Download Neural networks applied to the design of dry

Received: 10 November 2015
Revised: 22 May 2016
Accepted: 10 July 2016
DOI 10.1002/etep.2257
RESEARCH ARTICLE
Neural networks applied to the design of dry‐type transformers:
an example to analyze the winding temperature and elevate the
thermal quality
Marco A. Ferreira Finocchio1 | José Joelmir Lopes2 | José Alexandre de França2* |
Juliani Chico Piai2 | José Fernando Mangili Jr2
1
Universidade Tecnológica do Paraná, Cornélio
Procópio, Paraná, Brazil
2
Universidade Estadual de Londrina,
Departamento de Engenharia Elétrica, Laboratório
de Automação e Instrumentação Inteligente,
Londrina, Paraná, Brazil
Correspondence
José Alexandre de França Departamento de
Engenharia Elétrica, Universidade Estadual de
Londrina, Caixa Postal 10011, Londrina, Paraná,
Brazil.
Email: [email protected]
Summary
Much of the cost of manufacturing transformers is related to the complexity in the
development of its project, which involves many variables, such as different types
of materials, methods, and manufacturing processes. Also, it is notoriously difficult
to establish standards that relate the transformer characteristics with these design variables, since each, almost always, variable is calculated empirically. One of the most
important design variables is the internal temperature of the transformers, which
directly influences the lifetime of such equipments. So, a computational tool is under
development whose purpose is to automate the design of cast resin dry‐type transformers and minimize the time taken for its completion, by means of artificial neural
networks. In this article, we present some results of this tool, which relate some geometrical parameters of the specific transformer with the temperature of its windings.
For this, the winding losses and total losses are also estimated. The training of the
neural network was done with the test data of about 300 dry‐type transformers from
the same manufacturer, so, with the same constructive features. Nevertheless, our
results show that the technique is promising because the neural networks were well
fitted and its results present errors lower than 1% when compared with data from tests
with real transformers. Evidently, the proposed methodology is dependent on the
constructive technique; that is, once trained, the neural network can be used only
in the design of dry‐type transformers of the same constructive technique as those
whose results of the tests were used to train the network. Nevertheless, even if only
used to provide the initial parameters for a more complex design tool, the proposed
method is useful since it is rapid and has low computational cost.
KE YWO RD S
artificial intelligence, epoxy resin, magnetic losses, transformers design automation,
winding losses, winding temperature
1 | IN T RO D U C T IO N
Power transformers are essential components in electric
power distribution systems and industries. Suitable for indoor
installations that require safety and reliability, the dry‐type
transformers are the focus of this paper. These transformers
have smaller dimensions than the ones with oil insulation.
Also, they have advantages such as a higher mechanical
Int. Trans. Electr. Energ. Syst. 2016; 1–10
robustness; the possibility of installation close to the load
point; does not require transformation wells for oil collection,
fire fighting systems, fireguard walls, and oil supervision
accessories; and the lowest level of internal partial discharges
(owing to its vacuum encapsulation).1 Therefore, they are
typically used where there is the presence of people, for
example, in industries, residential buildings, and hospitals.2
However, dry‐type transformers are more expensive and have
wileyonlinelibrary.com/journal/etep
Copyright © 2016 John Wiley & Sons, Ltd.
1
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power and voltage limitations. In this sense, failures on these
equipments imply great financial and social losses. So, they
must be designed and manufactured so as to meet the specific
usage needs, respecting the limits set in international standards such as the IEC‐76 and the IEEE C57.12.50‐1981.
Unfortunately, designing a transformer is not trivial. The
importance of balancing lifetime, performance, and costs,
besides the increasing development and evolution of various
materials, methods, and manufacturing processes, cause the
design of power transformers to be a highly complex process.
