Frequency domain transient analysis of electrical

networks including non-linear conditions

Pablo Morenoa,*, Pablo Gomeza, Jose L. Naredoa, J.L. Guardadob

aCentro de Investigacion y de Estudios Avanzados del IPN, A.P. 31-438, Plaza la Luna, 44550 Guadalajara, Jal., MexicobInstituto Tecnologico de Morelia, A.P. 262, Morelia, Mich., Mexico

Received 4 April 2003; revised 11 August 2004; accepted 24 September 2004

Abstract

This paper describes a method for the analysis of electromagnetic transients in multiphase transmission networks using the Numerical

Laplace Transform. The proposed procedure is based on the superposition principle and is applied to switching and non-linear elements

modeling. Switching operations are modeled as initial condition problems by means of injected current sources. In the case of non-linear

elements, a piece-wise linear approximation is made, which reduces the problem to a sequence of switching operations. Several applications

and comparisons with results obtained with the EMTDC and ATP programs are presented.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Electromagnetic transient analysis; Numerical Laplace transform; Nonlinearities; Power system transients; Switching transients

1. Introduction

Transient overvoltages are commonly caused by energiza-

tion or reclosure of transmission lines, as well as by faults

occurrence and clearance. Time domain methods represent the

current trend of modeling for electromagnetic transient

analysis. These methods possess great versatility simulating

sequential changes of electrical networks topology and non-

linear elements. Among time domain methods the approach

introduced by Dommel [1] is nowadays the most powerful

existing tool for transient analysis in power systems. As an

inherent nature of time domain methods dealing with elements

with frequency dependent parameters introduces complicated

convolution procedures. Although these methods have

recently incorporated frequency dependent representations

of overhead lines and cables, these procedures have not been

extensively tested [2]. On the other hand, when using

frequency domain techniques to analyze electromagnetic

transients, the frequency dependence of the electrical

0142-0615/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijepes.2004.09.003

* Corresponding author. Tel.: C52 33 3134 5570; fax: C52 33 3134

5579.

E-mail address: pmoreno@gdl.cinvestav.mx (P. Moreno).

parameters can be taken into account very easily [3,4].

In spite of the fact that this later technique has been well

established for several years, a computer program of general

access is still missing. One problem here has been that dealing

with changes in network topology and with non-linear

elements presents some difficulties.

In order to develop a frequency domain method able to

solve practical problems, the superposition principle can be

used [4,5]. When applying this principle, discontinuities

(i.e. switch maneuvers) are treated as initial conditions

problems and non-linear elements (i.e. surge arresters) are

reduced to a series of sequential discontinuities. The general

procedure consists of adding the response due to certain

initial conditions to that due to the injection of some voltage

or current source.

In this paper, a technique for modeling switches and non-

linear resistors when using the numerical Laplace transform

method (NLT) is proposed. Numerical inversion of the

Laplace transform produces errors due to truncation of

integration range and discretization of the frequency

spectrum [6]. Practical insight on reducing these errors is

also presented. Three application examples are used for

testing the method and comparisons with EMTDC and ATP

results are provided.

Electrical Power and Energy Systems 27 (2005) 139–146

www.elsevier.com/locate/ijepes

P. Moreno et al. / Electrical Power and Energy Systems 27 (2005) 139–146140

2. Frequency response of a network

In this work, the nodal analysis method is adopted to deal

with transmission networks. The bus admittance matrix is

built from the admittance representation of the network’s

elements. In the case of multiphase transmission lines the

two-port admittance model is described by the following

equation

I0

Il

" #Z

Y0ðsÞcothðjðsÞlÞ KY0ðsÞcschðjðsÞlÞ

KY0ðsÞcschðjðsÞlÞ Y0ðsÞcothðjðsÞlÞ

" #V0

Vl

" #

(1)

where sZcCju; u is the angular frequency and c is a real

finite constant with a value greater or equal to zero; l is the

line length and I0, V0 are the sending end nodal current and

voltage vectors; Il, Vl are the receiving end nodal current

and voltage vectors

jðsÞ Z Mffiffiffil

pMK1 Z voltage propagation matrix (1a)

M, l are the eigenvector and eigenvalue matrices of

Z(s)Y(s)

Y0ðsÞ Z ZðsÞK1jðsÞ Z characteristic admittance (1b)

Z(s), Y(s) are per unit length impedance and admittance

matrices.

