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A Novel Approach for Evaluating Eddy Current Loss in Wind Turbine Generator StepUp Transformers
Bonginkosi Allen Thango*, Jacobus Andries Jordaan, Agha Francis Nnachi
Department of Electrical Engineering, Tshwane University of Technology, Emalahleni, 1034, South Africa
A R T I C L E I N F O
A B S T R A C T
Article history:
Received: 25 December, 2020
Accepted: 03 February, 2021
Online: 17 March, 2021
South Africa is aiming to achieve a generation capacity of about 11.4GW through wind
energy systems, which will contribute nearly 15.1% of the country’s energy mix by 2030.
Wind energy is one of the principal renewable energy determinations by the South African
government, owing to affluent heavy winds in vast and remote coastal areas. In the design
of newfangled Wind Turbine Generator StepUp (WTGSU) transformers, all feasible
measures are now being made to drive the optimal use of active components with the purpose
to raise frugality and to lighten the weight of these transformers. This undertaking is allied
with numerous challenges and one of them, which is particularly theoretical, is delineated
by the Eddy currents. Many times the transformer manufacturer and also the buyer will be
inclined to come to terms with some shortcomings triggered by Eddy currents. Still and all,
it is critical to understand where Eddy currents emanate and the amount of losses and
wherefore the temperature rise that may be produced in various active part components of
WTGSU transformers. This is the most ideal choice to inhibit potential failure of WTGSU
transformers arising from excessive heating especially under distorted harmonic load
conditions. In the current work, an extension of the author’s previous work, new analytical
formulae for the Eddy loss computation in WTGSU transformer winding conductors have
been explicitly derived, with appropriate contemplation of the fundamental and harmonic
load current. These formulae allow the distribution of the skin effect and computation of the
winding Eddy losses as a result of individual harmonics in the winding conductors. These
results can be utilized to enhance the design of WTGSU transformers and consequently
minimize the generation of hotspots in metallic structures.
Keywords:
Wind energy
Transformer
Eddy currents
Harmonics
Temperature rise
1. Introduction
South Africa remains engaged in diversifying the power mix
that will make it possible to reduce the subjection to coal power
generation. The ongoing decommissioning of various coal power
plants resulting from end of service life, can potentially make way
for a radically distinct power mix as opposed to the current
monopolization by coal power plants which have an installed
capacity of 40GW [1]. At the present day, South Africa’s
Department of Energy through the Renewable Energy Independent
Power Producer’s Procurement Programme (REI4P) has deployed
a total of 64 renewable energy facilities, of which 25 are wind
energy constituting of 961 wind turbines with a total generation
capacity of 2,1 GW [2]. Wind energy currently contributes 52% of
the country’s renewable energy capacity [3]. The wind energy
tariff is currently at about R0.62/kWh, in which is about 45% less
than the tariff for a coal plant. Each turbine generator in a wind
energy facility is furnished with a stepup transformer at the
bottom of the tower as shown in Figure 1 (a), which transforms its
output voltage to a desired medium voltage (MV) collector level.
In South Africa, these Wind Turbine Generator StepUp
(WTGSU) transformers are failing at an unnerving rate and utility
owners are confound to ascertain the potential causes for these
prevailing deficiencies. A typical WTGSU transformer failure
arising from an insulation failure which is attributable to a discto
disc short circuit failure to generate a spark that ignites the
transformer oil as shown in Figure 1 (b).
Field experience is indicating that these failures are triggered
by the use of regular distribution (RD) transformers that cannot
meet the required operational requirements of the wind energy
environment. With the objective to pledge future dependability of
wind energy in South Africa, WTGSU transformers must integrate
these requirements particularly with regard to a deformed loading
ASTESJ
ISSN: 24156698
*Corresponding Author: Bonginkosi Allen Thango, thangotech@gmail.com
Advances in Science, Technology and Engineering Systems Journal Vol. 6, No. 2, 488498 (2021)
www.astesj.com
Special Issue on Multidisciplinary Sciences and Engineering
https://dx.doi.org/10.25046/aj060256
B.A. Thango et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 6, No. 2, 488498 (2021)
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cycle, harmonics and distortion, derating, switching overvoltages
and stray gassing.
