REV2009003 (APIK)

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 2, MAY 2009 705

Distribution Transformer Losses Evaluation:A New Analytical Methodology and Artificial

Neural Network ApproachAdriano 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 analyticalmethodology and discuss some alternatives of artificial neuralnetwork models in order to evaluate losses in distribution systems,particularly in distribution transformers.

The procedure can also be extended to other components of thedistribution system (secondary and primary network and HV/MVtransformers). This is accomplished by using the utility’s databasesuch as the consumers’ monthly energy consumption and the typ-ical load curves of each class of consumption and type of activitydeveloped.

Index Terms—Information systems, neural networks, power dis-tribution, power transformer losses.

I. INTRODUCTION

C OMMONLY, distribution system losses are estimated be-cause of the unavailability of suitable metering systems.

For billing purposes, only energy meters are installed at theconsumer’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 ofthem, including ours, have a lack of accuracy mainly becausethe consumer load profiles are different on weekends and evenduring the weekdays as appliances are turned on/off in a randomway [1]. These inaccuracies have always been present, even inthe procedures described in this paper; although it showed sig-nificant progress in the treatment of the random variation.

In [2] and [3], the primary feeders, the distribution trans-formers and the secondary network for three-phase load flowcalculation are modeled. The consumers’ load profiles arerepresented on an hourly basis for power and then the lossesare calculated. Several calculations were done varying the loadlevel (FL), the transformer capacity (XFCAP), and the totalconductor length (CL). The results were then used for trainingan artificial neural network (ANN) to thereafter estimate thelosses in actual feeders considering its proper FL, XFCAP, andCL. This is an improvement of other methods, because it usesthe consumer’s daily load profile, and avoids the use of loss

Manuscript received October 14, 2007; revised August 15, 2008. First pub-lished 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: [email protected]).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: [email protected]).S. U. Ahn is with CPFL—Companhia Piratininga de Forca e Luz, Campinas,

Brazil (e-mail: [email protected]).Digital Object Identifier 10.1109/TPWRS.2008.2012178

and diversity factors. Although, as in all methods inaccuraciesstill exist.

Three-phase load flow is a suitable tool for the calculation ofunbalanced load/lines conditions; nevertheless, the authors hereconsider this unnecessarily complex to be used, on account ofthe inaccuracies existing in the load profile. In fact, at least intropical countries, the load profiles of the weekdays are differentand there is no time correlation among the consumers loading,which makes the task of setting up the load flow a source oferror.

The measurements of the consumers’ daily load profiles re-ported in [1] led to the evaluation of the mean and stan-dard deviation profiles in several type of consumers anddistribution transformers using sets of 15 to 30 measured dailyload profiles. The values of various type of consumers (res-idential, small/medium size commercial and industrial) were ofthe same size as , which clearly indicates a large varia-tion of the load at any time of the day. This is because the totalconsumer load is composed of energy uses of almost equal sizeand they are not turned on/off at the same time every day. Onthe other hand, the measurements carried in distribution trans-formers indicated an of about 20% of . However, forlarge consumers such as shopping malls, medium/heavy size in-dustries the values of are very low [4].

In [5], measurements in selected feeders were carried out.These results were then extrapolated to other feeders using prop-erties like the installed capacity of the distribution transformersand the feeders’ length. A similar procedure was adopted by[6] and [7], where instead of measurements, load flow modelingwas used for the calculation of selected feeders. Their extrapo-lation to other feeders followed a similar approach to that pre-sented in [5]. As discussed above, the load profile variation wasnot considered in all these methods.

Analyses to establish load profiles for engineering studies,supported by ANN, wavelets theory and clustering processeswere conducted in [8] and [9].

In this paper, a method that includes the load variabilityaimed at obtaining improvements for the estimation of distri-bution losses is presented.

The organization of this paper is as follows: initially, a pro-cedure to obtain the transformer daily load profile, based on theconsumer’s data, is described. Then, the losses calculation re-garding the load variability is presented. Finally, a method con-sidering the ANN technique as an alternative procedure is alsopresented.

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This work started out by using the contributions presentedin [1] and [10]–[16]. The consumer and transformer daily loadprofiles (active power) were represented by their average (M)and standard deviation (S) profiles.