With the admittance models of all the elements of the

network, the total admittance matrix in the Laplace domain

can be formed

I1ðsÞ

«

IjðsÞ

«

INðsÞ

266666664

377777775

Z

Y11ðsÞ . YijðsÞ . Y1NðsÞ

« 1 « 1 «

Yj1ðsÞ . YjjðsÞ . YjNðsÞ

« 1 « 1 «

YN1ðsÞ / YNjðsÞ . YNNðsÞ

266666664

377777775

V1ðsÞ

«

VjðsÞ

«

VNðsÞ

266666664

377777775

(2a)

or in reduced form

IðsÞ Z YbusðsÞVðsÞ (2b)

where V(s) is the nodal voltage vector and I(s) the injected

current vector.

Expression (2a) must be solved to get the node voltages

in the Laplace domain and the waveforms in the time

domain can be obtained using the inverse Laplace trans-

form:

vðtÞ Z1

2pj

ðcCjN

cKjNVðsÞest ds (3a)

The corresponding direct Laplace transform is given by

VðsÞ Z

ðN

0vðtÞeKst ds (3b)

For practical cases (2a) cannot be solved analytically,

even for small networks. Thus, (3a) and (3b) have to be

evaluated numerically. Numerical evaluation of these

equations gives rise to truncation and discretization errors.

Practical techniques for reducing numerical errors when

inverting from the Laplace domain to the time domain are

addressed in Section 3.

3. Numerical Laplace transform

3.1. Numerical algorithm

Let v(t) be a real and causal function of time and V(s) its

image in the Laplace domain. Considering a finite

integration range, the inverse Laplace transform (3a) can

be written as

vðtÞ Z Reect

p

ðU

0Vðc C juÞejut du

� �(4a)

where U is the maximum frequency. The corresponding

direct Laplace transform is

Vðc C juÞ Z

ðT

0½vðtÞeKct�eKjut dt (4b)

where T is the observation time.

The numerical forms of Eq. (4), with odd sampling in the

frequency domain, that allow using the Fast Fourier

Transform algorithm [6] are as follows

vn Z Re Cn

XNK1

mZ0

Vm expj2pmn

N

� �( ); n Z 1; 2;.;N K1

(5a)

Vm ZXNK1

nZ0

fnDn expKj2pmn

N

� �; m Z 1; 2;.;N K1

(5b)

where

Vm Z V½c C jð2m C1ÞDu� (5c)

vn Z vðnDtÞ (5d)

Dn Z Dt expðKcnDt K jpn=NÞ (5e)

Cn Z ð2Du=pÞexpðcnDt C jpn=NÞ (5f)

Du Zp

T(5g)

Dt ZT

N(5h)

U Z2p

Dt(5i)

being Du the spectrum integration step, Dt the time

discretization step and N the number of samples.

P. Moreno et al. / Electrical Power and Energy Systems 27 (2005) 139–146 141

When performing a transient study, T is simply chosen as

the time duration of interest in the analysis. Once this has

been done the value of Du is automatically fixed according

to (5g). The time step Dt and therefore the maximum

frequency U are determined by the election on the number

of samples N, according to (5h) and (5i). A good choice of N

can be done by using the sampling theorem and considering

the maximum bandwidth required for a particular study.

Fig. 1. Errors obtained with four different windows (three cycles).

Fig. 2. Errors obtained with cZ2Du (Wilcox).

3.2. Analysis of errors

Numerical evaluation of (4a) introduces two kinds of

errors: Gibbs oscillations due to truncation of integration

range and aliasing due to discretization of the continuous

variables.

