IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009
705
Distribution Transformer Losses Evaluation:
A New Analytical Methodology and Artificial
Neural Network Approach
Adriano Galindo Leal, Member, IEEE, José Antonio Jardini, Fellow, IEEE, Luiz Carlos Magrini, and Se Un Ahn
Abstract—The aim of this paper is to propose an analytical
methodology and discuss some alternatives of artificial neural
network models in order to evaluate losses in distribution systems,
particularly in distribution transformers.
The procedure can also be extended to other components of the
distribution system (secondary and primary network and HV/MV
transformers). This is accomplished by using the utility’s database
such as the consumers’ monthly energy consumption and the typical load curves of each class of consumption and type of activity
developed.
Index Terms—Information systems, neural networks, power distribution, power transformer losses.
I. INTRODUCTION
C
OMMONLY, distribution system losses are estimated because of the unavailability of suitable metering systems.
For billing purposes, only energy meters are installed at the
consumer’s residence or commerce rather than demand meters,
which have a high cost when compared to the consumer’s bill.
For losses estimation many proposals were put forward. All of
them, including ours, have a lack of accuracy mainly because
the consumer load profiles are different on weekends and even
during the weekdays as appliances are turned on/off in a random
way [1]. These inaccuracies have always been present, even in
the procedures described in this paper; although it showed significant progress in the treatment of the random variation.
In [2] and [3], the primary feeders, the distribution transformers and the secondary network for three-phase load flow
calculation are modeled. The consumers’ load profiles are
represented on an hourly basis for power and then the losses
are calculated. Several calculations were done varying the load
level (FL), the transformer capacity (XFCAP), and the total
conductor length (CL). The results were then used for training
an artificial neural network (ANN) to thereafter estimate the
losses in actual feeders considering its proper FL, XFCAP, and
CL. This is an improvement of other methods, because it uses
the consumer’s daily load profile, and avoids the use of loss
Manuscript received October 14, 2007; revised August 15, 2008. First published February 27, 2009; current version published April 22, 2009. Paper no.
TPWRS-00728-2007.
A. G. Leal is with Elucid Solutions, São Paulo, Brazil (e-mail: leal@ieee.org).
J. A. Jardini is with EPUSP-PEA, São Paulo, Brazil (e-mail: jardini@pea.
usp.br).
L. C. Magrini is with UNIP, São Paulo, Brazil (e-mail: magrini@pea.usp.br).
S. U. Ahn is with CPFL—Companhia Piratininga de Forca e Luz, Campinas,
Brazil (e-mail: seun@cpfl.com.br).
Digital Object Identifier 10.1109/TPWRS.2008.2012178
and diversity factors. Although, as in all methods inaccuracies
still exist.
Three-phase load flow is a suitable tool for the calculation of
unbalanced load/lines conditions; nevertheless, the authors here
consider this unnecessarily complex to be used, on account of
the inaccuracies existing in the load profile. In fact, at least in
tropical countries, the load profiles of the weekdays are different
and there is no time correlation among the consumers loading,
which makes the task of setting up the load flow a source of
error.
The measurements of the consumers’ daily load profiles reand stanported in [1] led to the evaluation of the mean
profiles in several type of consumers and
dard deviation
distribution transformers using sets of 15 to 30 measured daily
values of various type of consumers (resload profiles. The
idential, small/medium size commercial and industrial) were of
, which clearly indicates a large variathe same size as
tion of the load at any time of the day. This is because the total
consumer load is composed of energy uses of almost equal size
and they are not turned on/off at the same time every day. On
the other hand, the measurements carried in distribution transof about 20% of
. However, for
formers indicated an
large consumers such as shopping malls, medium/heavy size inare very low [4].
dustries the values of
In [5], measurements in selected feeders were carried out.
These results were then extrapolated to other feeders using properties like the installed capacity of the distribution transformers
and the feeders’ length. A similar procedure was adopted by
[6] and [7], where instead of measurements, load flow modeling
was used for the calculation of selected feeders. Their extrapolation to other feeders followed a similar approach to that presented in [5]. As discussed above, the load profile variation was
not considered in all these methods.
Analyses to establish load profiles for engineering studies,
supported by ANN, wavelets theory and clustering processes
were conducted in [8] and [9].
In this paper, a method that includes the load variability
aimed at obtaining improvements for the estimation of distribution losses is presented.
