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Application of Levinson algorithm for fault calculations in high phase order transmission systems
Math1 Comput. Modelling, Vol. 11, pp. 321-324,
0895.7177/88 $3.00 + 0.00
1988
Printed in Great Britain
Pergamon
Press plc
APPLICATION OF LEVINSON ALGORITHM FOR FAULT CALCULATIONS IN HIGH PHASE ORDER TRANSMISSION SYSTEMS
R. P. Broadwater
A. Chandrasekaran Center for Electric Power Tennessee Technological University Cookeville, TN 38505 USA
Abstract. Ever increasing pressures against the construction of new high voltage -ion lines are forcing electric utilities to seek new ways of augmenting the power transmission capabilities of existing rights-of-way. One of the alternatives being contemplated is high phase order transmission, where the number of phases In order to make full utilization of is raised from three to six, twelve or higher. the idea, transmission line design with the phase conductors being located on the circumference of a circle is advocated. The primitive impedance matrix of such a configuration of phase conductors is a Toeplitz matrix and is also symmetric. A generalized approach based on the application of Levinson algorithm is developed in the present paper for the calculation of fault currents in an arbitrary N-phase transmission line. The fault currents due to faults involving any number of phases to ground are calculated using the algorithm. Results for six and twelve phase systems obtained using the method are included in the paper.
Keywords.
Power
Systems;
High Phase
Order
Transmission;
Fault
Calculations
1NTRODUCTION With the ever increasing difficulties in securing new rights of way, all possible attempts have to be made for augmenting the power transfer capabilities of existing transmission corridors. High phase order transmission proposed originally by Barthold has received wide consideration and Barnes (1972) during the last decade and feasibility reports have been prepared for Allegheny Power by Venkata and, Stewart and Wilson have others (1976-1979). made extensive studies on many aspects of high Experimental phase order transmission (1978) . six and twelve phase systems are being studied under contracts from the Department of Energy (Stewart, 1982-1987).
-
N-Phase Line
Sl, S2 Equivalent
N-phase
zl, 22, 23 Equivalent
Detailed fault analysis forms an important aspect in the design of high phase order systems. Venkata et al used symmetrical component analysis and phase variable analysis first (Bhatt, 1977) for calculating fault currents later (Venkata, 1982) However, the and voltages in six phase systems. model used was a ‘perfectly’ transposed impedance whereas the more practical configuration matrix, Bergseth is ‘roll transposed’ and compact design. (Discussion of Ref. 6) reported errors of 13% due The to the assumption of ‘perfect transposition’. EPRI project on ‘Fault Protection for High Phase Order Transmission Lines’ (Auburn University, 1984) also uses symmetrical component analysis.
FIG. 1. Equivalent
with those transposition.
System
obtained
Sources
N-phase
Impedances
with High Phase
on
the
basis
Order
of
Line
‘perfect’
SYSTEM MODEL The system equivalent model used in the following is shown in Figure 1. The high phase order transmission line (order N) is retained in full detail and the remaining parts of the system are replaced by N-phase equivalents at the terminals of the N-phase line. The sources are also converted into equivalent N-phase sources. Methods of obtaining the LJ-phase equivalent are discussed in Reference (7). As is cl stomary in fault analysis, the system is assumed ur!oaded and the resulting Thevenin Equivalent is shown in Figure 2. The source Dhase voltaaes ai balanced and El, E2, E3, . .-., EN are assumed given by El, E2 = El /-a, E3 = El/2a,. .. EN = ElkN-l)a, where a = PT/N.
The aim of the present paper is to generalize the fault analysis of a high phase order transmission Roll system for all types of faults to ground. traltsposition is assumed and the phase conductors are located on the circumference of a circle. Levinson solving a system with algorithm (1947) for symmetric Toeplitz coefficient matrix is used for calculating the total and phase fault currents. In more general, order to make the analysis normalization of fault current and system parameters Actual results calculated are shown for is used. six and twelve phase systems. Results are compared
321
Proc. 6th Int. Conf. on Mathematical
322
Modelling
Lz]12 phase
=
%
‘El, E2. “1, “2,
. . EN Source Voltages . . . VN Fault Voltages
Il, 12, . . . IN Fault
c1 zA
zB
%
zA
Zml
Zm2
Zm3
zm4
1
Zm5
[ZBI
Currents
FIG. 2. Equivalent Circuit of Unloaded HiPh Phase Order System
MATHEMATICAL
EQUATIONS FIG. 3. System
The general voltage equations can be written in matrix form as
for
(1)
[“I = [El - [Zl[Il
where [VI is the terminal voltage vector, [El the source voltage vector, [II the phase current vector If k phases (1 5 k and [Zl the impedance matrix. < N) are connected to ground at the fault terminals, Equation (1) can be partitioned into two sets of faulted and unfaulted phases represented by F and U, respectively.
