Symmetrical Components for Power Systems Engineering

J. Lewis Blackburn

• Publisher: CRC Press
• Year: 2017
• ISBN:
  9781351838245
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Electrical Engineering & Telecommunications
• Index: Technology & Engineering

Emphasizing a practical conception of system unbalances, basic circuits, and calculations, this essential reference/text presents the foundations of symmetrical components with a review of per unit (percent), phasors, and polarity--keeping the mathematics as simple as possible throughout. According to IEEE Electrical Insulation Magazine, this book "…provides students and practicing engineers with a fundamental understanding of the method of symmetrical components and its applications in three-phase electrical systems...A useful feature of this book...is the incorporation of numerous examples in the text and 30 pages of problems."

1
Introduction and Historical Background

1.1 Introduction and General Aims

The method of symmetrical components provides a practical technology for understanding and analyzing electric power system operation during unbalanced conditions. Typical unbalances are those caused by faults between the phases and/or ground (phase to phase, double phase to ground, phase to ground), open phases, unbalanced impedances, and combinations of these. Balanced three-phase faults are included. Also, many protective relays operate from symmetrical component quantities. For example, all ground relays operate from zero-sequence quantities, which are normally not present in the power system. Therefore, a good understanding of this subject is of great importance and is a very important “tool” in system protection.

This discussion is for three-phase electric systems, which are assumed from a practical standpoint to be balanced or symmetrical up to a point or area of unbalance. A normal area or point of unbalance in a power system will usually be down in the low-voltage or distribution area, where single-phase loads are connected or where nonsymmetrical equipment is used.

In a symmetrical or balanced system the source voltages (generators) are equal in magnitude and are in phase, with their three phases displaced 120° in phase relations. Also, the impedances of the three-phase circuits and equipment are of equal magnitude and phase angle.

In a sense, symmetrical components can be called the “language” of the protection engineer or technician. Its great value is both in thinking or visualizing system unbalances, and as a means of detailed analysis of them from the system parameters. In this simile it is like a language, in that it requires experience and practice for easy access and application. Fortunately, faults and unbalances occur infrequently, and many do not require detailed analysis. Thus it becomes difficult to “practice the language.” This difficulty has increased significantly with the ready availability of fault studies via computers. These provide rapid access to voluminous data, frequently with little user understanding of the information, the background, or the method that provides the data.

The goal of this book is to provide (1) a practical understanding and appreciation of the fundamentals, basic circuits, and calculations; (2) an overview directed toward a clear visualization of faults and system unbalances; (3) where access to a computer or a proper fault program may not be available, the means of making fault and unbalanced calculations “by hand”; and (4) determination of the necessary system parameters for calculations or computer programs. Throughout, the math is kept as simple as possible.

Although computers and hand calculators provide great accuracy, fault and unbalance calculations will not be the same as will be experienced for real-life occurrences. The principal reasons are (1) approximations involved in the determination of many of the system parameters, especially for lines; (2) parameter changes resulting from temperature variations; (3) highly variable and relatively unknown fault resistance; and (4) variable generator impedances with time. Thus high accuracy for fault and unbalance calculations is not possible and should not be expected, and from a practical standpoint is not really needed.

Surprisingly, the calculated values often are quite close to actual values, and in general the calculations are very practical for system design and equipment selection and for the application and setting of protective relays. The general practice is to calculate only solid faults. This provides maximum values important for system design. Fault resistance will reduce the current values. However, as indicated above, it is highly variable and relatively unknown. Thus it becomes impossible practically to select the “correct” value for fault and unbalance studies.

In protective relaying, using solid fault data, overcurrent relays should be set so that their minimum pickup is at least one-half of the minimum fault currents and, hopefully, more sensitive as long as the phase relays do not operate on the maximum short-time load current and ground relays do not operate on the maximum tolerable zero-sequence unbalance. This generally provides good protection for system arcing faults, except for possible very high resistance faults in low-voltage distribution.

The method of symmetrical components is applicable to multiphase systems, but only symmetrical components for three-phase systems are covered in this book.

