Advanced Transmission Line Parameters and Electric and Magnetic Fields Computation Program

Advanced Transmission Line Parameters and

Electric and Magnetic Fields Computation Program

EDSA MICRO CORPORATION 16870 West Bernardo Drive, Suite 330

San Diego, CA 92127 U.S.A.

© Copyright 2008

All Rights Reserved

Version 3.60.00 October 2008

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EDSA MICRO CORPORATION

WARRANTY INFORMATION

There is no warranty, implied or otherwise, on EDSA software. EDSA software is licensed to you as is. This program license provides a ninety (90) day limited warranty on the diskette that contains the program. This, the EDSA User’s Guide, is not meant to alter the warranty situation described above. That is, the content of this document is not intended to, and does not, constitute a warranty of any sort, including warranty of merchantability or fitness for any particular purpose on your EDSA software package. EDSA Micro Corporation reserves the right to revise and make changes to this User's Guide and to the EDSA software without obligation to notify any person of, or provide any person with, such revision or change. EDSA programs come with verification and validation of methodology of calculation based on EDSA Micro Corporation's inhouse software development standards. EDSA performs longhand calculation and checks the programs’ results against published samples. However, we do not guarantee, or warranty, any program outputs, results, or conclusions reached from data generated by any programs which are all sold "as is". Since the meaning of QA/QC and the verification and validation of a program methodology are domains of vast interpretation, users are encouraged to perform their own inhouse verification and validation based on their own inhouse quality assurance, quality control policies and standards. Such operations - performed at the user's expense - will meet the user's specific needs. EDSA Micro Corporation does not accept, or acknowledge, purchase instructions based on a buyer's QA/QC and/or a buyer's verification and validation standards. Therefore, purchase orders instructions are considered to be uniquely based on EDSA's own QA/QC verification and validation standards and test systems. TRADEMARK EDSA is a trademark of EDSA Micro Corporation. COPYRIGHT © Copyright 2001 - 2008 by EDSA Micro Corporation. Please accept and respect the fact that EDSA Micro Corporation has enabled you to make an authorized disk as a backup to prevent losing the contents that might occur to your original disk drive. DO NOT sell, lend, lease, give, rent or otherwise distribute EDSA programs / User's Guides to anyone without prior written permission from EDSA Micro Corporation. All Rights Reserved. No part of this publication may be reproduced without prior written consent from EDSA Micro Corporation.

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TABLE OF CONTENTS

Foreword..........................................................................................................................................1

Introduction .....................................................................................................................................2

TDR and GMR .................................................................................................................................3

Supplied Sample Cases ....................................................................................................................5

Objective ..........................................................................................................................................5

Program Theory...............................................................................................................................5

Program Functions and Capabilities ..............................................................................................6

Required Input Data ........................................................................................................................8

Transposed vs. Un-transposed.........................................................................................................9

Tutorial – How to Use the Program ..............................................................................................17

Validation and Verification – Part I (EDSA vs IEEE)...................................................................30

Electric and Magnetic Fields.........................................................................................................30

Computation of Electric Field .......................................................................................................30

Computation of Magnetic Field.....................................................................................................34

Validation and Verification of Magnetic Field Computation........................................................36

Line Loading and Degree of Unbalance........................................................................................38

Example – Double Circuit Line with Ground Wire .......................................................................40

Validation and Verification – Part II (EDSA vs EMTP) ...............................................................49

References ......................................................................................................................................50

Appendix A: Overhead Line Parameters from Handbook Formulas and Computer Programs........................................................................................................................................51

A1. Introduction....................................................................................................................51

A2. Computer-Oriented Method...........................................................................................51

A3. Matrix Reduction and Transformation ..........................................................................54

A4. Comparison Between Bundling Procedures ..................................................................54

A5. Influence of Ground Wires on Positive Sequence Resistance.......................................56

A6. Comparison for Sequence Capacitances........................................................................57

A7. Comparison for Sequence Impedance ...........................................................................58

A8. Conclusions....................................................................................................................61

A9. References......................................................................................................................62

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List of Tables

Table 1: Comparison of positive sequence impedances using EDSA and IEEE.......................... 30

Table 2: Comparison of zero sequence impedances using EDSA and IEEE ............................... 30

Table 3: Comparison of positive sequence impedances using EDSA and EMTP........................ 49

Table 4: Comparison of zero sequence impedances using EDSA and EMTP ............................. 49

List of Figures

Figure 1: General Input Dialog........................................................................................................ 8

Figure 2: Input Dialog for Circuit Data ........................................................................................ 10

Figure 3: Input Dialog for Phase Conductor Data ........................................................................ 12

Figure 4: A Bundle With Four Conductors .................................................................................. 13

Figure 5: Input Dialog for Ground Conductor.............................................................................. 15

Figure 6: A Line With Continuous Ground Wires........................................................................ 16

Figure 7: A Line With Segmented Ground Wires ........................................................................ 16

Figure 8: Geometrical Parameters for the Sample Case ............................................................... 17 Note: You can view this manual on your CD as an Adobe Acrobat PDF file. The file name is:

Advanced Transmission Line Constants Trans_Line.pdf You will find the Test/Job files used in this tutorial in the following location:

C:\DesignBase\Samples\LineConstant = Adv. Transmission Line Constant Test Files: m2-2, m1-2, emtp, double, dommel

EDSA MICRO CORPORATION COPYRIGHT 2001-2008

ALL RIGHTS RESERVED

Computation of Transmission Line Parameters and Electric and Magnetic Fields

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Foreword EDSA Transmission Line Constants program is designed to calculate electrical parameters of

overhead transmission lines. The program also computes the electric and magnetic fields surrounding the transmission lines. It is assumed that the user is a professional engineer familiar with the concept under consideration. The interpretation and use of the results of the calculations encompassed by this program are the sole responsibility of the user.

All EDSA programs and related documents are the sole property of EDSA MICRO CORPORATION,

and are provided to the user’s company subject to the EDSA LICENSE AGREEMENT for the company’s use only. None of these programs should be supplied or loaned to any third party, or copied, or reproduced in any form without the express, written permission of EDSA MICRO CORPORATION. All copies and reproductions shall be the property of EDSA MICRO CORPORATION and must bear the copyright notice and ownership statement in their entirety.

The information contained in this document is subject to change without notice.


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Introduction This program computes electrical parameters of overhead transmission lines based on the

characteristics and positions of the conductors, and applies the matrix-oriented method using Carson's formula. The program is user friendly and a very powerful tool for practicing power system engineers, and for designers who would like to avoid time consuming and complex longhand calculations of overhead transmission lines based on electrical characteristics and positions of the wires.

It is import that a few program data requirements be discussed before the program usage is described. These data requirements will be illustrated through examples. A double circuit transmission line configuration having bundle conductors is shown below. This line parameters program computes the average height of a conductor based on the user selection of any of the two following methods:

1) [Height of the conductor at the tower - (2/3) x (Catenary sag at mid-point between towers)]. 2)

[Height of the conductor at the tower - (1/3) x (Catenary sag at mid-point between towers)].

For example, for the above example, the average height can be computed: Average height =Yaverage= Ytower –(2/3)*Sag or (1) Average height =Yaverage= Ytower –(1/3)*Sag (2) The user should provide the height of the conductor at the tower (Ytower ) and Catenary Sag and the program will compute the average height based on the user selected method (equation 1 or 2 above). The program is capable of calculating electrical parameters of multiple circuits, bundled phases, continuous or segmented ground wires, solid or hollow conductors.


