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Annals of the „Constantin Brancusi” University of Targu Jiu, Engineering Series , No. 2/2017
36
POWER TRANSFORMERS VOLTAGE OPTIMIZATION IN ORDER TO
MINIMIZE ACTIVE POWER AND ENERGY LOSSES AND MAXIMIZE
EFFICIENCY
Georgi Tsonev Velev, Technical University of Gabrovo, Gabrovo, BULGARIA
Krasimir Marinov Ivanov, Technical University of Gabrovo, Gabrovo, BULGARIA
Adriana Foanene, University „Constantin Brȃncuşi” of Tg Jiu, ROMANIA
ABSTRACT: The paper inhere derives mathematical models for optimization of the operating
voltage of distribution power transformers in order to minimize their active power losses and
maximize their efficiency in variable loading conditions. A comparative study has been made
between oil and dry-type of power transformers. The optimization models, that are built, are
implemented in the Microsoft Excel’s module “Solver” in order to process the calculations and
present them in graphical appearance. Appropriate conclusions are made in regard with the
optimal voltage of the different power transformers in dependence on the secondary loading.
KEY WORDS: distribution power transformers, transformer power losses, transformer efficiency,
voltage optimization.
1.INTRODUCTION In the process of electrical energy
transformation, a part of the energy has been
wasted as active power loss in the power
transformer. Losses of active power and
energy in power transformers are divided into
two types – losses in the transformer’s
windings (copper losses), which are
dependent on the transformer’s loading; and
magnetic losses (iron losses of hysteresis and
eddy currents), which are independent on the
load.
Up to now many researchers have studied
the influence of the load and the power factor
upon transformers’ active power and energy
losses, neglecting the operating voltage
impact [1, 2, 3].
Operating voltage has been introduced in
the mathematical expressions for calculation
of transformer’s active power losses and
efficiency, in order to reflect its influence.
Also, operating voltage is used as an
optimization variable in the optimization
procedure for minimizing the losses and
maximizing the efficiency of the transformer.
2.ACTIVE POWER LOSSES AND
EFFICIENCY OF POWER
TRANSFORMERS, TAKING
ACCOUNT OF THE OPERATING
VOLTAGE
According to the theory, the equivalent
electrical circuit of a 2-winding distribution
power transformer could be presented in two
ways (fig.1) [4, 5].
a)
0 0S P j Q
ТR ТX
ТG ТB
ТRТX
b) Figure 1. Equivalent circuits of a power
transformer, where:
Annals of the „Constantin Brancusi” University of Targu Jiu, Engineering Series , No. 2/2017
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ТR - series resistance , ; ТX - series
reactance , ; ТG - parallel conductance , S ;
ТB - parallel susceptance , S ; 0S – complex
power losses, VA; 0P - No-load power loss,
W; Q - reactive power loss , VAr .
All parameters of the equivalent circuit can
be calculated if the specification data
parameters are known, normally they are
provided by the transformer’s manufacturer:
rS - rated apparent transformer power
(capacity), kVA;
rU - rated voltage (for each winding), kV;
Ku - short circuit voltage, %;
0I - no-load current, %;
0P - no-load power losses, kW;
kP - short-circuit power losses, kW;
According to [4], the active power losses
of a transformer are: 2
2
2 T T
SP R U G
U (1)
,where:
U - Operating voltage at the primary winding,
V; S - Apparent power of transformer’s
load, VA;
Series resistance ТR and parallel
conductance ТG of a transformer are
calculated by well-known expressions [4, 5]: 2
2, k r
Т
r
P UR
S
(2)
0
2, SТ
r
PG
U
(3)
Expressions (2) and (3) are substituted in
(1): 2
2 0
2 2 2
k r
r r
P U PSP U
U S U
22 2
02 2 2
rk
r r
US UP P P
S U U
2 22
0r
k
r r
US UP P P
S U U
(4)
The following substitutions are made in (4):
Transformer loading factor:
L
r
Sk
S (5)
Operating voltage in per-unit values:
[pu]r
UU
U (6)
In that way expression (4) will be
transformed finally into:
2 2
02
1L kP k P U pu P
U pu (7)
Equation (7) gives the dependence of the
active power losses of a transformer P on
the loading Lk and the operating
voltage U pu .
