real-time estimation of transmission line impedance based
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Real-time estimation of transmission line impedance based on modal analysis
theory
Conference Paper · July 2011
DOI: 10.1109/PES.2011.6038983
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Asti G. Aparecida
São Paulo State University
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Sergio Kurokawa
São Paulo State University
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Eduardo C. M. Costa
University of São Paulo
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J. Pissolato
University of Campinas
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1
Abstract--The objective of this paper is to show a methodology
to estimate the longitudinal parameters of transmission lines. The method is based on the modal analysis theory and developed from the currents and voltages measured at the sending and receiving ends of the line. Another proposal is to estimate the line impedance in function of the real-time load apparent power and power factor. The procedure is applied for a non-transposed 440 kV three-phase line.
Index Terms—Transmission lines, Parameters estimation, Transmission line modeling.
I. INTRODUCTION T is known that the self and mutual impedances of overhead lines can be calculated from the solutions of the Maxwell’s
equations for the boundary conditions at the contact surfaces of the conductor, air and ground. The expressions obtained to calculate self and mutual impedances are functions of the frequency, specific resistances, magnetic permeabilities and dielectric permittivities [1]. In the most usual procedures to calculate the line parameters, there are some explicit or implicit aspects which infer in physical approximations. These approximations are usually based on geometry simplifying or electromagnetic field behavior. The first one consists of assuming that the soil surface is plane, the line cables are horizontal and parallel among themselves, the distance between any pair of conductors is much higher than the sum of their radii and the electromagnetic effects of structures and insulators are neglected. Some simplifying assumptions are usually made about electromagnetic field behavior and these approximations imply that, in what concerns the transversal behavior of the line, it is assumed the quasi-stationary electromagnetic field simplification [2].
Furthermore, from historical and cultural reasons, the most used procedures to represent the ground assume that the ground has a constant and frequency-independent conductivity and has a dielectric permittivity that can be neglected. These assumptions are quite far from reality. Except for very high electric fields, where significant soil ionization originates, soil electromagnetic behavior is
G. A. Asti ([email protected]) and S. Kurokawa
([email protected]) are with Unesp - Univ. Estadual Paulista, Ilha Solteira, Brazil.
E. C. M. Costa ([email protected]) and J. Pissolato ([email protected]) are with Unicamp - Universidade Estadual de Campinas, Brazil.
essentially linear, however the electric conductivity and electric permittivity are strongly frequency dependent and non-linear [2].
As mentioned in [3], transmission lines are running at the same time through plain regions, coastlines or mountainous regions and the soil resistivity of these regions are variable. The soil resistivity is totally different not only from region to region, but also within the same region during the year [3]. In the same region, the soil resistivity has significantly higher values during summer months in contrast to winter months due to high temperatures and low rainfalls dry up at least the upper layer of the ground [3].
Classical methodologies used to calculate the transmission line parameters consider a constant value of the soil resistivity, without consider the variations of the soil characteristics where the line is running over. In short, several approximations are considered in function of the line geometry, electromagnetic field, soil resistivity and other characteristics. Hence, the methodology to calculate and evaluate the line parameters can result in several inaccuracies.
In situations where it is necessary to known the line parameters taking into account the weather conditions, it is possible to use procedures to estimate these parameters from measured complex currents and voltages at sending and receiving ends of the line [4]. The simultaneity of the measurements can be done by using synchronized phasor measurements units (PMUs) [4]. Furthermore, nowadays, there are several new technologies that carry out real-time simulations, such as the RTDS (Real-Time Digital Simulator) and the OVNI (Object Virtual Network Integrator), which also simulate the measurements obtained via PMU [4-6].
In front of the constant evolution of real-time processes applied in measurements/simulations and the actual state of the art, involving recent concepts and tendencies, such as the smart grids, this paper proposes an alternative procedure to estimate transmission line parameters, taking into account real-time measurements of currents and voltages at the sending and receiving ends of the line based on the modal transformation theory. In short, the method consists in several manipulations of the measurements and line parameters in phase and modal domain, resulting in an estimated impedance matrix of the line.
The procedure is used to estimate the longitudinal impedance of a real non-transposed 440 kV three-phase line in function of the load profile.
