Energy and Power Engineering, 2013, 5, 1490-1496
doi:10.4236/epe.2013.54B282 Published Online July 2013 (http://www.scirp.org/journal/epe)
An Alternative Three-phase Transmission Lines
Model in Phase Domain
C. G. Carvalho1, S. Kurokawa1, J. Pissolato2
1Universidade Estadual Paulista (Unesp), Electrical Engineering Department, Ilha Solteira, Brazil
2Universidade Estadual de Campinas (Unicamp), Electrical Engineering Department, Campinas, Brazil
Email: carol.gcarvalho@hotmail.com
Received 2013
ABSTRACT
The objective of this paper is to show an alternative model of a non-transposed three-phase transmission line with a
vertical symmetry plane in phase domain. Due the line physical characteristics, it can be represented by a system
consisting of a single-phase and a two-phase line. In this system, the equations describing the behavior of the values in
single-phase line terminals are known, while the equations for two-phase line to be obtained. Using a transformation
matrix written explicitly according to three-phase line parameters, it is possible to obtain the currents and voltages in
phase domain of two-phase line. Then, modal values of three-phase line are converted into phase domain and thus ob-
tain the analytical model for this line. To verify the performance of this model, it was used to simulate the energization
of a 440 kV three-phase line and the results were compared to results obtained using a classical model.
Keywords: Transmission Lines; Frequency Dependence; Modal Representation; Transformation Matrix;
Newton-Raphson; Phase Domain
1. Introduction
It is known that methods of electromagnetic transient
analysis in power systems can consider transmission line
models written in phase domain or in modal domain [1].
If the transmission line is represented by its modes, the
line operations are effectuated in modal domain and after
that transformed back to phase domain. This way a
transmission line with n phases can be decoupled in its n
exact modes and each mode can be represented as being
a single-phase line transmission.
In multiphase lines, transformation phase-mode-phase
is done by a modal transformation matrix [2]. Because
per unit length longitudinal impedance matrix and shunt
admittance matrix are frequency dependence, modal
transformation matrices are frequency dependence too.
Therefore the modal transformation matrices are usually
obtained by numerical methods that not permitting the
development of an analytical model this line for.
Thereby, if it is obtained explicitly a function that lists
the elements of the transformation matrix with the line
parameters, it is possible to develop a direct model in
phase’s domain of a transmission line, depending only of
its parameters.
Advantages principal of a transmission line model de-
veloped directly in phase domain are: the model devel-
oped in phases domain, the currents and voltages can be
obtained in any situation of analysis line (e.g. in analysis
line considering the line in phase-ground and\or phase-
phase short circuit), which cannot be easily done in mo-
dal model; all terms present in equations that represent
this model are functions only of [Z] and [Y] parameters,
which will allow representing the phases in time domain.
This paper proposes to show the development of a
model for a non-transposed three-phase transmission line
with a vertical symmetry plane in phase modal.
2. Alternative Model for Three-Phase
Transmission lines
A transmission lines with phases are characterized by it’s
per unit length longitudinal impedance, matrix [Z], and
shunt admittance, matrix [Y]. If this line is represented in
modal domain, has the following relationship [2]:
[][][][]
t
mI I
Z
TZT
(1)
-1
[ ][][][]
t
mI I
YTYT
(2)
In (1) and (2), the matrices [Zm] and [Ym] are per unit
length impedance and admittance modal matrices, respec-
tively. Matrices [Zm] and [Ym] are diagonal matrices and
[TI] is a transformation matrix whose columns are the
eigenvectors associated with the eigenvalues of the
product [Y][Z]. In (1) [TI]t is the transpose of [TI] and in
(2) [TI]-t is the inverse of [TI]t.
Copyright © 2013 SciRes. EPE
C. G. CARVALHO ET AL. 1491
In modal domain, a multiphase line with n phases is
represented by their n propagation modes which behave
as n decoupled single-phase lines. Figure 1 shows a
generic mode of the multiphase transmission line.
In Figure 1, EA e EB are, respectively, modal voltages
at A and B terminals, respectively, while ImA and ImB are,
respectively, modal currents at A and B terminals,
written in modal domain.
The relationship between phase and modal values are
obtained as follows [3]:
mB
cosh() -Isinh()
AB C
EEd Zd
(3)
mB
-I cosh()sinh( )
B
mA C
E
I
dd
Z
 (4)
In (3) and (4), the terms γ and ZC are, respectively,
modal propagation function and modal characteristic
impedance [4, 5].
Considering a generic three-phase transmission line,
this line will be represented in modal domain by its three
propagation modes. Thus, applying (3) and (4) for each
mode, have the following voltage and current vectors in
modal domain:
mB
[][][]-[B][I]
AB
EAE (5)
B
[][C][E ][][]
mA mB
I
DI (6)
In (5) e (6), the matrix [D] is equal to the matrix [A]
and the matrices [A], [B] and [C] are written as:
1
2
3
cosh ()00
[]0cosh()0
00cosh(
m
m
m
d
Ad
d
)
)
(7)
11
22
33
sinh ()00
[]0sinh()0
00sinh(
Cm m
Cm m
Cm m
Zd
BZd
Z
d

