Energy and Power E ngineering, 2013, 5, 1449-1455
doi:10.4236/epe.2013.54B275 Published Online July 2013 (http://www.scirp.org/journal/epe)
Copyright © 2013 S ciRes. EPE
Equivalent for Electromagnetic Transient Calculation in
Power System with Multiple Transmission Line
Paweł Sowa, Katarzyna Łuszcz
Silesian University of Technology, Electrical Faculty, Bolesława Krzywoustego 2, 44-100 Gliwice, Poland
Received April, 2013
ABSTRACT
The results of searching o f equivalent for electro magnetic transient calcula tion in power system with the big number of
transmission lines are shown. Brief information on the proper transmission line model is given. Comparison of results
of simulation in real and reduced system is presented taking into consideration three methods of identification. Some
example are examined for different number of transmission lines in reduced system and co nsequentl y the re co mme nda-
tion are given for sea rching of equivalents of s ys tems with many lines.
Keywords: Electromagnetic T r ansients; Equivalents; Tra nsmission Line Representation
1. Introduction
For a large complex system the full representation for
transient electromagnetic analysis is not necessary or
practicable. The main interest during the study of dy-
namic phenomena in power system is directed to the
waveform occurring only in certain parts of the system.
Very often, this is defined as the internal syste m, and t he
remaining part is referred to as an external keyboard.
Internal networks are modeled with an accurate repre-
sentation of all the properties that are important in the
analysis of the p henomenon. The external s ystem can be
mapped on the basis of reduced structure or can be re-
placed by equivalent defined on the basis of one of the
available methods in the time or frequency domain.
The network models are replacement parts, corres-
ponding to the reduced power of the system. Very often
there are mistakes, involving the creation of a model too
careful or too radical for its simpli fication.
There is no generalized criterion by which it can be
determined in advance the structure of the reduced
equiva lent cir cuit for systems in which t he stud y o f ele c-
tromagnetic transients will be made. Despite many years
of research, as yet, the universal equivalent was not
found that would faithfully reproduce the behavior of the
power syst em during these eve nts.
Of course, the most reliable analytical results can be
obtained from measurements in a real power system.
Unfortunately, in the case of the study of electromagnetic
transient phenomena, which usually are the result of
faults, conducting measurements is due to economic,
technical and organizational reasons very rare.
These measurements are carried out in two ways:
• Carried out at various points in the system, eg at
protection location, during short-circuit i n the line
• Measuring devices are connected to the selected
node and recording waveform during faults (for which
"waiting").
In the first case, the choice of specific situation (the
type and location of fault) is possible, but safety consid-
erations require the introduction of a number of restric-
tions in order to avoid the possible consequences of the
development of the faults. Limitations cause disconnec-
tion of the system, some of the load off, carrying out tests
on specified dates, etc.
In the second case, the recording of the actual events
in the original system without restrictions is possible.
However there are “accidental faults”, which recorded
even within a fe w years, are not sufficient for a compre-
hensive analysis.
Computer simulations have the advantage that it is
possible their multi variability forcing different types of
disturbances in the power system anywhere, without any
risk to its future work.
Recently, most studies were performed in systems to
replace a large part of a complex system using simple
impedance calculated on the basis of short-circuit power
of these systems. This approach is acceptable only for
very simplified analysis of transients. The biggest prob-
lem in such cases is to determine - based usually on the
expe riment, how is t he part o f the s ystem wh ich mus t be
modeled in detail and which is the remaining part must
be replaced by equivalent.

P. SOWA, K. ŁUSZCZ
Copyright © 2013 S ciRes. EPE
1450
Since many years, there are a lot of approaches trying
to find a universal solutio n for this issue. Accordin g to [1]
is recommended taking into account the elements in the
branches of the two neighboring nodes in relation to the
node to which is attached the object studied. However
such solution, in the case of a very complex network
comprising a many transmission line give not guarantee
receiving exactly the same results as in the real system.
This is due to the specific requirements that are placed on
the line models in the calculation of electromagnetic
waveforms.
Thi s pape r pre sents the r esul ts of t he anal ysis o f str uc-
tures and patterns of substitution parameters for systems
containing a large number of transmi ssio n lines.
2. Transmission Line Representation
Transmission line is the only element of power system
with spatially distributed evenly parameters. Transmis-
sion line representation with concentrated parameters
(multiphase Π, T, Γ circuits) used during steady state and
dynamic electromechanical investigation, is a mistake,
especially when in many computer programs there are
distributed-parameter line models at disposal which for
transient solutions, are usually better. In addition fre-
quency dependent line model must be taken into account.
As demonstrated [2 ], the resistance, in particular for zero
sequence strongly increases with increasing frequency.
This includes the motivation was to create a so-called.