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It should be noted that some of the required transmission line data in this program may be obtained from the conductor manufacturers. For example, computation of DC resistance can be quit involved especially for conductor constructions such as ACSR. However, manufacturer commonly provides the effective DC resistance (see the “Transmission and Distribution Reference Book” by Westinghouse for the ACSR characteristics tables). Another example is the knowledge of the thickness to diameter ratio TDR. For hollow conductors, TDR can be computed from the internal and external radius of the conductors. However, for ACSR conductors that are stranded the thickness to diameter ratio should be obtained from manufacturer. TDR=0.5 for solid conductors. Value of 0.375 is recommended for most ACSR conductors.

TDR and GMR Conductor Thickness to Diameter Ratio TDR is define for a tubular conductor as shown below:

D

T

For the above tubular conductor TDR = T/D. Computation of TDR for stranded conductor is not

straightforward. The stranded conductors are normally:

Copper (not often used due to expense) All aluminum conductor (AAC) Aluminum conductor, alloy reinforced (ACAR) Aluminum conductor, steel reinforced (ACSR) Others

For example, ACSR has central strands of steel for mechanical strength, with outer strands of

aluminum for electrical conductivity as shown below:

For the stranded conductor like ACSR, TDR does not apply unless TDR for an equivalent tubular

conductor that approximates ACSR is computed. It is suggested that to obtain an equivalent tubular conductor to represent ACSR, the steel strands should be ignored and space occupied by steel to represent the hollow part.


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If the conductor GMR (geometric mean radius or equivalent conductor radius) is known (is usually

available from the manufacturer), then, it is recommended that the user supply the GMR rather than TDR. For practical stranded conductors, look up the resistance and the conductor GMR from tables supplied by the manufacturer. Geometric Mean Radius (GMR) value:

For stranded conductor is obtained from the manufacturer’s data

For solid conductor, reGMR *4/1−=0.7788*r (r is the conductor radius) =


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Supplied Sample Cases For the convenience of the users, several sample cases have been prepared. The sample cases are: DOMMEL.LC2 This example is fully described in the “Tutorial Section” M2-2.LC2 Described in section “EXAMPLE - DOUBLE CIRCUIT LINE WITH GROUND WIRE” EMTP.LC2 This example is used in the “Validation And Verification Part II” DOUBLE.LC2 This is another example of a double circuit lines M1-2.LC2 Example of a transmission line with one circuit and two ground wires

The above sample data files can be found in the “\EDSA2004\samples\ LineConstant” directory.

Objective The objective of this program is to provide an easy-to-use program with a proven methodology for

calculating the electrical parameters and electric and magnetic fields of overhead transmission lines.

Program Theory The program is based on the matrix-oriented method [1,3,6] using Gary, Deri, Tevan, Semlyen and

Castanheira [4, 5]. It uses the bundling procedure and ground wire elimination procedure of reference [6]. The program also incorporates skin effect [4] based on non-magnetic tubular conductors. Bessel functions in the skin effect formula are evaluated by using polynomial approximations [5].

Though the program has been developed basically for power frequencies, it can also be used for

higher frequencies. With Z = Series impedance matrix in phase domain, and C = Nodal capacitance matrix in phase domain, the admittance matrix Y and susceptance matrix B are given by Y = Z-1, and B = wC. The sequence parameters are obtained through the transformation Zs=AZA-1, and Cs=ACA-1 where A is the transformation matrix from phase domain to sequence domain, and Zs and Cs

respectively are impedance and capacitance matrices in sequence domain. It may be noted that the zero-sequence coupling capacitance between two circuits is the negative of

respective element in Cs.


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The transformation matrix A for a general configuration with multiple circuits is a block-diagonal matrix. The diagonal block corresponding to a circuit is the normalized (power invariant) transformation matrix from phase domain to sequence domain for that circuit.

In order to avoid mutual coupling of sequence networks in the case of untransposed circuits, the

program averages appropriate elements within Z and C before applying the transformation from phase domain to sequence domain.

Segmented ground wires are considered in capacitance calculation, but are ignored in impedance

calculation.

Program Functions and Capabilities The program has the following modeling capabilities:

a. One, or more, three-, two-, or single-phase circuits on the same or adjacent right of ways.

b. Up to 100 physical wires (conductors).

c. Bundling of conductors.

d. Computation of zero sequence mutual coupling for up to 5 circuits

e. Ground wires being either continuous or segmented.

f. Power frequencies as well as higher frequencies.

g. Un-transposed or continuously transposed circuits.

h. Compute electric field.

i. Compute magnetic field.

j. Graphical display of electric and magnetic fields in 1-D, 2-D and 3-D

k. Export of electric and magnetic fields result into text format The program can compute the following data: a. Series impedance & admittance matrices in phase domain. b. Nodal capacitance and susceptance matrices in phase domain. c. Positive-sequence self-impedance and capacitance. d. Zero-sequence self-impedance and capacitance. e. Zero-sequence mutual impedance and capacitance between circuits.

f. Line Loading Analysis and Degree of unbalance calculation


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The program is able to compute electrical parameters of overhead transmission lines with the

following system of unit options: a. English system; b. Metric system. Ground wire type options: a. Continuous; b. Segmented. Output options: Any combination of a. Impedance and capacitance matrices in phase domain; b. Admittance and susceptance matrices in phase domain; c. Sequence parameters.


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Required Input Data General: Frequency in c/s or Hz. Earth resistivity in meter-ohm (for English as well as Metric units). Number of circuits Number of ground wires System unit options: English or Metric units by checking the desired radio button. Average Height Calculation option (Please refer to page Error! Bookmark not defined.) Transposition

Figure 1: General Input Dialog


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Transposed vs. Un-transposed The transmission lines are normally transposed to achieve balanced condition. This program computes the transmission line series impedance and shunt capacitance matrices based on the provided line configuration. It is important to note that the so-called “sequence impedances” (positive, negative and zero) are actually defined for transposed (balanced) configuration. If the line is not transposed, then, the sequence impedances do not apply and “modal impedances” definition should replace them. This program provides the following output as far as transposition is concerned:

1) The series impedance matrix as well as shunt capacitance matrix is always computed and reported for the un-transposed configuration. The transposed series impedance matrix can easily be obtained by averaging as follows:

Zaa=Zbb=Zcc=1/3[Z(1,1)+Z(2,2)+Z(3,3)] (1) Zab=Zac=Zbc=1/6[Z(1,2)+Z(1,3)+Z(2,1)+Z(2,3)+Z(3,1)+Z(3,2)] (2)

2) The modal impedances are computed and reported 3) The sequence impedances are computed assuming perfectly transposed configuration (using equations 1 and 2

above) as follows:

Z0=Zaa+2*Zab Z1=Zaa-Zab


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Input variables for a circuit: Circuit Name: Circuit identification/name; can be linked to the EDSA2004 Analysis program such as powerflow, short circuit, etc. (optional data) From Bus Name: Substation name associated with the above-defined circuit (optional data) To Bus Name: Substation name associated with the above-defined circuit (optional data)