(8)
1
1
2
2
3
3
1sinh ()00
1
[]0sinh( )0
1
00sinh(
m
Cm
m
Cm
m
Cm
d
Z
Cd
Z
d
Z
)








(9)
Figure 1. A generic mode multiphase transmission line.
It is known that phase and mode values so satisfy the
following relationships [3]:
[][ ][]
t
I
ETV
(10)
1
[][][]
mI
TI
(11)
In (10), the vectors [V] and [E], are transversal
voltages written in phase and modal domain, respectively,
an l
cu
(5) and (6), and making the
appropriate mathematical manipul
d in (11), the vectors [I] and [Im], are the longitudina
rrents written in phase and modal domain, respectively.
Replacing (10) and (11) in
ations, are obtained:
1
IB
[ ][][][][ ][][][][]
tt t
AI IBI
VTATVTBTI

(12)
1
[][][][][ ][][][][]
t
AI IBIIB
I
TCTVTDTI
(13)
Equations (12) and (13) show the relationships between
phase values at A and B terminals of a generic three-
phase line. The vectors [VA] and [VB] contain the phase
vo
s,
respectively. Matrices [TI], [A], [B], [C] and [
obtained according to three-phase line parameters.
Equations (12) and (13) represent an equat
th
ltages in A and B terminals, respectively, while [IA]
and [IB] contain the phase currents in A and B terminal
D] are
ion model
at possibility to obtain the phase values for a generic
three-phase transmission line. However, the development
of this model is only possible if the matrix [TI] is
obtained analytically.
It is known that [TI] columns are eigenvectors
associated to the product [Y][Z] [2]. Therefore, it is
possible to obtain explicitly a function that represents the
elements of each column of the matrix [TI] according to
the three-phase line parameters. In this case, to get the
elements of the matrix [TI] is necessary, first, calculate
the eigenvalues corresponding to the product [Y][Z]
through [2].
However, for a generic three-phase transmission line,
the obtaining these eigenvalues depends on the polynomial
solution of degree 3, which is not easily found.
A solution in this case is to consider a non-transposed
three-phase transmission line with a vertical symmetry
plane. This line can be represented by a system
consisting of a single-phase and two-phase line [6]. Then
the three-phase line is separated in its propagation modes
by using two modal transformation matrices: the first is
the Clarke’s matrix [7]; the second is an adequate modal
transformation matrix whose elements are frequency
dependences [2, 6].
This way, to represent a three-phase line directly in
phase domain, using (12) and (13), it is need to obtain,
analytically, the polynomial solution of degree 2. There-
fore, as well as in single-phase line, the currents and vol-
tages of a three-phase line are obtained directly in the
phase domain.
Figure 2 shows a non-transposed three-phase trans-
Copyright © 2013 SciRes. EPE
C. G. CARVALHO ET AL.
1492
mission line with a vertical symmetry plane.
For this line, the longitudinal impedance [Z] and tran-
sverse admittance [Y] are written as [7]:
11 12 12
12 22 23
12 23 22
[]
zzz
Z
zzz
zzz




