IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 3, JULY 2009
1257
A Simple Model for Calculating Transformer
Hot-Spot Temperature
Dejan Susa, Member, IEEE, and Hasse Nordman
Abstract—A simple model for calculating the hot-spot temperature is introduced. The model is based on the hot-spot to ambient gradient. The model considers the changes of the oil viscosity
and winding losses with temperature. The results are compared
with temperatures calculated by IEEE Annex G method and measured results at varying load for the following transformer units:
250-MVA ONAF, 400-MVA ONAF, and 605-MVA OFAF.
Index Terms—Hot-spot temperature, oil viscosity, top-oil temperature, winding losses.
Weight of core and coil assembly (in
kilograms).
Weight of the oil (in kilograms).
Weight of the tank and fittings (in kilograms).
Correction factor of oil.
No-load losses.
DC losses per unit value.
Relative winding eddy losses, per unit of dc
loss.
NOMENCLATURE
Specific heat capacity of winding material.
Eddy losses (per unit value).
Stray losses (in watts).
Specific heat capacity of core.
Specific heat capacity of the tank and fittings.
DC losses (in watts).
Heat generation.
Specific heat capacity of oil.
Heat generated by total losses.
A constant.
Heat generated by winding losses.
Oil thermal capacitance.
Nonlinear winding to ambient thermal
resistance.
Winding thermal capacitance.
Thermal capacitance of the core.
Current density at rated load.
Thermal capacitance of the tank and other
metal parts.
Subscript indicates the ultimate value.
Constant.
Load current.
Portion of the core losses in the total
transformer losses.
Subscript indicates initial.
Load factor.
Portion of the stray losses in the total
transformer losses.
Weight of core (in kilograms).
Minute.
Weight of the tank and fittings (in kilograms).
Portion of the winding losses in the total
transformer losses.
Weight of oil (in kilograms).
Oil viscosity.
Weight of winding material (in kilograms).
Ambient temperature.
Top-oil temperature.
Top-oil temperature rise over ambient.
Manuscript received January 25, 2008; revised January 16, 2009. Current version published June 24, 2009. This work was supported in part by the SINTEF
Energy Research Department, Trondheim, Norway. Paper no. TPWRD-000362008.
D. Susa is with the SINTEF Energy Research Department, Trondheim
NO-7465 , Norway (e-mail: dejan.susa@ sintef.no).
H. Nordman is with the ABB, Power Transformers, Vaasa 65101, Finland
(e-mail: hase.nordman@fi.abb.com).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPWRD.2009.2022670
0885-8977/$25.00 © 2009 IEEE
Hot-spot temperature.
Rated hot-spot to ambient temperature
gradient.W
Winding time constant.
Winding time constant.
Oil time constant.
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 3, JULY 2009
Winding time constant.
R
Subscript indicates rated value.
pu
Subscript indicates per unit value.
I. INTRODUCTION
P
OWER transformers represent the largest portion of
capital investment in transmission and distribution
substations. In addition, power transformer outages have a
considerable economic impact on the operation of an electrical
network. One of the most important parameters governing a
transformer’s life expectancy is the hot-spot temperature value.
The classical approach has been to consider the hot-spot temperature as the sum of the ambient temperature, the top-oil temperature rise in tank, and the hot-spot-to-top-oil (in tank) gradient [1], [2]. During the last 20 years, fiber-optic probes have
been used by many authors [3]–[5] in order to obtain as accurate
values for transformer temperatures as possible. When the load
is increased, it takes some time before the corresponding oil circulation adapts its speed [4] due to lower temperature and higher
oil viscosity at the preceding loads. Consequently, the hot-spot
temperature rises rapidly during the first 10–20 min with a time
constant that is equivalent approximately to the winding time
constant [6] (Fig. 1). Nevertheless, this time period is different
for each transformer and it is very dependent on the transformer
design. Conveniently, it has been further observed that 50% of
the temperature change occurs during the rapid rise. When the
temperature threshold level is reached, the oil circulation is established at a rate, which is defined as critical, preventing further
rapid temperature rise. Therefore, the hot-spot temperature will
continue to rise slower with a time constant that is equivalent to
the top-oil time constant (Fig. 1).