(a)
(b)
Figure 1: (a) WTGSU transformer during service; (b) Ignition of a
WTGSU transformer.
Nonlinear loads and sporadic wind turbine generators are
infamous for producing harmonic load currents and call for
WTGSU transformers to connect them safely to the electrical
network [45]. Transformers are commonly designed under the
presumption of sinusoidal loading conditions [67]. It is widely
known that distorted harmonic load currents accentuates the
service losses in transformer and ultimately the temperature rise
and generation of hotspots in metallic structures, resulting in
degradation of insulation materials and untimely failures as shown
in Figure 1 (b) [810].
Transformer windings are conceivably the most crucial part of
WTGSU transformers, electrically and mechanically, and they are
one of the major contribution of untimely failures in WTGSU
transformers [11]. The contribution of winding conductors to
WTGSU transformer failures is about 80%, which is a significant
portion among the failures caused by other transformer parts [11].
In view of the increasing proclivity of nonlinear loads in the power
system, the WTGSU transformers are more notably susceptible to
fail. As a result, to minimize power losses and excessive heating in
winding conductors, magnetic shielding [1213], and laminating
the core material [1415] has been proposed in the literature.
At tender stage, it is essential to evaluate the transformer
winding stray losses triggered by the EMFs. In this way, the
inception of new formulae is recommendable and endeavor to
enhance the computation and evaluation methods. A number of
studies have been undertaken recently which take these objectives
into account [1620]. Thus, analytical techniques have
demonstrated to be efficacious in computing the electromagnetic
service losses. In a broader sense, there are two commonly used
techniques in the transformer manufacturing industry to evaluate
the winding Eddy current losses: (i) direct computation of the EMF
using Maxwell’s equations in [2122] and (ii) use of Poynting’s
theorem in [2122]. Early attempts to compute the EMF using both
these methods are presented by authors in [23]. In the book
Transformer Engineering: Design and Practice [24], the authors
described these methods and how they can be embedded into
manufacturers internal design programs to compute the Eddy
currents. Design programs based on the Finite element Method to
compute the Eddy currents are witnessed in the publications [25
28].
Owing to the increasing renewable energy market and
applications of nonlinear loads, the evaluation of the winding stray
losses in WTGSU transformer necessitates the consideration of the
harmonic load current spectrum (HLCS), with appropriate
computation of the electromagnetic fields (EMFs). Consequently,
it is essential to study the impact of the HLCS on the winding stray
losses and associated temperature rise of active part components.
In the current work, an extension of the author’s previous work
[29], comprehensive analytical formulae to calculate the winding
Eddy losses in the presence of harmonics by taking into account of
the winding conductor dimensions, EMF and skin effect are
derived. The acquired formulae ensure that the examination of the
contribution of individual harmonics to the power losses is done.
Therefore, these are effective formulae that can produce rapid
results for electrical designers without the requirement of costly
and highclass computational resources.
The remainder of this work is structured as follows. Section 2
outlines the challenges and electrical design considerations of
WTGSU transformers. In Section 3, a winding Eddy loss formula
is derived under normal operating conditions by taking into
account the fundamental frequency, EMF, local flux density and
winding conductor material properties. In section 4, a winding
Eddy loss formula is derived under harmonic load conditions and
takes into account the winding conductors, EMFs, skin effect,
additionally, a new and simplified harmonic loss factor (HLF) is
derived to account for the additional losses that will be seen by the
WTGSU transformer during service. In Section 5, a case scenario
of a WTGSU transformer supplying a distorted harmonic load is
presented and corresponding losses are computed using proposed
formulae. These results are then compared with the method for
calculating transformer losses recommended by the IEC standard
and simulation results obtained by Finite Element Method. Finally,
the conclusion is presented in Section 6.