Part of this complexity is also due to the large number of variables and their interrelationships.3,4
In the design of transformers, a major goal is to reduce
the internal power losses. These are related to the losses in
the winding and iron and are dependent on parameters such
as internal temperature, resistance of windings, working
current, nominal voltage, load, and quality of the aluminum
used. Evidently, these losses affect the machine's temperature
regime, which directly influences the efficiency and lifetime
of the transformer.5 This is because the thermal stress is a
major cause of failures, since it causes the deterioration of
the insulating material.6 Therefore, estimating the value of
the internal temperature (or the value of the temperature
increase) is one of the most complex tasks in the transformer
design.3,7,8
Usually, transformers designers use either some design
patterns curves or the thermal network method to estimate
the temperature increase in the windings of this kind of
equipment. Also, several studies have attempted to predict
the internal temperature of the windings using empirical
and mathematical models and computational intelligence
techniques, which have obtained different degrees of success.9 In these cases, the advantage is that the temperatures
can be foreseen when the operating conditions change, for
example, with the design of a new transformer. In this sense,
several studies deal with oil transformers, so as to minimize
production costs and find core losses, winding losses, and
no‐load losses.10–20 However, in the case of dry‐type transformers, there are only a few thermal models available and
related researches.21 Besides, for example, Srinivasan and
Krishnan22 also point out that there are few theoretical and
practical studies on estimating hot spot temperature and
calculating the average temperature of the windings of dry‐
type transformers. So they developed a computational
modeling for this purpose using statistical methods, such as
multiple variable regression, multiple polynomial regression,
and computational techniques, such as artificial neural networks (ANN) and adaptive neuro‐fuzzy inference system.
In that paper, Srinivasan and Krishnan22 used as input
variables cold resistance, ambient temperature, and temperature rise. The results were verified based on temperature rise
tests in a dry‐type transformer of 55 kVA. In another
example, Lee et al21 developed a thermal model for foil‐type
winding for ventilated dry‐type power transformers rated at
2000 kVA, to calculate the electrical losses. Temperature
ET AL.
distributions were obtained in 3 different loads by the
coupled electromagnetic‐thermal model with the aid of a
2D transient finite element analysis, determined by the finite
element method.
This study proposes a tool to obtain the internal temperature of the windings, as well as winding losses and the total
losses, of dry‐type transformers. The applied methodology
uses ANN in parts of the transformer design, where the relationship between the variables is not well defined and the
design parameters are obtained empirically.23 The main
justification for using ANN is its ability to learn to relate a
complex set of data, which may be unnoticed by the
designer.5 Moreover, these networks have a quick assessment
and precision to resolve problems.24 The type of ANN
chosen was the feed‐forward multilayer perceptron. To train
the ANN, we used a database with tests results of 225 similar
dry‐type transformers with power outputs between 15 and
150 kVA and voltage classes equal to either 15 or 25 kV.
Building data of the transformer under design and the
environment temperature conditions, to which it will be
subject to during operation, are inputs of the neural networks.
In the data validation step, we used another 75 transformers
with the same characteristics of those used in the training.
The results show that the technique is very promising. This
is because the results show that the proposed tool provides
the designer the possibility of testing various design conditions prior to manufacturing the transformer. Thus, the generated results allow the adequacy of the constructive variables
to the environmental condition of the equipment operation,
ensuring the construction of dry‐type transformers with better
performance and lifetime. Evidently, this makes the design
more efficient when it takes to preparation time and
specificity.
The model presented is simple. However, it works
because it applies only to the design of dry‐type transformers
based on the constructive technique of those transformers of
the tests, ie, transformers of which test results were used to
train the neural network. In this case, the neural network
was shown to be able to model the constructive technique
very well. Furthermore, by following the methodology
discussed here, the readers may conduct their own tests and
train a neural network that models their own constructive
technique, which can lead to great results. Also, another
application of the proposed tool is to provide the initial
parameters for a more complex design software. This can
be done rapidly and with low computational cost.
