Electr Eng (2007) 89: 301–309
DOI 10.1007/s00202-006-0008-4
O R I G I NA L PA P E R
Ebrahim Rahimpour · Davood Azizian
Analysis of temperature distribution in cast-resin dry-type
transformers
Received: 17 September 2005 / Accepted: 16 January 2006 / Published online: 11 April 2006
© Springer-Verlag 2006
Abstract Non-flammable characteristic of dry-type castresin transformers make them suitable for residential and hospital usages. However, because of resin’s property, thermal
behavior of these transformers is undesirable, so it is important to analyze their thermal behavior. Temperature distribution of cast-resin transformers is mathematically modeled in
this paper. The solution of the model was carried out successfully by finite difference method. In order to validate the
model, the simulation results were compared with the experimental data measured from an 800 kVA transformer. Finally,
the influences of some construction parameters and environmental conditions on temperature distribution of cast-resin
transformers were discussed.
Keywords Thermal modeling · Dry-type transformer ·
Cast-resin · Finite difference
1 Introduction
The transformer is a key component in the transmission and
distribution system of electrical energy. In some cases, as
in hospitals and residential areas, transformers must be protected against flame and explosion. Therefore, specialists
and designers tend towards new kinds of transformers. Thus,
non-flammable transformers have been built [1]. These transformers employ non-flammable insulations in their structure.
One of such insulations is askarel, which includes polychlorinated biphenyls (PCBs). However, in the mid-1970s, it was
determined that PCBs were not environmentally acceptable.
E. Rahimpour (B)
Electrical Engineering Department, Faculty of Engineering,
University of Zanjan,
Daneshgah road (Tabriz road km 5),
P. O. Box 45195-313, Zanjan, Iran
E-mail: ebrahim.rahimpour@web.de
Tel.: +98-241-5152678
D. Azizian
Iran Transformer Research Institute (ITRI),
Zanjan, Iran
Therefore, the usage of askarel in existing transformers has
gradually been phased out [2,3]. As a consequence, some
kind of new transformers have been developed among which
dry-type transformers are a common type.
One of the most common kinds of dry-type transformers
is cast-resin type which is investigated in this paper. These
transformers are protected against flammability and moisture attraction but they lack some advantages of oil-filled
transformers. Oil in the oil-filled transformers is not only
an electrical insulation but also works as a cooling matter,
while the cast-resin transformers lack any fluid for cooling.
In addition, resin used by these transformers is class F insulation with insulation system temperature of 155◦ C [4]. Thus,
analyzing the temperature behavior in such transformers is
crucially important.
A lot of papers have investigated experimental results of
temperature distribution in some kinds of dry-type transformers [5–8] and even Pierce in [9] has modeled the temperature
behavior of ventilated dry-type transformers. However, no
mathematical model has been presented for the cast-resin
transformers. Hence, thermal behavior of these transformers
is modeled in this paper. The modeling results are compared
with the experimental data collected from an 800 kVA and
20/0.4 kV transformer and accuracy of the model is validated.
Moreover, the influences of some construction parameters
and environmental conditions on temperature distribution are
discussed.
2 Heat transfer in cast-resin transformer
A schematic view of cast-resin transformer is shown in Fig. 1.
In this figure, LV and HV denote low voltage and high voltage, respectively. Regarding the cylindrical structure of the
transformer, temperature in φ direction can be assumed to be
symmetric, so the heat transfer can be taken into two-dimensional coordinates. In addition, we can divide this transformer
into four areas in terms of heat transfer (Fig. 2), described as
follows.
302
E. Rahimpour and D. Azizian
where,
qro : local radiation heat, transferred in height z in outer
surface (W),
:
qro
local radiation heat flux in height z in outer surface
(W/m2 ),
h ro : local heat transfer coefficient for radiation from outer
surface [W/(m2 K)],
ε:
emissivity coefficient of surface,
Ts :
local temperature of surface (K),
Tair : ambient temperature (K),
A:
the surface on which radiation occurs (m2 ), and
σ:
Stephan Boltzman’s coefficient (5.67 × 10−8 W/
(m2 K4 )).
Fig. 1 Schematic view of cast-resin transformer
Natural convection Theoretical equations for convection [10,
11] are very complex and unimportant to use here. Therefore,
some experimental equations are used for thermal modeling.
