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International Journal of Latest Research in Science and Technology
Volume 4, Issue 5: Page No.4-8, September-October 2015
ISSN (Online):2278-5299
G.Radhika, 2Dr.M.Suryakalavathi & 3K.Vamshi
Sr.Assistant Professor, VNRVJIET, Hyderabad, India
Professor, JNTUH, Hyderabad, India
VNRVJIET, Hyderabad, India
Email: [email protected], [email protected], [email protected]
Abstract- Economy is becoming increasingly dependent on electricity as a basic input. Transmission lines provide the means of
connecting the generating capacities to the load centres.Knowledge of the parameters of multi-conductor transmission lines is necessary in
analysing a number of problems in power-transmission systems. These parameters under power frequency i.e. 50Hz are required in order
to study load flow, system stability and fault levels. Transmission lines are electrically short at power frequency and it is permissible to
calculate series and shunt parameters separately. As the number of conductors increase, the complexity in calculating the matrices
increases with the use of classical methods. The complexity of the mathematical expressions involved in multi-conductor line problems is
reduced with the aid of digital computer program. The objective of this paper is to study the effect of transposition for a chosen 220kV twin
bundled over head transmission line with two ground wires having asymmetrical spacing between the conductors by representing with a
digital computer program using MATLAB. The Phase impedance matrix (3X3) calculated by eliminating earth wires contains unbalanced
impedances of (5x5) and also its corresponding sequence impedance matrix (3X3) contains intersequence mutual couplings and are
eliminated by doing Perfect Phase Transposition. The comparison shows the closer agreement between calculated values and those
obtained by the digital program. The same technique is valid for tower with any number of conductors.
Keywords - Transmission Line, Transposition, Phase Impedance Matrix, Sequence Impedance Matrix, MATLAB
Untransposed lines are reckoned to be the main source of
current unbalance in the transmission systems. To evaluate
the unbalanced conditions on transmission lines, method of
symmetrical components [4] is applied by expressing the
impedance of a transmission line as a positive, negative and
zero sequence components. The advantage of the application
of sequence components is that the size of the problem is
effectively reduced in compared to phase components
The asymmetrical spacing of lines results in the
unbalance of line self and mutual impedances in phase frame
of reference and hence unequal voltage drops even though the
currents are balanced and they can be made equal with the
help of transposition. It is defined as interchanging the
position of the line conductors at regular intervals along the
line so that each conductor occupies the original position of
every other conductor [1]. It is an effective way to equalize
the phase impedances and reduce the induction from normal
operating currents and voltages
In this paper, 220kV Overhead line with two ground
wires is presented with a horizontal phase conductor
configuration. Generally, Bundled conductors with two, four,
six and eight sub conductors are used for many reasons for
voltages exceeding 220kV.Twin Bundled Conductors are
chosen for this type of transmission line to present lesser
reactance per phase.
Modern computer technology gives an opportunity to
prevent the difficulties occurred during the calculation of
unbalance caused by transmission lines as well as to
implement the new approach in relation to operational speed
and accuracy of calculations [2].
The average height of the conductor including sag with
respect to earth is calculated by using [1].
Where, Hav is the average Height in mt
Ht is the total Height in mt
Sav is the average sag in mt
Figure shows 220kV single-circuit overhead line with a
symmetrical horizontal phase conductor configuration and
two earth wires. The cross numbered 1-5 represents the mid
span conductor positions due to conductor sag.
Figure 1: The 220KV S/C OH line geometry
International Journal of Latest Research in Science and Technology.
The spacing’s of the conductors relative to the centre of the
tower and earth [3], are given in the table.
So, after eliminating the earth wires the matrix size reduces to
3x3 as shown,
Table 1: Spacing’s of the Conductors with respect to
tower and earth
Average sag for
phase conductors
Average sag for earth
Average Height of phase conductors is calculated using
equation (1)
ZPhase is the phase impedance matrix of the unbalanced
three phase mutually coupled line. The matrix is symmetric
but the self or diagonal terms are unequal to each other, and
the mutual or off-diagonal terms on a given row or column
are unequal to each other. The impedance matrix in the
sequence frame of reference obtained by transforming the
phase impedance matrix by using eq (4).The Sequence
impedance matrix is calculated by assuming phase rotation of
RYB by using
Where H is the sequence to phase transformation matrix.
Average Height of earth conductors=
The Self impedance of every individual conductor and
ground wire is given as,
And the mutual impedances between the conductors ‘i’ & ‘j’
is given as,
Where, Derc is depth of equivalent earth return conductor.
The calculated self and mutual impedances are shown in a
matrix form,
Currents flowing in any one conductor will induce
voltage drops in the other two conductors and these may be
unequal even if the currents are balanced. This is because the
mutual impedances, which are dependent on the physical
spacing’s of the conductors, are unequal.
The 5x5 partitioned Z matrix obtained as shown below
including earth wires,
The sequence impedance matrix is full, non-diagonal and
non-symmetric. A non diagonal matrix means that mutual
coupling in the sequence circuits exist.
The conversion to sequence reference frame still
produces a full and even asymmetric sequence impedance
matrix that includes intersequence mutual couplings. Where
this intersequence mutual coupling is to be eliminated, the
circuit has to be perfectly transposed.
The calculated sequence series impedance matrices
include full intersequence mutual couplings. To eliminate the
intersequence mutual couplings, the line is to be perfectly
transposed. The objective of transposition is to produce equal
series self-impedances, and equal series mutual impedances
in the phase frame of reference.