2 | M AT E R I ALS AN D M E T H O DS
Not all steps of a dry‐type transformer project have the same
degree of subjectivity or uncertainty. So, in this paper, to test
the use of ANN in modeling this type of problem, we opted
to model the temperature in the windings (referred to as T)
of this type of transformer. Such quantity is a function of
FINOCCHIO
ET AL.
3
the following variables: area of the conductors (in this paper,
referred to as Sections and given in mm2); thickness of the
windings sheets (referred to as Thickness and given in mm);
number of winding channels (Channels), and the electrical
losses (referred to as LT and given in W). So, to calculate T,
first it is necessary to know LT. In turn, the quantities that
are part of the process of LT are defined as follows: high‐voltage coil resistance (referred to as RHV and given in Ω); low‐
voltage coil resistance (RLV, Ω); the environment temperature
(TA, °C); the excitation current (IEX, A); the core losses or the
no‐load losses (referred to as LC, given in W); and the
winding losses (LAl, W). Thus, to calculate LT, it is necessary
first to calculate LAl. In turn, the main variables that influence
LAl are as follows: the resistances of the high‐voltage and
low‐voltage coils, the temperature TA, and the short‐circuit
impedance (ZCC, given in %).
Based on the above, the developed tool as well as the
stages where the neural networks are applied and their
purposes can be seen in Figure 1. The neural network on
the left (Winding Losses Network) is intended to estimate
the winding losses (LAl), a variable that is used in the next
neural network (Total Losses Network), so as to estimate the
total losses (LT). Since the system aims to determine the internal temperature of the transformer, the Temperature Network
takes as inputs, in addition to the conductor section, the thickness of the core constituent plates and the number and dimensions of ventilation channels, besides the output of the Total
Losses Network. All these networks are discussed in detail
in Section 3.
For the training and validation of the 3 neural networks
developed in this paper, we used a database with tests results
of 300 dry‐type transformers, with power outputs from 15 to
150 kVA and voltages classes equal to either 15 or 25 kV.7 To
prepare the database, we performed the following tests with a
3‐phase transformer in a laboratory25: no‐load test, short‐circuit test, and winding resistance test. These tests are
discussed in detail below.
2.1 | No‐load test
No‐load tests of transformers aim to determine the following
quantities: core losses (LC), no‐load excitation currents (IEX),
transformation ratio, and parameters related to the magnetizing branch. Also, it is possible to analyze the nonsinusoidal
shape of the no‐load current and of the transient magnetizing
current. It is the safest test to determine those characteristics,
by applying the nominal operating voltage of the transformer. Figure 2 shows the block diagram of the no‐load
test, characterized by the Aron connection (2‐wattmeters
method). This test is characterized by open high‐voltage terminals (VH). VL are the low‐voltage terminals; W1 and W2
are wattmeters (which indicate the losses in W); V represents
the voltmeter; A1, A2, and A3 are ammeters; and F is a frequency meter.
2.2 | Short‐circuit test
This test is normally used to determine stray losses, but its
results also allow to determine the short‐circuit impedance
(ZCC), as well as the internal resistance and reactance of
transformers. For this, the low‐voltage terminals (VL) are
short circuited, and a voltage applied to the high‐voltage
terminal is varied so that the primary current observed on
the ammeter (Figure 3) is equal to the nominal current. This
voltage, referred to as VCC, represents from 10% to 15% of
the rated voltage (VR) of the high‐voltage side (VH). Under
these conditions, the core losses, which vary with the square
of the voltage, can be neglected. Thus, through the readings
of power, voltage, and short‐circuit current, the variables of
interest may be determined and applied in the determination
of the winding losses. More specifically, the impedance ZCC
will be considered in percentage, where ZCC = 100VCC/VR.
2.3 | Winding resistance test
The windings of the primary and the secondary of a transformer have an electrical resistance, as any conductor.