Figure 2: Input Dialog for Circuit Data


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Input variables for a conductor of a circuit: Phase Identification, an id assigned to a phase (e.g. A, B, R, S, T) Height, height of the phase conductor at the tower (Ytower see Introduction), in feet (English) or meters (Metric). (Please refer to page 2) Catenary Sag, The phase conductor sag (see Introduction), in feet (English) or meters (Metric). (Please refer to page 2) Horizontal Position, position from an arbitrary reference vertical plane, in feet (English units) or meters (Metric). Outside diameter in inches (English) or centimeters (Metric). Thickness/diameter, Thickness to diameter ratio. Note that TDR=0.5 for solid conductors. (See page 3) or Geometric Mean Radius (GMR), Alternative to providing TDR, the user can supply the GMR which is normally available from the manufacturer (See page 3) DC resistance in ohms per mile (English) or per kilometer (Metric). Computation of DC resistance can be quit involved especially for conductor constructions such as ACSR. However, manufacturer commonly provides the effective DC resistance (for example, see the “Transmission and Distribution Reference Book” by Westinghouse for table of ACSR characteristics). Phase to Phase Voltage (kV), this is only used in the electric field calculation Voltage Angle (degrees), this is only used in the electric field calculation Phase Current (Amps), this is used in the magnetic field calculation and line loading and degree of unbalance evaluation Current Angle (degrees), this is used in the magnetic field calculation and line loading and degree of unbalance evaluation Number of wires (conductors) in a bundle; enter 1 if conductor is not a bundled conductor (see Figure 4) Separation between Wires, in inches (English) or in centimeters (Metric). Angular Position of Wires in the bundle, in degree; positive angles are measured counter-clockwise.


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Figure 3: Input Dialog for Phase Conductor Data


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A regular bundle of four wires (conductors) is shown below. In a regular bundle, all the component wires are identical and the wires are uniformly spaced around the circumference of a circle.

Figure 4: A Bundle With Four Conductors


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Input variables for a ground wire: Ground Wire Identification, an id assigned to a ground wire (e.g. G, G1) Type, either the ground wire is continuous or the ground wire is segmented (see Figure 6 and Figure 7). Height, height of the ground conductor at the tower (Ytower see Introduction), in feet (English) or meters (Metric). (Please refer to page Error! Bookmark not defined.) Catenary Sag, The ground conductor sag (see Introduction), in feet (English) or meters (Metric). (Please refer to page Error! Bookmark not defined.) Horizontal Position, position from an arbitrary reference vertical plane, in feet (English units) or Meters (Metric). Outside diameter in inches (English) or centimeters (Metric). Thickness/diameter, Thickness to diameter ratio. Note that TDR=0.5 for solid conductors. (Please refer to page 3) or Geometric Mean Radius (GMR), Alternative to providing TDR, the user can supply the GMR which is normally available from the manufacturer (Please refer to page 3) DC resistance in ohms per mile (English) or per kilometer (Metric). Phase to Phase Voltage (kV), this is only used in the electric field calculation Voltage Angle (degrees), this is only used in the electric field calculation Phase Current (Amps), this is only used in the magnetic field calculation Current Angle (degrees), this is only used in the magnetic field calculation


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Figure 5: Input Dialog for Ground Conductor


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Figure 6: A Line With Continuous Ground Wires

Figure 7: A Line With Segmented Ground Wires


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Tutorial – How to Use the Program This tutorial will illustrate the step-by-step procedure to calculate the electrical parameters of the following transmission line:

0

10

20

30

40

50

60

-50 -40 -30 -20 -10 0 10 20 30 40 50

Conductor Diameter = 0.9"Bundle Spacing = 18"DC Resistance = 0.1686

Earth Resistivity = 100 ohm/mNo Ground wires

Spacing between phases = 40'Average Hieght above ground = 50'

500 kV

Figure 8: Geometrical Parameters for the Sample Case

The tutorial is based on line configuration shown above which includes single three-phase circuit with no ground wires but bundled phase conductors (see Appendix A or IEEE Trans. PAS, vol. 104, pp. 366-370, 1985) Figure 8 illustrates the information that pertains to the example used. All the measurements are given in feet. Details of how to enter data for each circuit and conductors are given in the tutorial steps that follow.


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1.1 From the EDSA main menu screen, invoke the Transmission Line Parameters program as follows:

> Select Analysis / Additional Calculations/ Transmission Line Parameters

1.2 Once in the program main menu, proceed to create the new file as follows:

Select File -> New or select the “New” icon as shown below:


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1.3 Proceed to assign a name to the file and save it.

> In the File name field, type "newcase.lc2" and then Press “Open” as shown above.


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1.4 Proceed to define the configuration of the circuit under study. For this tutorial, we will consider one circuit only. We will not include any ground wires in this example. The rest of general data are shown in the dialog below (Note that in this example, we have selected the US standard unit): > In the Frequency field type 60 Hz. > In the Number of Circuits field type 1 > In the Earth Resistivity field type 100 Ohm-m

> In the Number of Ground Wires field type 0 > Select US Standard Unit

1.5 Once the general data is specified, the program will direct the user to all of the necessary data

dialogs through a “Wizard” as shown below:


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1.6 Next proceed to enter the circuit identification for each circuit as shown below:

Circuit Name: Enter C1 From Bus Name: Enter 1 To Bus Name: Enter 2


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1.7 Once the circuit identification data has been entered, proceed with the Phase Input Data as

follows (note the below dialog is for the Phase 1 or Circuit 1):

Change the default data as follows:

Phase Identification A Height 50 feet Horizontal Position 40 feet Outside diameter 0.9 inches Thickness/diameter 0.5 (for solid conductor)


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DC resistance 0.1686 ohm/mile Phase to Phase Voltage (kV) 400 kV Voltage Angle (degrees) 0 degree Number of wires (conductors) in a bundle 4 Separation between Wires 18 inches Angular Position of Wires 45 degrees

The completed data dialog for phase 1 of circuit 1 is shown below:

1.8 Now proceed to enter the data for other two phases as shown below:


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1.9 Since no more data is required (based on the general data specified), select “Finish” as shown

below:


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1.10 To compute the transmission line parameters, select “Run->Impedance Calculation” as shown

below:


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1.11 Before proceeding with the calculation, select if the report should contain full results as shown

below:

The report generated for the above example is shown below:

Computation of Transmission Line Parameters and Electric and Magnetic Fields Computation of Transmission Line Parameters and Electric and Magnetic Fields

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Once the calculations are completed the results are shown in the results output screen (see above figure). With the aid of the tool bar menu, the user has the following options: Scroll up and down to read the results, Print the results, and Copy the results into the clipboard for importing purposes. From here, press DONE to return to the main menu. The user can save the result to a text file for later printing or inclusion in any other documents.


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Validation and Verification – Part I (EDSA vs IEEE) The sequence impedances computed for the above example are compared with those results published in the IEEE paper: IEEE Trans. PAS, vol. 104, pp. 366-370, 1985 (or see Appendix A using Bundling by Matrix Reduction). In the previous example, we assumed a TDR of 0.5. Since the aforementioned IEEE paper does provide GMR values, then, a new jobfile is created named dommel.lc2 which uses a GMR of 0.3672 inch as specified in the paper. The summary of the comparison is shown in the table below:

Table 1: Comparison of positive sequence impedances using EDSA and IEEE Resistance (ohm/mi) Reactance (ohm/mi) susceptance (µSiemens/mi) Capacitance µF/mi)EDSA 0.0428 0.5353 8.0560 0.02137

IEEE Paper 0.0422 0.5339 8.0672 0.0214

Difference (%) -0.23 0.67 0.14 0.14

Table 2: Comparison of zero sequence impedances using EDSA and IEEE Resistance (ohm/mi) Reactance (ohm/mi) susceptance (µSiemens/mi) Capacitance µF/mi)EDSA 0.3203 2.0331 5.0660 0.01344

IEEE Paper 0.3174 2.0060 5.0728 0.01346

Difference (%) -0.75 -1.1 0.13 0.13

Electric and Magnetic Fields Electric and magnetic fields are generally created by combination of current and voltage. Electric field is produced by voltage and current produces magnetic field. The word EMF refers to Electric and Magnetic Fields and not Electromagnetic. When the distances from source is large as compared to wavelength, electric and magnetic fields are linked and should be considered together. However, when the distance from source is small, such as in the case of magnetic field in the vicinity of transmission lines, the fields are independent and are considered independent and should be considered separately as electric and magnetic fields and not electromagnetic fields or radiation. The following sections show how to perform the electric and magnetic fields using EDSA’s program.

