Page 1

TLM modeling of transformer

with internal short circuit faults

Okan Ozgonenel

Department of Electrical and Electronics Engineering,

Ondokuz Mayis University, Samsun, Turkey, and

David W.P. Thomas and Christos Christopoulos

School of Electrical and Electronics Engineering, The University of Nottingham,

Nottingham, UK

Abstract

Purpose – The purpose of this paper is to describe a technique for modeling transformer internal

faults using transmission line modeling (TLM) method. In this technique, a model for simulating a two

winding single phase transformer is modified to be suitable for simulating an internal fault in both

windings.

Design/methodology/approach – TLM technique is mainly used for modeling transformer

internal faults. This was first developed in early 1970s for modeling two-dimensional field problems.

Since, then, it has been extended to cover three dimensional problems and circuit simulations. This

technique helps to solve integro-differential equations of the analyzed circuit. TLM simulations of a

single phase transformer are compared to a custom built transformer in laboratory environment.

Findings – It has been concluded from the real time studies that if an internal fault occurs on the

primary or secondary winding, the primary current will increase a bit and secondary current does not

change much. However, a very big circulating current flows in the shorted turns. This phenomenon

requires a detailed modeling aspect in TLM simulations. Therefore, a detailed inductance calculation

including leakages is included in the simulations. This is a very important point in testing and

evaluating protective relays. Since, the remnant flux in the transformer core is unknown at the

beginning of the TLM simulation, all TLM initial conditions are accepted as zero.

Research limitations/implications – The modeling technique presented in this paper is based on

a low frequency (up to a few kHz) model of the custom-built transformer. A detailed capacitance model

must be added to obtain a high-frequency model of the transformer. A detailed arc model, aging

problem of the windings will be applied to model with TLM þ finite element method.

Originality/value – Using TLM technique for dynamical modeling of transformer internal faults is

the main contribution. This is an extended version of an earlier referenced paper of the authors and

includes inductance calculation, leakages calculation, and BH curve simulation while the referenced

paper only includes piecewise linear inductance values. This modeling approach may help power

engineers and power system experts understand the behavior of the transformer under internal faults.

Keywords Transformers, Electrical faults, Inductance electric power transmission, Modelling

Paper type Research paper

1. Introduction

Large power transformers are a class of very expensive and vital components of

electric power systems. Since, it is very important to minimize the frequency and

duration of unwanted outages, there is a high demand imposed on power transformer

protective relays; this includes the requirements of dependability associated with no

mal-operations, security associated with no false tripping, and operating speed

The current issue and full text archive of this journal is available at

www.emeraldinsight.com/0332-1649.htm

COMPEL

26,5

1304

Received January 2006

Revised November 2006

Accepted November 2006

COMPEL: The International Journal

for Computation and Mathematics in

Electrical and Electronic Engineering

Vol. 26 No. 5, 2007

pp. 1304-1323

q Emerald Group Publishing Limited

0332-1649

DOI 10.1108/03321640710823037

Page 2

associated with short-fault clearing time. Protection of large power transformers is a

very challenging problem in power system relaying (Golshan et al., 2004; Mao and

Aggarwal, 2001).

The development and the validation of algorithms for transformer protection

require the preliminary determination of a power transformer model. This model must

simulate all the situations that will be chosen to study the behavior of the protection

algorithm. In particular, it must allow for the simulation of internal and external faults.

Most of the electromagnetic transient programs available are able to accurately

simulate other phenomena occurring in the transformer like inrush magnetizing

current, exciting current and transformer saturation (Bastard et al., 1994). Developing a

model to simulate internal and external faults of a power transformer, and applying it

to test a transformer protection algorithm, would provide a true validation (Leibfried

and Feser, 1999; Wilcox, 1997; Megahed, 2001).