Respectively, such resistances are usually indicated by R1,
R2, R3 and R4, R5, R6 (1 for each phase). In this paper, we
consider R1, R2, and R3 as being equal to each other and
referred to as RLV. Similarly, R4, R5, and R6 are referred to
as RHV. These resistances exert a double effect on the operation of the transformer: determining an ohmic power loss in
the primary and the secondary and producing a loss of energy
Sections
Thickness
Temperature
Network
Channels
RHV = RLV
Total Losses
Network
IEX
Lc
T
LT
RHV
RLV
TA
Winding
Losses
Network
LAl
ZCC
FIGURE 1
Proposed system for the neural network
FIGURE 2
Scheme of the no‐load test
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FIGURE 3
ET AL.
Scheme of the short‐circuit test
FIGURE 4
by Joule effect. To reduce this winding loss to limits
established by technical standards, it is necessary to make
such resistances small enough. This is done by appropriately
choosing the section of the winding conductors. In
conducting this test, the electrical resistances of each winding
must be registered, phase to phase, as well as the terminals
where they have been measured and their respective temperatures. This procedure is performed by means of the Wheatstone bridge. It is also important to note that the resistance
values should always be obtained at the reference temperature, which is generally set at 115°C for dry‐type
transformers.
After obtaining the experimental data and preparing the
database, these data have been used for training the neural
networks, as shown below.
3 | IMP LEM ENTAT ION OF THE PRO POSED
TOOL
All neural networks implemented in this work have an input
layer, a single hidden layer, and 1 output layer with a single
neuron. In addition, for each of the neural networks, the heuristic defined by Kolmogorov26 was applied to determine the
maximum number of neurons in the hidden layer, considering
a feed‐forward multilayer perceptron network. Since the
structure of Figure 1 has been defined, it was necessary to
train each neural network individually. For this, from the total
of the transformers design data, only 75% were used. The
remaining 25% were used for validation of this training. Also,
this training was performed by the modified Levenberg‐
Marquardt algorithm as learning rule.27,28,24 In the Winding
Losses Network and Total Losses Network (Figure 1), we
used the tangent sigmoid activation function.5 In the Temperature Network (Figure 1), in turn, we used the hyperbolic tangent activation function.29
3.1 | Winding losses network
The architecture of the neural network responsible for identifying the processes related to the transformer winding losses
is shown in Figure 4. The variables that make up the input
vectors of this neural network were defined through the
Neural network architecture for the transformer winding losses
short‐circuit and windings resistance tests. Among these variables, TA represents the environment temperature in Celsius.
On the other hand, the output vector of the neural network
consisted of a single variable, LAL, which represents the
winding loss in watts. Based on the method of Kolmogorov
and considering this number of inputs, the maximum number
of neurons in the hidden layer was set equal to 9. However, it
was empirically determined that the best results were
obtained when we used only 5 neurons in the hidden layer.
3.2 | Total losses network
This neural network was used to identify processes related to
the transformer's total losses, which include both the winding
losses (considered variables) and the core losses (considered
fixed). Its architecture is shown in Figure 5. The network
input vectors were defined from the short‐circuit and no‐load
tests, besides the power lost due to the winding losses, estimated by neural network discussed in the previous section.
More specifically, the input variables of this network were
defined as follows: RHV, RLV, TA, IEX (which represent the
excitation current in amperes), and LAl and LC (which represent the core loss in W). However, owing to their similarities,
RHV and RLV were considered equal.1 In turn, the output vector of the neural network has a single output, LT, which represents the total loss. Finally, based on the Kolmogorov
method, considering the number of inputs, the maximum
number of neurons in the hidden layer was set to 11. However, empirically, it was determined that the best results were
obtained when we used 10 neurons in this layer.
3.3 | Temperature network
Figure 6 shows the neural network used to estimate the
temperature. The input variables of this neural network are
as follows: the cross‐sectional area of the conductors used in
the windings (in mm2), referred to as Sections; thickness of
the winding sheet (in mm), referred to as Thickness;
1
It is possible because the resistances of the windings are balanced per phase
in both high and low voltages, since it is required by standard. The transformer leaves the factory with balanced windings. Their imbalance is generated by the load, since the system phases may be unbalanced.