Marti transmission line model [3], in which the accurate
modeling of transmission lines over the entire frequency
range using the Foster I network realization is developed.
The biggest limitation (and disadvantage) of Marti
model is the need for a transformation matrix which is
also depended on frequency. The importance of this fac-
tor increases with the degree of asymmetry of the line
configuration.
Transformation matrix, shown symbolically in block
form in Figure 2, between the phase and the modal
components is independent on frequency.
A1
A2
B1
B2
Internal object
EXTERNAL SYSTEM
(replaced or reduced)
Figure 1 . Test object as internal system.
This matrix should include elements of fixed and real,
and the approximation will be with errors that for some
frequency range may be negligible, but for others too
much. Representation of multiphase lines in modal com-
ponents is therefore a great simplification that can be
used for most analyzes of electromagnetic transients,
however, the results of calculations to be verified by
measurement or different modeling of transmission line.
The new representation: Z-line (shown schematically
in Figure 3), consisting of two main elements: the ideal
line represents only the external flux and adjustment
represents the external inductance and resistance is in [4]
proposed.
3. The Choice of Equivalent Circuit
Structu re
In most cases the calculations in equivalent system are
necessary. The results of analysis are depend on a num-
ber of factors and must be appropriately validated by
comparing the results obtained in the real system (pre-
ferably measuring). The use of equivalents makes sense
because of difficulty in the access to measurement data
and ta king i nto acc ount t he p ossib ilit y of a safe and rapid
computing verified result.
The determination of structure and parameter of
equivalent circuit diagram, using traditional methods is
very difficult, and in reality non possible taking into ac-
count t he co mplexit y of the sys tem, no n-linear properties
of individual objects, the spatial distribution of the para-
meters in the model line, as well as its frequency depen-
dence.
Figure 2 . Line model in mod al components.
Figure 3 . Z-Lin e mod e l.

P. SOWA, K. ŁUSZCZ
Copyright © 2013 S ciRes. EPE
1451
In many studies of electromagnetic phenomena as-
sumes a “double standard” during computer simulations.
The test object (e.g. transmission line) representation
takes into account all possible requirements, but the re-
maining system is reduced to a simple structure consist-
ing of few eleme nts.
A classic example is the diagram shown in Figure 4.
Transmission line model as the test object is supplied
from both sides systems, which are represented by
lumped parameters R, L. The irrationality of such ap-
proach is the fact that to the same node, which is con-
nected to the test line, may be attached other transmis-
sion lines, of which the parameters have the same impor-
tance as the test object. The question is: why the “object”
line is modeled with distributed, frequency depended
parameters, but the other lines are modeled with the help
of lumped parameters R and L.
Elements of equivalents in Figure 4 are determined on
the basis of short-circuit power system (calculation of
reactance) and the estimated time constant (dete rmin a-
tion of resistance).
Among supporter of such simplification is the simple
idea that systems characterized by high power short cir-
cuit do not substantially affect the electromagnetic
waveforms - espe ciall y the hi gher fr equenc y co mpone nts,
which quickly disappear and therefore do not play a ma-
jor role.
Very interesting is the proposal to CIGRE [5], a sim-
ple equivalent circuit (Figure 5), which may reflect the
impact of adjacent lines connected to the common node
of the test object. Additional parallel system provides
impedance Zs, the value of which is determined by the
ratio of wave impedance Zf and the number of transmis-
sion lines n operating in the system that is being re-
placed.
~
~
R
zs1
L
zs2
R
zs2
L
zs1
EQUIVALENT 1EQUIVALENT 2
Transmission
line model
Figure 4. Simply representation of system during simula-
tion.
~
~
R
zs1 X zs2 R
zs2
X
zs1
EQUIVALENT 1EQUIVALENT 2
Z
v1 = Z
f/n 1
Z
v1 = Z f/n
2
Transmission
line model
Figure 5. Equivalent of systems with multiple transmission
lines.
The search for optimal equivalent circuit for electro-
magnetic waveform is the most difficult task, due to the
fact that there are components of high (but also low) fre-
quency and the periodic component. For this reason, a
replacement system must take into consideration fre-
quency-dependent parameters. The search for alternative
structures is possible patterns in the time and frequency
domain. Mostly the method in the frequency domain is
preferred. In such case, the external system is replaced by
an equivalent, consisting of elements of R, L, C, whose
frequency response is the same as the original system.
The methods developed in the time domain these ele-
ments play not important role in the equivalent.
In some publications the combination of modeling in
the ti me and freq ue ncy d o main ha s be en. Suc h a sol ution
- the hybrid system proposed in [8], where a replacement
of external system is defined in the frequency domain,
and the anal ysis of the test object in the time domain.
4. Search for Structures
For the analysis of searching for structures of equivalent
the real system was selected and the network topology
was chosen similar to 400 kV Polish transmission sys-
tem.