The main problem which may be seen with computer monitoring systems is that of

detecting incipient (or internal) faults in transformers. It is well known that most

internal faults in a power transformer begin as small discharge inside the transformer

tank. As these currents continue to flow, they cause further damage, accelerate the

insulation breakdown, and lead to more serious permanent faults. At present, the

incipient faults are detected by analyzing the gases collected in the transformer tank as

a by-product of combustion process. It may be possible to develop a technique to

recognize the incipient fault condition by detecting certain features of the transformer

current. It would be very difficult to detect a change in current caused by the incipient

fault; it is too small compared to the transformer load current. Perhaps, the frequency

content of the transformer current may be a unique feature of the incipient fault. In any

case, it would be necessary to suppress the steady state components from such

consideration, and also rather high-sampling rates may be needed to detect the

expected high-frequency components in the incipient fault discharge current. However,

the implementations of the existing monitoring methods (Leibfried and Feser, 1999;

Megahed, 2001) tend to cost too much to be applied to distribution transformers. A

study of the records of modern transformer breakdowns which occur over a period of

years shows that between 70 and 80 percent of the number of failures are finally caused

by short-circuits between turns (Wilcox, 1997; Wang and Butler, 2001)

2. Developing of the proposed model

The TLM method was first developed in early 1970s for modeling two-dimensional

field problems. Since, then, it has been extended to cover three dimensional problems

and circuit simulations. For circuit simulation, the TLM method can be used to develop

a discrete circuit model directly from the system without setting up any

integro-differential equations. The TLM model algorithm is discrete in nature and

ideally suited for implementation on computers (Hui and Christopolus, 1992).

The basic TLM technique models linear reactive components as transmission line

segments (called stubs). The stub model representing the inductor is terminated by a

short circuit to emphasize inductive behavior; hence, storage in the magnetic field must

be maximized. The TLM model for a capacitor is a stub with its far end and open

circuit. It emphasizes voltage differences, storage in the electric fieldand,hence, mainly

capacitive behavior (Ozgonenel et al., 2005; Hui and Christopolus, 1991).

TLM modeling

of transformer

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Inductance calculation is essential for modeling aspect and internal fault analysis.

The existing analysis models of internal short-circuit fault of power transformer do

have some shortcomings such as defining the winding inductance as linear or is time

consuming (i.e. finite element-based methods).

If an internal fault occurs in the winding, the winding is sub-divided into three

sections (Figure 6). Each sub-divided part is represented by an inductance and

subsequently its TLM equivalent. The resistance of the short circuited stub can be

neglected since the reactance of this stub is much bigger than it. For an internal fault

condition, the system matrices are updated according to the new system state. Some

modifications to these matrices are performed in this study.

According to the analysis procedure described above, the proposed model for

analyzing internal short circuit fault satisfies two requirements:

(1) The leakage magnetic field should be taken into considerations; and

(2) The inductance of a winding is calculated directly according to the BH curve at

the each time step.

The two steps are the origin of the problem and will be discussed in this paper.

2.1 Inductance modeling

The TLM model of a linear inductance is shown in Figure 1.

Any differential terms in the form of Ldi=dt can be replaced by the discrete

transform ZLi þ 2Vi

pulse. It is assumed that it takes one time step Dt for the pulse to make a round-trip to

travel to one end and be reflected back as the incident pulse in the next time step.

In many applications, it is necessary to account for the non-linear behavior of

inductors and capacitors. Non-linear inductor and capacitor can be modeled as

non-linear stubs. Because of flux leakage, eddy currents, etc. it is not known exactly

what relationship is between voltages and currents. One of the objectives doing a

non-linear analysis is to resolve this sort of effect. Figure 2 shows a non-linear

inductance modeling and its Thevenin equivalent circuit in TLM.

The voltage drop across a current-dependent inductor LðiÞ ¼ dl=dt is described in

equation (1):

L, where ZL¼ 2L=Dt, Dt is the time step and Vi

Lis the incident

VL ¼dl

dt¼ LðiÞdi

dt¼ LðiÞ 1di

dt

??