These equations were used by Pierce in [9] and are given
below:
qco
qco
(z) =
(4)
= h co (Ts − Tair )
A
h co (z) =
Nuz =
Fig. 2 Different parts of heat transfer in cast-resin transformer
2.1 Part 1: windings (combination of conductors and
insulations)
Considering the solid structure of this part, the heat transfer mechanism is conduction. The heat transfer equation in
cylindrical coordinate can be written as
1 ∂
q̇
∂T
∂2T
r
+ 2 =−
r ∂r
∂r
∂z
ku
(1)
wherein, the quantities are as follows:
q̇:
T:
ku :
specific loss density (W/m3 ),
temperature (K), and
thermal conductivity of unit [W/(m K)].
2.2 Part 2: outer surface
Outer surface is a vertical cylindrical surface and can be assumed as a vertical plate. Two types of heat transfer can occur
in the outer surface, including radiation and natural convection, which are described as follows.
Radiation For the outer surface, radiation can be expressed
as the equation below [10,11]:
qro
4
qro
(z) =
) = h ro (Ts − Tair )
(2)
= εσ (Ts4 − Tair
A
3
3
h ro = σ ε(Ts3 + Ts Tair
+ Ts3 Tair + Tair
)
(3)
Gr∗ =
Nuz kair
Z
4 Pr 2 Gr∗
36 + 45 Pr
gβq co Z 4
kair ν 2
(5)
1/5
(6)
(7)
where,
qco :
local convection heat, transferred in height z in outer
surface (W),
:
qco
local convection heat flux at location z in outer surface (W/m2 ),
q co : average heat flux in outer surface (W/m2 ),
h co :
local heat transfer coefficient for convection from
outer surface [W/(m2 K)],
kair :
thermal conductivity of air [W/(m K)],
Gr∗ : Grashoff number for uniform heat flux,
Nuz : Nusselt number,
Pr:
Prandtl number,
β:
volumetric expansion of air (1/K),
ν:
kinematical viscosity of air (m2 /sec),
g:
acceleration of gravity (m/sec2 ), and
Z:
vertical distance (m).
The aforementioned equations are valid over the Grashoff’s range (105 ≤ Gr∗ ≤ 1010 ) [9]. Note that β, ν, and kair
are dependent on unknown temperatures [10,11].
2.3 Part 3: air ducts
In this part, the radiation and conduction heat transfers are
negligible in comparison to the convection.
Analysis of temperature distribution in cast-resin dry-type transformers
303
Natural convection As before, it is preferable to use the
experimental equations which are used by Pierce [9], these
are given below:
qcd
= h cd (Ts − Tair )
(8)
Nuz kair
(9)
h cd (z) =
b
wherein,
1/3
Nu Z = C1 (1 + R)1/6 ψ Z , for ψ Z ≤ 60◦
(10)
−1
1/2
1
9
1
24
−
,
+
Nu Z = C2 (1 + R)
ψZ 1 + R
70
2
2.4 Part 4: top and bottom surfaces
for ψ Z ≥ 60◦
wherein,
(b/Z )Gr∗ Pr
ψZ = 1/2
(b/L)Gr∗ Pr
3 Mathematical modeling with finite difference method
(11)
(12)
Gr∗ =
gβq Dh4
kair ν 2
(13)
Dh = 2
π(ro2 − ri2 )/N − a · b
Duct area
=2
Duct perimeter
2π(ro + ri )/N − 2a + 2b
(14)
in which C1 = 0.697, C2 = 1, and
heat transferred by convection at location z in
qcd :
inner or outer wall (W/m2 ),
qcd :
local convection heat flux at location z in inner
or outer wall (W/m2 ),
average heat flux in inner wall (W/m2 ) see
q̄i :
Fig. 3,
q̄o :
average heat flux in outer wall (W/m2 ) see
Fig. 3,
q :
is equal to q̄i and q̄o for inner wall and outer
wall, respectively,
R:
is equal to q̄o /q̄i for inner wall, and q̄i /q̄o for
outer wall,
h cd :
local heat transfer coefficient for convection
from inner or outer wall [W/(m2 K)],
Ts :
local temperature in inner or outer wall (K),
a:
width of dog bones (m), see Fig. 3,
b:
width of the air duct, which is equal to the length
of dog bones (m) see Fig. 3,
ri and ro : radii of inner and outer walls (m),
L:
total height of duct (m), and
N:
number of dog bones.