2. Electrical design concerns
A large portion of wind turbine transformers currently in
service are afflicted with numerous electrical concerns. These are
prevalently on account of the unpredictable wind speed round the
clock during their service. These inefficacies are the main drivers
of the premature failures of these transformers.
2.1. Distorted load cycle
Wind turbines depend to a large extent upon regional wind and
other weather patterns, and their annual average capacity factor is
close to 40% [30]. A majority of the power producers in the past
projected the operational loading would be more than 50%. The
comparatively light loading of WTGSU transformer present two
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distinct and operationally momentous challenges that must be
embedded into the design philosophy of WTGSU transformer.
The foremost challenge is that the WTGSU transformer’s
relatively low average capacity factor falsify purchasing
adjudication and invalidate classical economic models for regular
distribution transformers. On account of the ideational nature of
wind energy facility developments in South Africa, the
Engineering, Procurement and Construction (EPC) developer is
most frequently concerned with a cheaper initial transformer price.
The utility owner who takes over responsibility of the facility from
the EPC take a greater interest in the Total ownership cost (TOC)
of the WTGSU transformer over its designed service lifetime.
These costs include the initial purchase cost of the transformer, the
cost of service noload and load losses and routine maintenance
costs [3132]. The TOC models developed by authors in [3134]
do not take into consideration the relatively low average capacity
factor of WTGSU transformers. Bearing in mind that the loading
cycle of WTGSU transformers is so distinct from regular
distribution transformers, EPCs must be mindful of the obsolete
loss capitalization formulae when evaluating the TOC for WTGSU
transformers.
Another challenge is that the WTGSU transformers are
susceptible to regular thermal load cycling on account of the
sporadic characteristic of the wind turbines. The latter give rise to
the generation of hotspots in the transformer active part
components including core, windings, clamping structure, flitch
plates, tank walls et cetera. Recurrent thermal stresses drives the
immersion of nitrogen gas into the hot insulating oil and it is
emancipated as the insulating oil cools down. This phenomena
results in the production of combustible gaseous bubbles that can
potentially generate partial discharges and consequently damage
the cellulose insulation. Additionally, sporadic thermal cycling can
expedite the aging of electrical connections within the transformer
tank.
2.2. Harmonics and distortion
The transformer design principle is formed on the rationale of
generating an alternating EMF from a steady sinusoidal power
supply voltage to instigate the flow of current and voltage potential
in the winding conductors across that EMF.
During service, a steady sinusoidal wave shape is not practical.
In the power system, the voltage and current wave shapes are
distorted from the theoretical sinusoidal wave shapes. In practical
terms, the total harmonic distortion (THD) ranging between 1 and
2 is prevalent at the point of common coupling (PCC). The
application of nonlinear loads in the facility can further contribute
to distortion of the voltage and current wave shapes. These
accumulative distortions reiterate very cycle, embellishing peaks
that mount the voltage and current wave shapes and arise at distinct
frequencies from the fundamental frequency of 50 Hertz (Hz) as
shown in Figure 2.
The adverse effects of harmonics is that they trigger an upsurge
in the copper loss, winding Eddy loss and stray loss in other
metallic components. Eddy current and circulating current
generate overheating in the transformer active part components,
which must be treated by a sufficient cooling system design to
thermal aging and untimely failures of WTGSU transformers
during service. The wind turbine generator output, much like other
renewable energy sources is intermittent and will produce distorted
harmonic wave shapes. In this regard, the WTGSU transformer is
operated with solid state controls which furnish additional
harmonics and distortion.
Figure 2: Current wave shapes: With and without distortion
Electrical generator systems that employ electronic switching
devices and circuits also present specific problems for WTGSU
transformers. From a manufacturer perspective, design
philosophies to minimize the Eddy current loss to redress for
harmonic load currents are imperative. While harmonic filtering is
particularly not a function of a WTGSU transformer, magnetic
shields strategically positioned between the windings may act as
filters to mitigate the distorted harmonic currents into the collector
bus. As a result, magnetic shielding should be deemed compulsory
in the design of WTGSU transformers.