On the other hand, at the transformer cold start, 75% of the
temperature change occurs during the rapid rise period due to
more harsh initial oil conditions (i.e., initial oil speed is zero).
In contrast, the initial oil circulation prior to the load decrease
is faster than it will be under the load considered. Therefore, the
temperature will now decrease rapidly with a time constant that
is equivalent to approximately the winding time constant. Once
50% of the final temperature drop is reached, the oil velocity
will be much lower than initially. As a result, the temperature
starts decreasing slowly with a time constant equivalent to the
top-oil time constant.
This paper presents a simpler but still accurate temperature
calculation method, taking into account the findings mentioned
before. With alternate switching between two different time
constants, (the short one being equal to the winding time constant and the long one being equal to the top-oil time constant),
the model is based only on the hot-spot-to-ambient air gradient.
The thermal model is based on heat-transfer theory [7]–[10],
numerous transformer thermal tests and reports [3]–[6] and
[11]–[38], application of the lumped capacitance method, the
thermal-electrical analogy, and the concept of thermal resistance between winding insulation surface and ambient (i.e.,
air).
Fig. 1. Hot-spot and top-oil temperatures of the 120-kV winding in the
400-MVA ONAF-cooled transformer at 1.29-p.u. constant load.
The model presented in this paper takes oil viscosity changes
and loss variation with temperature into account. The changes in
transformer time constants due to changes in the oil viscosity are
also accounted for. The model requires an iterative calculation
procedure.
The models are validated by using experimental results,
which have been obtained from a series of thermal tests performed on three different power transformers: (250-MVA
ONAF, 400-MVA ONAF, and a 605-MVA OFAF-cooled unit),
[3], [4]. The model is tied to measured parameters that are
readily available (i.e., data obtained from a normal heat-run test
performed by the transformer manufacturer at commissioning).
II. THERMAL MODEL
A. Thermal Circuit
The thermal circuit for the hot-spot temperature rise over
the ambient based on heat-transfer theory, [7]–[11], [19] and
thermal-electrical analogy [30]–[36] is given in Fig. 2 with the
following elements:
heat generated by winding losses;
thermal capacitance of the winding;
hot-spot temperature rise over ambient;
winding to ambient nonlinear thermal resistance.
The steady-state temperature rise equation for the natural
convection and heat-exchange phenomena between the winding
insulation surface, (where the sensors are located), and the ambient is given as follows:
(1)
SUSA AND NORDMAN: SIMPLE MODEL FOR CALCULATING TRANSFORMER HOT-SPOT TEMPERATURE
1259
TABLE I
THRESHOLDS FOR ALL COOLING MODES
Fig. 2. Hot-spot temperature rise thermal circuit.
valid only for ONAN and ONAF cooling mode
where
is the hot-spot-to-ambient temperature gradient.
is a function of the fluid properties and winding characteristic
dimensions and is considered to be a constant [32]. is constant that is partly based on experimental results obtained from
thermal tests [3], [4]. The sensors locations have been discussed
in [3], [4], and [20]. is the viscosity variation with temperature
(in kilograms per millisecond), given by the following equation
[6]:
(2)
where the viscosity is evaluated at the value of
where the load current is given as and the rated current
is given as ;
the rated hot-spot to ambient gradient
(9)
;
the heat generated by the winding losses
(10)
where
is the loss per unit value given as
given by
(11)
(3)
is the hot-spot temperature and
is the ambient
temperature.
The nonlinear thermal resistance between the winding insulation surface and the ambient of the transformer is characterized
by (4), which is derived from (1)
(4)
The differential equation for the thermal circuit in Fig. 2 is
given as follows:
is the winding loss’s dependence on the hotwhere
spot temperature
(12)
and
describe the behavior
where
of the dc and eddy losses as a function of temperature [6].