If we consider the efficiency of a power
transformer, it is given by the expression:
2 2
1 2
P P
P P P
(8)
, where:
2P - active power at the transformer’s
secondary;
1P - active power at the transformer’s primary
side;
The active power of the secondary coil
could be expressed using the apparent power
of the load S and its power factor cos :
2 cos cosr
r
SP S S
S
2 cosL rP k S (9)
But we already have an expression for the
power losses of a transformer, according (7),
so expressions (7) and (9) are substituted in
(8):
2
2
2 2
02
cos
1cos
L r
L r L k
P
P P
k S
k S k P U pu PU pu
(10)
Equation (10) gives the dependence of the
efficiency of a transformer on the loading
Lk , the operating voltage U pu and the
power factor of the load cos .
3.MATHEMATICAL MODELS
FOR MINIMIZING THE POWER
LOSSES AND MAXIMIZING
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TRANSFORMER’S EFFICIENCY
BY OPTIMIZING OPERATING
VOLTAGE
A. Optimization model for
minimizing the power losses of a
transformer An optimization model has been compiled
in order to minimize the active power losses
of a transformer, by varying transformer’s
loading Lk in the range (0,1 - 1,3) pu in equal
steps of 0,05 pu. For optimization variable
has been chosen the operating voltage at the
transformer’s primary U pu . The deviation
of operating voltage has been limited by the
range (0,9 - 1,1) pu, according to the
requirements for voltage quality of EU [9].
The mathematical model of this
optimization problem is listed below:
Objective function, according to (7) is:
2 2
02
1MinL kP k P U pu P
U pu (11)
Constraints defined for the optimization
variable i.e. the operating voltage U pu
are:
0,9 1,1U pu
B. Optimization model for
maximizing efficiency of a
transformer An optimization model has been compiled
in order to maximize the efficiency of a
transformer, by varying transformer’s loading
Lk in the range (0,1 - 1,3) pu and the power
factor of the load cos in the range (0,7 - 1)
in equal steps of 0,05 pu. For optimization
variable has been chosen the operating
voltage at the transformer’s primary U pu .
The deviation of operating voltage has been
limited by the range (0,9 - 1,1) pu, according
to the requirement for voltage quality of EU
[9].
The mathematical model of this
optimization problem is listed below:
Objective function, according to (10) is:
2 2
02
cosMax
1cos
L r
L r L k
k S
k S k P U pu PU pu
(12)
Constraints defined for the optimization
variable i.e. the operating voltage U pu
are:
0,9 1,1U pu
4.CASE STUDIES IN ORDER TO
OPTIMIZE POWER TRANSFOR-
MERS’ OPERATING VOLTAGE
BY MINIMIZING LOSSES AND
MAXIMIZING EFFICIENCY In regard with the two optimization
problems (A and B) defined in the above
chapter, a case study has been performed for
oil-immersed and dry-type distribution power
transformers 20/0,4 kV, with four rated
capacities along the gamma. The
manufacturer specification parameters of
these transformers are given in table 1,
according to the EU “Green” requirements for
production of energy efficient power
transformers, stated in [6, 7, 8]:
Table 1
rS Oil-immersed Dry-type
0P kP Ku 0P kP Ku
kVA kW kW % kW kW %
250 0,3 3,2 4,5 0,52 3,8 6
630 0,6 6,5 6 1,1 7,6 6
1600 1,2 14 6 2,2 13 6
2500 1,75 22 6,5 3,1 19 6
The performed multi-case study in regard
with the two predefined optimization models,
as stated above, have been fulfilled in the
environment of the software product
Microsoft Excel, using its add-on for solving
of optimization problems “Solver”.
In regard with minimizing transformer’s
active power losses by optimization of its
operating voltage, the following results have
been obtained and presented graphically in
fig. 2 for oil-immersed transformers and in
fig. 3 for dry-type transformers.
