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ANN-based Short-Term Load Forecasting in Bogotá
Joaquin E. Mejia and Maria E. Correal.

Abstract--This paper proposes four different models for an Artificial Neural Network (ANN) based on short term load forecasting. Historical load data from Bogotá from 2000 to 2007 is used for testing, showing the good performance of the different methods.

Index Terms—ANN, Articial network, Short term load forecasting.

Introduction

During the last years the energetic markets in the world have been evolving from great monopolistic companies vertically integrated to new non-regulated systems, where competence has become an essential factor in the energetic distribution system. In the current model the resources optimization has turned into the competing companies’ economical advantage. Energy unlike the majority of products characterizes for not being storable, creating the need for the most possibly accurate demand forecasts, since that allows doing an adequate planning in the generation systems. It is possible to estimate the necessary reserves and the flow levels, in the same way it is possible to increase the energetic system’s security and generally they allow doing an adequate management of the power system.

Several forecast techniques have been used in the case of short term energetic demand, among which they are included time series, Kalman filters, exponential smoothing and pattern recognition, among others. These models have obtained adequate results, but it is not possible to represent with them the complex non-lineal relation existing between the influential factors and the charge curve.

The current development of the neuronal networks permits to solve this kind of problems with more accuracy and speed. The majority of the models used currently are based on time series, which suppose that the energetic demand is produced in a lineal fashion reason why the obtained results are not the best ones, mistake that can be reduced through the learning developed with the neuronal networks.

In this work a short term energetic demand forecast for Bogotá is sought, through the neuronal networks.

The new forecast techniques that have been developed in recent years, with amazing results, are the neuronal networks and the diffuse logic, which can represent the complex and non-lineal relation between the curve and the factors that affect the charge. Unlike from the diffuse logic, the neuronal networks allow us to represent the complex relation and non-lineal relation between factors and the charge curve without having a specific model, just using the historical demand data and the factors which may affect the charge. In this work different neuronal network models will be brought up, in order to observe which factors and which models are the most convenient to carry out a adequate forecast of the short term energy demand [1]-[2]-[3].

This paper is structured in the following way: the first part has a brief introduction to the subject; the second one talks about the hour by hour energy demand curve in Bogotá and the different factors that affect it; in the third part there is a brief theoretical framework about neuronal networks; in the fourth part the models used in this work and the way in which they were developed; the fifth part contains the results of the developed models, and the last part will give some conclusions and recommendations for posterior works on this interesting subject.

Load forecasting

For the execution of an adequate forecast it is necessary to clearly define the hour by hour energy demand curve in Bogotá, before establishing any prediction model, so that it can be clearly obtained the factors that may affect the demand curve before the model development. The first clarification that must be done is that the information used in this work refers to the hour charge average in megawatts hour in Bogotá since the year 2000 until August 2007. In some of the models 6 years of information were used for the networks training and in other cases it will be of 6 years and 7 months, depending on each one of the models that will be explained in the fourth section of this document.

Traditionally it has been considered that the demand curve is composed of 5 combined factors mainly [1]-[4].

[pic]n+Lw+Lr+Ls+Le (1)

Where:

L is the total load in the system [1].

Ln represents the normal curve for the different types of days in the year [1].

Lw represent the changes of the load curve for environment and meteorological conditions [1]. .

Lr represent the changes of the curve of demand generated by temporary or stationary factors [1].

Ls represent the random factors that can affect the load curve, which are inexplicable or depend generally on many factors[1].

Le represent the effects on the load that are generated per special days or improbable events that they affect the behavior of the curve [1].

When a short term energy demand forecast is done it is not necessary to include the economical factors such as economy growth or the demographic decrease, since on the short term these factors do not have a great over the charge.

In Fig. 1 the charge curve during year 2006 can be seen, and we can observe that, unlike the majority of countries and cities in the world, in the energy demand curve in Bogotá an impact of the seasons in the curve is not observed, but it can be seen that the demand is stable along the months. It is necessary to remind the reader that Bogotá, due to its geographic position, does not have defined seasons, but rain and sun periods. It is not possible to specify between these periods in the curve. In Fig. 1 it is observed that the demand is constant through the months, but it is possible to see some cyclic behavior in the days and the weeks.

[pic]
Fig. 1. Load curve in Bogotá 2006.
.

On Fig. 2 Bogota’s charge curve during the month of March 2006 is observed. It is possible to identify in the graphic four cycles and a half that correspond to the charge of four and a half weeks, where the weekly generated cycles can be seen, this means that the charge curve it characterized by a seasonality every 168 hours. On the other hand it can be observed that during the weekends the charge decreases in comparison to the weekdays, and this can easily be observed in Fig. 3.

[pic]Fig. 2. Load curve in Bogotá March 2006.

In Fig. 3 the curve of a week that begins on a Sunday and ends on a Saturday can be observed, and also the daily periodicity with similar parameters, that is, there is seasonality every 24 hours.

