Analytical Analysis of Transformer Interaction
Caused by Inrush Current
P. Heretík, M. Kováč, A. Beláň, V. Volčko and M. Koníček
Institute of Power and Applied Electrical Engineering
Slovak University of Technology in Bratislava
Bratislava, Slovakia
Pavol.Heretik@stuba.sk, Matus.Kovac@stuba.sk, Anton.Belan@stuba.sk,
Vladimir.Volcko@stuba.sk, Michal.Konicek@stuba.sk
Abstract— This paper analyzes the phenomenon of
sympathetic interaction between transformers which is very
likely to occur when a transformer is energized on to a system
where are other transformers already connected. This interaction
significantly changes the duration and the magnitude of the
transient magnetizing currents in the transformers involved and
may lead to maloperation of some power protections.
This article presents the formation process of sympathetic inrush
current through the flux analytical expression and bias flux
attenuation between two parallel operation transformers. After
that, the reasons of sympathetic inrush current generation are
illustrated. Presented analytical analysis is valuable for
comprehensive understanding of sympathetic inrush current,
estimating the reasons of generation and eliminating it.
Keywords—
Inrush
Current,
Power
Transformer Saturation, Transient Phenomenon
I.
Protections,
INTRODUCTION
Magnetizing inrush current occurs during the energization
of a single transformer connected to a power system network
with no other transformers. However, the energization of a
transformer connected to a network in the presence of other
transformers, as shown in Fig. 1, which are already in
operation, leads to the phenomenon of sympathetic inrush
current. Although, severity of the inrush current is higher
during single transformer energization, the sympathetic inrush
current is special important due to its unusual characteristics.
The inrush current in a transformer decays, usually, within a
few cycles, but the sympathetic inrush current persists in the
network for a relatively longer duration. In recent years, there
are several reports of closing a transformer without load which
led to the mal-operation of the adjacent transformer differential
protection or generator differential protection.
Many research addressed the transformer inrush transients
caused by energizing transformers into a system assuming that
there is no other transformers connected to the same system. In
practice, however, energisation of transformers is normally
conducted either in parallel or in series with other adjacent
transformers that are already in operation. By recent
observations [1,2,3] has been found out that such elements may
significantly influence the magnitude and duration of the inrush
current and therefore a comprehensive view of the switching
This work was supported by the Slovak Research and Development
Agency under the contract No. APVV-0280-10 Integrated Analysis of the
Solar Power Plants.
978-1-4799-3807-0/14/$31.00 ©2014 IEEE
transients from the perspective of a complete electricity
network is necessary. Moreover, these already connected
transformers may experience unexpected saturation during
energization of the adjacent transformer. This sharing of the
transient inrush current by already connected transformers is
called sympathetic inrush [2,4,5].
The available literature includes numerous investigations
on the inrush current phenomenon in power transformers and
its impact on the design and operation of protection schemes
[4,5,6]. Most of the approaches are based on the electrical
equivalent circuit [6-8] or magnetic equivalent circuit [9–11]
but they neglect certain surrounding devices in the electrical
system which could have some influence on the transients.
Some of them present analysis within the complete electrical
system but they are not comprehensive. In this work, in order
to solve the problems brought by the sympathetic inrush
current, we deeply expound mathematical theory of
sympathetic inrush current of two shunt wound transformer
which understanding is very meaningful in an effort to avoid
mal-operation of differential and generator protections.
II.
MEASUREMENTS OF SYMPATHETIC INRUSH IN REAL
SYSTEMS
Sympathetic inrush has been recently encountered in some
practical systems and caused significant concerns. In [12],
sympathetic interaction between transformers in a 20 kV
converter test facility was reported. The configuration of the
test facility is shown in Fig. 1. During converter testing, the
active power is circulating between S1-T1-TO-T2-S2-S1 and
only the losses are compensated from a 20 kV grid (with 160
MVA short-circuit level). The two transformers T1 and T2
need to be energised on a daily basis for carrying out tests. To
reduce inrush current, a 100 Ohm short-circuit current limiting
resistor was connected, which is large enough to limit the
inrush current magnitude to values below 150 A and to damp
out inrush current in less than 50 ms. However, subsequent to
the inclusion of resistance sustaining sympathetic inrush
currents were encountered. As shown in Fig. 2, T1 was
energised at 52.2 s and then T2 was energised at 52.26 s;
although inrush current caused by energising T1 was damped
out in one cycle, long-duration sympathetic inrush currents
were induced after energising T2. The damping resistor, which
helped reduce inrush current in the case of energising T1,
caused voltage asymmetry which resulted in sympathetic
inrush when T2 was energised.
Fig. 3. Simplified electrical system circuit diagram [13]
Fig. 1. Diagram of 20 kV supply system of Converter Test Facility [12]
Fig. 4. Measured voltage dips at 23 kV busbar according to Fig. 3 [13]
III.