Since
[Iul = [Ol = WFI, the
fault
[IFI = [ZFFI-1 The vector [IFI gives The total fault current
phase is
current
IEFI
currents
Impedance
Matrices
Figure
vector
.
is
The impedance matrix [ZFFI in Equation (3) is a principal submatrix of [Zl and is of order k. It can be seen from Figure 3 that the impedance matrix of a ‘roll transposed’ line is symmetric Toeplitz type, where only one row (column) need be specified to define the entire matrix. Further any principal submatrix is also a symmetric Toeplitz matrix. Toeplitz matrices are encountered in various applications like signal processing, spectral estimation, linear estimation and error control codes. Levinson Algorithm is an efficient method of solving symmetric Toeplitz equations (Bose, 1985) [9] and is used to solve equation (3) for various types of faults corresponding to I: varying from 1 to N. Before applying the algorithm, the system impedance matrix is normalized for getting general results. NORMALIZATION
(3) during
the fault. as the various
OF PARAMETERS
Single line to ground fault current is taken base for comparing the fault currents in the cases. This current is given by
k Ifault
=i:l Ii
SYSTEM IMPEDANCE
.
where Ephase is the phase impedance of each phase.
MATRIX
The structure of the system impedance matrix [Zl is assumed for a ‘roll transposed’ line in this Typical structures for six and twelve phase paper. systems are shown in Figure 3. zm2
zm3
Zm2
Zml
Zml zs
Zml
Zm2
Zm3
Zm2
Zm2zml
zs
Zml
Zm2
Zm3
Zm3 zm2
Zml
zs
Zml
Zm2
Zm2zm3
Zm2
Zml
zs
Zml
zmlzm2
Zm3
Zm2
zml
zs
ZS
[Zl 6ph =
zml
(a)
(5)
IL-G = Ephase/zs
_
voltage
and
zs the
self
For normalizing the system constants, the ratio zml/zs is chosen as a parameter where zml is the mutual impedance between two neighboring phase conductors on the circle. Other ratios like zm2/zs, zm3/zs can be expressed in terms of zml/zs by the following equations. and zs as explained Assume the resistances are negligible. zs = xs
(6)
z m = xm
(71
zm2 _=_=_
Xm2
ZS
xs
xml
+ p2 xs
(8)
Proc.
Here P2 is a constant the distances between zm3/zs, zm41zs, etc., of xmI/xs and P3/xs, P2, Pa, P4l . * .I depend distances. An example for a three phase line.
6th Int. Conf. on Mathematical
323
Modelling
determined by the ratio of the conductors. Similarly, can be expressed in terms P4/xs, etc. The constants upon the ratiosinterconductor is given in the Appendix 2.0
The ratio zmI/zs depends on distance between adjacent conductors and a range of 0.2 to 0.6 is chosen as typical for high voltage lines. The self reactance xs has a range of 0.345 ohms/mile to 0.476 ohms/mile (Stevenson, 1982) [lOI. The matrix
. . . -z, zs1
[ZFFI is written
1
[ZFFI =
Zml
as follows:
Zm2
1.5 2 .E E e ; ” f ?I L 0.5
1
zs
1.0
Toeplitz Symmetric
Plus
Zm/Zs=0.2
Cross
Zm/Zs=0.4
Triangle
Zm/Zs=O.6
\
‘,
‘*,
‘,
‘,
‘,
1
‘a
\ ‘\.
0.c
2
Number
-
+, \
4
3
\
‘\
5
of faulted
6
phases
Choosing zmI/zs as 0.2, 0.4, and 0.6 and z, as average value of 0.410, various cases are calculated. CALCULATION
OF FAULT
FIG. 4. Fault Currents an Six Phase System Solid Line: Perfectly Transposed Dot Line: ‘Roll’ Transposed Phase Current DashRoll’ Transposed Total Current.
