Equivalent for Electromagnetic Transient Calculation in Power System with Multiple Transmission Line

doi:10.4236/epe.2013.54B275

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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)

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

Email: pawel.sowa@polsl.pl, [email protected]

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

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.

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

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.

zs1

zs2

zs1

EQUIVALENT 1EQUIVALENT 2

Transmission

line model

Figure 4. Simply representation of system during simula-

tion.

zs1 X

zs2 R

zs2

zs1

EQUIVALENT 1EQUIVALENT 2

v1 = Z

f/n

v1 = Z

f/n

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

System SPM

System E

Sys tem BJSy st em A

Sy st em VZ

zs1

zs2

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

1452

System SPM

System ESystem B JSystem AZ

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

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.

System SPM

System E

System BJSystem AZ

LEAST-SQUAR ES METHOD (LS)

zs1

zs2

zs1

EQUIVALENT 1EQUIVALENT 2

= Z

Transmission

li ne model

=10 n

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.

-0,0025

-0,002

-0,0015

-0,001

-0,0005

0,0005

0,001

0,0015

-0,0008

-0,0006

-0,0004

-0,0002

0,0002

0,0004

0,0006

0,0008

246t[ms]

/dRzs2

/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

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.

System SPM

System E

Sy st em B JSy st em A

Sy st em V Z

zs1

zs2

zs1

EQUIVALENT 1EQUIVALENT 2

= Z

Transmission

line model

Powell- method

p.u.

=10 n

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

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 .

REFERENCES

[1] GIGRE WG 13-05 III, “Transmission Line Representa-

tion for Energization and Reenergization Studies with

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[2] P. Sowa, “Over voltage and Overcorrect During

Non-Simultaneous Faults in Transmission Lines,” IPST

95 International Conference on Power Systems Tran-

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[3] J. R. Marti, “Accurate Modeling of F requency-Dependent

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P. SOWA, K. ŁUSZCZ

1455

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