The dc losses vary directly with temperature, whereas the
is the temeddy losses vary inversely with temperature.
perature factor for the loss correction, equal to 225 for aluis the hot-spot temperaminium and 235 for copper.
ture.
Finally, (5) becomes
(5)
(13)
Now, if the following parameters are defined:
the thermal resistance
as
(6)
stand for the rated and relative
where subscripts and
values, respectively,
the winding time constant
(7)
stands for the rated winding time constant
where
given in Section II-C and it is assumed that
, [32];
the load factor
(8)
The final solution of (13) for the load increase and decrease
is given in the following section. The corresponding values for
the temperature threshold level and oil viscosity exponent are
given in Tables I and II.
B. Complete Model
1) Load Increase: The hot-spot temperature increases to a
level corresponding to a load factor of K
(14)
The initial hot-spot rise over ambient is
. The hot-spot
rise calculated for the end of previous load step is used as the
is
initial hot-spot rise for the next load step calculation.
the ultimate hot-spot rise given by the following equation:
(15)
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 3, JULY 2009
is the time delay correction for the following condition:
TABLE II
LOAD STEPS FOR THE 250-MVA TRANSFORMER
(25)
Equation (21) is applicable here as well.
Equations require an iterative calculation procedure.
3) Winding Exponent and Change Levels: The viscosity ex0.5). In
ponent for all cooling modes is equal to 0.5 (i.e.,
addition, the temperature change thresholds are given in Table I.
C. Time Constants
1) Winding Time Constant: The winding time constant [19]
is as follows:
The function
describes the relative increase of the hot-spot
temperature rise until the corresponding threshold is reached
(16)
for
(26)
for
(27)
where
is the winding time constant prior to the change
winding time constant in minutes at the rated
load;
(17)
hot-spot to ambient temperature gradient at the
rated load;
is the rated winding time constant given in Section II-C.
will be replaced by function
once
The function
the threshold level for a given load is reached as suggested in
Table I
relative winding eddy losses, per unit of dc loss,
corrected for the hot-spot temperature;
current density in A/mm2 at the rated load.
(18)
is the winding time constant after the change
2) Top-Oil Time Constant: The top-oil time constant at the
rated load is given as follows:
(28)
(19)
is the rated oil time constant given in Section II-C.
is the time delay correction for the following condition:
where
rated top-oil time constant (in minutes);
(20)
rated top-oil temperature rise over ambient
temperature (in Kelvin (K));
(21)
is the time when the function
reaches the corresponding
threshold level.
Equations require an iterative calculation procedure.
2) Load Decrease: The hot-spot temperature decreases to a
level corresponding to a load factor of K
(22)
The function
describes the relative decrease of the
hot-spot temperature rise until the corresponding threshold is
reached
(23)
The function
will be replaced by function
once the
change level for a given load is reached as suggested in Table I
(24)
total supplied losses (total losses) (in watts (W))
a the rated load;
equivalent thermal capacitance of the
transformer oil
.
The equivalent thermal capacitance of the transformer oil for
transformers with external cooling and a zigzag oil flow through
the windings is given by
(29)
where
weight of the winding material (use only the
excited parts) (in kilograms);
weight of the core (in kilograms);
weight of the tank and fittings (in kilograms);
SUSA AND NORDMAN: SIMPLE MODEL FOR CALCULATING TRANSFORMER HOT-SPOT TEMPERATURE
1261
TABLE III
MAXIMUM AND AVERAGE ERROR FOR THE 250-MVA TRANSFORMER
SM: simple model;
IEEE: IEEE-Annex G
weight of the oil (in kilograms);
specific heat capacity of the winding material
and
) in
, [32];
(
specific heat capacity of the core
, [32];
0.13) in
specific heat capacity of the tank and fittings
0.13) in
, [32];
specific heat capacity of the oil
, [32];
Fig. 3. Hot-spot temperature of the 118-kV winding in the 250-MVA ONAFcooled transformer.