Analyzing figure 2, for oil-immersed
distribution transformers, minimal power
losses are obtained in the following
conditions:
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Transformers with loading under 23% of
their rated capacity should operate at
minimal allowable voltage;
In the loading range of 23% - 43%,
operating voltage must be regulated to
increase linearly with load.
For loadings over 43 %, oil-immersed
transformers must be operated at maximal
allowable voltage.
0
0,2
0,4
0,6
0,8
1
1,2
Op
era
tin
g v
olt
ag
e
Loading factor
Sr=250kVA(oil)
Sr=630kVA(oil)
Sr=1600kVA(oil)
Sr=2500kVA(oil)
0,23 0,43Lk
Lk
, puU
Figure 2. Optimal operating voltage of oil-
immersed distribution transformers in p.u.
units as a function of transformer’s loading in
order to maintain minimal power losses.
0
0,2
0,4
0,6
0,8
1
1,2
Op
era
tin
g v
olt
ag
e
Loading factor
Sr=250kVA(dry)
Sr=630kVA(dry)
Sr=1600kVA(dry)
Sr=2500kVA(dry)
0,33 0,53Lk
Lk
, puU
Figure 3. Optimal operating voltage of dry-
type distribution transformers in p.u. units as
a function of transformer’s loading in order to
maintain minimal power losses.
Analyzing figure 3, for dry-type
distribution transformers, minimal power
losses are obtained in the following
conditions:
Transformers with loading under 33% of
their rated capacity should operate at
minimal allowable voltage;
In the loading range of 33% - 53%,
operating voltage must be regulated to
increase linearly with load.
For loadings over 53 %, dry-type
transformers must be operated at maximal
allowable voltage.
In regard with maximizing transformer’s
efficiency by optimization of its operating
voltage, the following results have been
obtained and presented graphically for oil-
immersed transformers in fig. 4 and fig. 5 and
for dry-type distribution transformers in fig.
6.
0,97
0,975
0,98
0,985
0,99
0,995
Eff
icie
ncy
Loading factor
Sr=250kVA(oil, PF=0,7)
Sr=250kVA(oil, PF=0,8)
Sr=250kVA(oil, PF=0,9)
Sr=250kVA(oil, PF=1)
Lk
Loading range with
maximum efficiency
Figure 4. Optimal loading range for
maintaining of maximal efficiency of oil-
immersed distribution transformers with rated
capacity of 250 kVA, varying load’s power
factor.
From figure 4 it is clearly obvious that a
certain power transform will have maximum
efficiency in a specific loading range, which
is independent on the load’s power factor.
Efficiency increases along with load’s power
factor but the trend of the curve remains
unchanged for the different cases.
0,97
0,975
0,98
0,985
0,99
0,995
Eff
icie
ncy
Loading factor
Sr=250kVA(oil)
Sr=630kVA(oil)
Sr=1600kVA(oil)
Sr=2500kVA(oil)
Lk
Optimal loading range
0,25 0,45Lk
0,8PF
Figure 5. Optimal loading range of oil-
immersed distribution transformers, having
different capacities, in order to maintain
maximal efficiency.
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0,965
0,97
0,975
0,98
0,985
0,99
0,995Ef
fici
en
cy
Loading factor
Sr=250kVA(dry)
Sr=630kVA(dry)
Sr=1600kVA(dry)
Sr=2500kVA(dry)
Lk
Optimal loading range
0,30 0,55Lk
0,8PF
Figure 6. Optimal loading range of dry-type
distribution transformers, having different
capacities, in order to maintain maximal
efficiency.
As seen in figure 5, oil-immersed power
transformers will have maximal efficiency in
the loading range between 25 and 45 % of
their capacity, independently on their power
rating.
Considering the dry-type distribution
transformers (fig. 6), they will have maximal
efficiency in the loading range between 30
and 55 % of their capacity, independently on
their power rating. It is obvious that dry-type
transformers have optimal loading range,
wider with almost 5%, compared to oil-
immersed transformers.