CURRENTS
In Equation (31, the inverse of the matrix [ZFF] multiplies the source voltages corresponding to Since adjacent phases have the faulted phases. the minimum phase diffeence between them, it can be assumed that the corresponding fault currents Hence for a k-phase fault only would be maximum. adjacent phases are taken into consideration in the calculation of fault currents.
4.0
A computer program is written for the solution of Equation (3) using Levinson Algorithm. For any k-phase fault, the currents are calculated for the range of the parameter values mentioned in the previous section. RESULTS
3.5
Plus Cross
Zm/Zs=02 Zm/Zs = 0.4
Tnangle
Zm/Zs
/. 3.0
\\
\\
,+’
-
\
/’
+
I’
2 .s
2.5
\ , \
I+
-
I’ ,’
E EI ; 2.0u
\ \
,4
\ \ \
,’
=
+
1.5 -
“6
1.0 -
phase systems are transposed values
It may be seen from Figures 4 and 5 that the phase fault currents calculated on the assumptions of ‘perfect’ transposition are in general higher than for ‘roll’ transposition. The error can be as high as 60%. The effect of zm/zs ratio is not significant in twelve phase systems in the chosen range. The total fault currents can be quite high for low values of zr&s.
%
I’
0.5
Similar graphs for twelve Perfectly shown in Figure 5. for phase currents are also given.
‘\ f’
AND CONCLUSIONS
Six and twleve phase systems have been analysed The magnitudes of using the method described. the maximum phase current and total fault current are shown in Figure 4 for various types of ground faults in a six phase system. An average phase conductor radius of 0.0373’ corresponding to xs = 0.41 ohm/mile is chosen. For three values of x,/x,, For comparison the phase the results are plotted. fault currents are also plotted for the ‘perfectly’ transposed case.
,,+---+\
: 0.6
-
o,o’
-
\
,
,
I
I
I
I
I
123456789
Number
of faulted
a. .‘x.. i s--,
‘,
10
12
II
phases
FIG. 5. Fault Currents in a la-Phase System Solid Line: ‘Perfectly’ Transposed (Phase Current) Dot Line: ‘Roll’ Transposed (Phase Current) Dashmoll’ Transposed (Total Current).
Proc. 6th Int. Conf. on Muthematical
324
TABLE 1. Effect of Condtictor Radius on Total Fault Current--Six Phase System
No. of Phases
Range
to Ground
of Total
z,/zs=o.
3 4 5
No. of Phases to Grounl
2.
Effect Fault
4
7.
Stewart, J.R. :tnd Wilson, D.D., “High Phase Feasibility Analysis, Transmission--A Order Pt I: Steady State Considerations; Pt II: Requirements”, and Insulation Overvoltages ILEE Transactions on Power Apparatus and Systems vol. PAS-97, Nov./Dee. 1978, op. 2300-2317.
8.
Stewart, J.R., “High Phase Order Transmission”, Department of Energy, ORNL-00212, Dec. 84April 87.
to Ground zm/zs=O.
6
0.88-0.89 0.52-0.76 0.17-0.32
S.S., “Feasibility Studies of Higher 9. Venkata, Transmission Order Electrical Phase Systems-Phases I, 11 and 111 Reports to Allegheny Pennsylvania, System, Greensburg, Power 1976-1979.
of Conductor Radius on Total Current--l2 Phase System
10.
B
ange
3 4 5 6 7 8 9 10 11
Current
1.12-1.13 0.76-0.76 0.30-0.32
1.49-1.52 1.14-1.17 0.54-0.54
TABLE
Fault z,/zs=o.
2
of Total
zm/zs=o.
2.12-2.20 2.54-2.81 2.86-3.44 3.03-4.14 3.01-4.97 2.76-6.18 2.24-8.98 1.50-6.99 0.66-3.00
2
Fault
Current
to Ground
z,/z,=o.4
z,/z,=O.
1.61-1.66 1.73-1.84 1.72-1.90 1.60-1.84 1.37-1.64 1.07-1.32 0.74-0.92 0.41-0.51 0.15-0.18
1.30-1.33 1.29-1.35 1.21-1.29 1.05-1.14 0.85-0.93 0.62-0.68 0.40-0.43 0.20-0.21 0.06-0.07
Modelling
6
W.C., Booth, W.H., Venkata, S.S., Guyker, Kondranunta. J., Saini. N.K.. Stanek. E.K.. Fault “138-kc Six’ Phase Transmission System: Analysis”, IEEE Transactions on Power Apparatus May 1982, pp. vol. PAS-101, and Systems, 1203-1218.
