuit-br
eakers—
Inductive load switching, 1994.
IEEE Std. C37.083-1999, IEEE Guide for Synthetic Capacitive Curr
ent Switching T ests of AC High-V oltage Cir cuit Br eakers.
IEC 61633, High-voltage alternating curr
ent cir
cuit-br
eakers—
Guide for short cir
cuit and switching test pr
ocedur
es for
metal-enclosed and dead tank cir
cuit-br
eakers, 1995.
Sour
ce:
Martinez, J.A. et al.,
IEEE Power Ener
gy Mag
., 3, 16, 2005. W
T ABLE 1.4 Para me ter D eter m in at ion o f P o w er C o m p one n ts i n Pr es en t Si m u la ti on T o o ls Component Parameter Determination
Status in Current Simulation T
ools
Over
head
line
If only the model of the conductors and shield wir
es is r
equir
ed, parameters ar
e
determined (as a function of fr
equency) fr
om the line geometry and fr
om the
physical pr
operties of the conductors and wir
es. However
, line models in fast-fr
ont
transient simulations must include insulators, towers, footing impedances, or even cor
ona ef fect. A dif fer ent appr
oach can be used for each part, although models based
on geometry and physical pr
operties of each part have been pr
oposed.
Over
head line parameters ar
e usually obtained by means of
a LINE CONST
ANTS supporting r
outine, which is
implemented in most transients simulation tools [7].
Insulated cable
Parameters ar
e determined (as a function of fr
equency) fr
om the cable geometry and
fr
om the physical pr
operties of the dif
fer
ent parts (conductor
, insulation, sheath,
armor) of the cable.
Insulated cable parameters ar
e usually obtained by means
of a CABLE CONST
ANTS supporting r
outine, which is
implemented in most transients simulation tools [1
1].
T
ransformer
Parameters for low- and midfr
equency models ar
e usually derived fr
om nameplate
data and the anhyster
etic curve pr
ovided by the manufactur
er . These values ar e obtained fr om standar d tests. Ther e ar e, however
, transformer models whose
parameters cannot be derived fr
om this standar
d information. Users can also
consider the possibility of estimating parameters fr
om transformer geometry and
physical pr
operties (e.g., magnetic saturation).
A
supporting r
outine is usually available in most transient
tools to obtain an unsaturated transformer model fr
om
nameplate data [6]. Fitting r
outines have been also implemented in some tools
to obtain unsaturated high-fr
equency models fr om fr equency r esponse data. High-fr
equency unsaturated models may be derived fr
om fr
equency r
esponse tests.
Rotating machine
Electrical parameters to be specifi
ed in the low- and midfr
equency model of a
synchr
onous machine ar
e derived fr
om standar
d tests (e.g., open-cir
cuit tests, short
cir
cuit tests, SSFR tests).
Parameters for high-fr
equency transient models (e.g., voltage distribution in stator
winding caused by a steep-fr
onted wave) ar
e derived fr
om machine geometry and
physical pr
operties.
A
supporting r
outine is usually available in most transients
tools to estimate parameters of a low-fr
equency
synchr
onous machine model fr
om values obtained in
standar
d open cir
cuit and short cir
cuit tests.
Fitting r
outines have been also implemented in some tools
to obtain an unsaturated model fr
om a SSFR.
Metal oxide sur
ge
arr
ester
Parameters of a standar
d model, acceptable for a wide range of fr
equencies, can be
derived fr
om manufactur
er
’s data and the arr
ester geometry
.
No arr
ester model has been implemented in any transient
tool, although users can cr
eate a module based on
standar d models. A simple iterative pr ocedur e is usually needed to fi t parameters. Cir cuit br eaker Parameters to be specifi
ed in black-box models, suitable for r
epr
esenting dynamic ar
c
during a thermal br
eakdown, can be derived fr
om cir
cuit tests using an evaluation
or fi
tting
pr
ocedur
e.
Ideal statistical switch models have been implemented in most transient tools. However
, a dynamic ar
c model is
only available in very few tools, although users can cr
eate
modules to r
epr
esent Mayr
- and Cassie-type models.