The organization of this paper is as follows: initially, a procedure to obtain the transformer daily load profile, based on the
consumer’s data, is described. Then, the losses calculation regarding the load variability is presented. Finally, a method considering the ANN technique as an alternative procedure is also
presented.
0885-8950/$25.00 © 2009 IEEE
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009
This work started out by using the contributions presented
in [1] and [10]–[16]. The consumer and transformer daily load
profiles (active power) were represented by their average (M)
and standard deviation (S) profiles.
It should be noted that nearly all calculations in a distribution
system present inaccuracies as they are based on the best data
available, which may be imprecise and incomplete, and statistical behavior. In order to fill in the incomplete data, some simplifications and assumptions are made. In this case, for instance,
losses in the transformer are evaluated using active power load
profiles instead of apparent load profiles, which must be adjusted by using a certain factor. In addition, the profiles of the
consumers are chosen from a statistic subset of representative
consumers, the weekend and working day profiles are in general
different, thus including another source of inaccuracy. Because
of that, errors in the losses calculation, even at the level of 25%
may be considered acceptable. Another source of error comes
from the billed energy metering that has some inaccuracies due
to measuring transformers and metering, which are in the order
of 0.5% to 1.0%. This is nearly at the same level of the transformer total rated losses (1% to 2%).
Fig. 1. Mean and standard deviation daily curve of a transformer (p.u.).
II. TRANSFORMER LOADING
The methodology used in this paper to define the transformer
loading is based on the mean and standard deviation load profiles, a procedure described in [1] and partially summarized
herein.
Studies characterizing the consumers daily load profiles were
reported in [1] and [10]–[12]. In these references, the demand
measurements of several types of consumers (residential, commercial, and low voltage industrial consumers) were carried out.
The consumers’ daily load profiles were set up so as to record
96 points (i.e., active power was recorded for each minute and
then averaged at intervals of 15 min).
For each consumer and distribution transformer about 15
and standard
daily profiles were considered. The mean
profiles were determined and set to characterize
deviation
the consumer and the distribution transformers.
For ease of manipulation, the demand values were normalized (per unit) by the monthly average demand Dav (i.e.,
monthly energy, , divided by the number of hours/month,
). Fig. 1 shows the mean and average profiles of
a distribution transformer (in p.u.).
and
For an th consumer
, where
and
are the mean values of
the demand at time , in p.u. of the monthly average demand
and
are
(Davi) and in kW, respectively. Similarly,
the standard deviation in p.u. and real values, respectively.
A procedure to aggregate (add) the consumers’ demand in a
distribution transformer was also developed [1]. If it is considered a distribution transformer with an number of consumers
of the type and of the type, the aggregated demand values
(in kW) can be calculated using the following:
Fig. 2. Transformer’s daily load curve stratified in 11 curves.
(1)
are the mean demand of the consumers
where
aggregation, and of the p, q consumers, respectively. Similarly,
are the standard deviations.
Both mean and standard deviation of the aggregated values
may be translated into p.u.
by simply dividing the real (kW) values by the transformer’s rated power.
The demand value within an interval is assumed to follow a
Gaussian distribution, so the figure with a certain non-exceeding
probability can be calculated through
(2)
is a constant that defines the probability in a Normal
where
90% of the
Distribution [18]. For example, for
values will be below
and 10% above.
Fig. 2 shows a set of 11 profiles (in p.u.) with probabilities
%, and
%, re(from bottom to top) % % %
spectively. These sets of profiles were used in [10] to evaluate
the distribution transformer’s loss of life due to loading. The
same set of profiles is used here to evaluate the losses in distribution transformers.
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LEAL et al.: DISTRIBUTION TRANSFORMER LOSSES EVALUATION
707
TABLE I
STRATIFICATION PROFILES
Fig. 3. Load losses calculation. Analytical procedure.
In (4), 96 multiplication operations are needed to evaluate the
term, 11 more multiplications to consider
,
to evaluate the series losses in
and one multiplication
one single transformer. This gives an idea of the computation
time required in the whole process.
being
The total losses of the distribution transformer
and the above
composed by both the no-load losses
on-load losses
III. DISTRIBUTION TRANSFORMER LOSSES
CALCULATION—GENERAL APPROACH
A. Main Equations Used in the Approach
It is well established that losses in a distribution transformer
are produced when the current flows through the coils. They also
appear whenever a magnetic field circulates around the core. So,
they can be classified into on-load losses and no-load losses [9],
[17].
do not vary according to the transformer
No-load losses
loading but according to the voltage; thus, it may be considered
constant for losses calculation purposes.
vary according to the transformer
On-load losses
loading and are responsible for the largest part of the load
losses. This work will mainly focus on such losses.
represents the transformer loading at the interval of
As
one profile, then, the series losses
can be written as [14],
[15]
(3)
where
is the transformer winding resistance and represents
. Should more precision
the series losses of the rated power
be needed, this value will have to be corrected with the variation
of the transformer’s internal temperature (this may be of particular interest in countries were the transformers are loaded above
their rated capacity).