¼ LðiÞVLu

ð1Þ

Figure 1.

(a) Inductance; (b) stub

model of inductance;

(c) TLM equivalent circuit

Vi

L

2Vi

L

Vr

L

VL

ZL

ZL

L

(a)(b)(c)

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where the expression in brackets represents voltage drop VLuacross a 1H inductor in

which current i flows. A stub model of the unit inductor may be constructed in the

usual way to give VLu¼ LðiÞ ZLui þ 2Vi

inductance. Thus, the voltage across the inductor L(i) is given in equation (2):

h

equation (2) is used with the circuit equations to find the current. At each time-step, the

value of L(i) is updated, and VLuis calculated. The new incident voltage is then

calculated in equation (3).

Lu

h i

, where ZLu¼ 2=Dt and is called unit

VL ¼ LðiÞ ZLui þ 2Vi

Lu

i

ð2Þ

kþ1Vi

Lu¼kVi

Lu2kVLu

ð3Þ

This modeling approach (equations (1)-(3)) is valid if the skin effect is not important

due to DC resistance of the coil.

Magnetic coupling between components is also described by using TLM technique.

A typical configuration by using non-linear inductances described above is shown in

Figure 3(a). Mutual coupling is modeled by a current controlled voltage source. The

resulting Thevenin equivalent is shown in Figure 3(b).

The controlled sources representing terms of the type Mðdi=dtÞ with varying

coefficients are described in equations (4a) and (4b):

h

h

The solution proceeds by writing KVL for the two circuits in Figure 3(b). Those

equations are derived from equations (2), (4a) and (4b):

h

h

Vm12¼ M12ði2Þ i2ZLmuþ 2Vi

Lmu

i

i

ð4aÞ

Vm21¼ M21ði1Þ i1ZLmuþ 2Vi

Lmu

ð4bÞ

Vs¼ i1ðRsþ R1Þ þ L1ði1Þ ZL1ui1þ 2Vi

L1u

i

þ

M12ði2Þ i2ZLmuþ 2Vi

Lmu

hi

ð5aÞ

0 ¼ i2ðR2þ RLÞ þ L2ði2Þ ZL2ui2þ 2Vi

L2u

i

þ

M21ði1Þ i1ZLmuþ 2Vi

Lmu

hi

ð5bÞ

Figure 2.

(a) A non-linear inductor;

and (b) its TLM

representation

VL

+

–

ZLu

2VLu

i

L(i)

i(t)

(a)(b)

TLM modeling

of transformer

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where Rsand R1are the source and primary side resistances, R2and RLare the

secondary side and load resistances, M12and M21are the mutual inductances, L1and

L2are the inductances of the primary and the secondary side, respectively. At the

beginning of the simulation, all initials values are set to zero. This assumes that there is

no remnant flux in the transformer core. The equations are solved to find i1and i2and

thus the total TLM voltages are calculated through equations (4a), (4b), (5a) and (5b).

L1and L2are current dependent inductances and shows non-linear characteristics

due to magnetic core material. Therefore, determination ofthe primary side inductance,

L1, is calculated directly according to the BH curve at the each simulation time step.

Then, L2is calculated according to the turn ratio of the transformer. i1and i2are

calculated as below:

i1¼ 2

?

ð2M12M21Vi

þ M12Vi

2 2R2Þ

þ RsðL2ZL2u2 RL2 R2Þ þ R1ðL2ZL2u2 RL2 R2Þ

Lmuþ 2i2M12Vi

Lmuð22L2ZL2u2 2RL2 2R2Þ þ L1Vi

?.?

L2uÞZLmuþ VsðL2ZL2uþ RLþ R2Þ

L1uð22L2ZL2u2 2RL

Lmuþ L1ZL1uð2L2ZL2u2 RL2 R2Þ

M12M21Z2

?

ð6aÞ

Figure 3.

