Inter
national
J
our
nal
of
P
o
wer
Electr
onics
and
Dri
v
e
Systems
(IJPEDS)
V
ol.
12,
No.
4,
December
2021,
pp.
2423
2434
ISSN:
2088-8694,
DOI:
10.11591/ijpeds.v12.i4.pp2423-2434
r
2423
Characterizing
po
wer
transf
ormer
fr
equency
r
esponses
using
bipolar
pseudo-random
curr
ent
impulses
Fr
edrick
Mwaniki
1
,
Ahmed
A.
Sayyid
2
1
Department
of
Electrical
and
Electronic
Engineering,
Uni
v
ersity
of
Stellenbosch,
Stellenbosch,
South
Africa
2
Department
of
Electrical
and
Information
Engineering,
Uni
v
ersity
of
Nairobi,
Nairobi,
K
en
ya
Article
Inf
o
Article
history:
Recei
v
ed
Apr
22,
2021
Re
vised
Sep
23,
2021
Accepted
Sep
27,
2021
K
eyw
ords:
Excitation
signals
Frequenc
y
response
Harmonics
P
arameter
estimation
Po
wer
quality
Po
wer
transformer
ABSTRA
CT
The
beha
viour
of
a
po
wer
tr
ansformer
is
comple
x
and
dif
ficult
to
predict
during
tran-
sient
conditions
or
during
operation
at
frequencies
belo
w
or
abo
v
e
its
nominal
fre-
quenc
y
,
a
phenomenon
common
in
rene
w
able
ener
gy
plants
due
to
harmonic
distor
-
tion.
Furthermore,
the
accurac
y
of
a
po
wer
system
simulation
depends
on
the
models
of
critical
subsystems
such
as
the
po
wer
transformers.
This
paper
presents
the
use
of
a
unique
e
xcitation
w
a
v
eform
com
prising
of
pseudo-random
current
impulses
to
ac-
curately
identify
the
wideband
characteristics
of
a
po
wer
transformer
.
By
injecting
the
e
xcitation
w
a
v
eform
to
the
rele
v
ant
transformer
terminals,
frequenc
y
responses
are
determined
by
cross-correlation
of
the
perturbation
signal,
and
the
measured
response.
Compared
to
the
traditional
t
ransformer
identification
methods,
the
pseudo-random
current
impulses
of
fer
a
wideband
e
xcitation
with
a
higher
de
gree
of
controllability
such
that
its
spectral
ener
gy
can
be
focused
in
the
frequenc
y
band
of
interest.
The
pro-
posed
method
w
as
in
v
estig
ated
on
a
16
kV
A,
22
kV/240
V
single-phase
transformer
.
The
obtained
wideband
frequenc
y
responses
pro
vide
useful
information
in
harmonic
penetration
and
o
v
er
-v
oltage
studies
and
are
also
used
to
estimate,
with
a
high
de
gree
of
accurac
y
,
the
lumped
parameters
of
the
equi
v
al
ent
transformer
broadband
circuit
model.
This
is
an
open
access
article
under
the
CC
BY
-SA
license
.
Corresponding
A
uthor:
Fredrick
Mw
aniki
Department
of
Electrical
and
Electronic
Engineering
Stellenbosch
Uni
v
ersity
Banghoek
Road,
Stellenbosch,
7600,
South
Africa
Email:
fmmw
aniki@sun.ac.za
1.
INTR
ODUCTION
In
recent
years,
there
has
been
a
rapid
increase
in
rene
w
able
ener
gy
sources
such
as
photo
v
oltaic
and
wind
generators
[1]-[3].
These
ener
gy
sources
ha
v
e
presented
ne
w
challenges
such
as
the
risk
of
high
harmonic
v
oltage
distortion
due
to
their
use
of
po
wer
electronic
con
v
erters
[1],
[4],
[5].
These
po
wer
system
harmonics
ha
v
e
increased
in
both
amplitude
and
frequenc
y
bandwidth.
Step-up
transformers
that
are
used
to
connect
these
rene
w
able
plants
to
the
grid
are
adv
ersely
af
fected
by
the
harmonics
from
these
rene
w
able
plants
[6].
Po
wer
transformers
do
not
ha
v
e
a
flat
frequenc
y
response,
which
mak
es
them
unsuitable
for
use
in
applications
with
high
harmonic
components
and
high
frequenc
y
transient
signals.
T
ransformer
impedances
are
functions
of
the
input
signal
frequencies
and
v
oltage,
as
well
as
the
loading
conditions
[7],
[8].
T
ransformer
parameters
such
as
leakage
i
n
duct
ances,
winding
capacitances,
and
the
non-linear
core
components
interact
to
cause
comple
x
series
and/or
parallel
resonant
phenomena
in
the
transformer
frequenc
y
characteristics
[9].
J
ournal
homepage:
http://ijpeds.iaescor
e
.com
Evaluation Warning : The document was created with Spire.PDF for Python.
2424
r
ISSN:
2088-8694
When
a
minimally
damped
resonant
point
in
the
po
wer
tr
ansformer’
s
wideband
transformation
ratio
charac-
teristics
is
e
xcited
by
a
harmonic
v
oltage
or
a
transient
e
v
ent,
e
xcessi
v
e
v
oltage
and/or
current
amplification
occurs.
Ov
er
-v
oltages
and
o
v
er
-currents
may
cause
protection
de
vices
to
acti
v
ate,
or
cause
damage
to
the
po
wer
transformer
and
other
equipment
connected
to
the
netw
ork,
resulting
to
unnecessary
do
wntime.
