Ali Abur Accurate modeling and simulation of transmission line transients using frequency dependent...
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Accurate Modeling and Simulation of Transmission Line Transients
Using Frequency Dependent Modal Transformations
Ali Abur Omer Ozguu
[email protected] .edu ozgun@)ee.tamu.edu
Department of Electrical Engineering
Texas A & M University
(CollegeStation, TX 778433128
Abstract :
Frequencydependentlinemodel(alsoknownas the J.
Marti model) which is currently used in most electromagnetic
transient programs [1], is very efficient and accurate for most
simulation cases. However, it makes an approximation in choosing
the modal transforruiition matrix that is used to switch variables
between the phase and modal domains at each simulation time step.
This approximation may not hold true for certain tower
configurations rind/or conductor types where line parameters vary
drastically with frequency. In this paper, a wavelet based alternative
sohrtion, which incorporates
frequency dependence of
transformation matrices into the simulation process will be
presented.
Keywords: Electromagnetic transients simulations, frequency
dependent transmission line parameters, modal trrmsfonnations,
wavelet transform.
I. INTRODUCTION
Simulation of large electric power systems during system
dis~bances, such as short circuits, switching of
loads
capacitors or other devices, line or transformer energization,
motor starting, etc. has been an active area of research for the
past several decades following the rapid improvements in
computer technolob~.
Power systems contain components
such as transmission lines whose model parameters vary as a
function of frequency and consequently lend themselves best
to frequency domain modeling and simulation. On the other
hand, there are devices with time varying andJor nonlinear
operating characteristics such as solid state rectifiers,
saturated transformers, surge arresters, metal oxide varistors,
etc. that exist in power systems and their models are typically
best realized in time domain due to their nonlinear
characteristics.
Reconciling the simulation and modeling requirements of
these mixed set of components has been one of the challenges
faced in the analysis of transients so far. This paper
addresses this challenge by presenting an alternative
simulation method, which is motivated by the unique
properties of the wavelet transform.
Fernando H Magnago
PCA Corporation
1921 S. Alma School Rd. 207
Mesa, Arizona 85210
Use of wavelet transform for simulation of power system
transients is investigated by Meliopoulos and Lee in [2],
where wavelet domain equivalent circuits of R,L and C
components are utilized to compute the transients in the
wavelet domain and recover the time domain solution via
inverse transform. Application of wavelet domain
equivalents to carry out harmonic analysis of nonlinear and
time varying loads is reported in [3] by Zheng et al. These
papers discuss the simulation and modeling of lumped
elements, which can be used to synthesize cascaded pi
sections to represent lines. Similar studies can also be found
in [4], [5] and [6], where spatial distribution of voltages along
non-uniform multi-conductor transmission lines is simulated
via the wavelet transform of the resulting differential
equations. The authors assume frequency independent line
parameters in these studies. In [7], use of the wavelet
transform for representation of frequency dependent
parameter transmission lines, with constant modal
transformation matrices, is discussed. Modeling of lossy
transmission lines with frequency dependent parameters can
also be accomplished by direct application of the wavelet
transform. One approach is to start with the general form of
the multi-conductor transmission line partial differential
equations expressed in the spatial distance z, and time t, for
the voltages and currents along the line. Then, use the
wavelet transform to convert them into large sparse algebraic
equations whose solutions will yield coefficients of the
wavelet transform of the voltages and currents of interest.
While this is a viable approach, integration of such a
computational procedure into an existing transients simulator
may not be trivial if possible at all. Instead, what is proposed
in this paper, is a fairly simple modification of the well
known constant but distributed parameter line model, or the
so called the Bergerons model [8], to incorporate frequency
dependence of line parameters using the wavelet transform.