A transformer example

with non-linear

inductances

2VL1u

i

i1

i1

i2

i2

R2

R2

Rs+R1

Rs+R1

RL

RL

2VL2u

i

ZL2u

ZL1u

Vm12

Vm21

VL1VL2

+

–

+

–

+

–

+

–

Vs

Vs

L(i)

M

(a)

(b)

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i2¼

?

þ 2L1L2Vi

ðM12Vs2 2M12M21Vi

Lmu2 2L1M12Vi

L1uÞZLmuþ ð2L1M21Vi

Lmu

L2uÞZL1u

þð2M21Rsþ M21R1ÞVi

Lmuþ ð2L2RsVi

L2uþ 2L2R1Vi

L2uÞ

?.

ðM12M21Z2

Lmu

þ L1ZL1uð2L2ZL2u2 RL2 R2Þ

þ RsðL2ZL2u2 RL2 R2Þ þ R1ðL2ZL2u2 RL2 R2Þ

The new incident voltages at the next time step are calculated in equations (7a)-(7d) by

using equation (3):

?

ð6bÞ

kþ1Vi

L1¼kVi

L12kVL1

ð7aÞ

kþ1Vi

L2¼kVi

L22kVL2

ð7bÞ

kþ1Vi

M12¼kVi

M122kVM12

ð7cÞ

kþ1Vi

M21¼kVi

M212kVM21

ð7dÞ

where k is the simulation time steps.

These values are substituted into equations (5a) and (5b) to calculate the new

currents at the next time steps.

Leakage inductances can be added to have a more complete transformer model.

Figure 4 shows the TLM transformer modeling by taking into account non-linear

leakage inductances of both sides. VLe1and VLe2are the leakage voltages of primary

side and secondary side, respectively.

Figure 4.

Transformer modeling

with non-linear leakages

inductances

2VL1u

i

2VLe1u

i

VLe1

VLe2

ZLe1u

ZLe2u

2VLe2u

i

i1

i2

R2

Rs+R1

RL

2VL2u

i

ZL2u

ZL1u

Vm12

Vm21

VL1

VL2

+

–

+

–

+

–

+

–

+

–

+

–

Vs

TLM modeling

of transformer

1309

Page 7

If leakage inductances of both primary and secondary sides are taken into account,

following equations, which are similar to equations (5a) and (5b), are derived from the

circuit topology:

Vs¼ i1ðRsþ R1Þ þ L1ði1Þ ZL1ui1þ 2Vi

h

L1u

h i

þ

M12ði2Þ i2ZLmuþ 2Vi

Lmu

i

þ i1ZLe1uþ 2Vi

Le1u

ð8aÞ

0 ¼ i2ðR2þ RLÞ þ L2ði2Þ ZL2ui2þ 2Vi

h

L2u

hi

þ

M21ði1Þ i1ZLmuþ 2Vi

Lmu

i

þ i2ZLe2uþ 2Vi

Le2u

ð8bÞ

where ZLe1uand ZLe2uare the unit inductances of the leakage inductances and Vi

and Vi

Le2uare the incident voltages across the leakage inductances. Generally, the

leakage inductance between two coils is calculated by using equation (9) from the

electromagnetic energy (W) stored in the coil:

Le1u

W ¼m0

2

v

ððð

H2dV

ð9Þ

The three dimensional integral is reduced to one dimensional integral since the

windings are the same height. In this case, the following hypotheses are assumed:

H1.

Magnetic field is parallel to the axis of the core.

H2.

Magnetic field is symmetrical in the relation to the core axis.

The total leakage inductance of the winding is calculated using the equation (10):

W ¼1

2Lei2

ð10Þ

At the beginning of the simulation, the leakage inductance, Leis assumed as zero and

the terms in equation (11a) and (11b) relating to characteristic impedances (ZLe) of the

Leare replaced by zero. According to the equations (8)-(10), those equations are again

solved to find i1and i2in equations (11a) and (11b):

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i1¼

?