APPENDIX Consider a in a horizontal between adjacent
3-phase line configuration conductors.
We can write
the following
The effect of conductor radius, i.e., the value of zs on the total fault currents for various types of faults is shown in Tables 1 and 2. It maybe observed that the spreads about the average conductor radius are fairly small for both six phase and twelve phase systems.
REFERENCES 1.
xml
2.
Bhatt, N.B., Venkata, S.S., Guyker, W.C., Booth, (multi-DhaSe) Power W.H., “Six-Phase Fault ‘Analysis”, IEEE Transmission Systems: Transactions on Power Apparatus and Systems, vol. PAS-96, May/June 1977, pp. 758-767. and
Bose, N.K., Applications”,
4.
EPRI EL-3316, Project 1764-5, “Fault Protection for Higher Phase Order Transmission Line--vol. 1, Basic Concepts, Models and Computations; vol. 2, Computer Codes”, Auburn University, 1984.
5.
Levinson, N., “The Wiener RMS (root mear, error in filter design”, Journal 01 square) Mathematical Physics, vol. 25, 1947, pp. 261-278.-
6.
Stevenson, Analysis”,
W.D., “Elements 4th Ed., McGraw-Hill,
of
Power System 1982 (book).
=
Here K is an appropriate mean radius.
K log
1 ?iYY
K log1
‘%
K log
K log1
xs
xs
K log; D
+K log1 r
P p2
and r the geometric
K log1
K log1
Xml =-+_
1 . -55
constant
=-
-=
Here P2 = K log l/2, Filters, Theory “Digital North-Holland, 1985 (book).
3.
=
D
Xm2
Barnes, H.C., Barthold, L.O., “High Phase Order Power Transmission”, CIGRE Report No. 31 ELECTRA No. 24, October 1972, pp. 139-153.
equation:
xs = K log1, r
xm2
The general conclusion drawn is that ground fault currents obtained on the assumption of ‘perfect’ transposition are higher than those calculated for ‘roll’ transposed systems.
with the conductors at a distance of D
a constant.
r
0895.7177/88 $3.00 + 0.00
1988
Printed in Great Britain
Pergamon
Press plc
APPLICATION OF LEVINSON ALGORITHM FOR FAULT CALCULATIONS IN HIGH PHASE ORDER TRANSMISSION SYSTEMS
R. P. Broadwater
A. Chandrasekaran Center for Electric Power Tennessee Technological University Cookeville, TN 38505 USA
Abstract. Ever increasing pressures against the construction of new high voltage -ion lines are forcing electric utilities to seek new ways of augmenting the power transmission capabilities of existing rights-of-way. One of the alternatives being contemplated is high phase order transmission, where the number of phases In order to make full utilization of is raised from three to six, twelve or higher. the idea, transmission line design with the phase conductors being located on the circumference of a circle is advocated. The primitive impedance matrix of such a configuration of phase conductors is a Toeplitz matrix and is also symmetric. A generalized approach based on the application of Levinson algorithm is developed in the present paper for the calculation of fault currents in an arbitrary N-phase transmission line. The fault currents due to faults involving any number of phases to ground are calculated using the algorithm. Results for six and twelve phase systems obtained using the method are included in the paper.
Keywords.
Power
Systems;
High Phase
Order
Transmission;
Fault
Calculations
1NTRODUCTION With the ever increasing difficulties in securing new rights of way, all possible attempts have to be made for augmenting the power transfer capabilities of existing transmission corridors. High phase order transmission proposed originally by Barthold has received wide consideration and Barnes (1972) during the last decade and feasibility reports have been prepared for Allegheny Power by Venkata and, Stewart and Wilson have others (1976-1979). made extensive studies on many aspects of high Experimental phase order transmission (1978) . six and twelve phase systems are being studied under contracts from the Department of Energy (Stewart, 1982-1987).