Heat transfer from bottom surfaces can be neglected with a
good precision (this assumption is because of the low quantity of Ts − Tair ). For top surfaces, heat transfer is unknown
and depends on many parameters such as air speed, color of
surface and so on, so the worst case in terms of the calculated
temperature was assumed and this heat transfer is neglected.
If it is necessary to consider the heat transfer from this part,
two equations similar to Eqs. (2) and (4) must be used.
Finite difference method, in which the so called resistive elements are employed, is used to solve the Eq. (1). This method
applies the energy conversion law to state the Eq. (1) in the
numerical form. The energy conversion law in the steadystate expresses:
Heat transferd to other
Heat generated
(15)
=
units or ambient
in the unit
In finite difference method, the solid parts are divided into a
number of units properly (see Fig. 4). Each unit is represented
by a node which is related to other nodes by thermal resistances. Heat transfer between two nodes can be expressed as
follows:
(T1 − T2 )
q12 =
(16)
R12
wherein,
T1 and T2 :
q12 :
R12 :
temperatures of the nodes 1 and 2,
heat transferred from node 1 to node 2 (negative value means that the heat is transferred from
node 2 to node 1), and
thermal resistance between nodes 1 and 2.
From Eq. (16), it is obvious that there is a duality between heat
flow and electrical current and also between energy conversion and kirchoff’s laws. By using relation (15), the following
equation results:
(A) Δ ro
Δ ri
(B)
m ,n+1
Δr
R+n
a
Δz
Δ zu
Δ zd
qi
m,n
b
Δ z = Δ zu /2+ Δ zd /2
Δ r = Δ ri /2+ Δ ro /2
qo
Fig. 3 Schematic view of a duct (including four dog bones) from top
of the transformer
m-1,n
Rm-
m+1,n
R+m
Rnm,n-1
Fig. 4 a Units in radial and vertical directions (2 nodes in radial direction and 13 nodes in axial direction are shown in this division). b
Schematic view of a unit, which is represented by a node named m, n
304
E. Rahimpour and D. Azizian
Table 1 Thermal resistances
otherwise
otherwise
Heat transfer from top surface
(in nodes of top surface)
otherwise
Heat transfer from bottom surface
(in nodes of bottom surface)
otherwise
(Tm,n − Tm−1,n ) (Tm,n − Tm+1,n ) (Tm,n − Tm,n−1 )
+
+
Rm−
Rm+
Rn−
(Tm,n − Tm,n+1 )
+
= qm,n
(17)
Rn+
wherein Rm− , Rm+ , Rn− , and Rn+ are thermal resistances
between nodes which m,n and inner, outer, lower and upper
nodes (Fig. 4) which are given in Table 1.
Note that, Tm−1,n , Tm+1,n , Tm,n−1 , and Tm,n+1 in the
nodes belong to outer surface or inner wall of air ducts, the
outer wall of the air ducts, the bottom surfaces and the top
surfaces are, respectively, replaced by ambient temperature
(Tair ). Obviously these replacements are not equalities and
just resulted from Eqs. (2), (4), and (8). The qm,n in Eq. (17)
is the total losses in the unit m,n and its value is 0 for the
insulation units, with ignoring very small dielectric losses.
For windings, qm,n can be calculated as explained in the next
subsection.
K eddy in Eq. (18) is given in the references [12–14] and:
Pdc = Rdc |I |2 =
2πr m ρ 2
|I |
hw
(19)
where ρ, rm , and I represent electrical resistivity, mean radius, and current of conductor, respectively. Now assume that
the nodes in the radial direction are selected in the middle of
insulation (Fig. 6), so that the related unit contains fourth part
of four conductors beside insulation. In this case, losses of
each unit (qm,n ) can be calculated as shown in Fig. 6, as the
summation of the losses in its conducting parts.
Note that, winding losses depend on temperature, as it
can be seen in the following equation.
K eddy
Ptotal−new = Pdc K E +
(20)
KE
with,
KE =
3.1 Winding losses
Figure 5 shows the cross-section of a conductor with the insulation around it. For the unit shown in Fig. 5, the total losses
can be expressed as follows:
Ptotal = (1 + K eddy )Pdc
(18)
Tnew + Tk
Tbase + Tk
(21)
and
Tk :
−48 K for Aluminum and −38.5 K for Copper,
Pdc :
dc losses calculated in the temperature Tbase , and
Ptotal−new : total losses in the new temperature Tnew .