ð2M12M21Vi

Lmuþ 2M12Vi

Le2uþ 2L2M12Vi

L2uZLmu

þ ðVs2 2M12Vi

þ ðL2Vs2 2L2M12Vi

þ ðRLþ R2ÞVsþ ð22M12RL2 2M12R2ÞVi

Lmu2 2Vi

Le1u12 2L1Vi

L1uÞZLe2u

Lmu2 2L2Vi

Le1u2 2L1L2Vi

L1uÞZL2u

Lmu

þð22RL2 2R2ÞVi

Le1uþ ð22L1RL2 2L1R2ÞVi

L1u

?.?

M12M21Z2

Lmu

þ ð2ZLe1u2 L1ZL1u2 Rs2 R1ÞZLe2u

þ ð2L2ZL2u2 RL2 R2ÞZLe1uþ ð2L1L2ZL1u2 L2Rs2 L2R1ÞZL2u

þ ð2L1RL2 L1R2ÞZL1uþ ð2RL2 R2ÞRs2 R1RL2 R1R2

?

ð11aÞ

i2¼

?

ðM21Vs2 M12M21Vi

Lmu2 2M21Vi

Le1u2 2L1M21Vi

L1uÞZLmu

þ ð2M21Vi

þð2M21Rsþ 2M21R1ÞVi

Lmuþ 2Vi

Le2uþ 2L2Vi

L2uÞZLe1u

Lmuþ ð2Rsþ 2R1ÞVi

Le2uþ ð2L2Rs

þ 2L2R1ÞVi

L2u

?.?

M12M21Z2

Lmuþ ð2ZLe1u2 L1ZL1u2 Rs2 R1ÞZLe2u

þ ð2L2ZL2u2 RL2 R2ÞZLe1uþ ð2L1L2ZL1u2 L2Rs2 L2R1ÞZL2u

þ ð2L1RL2 L1R2ÞZL1uþ ð2RL2 R2ÞRs2 R1RL2 R1R2

?

ð11bÞ

As soon as i1and i2are calculated, TLM voltages are then calculated and the whole

procedure continues repetitively. In equations (11a) and (11b), L1and L2are non-linear

inductances and their values are changed in each simulation time step. Therefore, the

values of M12and M21are also changed during the simulation since L1and L2are

changed.

3. Laboratory experiments

A single phase, 600VA, 220/110V, 50Hz, B ¼ 1T transformer with different tap

connections is used for modeling and real time applications. Primary side has a total of

440 turns with 200, 250, 260, 270, 300, 420, 430 tap connections and secondary side has

a total of 230 turns with 100, 150, 160, 180, 190 tap connections. Figure 5(a) and (b)

show the cross-sectional view of the transformer layout and the winding layout,

respectively.

In the TLM analysis, it is assumed that the coupling coefficient, k, between the two

inductors is the same in both directions, i.e. k12¼ k21¼ k. This assumption is valid

TLM modeling

of transformer

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Page 9

since the transformer physical structure is symmetrical with respect to the windings

(the two coils are geometrically identical). In Figure 5(b), inner parts (primary) are

inside of outer parts (secondary).

Real time simulations performed at the laboratory consist of the following topics:

. Unloaded and loaded healthy conditions;

. Magnetizing inrush conditions with different load variations;

. Load variations (quarter, half, full, and over-loaded); and

. Short circuit studies performed on both primary and secondary sides.

In the laboratory studies, to simulate internal short-circuit faults, fault resistance RF(at

fault location) is assumed to be zero and fault is switched on-off during the simulation.

To simulate incipient transformer faults and also limit the fault current, an RF(greater

than zero) is added as 1 and 2V.

As an example, Figure 6 shows the modeling of an internal fault on the secondary

side. The following modeling equations are derived for the Figure 6.

