CHAPTER2

ESTIMATION OF INRUSH CURRENT

2.1 INTRODUCTION

Inrush currents under consideration are high magnitude transient currents that flow

in the primary circuit of a transformer when it is energized under no-load or lightly

loaded condition. The peak of the inrush current is found to be as high as 10 times that of

the rated current of the transformer lasting for a few cycles. Inrush currents are

unsymmetrical, contains harmonics and de component. The decay rate of inrush current is

determined by the ratio of resistance to inductance of the primary winding and source.

The knowledge of peak inrush currents, decaying time, its harmonic order and magnitude

are required to avoid mal-operation of protective relays and temporary over voltages

during commissioning of cable connected transformers. In this chapter a numerical

method to estimate the worst case inrush current is described which presents satisfactory

results of inrush current calculated over a longer period of time by considering saturation

and residual flux. With this, remedial measures can be adopted to avoid the problem of

temporary over voltages during transformer energization. The equation for inrush current

is developed based on the characteristics of the inrush current present in a laboratory

type transformer both by simulation and experiment.

2.2 FACTORS INFLUENCING INRUSH CURRENT

The magnitude and duration of transient inrush current depend on 1) the point-on­

voltage wave at the instant of switching 2) the magnitude and polarity of the residual flux

in the transformer core at the instant of energization 3) the inherent primary winding air­

core inductance which depends on volts I turn , that is energization on low voltage side

causes more inrush current compared to high voltage side energization 4) the primary

winding resistance and the impedance of the circuit supplying the transformer 5) the

10

maximum flux-carrying capability of the core material. Inrush currents are originated by

the high saturation of iron core during switching- in of transformers [I].

Switching- on angle

Inrush current decreases when switching-on angle on the voltage wave increases.

The current response of RL circuit when switched at an angle (a.) is given by WR

i(t)=lm (sin(rot+o.-8)-sin (o.-S))e-Cx)t (2.1)

where i(t) : instantaneous current in ampere

a switching-on angle in degree

Im the maximum steady-state current in ampere

e power factor angle in degree

R resistance of the winding in ohm

X reactance of the winding in ohm

t : time in seconds

rot: in degree

The transient vanishes when a = e and therefore doubling effect will not take

place for highly reactive RL circuits when switched at a = 90° (8=90°). Moreover for any

RL circuit the peak switching current Ip, also confines to steady-state maximum level Im

when a = ± 90° . Thus the inrush current of a single-phase transformer can easily be

eliminated when switched at a± 90°, using an instant-controlled switching circuit.

Residual flux

This parameter is the most difficult to determine exactly. Inrush current is

significantly aggravated by the magnitude and polarity of residual flux density. This in

turn depends on core material characteristics, power factor of the load at interruption and

the angle at which the transformer was switched off. The total current i0 is made up of the

magnetizing current component im and the hysteresis loss component ih. The current

interruption generally occurs at or near zero of the total current waveform . The

magnetizing current passes through its maximum value before the instant at which the

total current is switched off for no load, lagging load and unity power factor load

11

conditions, resulting in maximum value of residual flux as per the flux- time curve of

Fig.2.1. For leading loads, if the leading component is less than the magnetizing

Fig.2.1 Flux-time curve

component, at zero of the resultant current the magnetizing component will have reached

the maximum value resulting in the maximum residual flux. On the contrary, if the

leading current component is more than the magnetizing component, the angle between

maximum of the magnetizing current and zero of the resultant current will be more than

90°. Hence, at the interruption of the resultant current, the magnetizing component will

not have reached its maximum resulting in a lower value of residual flux density. The

point on voltage waveform at which a transformer is switched-off influences the amount

of flux that remains in the core. Typically, the amount of flux that remains in the core is

anywhere from 30% to 80% of the maximum core flux and can be positive or negative.

Its maximum value is usually taken as about 80% and 60% of the saturation value for

cold rolled and hot rolled materials respectively. It is also a function of joint

characteristics. In practice the transformer supplying de load through rectifier circuits

appear to have a remnant flux close to the minimum value, when switched on no-load.

The value will depend on the transformer magnetizing inductance, the capacitance and

the resistance of the bridge.

Saturation flux

Saturation flux density is how much magnetic flux the magnetic core can handle

before becoming saturation and not able to hold any more. This depends on several

factors including core material temperature, electrical and magnetic condition on the

12

transformers. When the core saturates, the transformer no longer acts like an inductor

with a linear increase in current over time. The magnetic field cannot increase further and

current is limited by the source impedance of the power supply and the resistance of the

transformer wire. This leads to a very large current. It is usually essential to avoid

reaching saturation since it is accompanied by drop in inductance. The rate at which

current in the coil increases is inversely proportional to the inductance (di/dt= v/L). Any

drop in inductance therefore causes the current to rise faster, increasing the field strength

and so the core is driven even further into saturation causing fall in inductance.

