Ln, Hubert
`Ai. Rohrer,
`axal, Ernst
`
`ELEMENTS
`OF POWER
`SYSTEM ANALYSIS
`Fourth Edition
`
`William D. Stevenson, Jr.
`Professor of Electrical Engineering, Emeritus
`North Carolina State University
`
`McGraw-Hill Publishing Company
`New York St. Louis San Francisco Auckland Bogod Caracas
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`
`Titeflex - Exhibit 1030, cover
`
`

`

`This book was set in Times Roman.
`The editor was Frank J. Cerra;
`the production supervisor was Diane Renda.
`The cover was designed by Infield, D'Astolfo Associates.
`
`ELEMENTS OF POWER SYSTEM ANALYSIS
`
`Copyright © 1982, 1975, 1962, 1955 by McGraw-Hill, Inc. All rights reserved.
`Printed in the United States of America. No part of this publication
`may be reproduced, stored in a retrieval system, or transmitted, in any
`form or by any means, electronic, mechanical, photocopying, recording, or
`otherwise, without the prior written permission of the publisher.
`
`1 2 1 3 1 4 1 5 IIDIID 9987654321
`
`Library of Congress Cataloging in Publication Data
`
`Stevenson, William D.
`Elements of power system analysis.
`
`(McGraw-Hill series in electrical engineering.
`Power and energy)
`Includes index.
`1. Electric power distribution. 2. Electric
`power systems.
`I. Title. II. Series.
`TK3001.S85 1982
`621.319
`81-3741
`ISBN 0-07-061278-1 (Text)
`AACR2
`ISBN 0-07-061279-X (Solutions manual)
`
`Chapter 1
`1.1 1
`1.2
`1
`1.3
`1
`1.4
`I
`,1.5
`1.6
`1.7
`1.8
`1.9
`1.10
`
`S
`1
`
`Chapter 2
`2.1
`2.2
`2.3
`2.4
`2.5
`2.6
`2.7
`2.8
`2.9
`2.10
`2.11
`
`I
`S
`I
`
`(
`'I
`I
`
`(
`I
`
`Chapter 3
`3.1 1
`3.2
`F
`3.3 1
`3.4
`I
`
`Titeflex - Exhibit 1030, page i
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`Titeflex - Exhibit 1030, page i
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`

`

`SERIES IMPEDANCE OF TRANSMISSION LINES 39
`
`Alternate layers of wire of a stranded conductor are spiraled in opposite
`directions to prevent unwinding and make the outer radius of one layer coincide
`with the inner radius of the next. Stranding provides flexibility for a large cross-
`sectional area. The number of strands depends on the number of layers and on
`Whether -alr lie strands are the same diameter. The total number of strands in
`concentrically stranded cables, where the total annular space is filled with
`strands of uniform diameter, is 7, 19, 37, 61, 91, or more.
`Figure 3.2 shows the cross section of a typical steel-reinforced aluminum
`cable (ACSR). The conductor shown has 7 steel strands forming a central core,
`around which are two layers of aluminum strands. There are 24 aluminum strands
`in the two outer layers. The conductor stranding is specified as 24 Al/7 St, or
`simply 24/7. Various tensile strengths, current capacities, and conductor sizes are
`obtained by using different combinations of steel and aluminum.
`Appendix Table A.1 gives some electrical characteristics of ACSR. Code
`names, uniform throughout the aluminum industry, have been assigned to each
`conductor for easy reference.
`A type of conductor known as expanded ACSR has a filler such as paper
`separating the inner steel strands from the outer aluminum strands. The paper
`gives a larger diameter (and hence, lower corona) for a given conductivity and
`tensile strength. Expanded ACSR is used for some extra-high-voltage (EHV)
`lines.
`Cables for underground transmission are usually made with stranded copper
`conductors rather than aluminum. The conductors are insulated with oil-
`impregnated paper. Up to voltages of 46 kV the cables are of the solid type
`which means that the only insulating oil in the cable is that which is im-
`pregnated during manufacture. The voltage rating of this type of cable is limited
`by the tendency of voids to develop between the layers of insulation. Voids cause
`early breakdown of the insulation. A lead sheath surrounds the cable which may
`consist of a single conductor or three conductors.
`At voltages from 46 to 345 kV low-pressure oil-filled cables are available.
`Oil reservoirs at intervals along the length of the cable supply the oil to ducts in
`the center of single-conductor cables or to the spaces between the insulated
`conductors of the three-phase type. These conductors are also enclosed in a lead
`sheath.
`
`Figure 3.2 Cross section of a steel-reinforced conductor, 7 steel
`strands, 24 aluminum strands.
`
`tg the line form the
`between conductors
`Lree-phase line form
`ind capacitance are
`wed parameters, as
`
`luctors were usually
`1 copper because of
`onductor compared
`that an aluminum
`the same resistance
`tric flux originating
`urface for the same
`;tor surface and less
`ation produces the
`
`tors are as follows :
`
`than the ordinary
`central core of steel
`has a central core of
`cal-conductor-grade
`
`Titeflex - Exhibit 1030, page 39
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`