1.2 Historical Background

The method of symmetrical components was developed late in 1913 by Charles L. Fortescue of Westinghouse when investigating mathematically the operation of induction motors under unbalanced conditions. At the 34th Annual Convention of the AIEE on June 28, 1918, in Atlantic City, he presented a paper entitled “Method of Symmetrical Co-ordinates Applied to the Solution of Polyphase Networks.” This was published in the AIEE Transactions, Volume 37, Fart II, pages 1027–1140. This published paper was 89 pages (5 by 8 in.) in length, with 25 pages of discussion by six well-known “giants” of electric power engineering: J. Slepian, C. P. Steinmetz, V. Karapetoff, A. M. Dudley, Charles F. Scott, and C. O. Mailloux.

Application of the method “to the study of short circuits and system disturbances,” and the method as we know it today, was made by C. F. Wagner and R. D. Evans. They began a series of articles in the Westinghouse magazine The Electric Journal that ran for 10 issues, from March 1928 through November 1931. This series was enlarged by the two authors and published in the classic and still very useful book Symmetrical Components, published by McGraw-Hill Book Company, New York, 1933.

Just as the Wagner–Evans book was about to be printed, another Westinghouse engineer, W. A. Lewis, developed the concept of splitting the line reactance into components: one associated with the conductor reactance (X_a), one associated with the spacing to the return conductor(s) (X_d), and one associated with the depth of the earth return (X_e). This was added to the book as an appendix (VII) and is covered in this book in Chapter 11.

Another Westinghouse engineer, E. L. Harder, provided very useful tables of fault and unbalance connections that were presented in his paper “Sequence Network Connections” published in the December 1937 issue of The Electric Journal. This is covered in Chapter 5.

During this time, Edith Clarke of General Electric was developing notes and lecturing in this area. However, formal publication of her work did not occur until 1943.

On a personal note it was my privilege to meet Dr. Fortescue and Mr. Evans several times during my student assignment on the AC Network Analyzer in 1937.1 knew C. F. Wagner through his son Chuck (C. L. Wagner). Chuck, past President of the IEEE Power Engineering Society, and I have had a long association together in protective relaying.

I took my first course in symmetrical components under Dr. Lewis and then applied it while working for Dr. Harder. In fact, Ed Harder was instrumental in my obtaining a permanent job in relaying. At that point I was totally unfamiliar with relays, but it was a job and in the New York area where I wanted to be. And 55 years have quickly slipped by!

2
Per Unit and Percent Values

2.1 Introduction

Power systems operate at voltages where kilovolt (kV) is the most convenient unit for expressing voltage. Also, these systems transmit large amounts of power, so that kilovolt-ampere (kVA) and megavolt-ampere (MVA) are used to express the total (general or apparent) three-phase power. These quantities, together with kilowatts, kilovars, amperes, ohms, flux, and so on, are usually expressed as a per unit or percent of a reference or base value. The per unit and percent nomenclatures are widely used because they simplify specification and computations, especially where different voltage levels and equipment sizes are involved.

2.2 Per Unit and Percent Definitions

Percent is 100 times per unit. Both are used as a matter of convenience or of personal choice and it is important to designate either percent (%) or per unit (pu).

The per unit value of any quantity is the ratio of that quantity to its base value, the ratio expressed as a nondimensional decimal number. Thus actual quantities, such as voltage (V), current (I), power (P), reactive power (Q), volt-amperes (VA), resistance (R), reactance (X), a...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Table of Contents
• Preface
• 1. Introduction and Historical Background
• 2. Per Unit and Percent Values
• 3. Phasors, Polarity, and System Harmonics
• 4. Basic Fundamentals and the Sequence Networks
• 5. Shunt Unbalance Sequence Network Interconnections
• 6. Fault Calculation Examples for Shunt-Type Faults
• 7. Series and Simultaneous Unbalance Sequence Network Interconnections
• 8. Overview of Sequence Currents and Voltages During Faults
• 9. Transformer, Reactor, and Capacitor Characteristics
• 10. Generator and Motor Characteristics
• 11. Overhead Line Characteristics: Inductive Impedance
• 12. Overhead Line Characteristics: Mutual Impedance
• 13. Overhead Line Characteristics: Capacitive Reactance
• 14. Cable Characteristics
• Problems
• Appendix: Overhead Line Conductor Characteristics
• Bibliography
• Index