Truncation errors are reduced by introducing a ‘window’

function in the integrand of Eq. (4a). This is, V(cCju) is

multiplied by a window function s(u) and therefore it can

be written

vn Z Re Cn

XNK1

mZ0

Vmsm expj2pmn

N

� �( ); n Z 1;.;N K1

(6a)

where

sm Z ½ð2m C1ÞDu� (6b)

An evaluation of the effectiveness of four of the most

common windows, Hanning, Lanczos, Blackman and Riesz

[7], in reducing truncation errors was performed. The

window functions were applied to the numerical inversion

of the Laplace transform of a delayed cosine function

given by

f ðtÞ Z uðt KtÞcos½uðt KtÞ� (7a)

with uZ377 rad/s and tZ2 ms. The corresponding Laplace

transform of (7a) is

FðsÞ Z expðKtsÞs

s2 Cu(7b)

A number of 256 samples was used and observation

times of 1 and 3 cycles were tested. Relative errors due to

the inclusion of each window were calculated as follows

err Zf2ðtÞK f ðtÞ

max½f ðtÞ�

�������� (7c)

where f2(t) is the numerical approximation of f(t) using any

of the windows and computed according to (6a). All four

windows gave excellent results when used in one cycle.

However, as seen in Fig. 1, Lanczos and Riesz windows

gave poor results for an observation time of three cycles.

Aliasing errors can be reduced by ‘smoothing’ the

frequency response of the system. This is done by a

proper choice of the convergence factor c. Two formulas

for calculating this factor were tested. The first one was

proposed by Wilcox [6]

c Z 2Du (8a)

and the second one by Wedepohl [8]

c ZlnðN2Þ

T(8b)

The same cosine function given by (7a) was used in this

analysis. The evaluation was performed for three different

numbers of samples: 28, 210 and 212. Figures of error

obtained in this study are shown in Figs. 2 and 3. As can be

seen in Fig. 2, using the first formula provides good results,

however, the error remains almost constant when the

number of samples is varied and the observation time is

fixed to the same value. In the case of the second formula,

the error decreases as the number of samples used in the

simulation increases, as shown in Fig. 3. Therefore, with this

later formula, given the required observation time for a

particular study, aliasing errors can be reduced by increas-

ing the number of samples.

It is important to notice in Figs. 1–3, the presence of an

error of almost 100% at 2 ms. This time instant corresponds

to the cosine function delay. This is explained by the finite

elevation time that results from the numerical evaluation of

the inverse Laplace integral.

Fig. 3. Errors obtained with cZ lnðN2Þ=T (Wedepohl).

P. Moreno et al. / Electrical Power and Energy Systems 27 (2005) 139–146142

4. Switch model

Fig. 4. (a) Ideal switch and (b) practical switch model.

4.1. Ideal switch

Switching operations produce changes in network

topology that turn the network into a time variant system

precluding, apparently, the use of frequency domain

methods. However the Superposition Principle can still be

applied to overcome these problems.

An open switch can be represented by a voltage source

Vsw equal to the potential difference between its terminals.

Switch closure is accomplished by the series connection of a

voltage source Vsw2 with equal magnitude but opposite

sense to Vsw. The voltage source required to close the switch

at time tcO0 is given by

Vsw2 Z L KvswðtÞuðt K tcÞf g (9)

where vsw(t) is the time domain waveform of the voltage

between the switch terminals for the whole observation time

with the switch opened and L indicates the Laplace transform.

The total response of the electrical network is obtained

by superimposing the response due to Vsw, which is the

network operation condition existing before the switch

closure, to the response due to Vsw2.

On the other hand, a closed switch is represented by a

current source Isw, equal to the current flowing across it.

Switch opening is performed by connecting in parallel to

Isw, a current source of equal magnitude but opposite sense.

Opening the switch when the current reaches its first zero

value after a specified opening time is done by injecting a

current source Isw2 given by

Isw2 Z L KiswðtÞuðt K tzcÞf g (10)

where tzc is the current zero-crossing time and isw(t) is the

time domain waveform of the current flowing through the

closed switch for the whole observation time.

Similarly to the case of a closure the total response of the

electrical network is obtained by superimposing the

response due to Isw, to the response due to Isw2.