2.3 PHYSICAL PHENOMINA

When a transformer is re-energized by a voltage source, the flux linkage must

match the voltage change according to Faraday's law

Ym sin (rot+a) = i0R+ N d:tm

where , Vm

<Dm

N

maximum value of steady state voltage in volt

instantaneous value of flux in weber

primary winding turns

10 no load transformer current in ampere

The solution of the equation assuming linear magnetic characteristics [1] is

R

<Dm = (<Dmp COS a± <Dr) e-(x:-)t _ (<Dmp COS (rot+a))

where <Dmp: peak flux

<Dr : residual flux

(2.2)

(2.3)

The equation (2.3) shows that the first component is a flux wave of transient de

component, which decays at a rate determined by the ratio of resistance to inductance of

primary winding and a steady state ac component. The de component drives the core

strongly into saturation, which requires a high exciting current. Inrush current waveform

is completely offset in first few cycles with wiping out of alternate half cycles because

the flux density is below saturation value for these half cycles. Hence, the inrush current

is highly asymmetrical and has a predominant second harmonic component.

13

Time constant (L/R) of the circuit is not constant; the value of L changes

depending on the extent of core saturation. During the first few cycles, saturation is high

and L is low. Hence, initial rate of decay of inrush current is high. As the losses damp the

circuit, saturation drops, L increases slowing down the decay. Therefore the decay of

inrush current starts with a high initial rate and progressively reduces; the total

phenomenon lasts for a few seconds. Inrush currents are generated when transformer is

switched on at an instant of voltage wave, which does not correspond to the actual flux

density in the core at that instant. When a transformer is switched off, the excitation

· current follows the hysteresis curve to zero. The flux density value will change to a non­

zero value or zero corresponding on the material of the core. Suppose the transformer is

switched on again at the zero crossing instant of voltage, then flux should be at its

negative peak as flux lags behind applied voltage by 90°. In the subsequent half cycle,

flux varies from -<l>m to +<l>m , But as per the constant flux linkage theorem, magnetic flux

in an inductive cannot change instantaneously. Hence, the flux instead of starting from

negative maximum value starts from 0. Then it will rise to 2 <I>m in the subsequent half

cycle.

The flux will be unsymmetrical at the time of turn on. It will rise to 2 <I>m initially

and then depending upon the circuit parameters will reduce to <l>m , The flux value of 2 <I>m

will drive the core into saturation thereby causing currents of very high magnitude to

flow in the primary circuit. The inrush currents are highly unsymmetrical because in the

positive half cycle the flux has a very high value, thus the induced currents will be very

high. But in the negative half cycle, the flux has a very low value, thus the currents are

also low. As the flux falls to normal steady state value, the magnitude of inrush current

also decreases. The explanation given in the section can be justified mathematically using

the following equations:

Let the source voltage be given by

v(t) = V m sin ( rot+a)

Then the instantaneous flux in the circuit is given by 1 rt

<I>m = N Jo v(t)dt

Substituting (2.4) in (2.5)

14

(2.4)

(2.5)

<I>m = vm

[cosa - cos(wt + a)] Nw

(2.6)

Maximum flux that can be build up in the core when a voltage is applied at its zero

crossing instant is,

<I>mp = 2 <I>m (2.7)

If there is some amount of residual flux, say <I>r , in the transformer core at thetime of

switch on, then the flux will rise to a value of 2<I>m + <l>r . This will drive the core into

deep saturation, thereby causing the currents even higher than that of earlier case as

shown in Fig.2.2. The figure shows inrush currents produced in the transformer core with

and without residual flux. In the presence of remenance of flux

l(l)

Inrush current (with remanance)

Fig.2.2 Graphical description of inrush current phenomena

2.4 EXPERIMENTAL DETERMINATION

(2.8)

In order to study the different behaviour of transient magnetising current of a

transformer during energization on no load first it is determined experimentally. For this

a laboratory type single phase 230V /230V, l kV A 50 Hz transformer with a tertiary

winding of 115V is considered. The parameters of the transformer are determined from

open and short circuit tests. The values are calculated for a base current of Ibase =4.348A

and Zbase =52.89 and are shown in Table I of Appendix. The experimental set up for

measuring the primary current Ia and energization voltage Ea of the transformer is shown

in Fig.2.3. As the rated current of the transformer is 4A inrush current is assumed to be