`

`Figure 3.8 Single-phase li
`posite conductors.
`
`Conductor X is corn
`the current I/n. Co:
`conductor X, is corm
`the current -1/m. I
`letter D with appropi
`tor X, we obtain for
`
`tp„ = 2 x 10-7 -1- (ln
`
`from which
`
`- 2 x
`
`a = 2
`
`Dividing Eq. (147) b:
`
`1Pct
`La = I/n
`Similarly, the inductz
`
`b
`
`7-
`
`0
`1)
`= I/n
`The average inductor
`
`Conductor X is co]
`filaments had the san
`times the inductance
`tances, but the indu
`inductance. Thus the
`
`Substituting the log
`
`52 ELEMENTS OF POWER SYSTEM ANALYSIS
`
`Substituting Eq. (3.43) in the second term containing I, in Eq. (3.42) and recom-
`bining some logarithmic terms, we have
`
`1 1
`
`+ /3 In D
`
`+ • • • + ln
`
`3
`
`1
`
`Din
`
`tiip= 2 x 101/, In —1 + /2 In 1
`Di 2
`
`+
`
`In
`
`Dip
`+ I
`
`1-"np
`
`ln
`
`D2p
`
`
`DnP
`
`+ / ln D3P ± • • • + /n _j In D(a- 1)P
`D„p
`D,,p
`
`(3 A4 )
`
`Now letting the point P move infinitely far away so that the set of terms contain-
`ing logarithms of ratios of distances from P becomes infinitesimal, since the
`ratios of the distances approach 1, we obtain
`
`= 2 x 10- (/1 ln -,-1 + /2 ln 1 +131n 1 + • • • + /„ ln 1
`Din)
`ri
`D12
`D13
`
`Wbt/m
`
`(3.45)
`
`By letting point P move infinitely far away we have included all the flux linkages
`of conductor 1 in our derivation. Therefore, Eq. (3.45) expresses all the flux
`linkages of conductor 1 in a group of conductors, provided the sum of all the
`currents is zero. If the currents are alternating, they must be expressed as instan-
`taneous currents to obtain instantaneous flux linkages or as complex rms values
`to obtain the rms value of flux linkages as a complex number.
`
`3.9 INDUCTANCE OF COMPOSITE-CONDUCTOR LINES
`
`Stranded conductors come under the general classification of composite conduc-
`tors, which means conductors composed of two or more elements or strands
`electrically in parallel. We are now ready to study the inductance of a transmis-
`sion line composed of composite conductors, but we shall limit ourselves to the
`case where all the strands are identical and share the current equally. The
`method can be expanded to apply to all types of conductors containing strands
`of different sizes and conductivities, but this will not be done here since
`the values of internal inductance of s_pecific conductors are _generally available
`- from the various manufacturers and can Joeiound in handbooks. The method to
`
`be developed indicates the approach to the moreCOinprkited-fioblems of non-
`homogeneous conductors and unequal division of current between strands. The
`method is applicable to the determination of inductance of lines consisting of
`circuits electrically in parallel since two conductors in parallel can be treated as
`strands of a single composite conductor.
`Figure 3.8 shows a single-phase line composed of two conductors. In order
`to be more general, each conductor forming one side of the line is shown as an ,
`arbitrary arrangement of an indefinite number of conductors. The only restric-
`tions are that the parallel filaments are cylindrical and share the current equally.
`
`Titeflex - Exhibit 1030, page 52
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`Titeflex - Exhibit 1030, page 52
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`