The current flowing through the switch, isw(t), can be

calculated employing the currents injected into one of the

switch nodes. If the switch is connected between the jth and

the kth nodes the current across it is

iswðtÞZL YjjðsÞCXn

iZ1isj

YjiðsÞ

26664

37775Vj C

Xn

iZ1isj;k

YjiðVi KVjÞ

8>>><>>>:

9>>>=>>>;

(11)

In (11), Yji is the element corresponding to the jth row and

ith column of the admittance matrix Ybus and Vj is the jth

element of the nodal voltage vector V.

4.2. Practical switch model

To deal with electrical networks the nodal analysis

method was chosen in this work. Thus ideal voltage sources

cannot be used to simulate switch closures. The injection of

voltage Vsw2 must be accomplished by means of a Norton

equivalent with current source given by

Jsw2 ZVsw2

Rx

(12)

where Rx is a resistance needed to perform the source

transformation. Rx must be small to approximate an ideal

source or it can take some particular value for representing a

contact resistance.

A practical switch model suitable for simulating closures

and openings is shown in Fig. 4. The injected current Jsw is

given by

Jsw ZVsw2

Rx

; closure

Isw2; opening

((13)

and the Norton conductance Gx is given by

Gx Z

1

Rx

; closure

K1

Rx

; opening

8>><>>: (14)

Fig. 6. Circuit for a non-linear resistance modeled with N linear segments.

P. Moreno et al. / Electrical Power and Energy Systems 27 (2005) 139–146 143

The conductance between the switch nodes must be of a

large value when it is closed or zero when it is open. This

means that a topological change of the network nodal

matrix must be done when switching occurs. Assuming

that the switch is connected between the jth and kth nodes,

modification of the nodal matrix is accomplished as

follows:

yjj / yjk

« 1 «

ykj . ykk

264

375/

yjj CGx / yjk KGx

« 1 «

ykj KGx . ykk CGx

264

375 (15)

The topological effect due to the modification represented

in (15) given the values of (14) is introducing (closure) or

extracting (opening) Rx from the network.

The complete voltage response is obtained by adding the

system response existing before switching to that resulting

from applying the current source that performs the switch

maneuver. Therefore, the complete solution corresponding

to a maneuver can be expressed as follows

V Z Vð0Þ C ðYð1ÞbusÞ

K1Ið1Þ (16)

where V(0) is the node voltages before switching, Yð1Þbus is the

admittance matrix modified according to (15) and

Ið1Þ Z½0 / Jsw / KJsw / 0�T

1 j k N(17)

5. Non-linear elements

In order to include non-linear elements in frequency

domain techniques, it is necessary to approximate the non-

linear characteristics in piece-wise linear forms. Once this

approximation has been made, the simulation procedure is

reduced to a sequence of switching operations. Fig. 5 shows

the v–i characteristic of a non-linear element approximated

by N linear segments with slopes Rn. Each segment

represents the Thevenin equivalent that the network sees

toward the non-linear element over the corresponding

operation range. The voltage between nodes j and k for

Fig. 5. Piece-wise linear approximation of a non-linear resistance.

any operation point is given by

v Z Vn CRni (18)

where Vn is the crossing point of the line with slope Rn with

the vertical axis and i is the current through the element.

The circuit that represents the piece-wise linear charac-

teristic of Fig. 5 is shown in Fig. 6. Depending on the

voltage value the branches will be connected or discon-

nected from the network using switches, as described in

Section 4. The values of Rxn and Vxn that correspond to the

Thevenin equivalent of the nth segment are given by

VXn ZRnK1Vn KVnK1Rn

RnK1 KRn

(19a)

RXn ZRnRn

RnK1 KRn

(19b)

Notice that in the simulation procedure the branches

corresponding to the linear segments have to be introduced

or extracted in a sequential order. When switch n is open

it can be closed only after switch nK1 has been closed.

On the other hand, if switch n is closed it cannot be opened

if switch nC1 is still closed. To accomplish this, the time

step Dt must be small enough to prevent jumps between

non-contiguous segments of the linear piece-wise

approximation.

6. Application examples

The proposed procedures were validated by simulating

several examples and comparing the results with those

obtained using the EMTDC and ATP programs with

frequency dependent line models. The Phase Domain Line

Model [2,9] and the J. Marti Set up [10] were used for the

EMTDC and ATP simulations, respectively. For the

frequency domain calculation, the 2-Port admittance

representation for multiphase transmission lines was

used [3,4].

