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ISBN: 978-1-119-22685-7

CONTENTS

About the Author

Foreword

Preface and Acknowledgments

Chapter 1 Symmetrical Components Using Matrix Methods

1.1 Transformations
1.2 Characteristic Roots, Eigenvalues, and Eigenvectors
1.3 Diagonalization of a Matrix
1.4 Similarity Transformation
1.5 Decoupling a Three-Phase Symmetrical System
1.6 Symmetrical Component Transformation
1.7 Decoupling a Three-Phase Unsymmetrical System
1.8 Clarke Component Transformation
1.9 Significance of Selection of Eigenvectors in Symmetrical Components
References

Chapter 2 Fundamental Concepts of Symmetrical Components

2.1 Characteristics of Symmetrical Components
2.2 Characteristics of Sequence Networks
2.3 Sequence Impedance of Network Components
2.4 Construction of Sequence Networks
2.5 Sequence Components of Transformers
2.6 Example of Construction of Sequence Networks
References

Chapter 3 Symmetrical Components-Transmission Lines and Cables

3.1 Impedance Matrix of Three-Phase Symmetrical Line
3.2 Three-Phase Line with Ground Conductors
3.3 Bundle Conductors
3.4 Carson's Formula
3.5 Capacitance of Lines
3.6 Cable Constants
3.7 EMTP Models
3.8 Effect of Harmonics on Line Models
3.9 Transmission Line Equations with Harmonics
References

Chapter 4 Sequence Impedances of Rotating Equipment and Static Load

4.1 Synchronous Generators
4.2 Induction Motors
4.3 Static Loads
4.4 Harmonics and Sequence Components
References
Further Reading

Chapter 5 Three-Phase Models of Transformers and Conductors

5.1 Three-Phase Models
5.2 Three-Phase Transformer Models
5.3 Conductors
References

Chapter 6 Unsymmetrical Fault Calculations

6.1 Line-to-Ground Fault
6.2 Line-to-Line Fault
6.3 Double Line-to-Ground Fault
6.4 Three-Phase Fault
6.5 Phase Shift in Three-Phase Transformer Windings
6.6 Unsymmetrical Long Hand Fault Calculations
6.7 Open Conductor Faults
6.8 Short-Circuit Calculations with Bus Impedance Matrix
6.9 System Grounding
References
Further Reading

Chapter 7 Some Limitations of Symmetrical Components

7.1 Phase Coordinate Method
7.2 Three-Phase Models
7.3 Multiple Grounded Systems
References

Index

IEEE Press Series on Power Engineering

EULA

List of Illustrations

Chapter 1

Figure 1.1 (a) An unbalanced three-phase section of a transmission line; (b) a balanced section.
Figure 1.2 Decoupled system of Figure 1.1b.

Chapter 2

Figure 2.1 Conversion of an unbalanced system into symmetrical components and from symmetrical components back to original unbalanced system. (a) Original unbalanced system; (b) conversion to symmetrical components.
Figure 2.2 Positive, negative, and zero sequence network representation.
Figure 2.3 (a) Core form of three-phase transformers, flux paths for phase, and zero sequence currents; (b) shell form of three-phase transformers.
Figure 2.4 (a) Equivalent zero sequence circuit for delta-wye transformer, neutral solidly grounded; (b) zero sequence circuit of delta-wye transformer, wye neutral isolated.
Figure 2.5 Phenomenon of oscillating neutral in wye-wye connected transformer, both neutrals isolated.
Figure 2.6 Current distribution in a delta-delta system with zigzag grounding transformer for a single line-to-ground fault; (b) zigzag transformer winding connections.
Figure 2.7 Wye equivalent circuit of a three-winding transformer.
Figure 2.8 (a) Wye connected autotransformer equivalent circuit; (b, c) equivalent positive sequence and zero sequence circuits.
Figure 2.9 (a) A single-line diagram of a distribution system; (b–d) positive, negative, and zero sequence networks for a fault at point F.
Figure 2.10 Equivalent circuit reduction for the positive sequence network for a fault at point F.
Figure 2.11 Wye-delta and delta-wye transformations of impedances.

