Ali Abur Accurate modeling and simulation of transmission line transients using frequency dependent modal transformations

8/11/2019 Ali Abur Accurate modeling and simulation of transmission line transients using frequency dependent modal transf

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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

[email protected]

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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Ali Abur Accurate modeling and simulation of transmission line transients using frequency dependent modal transformations