50A peak (1.4142*4*8) a hall sensor HE055T and a potential transformer 240V/6V

15

are used for converting the high level current Ia and voltage Ea of the transformer in the

range of± 15V for compatibility with the data acquisition card. As the output current of

the hall sensor is 50mA for an input current of 50A a resistor R1 (6 Ohm ) is connecte� to

the output of the hall sensor so that the maximum voltage drop across R1 is 300mA. In

order to limit the operational amplifier output voltage within the range of +/-15 V which

is compatible for data card, the operational amplifier gain is taken as 45. As the potential

transformer has a 240V /6V turns ratio voltage gain multiplier is taken as 40 ( 40*6=240).

The circuit diagram of the set up for plotting the inrush current and voltage signals using

N

p

Ia

�---ol---/ Ill TEST

TRANSFORMER

TRANSISTOR

PCl1711

Fig.2.3 Experimental diagram for measuring voltage and current

Analog

Input

i--------1� ln10ut1 Analog lnpJt1

Mv.anteoh PCl-1711 (a,uto)

Analog

Input

Analog Input .Qdvanteoh

PCl-1711 lautol

:.::od

S::ope3

Dlglta

Output

Digital o,tput .Adv ante ch

Cl-1711 [auto)

S::,opei!5

S::,opeO

Fig 2.4a) Block diagram for plotting voltage and current signals using the MATLAB-Simulink real time workshop

16

ln1

Constant3

Logical

Operator

Fig. 2.4(b) Switching circuit for energizing the transformer at different inception angles

Out1

MA TLAB-Simulink real time workshop is shown in Fig.2.4. A PCI 1711 data card is

used to interface the voltage and current signals with the host computer. In order to plot

the inrush current at different inception angle, a zero crossing detector (zed) is used. The

expanded circuit of zero crossing detectors is shown in Fig.2.4b. It uses two switches to

convert the sinusoidal voltage signal and the delayed voltage signal using the transport

delay into a square waveform. This in phase and delayed waveforms are connected to a

logical XOR gate and again passed to an AND gate with that of in phase square

waveform. The required delay is obtained in the transport delay block in Fig.2.4b and a

mono pulse is generated which is modified by the discrete monostable. The output of this

block generates the required digital output for switching the transformer through a relay.

As this signal is low to drive the relay, a NPN transistor TIP12 I is used to amplify the

signal as shown in Fig.2.3. The inrush currents for different energization angles are

determined and are shown in Table 2.1. The peak inrush current of 45A occurs at a

switching angle near to o0 and 180°. Fig.2.5 shows the waveform for a switching angle of

0°. First plot shows the switching signal to close the relay, second plot the energization

Table 2.1 Measured values of inrush currents

Switching Angle ou 90u 180° 270°

Peak current First cycle 45 -2.5 -25 5.2 (A) Second cycle 18.5 -1.26 -7.3 1.4

DC peak (A) 9.6 -0.2 -3.5 0.5 Settling time (s) 0.5 0.45 0.47 0.43

17

voltage and the third plot shows the inrush current. There is a delay between the instant

of switching signal and the actual time of relay operation. Inrush current for a switching

angle of 90° is shown in Fig.2.6 and peak of inrush current in this case is 2.5A. The

parameters of the PCI 1711 card, hall effect current sensor HE055T, TIP121 and relay

OEN 51 are given in Appendix.

J aJ I0 0.1

(a) 0.2 0.3

tlrwvvwwvwJWW.... ______ �------�-------'--0 0.1

(b) 0.2 0.3

0.1 (c) 0.2 0.3

Time (s)

Fig.2.5 Transformer energization at 0° a) Switching signal b) Voltage c) Inrush current

,. .. ...................................... T ................................................ T ....................... .., ....... .

,A 0

Q 1 --·-,---·-·-····-··-r····--·-·-·--,-------r-------..---

005

f 0

V

-005

.(I t 0 01

(b) t) H, 02

Fig.2.6 Transformer energization at 90° a) Voltage and switching signal b) Inrush current

18

Harmonic analysis

The inrush current Ia during the peak value is subjected to harmonic analysis by

passing the current signal to the discrete Fourier block in MA TLAB-Simulink. The

harmonic spectrum for the inrush current is shown in Fig.2. 7. The spectrum shows

the de component is larger than the fundamental component of current. Currents in the

Fundamental (50Hz) = 0.003704 , THO= 100.63%

140

·�f100

80

60

40 �

20 ::a:

0 0 200 400 600 800 1000

Frequency (Hz)

Fig.2. 7 Harmonic spectrum of inrush current

frequency range of 100,300,400,500 and 800 are stronger. Fig.2.8 shows the plot of de

component. The peak value reaches to 11.5A instantaneously and exit for a short duration

of approximately 0.05s, then decreases exponentially.

