Iranian Journal of Science & Technology, Transaction B, Vol. 28, No. B3
Printed in Islamic Republic of Iran, 2004
© Shiraz University
DERATING OF DISTRIBUTION TRANSFORMERS FOR NON-SINUSOIDAL
*
LOAD CURRENTS USING FINITE ELEMENT METHOD
J. FAIZ1,** M. B. B. SHARIFIAN2, S. A. FAKHERI3 AND E. SABET-MARZOOGHI1
1
Department of Electrical and Computer Engineering, University of Tehran, Tehran, I. R. of Iran
Email: jfaiz@ut.ac.ir
2
Department of Electrical and Engineering, University of Tabriz, Tabriz, I. R. of Iran
3
Tabriz Electrical Power Distribution Company, Tabriz, I. R. of Iran
Abstract– Transformers are normally designed and built for use at rated frequency and prefect
sinusoidal load current. Non-linear loads on a transformer lead to higher losses, early fatigue of
insulation, premature failure and reduction of the useful life of the transformer. To prevent these
problems, the rated capacity of a transformer, which supply non-linear loads must be reduced.
In this paper a 50-kVA three-phase distribution transformer is modeled using the finite element
(FE) method and its losses are estimated under rated frequency and load conditions, as well as under
non-linear loads. An equivalent rating of the transformer is estimated based on the harmonic loss
factor and is compared to the recommended standard rating. This comparison shows that the
estimation of derating of the transformer supplying non-linear loads using the standard
recommendations is acceptable, but slightly conservative.
Keywords – Transformer, nonlinear load, finite element method
1. INTRODUCTION
Nowadays, electricity distribution companies are concerned about assigning ratings to transformers for nonsinusoidal load current operation. The envisaged mass production of electric vehicles in future decades may
lead to increased non-linear domestic loads due to the large number of battery chargers. Uses of other nonlinear domestic loads such as variable speed thermal pumps are increasing. In addition, due to the widespread use of non-linear loads such as computers, variable speed drives in heating, ventilation and air
conditioning (HVAC) systems and electronic ballasts of fluorescent lamps, harmonic distortion is increasing
in the commercial user and services.
Additional load losses due to non-sinusoidal voltage yield higher hot spot temperatures in transformers.
The temperature rise of transformers due to non-sinusoidal load currents was discussed in the IEEE
transformer committee meeting, in March 1980. This meeting recommended providing a standard guide for
estimation of the loading capacity of the transformers with distorted currents. Kline presented a procedure in
which the eddy current losses vary with the square of the current and harmonic order [1]. Finally, an IEEE
C57-110 entitled “Recommended practice for establishing transformer capability when supplying nonsinusoidal load currents” has been published [2]. The aim in publishing this standard was to provide a
procedure for the determination of the capacity of a transformer under non-sinusoidal loads. This procedure
determines the reduction in the rated current for the harmonics present. Reducing the maximum apparent
power called derating has been proposed in many papers [3-10].
This paper reviews the effects of non-linear loads on a transformer, as well as the standard IEEE
procedure for derating a transformer that is under distorted currents. The equivalent capacity of a typical 50kVA transformer is then evaluated using analytical and finite element (FE) methods and the results are
compared.
∗
Received by the editors September 25, 2002 and in final revised form January 3, 2004
Corresponding author
∗∗
316
J. Faiz / et al.
2. EFFECT OF NON-LINEAR LOADS ON THE TRANSFORMER LOSSES
Transformer losses consist of no-load losses (or core losses) and load losses:
PT = PC + PL
(1)
The no-load losses are due to the core excitation. The harmonic currents passing through the
transformer leakage impedance and system impedance may distort the transformer output voltage slightly.
Experience shows that the temperature rise of the core is not the limiting factor in determining the
permissible current for non-linear loads. IEEE-C57-110 standard also ignores the increase in core losses due
to the non-linear loads. The load losses are
PL = I 2 R + PEC + PSL
(2)
2
The value of I R expresses the ohmic losses of the transformer windings. If the rms value of the load
current increases due to the harmonic components in the current, these losses also will increase. There is no
definite test procedure for determining eddy current and stray losses separately. However, the total losses
can be obtained by the impedance test. The eddy and stray losses may be obtained by subtracting the ohmic
losses from the total load losses.
With the rated eddy current losses known, the eddy current losses due to any non-sinusoidal load
current can be calculated [2]
PEC = PEC − R
hmax
2 2
∑ [I h / I R ] h
h =1
(3)
Stray losses or total eddy current losses plus other stray losses can be determined for any non-sinusoidal
load using a similar procedure. In the standard 1561, 1562 of UL laboratory, factor K is defined as follows:
∞
K _ Factor = ∑ [ I h / I R ] 2 h 2
h =1
(4)
K_Factor shows the influence of amplitude and frequency of the harmonic current upon the increase of
the eddy current losses of the transformer under non-sinusoidal loads. In the revised version of the standard
IEEE C57-110 [11], the factor of harmonic losses (FHL) has been defined as follows:
h = h max
FHL
P
= EC =
PEC − R
∑
I h2 h 2
h =1
h = h max
∑
h =1
I h2
(5)
If the numerator and denominator of Eq. 5 are divided by the rated rms current, the current values will
be in pu. The harmonic loss factors differ with K_Factor and the following equation holds:
h = hmax

