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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