2.3. Switching OverVoltages
A major concern with vast, wind turbine arrays is the necessity
for connecting individual WTGSU transformers to the utility
collector bus, leading to lengthy runs of cables. In largescale wind
facilities with a cluster of wind turbine towers, the WTGSU
transformers may be positioned near the tower base as shown in
Figure 1 in order to minimize the costs of large and lengthy runs
of copper cables. Problems including excessive voltage drop,
cable losses and risk of ground faults may arise from lengthy cable
runs.
In view of the fact that wind energy facilities are in remote
areas, they are more susceptible to lightning surges. As a result,
surge arrestors must be mandatory for all WTGSU transformers.
Other things being equal, the biggest concern about switching
overvoltages may be that generated by the wind turbines. On a 24
hours period, depending on the intermittent nature of wind; the
wind turbines may be connected and disconnected online and
offline when the wind speed increase and decline respectively. The
latter may also occur when circuit breakers opens and closes the
WTGSU transformer from the collector bus circuit. The switching
phenomena by circuit breaker operation trigger transient over
voltages into the WTGSU transformers. This occurrence is
aggravated by the application of vacuum circuit breakers, which
have highspeed switching times. The fusion of the transient over
voltages and capacitance of the copper cables either HV or LV
WTGSU transformer side may result in stationary waves and
ringing that exceed the sinusoidal voltage amplitude as shown in
Figure 3.
B.A. Thango et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 6, No. 2, 488498 (2021)
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Figure 3: Typical switching surge [35]
Excessive transient overvoltages can trigger WTGSU
transformer insulation failures. The transient overvoltages as
shown in Figure 3 contains fast risetime and high voltage
amplitudes which clash with the resonant frequencies on the
windings and can introduce electrical stresses exceeding the
dielectric capability of the windings.
3. Winding Eddy loss under normal conditions
During service the winding conductor is immersed in an
alternating magnetic field and subsequently the Eddy current start
to flow. The energy loss dissipated as thermal energy on account
of the Eddy current bring forth the Winding Eddy losses. The
phenomena in rectangular winding conductors is demonstrated in
Figure 4.
Figure 4: Harmonic current profiles [36]
The derivation of the winding Eddy loss formulae is reliant on
factors suchlike frequency, local flux density and the winding
conductor material properties. Provided that a winding conductor
has the parameters as indicated in Figure 4 namely the, length,
height and thickness are denoted by L, h and τ respectively. Then
the alternating magnetic flux characteristic that is perpendicular to
the height and thickness can be described by the eq. (1).
(1)
Taking into consideration a closedpath in a clockwise
direction as shown in Figure 4, of thickness dx and distance x, a
presentation of a single conductor with an induced voltage in
closedpath is established. The area of this closedpath is expressed
as follows in Eq. (2).
(2)
The amount of magnetic flux passing through the unit area in
eq. (2) is expressed as follows in Eq. (3).
(3)
It follows then, that the amount of magnetic flux characteristic
that is perpendicular to the height and thickness in the closedpath
can be described by the eq. (4).
(4)
The induced voltage in the closedpath can be expressed in
accordance with the Faradays law of induction as expressed in Eq.
(5).
(5)
Equating the constant 4.44=√2 π yields the formula in Eq. (6)
(6)
Taking into account the amount of magnetic flux passing
through the closedpath unit area the eq. (6) yields the formula
expressed in Eq.(7).
(7)
Substituting the area of the closedpath in the induced voltage
formula in Eq. (7) yields eq. (8).
(8)
The resistance of the closedpath can then be expressed as
follows in Eq. (9).
(9)
The winding Eddy loss in the closedpath dx is then expressed
as shown in Eq. (10). Considering that the thickness of the closed
path is extremely small in relation to the height, then the distance
x I negligible.
(10)
Substituting eq. (7) into eq. (10) yields the expression in Eq.
(11).
(11)
Equating the constant √2 π=4.44 in Eq. (11) yields the formula
in Eq. (12).
(12)
Removing the square and expressing each terms to the
exponent 2 yields eq. (13).