Simulation of transients along multiphase transmission lines
has an additional drawback, which is the requirement that the
line equations ought to be decoupled into independent modal
equations, so that each one can be solved easily in the
respective modal domain. This decoupling is done through a
linear transformation matrix, which will be a function of
fi-equency if the corresponding line parameters also are. In
time domain simulations, due to the lack of a practical
alternative, a constant transformation matrix typically
evaluated at a chosen frequency is used as an approximation.
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So, even when using the advanced frequency dependent (FD-
method will be presented to accomplish this without ahering
model) model of [9], modal transformation matrix will have
the basic discrete time circuit model of Fig.1.
to be approximated. The simulation method, which will be
presented in this paper, provides a rather simple avenue to
Ik(t)
h(t) ~
improve this approximation by using the wavelet transform.
k-
The paper is organized such that a review of the FD-model
m
and the Bergerons constant parameter (CP-model) distributed
line model will be presented fwst. The proposed wavelet-
based simulation and modeling of transmission lines with
F TI
k(t) ZO
Jt
20 Vtn(t)
frequency dependent parameters will be discussed next.
Simulation results of some power system transients will then
Ikm
Imk
be shown followed by their discussion and conclusions.
II. REVIEW OF LINE MODELS
Multi-conductor transmission lines usually run distances
long enough to make their lumped parameter modeling
inaccurate. Approximate models that can fake the distributed
nature of the line parameters can be obtained by using several
cascaded lumped parameter pi section models. A more
accurate model, which is referred to as the constant parameter
(CP-model) line model, can be obtained by lumping the
resistance and modeling the remaining loss-less part, by using
the method of Bergeron.
This model incorporates traveling
wave delays via a simple equivalent circuit containing a
current source and a constant resistance (lines characteristic
impedance) at each end of the line. The current sources
depend upon the voltage and current values from the remote
end of the line, with a certain time delay that is determined by
the traveling wave velocity and the line length. This model is
shown in Fig.1.
Variations of line parameters such as R, L and C as a
function of frequency, are simply ignored when using the CP-
model of the line. In order to address this deficiency, a
frequency dependent line model (FD-model) is developed by
J. Marti in [9]. FD-model essentially uses the same equivalent
circuit as the CP-model shown in Fig. 1, except for the fact
that the characteristic impedance ZO,at each end of the line,
are replaced by properly chosen network equivalents that
have approximately the same flequency spectrum as that of
ZO. In addition, the current source values are no longer
simple time delayed functions of remote line end variables,
but involve more complicated convolutions [9]. Provided
that the required accuracy of the fitting fictions that
approximate the frequency response of Z. and the
propagation fimction are attained, FD-model of the line can
be used in transient simulation of single phase lines very
satisfactorily. When multiphase conductors are considered,
one is faced with the additional burden of decomposing the
line equations via a modal transformation matrix T,, which is
itself frequency dependent. In the current implementation of
FD-model, T. is computed at a suitable frequency and
maintained constant throughout the simulation period. While
for some tower configurations and conductor types, this
approximation is quite valid, certain cases may require
accurate incorporation of frequency effects on T, in the time
domain simulations. In the next section, a wavelet based
Fig 1
CP-model of a loss-less transmission line
III. WAVELET-BASED FD LINE MODEL
Wavelet transform facilitates time domain decomposition
of signals into a sub-band of frequency ranges. This implies
that the entire simulation can be decomposed into sub-bands
each of which can be calculated independent of the rest at a
given simulation time step. The advantage of this approach
will however be that line parameters as well as the modal
transformation matrices used in a particular sub-band of
frequencies can be properly chosen as the ones corresponding
to that fi-equency band.
These bands of fi-equencies are
referred to as scales of the wavelet transform due to their
special logarithmic structure [1O]. Frequency dependence of
the line parameters R, L and C as well as the resulting
transformation matrices which are fimctions of these
parameters, can be approximated by substituting
representative values calculated for each wavelet scale.
This will result in as many line models as the number of
chosen scales for a given line in each mode. Following the
CP-model of Fig. 1, the line model for scale k and mode i,
will look identical to the circuit in Fig. 1, except for the fact
that all variables, parameters and current source values will
correspond to that mode and scale.