-
N-Phase Line
Sl, S2 Equivalent
N-phase
zl, 22, 23 Equivalent
Detailed fault analysis forms an important aspect in the design of high phase order systems. Venkata et al used symmetrical component analysis and phase variable analysis first (Bhatt, 1977) for calculating fault currents later (Venkata, 1982) However, the and voltages in six phase systems. model used was a ‘perfectly’ transposed impedance whereas the more practical configuration matrix, Bergseth is ‘roll transposed’ and compact design. (Discussion of Ref. 6) reported errors of 13% due The to the assumption of ‘perfect transposition’. EPRI project on ‘Fault Protection for High Phase Order Transmission Lines’ (Auburn University, 1984) also uses symmetrical component analysis.
FIG. 1. Equivalent
with those transposition.
System
obtained
Sources
N-phase
Impedances
with High Phase
on
the
basis
Order
of
Line
‘perfect’
SYSTEM MODEL The system equivalent model used in the following is shown in Figure 1. The high phase order transmission line (order N) is retained in full detail and the remaining parts of the system are replaced by N-phase equivalents at the terminals of the N-phase line. The sources are also converted into equivalent N-phase sources. Methods of obtaining the LJ-phase equivalent are discussed in Reference (7). As is cl stomary in fault analysis, the system is assumed ur!oaded and the resulting Thevenin Equivalent is shown in Figure 2. The source Dhase voltaaes ai balanced and El, E2, E3, . .-., EN are assumed given by El, E2 = El /-a, E3 = El/2a,. .. EN = ElkN-l)a, where a = PT/N.
The aim of the present paper is to generalize the fault analysis of a high phase order transmission Roll system for all types of faults to ground. traltsposition is assumed and the phase conductors are located on the circumference of a circle. Levinson solving a system with algorithm (1947) for symmetric Toeplitz coefficient matrix is used for calculating the total and phase fault currents. In more general, order to make the analysis normalization of fault current and system parameters Actual results calculated are shown for is used. six and twelve phase systems. Results are compared
321
Proc. 6th Int. Conf. on Mathematical
322
Modelling
Lz]12 phase
=
%
‘El, E2. “1, “2,
. . EN Source Voltages . . . VN Fault Voltages
Il, 12, . . . IN Fault
c1 zA
zB
%
zA
Zml
Zm2
Zm3
zm4
1
Zm5
[ZBI
Currents
FIG. 2. Equivalent Circuit of Unloaded HiPh Phase Order System
MATHEMATICAL
EQUATIONS FIG. 3. System
The general voltage equations can be written in matrix form as
for
(1)
[“I = [El - [Zl[Il
where [VI is the terminal voltage vector, [El the source voltage vector, [II the phase current vector If k phases (1 5 k and [Zl the impedance matrix. < N) are connected to ground at the fault terminals, Equation (1) can be partitioned into two sets of faulted and unfaulted phases represented by F and U, respectively.
Since
[Iul = [Ol = WFI, the
fault
[IFI = [ZFFI-1 The vector [IFI gives The total fault current
phase is
current
IEFI
currents
Impedance
Matrices
Figure
vector
.
is
The impedance matrix [ZFFI in Equation (3) is a principal submatrix of [Zl and is of order k. It can be seen from Figure 3 that the impedance matrix of a ‘roll transposed’ line is symmetric Toeplitz type, where only one row (column) need be specified to define the entire matrix. Further any principal submatrix is also a symmetric Toeplitz matrix. Toeplitz matrices are encountered in various applications like signal processing, spectral estimation, linear estimation and error control codes. Levinson Algorithm is an efficient method of solving symmetric Toeplitz equations (Bose, 1985) [9] and is used to solve equation (3) for various types of faults corresponding to I: varying from 1 to N. Before applying the algorithm, the system impedance matrix is normalized for getting general results. NORMALIZATION
(3) during
the fault. as the various
OF PARAMETERS
Single line to ground fault current is taken base for comparing the fault currents in the cases. This current is given by
k Ifault
=i:l Ii
SYSTEM IMPEDANCE
.
where Ephase is the phase impedance of each phase.
MATRIX
The structure of the system impedance matrix [Zl is assumed for a ‘roll transposed’ line in this Typical structures for six and twelve phase paper. systems are shown in Figure 3. zm2
zm3
Zm2
Zml
Zml zs
Zml
Zm2
Zm3
Zm2
Zm2zml
zs
Zml
Zm2
Zm3
Zm3 zm2
Zml
zs
Zml
Zm2
Zm2zm3
Zm2
Zml
zs
Zml
zmlzm2
Zm3
Zm2
zml
zs
ZS
[Zl 6ph =
zml
(a)
(5)
IL-G = Ephase/zs
_
voltage
and
zs the
self
For normalizing the system constants, the ratio zml/zs is chosen as a parameter where zml is the mutual impedance between two neighboring phase conductors on the circle. Other ratios like zm2/zs, zm3/zs can be expressed in terms of zml/zs by the following equations. and zs as explained Assume the resistances are negligible. zs = xs
(6)
z m = xm
(71
zm2 _=_=_
Xm2
ZS
xs
xml
+ p2 xs
(8)
Proc.