B.A. Thango et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 6, No. 2, 488498 (2021)
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(13)
Integrating both sides of Eq. (13) and considering the closed
path then the Eddy loss can be derived using Eq. (14).
(14)
Subsequently, the solution can be expressed as follows in Eq.
(15).
(15)
Dividing the Eddy loss by hLτ equates to the area per unit
volume as expressed in eq. (16).
(16)
Finally, the Winding Eddy loss at fundamental frequency is
expressed as follows in Eq. (17).
(17)
Here,
The skin depth of penetration at fundamental frequency 50Hz
may be calculated as follows in Eq. (18).
(18)
The depth of penetration under HLC is then computed as
follows in Eq. (19).
(19)
In principle, the following conclusions can be drawn with
regards to the Eddy current loss in relation to the area per unit
volume of winding conductor based on this derivation:
• There is a direct proportion to the fundamental frequency,
conductor dimension and local flux density.
• There is an inversely proportional relationship to the
resistivity of the copper conductor.
4. Transformer winding Eddy loss weighting factor
4.1. Winding Eddy loss under harmonic current loading
The winding Eddy loss along a winding height (H) can be
expressed as follows in Eq. (20) [37].
(20)
The standardized winding Eddy loss is then expressed by
dividing eq. (18) with the rated copper loss to yield eq. (21).
(21)
Taking into consideration that the copper loss has a directly
proportional relationship with the root mean square (RMS) load
current under harmonic conditions [3840] then first part of eq.
(21) yields Eq. (22).
(22)
The contribution of the second part of Eq. (20) is as expressed
in Eq. (17). The magnetic flux leakage under harmonic conditions
has a directly proportional relationship to the RMS load current as
expressed in Eq. (23).
(23)
Under harmonic current conditions, the current I is expressed
in Eq. (24).
(24)
The harmonic current in Eq. (24) is expressed as. As such,
the Eq. (24) yields Eq. (25).
(25)
Here, the constant K is expressed in Eq. (17).
During service, when the transformer is operating at
fundamental frequency, the equivalent rated winding Eddy loss is
expressed as shown in Eq. (26).
(26)
Combining Eq. (25) and Eq. (26), the winding Eddy loss for a
transformer operating under harmonic current can be expressed as
shown in Eq. (27).
(27)
Subsequently, combining Eq. (22) and Eq. (27) yields Eq.
(28). This equation is formulated on the premise that induced
current is at fundamental load current of the transformer during
service.
(28)
Further simplification of Eq. (28) yields Eq. (29).
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(29)
The allowance made in order to take into account of the
harmonic load current in relation to their effects on transformer
thermal performance is then expressed as follows in Eq. (30).
(30)
In principle, the following conclusions can be drawn with
regards to the Eddy current loss under harmonic current loading:
• The winding Eddy loss is directly proportional to the RMS
load current and the harmonic order. This relationship is
similar in the computation of the stray loss in tank walls, core
clamps et cetera except that the harmonic order is expressed
to an exponent of 0.8 [35].
• The skin effect is not taken into consideration at high
harmonic order. The magnetic flux under these conditions
does not completely penetrate the surface of the winding
conductors.
In the next section, a weighted loss factor that considers the
skin effect is derived.
4.2. Winding Eddy loss under harmonic current loading
The Eddy current problem form part to the area of quasi
stationary electromagnetic effects of conductors. Insomuch that,
the displacement current enclosed by winding conductors may
incessantly be ignored in relation to the conductive current. This
is indeed the case even at high frequencies given that in practice
only winding conductors comprising high electric conductivity are
used. Eddy currents give rise to uneven dissemination of current
density in a studied cross sectional area of a conducting conductor.
This inherently leads to rise in joule heating as opposed to the state
produced by direct current (DC). The Eddy currents and related
uneven dissemination of the magnetic flux are known as the skin
effect. The rise in current density give rise to resistive heating as
opposed to the DC resistance as well as a reduction in the
inductance. In addressing the skin effect problem, this study
adopts the Maxwell equations in [38] and remodel these equations
to treat the quasistationary electromagnetic effects of conductors.