FINOCCHIO
ET AL.
5
do not influence the internal temperature of the transformers.
Instead, simply, it is not necessary to include them in the neural network input parameters because, as stated, these parameters correspond to the construction methodology of each
manufacturer. Thus, with respect to the ventilation ducts,
the only unknown is, in fact, the number of ducts.
In the following sections, the results of training of all
developed neural networks are presented.
4 | R E S U LTS
In this section, we present the results of the training and execution of the 3 neural networks implemented in the present
work.
4.1 | Winding losses network
FIGURE 5
Architecture of the neural network for the transformer total
losses
and the number and dimensions of ventilation channels
(Channels) and the output of the Total Losses Network
(Section 2), referred to as LT. In the output of this neural
network, the estimated variable is the temperature inside the
transformer, which indicates the temperature rise of the windings. This temperature must be added to the environment one,
thus determining the temperature of the winding in nominal
operating conditions. Again, using the Kolmogorov method,
the indicated maximum number of neurons in the hidden layer
is 9. However, the final implemented hidden layer has only
8 neurons.
Unfortunately, the presented model allows only to foresee
the temperature of a single point of the transformer. However,
when performing a test on the transformer, for example, a
short‐circuit test, one can verify that the temperature of the
transformer is not homogeneous, existing in higher‐temperature zones.4,1 Therefore, in the training of the neural network,
it is important not to consider an average temperature. This is
because during normal operation of the transformer, the
temperature in a hot spot may increase excessively, to the point
of damaging the equipment.8,2 So, naturally, in this work, for
the training of the neural network, the considered temperature
is the one with the hottest point in the transformer. In turn, the
hottest point was determined experimentally with the aid of a
set of thermocouples during short‐circuit tests.
It is also important to note that the proposed methodology
is dependent on the constructive method of the transformer.
Especially, with respect to the ventilation ducts, dry‐type
transformers based on a given construction method have no
ventilation ducts with different geometries and positions.
Normally, such parameters are well defined in the constructive methodology.6 This does not mean that these parameters
Figure 7 shows the results of the winding loss generalization
done by the Winding Losses Network. In this graph, the
values obtained experimentally and the ones obtained by
the neural network are compared. One can check that the
relative error is very small. In fact, it was calculated that the
average relative error μ = 0.267%, the standard deviation
σ = 0.204%, and the variance just σ2 = 0.042%. These results
show that the neural approach could adequately model the
winding losses processes.
After validating the neural network, we examined the
relationship between the winding losses process and the
short‐circuit impedance and temperature. The results are
shown in Figures 8 and 9. Figure 8 shows the behavior of
the winding loss relating to the short‐circuit impedance variation. In this simulation, the environment temperature was
considered constant, and mean values were used for the resistance of high‐voltage and low‐voltage bobbins (RHV and RLV,
see neural network architecture in Figure 4). One can observe
in this graph that in the intervals between 1.6% and 1.8% of
short‐circuit impedance, there is an increasing winding loss.
Beyond this range, the curve remains constant. That behavior
is due to the thermal effect inside the transformer. Generally,
FIGURE 6
Architecture of the feedforward multilayer perceptron network
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6
this slight variation in impedance is more difficult to be
observed experimentally, with this kind of detail, in traditional designs of transformers. However, this information is
FIGURE 7
Network behavior for the winding losses
ET AL.
important because this quantity numerically represents the
relationship between the circuit voltage and the nominal voltage of the transformer. In turn, Figure 9 shows the maximum
winding loss relating to the variation in the transformer temperature. To generate this graph, we considered average
values for ZCC and for the high‐voltage and low‐voltage
impedances, while varying the room temperature. This graph
shows that there is a direct dependency between the temperature and the winding losses. In the intervals between 12°C
and 15°C, the values of the winding loss have rapid and uniform increases, reaching 1250 W, which is an accepted value
according to standards.