The numbers in brackets in the circuit Figures 6-8 show the turns of the related

windings. For instance, in Figures 6, ten turns are {ð10ÞZL2þ ð10Þ2Vi

short-circuited and the remaining portions of the secondary winding after the short

circuit are shown between the brackets (150 þ 10 þ 70 ¼ 230 turns). Since, the

inductance of secondary side, L2, is changed because of the fault, mutual inductances

and leakage inductances are also changed during the fault condition. After the fault is

cleared, the whole system is turned to equations (8a) and (8b). Fault switch (FS) is used

for obtaining internal faults.

Switch model is also taken into account in TLM representation. The conventional

approach to switch modeling is then to represent the switch by a resistor with zero

value when it is closed and a very large value when it is open. The disadvantage of this

technique is that the network impedance matrices have to be recalculated each time

any switch changes state. An alternative TLM-based approach is possible that results

in a single, constant network matrices, irrespective of the state of the switches in the

network (Christopoulos, 1995). In implementing this method, a switch is modeled as a

L2}

Figure 5.

(a) Cross sectional view of

the transformer;

(b) winding layout

Plastic

200 turns

50 turns

10 turns

10 turns

30 turns

10 turns

10 turns

Paper

120 turns

(b)(a)

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small capacitance when open and a small inductance when closed. As is explained

above, using a constant system matrices yields an easy way of formulation and

solution of the circuit equations particularly.

In order to simulate internal fault current during the fault condition, the faulty

portion of the secondary winding is modeled in Figure 7.

As is shown in Figure 7, constant arc voltage magnitude is defined in equation (12):

Varc¼ 2Vi

Lswþ IarcðZLswþ RFÞð12Þ

where Iarcis the arc current during the internal short circuit. The addition of Iarcis

important since the short circuit current attempts to demagnetize the core and

increases the local heating. The value of Iarcis set to 10A to represent the worse short

circuit effects according to the laboratory experiments for a ten turns short-circuited.

The sign of the arc fault current is equalized to the sign of the calculated instantaneous

value of the primary current and the calculated arc voltage is added to the TLM

voltages. By means of the added arc voltage the primary current increases a bit. The

effect of the short circuit current can easily be shown in Figures 11-15 in BH curves.

During the fault condition, the TLM representation of the faulty portion,

ð10ÞZL2þ ð10Þ2Vi

Varc¼ 2Vi

L2, is replaced by:

Lswþ IarcðZLswþ RFÞ

Figure 6.

A short-circuit in

secondary side between

the turns 150 and 160

2VL1u

i

2VLe1u

i

2VLe2u

i

i1

i2

(220)R2

RF

Rs+R1

RL

(10)2VL2

i

(150)VL2

(150)2VL2u

i

(150)ZL2u

Fault

Switch

(70)2VL2u

i

(70)ZL2u

(10)ZL2

ZL1u

ZLe1u

ZLe2u

VLe1

VLe2

Vm12

Vm21

VL1

(70)VL2

+

–

+

–

+

–

+

–

+

–

+

–

+

–

+

–

Vs

TLM modeling

of transformer

1313

Page 11

Figure 8.

Internal short circuits

produced in both sides

(10)2VL1ui

RL

RF

RF

(70)2VL2ui

(420)2VL1ui

Rs+(420)R1

(10)2VL2i

(10)ZL2

(150)2VL2ui

(150)ZL2u

ZLe2u

ZLelu

2VLelui

+

–

VLe1

2VLe2ui

+

VLe2

(150)VL2

(220)R2

(70)ZL2u

(10)ZL1u

(420)ZL1u

Vm12

Vm21

VL1

i1

i2

(70)VL2

+

–

+

–

+

–

+

–

+

–

+

–

+

–

–

Vs

Fault

Switch

Fault

Switch

Figure 7.