Fig. 2. Inrush currents measured in the supply line during energization of T1
and T2 after inclusion of 100Ω resistor according to Fig. 1 [12]
Voltage dips caused by sympathetic inrush between 100
MVA 220/23 kV transformers were reported in [13]. The
configuration of the electrical system subjected to sympathetic
inrush is shown in Fig. 3. The substation is fed by two 220 kV
overhead lines with length of 178 km; it originally consisted of
two 100 MVA, 220/23 kV transformers (T1 and T2) connected
in parallel to supply power to mining facilities. A new
transformer T3 was added to meet increasing demands. When
energising T3, sympathetic inrush was induced in the two
already connected transformers T1 and T2. The energisation
also resulted in high distortion of voltages and caused tripping
of those equipment connected to 23 kV busbar due to
undervoltage protection relay. Field measurement of RMS
voltage dip waveforms are shown in Fig. 4. It can be seen that
in both cases, the maximum voltage dip magnitudes were no
more than 8%, however, the duration to achieve a recovery
were over 10 seconds. It shows that, despite of small voltage
dip magnitude, voltage dips accompanied by sympathetic
inrush lasts much longer and may still trip off sensitive
equipment.
PROCESS OF SYMPATHETIC INRUSH BETWEEN
PARALLEL TRANSFORMERS
To deeply illustrate the process of mutual interaction
between transformers "Sympathetic Inrush" is used an example
of parallel transformers according to Fig. 5. In this case T1 and
T2 are two identical 500/230 kV, 150 MVA single-phase
transformers.
Fig. 5. Equivalent circuit used to analyze sympathetic inrush of transformers
in parallel connection
The simulated currents i1, i2 and is are displayed in
following Fig. 6. As it can be seen, the inrush current i2 reached
maximum peak right after the energisation of T2 and then
decayed gradually, while the sympathetic inrush current i1 built
up in T1 gradually reached its maximum peak and then
gradually decayed; the supply current is is the sum of the
currents i1 and i2, the peaks of sympathetic inrush current i1 and
of the inrush current i2 occur in direction opposite to each
other, on alternate half cycles.
The reason of saturation transformer T1, despite the longterm operation and so generation of sympathetic inrush in
electrical circuit according to Fig. 5 is that sudden connection
of transformer T2 without load will produce excitation current
which passes the system resistance. Subsequently, the system
current causes asymmetrical fluctuation of the busbar voltage
(Fig. 7), resulting in saturations and generating sympathetic
inrush current in adjacent transformer T1.
The system impedance is given by parameters Rs = 6.55Ω, Ls
= 0.36H. The two transformers are exactly the same, having
same parameters: the primary, the secondary and tertiary
winding resistance are 0.002(pu), the leakage inductance are
0.08(pu), the excitation resistance are 500(pu), the basic
magnetization curve using two broken line linearization
processing, two broken line determined by the three points
(0,0; 0.0024,1.2; 1.0,1.52).
IV.
ANALYTICAL ANALYSIS OF SYMPATHETIC INRUSH
CURRENT
For the comprehensive analysis of the various influencing
factors of sympathetic inrush current, it is necessary to
understand the generation mechanism of it. This section will
present the formation process of sympathetic inrush current
through the flux analytical expression and bias flux attenuation
between two parallel operation transformers.
Operating one transformer, and closing the other without
loads, generally, has two connection modes: parallel or series.
In this work we will focus on parallel connection which is
shown in Fig. 5 whereas in an article [14] is derived that the
size of inrush currents during switching series transformers are
generally smaller than in the case of parallel. In the scheme in
Fig. 5 Us(t) is the system source voltage; RS & Ls are respective
resistance and inductance of electrical system, R11σ & L11σ, R12σ
& L12σ, R1m & L1m, are respective resistance and leakage
inductance of transformer T1’s primary and secondary winding
and also the excitation resistance and excitation inductance;
R21σ & L21σ, R22σ & L22σ, R2m & L2m, are respective resistance
and inductance of transformer T2 similarly to the transformer
T1. We will next consider i1(t) and i2(t) are respective inrush
current of operation T1 and the closing transformer T2 and
because of the fact that these transformers are operated at no
load we can write R1 = R11σ + R1m, L1 = L11σ + L1m, R2 = R21σ +
R2m, L2 = L21σ + L2m. The system voltage source is us(t) = Um
sin(ωt+α) and α is represented the close angle.