In
this
conte
xt,
the
response
of
transformers
to
input
signals
with
multiple
frequencies
needs
to
be
modeled
and
understood.
Therefore,
this
gi
v
es
rise
to
rene
wed
interes
t
in
performing
broad
frequenc
y
band
measurements,
especially
on
po
wer
transformers
that
are
connected
to
wind
and
photo
v
oltaic
(PV)
ener
gy
sources.
A
common
approach
in
transformer
identification
is
the
use
of
non-parametric
m
ethods
that
compute
the
transformer
frequenc
y
responses.
Frequenc
y
response
analysis
(FRA)
has
been
widely
used
for
condition
monitoring
of
transformers
[10]-[15].
Frequenc
y
response
measurements
ha
v
e
also
been
applied
in
transformer
parameter
estimation
[10],
[16]-[18].
Although
po
wer
transformer
frequenc
y
responses
ha
v
e
been
widely
used
for
condition
monitoring
and
parameter
estimation
[10]-[12],
[18]
their
use
in
po
wer
qual
ity
studies,
especially
in
cases
where
the
transformers
are
used
in
rene
w
able
ener
gy
sources,
has
not
been
gi
v
en
much
attention.
In
practice,
frequenc
y
response
measurements
are
conducted
by
applying
an
e
xcitation
signal
that
can
perturb
the
tar
get
system
at
the
frequencies
of
interest.
The
system
input
and
output
s
ignals
are
measured
o
v
er
an
interv
al
for
further
processing
to
o
bt
ain
the
rele
v
ant
frequenc
y
re
sponses.
The
e
xcitat
ion
signals
proposed
in
liter
ature
for
transformer
FRA
include
the
swept
sine
and
the
impuls
e
[11],
[19],
[20].
The
sine
sweep
in
v
olv
es
a
v
ariable-frequenc
y
sinusoidal
signal
which
is
simple
to
impl
ement
using
lo
w-v
oltage
sinusoidal
signal
generators.
In
practice,
ho
we
v
er
,
it
is
dif
ficult
to
design
and
implement
a
sinusoidal
source
to
perturb
a
lar
ge
po
wer
transformer
o
v
er
the
frequenc
y
range
of
interest,
especially
at
lo
w
and
high
ends
of
the
range.
This
is
due
to
reduced
magnetizing
reactance
and
winding
capaciti
v
e
reactance
at
these
frequencies
respecti
v
ely
.
Consequently
,
v
ery
lo
w
signal
le
v
els
result
in
measurement
errors
due
to
a
lo
w
signal
to
noise
ratio
(SNR).
The
impulse
method
in
v
olv
es
perturbing
the
transformer
with
a
classical
v
oltage
impulse
Which
has
a
lo
w
rise
time
and
a
higher
f
all
time.
Although
this
method
is
more
accurate,
the
spectral
ener
gy
of
t
he
classical
impulse
w
a
v
eform
drops
rapidly
with
frequenc
y
which
w
ould
result
to
limited
accurac
y
in
some
frequencies.
Controlled
signals
ha
v
e
been
proposed
for
use
in
po
wer
system
equipment
identification
[21]-[23],
and
ha
v
e
adv
antages
in
that
the
ir
spectral
characteristics
can
be
controlled
and
are
repeatable.
Therefore,
accurate
measurements
are
poss
ible
at
the
e
xpense
of
comple
x
instrumentation
[11].
Despite
the
adv
antages
presented
by
controlled
perturbation
signal
method,
there
has
been
limited
application
of
these
signals
for
transformer
FRA.
In
the
cas
e
of
a
pseudorandom
bi
nary
sequence
(PRBS),
the
clock
frequenc
y
and
sequence
length
can
be
controlled
to
induce
persistent
e
xcitation
for
the
dynamic
modes
associated
with
the
de
vice
under
test.
The
PRBS
has
a
flat
frequenc
y
spectrum
at
frequencies
belo
w
the
-3
dB
point
which
allo
ws
for
perturbation
with
uniform
ener
gy
[24].
The
PRBS
is
ho
we
v
er
not
suitable
as
a
perturbation
signal
for
transformers
due
to
a
possibility
of
injecting
lo
w
frequenc
y
components
that
w
ould
cause
core
saturation.
This
particularly
occurs
when
a
long
PRBS
is
used
[25].
The
main
contrib
ution
of
this
w
ork,
therefore,
includes
the
follo
wing:
The
use
of
a
wideband
controlled
perturbation
signal
to
obtain
accurate
transformer
frequenc
y
responses.
The
use
of
transformer
frequenc
y
responses
for
po
wer
quality
studies
and
parameter
estimation.
In
this
paper
the
use
of
a
unique
signal
consisting
of
bipolar
pseudo-random
current
impulses
as
a
broadband
e
xcitation
signal
to
determine
the
frequenc
y
responses
of
a
transformer
in
the
harmonic
frequenc
y
range
is
proposed.
The
obtained
responses
are
then
used
t
o
study
the
ef
fects
of
harmonics
especially
on
the
v
oltage
transfer
function,
to
characterise
the
po
wer
transformer
o
v
er
a
wide
frequenc
y
band
as
well
as
to
obtain
transformer
wideband
parameters.
The
sequence
combines
the
adv
antages
of
the
PRBS
and
the
classical
double
e
xponential
type
impulse.