From this temperature until 34°C, the losses remain constant. This is due to the inertia of the thermal process, in
which the saturation threshold of the material is reached
and then the loss becomes constant with the temperature
increase. This identification is very important for the designer
to select the most suitable type of aluminum, based on the
knowledge of the temperature maximum limit related to the
degree of purity of the material. However, it would be difficult to be observed in detail in experimental measurements
without the aid of the proposed tool.
4.2 | Total losses network
Figure 10 shows the results of the total loss generalization
done by the Total Losses Network, comparing the values
obtained experimentally with the ones obtained by the neural
network. One can see that the output values of the neural network approached the values obtained in laboratory tests,
which show the success of this application for this type of
problem. Comparing experimental data with the output of
the neural network, one can see that the relative error is very
small. In fact, it was calculated that the average relative error
μ = 0.774%, with standard deviation σ = 0.567%, and variance σ2 = 0.322%.
FIGURE 8
Variation of the winding loss versus impedance
FIGURE 9
Variation of the winding loss versus temperature
FIGURE 10
Behavior of the total loss network
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ET AL.
After validating the Total Losses Network, we examined
the dependence of the total losses regarding the no‐load
losses, winding losses, and excitation current, as presented
in Figures 11–13, respectively. For this, with the aid of the
validated neural network, we simulated the total losses
considering the minimum, average, and maximum values of
the following quantities: RHV, RLV, IEX, LAL, and LC (Section
2). In each graph, 3 curves are shown, which represent the
maximum, medium, and minimum of the input variables. It
is noteworthy that the simulations with the maximum and
average values are hypothetical, so as to identify possible
problems in the project, both in the constructive aspects and
in the quality of the material used.
Figure 11 shows the behavior of the total loss relating to
the no‐load loss, considering 3 different values for RHV, RLV,
TA, IEX, and LAL. In this graph, one can verify a linear trend
in all 3 curves, which indicates a thermal effect. It was also
observed that the curves of “maximum values” and “medium
values” show total loss values close to each other, even
considering the average and maximum values of the quantities already mentioned. This behavior shows that the estimation of phenomena was performed satisfactorily, because if
there were any nonlinearity in the simulations, it would
represent that the network had not learned the process for
various situations. In this regard, it is essential to mention that
the simulated values in the curves are in disagreement with
the values recommended by standard. However, this condition is an extreme project situation that provides valuable
information to the designer, since it identifies the worst
condition to the transformer's design. This kind of detail
would be almost impossible to verify through either conventional tools or experimental testing. This type of identification is also valuable so as to provide information to the
transformers designer so that the project will be set neither
below the blue curve (minimum values) nor above the black
curve (maximum values). This is because it is preferable to
dimension the transformer so that it has a better performance
FIGURE 11
Variation of the total loss versus no‐load loss
7
FIGURE 12
Variation of the total loss versus winding loss
FIGURE 13
Variation of the total loss versus excitation current
at an operation midpoint. Thus, it will continue to be efficient
at a point above or below the normal.
Figure 12 shows the behavior of the total loss relating to
the winding loss, considering different values for the quantities RHV, RLV, TA, IEX, and LC. One can observe that the
behavior of the total losses is similar to that in Figure 11.
The difference is that the average and maximum values are
farther. This analysis shows the sensitivity of the developed
neural tool. This observation is also very particular, because
it presents the contribution of the winding losses over the
total losses of a transformer.
In Figure 13, the total loss behavior with respect to the
excitation current is presented, considering 3 different values
of RHV, RLV, TA, LAl, and LC. In Figure 13, one can see the
influence of the excitation current in the overall result of
the total loss. This detail would be almost impossible to be
observed with conventional tools, owing to the complexity
found in the mathematical modeling of the phenomenon.
Generally, the weighing of the excitation current is restricted
only to no‐load core losses.