The transformation
matrix used to obtain the terminal currents and voltages for
this mode will be different for each scale. While this is still
an approximation due to the choice of discretely rather than
continuously changing matrices from one wavelet scale to the
next, proper choice of scales based on the observed variations
in the line parameters will improve this approximation
drastically.
Thus, the following itemized procedure is proposed for
simulating transients involving lines with fkequency
dependent parameters:
. Calculate the line parameters as a fimction of fi-equency
and select flequency ranges (scales) to properly
discretize the parameters.
Calculate the characteristic impedance Zi, the travel
delay jkand the modal transformation matrix T,k at scale
k and mode i for all modes and all chosen scales. These
values are calculated at Iiequencies within each scale k
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and are assumed constant for the whole range of
frequencies defried by that scale.
Use the discrete wavelet transform
(DWT) to decompose
three phase terminal input signals into the chosen scales
in the wavelet domain. Let of the DWT of the three
phase sending end voltage v, ~,ct), be given as:
WV ~,j(n)= DWT { v, ~,.t) }
where n represents the number of discrete time steps at
scale k.
Apply mc~daltransformations using the corresponding
Tvkmatrix for scale k, and calculate the modal voltages
[1
vak
=T k WV
Wvck
Solve the discrete time line equations at each scale for
each mode, and update the current sources.
Convert all modal voltages in each scale, into the phase
domain, using the inverse of T,k.
Reconstruct the multi-phase terminal voltage signals
from their discrete wavelet domain components ~a,b,,k
in each scaJe k,,by inverse wavelet transform. Wavelet
transform and its inverse are
accomplished
computationally quite efficiently via sparse matrix
operations.
Next section illustrates some practical cases where this
approach proved viable as evident from the comparison of
results with those of the well-established FD-model.
IV. SIMULATIONS
The sample power system used in transient simulations is
shown in Fig.2. A 50 mile transmission line whose conductor
data and tower geometry are shown in Appendix I, is used for
the study. The Line Constants auxiliary routine of ATP [11]
is used to calculate the line parameters at each frequency
level.
k
Z(w)
rl /
r
?
Zload
Fig.2. Studied power system
Several line energization cases are considered, including
direct current and alternate current sources for both balanced
and unbalanced loads. Sampling rate is chosen as 5psec for
all cases and the number of wavelet scales is chosen in order
to capture the entire frequency spectrum from highest to the
steady state frequency range. Wavelet decomposition is done
by using Daubechies wavelet as the mother wavelet, based on
our previous experience [7]. Open ended line energization
transients are simulated first.
In case 1, receiving end voltage signals in three phases for
a 100 Volts single step energization are simulated. The
results are shown in Fig.3 for both proposed and existing FD-
model.
Receiving End Voltage Waveform
~3m~
I
2
:0
So li d Wavel et Model -
: -I(SI -
dot ted FD Mode l
O 0.032 0.004 0.006 0.(08 0.01 0.012 0.014 0.016 0.018 0.02
-303, ,
: ~
O 0,0U2 0.004 O.OW 0.13W 0.01 0.012 0.014 0.016 0.018 0.02
~
-im
I
o o,m2 0.004 00C6 0,008 O,of 0.012 0.014 0,016 0.018 0.02
Time (Seconds)
Fig.3. Case 1: Single step energization of an open ended
transmission line.
In case 2, a set of unbalanced resistive loads given by
R,=2KQ, Rb=3K~, and ~=2KQ, is connected at the
receiving end of the line. Fig.4 shows the single step
energization transients for this case.