An
adv
antage
of
using
this
signal
is
that
its
po
wer
spectrum
can
be
controlled,
for
instance,
in
reducing
the
ener
gy
in
lo
w
frequencies
to
a
v
oid
transformer
saturation
during
measurements.
Properties
of
the
signal
such
as
the
clock
frequenc
y
and
time
const
ants
can
be
used
to
focus
the
spectral
ener
gy
to
the
frequenc
y
range
of
interest
and
impro
v
e
the
SNR.
The
bipolar
nature
of
the
sequence
ensures
that
the
transformer
is
not
dri
v
en
t
o
w
ards
a
biased
of
fset
point
from
the
normal
operating
point
whereas
its
wideband
nature
allo
ws
tests
to
be
conducted
within
short
time
periods
compared
to
con
v
entional
methods
such
as
the
sine
sweep.
In
section
2,
a
methodology
for
transformer
frequenc
y
response
measurements
is
discussed.
Experi-
mental
results
and
parameter
estimation
of
a
high
frequenc
y
po
wer
transformer
model
using
the
e
xperimental
frequenc
y
responses
is
discussed
in
section
3.
The
conclusion
is
presented
in
section
4.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
12,
No.
4,
December
2021
:
2423
–
2434
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
2425
2.
RESEARCH
METHOD
2.1.
T
ransf
ormer
testing
using
a
sequence
of
bipolar
pseudo-random
curr
ent
impulses
The
idea
of
using
a
bipolar
pseudo-random
impulse
sequence
as
a
perturbation
signal
in
the
ener
gy
systems
and
po
wer
equipment
w
as
first
proposed
in
[26].
The
bipolar
pseudo-random
impulse
sequence
is
a
wideband
signal
consisting
of
a
ser
ies
of
chopped
positi
v
e
and
ne
g
ati
v
e
impulses.
In
this
w
ork,
a
sequence
of
bipolar
pseudo-random
current
impulses
w
as
used
as
a
perturbation
signal
for
transformer
identification
and
modelling.
Figure
1
illustrates
the
bipolar
pseudo-random
impulse
sequence.
Bipolar
pseudo-random
current
impulses
can
be
generated
using
an
ef
ficient
circuit
topology
consisting
of
a
DC
source,
a
full-bridge
whose
switches
are
controlled
by
a
PRBS
g
ate
signal
and
a
series
RLC
netw
ork
as
sho
wn
in
Figure
2.
Time
0
PRIS (t)
Figure
1.
Bipolar
pseudo-random
impulse
sequence
Figure
2.
Circuit
diagram
for
the
bipolar
Pseudo-random
current
impulses
generator
The
PRBS
is
a
random,
deterministic
and
periodic
signal
that
can
be
produced
using
shift
re
gist
ers
dri
v
en
by
a
clock
sequence.
The
occurrence
and
duration
of
t
h
e
binary
states
is
random.
The
data
length
for
one
PRBS
period
is
defined
as
N
=
2
n
1
bits,
where
n
denotes
the
number
of
shift
re
gisters.
The
series
RLC
netw
ork
pro
vides
a
means
of
opt
imizing
the
time-constants
associated
with
the
rise-and
f
all-times
of
the
bipolar
Pseudo-random
current
impulses.
The
circuit
of
Figure
2
can
be
vie
wed
as
an
application
of
a
step
input
to
a
series
RLC
circuit,
wh
e
re
the
step
is
controlled
by
the
PRBS
signal.
The
sequence
of
bipolar
current
impulses
generated
by
the
circuit
in
Figure
2
o
v
er
one
period
can
be
e
xpressed
as:
i
p
(
t
)
=
((
N
+1)
=
2)
1
X
i
=0
;
2
;
4
:::
h
e
t=
1
(
V
0
V
dc
+
I
0
L
1
)
e
t=
2
(
V
0
V
dc
+
I
0
L
2
)
i
1
2
L
(
2
1
)
h
u
(
t
t
i
)
u
(
t
t
i
+1
)
i
((
N
+1)
=
2)
X
i
=1
;
3
;
5
:::
h
e
t=
1
(
V
0
V
dc
+
I
0
L
1
)
e
t=
2
(
V
0
V
dc
+
I
0
L
2
)
i
1
2
L
(
2
1
)
h
u
(
t
t
i
)
u
(
t
t
i
+1
)
i
:
(1)
where
u
(
t
t
i
)
u
(
t
t
i
+1
)
depict
random
switching
interv
als
in
the
full
bridge
circuit,
1
and
2
are
the
time
constants
associated
with
the
series
RLC
netw
ork,
I
0
and
V
0
denote
the
initial
current
through
the
inductor
and
the
initial
v
oltage
across
the
capacitor
respecti
v
ely
.
In
(1)
indicates
that
the
e
xcitation
current
depends
on
the
DC
source,
the
RLC
netw
ork
time
const
ants
and
the
PRBS
switching
interv
als
which
in
turn
are
dependent
on
the
PRBS
clock
frequenc
y
.
In
practice,
Char
acterizing
power
tr
ansformer
fr
equency
r
esponses
using
bipolar
...
(F
r
edric
k
Mwaniki)
Evaluation Warning : The document was created with Spire.PDF for Python.
2426
r
ISSN:
2088-8694
the
shape
of
the
perturbation
signal
is
also
af
fected
by
the
initial
ener
gy
in
the
inductor
and
capacitor
in
each
switching
c
ycle.