Real-Time Estimation of Transmission Line Impedance based on Modal Analysis Theory
G. A. Asti, S. Kurokawa, E. C. M. Costa and J. Pissolato
I
978-1-4577-1002-5/11/$26.00 ©2011 IEEE
2
II. TRANSMISSION LINE PARAMETERS Transmission lines are characterized by its distributed
longitudinal and transversal parameters. The longitudinal parameters are frequency dependent and
given per unit length (p.u.l.), they are represented by resistances and inductances. The transversal parameters p.u.l. are represented by conductances and capacitances [7]. The frequency dependence of the longitudinal parameters is due to ground and skin effects [8].
Considering a generic overhead transmission line with n phases, it is possible to define the p.u.l. longitudinal resistances and inductances matrices as being [7,9]:
Ω/km
)ωR()ωR()ωR(
)ωR()ωR()ωR()ωR()ωR()ωR(
)]ω[R(
21
22221
11211
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
nnnn
n
n
L
OM
L
L
(1)
H/km
)ωL()ωL()ωL(
)ωL()ωL()ωL()ωL()ωL()ωL(
)]ω[L(
21
22221
11211
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
nnnn
n
n
L
OM
L
L
(2)
In (1) and (2) matrices [R(ω)] and [L(ω)] are, respectively,
the p.u.l. longitudinal resistance and inductance matrices whereas ω denotes the angular frequency.
The generic terms R(ω)ii and R(ω)ij shown in (1) are real parts of the self impedance of the phase i and the real part of the mutual impedance between phases i and j, respectively [7]. In (2), terms L(ω)ii and L(ω)ij are, respectively, the self inductance of the phase i and mutual inductance between phases i and j.
For the n phase line above mentioned, the p.u.l. transversal capacitances matrix can be considered frequency independent and it is written as [7,9]:
F/km
CCC
CCCCCC
[C]
21
22221
11211
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
nnnn
n
n
L
OM
L
L
(3)
In (3) a generic term C11 is the apparent capacitance of the
phase i and a term Cij is the apparent capacitance between phases i and j.
For overhead lines, the transversal conductances can be usually neglected, except at very low frequencies [7, 9].
III. DESCRIPTION OF THE PROPOSED ESTIMATION PROCEDURE Consider a three-phase balanced power source connected
to three-phase balanced load where the connection between source and load is done by a three-phase line with a vertical symmetry plane as it is shown in Fig 1:
Fig. 1. Three-phase balanced power source connected to a three-phase balanced load.
First, it is considered the system operating in stead state
with nominal frequency and the complex voltages and currents can be measured by using synchronized phasor measurements units [4]. From these premises, the procedure to estimate transmission line parameters, taking into account its distributed nature, can be carried out adequately. Hence, to consider the distributed nature of the line parameters, the proposed procedure starts from the complex voltage and current equations obtained from the fundamental telegrapher’s equations for a multiphase line [10].
The complex voltage and current equations previously mentioned are well known for single phase lines, although they are not easily obtained for multiphase lines due to coupling between phases. To obtain the complex voltage and current equations for a multiphase line, they are represented in modal domain, where a multiphase line with n phases is represented by its n decoupled propagation modes.
The description in details of the procedure to estimate transmission line parameters using modal decoupling is carried out in six steps, as follows:
Step 1: Measurement of the phase currents and voltages, in
frequency domain, at two terminals of the line; Step 2: Converting of measured voltages and currents from
phase domain to modal domain; Step 3: From currents and voltages in modal domain, the
propagation function and the characteristic impedance of each modal component are calculated;
Step 4: Using the propagation function and the characteristic impedance, the longitudinal impedance and the transversal admittance matrices are calculated in modal domain;
Step 5: The longitudinal impedance and transversal admittance matrices, written in modal domain, are transformed back to phase domain;
Step 6: From the longitudinal impedance and transversal admittance matrices, it is possible to obtain the estimated longitudinal and transversal parameters of the line.