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ET AL.
5 | C O NC LU S I ON
FIGURE 14
Evolution of the error in the Temperature Network training
process
FIGURE 15 Comparison of internal temperature values so as to test the
generalization capacity
Currently, the design of dry‐type transformers is almost
entirely based on the designers' experience. Typically, several
attempts are necessary so as to achieve satisfactory results.
This is mainly due to the large number of variables involved
in calculations and to the fact that the relationships between
these variables are not well defined. In this article, we evaluated the use of ANN in modeling the influence of various
design parameters on the internal temperature of dry‐type
transformers. For this, we used results of practical tests with
300 transformers. Qualitative and quantitative results showed
that ANN were able to model, with excellent accuracy, the
relationship between various design parameters and the
losses and internal temperature rise in dry‐type transformers.
Thus, by means of simulations using the validated neural
networks, one could see details that are almost impossible
to be observed with conventional tools, owing to the
complexity in the mathematical modeling of phenomena
involved. Thus, the developed system can be used to study
the phenomena that define and relate the variables involved
in the design. Evidently, this will have a direct influence on
the quality of future transformers designs. This is because,
based on the notion of behavior extracted via neural
networks, it will be possible to obtain closer‐to‐real functions
to model the interrelationships between the design parameters. Evidently, this work can be extended so that not only
the internal temperature but also other dry‐type transformers
features could be modeled with the aid of ANN. Certainly,
this would minimize the time taken in the design and increase
the precision when estimating the final characteristics of this
type of transformer. Unfortunately, the proposed methodology is dependent on the constructive method of the transformer. However, by following the methodology discussed
here, the readers may conduct their own tests and train a
neural network that models their own constructive technique.
4.3 | Temperature network
The evolution of the error in the Temperature Network training process as a function of the number of iterations is shown
in Figure 14. The training stopping criterion was the stabilization of the mean square error. With the completion of the network training, the project initial parameters estimated by the
network are compared with the values actually used in the projects, through quantile‐quantile graphs.30,19 A comparison
between the results estimated by the Temperature Network
and the real values obtained in the tests with 75 transformers
(used to validate the proposed system) is shown in the
quantile‐quantile graphic of Figure 15, for the purpose of validating and testing the generalization quality of the network.
In this graph, the output of the neural network fits the experimental data with a coefficient of determination R2 = 0.988.
These results confirm that the neural network is well fitted,
having thus a good generalization. These facts demonstrate
the ability of the Temperature Network to solve the problem.
6 | L IST O F S Y M B O L S A N D
A B B R E V I AT IO N S
6.1 | Abbreviations
ANN
artificial neural networks
Channels
number of winding channels
FEA
finite element analysis
Sections
area of the conductors
Thickness
thickness of the windings sheets
6.2 | Symbols
μ
average relative error
A1, A2, A3
ammeters
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ET AL.
IEX
no‐load excitation current
LAL
winding losses
LC
core losses
LT
electrical losses
2
R
coefficient of determination
RHV
high‐voltage coil resistance
RLV
low‐voltage coil resistance
R1, R2, R3
electrical resistances of primary phase of the windings
R4, R5, R6
electrical resistances of secondary phase of the windings
σ
standard deviation
TA
environment temperature
σ2
variance
VCC
voltage at the common collector
VH
high‐voltage terminals
VL
low‐voltage terminals
VR
rated voltage
W 1 , W2
wattmeters
ZCC
short‐circuit impedance
F
frequency meter
kV
kilovolt amperes
kVA
kilovolt‐amps
T
Temperature
V
voltmeter
W
watts
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How to cite this article: Finocchio, M. A. F., Lopes,
J. J., de França, J. A., Piai, J. C., and Mangili, J. F., Jr
(2016), Neural networks applied to the design of
dry‐type transformers: an example for analysis of
winding temperature and increase in thermal quality, Int. Trans. Electr. Energ. Syst., doi: 10.1002/
etep.2257