It should be noted that nearly all calculations in a distributionsystem present inaccuracies as they are based on the best dataavailable, which may be imprecise and incomplete, and statis-tical behavior. In order to fill in the incomplete data, some sim-plifications and assumptions are made. In this case, for instance,losses in the transformer are evaluated using active power loadprofiles instead of apparent load profiles, which must be ad-justed by using a certain factor. In addition, the profiles of theconsumers are chosen from a statistic subset of representativeconsumers, the weekend and working day profiles are in generaldifferent, thus including another source of inaccuracy. Becauseof that, errors in the losses calculation, even at the level of 25%may be considered acceptable. Another source of error comesfrom the billed energy metering that has some inaccuracies dueto measuring transformers and metering, which are in the orderof 0.5% to 1.0%. This is nearly at the same level of the trans-former total rated losses (1% to 2%).

II. TRANSFORMER LOADING

The methodology used in this paper to define the transformerloading is based on the mean and standard deviation load pro-files, a procedure described in [1] and partially summarizedherein.

Studies characterizing the consumers daily load profiles werereported in [1] and [10]–[12]. In these references, the demandmeasurements of several types of consumers (residential, com-mercial, and low voltage industrial consumers) were carried out.The consumers’ daily load profiles were set up so as to record96 points (i.e., active power was recorded for each minute andthen averaged at intervals of 15 min).

For each consumer and distribution transformer about 15daily profiles were considered. The mean and standarddeviation profiles were determined and set to characterizethe consumer and the distribution transformers.

For ease of manipulation, the demand values were nor-malized (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 ofa distribution transformer (in p.u.).

For an th consumer and, where and are the mean values of

the demand at time , in p.u. of the monthly average demand(Davi) and in kW, respectively. Similarly, and arethe standard deviation in p.u. and real values, respectively.

A procedure to aggregate (add) the consumers’ demand in adistribution transformer was also developed [1]. If it is consid-ered a distribution transformer with an number of consumersof the type and of the type, the aggregated demand values(in kW) can be calculated using the following:

Fig. 1. Mean and standard deviation daily curve of a transformer (p.u.).

Fig. 2. Transformer’s daily load curve stratified in 11 curves.

(1)

where are the mean demand of the consumersaggregation, 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 di-viding the real (kW) values by the transformer’s rated power.

The demand value within an interval is assumed to follow aGaussian distribution, so the figure with a certain non-exceedingprobability can be calculated through

(2)

where is a constant that defines the probability in a NormalDistribution [18]. For example, for 90% of thevalues will be below and 10% above.

Fig. 2 shows a set of 11 profiles (in p.u.) with probabilities(from bottom to top) % % % %, and %, re-spectively. These sets of profiles were used in [10] to evaluatethe distribution transformer’s loss of life due to loading. Thesame set of profiles is used here to evaluate the losses in distri-bution transformers.


LEAL et al.: DISTRIBUTION TRANSFORMER LOSSES EVALUATION 707

TABLE ISTRATIFICATION PROFILES

III. DISTRIBUTION TRANSFORMER LOSSES

CALCULATION—GENERAL APPROACH

A. Main Equations Used in the Approach

It is well established that losses in a distribution transformerare produced when the current flows through the coils. They alsoappear whenever a magnetic field circulates around the core. So,they can be classified into on-load losses and no-load losses [9],[17].

No-load losses do not vary according to the transformerloading but according to the voltage; thus, it may be consideredconstant for losses calculation purposes.

On-load losses vary according to the transformerloading and are responsible for the largest part of the loadlosses. This work will mainly focus on such losses.

As represents the transformer loading at the interval ofone profile, then, the series losses can be written as [14],[15]

(3)

where is the transformer winding resistance and representsthe series losses of the rated power . Should more precisionbe needed, this value will have to be corrected with the variationof the transformer’s internal temperature (this may be of partic-ular interest in countries were the transformers are loaded abovetheir rated capacity).

Equation (3) can be applied to all 11 profiles . For instance,if %, then this profile can be the representative of allthe profiles with probability 15% to 25%, which represents aparticipation factor (kpf) of 10% .

Notice in Table I that nine profiles have , the re-maining two having profiles with .

Thus, the total average series losses can be expressedas

(4)

Fig. 3. Load losses calculation. Analytical procedure.