These equations are expressed as follows in Eq. (31) and Eq. (32).
(31)
(32)
On the above equations an addon of the Ohm’s law as
expressed in Eq. (33).
(33)
The electric field intensity must fulfil Eq. (34).
(34)
Similarly, the magnetic field must fulfil Eq. (35).
(35)
In investigating the dissemination of current, small but
sufficiently long conductors are considered. The location of the
conductors in a coordinate system as demonstrated in Figure 5. The
height and ratio of these conductors fulfil the condition
and the spatial dependence on the ycoordinate may be ignored
with insignificant error. When the current flows in the z
coordinate, the magnetic field has one component as described in
Eq. (36).
(36)
In the same order, the electric field intensity has a single
component as expressed in Eq. (37).
(37)
In the event that a winding conductor has the
conductivity
the permeability
and time
variant harmoniously at angular frequency
, by
elimination the ordinary secondorder differential Maxwell
equations for magnetic field intensity vector is expressed as
follows in Eq. (38).
(38)
Here, the constant k is expressed as shown in Eq. (39).
(39)
In investigating the skin effect in winding conductors with a
spasmodic magnetic flux, the rectangular copper conductors in
Figure 5 are considered.
Figure 5: Two winding conductors immersed in alternating EMFs
The spasmodic magnetic flux streams along in the zcoordinate
and the interaction of winding conductor 1 and 3 with respect to
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winding conductor 2 is investigated in terms of the winding Eddy
loss under harmonic conditions. The effect of the remainder of
winding conductors will be disregarded in order to presume that
the impact of winding conductor 1 and 3 will be completely offset
in the center winding conductor. Presupposing a harmonic time
response of the 2nd order differential equation in Eq. (38), with
the initial conditions necessitated for symmetry along the edges of
the winding conductors shall be valid the equation
and will yield the solutions in Eq. (40) and Eq. (41) as
expressed below. It is evident that the solution in this case scenario
is of nature.
(40)
(41)
Here, is the root mean square (RMS) magnetic field
strength. Additionally, the condition of is valid since the
impact along the edges of the winding conductors
is
negligible if is satisfied. The constants of integration from
Eq. (40) and Eq. (41) can then be expressed as follows in Eq. (42).
(42)
The solution of the magnetic field strength presume the format
expressed in Eq. (43).
(43)
By association, the magnetic flux density by Eq. (42) is
expressed as shown in Eq. (44).
(44)
Maxwell’s 1st equation in Eq. (33), can then be expressed as
follows in Eq. (45).
(45)
The constant k is introduced into the rationalization of the
equilateral hyperbola as shown in Eq. (46).
(46)
The skin depth of penetration a under harmonic conditions may
be calculated as follows in Eq. (47).
(47)
The skin depth at various harmonic orders is demonstrated in
Figure 6.
Figure 6: Depth of skin effect
The formulation of the current density is acquired and
expressed as shown in Eq. (48).
(48)
With the expansion of the equilateral hyperbola the modulus of
the current density is expressed shown in Eq. (49).
(49)
The dissemination of the magnetic flux density and current
density along the surface of the winding conductors is described
by Figure 7.
Figure 7: Distribution of the magnetic flux in the crosssection of a
winding conductor
The magnetic flux leakage impinging upon one conductor is
expressed by Eq. (50).
(50)
2
0
2
4
6
8
10
12
6 4 2 0 2 4 6
B
σ
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The peak value of the magnetic flux leakage is given by Eq.
(51).
(51)
Similarly, the magnetic flux density is expressed by Eq. (52).
(52)
By employing Eq. (49) and introducing Eq. (52), the formulation
of the current density is expressed as shown in Eq. (53).
(53)
The variable is introduced here to account for expression of
above as shown in Eq. (54).
(54)
Forthwith, the Eddy loss per unit of volume of a winding
conductor can be calculated using Eq. (17), where the weighted
factor will be employed as expressed in Eq. (55).