Perfect phase transposition means that each phase
conductor occupies successively the same physical positions
as the other two conductors in two successive line sections
for equal length.
Case-1: Forward Successive Phase Transposition:
Figure 2: Forward Phase Transposition
In large-scale power system short-circuit analysis, we are
interested in the calculation of short-circuit currents on the
faulted phases R, Y or B or a combination of these but
generally not in the currents flowing in the earth wires.
The Figure.2 shows three sections of a transposed line.
This represents a perfect transposed line where the three
sections have equal length. Perfect transposition results in the
same total voltage drop for each phase conductor and hence
equal average series self-impedances of each phase conductor
and series phase mutual impedances.
International Journal of Latest Research in Science and Technology.
Shows a complete forward transposition cycle, i.e. three
transpositions where the line is divided into three sections
and t (top), m (middle) and b (bottom) are used to designate
the conductor physical positions on the tower. If the three
conductors of the circuit are designated C1, C2 and C3, then
a forward transposition is defined as one where the conductor
positions for the three sections are C1C2C3, C3C1C2 then
C2C3C1 as shown.
ZPhase is the phase impedance matrix of the perfectly
transposed line, noting that the individual conductor
impedances are symmetric.
Figure 3: Reverse Phase Transposition
The sectional matrices formed by general matrix approach for
reverse transposition cycle are shown below,
The above section matrics are formed by general matrix
analysis approach which is more suitable for modern
calculations by digital computers.
Where T is the transposition matrix given by,
The effect of pre-multiplying matrix ZSection-1 by matrix
T is to shift its row 2elements up to row 1, row 3 elements
up to row 2 and row 1 elements to row 3.Also, the effect of
post-multiplying matrix ZSection-1 by matrix T is to shift its
column 2 elements to column 1, column 3 elements to
column 2 and column 1elements to column 3.
Using eq.7
ZPhase is the phase impedance matrix of a balanced or
perfectly transposed line.
Case-2: Reverse Successive Phase Transposition
The Figure.3 shows Reverse transposition cycle
which is defined as one where the conductor positions for the
three sections are C1C2C3, C2C3C1 and C3C1C2.
For R, Y, B electrical phase rotation of conductors 1, 2 and 3
respectively, the sequence impedance matrix is calculated
International Journal of Latest Research in Science and Technology.
H is the Sequence to phase transformation matrix given by,
The below Figure’s 4,5, represents the Untransposed
phase impedance matrix showing that, the diagonal and off
diagonal elements are not equal and their corresponding
sequence impedance matrix contains inter circuit mutual
couplings which are made to zero.
In the sequence reference frame, only diagonal elements
are present and off diagonal elements are all made to zero.
Also, the positive and negative sequence components are
equal in the sequence impedance matrix.
Figure 4: Untransposed phase impedance matrix
In this sequence impedance matrix, mutual sequence
coupling is eliminated completely i.e., all the off diagonal
elements are made to zero.
(i) As explained the input data are read in. These
include physical geometry of the line, earth
resistivity, stranding factors appropriate to the
bundling arrangements, sag, resistivity and
frequency. Let p be the number of phase conductors
and q be the number of earth wires.
(ii) Calculating the average heights with respect to earth
using the formula [eq.1].
(iii) Calculating the distances between the conductors
and substituting the values in self and mutual
(iv) The Z matrix of size 5x5 is formed and by
eliminating the earth wires get reduced to 3x3 by
using the formula [A]-[B-1] [C] [D].
(v) Sequence matrix is formed from the phase matrix by
(vi) Transposed phase and sequence impedance matrices
are formed by using [eq.7].
Figure 5: Corresponding sequence impedance matrix
Figure. 6 show the perfectly transposed or balanced
phase impedance matrix, showing the bars of equal length in
the self and off diagonal positions.
Figure 6: Perfectly transposed phase impedance matrix
And figure 7 contains only diagonal elements i.e.,
Positive, negative and zero sequence components only which
shows that the mutual coupling is eliminated completely.
To overcome the complexity of manual iterative
calculations, a program is coded to generate phase and
sequence impedance matrices showing the effect of
transposition of 220 kV, double ground wired, single circuit
transmission tower.
International Journal of Latest Research in Science and Technology.
Figure 7: Corresponding Sequence impedance matrix
Thus three transpositions are sufficient to produce
balanced mutual phase impedance matrix and mutual
sequence couplings are entirely eliminated in the sequence
impedance matrix.
The transposed phase and sequence impedance matrices
for both forward and reverse phase transpositions obtained
the same result.
The output and mathematical calculation hence shows an
excellent resemblance in similarity by which it is proved that
the impedance calculation has been reduced the complexity
by using programming technique. The same program can be
used for any tower configurations i.e. for multiple circuit
Nasser Tleis, “Power system modeling and fault analysis”, newness
publications, 2008.
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electrical parameters for short and long polyphase transmission
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G.Radhika, Dr.M.Suryakalavathi, “Back flashover analysis
improvement of a 220kV transmission line”, International journal of
Engineering Research and Applications, Vol.3, Issue.1, pp.533-536,
Fortescue.C.L, “Method of Symmetrical Coordinates applied to the
solution of polyphase networks”, Transactions of AIEE, Vol.37,
pp.1027-1140, 1918.
Arif M.Gashimov, Aytek R.Babayeva, Ahmet Nayir, “Transmission
Line Transposition”, International Conference on Electrical &
Electronics Engineering, pp.364-367, 2009.