Sour
ce:
Martinez, J.A.,
IEEE PES General Meeting
, T
ampa, 2007. W
updated. It is fi nally worth mentioning that some standard tests or measurements that can be important for the purpose of this book have not been detailed or even mentioned in any chapter. They are basically aimed at determining physical properties of some materials whose values can be crucial for the transient behavior of some components. For instance, no details are provided about the standard procedure that can be followed for estimating ground resistivity values, whose knowledge is of paramount importance for computing overvoltages caused by lightning.
Sophisticated numerical techniques (e.g., the FEM) have not been included in the book, but they are referred in some chapters. Although the FEM can be used for calculation of parameters and it has been extensively applied to most power components [12–19], it is a time-consuming approach that cannot be used in the transient analysis of many real world applications. There has been, however, a steady progress in both software and hard-ware, and some important experience is already available [20]. The increasing capabilities of simulation tools and computers are a challenge for developers, and more sophisticated and rigorous models will be developed and implemented at lower costs.
Other techniques applied in the estimation or identifi cation of system and component parameters [21], as well as more modern procedures, such as those based on genetic algo-rithms [22,23], or ant colony optimization [24], are out of the scope of this book.
The book includes two appendices. The fi rst appendix provides an update of techniques for identifi cation of linear systems from frequency responses; these techniques can be useful in parameter determination and for fi tting frequency-dependent models of sev-eral components analyzed in this book (e.g., lines, cables, transformers, and synchronous machines). The second appendix reviews simulation tools for electromagnetic transient analysis, considering both off-line and online (real time) tools, and discusses limitations and topics for practical simulation needs. Table 1.4 summarizes the capabilities of pres-ent simulation tools to obtain models considering the differpres-ent approaches for parameter determination.
References
1. J.A. Martinez, J. Mahseredjian, and R.A. Walling, Parameter determination for modeling sys-tem transients, IEEE Power Energy Magazine, 3(5), 16–28, Sepsys-tember/October 2005.
2. CIGRE WG 33.02, Guidelines for Representation of Network Elements when Calculating Transients, CIGRE Brochure 39, 1990.
3. IEC TR 60071-4, Insulation Co-ordination—Part 4: Computational Guide to Insulation Co-ordination and Modeling of Electrical Networks, IEC, 2004.
4. A. Gole, J.A. Martinez-Velasco, and A. Keri (eds.), Modeling and Analysis of Power System Transients Using Digital Programs, IEEE Special Publication TP-133–0, IEEE Catalog No. 99TP133-0, 1999. 5. J.A. Martinez, Parameter determination for power systems transients, IEEE PES General Meeting,
Tampa, June 24–28, 2007.
6. J.A. Martinez, R. Walling, B. Mork, J. Martin-Arnedo, and D. Durbak, Parameter determination for modeling systems transients. Part III: Transformers, IEEE Transactions on Power Delivery, 20(3), 2051–2062, July 2005.
7. J.A. Martinez, B. Gustavsen, and D. Durbak, Parameter determination for modeling systems transients. Part I: Overhead lines, IEEE Transactions on Power Delivery, 20(3), 2038–2044, July 2005.
8. J.A. Martinez, B. Johnson, and C. Grande-Morán, Parameter determination for modeling sys-tems transients. Part III: Rotating Machines, IEEE Transactions on Power Delivery, 20(3), 2063– 2072, July 2005.
9. J.A. Martinez, J. Mahseredjian, and B. Khodabakhchian, Parameter determination for mod-eling systems transients. Part VI: Circuit breakers, IEEE Transactions on Power Delivery, 20(3), 2079–2085, July 2005.
10. J.A. Martinez and D. Durbak, Parameter determination for modeling systems transients. Part V: Surge arresters, IEEE Transactions on Power Delivery, 20(3), 2073–2078, July 2005.
11. B. Gustavsen, J.A. Martinez, and D. Durbak, Parameter determination for modeling systems transients. Part II: Insulated cables, IEEE Transactions on Power Delivery, 20(3), 2045–2050, July 2005.