The investigation were started in simple system as
shown in Figure 4 and the single phase to ground fault
was simulated at the end of lines with a length of 100 km
but the cur rent wa s measured at the begi nning of the line.
In Figures 6 and 7 shows the comparison of the cal-
culated transient current at the beginning of the line to
the L1 phase, the short-circuit phase (L1 + e) at the end
of the l ine.
As can be seen the introduction of a parallel branch in
equivalent circuit reduces the error calculated from the
difference in waveforms of the real system and simpli-
fied. The calculations were performed using MicroTran
S
P
K
BE
A
JV
Z
System SPM
System E
Sys tem BJSy st em A
M
Sy st em VZ
~ ~
R
zs1
L
zs2
R
zs2
L
zs1
EQUIVALENT 1EQUIVALENT 2
Transmission
line model
i[A]
Figure 6. Current waveforms in the system without taking
into account the number of lines in external system.

P. SOWA, K. ŁUSZCZ
Copyright © 2013 S ciRes. EPE
1452
S
P
K
B
E
A
JV
Z
System SPM
System ESystem B JSystem AZ
M
~ ~
R
zs1 Lzs2 Lzs2
Lzs1
EQUIVALENT 1EQUIVALENT 2
Zv1 = Zf/n Zv1 = Zf/n
Tra nsmis s i on
li ne model
i[A]
Figure 7. Current waveforms in the system taking into ac-
count the l ine of n = 10 in external system.
[6], and for the equivalent circuit a different number of
transmission lines simplified was assumed. The best re-
sults are obtained for n = 10, which corresponds to a real
system connected to the line taking into account the 5
adjacent nodes.
5. Parameter Identification of Equivalent
Circuit
Regardless of how is determined equivalent circuit
structure (in the time or frequency domain, by reduction
of the original), it is always necessary to identify the pa-
rameters of components which are par t of the st ruct ure.
The choice of identification method - based on litera-
ture reviews is very difficult because the examples have
been shown for the well-defined problems associated
with adapted for this purpose simulation programs.
In practice, the universal methods of identification,
which would be accurate enough for all solved the prob-
lem has not been developed so far. All of the existing
methods have advantages and disadvantages, many dif-
ferent criteria are used, such as: possible convergence of
numerical solutions, the number of iterations required
computation time as well as the number of induced ob-
jective function. In Netomac-program [7] there are three
optimization methods: least squares, quasi-Newton and
Powell.
The search for the minimum of the objective function
using the least squares method is generally numerically
stable, but it req uires a long calc ulation time. I n addition,
in case of large number of identified parameters, may
arise similar linear relationship, which can cause non-
convergence of the solution. Therefore the similar gra-
dient method was implemented: instrumental variable
and maximum likelihood, which are applied particularly
in the regulation technology.
For this re ason, the so-called “quasi-Newton” group of
methods were introduced, where during simulation the
numerical calculation of second derivative of objective
function was avoid. Instead, they are looking for the ap-
proximation of the second derivative matrix inversion
(metric matrix).
Quasi-Newton method cannot be directly applied to
solving problems in the power system without appropri-
ate modificatio ns and a dditions. There are many tech nic-
al problems that cannot be ignored and which impose
limitations of this method, and in some cases prevent its
use. In Netomac-program the additional methods are
taken to eliminate all the disadvantages of “Newto n-
like”-methods (see Table 1).
Among proposed in the literature so-called coupled
optimization methods that do not require the calculation
of derivatives, there is a group called: “gradientless”
methods, based on the criterion of Powell [8], but very
often the combination of the original and the modified
method of Powell is used. Many additional modifications
have been forced, as the quasi-Newton method; the re-
strictions resulting fro m the imposed legal ranges identi-
fied parameters.
Table 1. Remedies for er r ors of "Newton-like" methods use d in Netomac program.
REMEDIES GENERAL NETOMAC
CRITER ION DOES NOT PROVIDE:
search function with a minimum of restric-
tions Stop changing par ameters on their limits i n each iteration use of the procedure for deter-
mining of the new search direction until th e
convergence of solutions
values of parameters with different ran ge Scaling parameter s at t he begin ning
of the ca l culation adaptive scaling
impact of the time-step on convergence solu-
tion
1) determining any change in the objective
function,
2) dete r m ination bas e d
on the va lue of the obj ec tive functi on
Search procedure using interpolation
rounding error during determination of the
positive definiteness of metric matrix constant examination of positive definiteness of the matrix The introduction of vector correct ion
problem s dur ing calculat i o n
of search direction of partial derivatives procedures to avoid th e impact of rounding and t runcation errors

P. SOWA, K. ŁUSZCZ
Copyright © 2013 S ciRes. EPE
1453
For the previous studied example, the identification
were made using three methods which are available in
Netomac-program. The subject of optimization was the
equivalent circuit where four parameters (two on each
side): resistance Rzs1, Rzs2, and reactance Xzs1, Xzs2 were
identified. The initial values for these parameters are
assumed, which are subject to change during the identi-
fication. In addition, the impedance Zs is varied by
chan ging t he number of line s .