In (4), 96 multiplication operations are needed to evaluate theterm, 11 more multiplications to consider ,

and one multiplication to evaluate the series losses inone single transformer. This gives an idea of the computationtime required in the whole process.

The total losses of the distribution transformer beingcomposed by both the no-load losses and the aboveon-load losses

(5)

B. Data Handling

From the Distribution Utility database the following infor-mation was used: and characteristic profile (in p.u. of themonthly average demand) of the representative type of con-sumers (see upper box in Fig. 3); the transformer parameters(rated power, series and no-load rated losses; left box at bottomof Fig. 3), the consumers’ type and energy consumption permonth connected to each transformer (bottom box in themiddle of Fig. 3).

The and profiles (in kW) of each consumer (in a trans-former) are calculated by multiplying the and values (inp.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 themiddle 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 trans-formers at CPFL (a Brazilian Distribution Utility) wererecorded. The average series losses for each load profile, aswell as their average values, were also determined .

Next, the and curves of each transformer were also de-termined. This was obtained using the same seven-day load pro-file curves. On the other hand, the series losses for the 57 trans-formers were determined using (4).

The error distribution betweenthose two calculations is shown in Fig. 4. As it can be seen, the



Fig. 4. Error distribution.

errors are small having a mean value of 0.3%, which means thatthe 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 re-sults. It is suitable in many situations such as when the mathe-matical relationship among variables is unknown, or when thereare few input data for interpolation. It can improve the timecomputation efficiency and simplicity while manipulating data.Aside of improving the computation efficiency, the accuracy ofthe results is reduced, a trade off to be searched. Therefore, aninitial ANN model was developed using the same input data ofthe analytical procedure. In such a model, there is no gain expec-tation neither in accuracy nor in computation efficiency; thus, al-ternative 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 thetransformers and profiles, the output being the series losses

. The number of neurons and layers were defined by trialand error tests in the MLP (multilayer perception) model. Thesupervisioned and back propagation training types were alsochosen.

The calculations presented in Section III (analytical proce-dure) were performed for a set of 61 485 transformers of thedistribution utility. The losses were regarded as “true values”mainly because the measurements of the losses were not avail-able. Part of these calculated values (losses) were used duringthe training stage and part of them for testing the training effi-ciency.

The system where the distribution transformers are locatedhas 608 primary feeders, 2.2 million consumers and suppliesaround 9 TWh/yr.

B. Clustering

The set of 61 485 transformers’ daily profiles ( and ) ob-tained through (1) and normalized in p.u. by the transformer’srated power, were put under cluster analysis. The number ofclusters specified, which followed the Euclidian distance crite-rion, was equal to 10.

Fig. 5. Result of Cluster 2 (commercial and industrial consumers).

Fig. 6. Result of Cluster 9 (residential consumers).

Note: Actually, several tests considering a different number ofclusters (up to 30 clusters) were carried out. It was continuouslydetermined the largest distance from one curve to the center ofthe cluster and also the distance among clusters. These distancesindicated that it would be necessary to have 30 clusters to getthem well grouped.

However, many of these clusters had a small number of trans-formers; thus, they were discarded. Those transformers withinthe discarded clusters were transferred to other clusters. At theend, a number of ten clusters whose mean values were kept forsubsequent calculations were adopted.

This means that transformers with similar profiles are as-signed 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 per-

tains to typical commercial/industrial loads (Fig. 5) whereas theother shows the characteristic peak (at 20:00 h) of a residentialload (Fig. 6).

Table II, shows the characteristic parameters of each cluster.The shape of Clusters 1 through 6 looked like Fig. 5, whereasClusters 7 through 10 resembled to the curve shown in Fig. 6.The profile patterns of the ten clusters correspond to the twoload types previously mentioned. The difference in clusters ismainly due to the peak value.

Since the profiles are normalized by the transformer’s ratedpower, apart from the shape, the main characteristic of thecluster will be its loading state.



TABLE IICHARACTERISTICS OF THE CLUSTERS

The objective of the clustering process was to evaluatewhether better results are obtained by training one ANN foreach 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 clusterrepresents transformers having commercial and/or residen-tial loads and which could be used for future reallocationduring the system’s expansion program;

• 46.2% of the transformers have typical load conditions ofareas with residential consumers (Clusters 7 through 10);

• 1.1% of the transformers, those pertaining to Clusters 5 and8, are much more loaded.