(14)
yyy

11 1212
12 22 23
12 23 22
[]Yyyy
yyy



(15)
The line in Figure 2 can be represented as a system
consisting by a single-phase line and two-phase line. In
this representation, the three-phase line can be decoupled
into exact modes by using two tra
[6]. Initially, Clarke’s matrix is used
line in their quasi-modes α and 0 and its exact mode β
and, after that, an adequate modal tran
decouples the quasi-modes α and 0
Figure 3 shows a schematic representation of the modal
de
(17)
nsformation matrices
for decoupling the
sformation matrix
into exact mode.
composition process by using two transformation
matrices.
If in (1) and (2) the transformation matrix [TI] is
substituted by Clarke’s matrix, the impedance and
admittance, in this case, are written, respectively, as being:
0
[][][][]
t
clarke clarke
ZTZT

(16)
-1
0
[][][][]
t
clarke clarke
YTYT

Considering the line parameters, equations (14) and
(15), the matrices [Zαβ0] and [Yαβ0], are rewritten, respec-
tively, as [6,7]:
1112 22 231112 22 23
02223
12
(24) 0()
33
[] 00
zzzz zzzz
Zzz


 



(18)
11 12 22 231112
21
2
()0(4)
333
zzzzz z
  


22 23
()zz
Figure 2. Non-transposed three-phase transmission line
with a vertical symmetry plane.
Figure 3. Modal representation by using two transfor-
mation matrices.
1112 22 231112 22 23
02223
1112 22 23111222 23
12
(24) 0()
33
[] 00
21
()0(4)(
33
yyyy yyyy
Yyy
yyyyy yyy

2
)
3
 

 
(19)
In (16) and (17), it is verified that the line shown in
Figure 2 can be represented as a single-phase and a two
t is excluded in (18) and (19), are ob-
tained the longitudinal impedance [Zα0] and transversal
admittance [Yα0] are given by:
-
phase line and there is not coupling between them [6].
If β componen
1112222311 122223
0
111222 23111222 23
33
3
12
(2 4)()
33
[] 212
()(4)(
zzzz zzzz
Z
zzzzz zz
 
)z
 
(20)
1112 222311122223
0
1112 22 23
12
(2 4)()
33
[] 212
()
33
yyyy yyyy
Y
yy yy
111222 23
(4)()
3
y yyy

 
(21)
These matrices represent a two-phase line without a
vertical symmetry plane [6].
In (20), impedance matrix elements are written as:
1112 22 23
1(2 4)
3
Z
zzzz
 (22)
0 11122223
2()
3
Z
zzzz
 (23)
011122223
(4
)( )
33
12
Z
zz zz
(24)
In (21), admittance matrix elements [Yα0] are written
as:
1112 22 23
1(2 4)
3
Yyyyy
 (25)
0111222
2()
3
Yyyy

23
y
(26)
01112222
12
(4)(3
Yy
y yy
itudinal and shunt three-phase line parameters.
In Figure 3, a modal decomposition of t
line is done by using modal transformation mα0
which diagonalizes the [Yα0][Zα0] [2]. Immediately, it can
be obtained a relationship between the elem
column of [Tα0] and the three-phase line parameters.
In this case, eigenvalues are obtained by the solution
of a polynomial of degree 2, whose roots are written as:
) (27)
33
In (22)-(27) all elements are written only according to
long
he two-phase
atrix [T],
ents of each
Copyright © 2013 SciRes. EPE
C. G. CARVALHO ET AL.
Copyright © 2013 SciRes. EPE
1493
writing in terms these parameters.
22
1122112211221221 )S (28)
1
()2 4
2
SSS SSSS
 
Obtained [Tα0] can be established an explicit relationship
between two-phase line values and three-phase line
parameters. Thereby, are obtained the currents and volt-
age phases of the two-phase line written according to
only three-phase line parameters (Figure 2).
22
11221122112212 21
2
()24)SSS SSSSS
 
2
In (28) and (29), S11, S12, S21 and S22 are the elements
of the matrix [S] obtained by calculating the product
[Yα0][Zα0]:
(29)
00000
0000000
[]
Z
YZYZYZY
S
Z
YZYZYZY
 
 



(30)