0.51) in
correction factor for the oil in the ONAF, ONAN,
and OFAF cooling modes;
correction factor for the oil in the ODAF cooling
mode;
portion of the stray losses in the total losses;
portion of the core losses in the total losses;
portion of the winding losses in the total losses.
Equation (29) is an empirical formula based on observations
from different thermal tests and the modeling that has already
been performed and validated in the author’s previous work
[33], [34].
The equivalent thermal capacitance of the transformer oil
for transformers without either external cooling or guided
horizontal oil flow through the windings (where the lack of
radiators and the lack of the horizontal oil flow through the
winding directly affects the oil flow inside the transformer tank,
thus slowing down the cooling process) is calculated according
to the IEEE Loading Guide-Annex G [2] and [6].
III. COMPARISON
The measured temperature results, which are recorded for
three different transformer units during different varying load
tests, are compared by the new calculation method presented in
this paper and the IEEE Annex G method. The input data for
both methods are given in the Appendix. The maximum and average errors obtained for both models are given in Tables III,
V, and VII. The maximum error is obtained as the maximum
difference between the measured and calculated curve. The average error is obtained as the sum of the data values divided by
the number of data values. The error plots are also shown in
Figs. 3–5.
A. The 250-MVA ONAF
The rated voltages of the 250-MVA transformer were
1.5%/118/21 kV. The windings were seen from
the limb side, the 118-kV and 230-kV main windings, the
regulating winding, and the 21-kV tertiary winding. The connection was YNyn0d11, and the short-circuit impedance in the
250/250-MVA main direction was 12%. The oil flow through
the windings was guided by oil guiding rings in a zigzag pattern.
The transformer was equipped with a total of 16 fiber-optic
sensors, eight in the 118-kV winding and eight in the 230-kV
winding, according to the principles explained in [4]. In total,
14 thermocouples were located in the tie plates and outer core
packets at the top level of the main windings of phase B.
In addition to the normal delivery tests, including the ONAN
and ONAF heat-run tests, the following load tests were performed on the unit operating in the ONAF cooling mode:
• constant load current; 1.28 p.u.; duration 13.5 h;
• constant load current; 1.49 p.u.; duration 15 h;
• varying load current Table II.
The measured hot-spot temperature results of the hottest
winding and sensor, recorded during the varying load current
test, are compared with the results obtained from the thermal
models in Fig. 3.
The maximum and average errors are given in Table III.
B. The 400-MVA ONAF
The rated voltages of the transformer were 410 6 1.33%/
120/21 kV. The windings were, seen from the limb side: 120-kV
and 410-kV main windings, a regulating winding, and a 21-kV
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 3, JULY 2009
TABLE IV
LOAD STEPS FOR THE 400-MVA TRANSFORMER
TABLE V
MAXIMUM AND AVERAGE ERROR FOR THE 400-MVA TRANSFORMER
SM: simple model;
IEEE: IEEE-Annex G
Fig. 4. Hot-spot temperature of the 410-kV winding in the 400-MVA ONAFcooled transformer.
probes (eight in each winding), and the tie plates, outer core
packets, and yoke clamps had a total of 37 thermocouples.
Additional load tests with ONAF cooling were the following:
• constant load current: 1.0 p.u.; duration 12 h:
• constant load current: 1.29 p.u.; duration 10 h;
• constant load current: 1.60 p.u.; duration 15 h;
• varying load current (Table IV).
The measured hot-spot temperature results of the hottest
winding and sensor, which were recorded during the varying
load current test, are compared with results obtained from the
thermal models in Fig. 4.
The maximum and average errors are given in Table V.
C. The 605-MVA OFAF
Fig. 5. Hot-spot temperature of the 362-kV winding in the 605-MVA OFAFcooled transformer.
tertiary winding. The connection was YNynd, and the short-circuit impedance in the 400/400-MVA main direction was 20%.