Faulty portion of

secondary winding

(10)ZL2

2VLswi

ZLsw

Varc

RF

Fault

Switch

Iarc

(10)2VL2i

+

–

+

–

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Total arc impedance is then defined as ZLswðorZCswÞ þ RF. Arc current, Iarc, is greater

than zero in case of ZLswþ RFand is zero for ZCswþ RF. External fault resistance, RF,

is chosen as 2V for the short circuits having 20 or more turns to limit the arc current.

In this case, TLM voltage equation of the primary side is the same as equation (8a).

However, equation (8b) is replaced by the following equation (referring to Figure 7).

0 ¼ i2{ð220ÞR2þ RL} þ {ð150ÞL2þ ð70ÞL2} ð220ÞZL2uþ ð70ÞZL2uÞi2þ 2Vi

h

þ 2Vi

L2u

hi

þ M21ði1Þ i1ZLmuþ 2Vi

n

where:

Lmu

i

þ i2ZLe2uþ 2Vi

o

Le2u

??

Lswþ IarcðZLswþ RFÞ

ð13Þ

ZLsw¼2Lsw

Dt

:

A more complicated model is also applied for the study of internal short circuits

occurring in both sides at the same time. A proposed model is shown in Figure 8. The

internal short circuit fault is between the turns 420 and 430 on primary side and

between the turns 150 and 160 in secondary side. For simplicity, a pure resistive load is

connected to secondary winding so that the secondary current and voltage have similar

behaviour.

As is shown in Figure 7, faulty portions of both primary and secondary windings

are replaced by arc model. TLM voltage equation of the secondary side is the same as

equation (13). TLM voltage equation of the secondary side is defined as below:

Vs¼ i1ðRsþ ð420ÞR1Þ þ ð420ÞL1ði1Þ½ð420ÞZL1ui1þ ð420Þ2Vi

þ ð420ÞM12ði2Þ½i2ZLmuþ 2Vi

þ {2Vi

Equations (13) and (14) are again solved to find i1and i2. Then these currents are used

to calculate both TLM parameters and magnetic flux intensity, H. H is simply

calculated by using the relationship of ampere-turns (n1·I1/length). Since, magnetic

flux density, B, is a function of voltage, it is directly calculated by using acquired

primary voltage samples. An error is calculated between the calculated and measured

primary current at each simulation step. If the error goes high at the beginning of the

simulation, the measured leakage and mutual inductance are used. This procedure is

used for only the first or (the second) simulation instant, if necessary. If the initials of

the transformer are more or less zero or the triggering angle of the source voltage is

around 90o, there is no need to use measured parameters at the beginning of the

simulation. The permeability of a magnetic material mris calculated by using the

known relationship between H and B. Finally, inductance of the primary winding L1is

determined and is transformed to inductance of secondary winding, L1by using turn

ratio.

L1u?

Lmu? þ ði1ZLe1uþ 2Vi

Le1uÞ

Lswþ Iarc1ðZLswþ RFÞ}

ð14Þ

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4. Simulation and results

4.1 Transformer parameters

Simulations were carried out based on custom-built 600VA, 50Hz, single phase,

shell-form transformer used for field experiments. The design data were as follows:

. The rated voltage ratio was 220/110V.

. The primary and secondary windings were made of copper with a diameter of 1.2

and 1.6mm, respectively.

. The insulation between the layers in the custom-built transformer was made of

insulation paper with a diameter of 0.2mm.

The custom-built transformer was equipped with various taps placed on both

windings so that internal faults could be performed by connecting two taps. A

combination of an electro-mechanic relay and a start switch is used for connecting two

taps. The start switch is controlled manually. The overall experimental set-up circuit is

shown in Figure 9.

4.2 Unloaded and loaded healthy conditions

A number of experiments and also TLM-based simulation studies were performed in

laboratory and on personal computer. As a primary voltage source, a variable

auto-transformer is used and manually controlled. The same acquired voltage samples

are also used in TLM simulations instead of using synthetic samples. In all real time

experiments and simulations, the time step was 0.5ms that corresponds 40 samples

per-period.