12. F. Piriou and A. Razek, Calculation of saturated inductances for numerical simulation of syn-chronous machines, IEEE Transactions on Magnetics, 19(6), 2628–2631, November 1983.
13. M.P. Krefta and O. Wasynezuk, A fi nite element based state model of solid rotor synchronous machines, IEEE Transactions on Energy Conversion, 2(1), 21–30, March 1987.
14. K. Shima, K. Ide, M. Takahashi, Y. Yoshinari, and M. Nitobe, Calculation of leakage induc-tances of a salient-pole synchronous machine using fi nite elements, IEEE Transactions on Energy Conversion, 14(4), 1156–1161, December 1999.
15. K.J. Meessen, P. Thelin, J. Soulard, and E.A. Lomonova, Inductance calculations of permanent-magnet synchronous machines including fl ux change and self- and cross-saturations, IEEE Transactions on Magnetics, 44(10), 2324–2331, October 2008.
16. O. Moreau, R. Michel, T. Chevalier, G. Meunier, M. Joan, and J.B. Delcroix, 3-D high frequency computation of transformer R–L parameters, IEEE Transactions on Magnetics, 41(5), 1364–1367, May 2005.
17. S.V. Kulkarni, J.C. Olivares, R. Escarela-Perez, V.K. Lakhiani, and J. Turowski, Evaluation of eddy current losses in the cover plates of distribution transformers, IEE Proceedings—Science, Measurement and Technology, 151(5), 313–318, September 2004.
18. Y. Yin and H.W. Dommel, Calculation of frequency-dependent impedances of underground power cables with fi nite element method, IEEE Transactions on Magnetics, 25(4), 3025–3027, July 1989.
19. H. Nam O, T.R. Blackburn, and B.T. Phung, Modeling propagation characteristics of power cables with fi nite element techniques and ATP, AUPEC 2007, December 9–12, 2007.
20. B. Asghari, V. Dinavahi, M. Rioual, J.A. Martinez, and R. Iravani, Interfacing techniques for electromagnetic fi eld and circuit simulation programs, IEEE Transactions on Power Delivery, 24(2), 939–950, April 2009.
21. A. van den Bos, Parameter Estimation for Scientists and Engineers, Wiley, Chichester, 2007.
22. R. Escarela-Perez, T. Niewierowicz, and E. Campero-Littlewood, Synchronous machine parameters from frequency-response fi nite-element simulations and genetic algorithms, IEEE Transactions on Energy Conversion, 16(2), 198–203, June 2001.
23. B. Abdelhadi, A. Benoudjit, and N. Nait-Said, Application of genetic algorithm with a novel adaptive scheme for the identifi cation of induction machine parameters, IEEE Transactions on Energy Conversion, 20(2), 284–291, June 2005.
24. L. Sun, P. Qu, Q. Huang, and P. Ju, Parameter identifi cation of synchronous generator by using ant colony optimization algorithm, 2nd IEEE Conference on Industrial Electronics and Applications, ICIEA 2007, Heilongjiang, China, pp. 2834–2838, May 23–25, 2007.