The oil flow through the windings was guided by the oil guiding
rings in a zigzag pattern. The main windings in this transformer
are representative of two basic cases: 1) “restricted oil flow”
(2-mm radial spacers in the 120-kV winding) and 2) “unrestricted oil flow” (3-mm radial spacers in the 410-kV winding).
The main windings were equipped with a total of 16 fiber-optic
The 605-MVA transformer was a generator stepup (GSU)
unit with the windings seen from the limb side: part of the HV
winding (i.e., 362 kV-winding), the double shell LV winding
(i.e., 22 kV-winding), and the main part of the HV winding.
The oil circulation through the windings was guided by the oil
guiding rings in a zigzag pattern in such a way that the oil flow
through the LV winding was restricted (2-mm radial spacer) and
through the HV winding unrestricted (3-mm radial spacer). The
transformer was not a sealed OD (i.e., the oil circulation was
not forced through the winding block). In total, 24 fiber-optic
sensors were installed in the top disc/turns of the outer shell of
the LV winding and the outer part of the HV winding.
In addition to the normal heat-run tests, the following load
tests were made with OFAF cooling:
• constant load current: 1.00 p.u.; duration 12 h;
• constant load current: 1.30 p.u.; duration 1.2 h;
• varying load current (Table VI).
The measured hot-spot temperature results of the hottest
winding and sensor, which were recorded during the varying
load current test, are compared with results obtained from the
thermal models and are shown in Fig 5.
SUSA AND NORDMAN: SIMPLE MODEL FOR CALCULATING TRANSFORMER HOT-SPOT TEMPERATURE
1263
TABLE VI
LOAD STEPS FOR THE 605-MVA TRANSFORMER
TABLE VII
MAXIMUM AND AVERAGE ERROR FOR 605-MVA TRANSFORMER
SM: simple model;
IEEE: IEEE-Annex G.
The maximum and average errors are given in Table VII.
IV. CONCLUSION
The athors have already developed a few transformer thermal
models [30]–[34]. All models take into account the oil viscosity
change with temperature as one of the parameters defining the
temperature curve. Also, the hot-spot to top-oil temperature gradient and the top-oil temperature rise are defined as two separate
systems but cascadely interconnected. Thus, any change of the
top-oil temperature will affect the hot-spot to top-oil gradient
and further on the hot-spot temperature.
Similarly, by switching from the winding time constant to the
top-oil time constant, the model presented in this paper takes
into account this additional top-oil effect on the hot-spot temperature rise.
A new feature in the thermal model developed in this paper
is that it is based directly on the hot-spot -to-ambient air gradient without splitting up this gradient into the two gradients
hot-spot-to-top oil and top oil-to-ambient air.
The oil viscosity effect and loss change with temperature are
also taken into account.
The model is based on an exponential iterative calculation
procedure. Nevertheless, more rigid and more precise mathematical procedures could be applied as well. The authors have
decided to use an exponential approach to follow well-known
temperature calculation procedures given in [1] and [2].
Comparably, the results obtained by the IEEE Annex G
method and the results plotted by the proposed model are in
good agreement with the measured values for the load increase
and load decrease. However, one could conclude that models
yield values either on the conservative side or with reasonable
accuracy. The main advantage of the proposed model is a
reduced number of the input data and its simplicity.
Nevertheless, both models develop higher error at the load
increase. This can be straightforwardly observed in Figs. 3–5.
The models simply predict a much faster initial temperature rise
Fig. 6. Computing algorithm for the simple model.
compared to the measured one. In addition, the higher overload
of the error is more pronounced as the oil viscosity effect is underestimated. In other words, the established oil circulation is
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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 24, NO. 3, JULY 2009
TABLE VIII
INPUT DATA FOR THE THERMAL MODELS
Fig. 7. Load profile during an extended heat-run test of a transformer.
the model could be used for the real-time online hot-spot
temperature computation as an integrated part of a monitoring
system, (i.e., indirect hot-spot measurement). Furthermore, the
model application will allow both transformer manufactures
and users to run different loading and ambient scenarios and,
by analyzing the results, improve the transformer design (costs,
size, and load carrying capacity).