K _ Factor =  ∑ I h2 ( Pu ) FHL
 h =1

(6)
If the rms current of the load is equal to the rated current of the transformer, the numerical value of factor K
will be equal to the harmonic loss factor.
3. EQUIVALENT POWER OF TRANSFORMER UNDER NON-LINEAR LOADS
The equivalent power of a transformer is the power that the transformer can supply to a non-linear load with
an arbitrary harmonic current content. This has been carried in such a way that the total load losses, load
losses in any winding and density of losses in the region with maximum eddy current losses, do not exceed
the designed values with a sinusoidal waveform at rated frequency and load. The equivalent power of a
transformer is determined by the maximum eddy current losses that can be dissipated in a region of the
winding.
Iranian Journal of Science & Technology, Volume 28, Number B3
June 2004
317
Derating of distribution transformers for non-sinusoidal…
There are two methods of determining the equivalent power of a transformer for non-linear load [12].
The first method is used where complete data concerning the density of the transformer losses are available.
The second method is less accurate and is employed where only the test data of the transformer exists.
Designers, therefore, generally use the first method, whilst users use the second method. The following
assumptions are made in the second method:
1. All stray losses are equal to the eddy current losses of winding.
2. Ohmic losses of any winding are distributed uniformly.
3. For all transformers having rated currents lower than 1000 A, the distribution of the eddy current losses
is 60% for the internal winding and 40% for the external winding. This distribution will be 70% and
30% for transformers having rated currents larger than 1000 A.
4. Maximum density of eddy current losses in any winding is equal to 400% of the average density of eddy
current losses.
Considering the above assumptions, the maximum density of the eddy current losses PEC-R in pu, for a
three-phase 20 kV/0.4 kV distribution transformer, under nominal conditions, is obtained as follows [2, 13]:
for S< 630 kVA:
PEC − R ( pu ) =
0.8PEC − R
I 22 R L
(7)
PEC − R ( pu ) =
2.8 PEC − R
3I 22 R L
(8)
for S> 630 kVA:
Maximum permissible non-sinusoidal load current of transformer (Imax) in pu is the current at which the
maximum loss density is equal to the designed loss density for the rated conditions. It is calculated as
follows [2]:
1/ 2






PLL−R( pu)
Imax( pu) = 


  h=hmax 2 2 h=hmax 2 


1+  ∑ fh h / ∑ fh .PEC−R( pu) 
h=1
 
  h=1

(9)
The I2R losses at the rated load are 1 pu, and it is assumed that all stray losses are equal to the eddy current
losses of the windings. Eq. (8) can be simplified as follows:
1/ 2
 1 + PEC− R ( pu) 
Imax( pu) = 