(55)
Here, the weighting function that takes into consideration the
skin effect in winding conductors under harmonic conditions is
expressed as follows in Eq. (56).
(56)
This function is presented graphically at different conductor
dimensions as shown in Figure 8.
Figure 8: The function and
It is evident from Figure 8, that at higher harmonic order, the
magnetic flux leakage does not completely penetrate upon the
surface of the conductors. The latter addresses the short coming of
Eq. (30).
5. Materials and Methods
In this section, a WTGSU transformer rated at 5000 KVA,
11000/33000 volts, 3phase, 50 cycles, oilimmersed, naturally
cooled (ONAN), core type, double wound with copper conductor
and fitted with on load tap changer is studied. The temperature rise
in oil and winding conductors is 50 ºC and 55 ºC respectively. The
rated losses under consideration are at 75ºC on normal Tap
position (in Watts) as shown in Table 1.
Table 1: Rated transformer losses
Type of loss
Rated Losses (Watts)
No load
4 500
Copper
25 000
Winding Eddy
938
Other Stray
1 253
Total
31 691
The studied transformer has a full load low voltage (LV)
winding current of 262,43A with a mean winding Eddy loss at the
highest loss ratio under rated conditions at 15% of the Copper
losses.
The load that will been seen by the studied WTGSU
transformer is described as shown in Figure 9. It is preferential that
this harmonic load spectrum (HLS) to which the WTGSU will be
susceptible be indicated to the transformer manufacturer during
bid stage. A precise evaluation for the appropriate sizing of
WTGSU transformers can only be designed by estimating the
particular HLS.
Figure 9: Harmonic Load Spectrum
The resultant HLFs and service losses are computed and
discussed in the next section.
6. Results
6.1. Computation of Harmonic Loss Factor
It is handy to specify a single digit that may have recourse to
evaluate the capability of the studied WTGSU transformer in
facilitating power to the supplied harmonic spectrum above. This
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6
F_H
בּ
F_H 3/z
0
0.2
0.4
0.6
0.8
1
1357911 13 15 17 19
Harmonic spectrum (p.u)
Harmonic Order
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value viz. the harmonic loss factor (HLF) illustrates the effective
RMS heating on account of the harmonic spectrum. The
calculation of the HLF is tabulated in Table 2. The HLF by FEM
is computed using the procedure described in [38].
Table 2: Computation of HLFs.
h
IEC
Proposed
FEM
1
0,97
0,941
0,941
0,941
0,941
3
0,37
0,137
1,232
1,090
1,213
5
0,35
0,123
3,063
2,569
2,924
7
0,1
0,010
0,490
0,100
0,448
9
0,028
0,001
0,064
0,063
0,055
11
0,11
0,012
1,464
1,455
1,184
13
0,071
0,005
0,852
0,807
0,640
15
0,026
0,001
0,152
0,135
0,106
17
0,057
0,003
0,939
0,925
0,600
19
0,047
0,002
0,797
0,774
0,468
Total
1,234
9,994
8,860
8,576
1.111
HLF
8,10
7,177
6,948
The ratio of the harmonic load current for each respective
method and the RMS current (p.u) yields the HLFs as indicated
previously.
6.2. Computation of Winding Eddy loss under harmonic current
loading
The evaluation of the total service losses as a result of the HLC
using the IEEE Std. C57.1102018 is tabulated as shown in Table
3.
Table 3: Computation of total service losses: IEEE Std. C57.110
2018.
Type of
loss
Rated
Losses
(W)
Load
Losses
(W)
HLF
Service Losses
(W)
No load
4 500
4 500

4 500
Copper
25 000
30 859

30 859
Winding
Eddy
938
1158
8,10
9 372
Other
Stray
1 253
1547
1,58
2 442
Total
31 691
38 064
47 173
Under the supplied HLS, the service losses are witnessed to
have a surge of about 33% and 19% from the rated and load losses
respectively. The computation of the HLF for other stray loss in
this work is similar to that of the winding Eddy loss as shown in
Eq. (27), however these losses generated by metallic components
such as tank walls, core clamp, windings, core, flitch plate et cetera
are proportional to the square of the harmonic load current and the
harmonic order to the exponent 0.8 as stated in [37]. The peak local
load loss ratio for the HLC is computed as follows:
(57)
Given that the mean winding Eddy loss at the highest loss ratio
under rated conditions is 15% of the Copper losses, then the load
loss will be 1.15 p.u as shown below. The maximum permissible
harmonic load current with the supplied harmonic spectrum is as
follows:
(58)
Consequently, with the supplied harmonic spectrum the
studied WTGSU transformer capability is about 261.048A.