0.25�-�--�-�--�-�---.-----,

0.2

0.15

o 0.1

0.05

0o 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Time(s)

Fig.2.8 de component of inrush current

19

Effect of power factor and angle of current interruption on inrush current

In order to find the effect of power factor of the load, first the test transformer is

demagnetized to bring the residual flux equal to zero. This is achieved by applying a 50

Hz low voltage to the secondary winding through an autotransformer and then decreasing

gradually to zero. Then the transformer is energized at different switching angles each

time connecting to a resistive load and demagnetizing the core before next energization.

In each case the peak inrush current is noted. Next an inductive load is connected instead

of resistive load and the experiment is repeated. To study the effect of current

interruption angle the residual flux is made zero. Then the transformer is energized and

connected to a resistive load and interrupted at a definite angle. Again the transformer is

energized and peak inrush current is noted. This is repeated for different instants of

current interruption with resistive and inductive loads. The results reveal that the nature

of load current and interruption angle has effect on inrush current. Since in real situations

it is difficult to control the switching off the load current, further investigations are not

carried out in this direction. The results are given in Table 2.2

Table 2.2 Variation of inrush current due different load interruption angle

le 0 42

2.5. SIMULATION RESULTS

90

-5

0 180 270 -30 6

In this section the investigation related to the determination of the inrush current

characteristics is carried out with time-domain simulations in PSCAD environment. The

transformer and the energization circuit modeling details are discussed in this section.

Transformer model

The transformer representation used for the investigations is the model available

in PSCAD. The value of parameters for simulation is those of a transformer used for

experimental determination which is given in Appendix. Magnetic core residual flux is

represented by a de current source in parallel with the low voltage winding; the current is

chosen to establish the required level of residual flux linkage [15]. The polarity of the

20

residual flux is changed by reversing the de source. The residual flux and air core

reactance in the model transformer is achieved by setting the de current source such that

both the simulation and experimental peak inrush currents are the same. In this thesis, this

transformer model is taken throughout for simulation studies and the results are validated.

Circuit model

The circuit diagram considered for inrush current investigations is shown in

Fig.2.9. The test transformer is connected to the line to ground voltage of the secondary

side of an 11 kV /415V delta-star solidly grounded distribution transformer. This is taken

as the voltage source and this transformer impedance is taken as the source impedance.

()

.i,. '*le

-=- ����

[;>

la

P = 5.256e-008 Q = 8.243e-01 o

V= 0.2268 BRK1

···· laR·��

f !: Cl.ClCl 1 ·· .... · I a R

I:: �1lo-' t-N \- -=-'. ...j

}J�o la

FFT

Ma 7 (7) HarmonicTotal .. :\'Wol ·

F = 50.0 [Hz]

s

Ph

(7)

Distortion 7Individual

Fig.2.9. Circuit for simulating transformer inrush current

,._JQ �aryvoltage

A',n;:::i

l�ary current h�

rtrut

The voltage, current and power signals are obtained from the power system meters. The

primary voltage is measured by the voltmeter Ea, current by ammeter Ia and the power by

the multimeter A/V. The medium voltage side line resistance and inductance are taken

from data book [69] which is I ohm and 0.0031 H respectively. In the figure, the primary

21

side switching angle is controlled by a timer switch such that voltage is applied at the

instant of its zero-crossing or peak of voltage in both positive and negative cycles. The

inrush current is analysed by Fast Fourier Transform (FFT) method.

Simulation of inrush current

Different cases of inrush current are simulated by varying those major parameters

that influence the characteristics of inrush current. These parameters are angle of

switching, magnitude and polarity of residual flux. First the primary side of the test

transformer is energized by a voltage of 230 V at a switching angle of o0 without any

residual flux. Fig.2.10 shows the variation of the applied voltage together with the inrush

current and core flux which is the integral of voltage. It is observed from the waveform

that at the time of switching on the primary voltage at zero cross over instant, without

residual flux, the flux rises from zero value to a maximum value of l .8pu and is highly

unsymmetrical. The reason for this is that the flux present in the transformer core before

the application of the primary voltage was zero. By virtue of constant flux linkage

theorem, the flux remains at the same value when the primary voltage is applied. Since

the flux is expected to lag the applied voltage, the flux is to be at a value of -lp.u,