The
properties
of
the
current
impulses
can
easily
be
controlled
by
v
arying
e
xcitation
source
parameters
namely
the
input
DC
source,
PRBS
clock
frequenc
y
f
c
,
and
the
impulses
time
constants.
This
high
de
gree
of
control
of
fers
capability
to
impro
v
e
the
signal’
s
SNR,
guarantee
repeatable
e
xperiments
and
allo
w
the
user
to
concentrate
the
signal’
s
ener
gy
in
the
required
frequenc
y
band.
The
broadband
nature
of
the
sequence
ensures
that
the
transformer
identifi
cation
procedure
is
conducted
f
ast
compared
to
using
con
v
entional
e
xcitation
signals
such
as
stepped
sine.
This
is
an
adv
antage
especially
in
in
situ
po
wer
system
measurements
where
parame
ters
such
as
harmonics
are
time-dependent.
The
bipolar
pseudo-random
impulse
sequence
can,
therefore,
be
adjusted
to
achie
v
e
persistent
e
xcitation
for
a
broad
range
of
frequencies
and
applications.
This
is
an
important
aspect
especially
in
identification
of
systems
in
v
olving
electromagnetic
components
such
as
transformers
where
superior
performance
is
achie
v
ed
when
e
xcitation
ener
gy
is
directed
in
the
higher
frequenc
y
band
and
the
lo
w
frequenc
y
perturbation
is
restricted.
2.2.
T
ransf
ormer
test
arrangement
2.2.1.
Offline
fr
equency
r
esponse
measur
ements
In
of
fline
frequenc
y
response
measurements,
the
transformer
is
disconnected
from
the
normal
A
C
supply
v
oltage.
A
perturbation
source
is
then
connected
to
the
transformer
terminal
s
thus
pro
viding
e
xcitation.
Figure
3
sho
ws
a
test
arrangement
that
can
be
used
to
obtain
the
frequenc
y
response
of
the
secondary
open-
circuit
input
impedance.
This
is
the
impedance
seen
from
the
secondary
terminals
of
transformer
T
r
with
primary
terminals
left
open.
A
perturbation
source
is
connected
to
the
secondary
terminals
as
sho
wn
in
Figure
3.
The
w
a
v
eforms
of
the
applied
perturbation
signal,
i
p
(
t
)
,
and
the
perturbed
v
oltage
across
the
secondary
winding,
v
s
(
t
)
,
are
then
recorded
o
v
er
a
time
interv
al.
Figure
3.
T
est
arrangement
for
secondary
open-circuit
input
impedance
frequenc
y
response
Re
g
arding
the
transformer
as
a
linear
time-in
v
ariant
system
and
ignoring
measurement
and
quantiza-
tion
noise,
the
sampled
v
oltage
is
gi
v
en
by
the
discrete
con
v
olution
of
the
impulse
response
with
the
perturba-
tion
signal
which
can
be
e
xpressed
as.
v
s
[
n
]
=
1
X
k
=1
z
[
k
]
i
p
[
n
k
]
(2)
where
v
s
(
n
)
is
the
discrete-time
v
oltage
signal,
i
p
(
k
)
is
the
discrete-time
perturbation
signal
and
z
(
k
)
is
the
sampled
system
impulse
response.
Con
v
olution
of
time
domain
signals
implies
multiplication
of
the
signals
in
frequenc
y-domain
and
therefore,
the
secondary
open-circuit
input
impedance
frequenc
y
response
Z
s
(
!
)
can
be
obtained
by
applying
the
F
ourier
transform
to
the
measured
v
oltage
and
current
signals.
Z
s
(
!
)
can
then
be
e
xpressed
as.
Z
s
(
!
)
=
F
f
v
s
(
t
)
g
F
f
i
p
(
t
)
g
(3)
where
F
denotes
the
F
ourier
transform.
The
primary
open-circuit
input
impedance
frequenc
y
response
Z
p
;open
(
!
)
can
be
obtained
from
the
test
arrangement
sho
wn
in
Figure
4.
The
e
xcitati
on
source
is
connected
to
the
primary
winding
terminals
and
the
perturbation
current,
i
p
(
t
)
,
and
the
perturbed
v
oltage
across
the
primary
winding,
v
p
(
t
)
,
are
then
recorded
o
v
er
a
time
interv
al.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
12,
No.
4,
December
2021
:
2423
–
2434
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
2427
Figure
4.
T
est
arrangement
for
primary
open-circuit
input
impedance
frequenc
y
response
The
frequenc
y
response
of
the
primary
open-circuit
input
impedance
can
be
obtained
from
the
e
xpres-
sion
Z
p
;open
(
!
)
=
F
f
v
p
(
t
)
g
F
f
i
p
(
t
)
g
(4)
The
frequenc
y
response
of
the
transformation
ratio
T
R
(
!
)
can
be
obtained
using
the
topology
sho
wn
in
[4]
by
recording
the
perturbed
v
oltages
across
both
primary
and
secondary
windings.
The
transformation
ratio
frequenc
y
response
can
be
e
xpressed
as.
T
R
(
!
)
=
F
f
v
p
(
t
)
g
F
f
v
s
(
t
)
g
(5)
By
short-circuiting
the
secondary
term
inals
of
the
test
arrangement
sho
wn
in
Figure
4
the
primary
short-circuit
input
impedance
frequenc
y
response
can
be
obtained
using
the
e
xpression
in
(4).
2.2.2.