Receiting End Voltage Waveform
~ 2001
I
sol id: Wavelet model
z
Q. .Irn
dottedJ. Mar t i model
i
O 0.002 0.004 0.036 0.008 0.01 0,012 0.014 0.016 0.018 0.02
:~
0.002 0,004 0,006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
- 203,
~ lml
O 0.002 0.004 0.856 0.008 0.01 0.012 0.014 0.016 0.018 0.02
lime (Seconds)
Fig.4. Case 2: Single step energization transients
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The differences between the proposed wavelet model and
existing FD mcldel simulations in both cases, are due to the
fact that, FD model uses a constant transformation matrix T,
whereas the wavelet model incorporates frequency
dependence of it into the simulations.
The rest of the simulations are carried out by using three
phase AC voltage source, where all the phases are energized
simultaneously. Fig.5 shows the results of case 3, where the
same line and load configuration as in case 2, is now
energized by a balanced three phase sinusoidal source.
Receiving End Voitage Wsveform
~l=~
f .mo~~
0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
o
0.02 0 03 0.04 0.05 0.Q3 0.07
O.oa
o 1 0s2
0.03 0.04 0.C5
0.06 0.07 0.08
Time (Seconds)
Fig.5. Case 3: Sinusoidal AC voltage source energization
transients
In case 4, the load resistance is chosen close to the
characteristic impedance of the transmission line to reduce
reflections fiorn the receiving end. Simulation results are
displayed in Fig.6 where reflections are significantly
diminished when compared to Fig.5, consistent with the
expectations fic~mthis model.
Receiving End Voltage Waveform
i:Fa
o 0.01 0.02 0.03 0.04 0.05 0.06
0.07 0.08
i:trd
0.01
0.02 0 03 0.04 0.05
0.06 0.07 0.08
o 1
2 o.m 0.04 0.05
0.06 0.07
0.08
Time (Seconds)
Fig 6
Case 4:
AC
voltage source energization transients
after changing load value
In
case 5, the effect of using frequency dependent modal
transformation matrix (Tv) is further illustrated. In order to
accomplish this, the tower geometry chosen for the line used
in the previous cases, is slightly modified (see Fig.A.11 in
Appendix I). Initially, the modal transformation matrices are
intentionally kept constant while simulating the transients
with the wavelet based model. The results of this case are
compqed with those of the FD-model. As shown in Fig.7,
they matched quite well. This is expected, since T, matrix is
assumed to be constant by the FD-model aswell.
Recehing End Voltage Wweform forConstant T Matr ices
n r I
i.l:~
, IDI So li dWevel et model
o
1 2 3 4 5 0.06
0.07 0.08
r~j
o 1 0.02 0.03
0.04 0.05 0.06 0.07 0.08
~ lm
=
5
00
:
-Im
o 0.01 0.02 0.03 O.M 0.05 0.06 0.07 0.08
Time (Seconds)
Fig.7. Untransposed line simulation with constant Tvmatrix.
Receiving End Voltaga Waveform
1 1
0.05
0 0 006 0 065 0.07
Tims(Seconds)
Fig 8 Effects of frequency dependent Modal Transformation
Matrices.
Next, modal transformation matrices are calculated for
each frequency level, and the wavelet based line model is
implemented per section 3.
The effect of this modeling
improvement is evident from the results shown in Fig.8
where both FD-model and wavelet based model simulations
are presented together. It is interesting to note that a rather
slight perturbation of the tower configuration may lead to
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noticeable changes in the frequency dependent behavior of a
given line.
This suggests that the difference between ignoring and
considering the flequency dependent nature of modal
transformation matrices may be significant when working
with untransposed lines. The differences naturally depend
heavily on the tower geometry, conductor configuration and
type in the case of overhead lines.
5. CONCLUSIONS
This paper extends the results of previous work [7] by
incorporating the effect of frequency dependence of modal
transformation matrices into the transient simulations. A
different approach to the simulation of frequency dependent,
untransposed transmission line transients is introduced. The
effect of strong frequency dependence of modal
transformation matrices on the transmission line transients is
accounted for in the time domain simulations via the use of
the wavelet transform of the signals. This allows the use of
accurate modal transformation matrices that vary with
frequency and yet still remain in the time domain during the
simulations. Comparative simulation results are presented for
the proposed and existing FD line models.