-400

30 11w

g :w

C: 10

� 0

2.0 •s;sJ

! 1.0

,.. 0.0

.1.0

n,ne<s) 0.100 0.150 0.200 O.Z50 0.300

•:

Fig.2.10 Transformer energization without residual flux at switching angle 0° a) Voltageb) Inrush current c) Core flux

22

which is to rise to I p.u. when the voltage is applied. However since the initial flux has a

value of zero, it will rise to a maximum value of 1.8 p.u. in the first half cycle and is

highly unsymmetrical. This causes inrush current of peak magnitude equal to 27 A and

14A to flow in primary circuit during first and second cycles respectively. Then due to

damping in the circuit the inrush current is found to reduce to their steady state value of

1. IA. The peak value of active power and reactive power demand read by the multimeter

is 350 W and 2075 VAr respectively. The worst case scenario when the transformer is

switched-in at a switching angle of 0° with positive residual flux of 0.8pu is shown in

Fig.2.11. In this case, at the zero voltage cross over the core flux starts from 0.8 pu and

reaches to a peak value of 2.2 pu. The peak value of inrush current for the two successive

cycles is 45A and 24A and settles at 0.3s. The transformer is also energized for a negative

residual flux at a switching angle of 0° and Fig.2.12 shows the plot of flux and inrush

current. The variation of active power and reactive power demand during energization at

o0 switching angle with positive residual flux is shown in Fig.2.13. The figure shows that

there is a peak demand of 850W active power and 3000 Y Ar reactive power from the

source which gradually decreases.

50�-"=----····--···-��--�-------����������--'

40 30 20 10 0

-10

-1.50

rune(s) 0.1 oo 0.150 0.200 0.250 0.300

Fig.2.11 Transformer energization with positive residual flux at switching angle 0°

(a) Core flux (b) Inrush current

23

•.oo. 2.5 ························· ·····

g:

-17.5

1.00 ��--

0.50.... 0.00

! -0.50 )( -1.00

-1.50-2.00

Time(s) 0.100 0.150 0.200 0.250

--�--�-..-.....

Fig.2.12 Transformer energization negative residual flux at switching angle 0° a) Inrush current

b) Core flux

................ ... ............................... ____ ... __ ........ _ _J

-100 ------·---···-·-···"'""'""' ......... -........ -.. -·-----------,---!

2.0k

�

-®. 1.0k .... 0.0

-1.0k -2.0k-3.0k-4.0k

•

... ___ ..................... _ ... "'-''' ·=·· -====-===="-;===�==·=···a,· .. ·=· .. ··= .. ·-=···=-.. ·=··"",'"''="·=· ..... = ...... =a,==,----; Time(s) 0.100 0.150 0.200 0.250 0.300 0.350 0.400

Fig.2.13 Transformer energization with positive residual flux at switching angle 0° a) Active power b) Reactive power

Harmonic analysis

The harmonic content of the transformer inrush current is calculated using Fast

Fourier Transform (FFT) method Fig.2.14 shows the waveform of the lower three

frequency components present in the inrush current. The results presented are for the

worst case switching of o0 with positive residual flux to get the maximum inrush current.

It shows that the peak value of any individual harmonic component during one cycle is

generally different from its peak during another cycle.

24

•

25.0 ---------------------------�

20.0

15.0

10.0

5.0

-------··�'-=r---'--�=r'-�-'-"-'....---�T-'-���-----�"-'-=--=-- ·=--· =, -·- -;= -=-=--�-T"""==r-=------i Time(s) 0.100 0.120 0.140 0.160 0.180 0.200 0.220 0.240 0.260

- ------------·-·-··-··-···--.. ---- ·-··-··--·---··---·---·---------------�--'

Fig.2.14. Harmonic components of inrush current.

Due to the non-symmetrical wave shape, the transformer inrush current contains

all harmonic components i.e. fundamental, 2n°, 3

rc1, 4

th, 5

th etc as well as a de (zero

frequency) component. The magnitude of the peak significant (fundamental) component

(blue) is 22 A, second (green), and third (red) order frequency components are 12A, and

9.2 respectively. The fourth, fifth, sixth and seventh order harmonics are 1.3A, 0.94A,

0.1 and 0.36A respectively. The spectrum also contains a de component of peak value

16A and this is shown in Fig.2.15. The second harmonic current creates negative

sequence current and the DC component causes the sympathy current in the power

system. The THO measured is 71 %. The frequency spectrum is shown in Fig. 2.16.