Equation (3) can be applied to all 11 profiles . For instance,
if
%, then this profile can be the representative of all
the profiles with probability 15% to 25%, which represents a
.
participation factor (kpf) of 10%
, the reNotice in Table I that nine profiles have
maining two having profiles with
.
can be expressed
Thus, the total average series losses
as
(4)
(5)
B. Data Handling
From the Distribution Utility database the following information was used: and characteristic profile (in p.u. of the
monthly average demand) of the representative type of consumers (see upper box in Fig. 3); the transformer parameters
(rated power, series and no-load rated losses; left box at bottom
of Fig. 3), the consumers’ type and energy consumption per
connected to each transformer (bottom box in the
month
middle of Fig. 3).
and profiles (in kW) of each consumer (in a transThe
and values (in
former) are calculated by multiplying the
p.u.), that represent the type of consumer, by its average demand
; see Section II).
The and profiles of the transformer are obtained through
(1), hence the 11 profiles depicted in Fig. 2 (right box in the
middle of Fig. 3).
Finally, the load losses can be evaluated using (4). Fig. 3,
shows the procedure used here termed “Analytical Procedure”.
C. Application
The seven-day load profiles of the 57 distribution transformers at CPFL (a Brazilian Distribution Utility) were
recorded. The average series losses for each load profile, as
well as their average values, were also determined
.
Next, the
and curves of each transformer were also determined. This was obtained using the same seven-day load profile curves. On the other hand, the series losses for the 57 transwere determined using (4).
formers
The error distribution
between
those two calculations is shown in Fig. 4. As it can be seen, the
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009
Fig. 4. Error distribution.
Fig. 5. Result of Cluster 2 (commercial and industrial consumers).
errors are small having a mean value of 0.3%, which means that
the analytical procedure led to right results.
IV. DISTRIBUTION TRANSFORMER SERIES LOSSES
CALCULATION—BASIC ANN APPROACH
A. ANN Model
ANN is a useful technology often used to get interpolated results. It is suitable in many situations such as when the mathematical relationship among variables is unknown, or when there
are few input data for interpolation. It can improve the time
computation efficiency and simplicity while manipulating data.
Aside of improving the computation efficiency, the accuracy of
the results is reduced, a trade off to be searched. Therefore, an
initial ANN model was developed using the same input data of
the analytical procedure. In such a model, there is no gain expectation neither in accuracy nor in computation efficiency; thus, alternative ANN models were developed to improve the process,
of course using the experience of previous models.
As mentioned, the inputs for the ANN model used are the
transformers and profiles, the output being the series losses
. The number of neurons and layers were defined by trial
and error tests in the MLP (multilayer perception) model. The
supervisioned and back propagation training types were also
chosen.
The calculations presented in Section III (analytical procedure) were performed for a set of 61 485 transformers of the
distribution utility. The losses were regarded as “true values”
mainly because the measurements of the losses were not available. Part of these calculated values (losses) were used during
the training stage and part of them for testing the training efficiency.
The system where the distribution transformers are located
has 608 primary feeders, 2.2 million consumers and supplies
around 9 TWh/yr.
B. Clustering
The set of 61 485 transformers’ daily profiles ( and ) obtained through (1) and normalized in p.u. by the transformer’s
rated power, were put under cluster analysis. The number of
clusters specified, which followed the Euclidian distance criterion, was equal to 10.
Fig. 6. Result of Cluster 9 (residential consumers).
Note: Actually, several tests considering a different number of
clusters (up to 30 clusters) were carried out. It was continuously
determined the largest distance from one curve to the center of
the cluster and also the distance among clusters. These distances
indicated that it would be necessary to have 30 clusters to get
them well grouped.
However, many of these clusters had a small number of transformers; thus, they were discarded. Those transformers within
the discarded clusters were transferred to other clusters. At the
end, a number of ten clusters whose mean values were kept for
subsequent calculations were adopted.