Chapter 3

Figure 3.1 Transmission line section with two ground conductors.
Figure 3.2 Transformation of bundle conductors to single conductors.
Figure 3.3 Conductors and their images in the earth, Carson's formula.
Figure 3.4 Distribution line configuration for calculations of line parameters.
Figure 3.5 Configuration of bundle conductors, Example 3.2.
Figure 3.6 Calculations of line capacitances, conductors mirror images, spacing, and charges.
Figure 3.7 (a) Capacitances of a three-phase line; (b) equivalent positive, negative, and zero sequence networks of capacitances.
Figure 3.8 (a) construction of a concentric neutral cable; (b) configuration for calculations of series inductance.
Figure 3.9 (a) Capacitance of a three-conductor cable; (b, c) equivalent circuits; (d) final capacitance circuit.
Figure 3.10 Explanation of J. Marti frequency dependent (FD) model of transmission line in EMTP; (a) voltage source connected through matching impedance to node m; (b) RC network; (c) circuit with equivalent resistance after applying implicit integration.
Figure 3.11 (a) frequency scan of a 400 kV line with bundle conductors; (b) corresponding phase angle of the impedance modulus; (c, d) corresponding plots with line loaded to 200 MVA, 0.9 lagging power factor. There are no harmonic generating loads, see text.

Chapter 4

Figure 4.1 Ratio K for average loss to maximum loss based on harmonic pair.
Figure 4.2 (a) Flow of zero sequence current in the phases and neutral of a synchronous generator; (b) equivalent circuit.
Figure 4.3 Sequence components of synchronous generator impedances.
Figure 4.4 Terminal short-circuit of a synchronous generator, ac component decay; subtransient, transient, and steady-state currents.
Figure 4.5 Short-circuit current profile of a synchronous generator with decaying ac and dc components.
Figure 4.6 Park's transformation—forward and reverse.
Figure 4.7 Equivalent circuit of an induction motor.
Figure 4.8 Equivalent circuit of an induction motor for negative sequence currents.

Chapter 5

Figure 5.1 Circuit of a grounded wye-delta transformer with voltage and current relations for derivation of connection matrix.
Figure 5.2 Circuit of a grounded wye-delta transformer with voltage and current relations for derivation of connection matrix.
Figure 5.3 Sequence impedances of a wye-grounded delta two-winding transformer.
Figure 5.4 (a) Mutual couplings between a line section with ground wire in the impedance form; (b) transformed network in impedance form; (c) equivalent admittance network of a line section.
Figure 5.5 (a) capacitance in a three-phase circuit; (b) equivalent current injections.

Chapter 6

Figure 6.1 (a) Line-to-ground fault in a three-phase system; (b) line-to-ground fault equivalent circuit; (c) sequence network interconnections.
Figure 6.2 (a) Line-to-line fault in a three-phase system; (b) line-to-line fault equivalent circuit; (c) sequence network interconnections.
Figure 6.3 (a) Double line-to-ground fault in a three-phase system; (b) double line-to-ground fault equivalent circuit; (c) sequence network interconnections.
Figure 6.4 (a) Three-phase fault; (b) equivalent circuit; (c) sequence network.
Figure 6.5 Winding connections and phase displacements of voltage vectors for transformers; (a) high-voltage winding in wye and low-voltage winding in delta; (b) high-voltage winding in delta and low-voltage winding in wye connection.
Figure 6.6 Phase displacements and terminal markings in three-phase transformers according to ANSI/IEEE standard.
Figure 6.7 Transformer vector groups, winding connections, and vector (phasor) diagrams.
Figure 6.8 Balanced delta connected load on an unbalanced three-phase power supply.
Figure 6.9 Three-phase transformer connections and fault current distributions for secondary faults.
Figure 6.10 A single line diagram of a small power system for Example 6.2.
Figure 6.11 Sequence network connections for single line-to-ground fault (Example 6.2).
Figure 6.12 Fault current distribution shown in a three-line diagram (Example 6.2).
Figure 6.13 (a) Two conductor open series fault; (b) connection of sequence networks.
Figure 6.14 (a) One conductor open series fault; (b) connection of sequence networks.
Figure 6.15 Equivalent circuit of an open circuit fault in Example 6.3.
Figure 6.16 Positive and zero sequence networks for Example 6.4.
Figure 6.17 Methods of system grounding.
Figure 6.18 (a, b) stray currents under no-fault conditions; (c) flow of stray capacitance current and resistor current for a single line-to-ground fault in phase a; (d) voltage to ground; (e) summation of resistive and capacitive currents.