Fig. 6. Inrush current (i2), sympathetic inrush current (i1) and system current
(iS) during energization parallel transformer T2 according to Fig. 5
In [14, 15] and [16], the interactions between parallel
transformers were simply analyzed using the voltage drop
across circuit resistances, with system and transformer winding
inductances neglected. Before closing S, only the magnetizing
current of the unloaded transformer T1 flows through the
system; application of kirchhoff's law we could write:
us (t ) = ( Rs + R1 ) i1 +
dψ 1
dt
(1)
The integration of us over one cycle gives:
t + Δt
∫
us (t )dt =
t
Fig. 7. Voltage drop across the system impedance according to Fig. 5
Note: In the simulation, we considered the voltage source
whose rated voltage Us is 500kV and rated frequency is 60Hz.
t +Δt
∫ ⎡⎣( R
s
+ R1 ) i1 ⎤⎦ dt + Δψ 1
(2)
t
Where ∆t is of one cycle interval and ∆ψ1 represents the
flux change per cycle in transformer T1. Considering the
system source us(t) is symmetric periodical function, so:
t +T
∫U
t
m
sin (ωt + α ) dt = 0
(3)
We could write the following relation for the flux change
per cycle in transformer T1:
Δψ 1 = −
t +Δt
∫ ⎡⎣( R
s
+ R1 ) i1 ⎤⎦ dt
(4)
t
According to previous equation if current of transformer i1
is symmetrical and transformer T2 is not connected, ∆ψ1 will
be zero. This situation corresponds to the state before switching
transformer T2.
After closing S, saturation of transformer T2 causes a
transient inrush current i2 which flows through Rs. Due to the
unidirectional characteristic of the inrush current, each cycle
transformer T1 experiences an offset flux by an amount of:
Δψ 1 = −
t +Δt
∫ ⎡⎣( R
s
+ R1 ) i1 + Rs i2 ⎤⎦ dt
(5)
be significantly prolonged in power systems with large
transformers energised, as the winding resistances of these
transformers are normally of relatively small value.
Note: The fluxes ψ1, ψ2 attenuate respectively by the value
of (10)(11) periodically and the damping role of the system
resistance disappears, and it is attenuating, in accordance with
the basic time constant τ1 and τ2 which results in longer
attenuation.
∫
⎡⎣( Rs + R2 ) i2 + Rs i1 ⎤⎦ dt
t
As the transformer T1 becomes more and more saturated, a
sympathetic inrush current i1 gradually increases from the
steady state magnetizing current to a considerable magnitude.
Noted that as the transformer T1 saturates with the polarity
opposite to that of transformer T2, the peaks of the sympathetic
inrush current i1 are with polarity opposite to that of inrush
current i2, on alternate half cycles. As a result, the voltage
asymmetry on transformer terminals caused by the inrush
current i2 during one half cycle is reduced by the voltage drop
produced by the sympathetic inrush current i1 during the
subsequent half cycle. This decreases both ∆ψ1 and ∆ψ2, and
therefore reduces the changing rate of the magnitude of both
the increasing sympathetic inrush current i1 and the decaying
inrush current i2.
After a certain time, the increase of i1 and decay of i2 can
reach a point that:
( Rs + R1 ) i1 = − Rs i2
(10)
Where ψ1 and ψ2 are respective fluxes of transformers T1
and T2. Obviously, the relation (10) is a nonlinear equation. In
order to analyse the flux relation of the two transformers, we
should get analytical formula because we need to do linear
process with relation (10). Here, the non-linear excitation
inductance L1m and L2m are replaced by the average inductance
of the transformer magnetizing circuit and we will further
consider the same transformers: L1 =L2 =L， R1m =R2m = R.
Then using Laplace transform and Laplace inverse
transform, we can get two transformers flux expression:
ψ 1 (t ) =
L
1
U m sin (ω t + α − ϕ ) − [ψ 1 ( 0 ) −
Z
2
−ψ 2 ( 0 )]e
ψ 2 (t ) =
−
R + 2 Rs
t
L + 2 Ls
R
− t
1
+ ⎡⎣ψ 1 ( 0 ) −ψ 2 ( 0 ) ⎤⎦ e L
2
L
1
U m sin (ωt + α − ϕ ) − [ψ 1 ( 0 ) −
Z
2
−ψ 2 ( 0 )]e
−
R + 2 Rs
t
L + 2 Ls
R
− t
1
− ⎡⎣ψ 1 ( 0 ) −ψ 2 ( 0 ) ⎤⎦ e L
2
(11)
(12)
Where:
(7)
At this point, the flux change per cycle ∆ψ1 is zero and
hence current i1 stops increasing. Thereafter, the polarity of
∆ψ1 reverses and starts to reduce the offset flux in the
transformer T1, as a result, the sympathetic inrush current i1
begins to decay (so does the inrush current i2). Since both
decaying currents have the same amplitude but with polarities
opposite to each other, no voltage asymmetry is produced on
the transformer terminals and the flux change per cycle in each
transformer only depends on the winding resistance of each
transformer. This is one of the reasons for the inrush current to
(9)
dis
dψ
+ R1i1 + 1 = U m sin (ω t + α )
dt
dt
dψ
dψ 2
R1i1 + 1 = R2 i2 +
dt
dt
is = i1 + i2
(6)
At the initial stage, both ∆ψ1 and ∆ψ2 are of the same
polarity and mainly depend on the voltage drop caused by the
inrush current i2. The accumulation of ∆ψ1 drives transformer
T1 into saturation, while the effect of ∆ψ2 is to reduce the
initial offset flux in transformer T2 so as to produce the decay
of inrush current i2.