In the Fig. 3 it can also be seen that the behavior between Monday and Friday is very similar. On the other hand, on weekend a similar behavior to Sunday to Saturday can be found. In general, it can be seen that on weekdays the energy consumption is higher than on weekend days, this can be explained because the majority of the factories and companies do not work on weekends and that is why the charge in these sectors will be less. In the models to be carried out it is necessary to take into account that there is a great difference between the weekdays and the weekends when doing the forecast.
[pic]
Fig. 3. Load curve in one week of May
Bogotá 2006.

In the Fig. 4 the curve during a week in which the days can be compared can be seen. In this graphic a lesser consumption on weekends can be demonstrated. It is possible to observe that the consumption peak comes at 8 pm every day. In general the daily curve represents the synchronism of the behavior of people who sleep at night, work during the day and have a greater consumption during lunch and dinner.
.
[pic]
Fig. 4. Load curve in one week of May
Bogotá 2006.

In Fig. 5 the behavior of the demand curve from Tuesday July 25 in the year 2000 and on year 2006 can be observed. With the graphic it can be demonstrated that year after year this curve maintains the same behavior, even though through the years it would move up, indicating that the consumption increases.
.

[pic]
Fig. 5. load curve in Bogotá, Tuesday 25 of July 2000 and 2006.

Colombia is one of the countries with more holidays in the world, making necessary to observe in general how the charge behaves during these special days. The majority of these holidays fall on Monday; nevertheless, some of them are celebrated on any weekday. In Fig. 6 the behavior of the charge curve of four different holidays can be seen, it is very similar to that of Sundays, since the charge is lesser than that of the other days and the curve behavior is almost identical. Without mattering if it is a holiday Monday as May 1st, August 7th or July 20th the behavior of the curve is alike to that of Sundays
[pic]
Fig. 6. Load curve in holly days, Bogotá, 2006.

Neural Network

Every time the technological development in the world comes faster and with smaller elements, nevertheless, despite these great advances it is not possible to simulate the basic processes that our brain does. The digital systems characterize for needing clear entries in order to give specific results, what does not happen with the brain, which has the capability of functioning with diffuse entries and arrive to specific results. An example of this is when a person is able of identifying a known voice over many sounds and to make a mental image of the person speaking [5]-[6].

The brain works with basic units called neurons, and even though its functioning is known, it is still hard to understand how the neuronal network allows having so many emerging properties. Seeking to model the brain functioning the artificial neuronal network concept has been developed during the recent years [2]-[4]-[7].

The neuronal networks, as well as the brain, characterize for being composed by small interconnected units. In Fig. 7 it is possible to see the model of an artificial neuron.
[pic]
Fig. 7. Artificial neuron model.

In general a neuron has an input vector x (x1, x2, x3,…xm), these entries generally come from connections with other neurons or it is information which enters directly into the network. Each one of these entries has an associated synaptic weight (wk1,wk2,…wkm), these weights change according to the network training and are the ones which determine the exit of the network in front of certain stimulus. After the entry stimuli pass through their corresponding weights a sumatory of all the entries with their respective weights which is denominated propagation, after the propagation process the information is passed by a transference function that does the filtering work to obtain the output [1]-[2]-[5].

In Matlab 7.1 [8] there are several transference functions to be used in the neuronal networks, the type of function depends on the entry data and the result which is sought to be obtained.

Generally a neuronal network is composed by three different neuron layers, where each layer has a specific function in the network [9]-[10]. The first layer, which is the entry one, is in charge of receiving the information that enters to the network so that it can be trained or evaluated. The second layer is the hidden layer and it is where the majority of the network processes take place. The last layer is the exit one and as it name indicates is the one in charge of the output of the network results.

In this work the Haykin formula (2) was used to obtain the number of neurons in the hidden layer [10].

H=A+S (2)

Where

H is the number of neurons in the hidden layer.

A is the number of neurons in the input layer.

S is the number of neurons in the output layer .

The type of network is determined by its architecture and learning method used. There are mainly four learning types [10]: supervised, not supervised, hybrid and reinforced. The supervised learning is when the network is given examples to adjust its synaptic weights; in this way to different entries expected exits are given to train the network. The not supervised learning is when the networks is given only entries, and in this why it tries to determine the data distribution. The hybrid learning is combination of the two preceding, where in some layers the supervised is used and in others the not supervised is used. The reinforced learning is when the network is given entries without a specific exit and information about the generated error is given to it [10].

The network architecture may be unidirectional or recurrent. In the unidirectional network there is no connection between a neuron and itself by any path. In the recurrent network, on the contrary, there is an interconnection path [10].