Chapter 7

Figure 7.1 (a) Three-phase network representation, primitive impedance matrix; (b, c) single-line representation of three-phase network.
Figure 7.2 Circuit of a three-phase element which can represent a generator.
Figure 7.3 (a) Norton equivalent circuit of a generator for distribution systems; (b) circuit for load flow calculations.
Figure 7.4 Representation of a load window.
Figure 7.5 (a) Three-phase load representation; (b) equivalent circuit injection.
Figure 7.6 Typical grounding practice for wye-service entrance served by multiple grounded, medium voltage system in North American systems.
Figure 7.7 Typical grounding practice for industrial systems, the transformer neutral is only grounded at the source in North American Systems.
Figure 7.8 An equivalent circuit of a multiple grounded system for a line-to-ground fault at the remoter node.
Figure 7.9 Line-to-ground overvoltage calculation, matrix method verses symmetrical component method; see text.

About the Author

J. C. DAS is an independent consultant at Power System Studies Inc., Snellville, GA. Earlier, he headed the electrical power systems department at Amec Foster Wheeler for the last 30 years. He has varied experience in the utility industry, industrial establishments, hydroelectric generation, and atomic energy. He is responsible for power system studies, including short-circuit, load flow, harmonics, stability, arc-flash hazard, grounding, switching transients, and protective relaying. He conducts courses for continuing education in power systems and has authored or coauthored about 68 technical publications nationally and internationally. He is author of the books:

Arc Flash Hazard Analysis and Mitigation, IEEE Press, 2012.
Power System Harmonics and Passive Filter Designs, IEEE Press, 2015.
Transients in Electrical Systems: Analysis, Recognition, and Mitigation, McGraw-Hill, 2010.
Power System Analysis: Short-Circuit Load Flow and Harmonics, Second Edition, CRC Press, 2011.

These books provide extensive converge, running into more than 3000 pages, and are well received in the technical circles. His interests include power system transients, EMTP simulations, harmonics, passive filter designs, power quality, protection, and relaying. He has published more than 200 electrical power system study reports for his clients.

Mr. Das is a life fellow of Institute of Electrical and Electronics Engineers, IEEE (USA); a member of the IEEE Industry Applications and IEEE Power Engineering societies; a fellow of Institution of Engineering Technology (UK); a life fellow of the Institution of Engineers (India); a member of the Federation of European Engineers (France); a member of CIGRE (France); etc. He is a registered Professional Engineer in the States of Georgia and Oklahoma, a Chartered Engineer (C. Eng.) in the United Kingdom, and a European Engineer (Eur. Ing.) in the European Union (EU). He received meritorious award in engineering, IEEE Pulp and Paper Industry in 2005.

He received MSEE degree from Tulsa University, Tulsa, Oklahoma; and BA (advanced mathematics) and BEE degrees from Punjab University, India.

Foreword

THIS BOOK BY J. C. DAS OFFERS AN IN-DEPTH, practical, yet intellectually appealing treatment of symmetrical components not seen since the late Paul M. Anderson's classic, Analysis of Faulted Power Systems, which was first published in 1995 by the Wiley-IEEE Press in the Power Engineering Series. The present book leverages the author's well over 30 years of experience in power system studies, and continues in his same tradition of attention to details, which should appeal to those professionals who benefitted from his writing style demonstrated in his four earlier books. The subject is taught at the undergraduate and graduate courses in most universities with a power systems option.