The best results were obtained for the least squares
method identification as shown by comparing the result-
ing vo ltage wa vefor m at t he begin ning of t he line dur ing
t wo-phase to ground short-circuit (L1 + L2 + e) simu-
lated at the end o f t his line.
In Figure 8 the voltage waveforms received in a real
and equivalent system are compared before and after the
parameter identification. The waveforms were calculated
for determined number of transmission lines connected to
both sides of the test line. The results are received using
least-squares method of parameter identification.
Throughout the identification process was observed
the domination o f individual parameters of both systems.
This allo ws accelerating in a rational manner by blocking
the iteration parameter changes, which do not have or
have very little effect on the change in the objective
function.
There is no doubt that, much easier and faster was to
identify the current signals without components of free
higher frequenc ies, which is c learly visible a fter compar-
ison of wave forms in Figures 7 and 8.
In Figures 9 and 10 s how the effect of changes in the
parameters identified by the replacement of one chosen
syste m on the curre nt and vo ltage wave form re specti vel y,
for the cases of identification from Figure 7 and 8.
The results obtained using Powell (gradientless) met ho d
is disappointing. As shown in Figure 11 the identifica-
tion of parameters takes effect op p o site than expected.
S
P
K
B E
A
JV
Z
System SPM
System E
System BJSystem AZ
M
LEAST-SQUAR ES METHOD (LS)
~ ~
R
zs1
L
zs2
R
zs2
L
zs1
EQUIVALENT 1EQUIVALENT 2
Z
v1
= Z
f
/n
1
Z
v1
= Z
f
/n
2
Transmission
li ne model
n
1
=10 n
2
=5
p.u.
Figure 8. Voltage waveforms compared before and after
identification (LS).
Figure 9. Influence of the identified parameters of equiva-
lent syste m on the chang e of the curr ent w aveform for cho-
sen wor king poi nt.
0
-0,0025
-0,002
-0,0015
-0,001
-0,0005
0
0,0005
0,001
0,0015
-0,0008
-0,0006
-0,0004
-0,0002
0
0,0002
0,0004
0,0006
0,0008
246t[ms]
d W /dRzs2
d W /dXzs2
Figure 10. Influe nce of t he iden tified paramet ers of e quiva-
lent system on the change of the voltage waveform for cho-
sen wor king poi nt.
On the basis of these results cannot be generalized,
however, derive recommendations. Very difficult to clear
even the qualitative opinion that the methods discussed
above allows for a greater approximation of passes in the
original and reduced systems. Comparison of the rate of
change in the parameters of identification may give false

P. SOWA, K. ŁUSZCZ
Copyright © 2013 S ciRes. EPE
1454
impression ab out the suitab ility of a particular method in
comparison to the others.
Pessimistic statement is that, despite many years of
research so far has not found a universal equivalent sys-
tem that would faithfully reproduce the behavior of the
electromagnetic power system transients. It seems that is
necessary to distinguish between the structure and the
search process parameter identification.
6. Final Remarks
To obtain reliable results in reduced system containing a
large number o f transmission lines will be subject to f ul-
fillment of conditions:
• Trans mission line, for which transient current
and/or voltage should be determined, must be modeled
taking into account the depending on the frequency pa-
rameters and withou t a ny simplifications.
S
P
K
BE
A
JV
Z
System SPM
System E
Sy st em B JSy st em A
M
Sy st em V Z
~ ~
R
zs1
X
zs2
R
zs2
X
zs1
EQUIVALENT 1EQUIVALENT 2
Z
v1
= Z
f
/n
1
Z
v1
= Z
f
/n
2
Transmission
line model
Powell- method
p.u.
n
1
=10 n
2
=5
Figure 11. Voltage waveforms compared before and after
identification (Powell).
Table 2. Recommendations for e quivalents during study of electromagnetic t ransient.
Action Knowledge of the structure
of the original system Finding structure in doma in Parameter identification Reducti on Starting struc ture
YES
Simply supply or
non-connected system
lm frequency
obligate
static
redundant
all time inadvisable
connected lm frequency dynamic
all time inadvisable
NO
Results f rom mea surements
redundant redundant
exact π system
or ANN
Only short circuit capacity
(determined number of lines)
consistent with Figure
5 or AN N
• Remaining part of the system which have to be re-
duced must be represented by equivalent whose parame-
ters are ide ntified by an appropriate optimization method.
Table 2 summarizes the recommended procedures in
the search for equivalents for the analysis of electromag-
netic t ransient phenomena .
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