17
Overhead Lines
Juan A. Martinez-Velasco, Abner I. Ramirez, and Marisol Dávila
CONTENTS
2.1 Introduction ... 18 2.2 Phase Conductors and Shield Wires ... 20 2.2.1 Line Equations ... 21 2.2.2 Calculation of Line Parameters ... 23 2.2.2.1 Shunt Capacitance Matrix ... 23 2.2.2.2 Series Impedance Matrix ... 24 2.2.3 Solution of Line Equations ... 28 2.2.4 Solution Techniques ... 29 2.2.4.1 Modal-Domain Techniques ... 29 2.2.4.2 Phase-Domain Techniques ... 31 2.2.4.3 Alternate Techniques ... 33 2.2.5 Data Input and Output ... 33 2.3 Corona Effect ...43 2.3.1 Introduction ...43 2.3.2 Corona Models ... 45 2.4 Transmission Line Towers ... 50 2.5 Transmission Line Grounding ... 61 2.5.1 Introduction ... 61 2.5.2 Grounding Impedance ... 66 2.5.2.1 Low-Frequency Models ... 66 2.5.2.2 High-Frequency Models ... 67 2.5.2.3 Discussion ... 68 2.5.3 Low-Frequency Models of Grounding Systems ... 68 2.5.3.1 Compact Grounding Systems ... 68 2.5.3.2 Extended Grounding Systems ... 71 2.5.3.3 Grounding Resistance in Nonhomogeneous Soils ... 72 2.5.4 High-Frequency Models of Grounding Systems ... 74 2.5.4.1 Distributed-Parameter Grounding Model ... 74 2.5.4.2 Lumped-Parameter Grounding Model ...77 2.5.4.3 Discussion ... 78 2.5.5 Treatment of Soil Ionization ... 81 2.5.6 Grounding Design ... 87 2.6 Transmission Line Insulation ... 89 2.6.1 Introduction ... 89 2.6.2 Defi nitions ... 90 2.6.2.1 Standard Waveshapes ... 90
2.6.2.2 Basic Impulse Insulation Levels ... 91 2.6.2.3 Statistical/Conventional Insulation Levels ... 91 2.6.3 Fundamentals of Discharge Mechanisms ... 93 2.6.3.1 Description of the Phenomena ... 93 2.6.3.2 Physical–Mathematical Models ... 93 2.6.4 Dielectric Strength for Switching Surges ... 98 2.6.4.1 Introduction ... 98 2.6.4.2 Switching Impulse Strength ... 102 2.6.4.3 Phase-to-Phase Strength ... 104 2.6.5 Dielectric Strength under Lightning Overvoltages ... 111 2.6.5.1 Introduction ... 111 2.6.5.2 Lightning Impulse Strength ... 112 2.6.5.3 Conclusions ... 115 2.6.6 Dielectric Strength for Power-Frequency Voltage ... 123 2.6.7 Atmospheric Effects ... 125 References ... 128
2.1 Introduction
Results derived from electromagnetic transients (EMT) simulations can be of vital impor-tance for overhead line design. Although the selection of an adequate line model is required in many transient studies (e.g., power quality, protection, or secondary arc stud-ies), it is probably in overvoltage calculations where adequate and accurate line models are crucial.
Voltage stresses to be considered in overhead line design are [1–3] 1. Normal power-frequency voltage in the presence of contamination
2. Temporary (low-frequency) overvoltages, produced by faults, load rejection, or ferroresonance
3. Slow front overvoltages, as produced by switching or disconnecting operations. 4. Fast front overvoltages, generally caused by lightning fl ashes
For some of the required specifi cations, only one of these stresses is of major importance. For example, lightning will dictate the location and number of shield wires and the design of tower grounding. The arrester rating is determined by temporary overvolt-ages, while the type of insulators will be dictated by the contamination. However, in other specifi cations, two or more of the overvoltages must be considered. For example, switching overvoltages, lightning, or contamination may dictate the strike distances and insulator string length. In transmission lines, contamination may determine the insula-tor string creepage length, which may be longer than that obtained from switching or lightning overvoltages.
In general, switching surges are important only for voltages of 345 kV and above; for lower voltages, lightning dictates larger clearances and insulator lengths than switching overvoltages do. However, this may not be always true for compact designs [3].
As a rule of thumb, distribution overhead line design is based on lightning stresses. By default, it is assumed that a distribution line fl ashovers every time it is impacted by
a lightning stroke. In addition, the selected distribution line insulation level is usually the highest one of standardized levels. Except when calculating overvoltages caused by nearby strokes to ground, there is little to do with EMT simulation of distribution lines in insulation coordination studies, although those simulations can be very important in other studies (e.g., power quality).
The rest of this chapter has been organized bearing in mind the models and parameters needed in the calculation of voltage stresses that can be required for transmission line design.