This means that transformers with similar profiles are assigned to the same “cluster box”. Figs. 5 and 6 show the average
curves of two different clusters.
It can be seen that the mean profiles differ because one pertains to typical commercial/industrial loads (Fig. 5) whereas the
other shows the characteristic peak (at 20:00 h) of a residential
load (Fig. 6).
Table II, shows the characteristic parameters of each cluster.
The shape of Clusters 1 through 6 looked like Fig. 5, whereas
Clusters 7 through 10 resembled to the curve shown in Fig. 6.
The profile patterns of the ten clusters correspond to the two
load types previously mentioned. The difference in clusters is
mainly due to the peak value.
Since the profiles are normalized by the transformer’s rated
power, apart from the shape, the main characteristic of the
cluster will be its loading state.
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LEAL et al.: DISTRIBUTION TRANSFORMER LOSSES EVALUATION
TABLE II
CHARACTERISTICS OF THE CLUSTERS
709
TABLE III
TRANSFORMERS DISTRIBUTION WITHIN THE TEST AND TRAINING VECTORS
TABLE IV
PERCENTAGE OF TRANSFORMERS WITH ERROR LESS THAN 10%
The objective of the clustering process was to evaluate
whether better results are obtained by training one ANN for
each cluster or only one for all the transformers.
From the clustering process it can be concluded that:
• 42.5% of the transformers, represented by cluster 1, are
operating under extremely low loading levels. This cluster
represents transformers having commercial and/or residential loads and which could be used for future reallocation
during the system’s expansion program;
• 46.2% of the transformers have typical load conditions of
areas with residential consumers (Clusters 7 through 10);
• 1.1% of the transformers, those pertaining to Clusters 5 and
8, are much more loaded.
C. Results
The training vector, as well as the test vector, are formed by
a certain group of inputs and one output, constituted by:
• 24 points of the transformer curve (in p.u., 1 point per
hour);
• 24 points of the transformer curve (in p.u., 1 point per
hour);
term, calculated through the method de• the
scribed in Section III, constitutes the output variable.
The parameters used in all the simulations of the ANN model,
will be described next. Table III shows the amount of elements
in both the training and test vectors used.
1) Architecture: The ANN architecture is composed by: four
layers, the input-layer having 48 neurons, the second and third
layers having 35 and 24 neurons, respectively; and the output
layer having only one neuron.
2) Training Process: The ANN model used was set up to
perform nine internal iterations and a total of 9000 iterations.
The training process finishes when the tolerance is below 0.15%
(or when the total number of iterations is reached).
The errors in each cluster as well as the percent of cases with
errors below 10%, here called “error index” were also evaluated.
The error indexes in all clusters are shown in Table IV. It can be
observed that when the E000 training is used (which would be
preferred due to its simplicity) the accuracy is not good, except
for clusters with a small number of transformers. The error in
around 92.5% of the transformers (for the ANN trained specifically for each cluster) was below 10%. This leaves 7.5% with
an error greater or equal to 10%. It is also shown (Table IV)
the breakdown of the clusters and the total (all transformers).
It should be noted that for all transformers 4.2% is within 10 to
30%, 2.3% is within 30 to 100% and 1% was greater than 100%.
The global error (sum of all 61 485 transformers) regarding
the analytical procedure as reference was 9.7%.
Although the results can be considered as satisfactory, the
procedure involved lots of multiplying operations, due mainly
to the number of layers and neurons: 48*35 (input to 2nd layer);
35*24 (2nd to 3rd layer), and 35*1 (3rd layer to output layer)
products. Therefore, the calculation being more time consuming
than the analytical procedure (see Section III-A).
In the next section, some ANN alternative architectures aimed
at reducing the computation time (although not strictly necessary as today’s microcomputer can handle this task), though at
the expense of reducing the accuracy, are presented. The first
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009
attempt is to reduce the input data and then reduce also the intermediate calculations.
Now, it is important to know the computation time involved
for the 61 485 transformers.
In the analytical procedure [with 96 points for (4)], it took
15.1 min to process (1) and 5 min to process (4), giving a total
time of 20.1 min.
If instead of 96 intervals it would be considered 24 points,
then, the time needed to process (4) would be 1.3 min.