In
situ
fr
equency
r
esponse
measur
ements
In
in
situ
measurements,
the
transformer
stays
connected
to
the
con
v
entional
supply
v
oltage
during
perturbation.
The
transformer
does
not
need
to
be
disconnected,
which
reduces
do
wn
time.
Furthermore,
during
in
situ
measurements,
the
ef
fects
of
the
normal
transformer
operat
ion
are
accounted
for
[26].
The
dynamic
range
of
the
e
xcitation
signal
ho
we
v
er
has
to
be
k
ept
lo
w
enough
to
ensure
that
the
transformer
is
not
perturbed
too
f
ar
from
its
optimal
operating
point.
Figure
5
sho
ws
a
connection
scheme
that
can
be
used
to
characterize,
in
situ
,
the
secondary
open-
circuit
input
impedance
Z
s
.
The
transformer
is
connected
to
its
normal
ac
v
oltage
supply
V
T
h
which
has
an
internal
impedance
Z
T
h
.
The
perturbation
source
is
then
connected
across
the
v
oltage
source,
thereby
injecting
the
e
xcitation
current
into
the
transformer
terminals.
The
perturbation
current,
i
p
(
t
)
,
causes
a
v
oltage
perturbation
to
be
superimposed
on
the
supply
v
oltage
V
T
h
due
to
a
v
oltage
drop
across
Z
T
h
.
Figure
5.
T
est
connection
for
in
situ
transformer
perturbation
The
perturbed
v
oltage
v
s
(
t
)
and
current
i
s
(
t
)
into
the
transformer
are
then
measured
o
v
er
a
time
interv
al.
The
secondary
open-circuit
input
impedance
frequenc
y
response
Z
s
(
!
)
can
be
obtained
from
the
e
xpression,
Z
s
(
!
)
=
F
f
v
s
(
t
)
g
F
f
i
s
(
t
)
g
(6)
A
similar
arrangement
can
be
used
to
obtain
the
v
oltage
transformation
ratio
whereas
the
s
econdary
short-circuit
input
impedance
measurement
can
be
done
with
the
primary-s
ide
terminals
shorted.
In
in
situ
measurements,
a
high
SNR
can
be
achie
v
ed
with
lo
wer
e
xcitation
compared
to
the
of
fline
measurements.
Char
acterizing
power
tr
ansformer
fr
equency
r
esponses
using
bipolar
...
(F
r
edric
k
Mwaniki)
Evaluation Warning : The document was created with Spire.PDF for Python.
2428
r
ISSN:
2088-8694
3.
RESUL
TS
AND
AN
AL
YSIS
3.1.
Fr
equency
r
esponse
analysis
V
erification
of
the
proposed
perturbation
method
is
done
by
perturbing,
of
fline,
a
16
kV
A,
22
kV/240
V
single-phase
distrib
ution
transformer
using
a
pseudo-random
current
impulse
described
in
section
2.1.
The
perturbation
signal
is
generated
by
controlling
the
H-bridge
switches
using
a
PRBS
g
ate
signal
with
f
c
set
to
15
kHz
thereby
switching
a
series
RLC
netw
ork
with
v
alues
R
=
50
;
L
=
100
H
;
C
=
2
F
.
A
DC
v
oltage
of
60
V
is
applied
to
the
H-bridge
input.
The
e
xcitation
current
generated
is
then
applied
to
the
transformer
terminals
usi
n
g
the
test
arrange-
ments
discussed
in
section
2.2
to
obtain
the
v
arious
frequenc
y
response
estimates.
Each
measurement
is
con-
ducted
for
a
duration
of
1
s
and
the
required
v
oltage
and
current
data
recorded.
V
oltage
and
current
w
a
v
eforms
are
recorded
using
the
GW
instek
GDP-025
dif
ferential
v
oltage
probe
and
the
A622
T
ektronix
current
probe
respecti
v
ely
.
The
National
Instruments
CompactD
A
Q
chassis
and
the
NI
9223
module
is
used
for
data
acqui-
sition.
Figure
6
sho
ws
the
measured
w
a
v
efor
ms
for
the
bipolar
Pseudo-random
perturbation
current
and
the
PRBS
g
ate
signal
obtained
for
the
circuit
sho
wn
in
Figure
2.
Figure
6.
Experimental
w
a
v
eforms
for
the
bipolar
Pseudo-random
perturbation
current
(top)
and
PRBS
g
ate
signal
(bottom)
T
o
eliminate
stochastic
measurement
noise
and
reduce
leakage
and
aliasing
problems,
the
acquired
data
is
processed
i
n
Matlab
using
the
W
el
ch
estimator
where
the
measured
data
is
di
vided
into
o
v
erlapped
and
windo
wed
se
gments.
The
transfer
function
of
each
se
gment
is
computed
and
then
a
v
eraged
to
generate
a
frequenc
y
response.
The
Hann
windo
w
w
as
sele
cted
as
it
of
fered
good
frequenc
y
resolution
and
an
o
v
erlap
percentage
of
50%
is
used.
The
lo
w
v
oltage
(L
V)
open
circuit
input
impedance
is
measured
by
using
the
test
arrangement
sho
wn
in
Figure
3.
In
this
case,
the
bipolar
pseudo-random
current
impulses
generator
is
connected
to
the
L
V
side
termi-
nals
of
the
transformer
while
the
high
v
oltage
(HV)
winding
is
left
open.
The
perturbed
L
V
V
oltage
and
the
perturbation
current
are
recorded
and
used
for
the
impedanc
e
estimation.