C. Results

The training vector, as well as the test vector, are formed bya certain group of inputs and one output, constituted by:

• 24 points of the transformer curve (in p.u., 1 point perhour);

• 24 points of the transformer curve (in p.u., 1 point perhour);

• the term, calculated through the method de-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 elementsin both the training and test vectors used.

1) Architecture: The ANN architecture is composed by: fourlayers, the input-layer having 48 neurons, the second and thirdlayers having 35 and 24 neurons, respectively; and the outputlayer having only one neuron.

2) Training Process: The ANN model used was set up toperform 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).

TABLE IIITRANSFORMERS DISTRIBUTION WITHIN THE TEST AND TRAINING VECTORS

TABLE IVPERCENTAGE OF TRANSFORMERS WITH ERROR LESS THAN 10%

The errors in each cluster as well as the percent of cases witherrors below 10%, here called “error index” were also evaluated.The error indexes in all clusters are shown in Table IV. It can beobserved that when the E000 training is used (which would bepreferred due to its simplicity) the accuracy is not good, exceptfor clusters with a small number of transformers. The error inaround 92.5% of the transformers (for the ANN trained specif-ically for each cluster) was below 10%. This leaves 7.5% withan 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 to30%, 2.3% is within 30 to 100% and 1% was greater than 100%.

The global error (sum of all 61 485 transformers) regardingthe analytical procedure as reference was 9.7%.

Although the results can be considered as satisfactory, theprocedure involved lots of multiplying operations, due mainlyto 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 consumingthan the analytical procedure (see Section III-A).

In the next section, some ANN alternative architectures aimedat reducing the computation time (although not strictly neces-sary as today’s microcomputer can handle this task), though atthe expense of reducing the accuracy, are presented. The first



attempt is to reduce the input data and then reduce also the in-termediate calculations.

Now, it is important to know the computation time involvedfor the 61 485 transformers.

In the analytical procedure [with 96 points for (4)], it took15.1 min to process (1) and 5 min to process (4), giving a totaltime 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 sfor the losses calculation (note that for the ANN procedure theprofile was scaled back from 15 min to 1 h, so the latter termof the computational time would become worse if the full 96points were considered.) The total time being 16.6 min. It shouldbe noted that the training process needed 1.1 h in each cluster.The losses calculations by most of the distribution utilities arecarried out on a monthly basis; conversely, there is no need toupdate the training process at that same basis (once a year wouldbe adequate).

V. LESS TIME CONSUMING ANN APPROACH

Two other architectures aiming at improving the calculationtime (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 ofinputs and one output, constituted by:

• four inputs of the transformer profile at around 03:00,14:00, 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 neu-rons are 6 for the first layer, 18 and 10 for the hidden layers, and1 for the output layer.

The same clusterization result of the initial approach wasused. In addition, for comparison purposes, the same amount ofelements in the training and test vectors was used (see Table III).

Now, from the total estimations 83.2% were obtained with er-rors below 15%. The results, here considered poor indicate thatprobably, at least from the authors’ viewpoint, a new clusteriza-tion 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 thereference.

The computation time was 1.5 min to process (1), 0.3 s tocalculate both losses and cluster process, making a total time of1.8 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 thedatabase) the amount of each type of consumer connected to thetransformer and its total energy consumption.

In this approach, the training vector and the test vector areformed by a group of inputs and one output, constituted by:

Fig. 7. Consumers’ daily load curves Type 1 (residential).

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 themean consumption of each consumers type;

• one input representing the transformer rated power;• the value of also constitutes the output variable.Again, the parameters used in all the simulations of the ANN

model are the same as before, except that the layer neurons are9 for the first layer, 16 and 8 for the hidden layers, and 1 for theoutput layer.

A new clusterization process whose results are presented inTable V, was performed. Here, Qty is the mean value of the



Fig. 10. Consumers’ daily load curves Type 4 (commercial).

TABLE VCHARACTERISTICS OF THE CLUSTERS

consumer type, considering all transformers within a cluster;whereas kWh represents, similarly, the mean consumption ofthe consumer.