-···-· - -·-- -·------·----··---· ----·-------·-----·-------··---·--------------------_:J

�

Q

- -

16.0

12.0

8.0

4.0

0.0

Tme(s)

• I •

- -·-··-·····--- ·- ·----··-·-·-··--····--------·-----------------·____j

0.100 0.150 0.200 0.250 0300 0.350

•

Fig, 2.15 de component of inrush current

25

1:50.0

0.0

Harmonic order (7) 2.76879

Fig.2.16 Frequency spectrum of inrush current

Comparative evaluation

In order to study the performance at various switching conditions, the

transformer is energized at different inception angles. Parameters like peak inrush

currents, peak demand of active power, peak demand of reactive power; peak de values,

second harmonic component, Total Harmonic Distortion (THO) and settling time are

determined. Since inrush current energy parameter J 12t is a measure of the temperature

rise it is also calculated for one cycle. Table 2.3 summarises the parameter values for

different switching instants without any residual flux. The effect of adding positive

residual flux and negative residual flux in the transformer core on inrush current is

represented in Table 2.4 and Table 2.5 respectively.

Table 2.3 Switching parameters for zero residual flux

Switching Angle ou 90° 180° 270°

Peak current First cycle 27 -12 -38 12 (A) Second cycle 14 -6 15 7

Harmonics Second 64 50 66 47 % THO 85 65 84 65

DC peak (A) 5.6 -2 -8.5 2.1 Peak reactive power (V Ar ) 2075 774 3082 776

Peak power (W) 350 157 591 157 JI2t 0.24 0.04 0.4 0.04

Peak flux First 1.8 -1.6 7.9 1.6 pu Second 1.7 -1.53 -1.7 1.5

Settling time (s) 0.9 0.9 0.9 0.9

26

Table 2.4 Parameters for 0.8pu positive residual flux

Switching Angle ou 90° 180° 270°

Peak current First cycle 45 -2.7 -20 30 (A) Second cycle 25 -1.2 -11 16

Harmonics Second 43 18.7 59 68 % THO 80 25 83 82

Peak DC(A) 10.7 0.137 -4 6

Peak reactive power (VAr) 3000 237 588 822 Peak power (W) 850 59 153 213

JI2t 0.78 0.004 0.12 0.34 Peak Flux First 2.03 1.3 -1.7 1.8

pu Second 1.76 1.29 -1.6 1.6 Settling time (s) 0.6 0.13 0.9 1

Table 2.5 Parameters for 0.8pu negative residual flux

Switching Angle au 90° 180v 270v

Peak current First cycle 9.5 30 -45 2.7 (A) Second cycle 6.2 -4 -17 2.5

Harmonics Second 36 42 62 17 % THO 45 54 69 22

Peak DC (A) 2 -4.7 -11 0.75 Peak reactive power (V Ar) 296 440 795 254

Peak power (W) 68 100 178 57 Jh 0.04 0.31 0.84 0.004

Peak Flux First 1.58 -1.9 -2 1.34 pu Second 1.29 -1.5 -1.7 1.3

Settling time(s) 0.4 I I 0.3

The Tables indicate that inrush currents are large during energization, depends on

polarity of residual flux and angle of switching. The large value of inrush currents and

reactive power demand in the system causes voltage sag problems. Various parameters

during transformer energization at different inception angle are compared from the tables

and a bar chart is drawn which is shown in Fig.2.17. In this figure x-axis is the switching

angle and y-axis is the magnitude of current. The bar chart shows that the most

favourable condition for switching a transformer is at 90 ° or 270°.

27

• For ;:ero residu"llIux

SO .Forpositiive re~du~fl

0

3t1

20

10~ 0'...;/

~Q) ·10t:: ·20;:J0 -30

...0

·50

o 90 130 2701

Switching angle (degrees)

Fig.2.17. Bar chart inrush currents for various switching instants.

Effect of harmonics

The effect of harmonics in the grid voltage during energization of transformer is

also studied. The point of coupling of the transformer is polluted by connecting a non

linear load to achieve a THD of 5%. The harmonically polluted voltage waveform is

shown in Fig.2.18. The values of inrush current for various switching angles and residual

flux are also simulated. Observation of the values shows that the value of inrush current

is slightly feduced and there is change in the harmonics content of inrush current.