Input data necessary for the suggested thermal model.
at a much higher rate than it is accounted for by the presented
models. The way to overcome the problem is in the further improvement of the winding time constant equation assuming that
the oil viscosity equation is correct.
Therefore, future work should consider additional thermal
tests and investigations to derive general and simpler winding
and top-oil time constants calculation procedures.
The concept for the model application in transformer monitoring, the model computation algorithm, Fig. 6, and the model
validation are given in the Appendix, respectively.
APPENDIX
A. Transformer Monitoring
A transformer online monitoring system, which collects
information from several measurable variables, should also
include real-time application of the thermal models to provide
an accurate picture of the operating condition of the transformer, allowing the operator to detect the early signs of faults
and correct them. In general, the monitoring system identifies
faults by comparing the results of measurements with prediction of the models. Consequently, the complete application of
the suggested model is only possible in the systems used for
transformers equipped with the fiber-optic sensors as is the
case with all developed hot-spot thermal models. However,
B. Model Algorithm
The algorithm that describes the steps to follow in order to
calculate the hot-spot temperature is given in Fig. 6.
C. Model Validation
The model can be validated in an extended heat-run test (Fig.
7) made on a transformer with installed fiber-optic sensors by
using fitting and extrapolation techniques. Note that these techniques should be applied in a manner consistent with the modeling presented in this paper. An application example as well as
a corresponding mathematical procedure are given in [35]. The
extended heat-run test consists of a regular heat-run test with
added an overload test (Fig. 7). The overload should be applied
three hours after the cooling curve is recorded in order to obtain
a prolonged cooling curve as well. In this way, different oil-flow
modes (i.e., due to cold start, load increase, and load decrease),
are considered as well as their effect on the model parameter.
The duration and size of the applied overload should be limited
with the maximum steady-state hot-spot temperature of 140 ,
[1].
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Dejan Susa (S’05–M’06) was born in Split, Croatia,
on May 22, 1972. He received the D.Eng. degree
in electrical engineering from the University of
Nis, Nis, Serbia, in 2000, and the M.Sc. and D.Sc.
degrees from the Helsinki University of Technology,
Espoo, Finland, in 2002 and 2005, respectively.
He was with the Power Systems Laboratory,
Helsinki University of Technology, from 2001
to 2006. He has been with the Center for Power
Transformer Monitoring, Diagnostic and Life Management, Monash University, Clayton, Australia,
since 2006. Currently, he is with SINTEF Energy Research Department,
Trondheim, Norway. He is working on different power transformer research
topics (losses, temperatures, moisture, gasses, online monitoring).
Dr. Susa is a member of Norwegian IEC National Committee and of IEC
MT1 (loading guide for oil-immersed power transformers), IEC MT2 (ability
to withstand short circuit), and IEC MT6 (temperature rise).
Hasse Nordman (M’08) was born in Overmark, Finland, in 1945. He received the Ph.D. degree in mathematics from the Abo Akademi University, Turku,
Finland, in 1977.
From 1970 to 1982, he was with ABB Corporate
Research (formerly Stromberg Research Centre),
Vaasa, Finland, working on current-related phenomena (losses, temperatures, short-circuit forces)
in electric power equipment. Since 1982, he has been
with the Development Engineering Department in
the Power Transformer Division of ABB, Vaasa. He
is also the leader of the global ABB R&D activity “Load Losses and Thermal
Performance.”
Dr. Nordman is a member of CIGRE, Chairman of the Finnish National Committee in the IEC Power Transformer Technical Committee (TC 14), and Convenor of the Maintenance Team MT1: Revision of IEC 354: Loading guide for
oil-immersed power transformers.