Computation of Electric Field The electric field in the surrounding area of the transmission lines can be computed for any number of conductors operating at different voltage levels. It should be noted that the computed electric field is the net electric field due to all of the conductors having non zero voltage. If the user wishes to examine electric field due to one or more conductors, then, voltages for other conductors should be set to zero. To compute the space electric field, select “Run Electric Field Contour Graph” as shown below (of course, it is assumed that the data for the transmission line circuits have already been entered as was demonstrated in the previous section):


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Next, the program prompts the user to specify the area where the space electric field should be computed.

The space electric field is computed in a vertical plane defined above. The equipotential contours of the space electric field is then computed and displayed for the area defined as shown below:


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The above graph can be saved or printed. The options are shown below:

It is also possible to compute the surface electric field along a user-defined path. To exercise this option, select “Run Electric Field Axis Graph” icon as shown below:


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Next, the beginning and end point of the computation axis should be defined in the dialog shown below:

The program will display the electric field computed along the specified path.


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Computation of Magnetic Field

The magnetic field in the surrounding area of the transmission lines can be computed for any number of conductors with different currents flow through them. It should be noted that the computed magnetic field is the net magnetic field due to all of the conductors having non zero current. If the user wishes to examine magnetic field due to one or more conductors, then, currents for other conductors should be set to zero. To compute the space magnetic field, select “Run Magnetic Field Contour Graph” as shown below (of course, it is assumed that the data for the transmission line circuits have already been entered as was demonstrated in the previous section):


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Next, the program prompts the user to specify the area where the space magnetic field should be computed.

The space magnetic field is computed in a vertical plane (perpendicular to the line) defined above. The magnetic fields contours (equal magnetic field magnitude) is then computed and displayed for the area defined as shown below:

The above graph can be saved as shown before.

It is also possible to compute the magnetic field along a user-defined path and the ground level. To exercise this option, select “Run Magnetic Field Axis Graph” icon as shown below:


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Next, the beginning and end point of the computation axis should be defined similar to the electric field computation. The program will display the magnetic field computed along the specified path.

Validation and Verification of Magnetic Field Computation


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The example of reference 9 (Magnetic Field Management for Overhead Transmission Lines: A Primer, Definitions, Methods of Performing Calculations, Field Management Options, and Other Issues. EPRI (Electric Power Research Institute report TR-103328 December 1994, page 1-21) is used to validate the magnetic field computation result. The line consists of three phases without ground wire. The coordinates of each phase are as follows:

Phase X (m) Height(m) Current (Amps) A -9.15 12.2 1000 B 0 12.2 1000 C 9.15 12.2 1000

The result of magnetic field calculation is shown in the below figure:

The only result reported in reference 9 corresponds to the value of magnetic field at the lower left hand corner of the above figure, i.e., X=-30.5, H=1.0

EDSA EPRI Deviance (%) 32.3 32.33 0.093%

The above shows an excellent agreement with the result of reference 9.


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Line Loading and Degree of Unbalance The program can compute the line loading and determine the degree of unbalance. The measure of unbalance is defined as follow:

100*

0voltagesequencepositive

voltagesequencezerou =

100* 1voltagesequencepositivevoltagesequencenegativeu =

Where u0 is the degree of zero sequence voltage unbalance and u1 is the degree of negative sequence voltage unbalance. Power utilities have defined different acceptable values for the degree of unbalance. To compute these indices the user should enter all phase voltage and current information correctly. The program first computes the line loading from the following equation:

−

= AAA IVS * Where:

AS is the total power on phase A

AV phase A line-ground voltage and −

AI is the conjugate of phase A current.` Similar calculation is performed for other phase. The program then computes the receiving end voltages assuming the sending end voltages are VA, VB, etc. and the line is loaded SA, SB, etc. The computation is also performed for different line length. To obtain the result for line loading analysis, select “Run->Impedance Calculation”. Sample output is shown below:

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39


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Example – Double Circuit Line with Ground Wire In this example we will show how to enter data for a double circuit transmission line having ground wire. The data with line has already been entered and saved in sample file named “M2-2.LC2”. The line geometry is shown in the below figure:

0

5

10

15

20

25

30

35

40

45

-10 -8 -6 -4 -2 0 2 4 6 8 10

Ground wireRDC = 3.8 ohm/kmDiameter = 1.05 cm

Circuit 1RDC = 0.066 ohm/kmDiameter = 2.952

Circuit 2RDC = 0.066 ohm/kmDiameter = 2.952

First, let’s examine how the general data is entered. After loading the above sample data file (“M2-2.LC2”), select “Edit->Edit Master File” as shown below:


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The general data shows that the system frequency in this example is 50 Hz. The system of unit is Metric and earth resistivity is 100 ohm-m. Since this is a double circuit line, we have specified Number of Circuits to be 2. There is only one ground wire in this example.


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The three phases of circuit one are shown in the figure below. To inspect the data for a phase, double left mouse click on the corresponding row. The data dialog for the selected phase/conductor will appear as shown on the next page:


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To examine the data for the second circuit, use the dropdown shown to the right of “Circuit” and select the desire circuit as shown below:


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To examine the data for ground wire(s), select the radio button to the left of “Ground Wires” as shown below:

Now, double left mouse click on the desired ground wire (see above figure) will open the data dialog for the selected ground wire:


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The calculated line parameters for the above example are shown below. Note that since a double circuit line is considered in this example, the sequence impedances for each circuit are reported followed by the mutual impedances between circuit 1 and circuit 2.

Overhead Configuration ====================== Frequency = 50.00 (Hz) Earth Resistivity = 100.00 (ohm-meter) Average Height Calculation : Method 1 Transposition : Yes Circuits ======== [ 1] Circuit Name : C1 From Bus : 1 To Bus : 2