As mentioned in Chapter 1, the simulation of transient phenomena may require a rep-resentation of network components valid for a frequency range that varies from DC to several MHz. Although an accurate and wideband representation of a transmission line is not impossible, it is more advisable to use and develop models appropriate for a specifi c range of frequencies. Each range of frequencies will correspond to a particular transient phenomenon (e.g., models for low-frequency oscillations will be adequate for calculation of temporary overvoltages).
Two types of time-domain models have been developed for overhead lines: lumped- and distributed-parameter models. The appropriate selection of a model depends on the line length and the highest frequency involved in the phenomenon.
Lumped-parameter line models represent transmission systems by lumped R, L, G, and C elements whose values are calculated at a single frequency. These models, known as pi-models, are adequate for steady-state calculations, although they can also be used for transient simulations in the neighborhood of the frequency at which the parameters were evaluated. The most accurate models for transient calculations are those that take into account the distributed nature of the line parameters [4–6]. Two categories can be distin-guished for these models: constant parameters and frequency-dependent parameters.
The number of spans and the different hardware of a transmission line, as well as the models required to represent each part (conductors and shield wires, towers, grounding, and insulation), depend on the voltage stress cause. The following rules summarize the modeling guidelines to be followed in each case (Section 1.1):
1. In power-frequency and temporary overvoltage calculations, the whole trans-mission line length must be included in the model, but only the representation of phase conductors is needed. A multiphase model with lumped and constant parameters, including conductor asymmetry, will generally suffi ce. For transients with a frequency range above 1 kHz, a frequency-dependent model could be needed to account for the ground propagation mode. Corona effect can be also important if phase conductor voltages exceed the corona inception voltage. 2. In switching overvoltage calculations, a multiphase distributed-parameter model
of the whole transmission line length, including conductor asymmetry, is in gen-eral required. As for temporary overvoltages, the frequency dependence of param-eters is important for the ground propagation mode, and only phase conductors need to be represented.
3. The calculation of lightning-caused overvoltages requires a more detailed model, in which towers, footing impedances, insulators, and tower clearances, in addition to phase conductors and shield wires, are represented. However, only a few spans at both sides of the point of impact must be considered in the line model. Since lightning is a fast-front transient phenomenon, a multiphase model with distrib-uted parameters, including conductor asymmetry and corona effect, is required for the representation of each span.
Note that the length extent of an overhead line that must be included in a model depends on the type of transient to be analyzed, or, more specifi cally, on the range of frequencies involved in the transient process. As a rule of thumb, the lower the frequencies, the more length of line to be represented. For low- and mid-frequency transients, the whole line length is included in the model. For fast and very fast transients, a few line spans will usually suffi ce. These guidelines are illustrated in Figure 2.1 and summarized in Table 2.1, which provide modeling guidelines for overhead lines proposed by CIGRE [4], IEEE [5], and IEC [6].
The rest of the chapter is dedicated to analyzing the models to be used in the repre-sentation of each part of a transmission line, as well as the parameters to be specifi ed or calculated for each part.
2.2 Phase Conductors and Shield Wires
Presently, overhead line parameters are calculated using supporting routines available in most EMT programs. The parameters to be calculated depend on the line and ground model to be applied, but they invariably involve the series impedance (longitudinal fi eld effects) and the shunt capacitance (transversal fi eld effects) of the line.
This section deals with, among other aspects, data input that is required for a proper modeling of overhead lines in transient simulations. To fully understand the meaning of the data input/output, the main theory of line modeling is introduced in this chapter. The concepts include the description of the time-domain and frequency-domain line equa-tions, the calculation of the line parameters, and the description of the adopted techniques for solving the line equations.
FIGURE 2.1
Line models for different ranges of frequency. (a) Steady-state and low-frequency transients, (b) switching (slow-front) transients, and (c) lightning (fast-front) transients.
3 (a) 2 1 3 2 1 Coupled phases Grounding impedance Tower Phase conductor Shield wire Stroke I Vs Vs Vc Vt Vc
Multiphase line with coupled phases and frequency-dependent distributed paramters
3 2 1 3 2 1 (c) (b)