•400

300

200

~100

Q) 00>

'"~

-100

-200

-300

Time(s) 0.100 0.110 0.120 0.130 0.140 0.150 0.160 0.170 0.180 0.190 0.200

Fig.2.18.Harmonically polluted grid voltage

28

2.6 MATHEMATICAL FORMULATION

The inrush current and the de component waveform obtained both by simulation

and experiment for an energization angle of o0 is shown in Fig.2.19 and Fig.2.20

respectively. It is assumed that the residual flux is in the same direction as that of the

................................. I ·�

•@ 14.0 ............... ·

..... . .................... ........................ .1

-2.0

Time(s) 0.100 0.150 0.200 0.250 0.300 0.350

Fig2.19 Simulated waveform (a) Inrush current (b) de component

g 1

= 0.5 0

0.1 0.2

(a)

0.3 Time (s)

0.25 ,----.--�--�---,---�---.-----.

0.2

0.05

I00!_-J---='o.�,---=o�.2=---�o�.3,-----�o.�4---:o�.s;:::--�o�.6=----:::-'o.7 Time(s)

(b)

Fig.2.20 Experimental waveforms a) Inrush current b) de component

29

. :

initial flux change, thus giving maximum possible value of inrush current. The

comparison of inrush current and de component waveform obtained by both experiment

and simulation are similar to that of an exponentially decaying half wave rectified

waveform. The inrush current is completely offset in first few cycles with wiping out of

alternate half cycles. Due to the non-symmetrical wave shape, this wave form contains,

besides the fundamental, harmonics of the order kn± 1 (k= l , n= l, 2, 3 ... ) where k is the

no. of pulses per cycle. The (1 n-1 t are of the negative-sequence type, whereas ( 1 n+ 1 )1h

harmonics are of the positive sequence type. Hence the inrush current contains all

harmonic components; even harmonics appear to be the dominant and a de (zero

frequency) component. The exponential decay of inrush current is taken same as that for

R

the transient flux from (2) which is proportional to e - ""It . The waveform of the de current

shows that the time constant (L/R) of the circuit is not constant; the value of L changes

depending on the extent of core saturation. During the first few cycles, saturation is high

and L is low. Hence the initial peak current is large and the rate of decay of inrush current

is quite high. As the losses damp the circuit saturation drops, L increases slowing down

the decay. Therefore the decay of inrush current starts with a high initial rate and

progressively reduces.

If a periodic wave form is represented by an analytical expression f (t), then a Fourier

series can be applied to obtain a descriptive equation for f(t). In order to interpret this, the

following mathematical model is considered.

1 J,Tao = T O

f(t). dt

2 J,Tan = T O

f(t). cos nwt. dt for n = 1,2,3 etc.

bn = � f0T f(t). sin nwt. dt for n = 1,2,3 etc.

where ao : de component

an : Fourier coefficient of cosine terms

bn : Fourier coefficient of sine terms

Then the periodic function is represented by the Fourier series [70]

f(t) = ao + i:�=1 Can cos nwt + bn sin nwt )

(2.9)

(2.10)

(2.11)

If the inrush current waveform is considered as a half wave rectified waveform

which is exponentially decaying, the equation satisfying Fourier series comprises of a de

30

term, cosine terms, bn coefficient for n = I only (for n greater than I, bn =O) and an exponential function to represent the decay. From these terms an approximate mathematical model to realize the inrush current waveform can be deduced as,

i(t) = (� + � cos(wt - 90) + L�=2

2102

cos nwt) e -It (2.12) n 2 n(l-n )

The first term of the equation inside the parenthesis ( 10 ) is the de component which is

Tr

the average value of the function, second term a sinusoid at the frequency of the waveform representing the fundamental component and the third term corresponds to the

R

harmonic components. The multiplying exponential term e -It is the time for which inrush current flows. Considering the initial steady de value and the non-linear inductance L 1 , more approximate equation is

i(t) = (1; ((u(t) - u(t - t0) + e-tiCt-to). u(t - t0

)) +

R

(�. cos(wt - 90) + L�=2 C 210

2) cos nwt) e -L1

t

2 1t 1-n (2.13)

where u(t) is the step input, to= duration of the steady de value. This is obtained from

the various simulation graphs of de current waveforms, the average value of which is

taken as 3ms. The non-linear inductor L I is represented as follows [9]

L 1 = Ls O<t<to = L to<t<oo

where Ls and Ln are saturation and nominal inductance. Normally nominal inductance L = V/2*3.14£n, Ls = V/2*3.14£5 where £n Normal flux

£5

Saturation flux

L 1 non linear inductance of transformer

Io is the peak inrush current of the first cycle given by [1]

where

Io = (K1 ..Jzv (1 - cos �))/Xs

air core reactance Xs = (1-1-0N2 Awfhw)2rrf

K 1 correction factor for the peak value = 1.1

Aw area inside the mean turn of excited winding

31

� absolute permeability

hw height of energized winding

f frequency

� instant at which the core saturates, is

� = K2cos- 1 {(85 -Bmp-Br)/Bmp}

where, 85 saturation flux density= 2.03 Tesla

Bmp peak value of steady state flux density in the core (1.7 T)

Br residual flux density (0.8 Bmp= l.36T)

For cold rolled material, maximum residual flux density is usually taken as 80% of the

rated peak flux density.