46

Number of Phases : 3 No. Phase Bdl DC RES Avg Horz DIA TDR GMR Voltage Bundle Wrs Hght Pos Sep Angle (ohm/km) (m) (m) (cm) (cm) (kV) (deg) (cm) (deg) --- ----- --- -------- ------- -------- -------- ----- ------- ------- ------ ----- ----- 1 A 1 0.06600 31.550 5.000 2.925 n/a 1.18400 500.0 0.0 40.0 45.0 2 B 1 0.06600 25.050 8.000 2.925 n/a 1.18400 500.0 240.0 40.0 45.0 3 C 1 0.06600 18.550 5.000 2.925 n/a 1.18400 500.0 120.0 40.0 45.0 [ 2] Circuit Name : C2 From Bus : 3 To Bus : 4 Number of Phases : 3 No. Phase Bdl DC RES Avg Horz DIA TDR GMR Voltage Bundle Wrs Hght Pos Sep Angle (ohm/km) (m) (m) (cm) (cm) (kV) (deg) (cm) (deg) --- ----- --- -------- ------- -------- -------- ----- ------- ------- ------ ----- ----- 1 A 1 0.06600 31.550 -5.000 2.925 n/a 1.18400 500.0 0.0 40.0 45.0 2 B 1 0.06600 25.050 -8.000 2.925 n/a 1.18400 500.0 240.0 40.0 45.0 3 C 1 0.06600 18.550 -5.000 2.925 n/a 1.18400 500.0 120.0 40.0 45.0 Ground Wires ============ No. Grnd SEG DC RES Avg Horz DIA TDR GMR Voltage Wire Hght Pos (ohm/km) (m) (m) (cm) (cm) (kV) (deg) --- ----- --- -------- ------- -------- -------- ----- ------- ------- ------ 1 G1 N 3.80000 41.400 0.000 1.050 n/a 0.40800 0.0 0.0 Impedance Calculation Results ============================= Series-impedance matrix(Ohm/km) 0.1316 0.6974 6.3418E-02 0.3034 6.2565E-02 0.2660 6.5353E-02 0.2819 6.3413E-02 0.2589 6.2562E-02 0.2514 6.3418E-02 0.3034 0.1280 0.6984 6.1011E-02 0.3040 6.3413E-02 0.2589 6.1726E-02 0.2533 6.1006E-02 0.2595 6.2565E-02 0.2660 6.1011E-02 0.3040 0.1266 0.6986 6.2562E-02 0.2514 6.1006E-02 0.2595 6.0352E-02 0.2830 6.5353E-02 0.2819 6.3413E-02 0.2589 6.2562E-02 0.2514 0.1316 0.6974 6.3418E-02 0.3034 6.2565E-02 0.2660 6.3413E-02 0.2589 6.1726E-02 0.2533 6.1006E-02 0.2595 6.3418E-02 0.3034 0.1280 0.6984 6.1011E-02 0.3040 6.2562E-02 0.2514 6.1006E-02 0.2595 6.0352E-02 0.2830 6.2565E-02 0.2660 6.1011E-02 0.3040 0.1266 0.6986 Shunt-admittance matrix(Mho/km) 0.000 2.4012E-06 0.000 -4.2648E-07 0.000 -1.8845E-07 0.000 -3.0699E-07 0.000 -1.6592E-07 0.000 -1.2646E-07 0.000 -4.2648E-07 0.000 2.4327E-06 0.000 -4.0285E-07 0.000 -1.6592E-07 0.000 -1.2907E-07 0.000 -1.4785E-07 0.000 -1.8845E-07 0.000 -4.0285E-07 0.000 2.4473E-06 0.000 -1.2646E-07 0.000 -1.4785E-07 0.000 -2.6913E-07


47

0.000 -3.0699E-07 0.000 -1.6592E-07 0.000 -1.2646E-07 0.000 2.4012E-06 0.000 -4.2648E-07 0.000 -1.8845E-07 0.000 -1.6592E-07 0.000 -1.2907E-07 0.000 -1.4785E-07 0.000 -4.2648E-07 0.000 2.4327E-06 0.000 -4.0285E-07 0.000 -1.2646E-07 0.000 -1.4785E-07 0.000 -2.6913E-07 0.000 -1.8845E-07 0.000 -4.0285E-07 0.000 2.4473E-06 Current Eigenvectors [Ti] 0.3828 -8.9127E-03 0.3283 3.5364E-02 0.5087 -6.0787E-03 -0.5030 1.5435E-02 -0.3731 4.0564E-03 -0.2846 -1.4423E-02 0.3799 2.3206E-03 0.5301 5.4938E-03 3.0209E-03 2.1352E-02 -2.9196E-03 4.0098E-02 0.4700 -4.2889E-03 0.5797 -8.8969E-04 0.4574 5.5305E-03 0.3381 -4.2948E-02 -0.4916 -6.1590E-03 0.4990 1.5793E-02 -0.3741 -9.4340E-03 -0.2885 1.2440E-02 0.3828 -8.9127E-03 -0.3283 -3.5364E-02 0.5087 -6.0787E-03 0.5030 -1.5435E-02 0.3731 -4.0564E-03 -0.2846 -1.4423E-02 0.3799 2.3206E-03 -0.5301 -5.4938E-03 3.0209E-03 2.1352E-02 2.9196E-03 -4.0098E-02 -0.4700 4.2889E-03 0.5797 -8.8969E-04 0.4574 5.5305E-03 -0.3381 4.2948E-02 -0.4916 -6.1590E-03 -0.4990 -1.5793E-02 0.3741 9.4340E-03 -0.2885 1.2440E-02 [Y'] Diag (Mho/km) -1.6095E-09 1.2286E-06 1.4025E-09 2.2362E-06 2.4398E-09 2.4448E-06 -1.3616E-09 2.7768E-06 -4.0949E-11 2.9734E-06 -8.3032E-10 2.9027E-06 [Z'] Diag (Ohm/km) 0.4395 2.053 6.6078E-02 0.5016 6.6671E-02 0.4635 6.6293E-02 0.4005 6.6221E-02 0.3742 6.6329E-02 0.3834 Eigenvalues of [Y].[Z] Mode 1: -.252289E-11 0.536732E-12 Mode 2: -.112157E-11 0.148470E-12 Mode 3: -.113310E-11 0.164125E-12 Mode 4: -.111234E-11 0.183539E-12 Mode 5: -.111272E-11 0.196886E-12 Mode 6: -.111290E-11 0.192214E-12 Modal propagation constants Mode 1: 0.168021E-06 0.159722E-05 Mode 2: 0.699439E-07 0.106135E-05 Mode 3: 0.768918E-07 0.106725E-05 Mode 4: 0.867194E-07 0.105823E-05 Mode 5: 0.929632E-07 0.105894E-05 Mode 6: 0.907668E-07 0.105884E-05 Surge impedances (Ohm) Mode 1: 1299.82 -138.457 Mode 2: 474.636 -30.9799 Mode 3: 436.576 -31.0160 Mode 4: 381.077 -31.4164 Mode 5: 356.138 -31.2698 Mode 6: 364.769 -31.3742 Sequence Impedances Assuming Complete Transposition Impedances for Circuit number 1 Resistance (ohm/km) Reactance (ohm/km) susceptance (Micro-Siemens/km) ------------------- ------------------ ------------------------------ Positive Sequence 0.0664 0.4070 2.7664 Zero Sequence 0.2534 1.2804 1.7486 Impedances for Circuit number 2 Resistance (ohm/km) Reactance (ohm/km) susceptance (Micro-Siemens/km) ------------------- ------------------ ------------------------------ Positive Sequence 0.0664 0.4070 2.7664 Zero Sequence 0.2534 1.2804 1.7486 Mutual Impedance between Circuit 1 and Circuit 2


48

Resistance (ohm/km) Reactance (ohm/km) susceptance (Micro-Siemens/km) ------------------- ------------------ ------------------------------ Zero Sequence 0.1871 0.7860 -0.5286


49

Validation and Verification – Part II (EDSA vs EMTP) In this validation we will compare the result of a transmission line parameters calculated using EDSA to the result obtained from the Electromagnetic Transient Program (EMTP). The line configuration for this example is shown below:

0

5

10

15

20

25

30

-6 -4 -2 0 2 4 6

Distance (meters)

Ave

rage

Hei

ght (

met

ers)

Diameter = 3.3 cmThickness/Diameter=0.375DC resistance = 0.04822 ohm/kmBase Frequency = 60 Hz

The sequence impedances computed for the above example using EDSA are compared with those results obtained by EMTP. The summary of the comparison is shown in the tables below:

Table 3: Comparison of positive sequence impedances using EDSA and EMTP Resistance (ohm/mi) Reactance (ohm/mi) susceptance (µSiemens/mi) Capacitance µF/mi)EDSA 0.05100 0.50120 3.28260 0.00871 EMTP 0.04998 0.50107 3.28714 0.00872 Difference (%) -2.05 -0.03 0.14 0.14

Table 4: Comparison of zero sequence impedances using EDSA and EMTP Resistance (ohm/mi) Reactance (ohm/mi) susceptance (µSiemens/mi) Capacitance µF/mi)EDSA 0.22260 1.52560 2.05300 0.00545 EMTP 0.21995 1.51000 2.05588 0.00545 Difference (%) -1.20 -1.03 0.14 0.14


50

References 1. Anderson, P.M., "Analysis of Faulted Power Systems", The Iowa State University Press/AMES, 1973. 2. Carson, J.R., "Wave Propagation in Overhead Wires with Ground Return", Bell System Technical Journal, vol. 5, pp. 539-554, 1926. 3. Dommel, H.W., "Overhead Line Parameters from Handbook Formulas and Computer Programs", IEEE Trans. PAS, vol. 104, pp. 366-370, 1985. 4. Dwight, H.B., "A Precise Method of Calculation of Skin Effect in Isolated Tubes", AIEE Journal, vol. 42, pp. 827-831, 1923. 5. "Handbook of Mathematical Functions", edited by Abramowitz, M., and I.A. Stegun, pp. 384-385, published by US Dept. of Commerce, 1964. 6. Hesse, M.H., "Electromagnetic and Electrostatic Transmission - Line Parameters by Digital Computer", IEEE Trans. PAS, vol. 82, pp. 282-291, 1963. 7. Shipley R.B., and Coleman D.W., "A New Direct Matrix Inversion Method", AIEE Trans, pt. I (Commun. and Electronics), vol. 78, pp. 568-572, 1959. 8. Woodruff, L.F., "Electric Power Transmission", John Wiley & Sons Inc., 1938. 9. Magnetic Field Management for Overhead Transmission Lines: A Primer, Definitions, Methods of Performing Calculations, Field Management Options, and Other Issues. EPRI (Electric Power Research Institute) report TR-103328 December 1994.


51

Appendix A: Overhead Line Parameters from Handbook Formulas and Computer Programs

* IEEE Transaction on Power Apparatus and Systems, Vol. PAS-104, No. 2. February 1985

OVERHEAD LINE PARAMETERS FROM HANDBOOK FORMULAS AND COMPUTER PROGRAMS

H.W. Dommel, Fellow, IEEE

The University of British Columbia 2356 Main Hall

Vancouver, B.C., V6T 1W5 Canada

Abstract - Overhead line parameters can be calculated from handbook formulas, or with more general computer-oriented methods. At power frequency, the differences between the two approaches are usually negligible, but they can become large at higher frequencies. This paper discusses the causes of these differences for the engineer who wants to compare results from computer programs with those obtained from handbook formulas. It contains no new theories, but simply summarizes the experience gained in analyzing such differences over many years.

A1. Introduction Nowadays, overhead line parameters are usually obtained with computer programs. Some of

these programs may still be based on handbook formulas, but most of them use more general computer-oriented methods which are valid for any number of phases and ground wires at any frequency. A prudent engineer may not want to trust the output of such general-purpose program blindly, but may want to compare at least some results with those obtained from handbook formulas. There are differences in the results from the two approaches, which this paper tries to explain. While the differences are sometimes of little practical importance, they must be understood if one wants to gain confidence in the results of a general-purpose program.

After a brief discussion of computer-oriented methods, the positive and zero sequence

parameters obtained with them are compared with those obtained from handbook formulas.

A2. Computer-Oriented Method A general method well suited for the calculation of overhead line parameters with computers was

described by M.H. Hesse more than 20 years ago [1]. To explain this method, a single-circuit, three-phase line with twin bundle conductors and two ground wires, as shown in Fig. 1, will be used as an example. It must be emphasized, however, that the method is completely general, and could as well be used for a double - circuit line, or for a single-circuit, three-phase line in parallel with a bipolar dc line, or for any other configuration of which one might think.

For the case of Fig. A-1, there are 8 parallel conductors. Two systems of equations describe the steady-state behavior of these 8 conductors, namely the system of phasor equations.

* This paper has been printed with the permission of The Institute of Electrical and electronics Engineers, Inc. Standards Activities. Please note: “ (OVERHEAD LINE PARAMETERS FROM HANDBOOK FORMULAS AND COMPUTER PROGRAMS IEEE, Transaction on Power Apparatus and Systems, Vol. PAS-104, No. 2. February 1985. ), Copyright © 1985. IEEE. All rights reserved. This is an unapproved IEEE Standards Draft, subject to change. The IEEE disclaims any responsibility or liability resulting from the placement and use in the described manner.”


52

[ ]− =dVdx

[Z][I] (1)

for the longitudinal voltage drops along the line, and the system of phasor equations

[ ]− =dIdx

j C ][ Vϖ [ ] (2)

for the current changes along the line ( shunt conductances, as usual, are ignored here ).

Fig. A-1

Tower Configuration

The elements of the 8 × 8 impedance matrix [ Z ] in Eq. (1) are usually calculated from Carson’s formula [ 2 ]. The diagonal element Zii is the series impedance per unit length of the loop formed by conductor i and ground return, and the off - diagonal element Zik = Zki is the series mutual impedance per unit length between the two loops conductor i / ground return and conductor k / ground return. Carson’s formula contains integrals which can not be solved in closed form. They have been developed into reasonably well converging infinite series for small arguments of the parameter a,

a 2.81 10 D f3= × −

ρ (3)

where: f = frequency in Hz,


53

ρ = earth resistivity in Ωm, D = distance in m between conductor i and image below earth surface of conductor k for

mutual impedance ( or twice conductor height in m for self impedance ). For large arguments a, asymptotical expansions are usually used. Most handbook formulas were

derived from these series, with only the first one or two terms retained. In computer-oriented methods, it is best to add as many terms as are necessary for obtaining a specified degree of accuracy. Fig. 2 shows that the errors with truncation of the series after the first or second term would be unacceptable for the mutual impedance between two conductors in cases of wide separation, or alternatively, in cases of less wide separation but higher frequency or lower earth resistivity [3].

Much simpler impedance formulas with closed-form solutions have recently been developed by

Gary, Deri, Tevan, Semlyen and Castanheira [4, 5]. They give results close to those obtained from Carson’s formula (largest differences approximately 10% in the range 100 Hz to 10 kHz, and smaller elsewhere). These new formulas may replace Carson’s formula one day, but they are not discussed here because they have been adequately described in [4, 5].

Fig. A-2

Mutual Reactance Between Two Parallel Conductors The elements of the 8 × 8 capacitance matrix [C] in Eq. (2) are easier to calculate, and are real

rather than complex. They are obtained indirectly, by first building a “potential coefficient” matrix [P], with


54

P 12

In 2hrii

i

i=

π ε 0, P 1

2In D

dikik

ik=

π ε 0 (4)

where: hi = average height above ground of conductor i,

ri = radius of conductor i, Dik = distance between conductor i and image below earth surface of conductor k, dik = direct distance between conductor i and k, ε0 = permittivity of free space. Once [P] is know, [C] is found by matrix inversion, [C] = [P]-1 (5)

A3. Matrix Reduction and Transformation Usually one is not interested in the details contained in the 8 × 8 matrices of Eq. (1) and (2). A

simpler description is obtained by reducing them to 3 × 3 matrices for the phase quantities, which still contains more detail than most handbook formulas would allow. The reduction is accomplished by first introducing the bundling conditions into the equations. For example, if conductors 1 and 2 form phase A, then V1 = V2 = VA and I1 + I2 = IA in Eq. (1). For continuous ground wires which are grounded at every tower, e.g., for conductors 7 and 8, one simply sets V7 = V8 = 0 in Eq. (1). The reduction to smaller matrices is then achieved, in the example, by introducing IA as a new variable, and by eliminating I1, I2, I7 and I8. The reduction procedure for ground wires is correct as long as the ground wire potential is continuously zero. For typical tower spans of 250 m to 350 m, this assumption is reasonable up to approximately 250 kHz [6]. For bundling, the reduction procedure is correct as long as the potentials on the subconductors are continuously equal, which is a reasonable assumption up to approximately 500 kHz with spacers typically 100 m apart.