Analysis of temperature distribution in cast-resin dry-type transformers
305
test object. Its geometrical construction is shown in Fig. 7
and its rating parameters are given in Table 2.
The LV terminals of the transformer were short-circuited
and the applied voltage to the HV terminals were adjusted
until the currents reached their rated values in three phases.
After reaching the steady-state condition of temperature, temperatures were measured in the outer surface of the outer legs
by infrared periscope in different angles (Fig. 8a). It can be
seen from Fig. 8a that the assumption of symmetry in φ direction, which is discussed in Sect. 2, might not occur in practice,
because of a number of reasons, such as influence of neighbor phases, terminal connections, production tolerances, and
non-uniform conditions of environment. Therefore, the temperature varies with φ, which was not considered in the modeling process. Hence, the average value of these temperatures
is considered as the experimental data and is compared with
the modeling results, as shown in Fig. 8b. It can be seen that
the modeling is valid with a good accuracy.
For further verification of the model, Table 3 shows the
average and the hottest spot temperatures, predicted by modeling and comparison of them with experimental and international electrical committee (IEC) rated values [4]. From
Table 3, it is seen that the ratio of the hottest spot temperature rise/the average temperature rise (HS/AV) predicted in
this study is near the value predicted by [9].
The causes of differences between modeling and experimental results can be grouped into experimental and modeling errors, as follows:
Fig. 5 Cross-section of a conductor with its insulation
Fig. 6 Losses of one unit (qm,n ) as the summation of the losses in its
conducting parts
3.2 Solving thermal equations
By using the above process, the matrix form of Eq. (17) can
be derived as follows:
[G]n∗n [T ]n∗1 = [Pc ]n∗1
(22)
wherein,
[G]: coefficients matrix,
[T ]: temperatures matrix, and
[Pc ]: heat matrix.
[G] and [Pc ] are dependent on matrix [T], so an iterative
process is necessary to solve the Eq. (22). For thermal
modeling of cast-resin dry-type transformer, 58 nodes are
selected in the radial direction and 55 nodes in the vertical
direction and temperatures are calculated by solving (22).
4 Verification of model
To evaluate the validity and accuracy of the discussed thermal model, a cast-resin dry-type transformer is chosen as a
Fig. 7 Geometrical construction of cast-resin dry-type transformer
(dimensions are in mm)
306
E. Rahimpour and D. Azizian
Table 2 Rating parameters of the used test object
LV terminal voltage
400 V
HV terminal voltage
20 kV
Core heat flux
667.54 watt/m2
LV line current
1154.7 A
HV line current
23.09 A
Power
800 kVA
LV connection
Y
HV connection
Δ
Frequency
50 Hz
LV turns
18
HV turns
1,559
120
120
(B)
110
110
Temperature ( C)
Temperature ( C)
(A)
º
º
100
ϕ =60ºC
ϕ =120ºC
ϕ =240ºC
ϕ =300ºC
90
80
70
100
90
80
70
60
0
200
400
600
Vertical Distances (mm)
Measured
Calculated
800
200
400
600
Vertical Distances (mm)
800
Fig. 8 Comparison of modeling and experimental results. a Measured temperature distribution in different angles. b Comparison between
measured and simulated results
Table 3 The average and the hottest spot temperature rises of the test object
Modeling
Experience
IEC std [4]
Reference [9]
The average temperature rise (AV) (◦ C)
The hottest spot temperature rise (HS) (◦ C)
HS/AV
LV
HV
LV
HV
LV
HV
88
85
100
–
95.8
93
100
–
103
Not measured
115
–
119.3
Not measured
115
–
1.17
1.24
1.15
1.26
1.15
1.26
1. Experimental errors
1. Some laboratory environmental conditions such as air
displacements, which cannot be determined precisely
and create inaccuracy.
2. Body of the infrared periscope may be warmed subject
to magnetic fields. This can cause an error in temperature reading [15].
A number of assumptions such as neglecting the heat
transfer from top and bottom surfaces can influence the modeling results too. In the next section, influence of some of
them on the modeling results will be discussed.