1 + (k / I ).PEC−R ( pu) 
(10)
4. ILLUSTRATIVE EXAMPLE
The equivalent kVA rating of a distribution transformer with specifications given in Table 1 is obtained for a
non-sinusoidal load current with an rms value of 1 pu and the harmonic distribution given in Table 2.
Table 1. Specifications of the proposed transformer
Rated power
HV voltage
LV voltage
HV current
LV current
Load losses
HV winding resistance
LV winding resistance
June 2004
50 kVA
20 kV
400 V
1.44 A
72 A
1250 W
121.5 Ω
0.03 Ω
Table 2. Distribution of load current harmonics
H
Ih (pu)
H
Ih (pu)
1
0.975
13
0.028
5
0.171
17
0.015
7
0.108
19
0.0098
11
0.044
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J. Faiz / et al.
a) Conventional method
The eddy current losses of the transformer can be obtained by subtracting the ohmic losses from the
load losses under nominal conditions as follows:
PEC-R=1250-3[1.442(121.5)+722(0.03)]=27.62W
The peak density of eddy current losses is calculated in pu using Eqs. (7-8) as follows:
PEC − R ( pu ) =
0.8 × 27.62
= 0.142 pu
155 .52
In order to compute K_Factor, Table 3 is obtained.
Table 3. Computation table for K_factors
h
1
5
7
11
13
17
18
Σ
Ih
0.978
0.171
0.111
0.044
0.080
0.015
0.0098
Ih2
0.957
0.929
0.012
0.002
0.00078
0.00023
0.0001
1.00
h2
1
25
49
121
169
289
361
Ih2h2
0.957
0.731
0.571
0.234
0.133
0.065
0.035
K=2.726
The permissible peak rms non-sinusoidal load current is determined using Eq. (10) as follows:
1/ 2


1 + 0.142
I max ( pu ) = 
= 0.9073

1 + ( 2.726 × 0.142 ) 
I max = 0 .9073 × 72 = 65 .32 A
and kVA can then be estimated as follows:
Equivalent kVA = 50 × 0.9073 = 45.36 kVA
The equivalent kVA is only an estimation of the total kVA of the transformer. If the harmonic content of the
load is changed considerably, this kVA must be re-evaluated.
b) Finite element method
The ANSYS 5.4 software for FE modeling of the proposed transformer is used. A two-dimensional
cross-section of the three-phase transformer is first introduced to the FE software. The transformer load
losses under linear and non-linear loads are then computed based on the circuit model of the transformer and
a short circuit test. As long as the size of the conductor is smaller than the local skin effect, the superposition
of the eddy currents and other losses is allowed [14, 15]. On the other hand, the flux level is lower than the
saturation level. So the principle of superposition is used to determine the load losses under non-sinusoidal
current. The total load losses are, thus, the sum of the transformer losses due to the different harmonic
current components of the non-linear load. For every component, the equivalent sinusoidal current of the
transformer and the transformer winding losses are evaluated. The FE software calculates these losses based
on the leakage flux and resistivity of the windings. Figures 1 and 2 show the two-dimensional cross-section
and meshed model of the transformer, respectively. Certainly finer mesh leads to a better result, but it takes a
longer computation time. Therefore, in the analysis for different harmonics a compromise has been made
between the accuracy and the computation time.
Iranian Journal of Science & Technology, Volume 28, Number B3
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Derating of distribution transformers for non-sinusoidal…
Fig. 1. Two-dimensional cross-section of the transformer
319
Fig. 2. Two-dimensional meshed model for the transformer
In order to obtain the FE results, the transformer is exited by the rms current of each harmonic. The
relevant flux distribution within the core and windings are then evaluated. The induced currents produced by
the distributed flux in each part of the core are determined next. The resistivity of the windings is introduced
to the software and the eddy current losses are then calculated based on the I2R equation.
If the transformer operates under rated frequency and load conditions, the load losses using the FE
method are as summarized in Table 4.
Table 4. Load losses using FE method
Load losses in
HV (W)
844.71
Load losses in
LV (W)
437.6
Total load
losses (W)
1282.3
Figure 3 shows the magnetic flux distribution in the transformer magnetic circuit. In this case, conditions are
very similar to the short circuit of the transformer (by harmonic or eddy current analysis in the ANSYS 5.4
software), and magnetization current and the flux amplitude are also very low. Due to the short circuit of the
transformer phases, considerable flux leaks through the windings. This represents a major factor in such
operations. For calculation of the losses under non-sinusoidal operation, the load losses for each current
harmonic is obtained and summarized in Table 5.
Fig. 3. Flux distribution of magnetic circuit of the transform
As indicated in Table 5, the total load losses for the given non-sinusoidal current are 1364.66 W. Thus
the increase in the losses compared to the rated frequency and load is
1364.66 − 1282.31 = 82.35 W
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J. Faiz / et al.
Table 5. Losses calculations for non-sinusoidal case
h
1
5
7
11
13
17
19
∑
Ih (pu)
0.9780
0.1710
0.1080
0.0440
0.0280
0.0150
0.0098
1
PH (W)
807.95
56.38
25.87
5.33
2.35
0.78
0.36
899.02
PL (W)
418.56
28.71
13.89
2.74
1.19
0.38
0.17
465.64
PL (W)
1226.51
86.09
39.76
8.07
3.54
1.16
0.53
1364.66
The winding eddy current losses for sinusoidal and non-sinusoidal conditions are obtained by
subtracting the winding dc losses from the total load losses. The total load current in both sinusoidal and
non-sinusoidal conditions are assumed 1 pu. The winding DC losses in sinusoidal conditions with rated
frequency and current (having total rms value of 1 pu and the given harmonic content) are as follows:
3[1.44 2 ×121.5 + 72 2 × 0.03] = 1222.38 W
The winding eddy current losses are
1282.31 − 1222.38 = 59.93 W
Eddy current losses of the winding at non-linear load current are as follows:
1364.66 − 1222.38 = 142.28 W
The factor of harmonic losses (FHL) based on Eq. (5) is equal to the ratio of winding total eddy current
losses at non-sinusoidal load current, and winding eddy current losses at the rated frequency and current
FHL =
142.28
= 2.37
59.93
Since the total rms load current is equal to the rated current of the transformer (1 pu), the numerical
value of the FHL is equal to the numerical value of K_Factor. Thus: K= FHL=2.37. The value obtained using
the FE method is compared to the analytical method in Table 6.
Table 6. Comparison of harmonic losses factor computed by FE and analytical methods
FHL
Analytical
2.726
FEM
2.37
A comparison of applying the two methods shows that the FE method predicts a smaller harmonic loss
factor than the analytical method. The reason is that in the analytical method the eddy current losses are
assumed proportional to the square of the harmonic orders. This represents a slightly conservative result.
The permissible peak rms current of the transformer based on Eq. (10) is