Even the 3 × 3 matrices are often too detailed. For example, only positive sequence parameters

are needed in power flow studies, or positive and zero sequence parameters in short-circuit studies. Sequence parameters are easily obtained from the 3 × 3 matrices for phase quantities by transforming them to 3 × 3 matrices for symmetrical components. The zero, positive and negative sequence parameters are simply the diagonal elements of these matrices, with Zneg = Zpos, while the off-diagonal elements are normally ignored. For untransposed lines, the off - diagonal elements, do contain useful information about coupling effects between sequences quantities, however, and are used in [7, pp.93 - 103] to derive unbalance factors.

A4. Comparison Between Bundling Procedures The bundling procedure by matrix reduction, as described in section 3, differs from the procedure

used in most handbook formulas and in some computer programs, where the bundle of subconductors is replaced by a single equivalent conductor from the beginning ( 7, pp. 111 - 114). Formulas are usually only given for the more important case of symmetrical bundles, even though they could be derived for asymmetrically bundled conductors as well. In Eq. (4), replacing the bundle of subconductors by one equivalent conductor located at the center of the bundle is achieved by using requiv in place of r,


55

requiv = N r A N 1N × × − (6) where: N = number of subconductors in bundle, r = radius of subconductor, A = radius of bundle. For the impedance calculation, the geometric mean radius GMR of one conductor is replaced by

GMRequiv, with the same formula as Eq. (6) ( except GMR in place of r ). In practice, the bundling procedures with matrix reduction and with equivalent conductors

produce almost identical results. For the case of a 500 kV three-phase line with the data of Table 1, the results from both procedures are shown in Table 2. At least in this case, they are practically identical at 60 Hz, though they would probably differ somewhat more at higher frequencies.

Table 1

Data for 500 kV Three - Phase Line

Phase arrangement: Horizontal tower configuration Spacing between phases = 40 feet Average height above ground = 50 feet Bundle with 4 subconductors, requiv = 7.80524 inches GMRequiv = 7.41838 inches Spacing between subconductors = 18 inches Subconductors r = 0.45 inches GMR = 0.3672 inches dc resistance = 0.1686 Ω/mile No ground wires Earth resistivity = 100 m


56

Table 2.

Comparison Between Bundling Procedures Positive and Zero Sequence Parameters at 60 Hz

Bundling by Matrix Reduction

Equivalent Conductors

Rpos ( Ω/mile ) Xpos ( Ω/mile ) Cpos ( μF/mile ) Rzero ( Ω/mile ) Xzero ( Ω/mile ) Czero ( μF/mile )

0.042223 0.53394 0.021399 0.31740 2.0065 0.013456

0.42205 0.53399 0.021397 0.31738 2.0065 0.013455

A5. Influence of Ground Wires on Positive Sequence Resistance While it is well known that ground wires have an influence on zero sequence parameters, it is

less well known that they can influence positive sequence parameters, too. Of practical importance is the increase in the positive sequence resistance Rpos if the line has ground wires which are grounded at every tower. Since the mutual impedances from the three phase conductors A, B, C to the ground wire G are never exactly equal, there is always a small longitudinal voltage induced in the ground wire, even for symmetrical positive sequence currents with IB = IA e -j120°C, IC = IA e +j120°C.

− = + +− +dVdx

( Z Z e Z e )IGAG BG

j120 CCG

j120 CA

o o (7)

With the ground wire grounded at every tower, this induced voltage produces a circulating current

which flows through the ground wire, towers and ground ( Fig. A-3 ). This circulating current produces additional losses, which show up as an increase in the value of the positive sequence resistance in computer-oriented methods. Handbook formulas would not show this increase. In one particular case of a single-circuit 500 kV line, this increase was 6.5% at 60 Hz.

To avoid the losses associated with these circulating currents, some utility companies use

“segmented” ground wires in an arrangement which has the form of a “ T”: The ground wire is grounded in the middle, and insulated at the adjacent towers to the left and right. At both ends of the segmentation section, the ground wire is interrupted as well, to prevent circulating currents from flowing. In computer-oriented methods, segmented ground wires are handled by ignoring them in the series impedance calculation (or by setting the mutual impedance to the other conductors to zero), but by taking them into account in the capacitance calculation.


57

Fig. A-3

Circulating Current in Ground Wire

A6. Comparison for Sequence Capacitances For positive sequence capacitance, most handbooks give the formula

C 2

In dr

pos0

m

equiv

=π ε (8)

with dm = geometric mean distance among the three phases. For the 500 kV line of Table 1, this

produces a value which is approximately 4% lower than that obtained from computer-oriented methods. The difference is caused by ignoring the influence of height above ground in Eq. (8), or more specifically, by assuming that the geometric mean distance Dm from one phase to the image of another phase is approximately equal to twice the geometric mean height. Almost identical results would be obtained with

C 2

In d hr D

pos0

m m

equiv m

=××

π ε2 (9)

where: hm = h h hA B C

3 ( geometric mean distance). dm = d d dAB AC BC

3 ( geometric mean distance ). Dm = D D DAB AC BC

3 ( geometric mean distance to images ). The differences would be less for lines of lower voltage ratings, because the phases would be closer

together. The formula for zero sequence capacitance in Eq. (8, 9).


58

C2

Inh D

r d

zero0

m m

equiv m

=×

×

π ε2 2

2

( Siemens ) (10)

can be derived by averaging the diagonal elements of Eq. (4) among themselves, as well as

averaging the off-diagonal elements among themselves, to account for transposition. Computer-oriented methods do the averaging in the elements of the [C] - matrix. Both give practically the same answer. For the line of Table 1, Eq. (10) produces a value for Czero which is 0.23% lower than the value obtained from computer-oriented methods. In [10], Eq. (10) is further simplified by assuming Dm = 2hm, or

C 2

In hr d

zero0

m

equiv m

=

×

π ε( )2 3

2

( Westinghouse ) (11)

which produces a value which is 4% higher than the value from computer-oriented methods for the

line of Table 1. While Eq. (11) is theoretically less accurate, the value obtained from it may actually be closer to measured values because the influence of towers on the zero sequence capacitance, which is neglected in all formulas, increases the calculated zero sequence capacitance. This increase is typically 8 to 9% on 110 kV lines, 6% on 220 and 380 kV lines, and 4% on 700 kV lines [ 11, p.218 ].