VI ACKNOWLEDGMENTS
Partial support provided by the NSF grant ESC-9821 090 is
gratefully acknowledged.
VII REFERENCES
[1] H.W. Dommel, Digital Computer Solution of
Electromagnetic Transients in Single and Multiphase
Networks, IEEE Trans. on Power App. and Systems, Vol.
PAS-88, No.4, April 1969, pp.388-399.
[2] A.P. Sakis Meliopoulos and Chien-Hsing Lee, Power
Disturbance Analysis via Wavelet Domain Equivalents,
Proc. of the 8ti Int. Conference on Harmonics and Quality of
Power, ICHQP, Athens, Greece, Oct. 14-16, 1998, pp.388-
394.
[3]
T. Zeng, E.G. Makram and A.A. Girgis, Power System
Transients and Harmonic Studies Using Wavelet Transform,
IEEE Trans. on Power Delivery, Vol. 14, No.4, Oct.1999,
pp.1461-1468.
[4] G. Antonini and A. Orlandi, Lightning-Induced Effects
on Lossy MTL Terminated on Arbitrary Loads: A Wavelet
Approach, IEEE Trans. on Electromagnetic Compatibility,
VO1.42,No.2, May 2000, pp.181-189.
[6] W. Raugi, Wavelet Transform Solution of
Multiconductor Transmission Line Transients, IEEE Trans.
on Magnetics, VO1.35,No.3, May 1999, pp. 1554-1557.
[7] F.H.
Magnago and A. Abur, Wavelet-Based Simulation
of Transients Along Transmission Lines with Frequency
Dependent Parameters, Proceeding Paper, IEEE PES
Summer Meeting, Seattle, WA, July 16-21,2000.
[8]
L. Bergeron, DUCoup de Belier en Hydraulique au Coup
de Foudre en Electricity, Dunod, France 1949, (English
translation: Water Hammer in Hydraulics and Wave Surges
in Electricity, ASME Committee, Wiley, NY 961.)
[9] J. R. Marti, Accurate Modeling of Frequency-Dependent
Transmission
Lines in Electromagnetic Transient
Simulation,
IEEE Trans. on Power App. and Systems,
VO1.PAS-101,No.1, Jan. 1982, pp.147-155.
[10] S.G. Mallat, A Theroy for Multiresolution Signal
Decomposition: The Wavelet Representation, IEEE Trans.
Pattern Anal. Machine Intel., 11(7), pp.674-693, 1989.
[11] Bonneville Power Administration, Altemateive
Transients Program (ATP) Reference Manual, Portland,
Oregon, 1986.
VIII BIOGRAPHIES
Ali Abur received his B.S. degree from METU, Turkey in
1979, M.S. and Ph.D. degrees from The Ohio State
University, Columbus, Ohio, in 1981 and 1985 respectively.
He is currently a Professor at the Department of Electrical
Engineering at Texas A&M University, College Station,
Texas.
Omer Ozgun received his B.S. and M.S. both in Electrical
and Electronics Engineering flom Bogazici University,
Istanbul, Turkey in 1995 and 1997 respectively. He is
currently a Ph.D. student in the Department of Electrical
Engineering at Texas A&M University.
F H Magnago
received his B.S. degree from UNRC,
Argentina in 1990, the M.S. and Ph.D. degrees from Texas
A&M University in 1997 and 2000 respectively. Since June
2000, he has been with PCA Corporation, Mesa, AZ.
[5] S. Grivet-Talocia and F.crmavero, Wavelet-Based
Adaptive Solution for the Nonuniform Multi-conductor
Transmission Lines, IEEE Microwave and Guided Wave
Letters, VO1.8, No.8, Aug.1998, pp.287-289.
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APPENDIX I
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