1 + 0.142
I max ( pu ) = 

1 + (2.37 )(0.142) 
1/ 2
= 0.9243
I max = 0.9243× 72 = 66.54 A
The equivalent kVA of the transformer is
0.9243 × 50 = 46.21 kVA
The results of two analytical and FE methods have been compared in Table 7. The comparison shows
that the predicted values using the analytical and FE methods are close, and although the recommended kVA
of the transformer based on the analytical method is slightly conservative, it is a reasonable estimation of the
derating of the transformer for non-linear loads.
Iranian Journal of Science & Technology, Volume 28, Number B3
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Derating of distribution transformers for non-sinusoidal…
321
Table 7. Comparison of calculated equivalent kVA transformer using analytical and FE methods
Imax (A)
kVA
Analytical
65.32
45.36
FEM
66.54
46.21
5. CONCLUSIONS
Effects of non-linear loads upon transformer losses based on the conventional method, have been studied for
a derating purpose. An FE method has been used to estimate the load losses of a three-phase 50 kVA
distribution transformer at sinusoidal and non-sinusoidal loads. The harmonic losses factor has been
computed in order to evaluate the equivalent kVA of the transformer for supplying non-linear loads. The
estimated kVA of the transformer has been compared to that calculated by the available standard method. A
summary of the results is as follows:
1. The most significant effect of non-linear loads upon the transformer is due to the winding eddy
current losses and load losses.
2. An increase in the transformer losses under non-sinusoidal loads leads to early fatigue of insulation,
premature failure and a reduction in the useful life of a transformer. The kVA of the transformer
must therefore be reduced under non-sinusoidal loads.
3. The assumption of an increase in the winding eddy current losses with the square of the frequency in
the analytical methods and the available standards is somehow less accurate.
4. The existing standard for derating a transformer provides a slightly conservative estimate of the
derating of the transformer under non-linear loads.
5. The FE method as a very precise method for transformer loss calculation under linear and non-linear
losses, and can be used for the final stage of the derating of a transformer.
Acknowledgment- The authors thank the Azarbaijan Regional Electricity Company, Tabriz, Iran for
financial support of the project.
NOMENCLATURES
Fh
H
I
I2
Ih
IR
PEC
PEC-R
PEC-R
PLL-R
PSL
PT, Pc, PL
R
R2
ratio of harmonic number h to fundamental component at nominal frequency
harmonic order
rms load current in A
rms current of LV winding in A
rms value of the hth harmonic current
rms value of sinusoidal load current at rated frequency in A
eddy current losses of winding in W
max. density of winding eddy current losses under rated conditions
rated eddy current losses of winding in W
max. density of load losses under rated conditions
stray losses in W
transformer, core and load losses in W
DC resistance in Ω
DC resistance of LV winding in Ω
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