Table 4: Computation of total service losses: Proposed method
Type of
loss
Rated
Losses
(W)
Load
Losses
(W)
HLF
Service
Losses (W)
No load
4 500
4 500

4 500
Copper
25 000
30 859

30 859
Winding
Eddy
938
1158
7,177
8 308
Other
Stray
1 253
1547
1,58
2 442
Total
31 691
38 064
46 110
Under the supplied HLS, the service losses are witnessed to
have a surge of about 31% and 17% from the rated and load losses
respectively. The peak local load loss ratio for the HLC is
computed as follows:
By applying proposed method, the maximum permissible
harmonic load current with the supplied harmonic spectrum is as
follows:
Consequently, with the supplied harmonic spectrum the
studied WTGSU transformer capability is about 259,85A. The
evaluation of the total service losses as a result of the HLC using
FEM [38] is tabulated as shown in Table 5.
Table 5: Computation of total service losses: FEM
Type of loss
Rated
Losses
(W)
Load
Losses
(W)
HLF
Service
Losses
(W)
No load
4 500
4 500

4 500
Copper
25 000
30 859

30 859
Winding
Eddy
938
1158
6,948
8 043
B.A. Thango et al. / Advances in Science, Technology and Engineering Systems Journal Vol. 6, No. 2, 488498 (2021)
www.astesj.com 497
Other Stray
1 253
1547
1,58
2 442
Total
31 691
38 064
45 844
Under the supplied HLS, the service losses are witnessed to
have a surge of about 31% and 17% from the rated and load losses
respectively. The peak local load loss ratio for the HLC is
computed as follows:
Based upon the FEM procedure [38], the maximum
permissible harmonic load current with the supplied harmonic
spectrum is as follows:
Consequently, with the supplied harmonic spectrum the
studied WTGSU transformer capability is about 258,70A.
7. Conclusions
In the current work, an extension of the author’s previous work
[29], new analytical formulae for the Eddy loss computation in
transformer winding conductors have been explicitly derived, with
appropriate contemplation of the fundamental and harmonic load
current. These formulae allow the distribution of the skin effect
and computation of the winding Eddy losses as a result of
individual harmonics in the winding conductors. The rectangular
winding conductors considered are based on real transformer
geometries and represents the configuration of the conductors
carrying harmonic load current.
The new formulae were triumphantly corroborated by
comparing their performance with computationally intensive FEM
simulations, authenticating their adequacy and efficacy. On that
account, our findings are handy for the transformer manufacturing
industry, where the transformer anatomy and design necessitate
specific performance and inexpensive computational resources.
Additionally, it has been demonstrated that the harmonic load
currents minimizes the depth of skin effect, lead to an increase in
the copper losses and winding stray losses and, thereby,
consideration must be given for proper evaluation and design of fir
for purpose transformers capable of facilitating wind power. These
formulae can be applied in the transformer design systems for
calculating the effects of harmonics on the transformer active part
components. These results can be utilized to enhance the design of
transformer s and consequently minimize the generation of
hotspots in metallic structures. In this way, the developed
formulae are a critical furtherance to the existing methodologies
employed by manufactures.
The method herein provides a fundamental basis for
subsequent development such as analysis of thermal performance
of transformers in wind power technologies.
Future work involves the development of TOC model for
WTGSU transformers to evaluate their economic life by
considering the intermittent nature of wind turbine generators.
Conflict of Interest
The authors declare no conflict of interest.
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