Here P2 is a constant the distances between zm3/zs, zm41zs, etc., of xmI/xs and P3/xs, P2, Pa, P4l . * .I depend distances. An example for a three phase line.
6th Int. Conf. on Mathematical
323
Modelling
determined by the ratio of the conductors. Similarly, can be expressed in terms P4/xs, etc. The constants upon the ratiosinterconductor is given in the Appendix 2.0
The ratio zmI/zs depends on distance between adjacent conductors and a range of 0.2 to 0.6 is chosen as typical for high voltage lines. The self reactance xs has a range of 0.345 ohms/mile to 0.476 ohms/mile (Stevenson, 1982) [lOI. The matrix
. . . -z, zs1
[ZFFI is written
1
[ZFFI =
Zml
as follows:
Zm2
1.5 2 .E E e ; ” f ?I L 0.5
1
zs
1.0
Toeplitz Symmetric
Plus
Zm/Zs=0.2
Cross
Zm/Zs=0.4
Triangle
Zm/Zs=O.6
\
‘,
‘*,
‘,
‘,
‘,
1
‘a
\ ‘\.
0.c
2
Number
-
+, \
4
3
\
‘\
5
of faulted
6
phases
Choosing zmI/zs as 0.2, 0.4, and 0.6 and z, as average value of 0.410, various cases are calculated. CALCULATION
OF FAULT
FIG. 4. Fault Currents an Six Phase System Solid Line: Perfectly Transposed Dot Line: ‘Roll’ Transposed Phase Current DashRoll’ Transposed Total Current.
CURRENTS
In Equation (31, the inverse of the matrix [ZFF] multiplies the source voltages corresponding to Since adjacent phases have the faulted phases. the minimum phase diffeence between them, it can be assumed that the corresponding fault currents Hence for a k-phase fault only would be maximum. adjacent phases are taken into consideration in the calculation of fault currents.
4.0
A computer program is written for the solution of Equation (3) using Levinson Algorithm. For any k-phase fault, the currents are calculated for the range of the parameter values mentioned in the previous section. RESULTS
3.5
Plus Cross
Zm/Zs=02 Zm/Zs = 0.4
Tnangle
Zm/Zs
/. 3.0
\\
\\
,+’
-
\
/’
+
I’
2 .s
2.5
\ , \
I+
-
I’ ,’
E EI ; 2.0u
\ \
,4
\ \ \
,’
=
+
1.5 -
“6
1.0 -
phase systems are transposed values
It may be seen from Figures 4 and 5 that the phase fault currents calculated on the assumptions of ‘perfect’ transposition are in general higher than for ‘roll’ transposition. The error can be as high as 60%. The effect of zm/zs ratio is not significant in twelve phase systems in the chosen range. The total fault currents can be quite high for low values of zr&s.
%
I’
0.5
Similar graphs for twelve Perfectly shown in Figure 5. for phase currents are also given.
‘\ f’
AND CONCLUSIONS
Six and twleve phase systems have been analysed The magnitudes of using the method described. the maximum phase current and total fault current are shown in Figure 4 for various types of ground faults in a six phase system. An average phase conductor radius of 0.0373’ corresponding to xs = 0.41 ohm/mile is chosen. For three values of x,/x,, For comparison the phase the results are plotted. fault currents are also plotted for the ‘perfectly’ transposed case.
,,+---+\
: 0.6
-
o,o’
-
\
,
,
I
I
I
I
I
123456789
Number
of faulted
a. .‘x.. i s--,
‘,
10
12
II
phases
FIG. 5. Fault Currents in a la-Phase System Solid Line: ‘Perfectly’ Transposed (Phase Current) Dot Line: ‘Roll’ Transposed (Phase Current) Dashmoll’ Transposed (Total Current).