VA1 VB 1
VA2
VA3
Three-phase balanced load
VB 2
VB 3
ground
Three-phase balanced source
Three-phase line
IB 3
IB 2
IB 1
IA3
IA2
IA1 Phase 1
Phase 2
Phase 3
3
A. Measurements of the phase currents and voltages In [4] is mentioned that complex voltages and currents at
two ends of the line can be measured simultaneously by using synchronized phasor measurements units. Thus, the proposed development is supposing that the voltages and currents are known or previously measured using PMUs.
This way, it is possible to define the vectors with complex phase voltages and currents, written as being:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
B3
B2
B1
B
A3
A2
A1
A
VVV
][V;VVV
][V (4)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
B3
B2
B1
B
A3
A2
A1
A
III
][I;III
][I (5)
In (4) [VA] and [IA] are vectors with complex phase
voltages and currents, respectively, at sending end of the line whereas in (5), [VB] and [IB] are vectors with phase voltages and currents at receiving end of the line.
B. Converting phase voltages and currents to modal domain The relationship of voltage and current, between phase and
modal domains, are given as follows [10]:
[V]][T[E] tI= (6)
[I]][T][I 1Im
−= (7)
In (6) and (7), [E] and [Im] are the vectors with complex voltages and currents, respectively, in modal domain. [TI]t and [TI]-1 are, respectively, the transposed and inverse matrices of the transformation matrix [TI]. The transformation matrix is frequency-dependent and calculated in function of the impedance and admittance matrices.
Then, the transformation from the phase domain to modal domain of the measured complex voltages and currents infers in the following expressions:
][V][T][E At
IA = (8)
][V][T][E Bt
IB = (9)
][I][T][I A1
IAm−= (10)
][I][T][I B1
ImB−= (11)
The vectors [EA] and [EB] are the measured complex
voltages, at terminals A and B of the line, in modal domain. The vectors [IAm] and [IBm] are composed of complex currents, written in modal domain, at sending and receiving ends.
C. Calculating the propagation function and characteristic impedance
A generic propagation mode can be represented as being a
single phase line, such as in Fig. 2.
Fig. 2. Representation of the k-th mode of the line.
The terms EAk and EBk are complex voltages at sending and
receiving end of the k-th mode, respectively. The elements IAk and IBk are complex currents at sending and receiving end of the k-th mode, respectively.
For the k-th mode in Fig. 2, the relationships between complex currents and voltages are given by [8]:
)(γsinhZI)(γcoshEE kckBkkBkAk dd −= (12)
)(γsinhZE)(γcoshII k
ck
BkkBkAk dd +−= (13)
In (12) and (13), γk and Zck are, respectively, the
propagation function and characteristic impedance of the k-th mode [9]. The term d is the line length in kilometers.
From (12) and (13), it is possible to write the propagation function and characteristic impedance for the k-th mode as being:
⎟⎟⎠
⎞⎜⎜⎝
⎛−−=
BkAkAkBk
BkBkAkAkk IEIE
IEIEcoshArc1γd
(14)
)(γcoshII)(γsinhEZ
kBkAk
kBkck +
= (15)
Using (14) and (15), it is possible to calculate the
propagation function and the characteristic impedance for all modes of the line.
D. Calculating the longitudinal impedance and transversal admittance matrices in modal domain
It is known that the propagation function and characteristic impedance of the k-th mode are defined as follows [11]:
mkmkk YZγ = (16)
mk
mkck Y
ZZ = (17)
where Zmk and Ymk are, respectively, the longitudinal impedance and transversal admittance of the k-th mode.
From (16) and (17), Zmk and Ymk can be written as follows:
EAk EBk
IAk IBk
ground
4
ckkmk ZγZ = (18)
ck
kmk Z
γY = (19)
Therefore, considering a three-phase line, the modal
parameters can be expressed in matrices Zm and Ym, longitudinal impedance and transversal capacitance in modal domain, respectively:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
m3
m2
m1
m
Z000Z000Z
][Z (20)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
m3
m2
m1
m
Y000Y000Y
][Y (21)
Emphasizing that the modal parameters are per unit of
length as well: ohms and siemens per kilometer, respectively.