The advent of the symmetrical components concept is due to the Westinghouse electrical engineer Charles LeGeyt Fortescue, who was born in 1876 at York Factory in Manitoba, Canada, who became the first electrical engineer to graduate from Queen's University at Kingston in Ontario, Canada, in 1898. In 1918, Fortescue contributed an 88 page, now classic, remarkable paper by the title “Method of Symmetrical Coordinates Applied to the Solution of Polyphase Networks” in the Transactions of the American Institute of Electrical Engineers (AIEE), one of the two predecessors of present day IEEE. This breakthrough is due to Fortesue's investigations of railway electrification problems which began in 1913. Following the paper's publication, the earlier name “Symmetrical Coordinates” was changed to “Symmetrical Components” and the approach gained in popularity ever since it was disclosed as an indispensable method of dealing with unbalanced three-phase operation problems of electric power systems. A thorough understanding of the application of symmetrical components is required for proper design of electric power protection systems.

Chapter 1 uses matrix algebra to demonstrate the non-uniqueness of symmetrical component transformations. Chapter 2 treats sequence impedances, their networks, and their reduction. Chapters 3 and 4 discuss symmetrical component applications in generating models for transmission lines, cables, synchronous generators, and induction motors. Chapter 3 notes that much of the theoretical underpinnings of the area discussed should be reviewed elsewhere. Prior to discussing three-phase models of two-winding three-phase transformers and conductors, Chapter 5 begins by advising the reader to study this chapter along with Chapter 7. Chapter 6 covers unsymmetrical shunt and series faults and also calculations of overvoltages at the fault plane.

M. E. El-Hawary

Preface and Acknowledgments

THIS SHORT BOOK consisting of seven chapters attempts to provide a clear understanding of the theory of the symmetrical component transformation and its applications in power system modeling.

Chapter 1 takes a mathematical approach to document that the symmetrical component eigenvectors are not unique and one can choose arbitrary vectors meeting the constraints, but these will not be very meaningful in the transformation—thus selection of vectors as they are forms a sound base of the transformation. This is followed by Chapter 2, which details the concepts of sequence impedances, their models, formation of sequence impedance networks and their reduction. Chapters 3 and 4 are devoted to symmetrical component applications in generating the models for transmission lines, cables, synchronous generators, and induction motors. Chapters 5 and 7 are meant to be read together and describe three-phase models and phase-coordinate method of solution where the phase-unbalance in the power system cannot be ignored and symmetrical components cannot be applied. Chapter 6 covers unsymmetrical shunt and series faults and also calculations of overvoltages at the fault point (COG). It has a worked out longhand example to illustrate the complexity of calculations even in a simple electrical distribution system. This is followed with the matrix methods of solution which have been adopted for calculations on digital computers. The author is thankful and appreciates all the cooperation and help received from Ms. Mary Hatcher, Wiley-IEEE and her staff in completing this publication. She rendered similar help and cooperation for the publications of author's other two books by IEEE Press (see Author's profile). An author cannot expect anything better than the help and cooperation rendered by Ms. Mary Hatcher.

The authors special thanks go to Dr. M.E. El-Hawary, Professor of Electrical and Computer Engineering, Dalhousie University, Canada for writing the Foreword to this book. He is a renowned authority on Electrical Power System; the author is grateful to him, and believes that this Foreword adds to the value and the marketability of the book.

J. C. Das

George W. Arnold	Xiaoou Li	Ray Perez
Giancarlo Fortino	Vladimir Lumelsky	Linda Shafer
Dmitry Goldgof	Pui-In Mak	Zidong Wang
Ekram Hossain	Jeffrey Nanzer	MengChu Zhou