K2 = correction factor for saturation angle = 0.9

The validity of the formulated inrush current equation (2.13) is established by

plotting in MATLAB and this is shown in Fig.2.21. The first plot shows the inrush

current and the second plot shows de component of the inrush current. The simulated

inrush current waveform (Fig.2.19), waveform obtained by experiment (Fig.2.20) and the

computed waveform (Fig.2.21) are compared. The inrush current waveform obtained

60 - - -, - - -- r · - --------.----- --- -----r- - --�-.----�

' I I -r - - - - - -r - - - - - -r -- - - -

1 I

I I I I I I - - - - - -� - - - - -r --- - - -� - - - - - -� - - - - - -r --- - - -� - - - - - -r - -- - -

0 ---

0 0.02

�20 __ __ �

QI C

8. 10E

1 I I I I I I

---- ·1 I I

0.06

- -· 1·

I

0.08 (a)

0.1 0.12 0.14

·- --------r- -· ------,--··-�----,-------,-

r - - - - - -r - - ----r - -----r -----

0.16

(.) I , I : C O _____ ____[__ ______ _J ______ _J___ ________ __L=-�--:: _::::::::::::::::::�==�=�_j

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 (b) Time (s)

Fig.2.21 a) Formulated equation of current waveform a) Inrush current b) de component

32

mathematically is practically matching with that of the simulated and experimental

waveforms. In all the cases duration of settling time of the inrush current is 0.15s, time

taken by the de component to settle to zero or the inrush current to reach the steady state

no-load current . Hence the validity of the equation for the calculation of inrush current

in single phase transformer is verified. Advantage of this equation is, for a given

transformer the peak inrush current, de component, their settling time and magnitudes of

different harmonic component can be estimated in terms of the energization voltage and

physical parameters of the transformer. Knowing the harmonic components total

harmonic distortion (THO) at the time of energization can be estimated. Various

parameters calculated during energization by the equation for the sample transformer is

given in Table 2.6. Similarly the parameters obtained by simulation are given in Table

2.7. Comparison of the values given in the Tables shows the validity of the formulated

equation. Photograph of the experimental set up is shown in Fig.2.22

Table 2.6. Parameters obtained by calculation due to energization of transformer

Funda Harmonics Inrush de

mental Peak current component 2nd 3rd 4th 5th THDvalue (A) 45 14 24 9.5 4.75 1.81 1.2 0.94

ettling 0.9 0.6 - 0.9 0.82 I

0.62 0.42 0.22 time(s)

Table 2.7. Parameters obtained by simulation due to energization of transformer

Funda Harmonics Inrush de

mental Peak current component 2nd

'

3rd 4th 5th THD

value (A) 45 16 22 12 9.2 1.3 0.94 0.8

settling 0.16 0.25 - O.ll6 0.16 0.16 0.16 0.16 time(s) I

33

Fig.2.22 Photograph of the experimental setup

2.7. CONCLUSION

In thi chapter the general characteristics of transformer energization current on no­

load both by experiment and simulation has been determined. A circuit for switching-in

the transformer at various inception angles using real time MATLAB-Simulink Lab work

shop has been developed. Maximum possible inrush current and its dependency both by

simulation and experiment are found out. The inrush current depends on the angle of

switching, load power factor, angle of load interruption and residual flux. The load

current interrupting time has small effect on inrush current. The harmonic analysis of

the inru h current shows the presence of all lower order harmonics and dc component.

During the period of inrush current, the demand of peak of reactive power is large and it

give an indication of transformer saturation. A mathematical equation, in terms of

physical parameters of transformer, which takes account of the presence of dc component

and harmonic components of the inrush current of a switch-in power transformer, has

been developed. The equation makes it possible to estimate the peaks of decaying inrush

currents, its duration, different harmonic components and the dc component. Networks

can be checked for danger of inrush current over voltages due to resonant conditions. It is

possible to foresee the eventual needed remedial measures when energizing or

commissioning off-shore transformers with limited generation and long sea

interconnecting cables.

34