Therefore, [Tα0] can be written as:
12 12
2
2
11 11211212
011 1112
2
22
11 11211212
()( )
[]
()( )
SS
SSSS
TSS
SSSS




 

 

2
2
2
tically according to three-phase line
parameters since the elements of the matrix [S] are
fB
Verifies in Figure 3 that β component is an exact
mode of three-phase line. Thus, the currents and voltages
in this mode are obtained from the equations (3) and (4).
Obtained the values in α, β and 0 components, the next
step is the conversion of these values to domain phases.
Process result is a series of equations that allows to ob-
tain currents and voltages phase of the three-phase line.
These equations represent a direct model developed in
phase domain of three-phase line in Figure 2.
Considering the matrix [Tα0] obtained in (31), the
equations (12) and (13) are rewritten as:
(31)
12
[][][][ ][]
AB
VNVNI
(32)
Considering the expression (31), can be say that [Tα0]
is writing analy34
[][][][][]
AfBB
I
NI NV
(33)
where:
1
1
222
1
[]2( )
da db db
Ncacbcb
da bc


1
1
cosh ()00
0 cosh( ) 0
00cosh()
cosh ()00
00 0
1
0110cosh()
2011 00cos
222
1
m
m
m
dd
ca cb cbd
dd
cb db db

















2
2
2
cosh ()00
o()0
0cosh()
m
m
m
ddd

0
h()d
2( )ca da da
da bcca da da
0c
sh
0


(34)


11
211
11
()
222
1Cm m
Zsenh d
de dfdf
 0 0
[] 0()0
2( )00 ()
() 00
00 0
1
011 0()0
2011 00 (
2
1
2( )
Cm m
Cm m
C
C
C
Nc
ecfcf Zsenhd
da bccecfcfZsenhd
Zsenh d
Zsenh d
Zsenh d
dabc


















22
22
22
() 00
22
0()0
00 ()
Cm m
Cm m
Cm m
Zsenhd
gb hb hb
ga hahaZsenhd
ga haha
)
Z
senh d





(35)
1
3 1
1
cosh ()00
222
1
[]0 cosh()0
2( )00cosh()
cosh ()00
00 0
1
0110cosh()0
201100cosh()
222
1
2( )
m
m
m
d
he hfhf
Ngegfgfd
he gfge gfgfd
ddd
gf hfhf
ge hehe
he gfg




















2
2
2
cosh ()00
0cosh()0
00cosh(
m
m
m
dd
ehehe d
)




(36)
C. G. CARVALHO ET AL.
1494
1
1
4 1
1
1
1
1() 00
222
11
[] 0()0
2( )
1
00 ()
1() 00
000
11
011 0()0
2011
1
00 ()
m
Cm
m
Cm
m
Cm
C
C
C
senh d
Z
ha hbhb
Ngagbgb senhd
he gfZ
ga gb gb
senh d
Z
senhd
Z
senhd
Z
senh d
Z
















 



2
2
2
2
2
2
1() 00
222
11
0()0
2( )
1()00
m
Cm
m
Cm
m
Cm
senh d
Z
cf df df
ce de desenhd
he gfZ
ce de de
s
enh d
Z