The global error (for all the 61 485 transformers) regardingthe analytical procedure as reference was 22.8%.

The computation time of the losses was 2 s whereas the clus-tering process 0.9 min. The calculation time [15.1 min, using(1)] was eliminated. In this procedure the accuracy may be im-proved by considering more than four types of consumers, ofcourse 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 isalso applicable to evaluate the series losses in the secondary andprimary network, as well as for the HV/MV transformers. Forinstance, for a section of a primary feeder, the and curvesof the transformers, beyond this section can be aggregated using(1). A specific ANN and test procedure shall be carried out totrain and calculate the primary feeder losses.

This methodology could be incorporated into a GeographicalInformation System (GIS) so as to turn its calculation proceduremore independent from the user interaction.

TABLE VICOMPARISON OF RESULTS

VII. COMPARISON OF THE PERFORMANCES—RESULTS

Table VI shows the global loss values of all the distributiontransformers.

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 errorswere less than 10%. The disadvantage of this method isthat the amount of mathematical operations, necessary toobtain this result, is greater than that needed in the analyt-ical procedure;

• the second and third architectures showed global errorsof 25.9% and 22.8, respectively. They were less accurate;however, their processing times were reduced as those ar-chitectures required less mathematical operations;

• it should be emphasized the fact that the “analyticalmethod” was assumed to lead to the correct results (truevalues). The use of the third architecture together with themeasured values of losses may constitute a method withreasonable precision and adequate processing time.

• The Alternative 2 model offered a global accuracy as goodas the second one. Another advantage is that it dispenseswith the calculation of the and curves of both con-sumers and transformers.

Through the present ANN application, in terms of accuracyand time computation, a reasonable estimation of the lossesin a distribution system can be achieved. However, it must bepointed out that the parameters used to train the ANN have notbeen exhaustively optimized, as that was not the main objectiveon this work. Therefore, there still are some improvements pos-sible on the accuracy.

The Alternative 2 ANN Architecture is even faster than theother two options, as it does not need to calculate the consumerprofiles and the aggregation of the distribution transformers.Another advantage, aside of the calculation speed, is that theutility does not need to perform measurements to evaluate theload profile for all the type of consumers, which is a costly op-eration.



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Adriano Galindo Leal (M’06) was born in São Paulo, Brazil, on September19, 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 groupin the Polytechnic School at University of São Paulo, where was responsiblefor the study and development of automation and information systems in thefields of generation, transmission, and distribution of electricity. Since April2007, he has been a Research and Development Coordinator for Electrical En-gineering and Business Intelligence Projects at Elucid Solutions, a consultingand TI company for several utilities companies in Brazil. His main research in-terests are power transformers, distribution system losses, remote terminal units,project management, geographical information systems, cloud computing, de-cision support systems, business intelligence, and artificial intelligent solutionsfor operation and maintenance of electric power systems.

José Antonio Jardini (M’66–SM’78–F’90) was born in São Paulo, Brazil, onMarch 27, 1941. He received the Electrical Engineering, M.Sc., and Ph.D. de-grees 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 consultingcompany in Brazil, where he conducted many power systems studies and par-ticipated 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 alsoleads the GAGTD group, which is responsible for the study and development ofautomation systems in the fields of generation, transmission, and distribution ofelectricity.

Dr. Jardini represented Brazil in the SC-38 of CIGRÉ and is a DistinguishedLecturer of IAS/IEEE.

Luiz Carlos Magrini was born in São Paulo, Brazil, on May 3, 1954. He re-ceived the Electrical Engineering, M.Sc., and Ph.D. degrees from the Poly-technic School at University of São Paulo in 1977, 1995, and 1999, respectively.

For 17 years, he worked at Themag Engenharia Ltda, a leading consultingcompany in Brazil. He is currently a researcher in the GAGTD group in thePolytechnic 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 andthe M.Sc. and Dr. degrees in electrical engineering from the Polytechnic Schoolat the University of São Paulo in 1993 and 1997, respectively.

He has worked since 1986 as a research engineer in distribution systems at thePiratininga CPFL company (former Eletropaulo and Bandeirantes), all of thembeing power concessionaries. His professional activities include load curves useof expansion planning of the electric system.


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REV2009003 (APIK)