The
obtained
e
xperimental
L
V
open
circuit
input
impedance
magnitude
and
phase
frequenc
y
responses
are
sho
wn
in
Figure
7.
The
impedance
frequenc
y
response
in
Figure
7
indicates
v
arious
resonance
peaks.
The
first
resonance
peak
indicates
a
parallel
resonance
at
approximately
100
Hz,
a
series
resonance
at
about
6
kHz
and
a
parallel
and
series
resonance
pair
between
20
kHz
and
30
kHz.
A
highly
damped
parallel
and
series
resonance
pair
is
also
noticeable
between
40
kHz
and
50
kHz.
Experiments
to
det
ermine
the
transformer
input
response
from
the
HV
side
are
conducted
using
the
test
arrangement
sho
wn
in
Figure
4.
T
o
determine
the
HV
short-circuit
input
impedance,
the
perturbation
current
is
applied
to
the
HV
terminals
with
the
L
V
terminals
shorted.
The
perturbed
HV
V
oltage
and
the
injected
current
are
recorded
and
used
in
the
estimation
of
the
impedance
response.
In
this
case,
a
lar
ge
current
is
dra
wn
by
the
winding
resistances
and
leakage
inductances
due
to
the
shorted
L
V
winding.
Figure
8
sho
ws
the
obtained
e
xperimental
short-circuit
input
impedance
magnitude
and
phase
frequenc
y
responses
of
the
HV
side
of
the
transformer
.
A
parallel
resonance
at
about
2.5
kHz
and
a
series
resonance
at
about
3
kHz
are
e
vident.
At
lo
w
frequencies,
up
to
about
2
kHz,
the
impedance
is
pr
edominantly
inducti
v
e
which
can
be
attrib
uted
to
the
leakage
inductance
of
the
transformer
.
Ho
we
v
er
,
at
higher
frequencies
be
yond
4
kHz,
the
impedance
turns
capaciti
v
e
which
can
be
attrib
uted
to
the
lar
ge
HV
winding
capacitor
.
The
transformation
ratio
is
obtained
by
applying
the
pseudo-random
current
impulse
sequence
to
the
HV
side
with
the
L
V
winding
open.
V
oltage
w
a
v
eforms
v
H
V
(
t
)
and
v
LV
(
t
)
are
recorded
and
us
ed
to
estimate
the
transformation
ratio
frequenc
y
response.
F
igure
9
sho
ws
the
obtained
e
xperimental
HV
to
L
V
transformation
ratio
magnitude
and
phase
frequenc
y
responses.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
12,
No.
4,
December
2021
:
2423
–
2434
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
2429
10
1
10
2
10
3
10
4
10
4
10
6
Magnitude [Ohms]
10
1
10
2
10
3
10
4
Frequency (Hz)
-200
-100
0
100
200
Angle [Degrees]
Figure
7.
Experimental
L
V
open-circuit
input
impedance
magnitude
(top)
and
phase
(bottom)
frequenc
y
responses
10
1
10
2
10
3
10
4
10
6
10
8
Magnitude [Ohms]
10
1
10
2
10
3
Frequency (Hz)
-200
-100
0
100
200
Angle [Degrees]
Figure
8.
Experimental
HV
short-circuit
input
impedance
magnitude
(top)
and
phase
(bottom)
frequenc
y
responses
10
1
10
2
10
3
10
4
10
1
10
2
Transformation Ratio
10
1
10
2
10
3
10
4
Frequency (Hz)
-80
-60
-40
-20
0
20
Angle [Degrees]
Figure
9.
Experimental
HV
-L
V
v
oltage
transformation
ratio
magnitude
(top)
and
phase
(bottom)
frequenc
y
responses
From
the
magnitude
res
pon
s
e
of
the
transformation
ratio,
it
can
be
seen
that
at
lo
w
fr
equencies
up
to
approximately
10
kHz,
the
ratio
V
H
V
(
f
)
=V
LV
(
f
)
is
flat
at
a
v
alue
of
about
91
as
e
xpected.
As
the
frequenc
y
increases
be
yond
10
kHz,
resonant
peak
pairs
are
observ
ed
to
appear
at
frequencies
be
yond
20
kHz.
These
Char
acterizing
power
tr
ansformer
fr
equency
r
esponses
using
bipolar
...
(F
r
edric
k
Mwaniki)
Evaluation Warning : The document was created with Spire.PDF for Python.
2430
r
ISSN:
2088-8694
peaks
in
the
v
oltage
transformation
ratio
are
attrib
uted
to
the
resonance
modes
caused
by
t
h
e
interaction
of
v
arious
transformer
elements.
It
is
seen
that
the
v
oltage
transfer
at
20
kHz
is
about
twice
the
v
oltage
transfer
at
the
fundamental
frequenc
y
of
50
Hz.
The
presence
of
resonance
characteristics
in
the
wideband
transformation
ratio
such
as
the
one
sho
wn
in
Figure
9
presents
challenges
to
the
use
of
classical
distrib
ution
transformers
in
applications
that
are
prone
to
harmonic
distortion
and
high
frequenc
y
transient
signals
such
as
rene
w
able
ener
gy
plants.
3.2.
T
ransf
ormer
equi
v
alent
model
and
parameter
estimation
A
transformer
model
with
parameters
representing
its
ph
ysical
beha
viour
is
important
for
designers
[27].
Such
a
model
can
also
be
used
for
po
wer
system
simulation.