With the ANN procedure it took also 15.1 min to process (1),
1.5 min to obtain the transformers clustering process. and 13 s
for the losses calculation (note that for the ANN procedure the
profile was scaled back from 15 min to 1 h, so the latter term
of the computational time would become worse if the full 96
points were considered.) The total time being 16.6 min. It should
be noted that the training process needed 1.1 h in each cluster.
The losses calculations by most of the distribution utilities are
carried out on a monthly basis; conversely, there is no need to
update the training process at that same basis (once a year would
be adequate).
Fig. 7. Consumers’ daily load curves Type 1 (residential).
V. LESS TIME CONSUMING ANN APPROACH
Two other architectures aiming at improving the calculation
time (efficiency), are put forward in this paper.
A. Alternative 1—Reduction of the Input Data
The training and the test vector are now formed by a group of
inputs and one output, constituted by:
profile at around 03:00,
• four inputs of the transformer
14:00, 19:00, and 21:00;
• two inputs of the profile at 12:00 and 18:00;
• the value of
constitutes the output variable.
The parameters used in all the simulations of the ANN model
are the same as those used previously, except that the layer neurons are 6 for the first layer, 18 and 10 for the hidden layers, and
1 for the output layer.
The same clusterization result of the initial approach was
used. In addition, for comparison purposes, the same amount of
elements in the training and test vectors was used (see Table III).
Now, from the total estimations 83.2% were obtained with errors below 15%. The results, here considered poor indicate that
probably, at least from the authors’ viewpoint, a new clusterization process should be done to improve the whole process.
The global error (for all the 61 485 transformers) was 25.9%.
The results of the analytical procedure were again taken as the
reference.
The computation time was 1.5 min to process (1), 0.3 s to
calculate both losses and cluster process, making a total time of
1.8 min.
B. Alternative 2—Reduction of the Intermediate Calculations
In this approach, all consumers were classified into four types
(Figs. 7–10). This classification enabled us to calculate (in the
database) the amount of each type of consumer connected to the
transformer and its total energy consumption.
In this approach, the training vector and the test vector are
formed by a group of inputs and one output, constituted by:
Fig. 8. Consumers’ daily load curves Type 2 (industrial).
Fig. 9. Consumers’ daily load curves Type 3 (flat).
• eight inputs representing the number of consumers and the
mean consumption of each consumers type;
• one input representing the transformer rated power;
also constitutes the output variable.
• the value of
Again, the parameters used in all the simulations of the ANN
model are the same as before, except that the layer neurons are
9 for the first layer, 16 and 8 for the hidden layers, and 1 for the
output layer.
A new clusterization process whose results are presented in
Table V, was performed. Here, Qty is the mean value of the
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LEAL et al.: DISTRIBUTION TRANSFORMER LOSSES EVALUATION
711
TABLE VI
COMPARISON OF RESULTS
Fig. 10. Consumers’ daily load curves Type 4 (commercial).
TABLE V
CHARACTERISTICS OF THE CLUSTERS
VII. COMPARISON OF THE PERFORMANCES—RESULTS
consumer type, considering all transformers within a cluster;
whereas kWh represents, similarly, the mean consumption of
the consumer.
The global error (for all the 61 485 transformers) regarding
the analytical procedure as reference was 22.8%.
The computation time of the losses was 2 s whereas the clustering process 0.9 min. The calculation time [15.1 min, using
(1)] was eliminated. In this procedure the accuracy may be improved by considering more than four types of consumers, of
course at the expense of increasing the computation time.
VI. EXTENSION TO OTHER PARTS
OF THE DISTRIBUTION SYSTEM
The same procedure used for the distribution transformers is
also applicable to evaluate the series losses in the secondary and
primary network, as well as for the HV/MV transformers. For
and curves
instance, for a section of a primary feeder, the
of the transformers, beyond this section can be aggregated using
(1). A specific ANN and test procedure shall be carried out to
train and calculate the primary feeder losses.
This methodology could be incorporated into a Geographical
Information System (GIS) so as to turn its calculation procedure
more independent from the user interaction.
Table VI shows the global loss values of all the distribution
transformers.
From the analysis presented it can be concluded that:
• as the load curve was better represented, the initial ANN
architecture should be the best one obtained. The errors
were less than 10%. The disadvantage of this method is
that the amount of mathematical operations, necessary to
obtain this result, is greater than that needed in the analytical procedure;
• the second and third architectures showed global errors
of 25.9% and 22.8, respectively. They were less accurate;
however, their processing times were reduced as those architectures required less mathematical operations;
• it should be emphasized the fact that the “analytical
method” was assumed to lead to the correct results (true
values). The use of the third architecture together with the
measured values of losses may constitute a method with
reasonable precision and adequate processing time.