5 Analyzing some factors affecting temperature
distribution
2. Modeling errors
1. Errors caused by numerical calculations due to some
computational operations such as rounding and estimation.
2. Errors due to limited number of nodes, increased number of nodes in radial and vertical directions can
improve the precision, but the time for running the
program outweigh the precision.
With the help of verified model, the influences of some parameters on temperature distribution are evaluated. These parameters and results of their analysis are given as follows:
1. Air ducts
There are three air ducts in the test object. The behavior
of all of them is similar. The influence of the air duct
between LV and HV windings (air duct 3 in Fig. 7) on
200
1.5
(A)
180
307
(B)
LV
HV
LV
HV
1.4
160
HS/AV
Average Temperature ( º C)
Analysis of temperature distribution in cast-resin dry-type transformers
140
120
100
1.3
1.2
1.1
80
0
20
40
60
80
Air duct's width (mm)
1
100
0
20
40
60
80
Air duct's width (mm)
100
Fig. 9 Effect of air duct’s width on temperature distribution of windings. a Average temperature. b Ratio of HS/AV
95
º
º
90
85
80
120
110
(A)
Temperature( C)
Temperature( C)
100
With eddy Losses
Without eddy Losses
140
160
180
(B)
105
100
95
210
With eddy losses
Without eddy losses
220
230
240
250
Radius(mm)
Radius(mm)
Fig. 10 Effect of eddy losses on temperature distribution in LV winding (a) and HV winding (b), in the middle of windings
average temperatures and HS/AVs of LV and HV windings is given in Fig. 9.
Along with the increment of air-duct width two phenomena occur:
(1) The convection in the air duct increases the total heat
transfer.
(2) The total HV winding losses increase due to rise in
conductors’ length of the HV winding.
The first factor decreases the temperature, while the second one increases it. For small widths, the convection
factor is dominant, while the losses factor is more efficient in high widths (Fig. 9).
2. Edge bands (see Fig. 7)
Edge bands cover the edge of conductor foils in order
to improve the insulation structure. The model shows no
visible effect on temperature distributions by edge bands.
3. Eddy losses
Fig. 10 shows the effect of eddy losses on temperature
distribution. As expected these losses increase the temperature in both windings.
While the temperature rise in HV winding is negligible,
it is considerable in LV winding. The reason is that eddy
losses in HV winding are small, since its conductor’s
cross-section is smaller than in LV winding.
4. Radiation from outer surface
Radiation from outer surface does not have any significant effect on temperature of LV winding. Nevertheless, it
has a considerable influence on temperature distribution
of HV winding (Fig. 11).
5. Heat transfer from top and bottom surfaces
The effect of radiation from top (εt ) and bottom (εb ) surfaces on temperature distribution are shown in Fig. 12, in
which only radiation is considered as heat transfer factor. This figure shows that heat transfer from bottom surfaces is negligible and from top surfaces has a small effect
on temperature. This behavior is in agreement with the
assumption made in Sect. 2.
6. Dog bones (see Fig. 3)
Only air ducts 1 and 2 (Fig. 7) contain dog bones. The
effect of dog bones on average and the hottest spot temperature is modeled through changing its number between
0 and 12. It was observed that the dog bones had no considerable influence on temperature of HV winding, while
they had a considerable influence on temperature of LV
winding (Fig. 13).
To give a better view of the effects of various parameters on temperature distribution of LV and HV windings,
Table 4 shows a summary of the results.
6 Conclusion
Cast-resin dry-type transformer is modeled in this paper in order to calculate temperature distribution in its windings. The
validity and accuracy of the modeling results is shown with
308
E. Rahimpour and D. Azizian
100
130
95
90
ε =0.8
ε =0.9
ε =0.95
85
120
140
160
Temperature( º C)
Temperature( º C)
(A)
(B)
120
110
ε =0.8
ε =0.9
ε =0.95
100
90
180
220
Radius(mm)
230
240
250
Radius (mm)
Temperature( º C)
Fig. 11 Effect of radiation of outer surface on temperature distribution in LV winding. (a) and HV winding (b), in the middle of windings
(ε denotes emissivity coefficient of outer surface)
150
Table 4 Effects of various factors on temperature distribution of LV
winding and HV winding
100
Air ducts
Edge bands
Eddy losses
Radiation from outer surface
Heat transfer from top surface
Heat transfer from bottom surface
Dog bones
50
0
εb=0.0 , εt=0.0
εb=0.8 , εt=0.0
εb=0.0 , εt=0.8
0
200
400
600
800
1000
Vertical Distances (mm)
Fig. 12 Variation of temperature in inner surface of HV winding due to
emissivity of top and bottom surfaces (εt denotes emissivity coefficient
of top surface and εb denotes emissivity coefficient of bottom surface)
the help of experimental results from an 800 kVA transformer
and comparing the calculated and measured temperatures.