Proc. 6th Int. Conf. on Muthematical
324
TABLE 1. Effect of Condtictor Radius on Total Fault Current--Six Phase System
No. of Phases
Range
to Ground
of Total
z,/zs=o.
3 4 5
No. of Phases to Grounl
2.
Effect Fault
4
7.
Stewart, J.R. :tnd Wilson, D.D., “High Phase Feasibility Analysis, Transmission--A Order Pt I: Steady State Considerations; Pt II: Requirements”, and Insulation Overvoltages ILEE Transactions on Power Apparatus and Systems vol. PAS-97, Nov./Dee. 1978, op. 2300-2317.
8.
Stewart, J.R., “High Phase Order Transmission”, Department of Energy, ORNL-00212, Dec. 84April 87.
to Ground zm/zs=O.
6
0.88-0.89 0.52-0.76 0.17-0.32
S.S., “Feasibility Studies of Higher 9. Venkata, Transmission Order Electrical Phase Systems-Phases I, 11 and 111 Reports to Allegheny Pennsylvania, System, Greensburg, Power 1976-1979.
of Conductor Radius on Total Current--l2 Phase System
10.
B
ange
3 4 5 6 7 8 9 10 11
Current
1.12-1.13 0.76-0.76 0.30-0.32
1.49-1.52 1.14-1.17 0.54-0.54
TABLE
Fault z,/zs=o.
2
of Total
zm/zs=o.
2.12-2.20 2.54-2.81 2.86-3.44 3.03-4.14 3.01-4.97 2.76-6.18 2.24-8.98 1.50-6.99 0.66-3.00
2
Fault
Current
to Ground
z,/z,=o.4
z,/z,=O.
1.61-1.66 1.73-1.84 1.72-1.90 1.60-1.84 1.37-1.64 1.07-1.32 0.74-0.92 0.41-0.51 0.15-0.18
1.30-1.33 1.29-1.35 1.21-1.29 1.05-1.14 0.85-0.93 0.62-0.68 0.40-0.43 0.20-0.21 0.06-0.07
Modelling
6
W.C., Booth, W.H., Venkata, S.S., Guyker, Kondranunta. J., Saini. N.K.. Stanek. E.K.. Fault “138-kc Six’ Phase Transmission System: Analysis”, IEEE Transactions on Power Apparatus May 1982, pp. vol. PAS-101, and Systems, 1203-1218.
APPENDIX Consider a in a horizontal between adjacent
3-phase line configuration conductors.
We can write
the following
The effect of conductor radius, i.e., the value of zs on the total fault currents for various types of faults is shown in Tables 1 and 2. It maybe observed that the spreads about the average conductor radius are fairly small for both six phase and twelve phase systems.
REFERENCES 1.
xml
2.
Bhatt, N.B., Venkata, S.S., Guyker, W.C., Booth, (multi-DhaSe) Power W.H., “Six-Phase Fault ‘Analysis”, IEEE Transmission Systems: Transactions on Power Apparatus and Systems, vol. PAS-96, May/June 1977, pp. 758-767. and
Bose, N.K., Applications”,
4.
EPRI EL-3316, Project 1764-5, “Fault Protection for Higher Phase Order Transmission Line--vol. 1, Basic Concepts, Models and Computations; vol. 2, Computer Codes”, Auburn University, 1984.
5.
Levinson, N., “The Wiener RMS (root mear, error in filter design”, Journal 01 square) Mathematical Physics, vol. 25, 1947, pp. 261-278.-
6.
Stevenson, Analysis”,
W.D., “Elements 4th Ed., McGraw-Hill,
of
Power System 1982 (book).
=
Here K is an appropriate mean radius.
K log
1 ?iYY
K log1
‘%
K log
K log1
xs
xs
K log; D
+K log1 r
P p2
and r the geometric
K log1
K log1
Xml =-+_
1 . -55
constant
=-
-=
Here P2 = K log l/2, Filters, Theory “Digital North-Holland, 1985 (book).
3.
=
D
Xm2
Barnes, H.C., Barthold, L.O., “High Phase Order Power Transmission”, CIGRE Report No. 31 ELECTRA No. 24, October 1972, pp. 139-153.
equation:
xs = K log1, r
xm2
The general conclusion drawn is that ground fault currents obtained on the assumption of ‘perfect’ transposition are higher than those calculated for ‘roll’ transposed systems.
with the conductors at a distance of D
a constant.
r