• The Alternative 2 model offered a global accuracy as good
as the second one. Another advantage is that it dispenses
and curves of both conwith the calculation of the
sumers and transformers.
Through the present ANN application, in terms of accuracy
and time computation, a reasonable estimation of the losses
in a distribution system can be achieved. However, it must be
pointed out that the parameters used to train the ANN have not
been exhaustively optimized, as that was not the main objective
on this work. Therefore, there still are some improvements possible on the accuracy.
The Alternative 2 ANN Architecture is even faster than the
other two options, as it does not need to calculate the consumer
profiles and the aggregation of the distribution transformers.
Another advantage, aside of the calculation speed, is that the
utility does not need to perform measurements to evaluate the
load profile for all the type of consumers, which is a costly operation.
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networks using typical demand curves of consumers and artificial
neural networks,” Ph.D. dissertation, Polytechnic School, Univ. São
Paulo, São Paulo, Brazil, 2006, 158 pp., in Portuguese.
[15] A. G. Leal, J. A. Jardini, L. C. Magrini, S. U. Ahn, H. P. Schmidt,
and R. P. Casolari, “Distribution system losses evaluation by ANN approach,” in Proc. 2006 IEEE PES Power Systems Conf. Expo., Atlanta,
GA, 2006.
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losses: The concept of equivalent current,” in Proc. Int. Conf. Electricity Distribution, Barcelona, Spain, 2003.
[17] B. C. Degeneff, “Power Transformers,” in The Electrical Engineering
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Adriano Galindo Leal (M’06) was born in São Paulo, Brazil, on September
19, 1971. He received the B.Sc. degree in electrical engineering and the M.Sc.
and Ph.D. degrees from Polytechnic School at University of São Paulo in 1996,
1999, and 2006, respectively.
For 11 years, he worked as a R&D Engineer for the GAGTD research group
in the Polytechnic School at University of São Paulo, where was responsible
for the study and development of automation and information systems in the
fields of generation, transmission, and distribution of electricity. Since April
2007, he has been a Research and Development Coordinator for Electrical Engineering and Business Intelligence Projects at Elucid Solutions, a consulting
and TI company for several utilities companies in Brazil. His main research interests are power transformers, distribution system losses, remote terminal units,
project management, geographical information systems, cloud computing, decision support systems, business intelligence, and artificial intelligent solutions
for operation and maintenance of electric power systems.
José Antonio Jardini (M’66–SM’78–F’90) was born in São Paulo, Brazil, on
March 27, 1941. He received the Electrical Engineering, M.Sc., and Ph.D. degrees from the Polytechnic School at University of São Paulo in 1963, 1971,
and 1973, respectively.
For 25 years, he worked at Themag Engenharia Ltda., a leading consulting
company in Brazil, where he conducted many power systems studies and participated in major power system projects such as the Itaipu hydroelectric plant.
He is currently a Professor in the Polytechnic School at São Paulo University,
where he teaches power system analysis and digital automation. There he also
leads the GAGTD group, which is responsible for the study and development of
automation systems in the fields of generation, transmission, and distribution of
electricity.
Dr. Jardini represented Brazil in the SC-38 of CIGRÉ and is a Distinguished
Lecturer of IAS/IEEE.
Luiz Carlos Magrini was born in São Paulo, Brazil, on May 3, 1954. He received the Electrical Engineering, M.Sc., and Ph.D. degrees from the Polytechnic School at University of São Paulo in 1977, 1995, and 1999, respectively.
For 17 years, he worked at Themag Engenharia Ltda, a leading consulting
company in Brazil. He is currently a researcher in the GAGTD group in the
Polytechnic School at São Paulo University.
Se Un Ahn was born in Inchon, South Korea, in 1957. He received the B.Sc.
degree from the Mackenzie Engineering School, São Paulo, Brazil, in 1981 and
the M.Sc. and Dr. degrees in electrical engineering from the Polytechnic School
at the University of São Paulo in 1993 and 1997, respectively.
He has worked since 1986 as a research engineer in distribution systems at the
Piratininga CPFL company (former Eletropaulo and Bandeirantes), all of them
being power concessionaries. His professional activities include load curves use
of expansion planning of the electric system.
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