τ 2 = L2 R2
Rs is + Ls
Meanwhile, an offset flux per cycle ∆ψ2 is produced in
transformer T2 by:
t +Δt
(8)
In order to express exact current waveforms of transformers
during sympathetic interaction we need to solve the complete
system of differential equations which define electrical grid in
Fig. 5. The application of the circuit principle we can write:
t
Δψ 2 = −
τ 1 = L1 R1
Z=
( R + 2 Rs ) + ( L + 2 Ls )
ϕ = arctan
2
ω ( L + 2 Ls )
R + 2 Rs
2
(13)
(14)
And ψ1(0) and ψ2(0) are the respective initial fluxes of
transformers T1 and T2.
From equations (11) and (12), it can be seen that both ψ1
and ψ2 consist of one sinusoidal component and two
exponential DC components. The AC component and the first
DC component are the same, but the second DC component in
ψ1 is opposite to that in ψ2, therefore i1 and i2 are opposite to
each other and appear alternately.
Also, because the both DC components in ψ2 are negative, so
the maximum peak of i2 would appear right after the
energisation of T2, whilst the DC components in ψ1 are of
opposite polarity and the time constant of the first DC
component τ1 = (L+2Ls)/(R+2Rs) is smaller than that of the
second DC component τ2 = L/R, so i1 will gradually reach the
maximum peak, and gradually decay afterwards as we could
see in Fig. 6 and Fig. 8.
The simplified analytical analysis shows in a general way the
variation of flux-linkages in T1 and T2 which depends on the
time constants formed by the inductances and resistances of the
circuit branches. In real situation, the core inductance is
nonlinear and therefore the time constants cannot be so readily
determined.
is ( t ) =
2
1
U m sin (ω t + α − ϕ ) − [ψ 1 ( 0 )
Z
L
−ψ 2 ( 0 )]e
−
R + 2 Rs
t
L + 2 Ls
(17)
Now is apparent that, one of two transient component from
(11,12) with time constant τ2 = L/R is circulating around the
loop formed by the two transformer windings in series without
flowing in the transmission line during the sympathetic
phenomenon (Fig. 9), whilst the second one with time constant
τ1 = (L+2Ls)/(R+2Rs) is created by system voltage source.
Fig. 9. Transient magnetic component of magnetic flux which represents
circulating current
It is interesting to note that the circulating current, both
with respect to magnitude and rate of decay, is entirely
independent of the characteristics of the transmission line
circuit, being determined solely by the inductance and
resistance of the transformers themselves in conjunction with
the initial bus voltage, the frequency, and the switching angle
and it causes relatively longer duration of transient
phenomenon of energization a parallel transformer in
comparison to energization of a single transformer.
V.
Fig. 8. Waveforms of magnetic fluxes of transformers T1 and T2
For aperiodic components of magnetic fluxes ψ1 and ψ2
from equations (11) (12) in the previous Fig. 8, we could
apply:
1
ψ a ( t ) = ⎣⎡ψ 1 ( 0 ) + ψ 2 ( 0 )⎦⎤ e
2
−
R + 2 Rs
t
L + 2 Ls
R
− t
1
ψ b ( t ) = ⎣⎡ψ 1 ( 0 ) + ψ 2 ( 0 ) ⎦⎤ e L
2
(15)
Summing the individual magnetic fluxes (ψ1, ψ2) we get
relation for total magnetic flux:
ψ (t ) =
2L
U m sin (ω t + α − ϕ ) − [ψ 1 ( 0 )
Z
−ψ 2 ( 0 )]e
−
R + 2 Rs
t
L + 2 Ls
In this paper, we have presented a detailed formulation for
the analysis of sympathetic inrush phenomenon for the
configuration of parallel transformers.
When the sympathetic interaction between transformers
happens, the operating transformer coud be saturated and
generate the sympathetic inrush, which could influence
transformer protections and therefore the exact understanding
of this phenomenon is very important. The analytical analysis
helped us to explain its behaviour and confirmed some real
cases of sympathetic interactions which are also shown.
Furthermore, a practical example of the phenomenon was
presented. The system was created and simulated in
Matlab/Simulink which enabled us to present instantaneous
current waveforms in detail.
(16)
Considering linear characteristic of transformers T1 and T2
(i1 = ψ1/L1, i2 = ψ2/L2) we could write equation represented line
current (is) based on relation (16):
CONCLUSION
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