The
tes
ted
transformer
w
as
modelled
using
the
circuit
presented
in
Figure
10
which
sho
ws
the
lumped-parameter
equi
v
alent
circuit
model.
The
circuit
has
been
simplified
by
referring
the
secondary
circuit
parameters
to
the
primary
side
of
the
transformer
.
Figure
10.
Lumped
parameter
circuit
transformer
model
C
p
,
C
s
and
C
ps
denote
the
transformer
primary
winding,
secondary
winding
and
inter
-winding
capac-
itances
respecti
v
ely
.
R
s
and
R
e
denote
the
transformer
winding
resistance
and
core
losses
resistor
respecti
v
ely
while
L
s
and
L
m
denote
the
leakage
inductance
and
magnetizing
inductance
respecti
v
ely
.
The
parameter
estimation
process
is
summarised
in
the
flo
wchart
sho
wn
in
Figure
11.
S
t
a
r
t
P
o
p
u
l
a
t
e
t
r
a
n
sf
o
r
me
r
mo
d
e
l
w
i
t
h
i
n
i
t
i
a
l
p
a
r
a
m
e
t
e
r
s,
θ
i
n
i
t
i
a
l
Est
i
m
a
t
e
t
h
e
m
o
d
e
l
i
m
p
e
d
a
n
c
e
f
r
e
q
u
e
n
c
y
r
e
sp
o
n
se
M
i
n
i
m
u
m
c
r
i
t
e
r
i
o
n
?
O
b
t
a
i
n
t
r
a
n
sf
o
r
m
e
r
e
x
p
e
r
i
m
e
n
t
a
l
f
r
e
q
u
e
n
c
y
r
e
s
p
o
n
s
e
O
b
j
e
c
t
i
v
e
f
u
n
c
t
i
o
n
c
a
l
c
u
l
a
t
i
o
n
,
ϵ
(
θ
)
O
p
t
i
m
i
z
a
t
i
o
n
a
l
g
o
r
i
t
h
m
En
d
Z
s
m
(
ω
)
Z
s
e
(
ω
)
θ
N
o
Y
e
s
+
-
Figure
11.
Flo
wchart
of
the
transformer
parameter
estimation
procedure
used
in
this
study
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
12,
No.
4,
December
2021
:
2423
–
2434
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Po
w
Elec
&
Dri
Syst
ISSN:
2088-8694
r
2431
The
current
signal,
i
p
(
t
)
,
that
w
as
measured
in
the
L
V
open
circuit
input
impedance
e
xperiment
is
transformed
to
the
intermediate
v
oltage
le
v
el
using
the
ratio
1
T
R
.
The
transf
o
r
med
current
signal
is
applied
to
a
Simulink
model
of
the
circuit
in
Figure
10,
whose
parameters
ha
v
e
been
initialised
with
v
alues
based
on
a
priori
kno
wledge
of
the
transformer
under
in
v
estig
ation.
The
secondary
v
oltage
signal
at
intermediate
v
oltage
le
v
el
is
then
simulated.
The
transformed
current
and
the
simulated
v
oltage
signals
are
used
to
estimate
the
frequenc
y
response
of
the
secondary
open-circuit
input
impedance,
Z
se
(
!
)
.
The
measured
and
estimated
responses,
Z
sm
(
!
)
and
Z
se
(
!
)
,
were
compared
to
yield
an
objecti
v
e
function.
The
objecti
v
e
function
used
in
this
estimation
can
be
stated
as.
(
)
=
M
X
k
=1
j
Z
sm
(
!
k
)
Z
se
(
!
k
)
j
2
;
(7)
where
M
indicates
the
number
of
discrete
frequenc
y
points
in
the
compared
frequenc
y
responses
and
(
)
is
the
objecti
v
e
function
and
is
the
v
ector
of
parameters
to
be
estimated.
The
considered
equi
v
alent
circuit
of
a
single-phase
po
wer
transformer
leads
to
a
parameter
v
ector
that
can
be
represented
as.
=
C
p
;
R
s
;
L
s
;
C
s
;
R
e
;
L
m
;
C
ps
(8)
The
optimization
procedure
w
as
used
to
minimize
the
objecti
v
e
function
by
repetiti
v
ely
changing
the
trans-
former
model
parameters,
such
that
the
dif
ference
between
Z
sm
(
!
)
and
Z
se
(
!
)
w
as
at
its
minimum
v
alue.
The
fmincon
optimization
procedure
w
as
used
as
the
optimization
algorithm
to
reduce
the
objecti
v
e
function
in
this
in
v
estig
ation.
T
able
1
presented
the
estimated
parameter
v
alues
obtained
for
the
test
transformer
.
T
able
1.
Estimated
parameter
v
alues
for
the
transformer
model
in
Figure
10
P
arameter
V
alue
P
arameter
V
alue
P
arameter
V
alue
C
p
2
nF
C
ps
0.1375
nF
C
s
31.75
pF
R
s
1
k
R
e
1
M
L
s
0.32
H
L
m
1350
H
The
measured
and
estimated
frequenc
y
responses
of
the
L
V
open-circuit
input
impedance
are
sho
wn
in
Figure
12.
It
can
be
seen
that
the
considered
transformer
model
pro
vides
a
reasonable
good
prediction
of
the
frequenc
y
response
up
to
20
kHz,
despite
its
relati
v
e
simplicity
.