E. Converting longitudinal impedance and transversal admittance matrices from modal domain to phase domain
The transformation of the longitudinal impedance and transversal admittance matrices from modal domain to phase domain is expressed as follows [12]:
1Im
tI ][T][Z][T[Z] −−= (22)
tImI ][T]][Y[T[Y] = (23)
Matrices [Z] and [Y] are, respectively, the longitudinal
impedance matrix and transversal admittance matrix of the line, in phase domain.
F. Obtaining the transmission line parameters From (22) and (23), it is possible to obtain matrices [R],
[L], [G] and [C] as follows:
[Z]Real[R] = (24)
[Z]Imagω1[L] = (25)
[Y]Real[G] = (26)
[Y]Imagω1[C] = (27)
In (24) and (25), [R] and [L] are resistance and inductance
matrices, respectively. The matrices [G] and [C], in (26) and (27), are transversal conductances and capacitances in phase domain, respectively.
The procedure proposed in this section is based on the previously knowledge of the modal transformation matrix. At first sight, if the line parameters are known, the transformation matrix is also obtained. However, there are situation where the modal transformation matrix is not function of the line
parameters and for this cases, this matrix can be obtained from the geometric characteristics of the line [13]. In transmission lines with these characteristics, the proposed methodology to estimate transmission line parameters can be applied [14].
The proposed methodology can be applied to estimate the parameters of ideally transposed three-phase lines. If the line is not ideally transposed or then non-transposed, but it has a vertical symmetry plane, the Clarke’s matrix can be used as an approximated transformation matrix [15]. This matrix has real and constant terms and it is written as follows [15]:
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−=
31
21
61
31
21
61
310
62
]T[ Clarke (28)
In this paper, it was analyzed the possibility to apply the
proposed methodology to estimate parameters of non- transposed three-phase lines with a vertical symmetry plane using Clarke’s matrix as being the modal transformation matrix.
IV. APPLICATION OF THE METHODOLOGY TO ESTIMATE THE TRANSMISSION LINE IMPEDANCE
The proposed procedure is applied to estimate the electric parameters of a conventional 440 kV transmission line, non-transposed and characterized by a vertical symmetry plane. The steel tower and line geometric configurations are described in Fig. 3.
1
2 3
4 5
(9.27; 24.4)
(7.51; 36)
3.6
m
Fig. 3. 440 kV three-phase transmission line.
In line shown in Fig. 3, the soil resistivity is 1000 Ω.m,
phase conductors consist of four Grosbeak subconductors and the ground wires are EHSW-3/8” conductors.
In order to validate the proposed methodology, the complex phase voltages and currents, at the terminals of the
0.4 m
5
line, are previously calculated using a digital line model, based on the distributed parameters. However, this voltages and currents can be measured by using of PMUs or then using a real-time simulator, such as RTDS (Real-Time Digital Simulator) and OVNI (Object Virtual Network Integrator), simulating the measurements obtained via PMU [4-6].
Therefore, the reference values of the voltages and currents, obtained from the digital model of the line, are calculated based on the previous knowing of the longitudinal and transversal parameters according with the configurations described before and by Fig. 3. These parameters are denominated, in this work, as being the “reference values”, for a posterior comparison between values estimated and calculated, taking into account the soil and skin effects [7-9]. The reference parameters are given by the following matrices:
Ω/km0604.00581.00580.00581.00604.00580.00580.00580.00603.0
]R[⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=ref (29)
mH/km9380.19986.01234.19986.09380.11234.11234.11234.19385.1
]L[⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=ref (30)
In (29)-(31), [Rref] and [Lref] are the calculated resistance
and inductance matrices of the line, respectively. Using a digital transmission line model to represent the
line, shown in Fig. 3, the complex phase currents and voltages are calculated in frequency domain at 60 Hz for several load conditions. Then, using the proposed methodology, the line parameters are estimated and compared to the calculated reference parameters.
From the variations between parameters calculated and estimated, a relative error is calculated. However, it is very important to note that the proposed comparison is not an exact measurement on the efficacy or precision of the proposed method, once that the calculation procedure, such as described before, has inaccuracies due to several approximations and simplifications. Thus, the calculation of the relative error is carried out in order to verify the coherence of the results estimated from the proposed method.