Fig. 7. Circuit for example A.

Fig. 8. Conductors’ arrangement for example A.

Fig. 10. Transient recovery voltage in phase B using ATP.

P. Moreno et al. / Electrical Power and Energy Systems 27 (2005) 139–146144

6.1. Transient recovery voltage

In this example, the transient recovery voltage is

obtained when a three-phase to ground fault at the receiving

end is cleared 2 ms after line energization. The circuit and

the conductors’ arrangement are shown in Figs. 7 and 8,

respectively. Note that the system of Fig. 7 does not

represent a real case due to the pure inductance on the

source side. However, this example illustrates some

problems that can appear when using programs based on

time domain methods.

Fig. 9 shows the transient recovery voltage in phase B of

the switch obtained with the NLT program and the EMTDC.

Fig. 9. Transient recovery voltage in phase B using EMTDC and NLT.

The EMTDC-1 waveform was obtained using the same Dt

as in the NLT, while in the EMTDC-2 the Dt used was

10 times smaller. A frequency range of 0.5 Hz–0.5 MHz

was used for the vector fitting, with a maximum number of

poles/zeros of 20.

This example was also simulated using the ATP and the

results are shown in Fig. 10. The ATP-1 waveform was

found with the same Dt as the NLT, while for the ATP-2 a

Dt 10 times smaller was used. Initial frequency in the J.

Marti Setup was 0.5 Hz, considering 10 decades and 10

points/decade. The transformation matrix was computed at

5 KHz.

6.2. Surge arrester

The second application example is a sequential energiza-

tion of the line shown in Fig. 11. A surge arrester was

connected on each phase at the receiving end of the line. The

arresters were represented as non-linear resistances with v–i

Fig. 11. Circuit for example B.

Table 1

V–I characteristic of arresters

Voltage (kV) Current (kA)

480 0.176

520 0.3226

560 0.7626

600 1.6426

620 12.6426

Fig. 12. Voltage in phase B at the receiving line end.

Fig. 14. Conductors’ arrangement for example C.

P. Moreno et al. / Electrical Power and Energy Systems 27 (2005) 139–146 145

curves approximated with five linear segments whose values

are presented in Table 1. Closing times for phases A, B and

C were 3, 6 and 9 ms, respectively. Fig. 12 shows the

voltage of phase B at the receiving end. Again, the time step

of the EMTDC simulations was 10 times smaller than that of

the NLT. The same data of example A was used in the

EMTDC vector fitting routine.

Fig. 15. Voltage in phase C at the receiving node of circuit 2.

6.3. Sequential energization of a highly asymmetrical line

As a final application example, the circuit shown in

Fig. 13 is considered [11]. The conductors arrangement

consists of two flat circuits separated by a significant

distance, as shown in Fig. 14. A sequential energization of

circuit 1 was simulated, with closing times of 1.95, 3.15 and

0.25 ms for phases A, B and C, respectively. Receiving ends

of both circuits were left open.

Fig. 15 shows the voltage induced on the phase C at the

receiving node of circuit 2. In this case, the time step needed

in the EMTDC was five times larger than in the NLT to get

similar results. As can be seen in Fig. 15, there still exists a

time delay and a difference in the spikes amplitude between

the waveforms. In this case it was not possible to obtain a

better match by increasing the number of samples beyond

five times in the EMTDC.

Fig. 13. Circuit for example C.

7. Conclusions

The results obtained show the effectiveness of the

proposed method for simulating a variety of events that

generate electromagnetic transients. It was found that in

some cases the number of points needed to get correct

results using the EMTDC can be greater than that employed

by the NLT. It was also found that numerical artifacts that

produce oscillations still exist in the ATP but not in the

EMTDC. The authors’ opinion is that, although time

domain methods are more versatile than the frequency

domain ones, these last are of great help in validating the

behavior of new models for time domain programs. In

addition, the NLT simulations help to determine whether the

oscillations presented in time domain analysis are part of the

transient phenomenon or due to the numerical computations

involved.

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