The
location
of
the
major
re
sonant
frequencies
of
the
response,
at
100
kHz
and
6.2
kHz,
and
the
amount
of
damping
associated
with
these
resonances
is
well
predicted.
Ho
we
v
er
,
the
order
of
the
model
is
too
lo
w
to
accurately
predict
the
minor
resonances
occurring
at
higher
frequencies.
The
transformer
model
can
be
impro
v
ed
by
taking
into
account
the
non-linear
phenomena
such
as
magnetic
saturation
and
frequenc
y-dependence
of
the
winding
resistances
due
to
skin
ef
fect.
The
inductance
of
the
measurement
cables
between
the
perturbation
source
and
the
transformer
terminals,
which
resonates
with
the
transformer
stray
capacitors,
also
need
to
be
tak
en
into
account
in
the
model.
Frequencies
at
which
the
resonance
occurs
can
be
confirmed
using
(9)
f
r
=
1
2
p
LC
(9)
where
f
r
is
the
resonant
frequenc
y
,
L
and
C
denote
the
inductance
and
capacitance
respecti
v
ely
.
F
or
instance,
the
firs
t
parallel
resonance
in
Figure
12
occurs
when
L
m
resonates
with
C
p
while
the
first
series
resonance
can
be
attrib
uted
to
L
s
resonating
with
C
p
.
T
o
v
alidate
the
obtained
parameters,
the
trans-
former
model
sho
wn
in
Figure
10
w
as
populated
with
the
parameters
in
T
able
1
and
simulated
by
perturbing
it
using
a
sine
sweep.
The
model
w
as
used
to
estimate
the
frequenc
y
response
of
the
transformation
ratio.
Figure
13
sho
ws
that
the
simulated
results
match
with
the
measured
results
in
Fi
gure
9
indicating
resonance
points
between
20-30
kHz.
Char
acterizing
power
tr
ansformer
fr
equency
r
esponses
using
bipolar
...
(F
r
edric
k
Mwaniki)
Evaluation Warning : The document was created with Spire.PDF for Python.
2432
r
ISSN:
2088-8694
10
1
10
2
10
3
10
4
Frequency [Hz]
10
3
10
4
10
5
10
6
Magnitude [Ohms]
Figure
12.
Measured
and
estimated
frequenc
y
responses
of
the
L
V
open-circuit
input
impedance;
1)
solid-measured,
2)
dashed-estimated
10
1
10
2
10
3
10
4
Frequency (Hz)
10
-1
10
0
10
1
Transformation Ratio
Figure
13.
T
ransformation
ratio
frequenc
y
response
simulated
using
the
estimated
parameters
4.
CONCLUSION
A
no
v
el
methodology
that
can
be
used
to
study
the
beha
viour
of
po
wer
transformers
not
only
at
the
fun-
damental
frequenc
y
b
ut
also
at
harmonic
frequencies
is
presented.
Accurate
frequenc
y
responses
are
obtained
by
perturbing
a
16
kV
A,
22
kV/240
V
po
wer
transformer
using
a
sequence
of
bipolar
pseudo-random
current
impulses.
The
time-and
frequenc
y-domain
properties
of
this
wideband
e
xcitation
signal
are
controllable
which
mak
es
it
particularly
suitable
in
transformer
identification
where
e
xcitation
ener
gy
should
be
concentrated
to-
w
ards
the
higher
frequenc
y
band
with
limited
lo
w
frequenc
y
components.
This
reduces
the
saturation
ef
fects
of
the
electromagnetic
system
and
is
a
major
adv
antage
of
the
proposed
perturbation
signal
for
transformer
iden-
tification,
compared
to
other
approaches
such
as
those
utilising
sine
sweep,
impulse
and
PRBS.
The
proposed
method
can
be
used
to
study
the
impact
of
harmonics
especially
on
transformers
connected
to
rene
w
able
ener
gy
sources.
Furthermore,
the
measured
frequenc
y
response
results
can
be
used
to
estimate
the
parameter
v
alues
of
a
wideband
lumped
parameter
equi
v
alent
circuit
model
of
the
transformer
.
As
future
w
ork,
the
accurac
y
of
the
estimated
wideband
transformer
parameters
can
be
impro
v
ed
by
considering
non-linear
phenomena
such
as
core
saturation
and
skin
ef
fect.
The
no
n-
linearity
will
result
in
additional
model
parameters
and
hence
a
higher
order
transformer
model.
REFERENCES
[1]
R.
Murray
and
J.
A.
de
K
ock,
“
A
proposed
method
for
e
v
aluating
the
frequenc
y
response
of
22
kV
outdoor
cur
-
rent
transformer
for
harmonic
measurements
in
rene
w
able
ener
gy
plant
applications,
”
2019
IEEE
10th
International
W
orkshop
on
Appli
ed
Measurements
for
Po
wer
Systems
(AMPS)
,
2019,
pp.
1-6,
doi:
10.1109/AMPS.2019.8897784.
[2]
S.
N.
Nik
olo
vski,
P
.
Z.
Maric,
and
L.
V
.
Majdandzic,
“Inte
gration
of
Solar
Po
wer
Plant
in
Distrib
ution
Netw
ork,
”
International
Journal
of
Electrical
and
Computer
Engineering
(IJECE)
,
v
ol.
5,
no.
4,
pp.
656–668,
Aug.
2015.
Int
J
Po
w
Elec
&
Dri
Syst,
V
ol.
12,
No.
4,
December
2021
:
2423
–
2434
Evaluation Warning : The document was created with Spire.PDF for Python.