In fact, the estimation and calculation of the line electric parameters is a very complex task, taking into account that it is difficult to obtain exact reference values to evaluate the new methodologies developed for these purposes. In other words, there is not a totally reliable method to serve as a reference.
Such as described before, the line longitudinal parameters are estimated in function of the load characteristic. By this means, it is considered several load magnitudes and values for the power factor (inductive).
First, the relative error between self and mutual resistances p.u.l. are shown in figures 4 and 5, as follows:
Fig. 4. Modulo of the relative error calculated from the estimated self resistance in function of the load profile.
Fig. 5. Modulo of the relative error calculated from the estimated mutual resistance in function of the load profile.
From the above figures, it is possible to verify that the self and mutual resistances have approximated values in comparison to the calculated ones. Furthermore, a non-linear behavior is observed for apparent power values above to 1 MVA, in function of the power factor variation. However, considering the power factor values approximately up to 0.8, the relative error can be less than 10%.
In Fig. 6, the relative error related to the self inductances/reactances is shown.
Fig. 6. Modulo of the relative error calculated from the estimated self inductance/reactance in function of the load profile.
The relative error related to the imaginary part of the impedances, calculated and estimated, is practically constant in function of the apparent power and power factor, around 27.5 and 28 %, approximately.
Figure 7 shows the relative error of the mutual inductance/reactance.
6
Fig. 7. Modulo of the relative error calculated from the estimated mutual inductance/reactance in function of the load profile.
Concerning the estimated inductance, a major variation in function of the calculated inductance is observed. In Fig. 7, the relative error is practically constant in function of the load profile.
V. DETERMINATION OF AN ESTIMATED IMPEDANCE MATRIX In order to define an estimated matrix for the resistance and
inductance p.u.l., an average relative error is calculated from all samplings estimated for each parameter, in function of the apparent power and power factor: mutual and self resistances and impedances. By this means, the impedance matrix estimated, considering the line in Fig. 3, is estimated and given by the following resistance and inductance matrices:
Ω/km0661.00616.00615.00616.00661.00615.00615.00615.00660.0
]R[⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡= (31)
mH/km4050.15543.06235.05543.04050.16235.06235.06235.04055.1
]L[⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡= (32)
The major variations, in comparison to the calculated
values given in (29) and (30), are observed in the mutual terms of the matrix [L].
VI. CONCLUSIONS This paper has shown a procedure to estimate transmission
line parameters taking into account its distributed nature. The methodology can be applied from the measured complex currents and voltages at the sending and receiving ends of the line, making possible a real-time application for measurement of the line electric parameters. However, the exact transformation matrix, in the proposed procedure or for any other method, is unknown. Hence, the Clarke’s matrix was considered as being a modal transformation matrix, once that it can be applied for transposed lines or then non-transposed lines with symmetry vertical plane.
The impedance estimation and evaluation is performed in function of the load profile, enabling the method be carried
out for several operation conditions. After, the values estimated were compared to those previously calculated by the classic procedure, using Bessel and Carson formulations. This comparison is carried out to verify the coherence of the values estimated, if they are approximately close of those calculated. In fact, the values calculated do not represent an ideal reference to measure the exact accuracy of the proposed method, considering that the usual method to calculate the line electric parameters has several approximations and simplifications, which are emphasized along the paper.
The relative error calculated for the self and mutual inductances are up to 27.5% and 44%, respectively. However, the relative error calculated for the self and mutual resistances, which represent the active losses of the line, are 10% and 6%, respectively.
Anyway, the proposed estimation method, from a practical point of view, shows to be a useful tool to obtain important information in transmission systems without the previous knowing of the line geometric and physical descriptions. Besides the inductances of the line were underestimated, it does not infer that the method is totally imprecise, taking into account that the calculation procedure presents several inaccuracies as well. Furthermore, the method has a significant precision on the resistance estimation, representing a possible way to measure the instantaneous power losses along the line with good accuracy.
ACKNOWLEDGMENTS This research was supported by Coordenação de
Aperfeiçoamento de Pessoal de Nível Superior (CAPES) and Conselho Nacional de Desenvolvimento Cientifico e Tecnológico (CNPq).
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