A7. Comparison for Sequence Impedance The formulas for zero and positive sequence impedances in most handbooks are based on the

assumption that parameter a in Eq. (3) is so small that only the first term in Caron’s infinite series need be retained. For normal phase spacings this is a reasonable assumption at power frequency (50 or 60 Hz). Then, after all diagonal and off-diagonal elements in [Z] of Eq. (1) have been averaged out among themselves, respectively, to account for transposition, the correction terms for the influence of the finite earth resistivity become

Δ ΔR Rself mutual= =× × −ϖ π 10

2

4 Ω/km (12)

and

ΔX In fself m= × − × × ×− −2 10 0 6159315 281 10 24 3ϖ

ρ[ . ( . )]h Ω/km

ΔX In fmutual m= × − × × ×− −2 10 0 6159315 2 81 104 3ϖ

ρ[ . ( . )]D Ω/km (13)

where: hm, Dm in m, f in Hz, and ρ in Ωm. With these correction terms, the zero and positive

sequence impedances can easily be derived from the self and mutual impedances, with Zpos = Zself - Zmutual (14) and Zzero = Zself + 2Zmutual (15)


59

Using the correction terms of Eq. (12) and (13) leads to the simple expression for the positive sequence impedance

Zpos = R j2 In dGMRac

m

equiv+ × −ϖ 10 4 in Ω/km (16)

which is found in all handbooks, with Rac = ac resistance of the bundle. It is surprising that the

influence of ground resistivity and of conductor height, which is present in the self and mutual impedances, disappears completely in Zpos of Eq. (16). This can easily be verified, however, if one knows that Zpos without earth resistivity correction terms* is

Zpos ( ΔR = ΔX = 0 ) = R j2 In2h d

GMR Dacm m

equiv m+ × −ϖ 10 4 in Ω/km (17)

Table 3 compares the results from the handbook formula (16) and from computer-oriented methods

with accurate earth resistivity correction terms for the 500 kV line of Table 1. In this comparison, skin effects within the conductors were intentionally ignored (Rac = Rdc), to clearly show the influence of earth resistivity. Table 3 shows that the handbook formulas are quite accurate for the inductance Lpos over a wide frequency range, whereas Rpos becomes less accurate as the frequency increases (0.33% difference at 100 Hz, but different by orders of magnitude at 100 kHz). The increases in Rpos for higher frequencies is caused by eddy currents in the ground, as indicated for a bipolar dc line or a single-phase ac line in Fig. A-4 (the phenomenon is similar in three-phase lines, but not as easy to illustrate as for a two-conductor line). Handbook formulas ignore this eddy current effect in the ground.

Fig. A-4

Eddy Currents in Earth


60

Table 3

Accurate and Approximate Positive Sequence Resistance and Inductance

ACCURATE

APPROXIMATE FROM EQ. (16)

f ( Hz )

Rpos ( Ω/mile )

Lpos ( mH/mile )

Rpos ( Ω/mile )

Lpos ( mH/mile )

10-6

0.04215

1.417

0.04215

1.417

10

0.04215

1.416

0.04215

1.417

100

0.4229

1.416

0.04215

1.417

1000

0.05003

1.416

0.04215

1.417

10000

0.3528

1.413

0.04215

1.417

100000

6.229

1.401

0.04215

1.417

The zero sequence impedance obtained with the correction terms of Eq. (12) and (13) is

Z ( R 3 102

) j6 10 In659 f

GMR dzero ac

44

equiv m23

= +×

+ ×

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

−−ϖ ϖ

ρ in Ω/km (18)

with f in Hz, ρ in Ωm, Rac in Ω/km, and all distances in m. This is the formula found in most

handbooks, including [8, 9, 10]. Table 4 compares the results from the handbook formula (18) with those obtained from computer-oriented methods with accurate earth resistivity correction terms. The zero-sequence resistance Rzero and inductance Lzero of the handbook formula are reasonably accurate in the lower frequency range (up to approximately 1 kHz), but differ appreciably at higher frequencies.


61

Table 4

Accurate and Approximate Zero Sequence Resistance and Inductance

ACCURATE

APPROXIMATE FROM EQ. (18)

f ( Hz )

Rzero ( Ω/mile )

Lzero ( mH/mile )

Rzero ( Ω/mile )

Lzero ( mH/mile )

10-6

0.04215

13.94

0.04215

13.94

10

0.08905

6.170

0.08980

6.158

100

0.4960

5.084

0.5187

5.046

1000

4.169

4.052

4.807

3.934

10000

32.12

3.164

47.69

2.823

100000

184.0

2.568

476.6

1.711

A8. Conclusions Differences in overhead line parameters obtained with computer-oriented methods and from

handbook formulas are usually small at power frequency (50 or 60 Hz). The bundling procedure based on matrix reduction and the use of equivalent conductors for bundles produce practically identical results. The zero and positive sequence capacitances may differ by approximately 4% for a typical 500 kV line. Of more practical importance is the increase in the value of the positive sequence resistance on lines with ground wires which are grounded at every tower. This increase reflects the losses caused by the circulating currents in the ground wire.

Line parameters at higher frequencies are required for switching and lightning surge studies, for

power line carrier studies, and for similar problems. Results in the high frequency range from handbook formulas and from computer-oriented methods can be quite different. This is not surprising, since most handbook formulas were not derived for frequencies much beyond power frequency.


62

A9. References [1] M.H. Hesse, “ Electromagnetic and electrostatic transmission - line parameters by digital

computer, ” IEEE Trans. Power App. Syst., vol. 82, pp. 282 - 291, June 1963. [2] J.R. Carson, “ Wave propagation in overhead wires with ground return, ” Bell System Techn.

Journal, vol. 5, pp. 539 - 554, 1926. [3] H.W. Dommel, discussion of “Electromagnetic Effects of Overhead Lines”, by IEEE Working

Group, IEEE Trans. Power App. Syst., vol. PAS-93, pp. 900-901, May/June 1974. [4] C. Gary, “Approche Complete de la Propagation Multifilaire en Haute Frequence per

Utilisation des Matrices Complexes” (“Complete Approach to Multiconductor Propagation at High Frequency with Complex Matrices”, in French), EdF Bulletin de la Direction des Etudes et Recherches, Serie B, no. 3 /4 pp. 5-20, 1976.

[5] A. Deri, G. Tevan, A. Semlyen and A. Castanheira, “The Complex Ground Return Plane, A

Simplified Model for Homogeneous and Multi-Layer Earth Return”, IEEE Trans. Power App. Syst., vol. PAS-100, pp. 3686-3693, Aug. 1981.

[6] L.M. Wedepohl and R.G. Wasley, “Wave Propagation in Polyphase Transmission Systems;

Resonance Effects Due to Discretely Bonded Earth Wires”, Proc. IEEE, vol. 112, pp. 2113-2119, Nov. 1965.

[7] General Electric Co., Transmission Line Reference Book 345kV and Above. New York: F.

Weidner & Son Printers, 1975. [8] E.V. Rziha, Starkstromtechnik-Taschenbuch fuer Elektrotechniker, (“Electric Power

Handbook for Electrical Engineers”, in German), Berlin: Wilhelm Ernst u. Sohn, 1960. [9] Siemens, Formel-und Taballenbuch fuer Starkstrom-Ingenieure, (“Handbook of Formulas

and Tables for Electric Power Engineers”, in German), Essen: Girardet 1965. [10] Westinghouse Electric Corp., Electrical Transmission and Distribution Reference Book.

Pittsburgh: Westinghouse Electric Corp., 1964. [11] H. Happoldt and D. Oeding, Elektrische Kraftwerke und Netze (“Electric Power Plants and

Networks”, in German), Berlin: Springer, 1978. * The self and mutual inductances without correction terms are calculated from formulas which have the same

form as Eq. (4), except that 12 0πε

is replaced by μπ2

, and r is replaced by GMR.

Advanced Transmission Line Parameters and Electric and Magnetic Fields Computation Program

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