Figure 10 shows primary current of the custom-built transformer. Since, all the

initial values are chosen as zero in TLM simulations for simplicity – this concept may

Figure 9.

Experimental set-up in

laboratory

Data

Acquistion Card

leakages

Custom-built TR

Electro-mechanical

relay

Fault

Switch

CT

CT

RF

Load

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not be true in real-time applications – the simulation converges after 2-3 time-steps

with a constant error. It may be due to defining leakages and not taking into account

the effect of eddy currents in simulations.

The error value is acceptable as is shown in Figure 10.

4.3 Magnetizing inrush conditions and BH curves

This section covers two different studies. In order to get BH curve of the transformer,

variable voltage source is controlled manually – 0...240V. Secondly, voltage source is

set to 220V and the transformer is switched on randomly. Figure 11 shows the

simulated BH curve of the custom-built transformer. To simulate unload condition in

TLM simulation, a secondary load of 10.68kV is connected to the terminals. This is

required for calculating magnetic coupling, M12¼ M21, in TLM simulation.

Since, the acquired voltage samples are directly used in the simulation, a number of

inner loops in BH curve are obtained.

Figure 12(a) and (b) show a typical magnetizing inrush current and BH curve of the

transformer during the instant switching.

Total simulation time is set to 1.5s. and a total of 3,000 samples is acquired for the

real time study. The transformer is switched on at 0.12s. instantly. As it is shown in

Figure 12(a), there is an acceptable error between real time and simulated primary

current due to the initial conditions of the transformer. All initials are set to zero in

TLM simulation.

4.4 Short circuit studies and simulations

Based on the simulation system in Figures 6 and 7, some typical fault cases were

simulated. In fault studies, we only concentrate on the primary and secondary current.

Figure 10.

Primary current of the

custom-built transformer

0.2

0.15

0.1

0.05

0

–0.05

–0.1

–0.15

–0.2

0 0.0050.01 0.0150.02 0.025

Time in sec

0.03 0.035 0.04 0.0450.05

Magnitude in A

real time

simulated

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When an internal fault occurs on the primary, the primary will increase a bit and

secondary current does not change much. However, a circulating short circuit current

flows in the shorted turns (Figure 7). For a ten turn short circuited (over 440) on

primary side, this circulating current is measured up to 10A. As was stated before, a

fault resistance, RF, was used to limit the circulating current. Since, short circuits were

performed manually over an electro-mechanic relay, duration of the fault lasted 3-6

periods.

Figure 13 shows the simulation of BH curve during an internal fault between 420

and 430 turns on primary side, real time and simulated primary current, respectively.

The fault starts at 0.54s. and ends 0.67s. In this case, transformer is loaded with an R-L

load.

As is shown in Figure 13, due to the transients during the internal fault, there are

some divergences in simulating BH curve. Before the fault, the whole simulated system

is working according to equations (8a), (8b), (9a) and (9b). When the time reaches at

beginning the fault, 0.54s, equation (8a) is replaced with equation (12) to simulate the

short circuit condition. After the clearance, the fault on primary side (at 0.67s),

equation (8a) is again used for simulating healthy condition.

Similarly, Figure 14 shows the simulation of BH curve during an internal fault

between 150 and 160 turns on secondary side, real time and simulated primary current,

respectively.

In Figure 14, the transformer is un-loaded and internal fault begins at 0.2205s and

ends 0.3435s. The effect of the circulating current on secondary side can easily be seen

in primary current. As in the previous case, before the fault, the whole simulated

system is working according to equations (8a), (8b), (9a) and (9b). When the time

reaches at beginning the fault, 0.2205s, equation (8b) is replaced with equation (11a)

Figure 11.

Typical BH curve of the

custom-built transformer

in TLM simulation

1

0.8

0.6

0.4

0.2

0

–0.2

–0.4

–0.6

–0.8

–1

–500

–400

–300

–200

–100

0

100

200

300

400

500

H-Amp-Turns

B in Tesla

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