(37)
In (34)-(37), the terms γm1 and γm2 are, respectively,
propagation functions in modes 1 and 2 of the two-phase
line, while ZCm1 and ZCm2 are, respectively, the characteristic
impedances in modes 1 and 2 this line. Also, in (34)-(37),
γβ and ZCβ are, respectively, propagation function and
characteristic impedance in β. All these elements are
calculated according to three-phase line parameters.
In (34)-(37) a, b, c and d elements are written as:
11 21
2
63
TT
a
(38)
21 11
36
TT
b (39)
12 22
2
63
TT
c
(40)
22 12
36
TT
d (41)
In (38)-(41), T11, T12, T21 and T22 are [Tα0] elements
obtained in (31).
In (34)-(37) e, f, g and h elements are written as:
11 21
2
63
X
X
e
(42)
21 11
36
X
X
f (43)
12 22
2
63
X
X
g
(44)
22 12
36
X
X
h
In (42)-(45), X11, X12, X21 and X22 are [Tα0]t elements
obtained by the inverse transpose of [Tα0].
It is observed in equations (32) and (33) that three-
phase line model obtained directly in phase domain is
calculated according to [N1], [N2], [N3] and [N4] which
are obtained only according to longitudinal and shunt
parameters of the three-phase line, show in Figure 2.
Therefore, equations (32) and (33) represent an analytical
model for three-phase lines developed directly in the
phase domain.
3. Results
Results obtained with the model deve
phase domain will be compared to results obtained using
a model developed modes domain [2].
The curves will be analyzed for the 200 km transmission
line in the interval 10-2 Hz 105 Hz. To analyze the behavior
of current curves were used the parameters of a 440 kV
three-phase line transmission line [6] (Figure 4).
Classical modal method used in simulations of the line is
Newton-Raphson method [2]. The line was energized by
a constant tension source in two situations, the open and
short circuit line [8].
(45)
loped directly in
Figure 4. 440 kV three-phase line transmission line.
Copyright © 2013 SciRes. EPE
C. G. CARVALHO ET AL. 1495
3.1. Open circuit Response
In this scheme a 1 p.u. step voltage is
while the phases 2 and 3 are shorted at the sending end.
All the phases are open circuited
described in Figure 5. Figure 6 shows the open circuit
current at the sending end in phase 1.
In Figure 6, the curve 1, whic
obtained by using the model propose has behavior
similar to curve 2, which repr
by using the classical model, fo
zed
at
this wave shape are widely used in the
high-voltage tests of power system devices, including
pacitor banks, oxide surge arresters, and
applied on phase 1,
at the receiving end, as
h represents the result
esents the result obtained
r all frequency range.
3.2. Performance Considering an Atmospheric
Surge
A frequency domain simulation considering an atmospheric
impulse signal is carried out based on the aforementioned
system configuration. The unitary impulse signal is
modeled in the frequency domain as a normali
mospheric impulse of 1.2/50µs, corresponding to the
technical descriptions provided by the IEC. This
procedure and
insulators, ca
power transformers [9].
Figure 5. Open circuit test.
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
-10
10
-8
10
-6
10
-4
10
-2
Frequency [Hz]
1 Proposed Model
2 Classical Model
Figure 6. Open circuit response: Absolu
rent in phase 1.
Cu rre
Figure 7. Atmospheric surge test.
10
-2
10
0
10
2
10
4
10
-7
10
-6
10
-5
10
-4
nt Absolute Value [p.u]
1
2
te value of the cur-
Frequency [Hz]
Vo
1 Proposed Model
2 Classical Model
Figure 8. Atmospheric surge response: Absolute value of
the voltage at the receiving end in phase 1.
The atmospheric surge is applied at the sending end of
phase 1, as described in Figure 7. The voltage profile on
the receiving end of the three-phase lines is shown in
Figure 8.
In Figure 8 that, for all
equency range, the curve 1 has behavior similar to of
the curve 2. Curve 1 shows results obtained with the
model developed in phase domain (proposed model) and
curve 2 shows results obtained with a model developed
in modal domain (classical model).
4. Conclusions
This paper was presented a model developed directly in
the phase domain of a non-transposed three-phase
transmission line with a vertical symmetry plane.
Simulations have shown that the model responds
adequately in the corresponding frequency range to
maneuver and switching procedures. It was verified that
for the open and short circuit test, the results wer
urrents and voltages can be obtained in
any line situation analysis, these equations do not require
knowledge of complex mathematical concepts (e.g.
eigenvector and eigenvalue concepts) and all terms
present in the equations are functions according to line
parameters, which will represent the line directly in time
domain.
ltage Absolute Value [p.u]
1
2
it is possible to verify
fr
e
consistent across the frequency range.
Considering the equations that describe the model
proposed, the c
Copyright © 2013 SciRes. EPE
C. G. CARVALHO ET AL.
Copyright © 2013 SciRes. EPE
1496
Therefore, the developed model can be used as an
alternative model in the analysis of transient phenomenons
that occur in transmission lines of electric power system.
5. Acknowledgements
The authors thank LETEL group – Electromagnetic
Transient Laboratory – throughout complicity
collaboration andlvim
ientífico e
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