The effects of some factors on temperature distribution are
investigated with the help of model and the following results
are obtained:
(1) By increasing the air-duct width the temperature decreases for small air-duct widths, while it increases for
large air-duct widths.
º
HV winding
Strong
Insignificant
Considerable
Insignificant
Weak
Insignificant
Considerable
Strong
Insignificant
Insignificant
Considerable
Weak
Insignificant
Insignificant
(2) The dog bones have no influence on temperature of HV
winding, while they cause a considerable increase on
temperature of LV winding.
(3) Edge bands have no evident effect on temperature distributions.
(4) Eddy losses increase the generated heat and therefore the
temperature of windings. They are dependent on crosssection of conductors. Consequently, the amount of temperature rise due to eddy losses depend on cross-section
of conductors.
(5) Radiation from outer surface influences only the temperature of outer winding considerably.
(6) Heat transfer from bottom surfaces is negligible and from
top surfaces has a small effect on temperature.
(7) It is observed that the ratio of HS/AV varies slightly
around a constant value.
(B) 1.175
90
89
HS/AV
Average
Temperature( C)
(A)
LV winding
88
87
1.17
1.165
86
85
0
5
10
Nomber of dog bones
1.16
0
5
10
Nomber of dog bones
Fig. 13 Effect of dog bones on temperature distribution of LV winding. a Average temperature b Ratio of HS/AVb
Analysis of temperature distribution in cast-resin dry-type transformers
References
1. Nunn T (2000) A comparison of liquid-filled and dry-type transformer technologies. IEEE-IAS Cement Industry Committee, pp
105–112
2. Claiborne CC, Pearce HA (1989) Transformer fluids. IEEE Electr
Insul Mag 5(4):16–19
3. McMahon MD, Rian BJ (1989) Reclassification of askarel transformers – Field application and results. IEEE Trans Ind Appl
25(2):371
4. Standard (2005) Dry type power transformers, IEC Standard Publication, 60076–11
5. Dianchun Z, Jiaxiang Y, Zhenghua W (1999) Thermal field and hottest spot of the ventilated dry-type transformer. In: IEEE Proceedings of the 6th international conference on properties and applications of dielectric materials, pp 141–143
6. Pierce LW (1994) Thermal consideration in specifying dry-type
transformers. IEEE Trans Ind Appl 30(4):1090
7. Cho HG, Lee UY, Kim SS, Park YD (2002) The temperature distribution and thermal stress analysis of pole Cast resin transformer for
power distribution. In: IEEE conference international symposium
on electrical insulation, pp 384–386
309
8. Pierce LW (1992) An investigation of the temperature distribution in cast-resin transformer windings. IEEE Trans Power Deliv,
7(2):920–926
9. Pierce LW (1994) Predicting hottest spot temperatures in ventilated dry type transformer windings. IEEE Trans Power Deliv
9(2):1160–1172
10. Holman JF (1934) Heat Transfer. McGraw-Hill, Newyork, ISBN
964-6283-12-8
11. Incropera FP, Dewitt DP (2001) Introduction to heat transfer. Wiley, Newyork, ISBN 0471386499
12. Dietrich W (1988) Berechnung der Wirbelstromverluste in
den Wicklungen von Mehrwicklungstransformatoren. etz Archiv
10(10):309–317
13. Dietrich W (1961) Berechnung der Wirkverluste von Transformatorwicklungen unter Berücksichtigung des tatsächlichen Streufeldverlaufs. Archiv für Elektrotechnik 46(4):209–222
14. Rahimpour E (2003) Hochfrequente Modellierung von Trnsformatoren zur Berechnung der Übertragungsfunktionen. Dissertation,
Stuttgrat, Shaker Verlag, ISBN 3826592271
15. Standard (2000) IEEE guide for determination of hottest spot temperature in dry type transformers. IEEE Std C57, pp 134–200