0,97
0,975
0,98
0,985
0,99
0,995
Ma
xim
al
eff
icie
ncy
Loading factor
Sr=630kVA(oil, PF=0,7)Sr=630kVA(dry, PF=0,7)
Lk
Efficiency difference 0,4%
a)
0,976
0,978
0,98
0,982
0,984
0,986
0,988
0,99
0,992
0,994
Ma
xim
al
eff
icie
ncy
Loading factor
Sr=630kVA(oil, PF=0,9)Sr=630kVA(dry, PF=0,9)
Lk
Efficiency difference 0,4%
b)
0,98
0,982
0,984
0,986
0,988
0,99
0,992
0,994
Ma
xim
al
eff
icie
ncy
Loading factor
Sr=2500kVA(oil, PF=0,7)Sr=2500kVA(dry, PF=0,7)
Lk
Efficiency difference 0,2%
c)
0,984
0,986
0,988
0,99
0,992
0,994
0,996
Ma
xim
al
eff
icie
ncy
Loading factor
Sr=2500kVA(oil, PF=0,9)Sr=2500kVA(dry, PF=0,9)
Lk
Efficiency difference 0,2%
d)
Figure 7. Maximal efficiency difference
between oil-immersed and dry-type
distribution transformers.
a) 630 kVArS , cos 0,7 ; b) 630 kVArS ,
cos 0,9 ; c) 2500 kVArS , cos 0,7 ; d)
2500 kVArS , cos 0,9 ;
If contemporary oil-immersed and dry-type
power transformers are compared in regard
with their efficiency, the following
conclusions can be made (fig.7):
Oil-immersed transformers have always
higher energy efficiency than dry-type
transformers;
In the low capacity range gamma, the
maximal efficiency difference is around
0,4 – 0,5 % in favor of oil transformers,
independently of the power factor(fig. 7 a,
b);
In the high capacity range, the maximal
efficiency difference decreases and is
around 0,2 – 0,3 %, again in favor of oil
transformers;
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CONCLUSIONS
Dry-type distribution transformers have
wide application and are preferred more and
more in urban areas, small companies, public
buildings etc., because of their fire and
explosion proof safety. Also they can be
installed directly inside the buildings saving
space and additional expenses.
However, dry transformers have higher
noise levels and operating temperature, higher
active power and energy losses, and lower
efficiency than oil-immersed transformers.
All this causes increased long-term expenses
for covering of their losses.
In order to decrease power losses and
increase transformers’ efficiency, the
following conclusions can be made in regard
with the optimal operating parameters of oil-
immersed and dry-type distribution
transformers:
A. By regulation of operating voltage at
the primary of the transformer:
Oil transformers with loading under 23%
of their rated capacity should operate at
minimal allowable voltage. In the loading
range of 23% - 43%, operating voltage
must be regulated to increase linearly with
load. For loadings over 43 %, oil-
immersed transformers must be operated at
maximal allowable voltage.
Dry-type transformers with loading under
33% of their rated capacity should operate
at minimal allowable voltage. In the
loading range of 33% - 53%, operating
voltage must be regulated to increase
linearly with load. For loadings over 53 %,
dry-type transformers must be operated at
maximal allowable voltage.
Operating voltage can be regulated
successfully in industrial electrical
distribution networks mostly and is hardly
applicable for public distribution power
networks;
B. By choosing the optimal rated
capacity and the correct type of power
transformer:
Oil-immersed transformers have always
better energy efficiency than dry-type
transformers;
Oil-immersed power transformers have
maximal efficiency in the loading range
between 25 and 45 % of their capacity,
independently on their power rating;
Dry-type distribution transformers have
maximal efficiency in the loading range
between 30 and 55 % of their capacity,
independently on their power rating.
Dry-type transformers have optimal
loading range, wider with almost 5%,
compared to oil-immersed transformers.
In the low capacity range gamma, the
maximal efficiency difference is around
0,4 – 0,5 % in favor of oil transformers,
independently of the power factor;
In the high capacity range, the maximal
efficiency difference decreases and is
around 0,2 – 0,3 %, again in favor of oil
transformers;
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