In the bibliographical information it can be observed that different network models have been being used for the short term energy demand prediction. Nevertheless, the one which has given the best results, and which for that reason will be used in this work, is the Backpropagation. The Backpropagation network model is characterized by a supervised learning in the following two phases [10]-[11]: in the first phase at the network entry a stimulus is applied which propagates forward all the way to the exit, the resulting signal is compared with the expected result and the error is calculated in each exit; in the second phase these errors propagate backwards to the hidden layers. The neurons in each hidden layer only receive a portion of the error signal depending on the signal each neuron has provided to obtain the exit, in this way each of the neurons adjusts its synaptic weights. By means of this two phases the network trains itself using the training data and repeating until finding the expected error for the network [2]-[4]-[10]-[11]:.

REALISED MODELS

In the literature there are different studies about the forecast of the short term energy demand where different models and variables have been used depending on the consumption type and the geographical and demographical conditions. In the majority of studied models networks with Backpropagation architecture networks are used, which was explained in the preceding section of this article.

In some models temperature variables [2] are used within the network, these studies generally are done in cities where there are seasons, reason why a relation between temperature and the charge curve may exist as it was explained in section 2. On the other hand, as it was observed in section 2, the charge curve in Bogotá is not affected by seasons, reason why this factor will not be taken into account in the model. In Bogotá the temperature does not vary much and the small variation does not affect the charge since heaters and air conditioning are not used customarily.

As it was seen in section 2, it is not necessary to use the economic variables such as economy growth, energy price, among others, since in short term forecasts these variables are not influential. Perhaps these variables may influence in long term forecasts.

In section 2 it was observed that there are some variables that do affect the charge curve, evidently one can think that the curve depends on the type of day being forecasted. Two classes of days can be clearly distinguished which have similar behavior: on one hand there are the weekdays (Monday to Friday, not holidays), and on the other hand the weekend days (Saturday and Sunday). It also was observed that the holidays have a similar behavior to Sundays.

From the monthly and weekly graphics of chapter 2 it can be observed that a periodicity in the curve behavior every 24 and 168 hours, reason why it is necessary to take into account these factors that condition the charge curve. Section 2 shows that in general year after year the data on a same day behave in similar way. In this section it can be observed, for example, that the behavior of Tuesday 25 of July 2000, is very to also Tuesday 25 of July 2006, so it can be assumed that year after year there is a similar behavior in a same date, which generates a periodicity of 8.760 hours which is the hour time existent in one year.

Taking into account these factors 4 different neuronal network models were designed in order to find the one which presents the best results for short term energy demand forecast. Each one of these models presents the architecture shown in Fig. 8 and in all cases Backpropagation was used.
.

[pic]

Fig. 8 Type of architecture in the models.

Model 1

Using the information obtained in other studies and the charge curve characterization in section 2 the first model was designed having the next 50 neurons in the entry layer: the 24 hours of the previous day, the 24 hours of the same day but a week back and 2 binary variables which represent if the day to be forecasted is a weekday or a weekend day, in this model the holidays will be treated as weekends.

This model has as an exit forecast 24 variables which represent the 24 hours of the following day. In this model the hidden layer includes 74 neurons.

Model 2

Model 2 has the same 50 entrances that model 1, but this model additionally contains 24 hours that represent the 24 hours of the year previous to the day that it looks for to foretell, these 24 additional hours were included for two reasons, first is that, as could be observed in chapter 2, the same days in different years can have a similar behavior; the second reason is that in Colombia some holidays like 1 May, the 7 of August, the 20 of July or the 1 of January are fixed days and have a very similar behavior. This model has 98 neurons in the hidden layer, and like the previous model, has 24 variables of result.

Model 3

Model 3, like model 1, has 48 entrances that represent the 24 hours of the previous day but the 24 hours of the same day of the previous week; additionally to these binary 7 entrances the holidays are included variable that represent the different days of the week, for this model will be catalogued as Sunday reason why were observed in section 2; additionally other 3 binary variables were included result of the analysis of the load curve. When studying the load curve during a year was possible to observe that three weeks in the year exist where the curve has an atypical behavior, these weeks are: the Easter, completes week of December and the first week of the year. Each these 3 binary variables represent that the day to foretell is within one of these special weeks.

This model has a total of 58 neurons in the entrance layer, 24 neurons in the exit layer, that represent the 24 hours of the day to foretell, and 82 neurons in the hidden layer

Model 4

Model 4 has the same 10 binary variables that model 3, but in this case is not going away to make the prognosis for the 24 hours of the following day, if it is not going away to realise the prognosis of one hour specific; additionally an entrance variable exists that represents the first hour that looks for to foretell for the previous day and another variable of entrance that represents the hour that is desired to foretell for one week previous. This model will be evaluated in 2 hours specific to 12 at night and the 8 at night that are the hours with minor and major variability respectively in the load curve.

This model counts on 12 entrances, 1 exit and 13 neurons in the hidden layer of the network. This design allows us to see if it is possible to realise the prognosis by hourly of individual form or is necessary to place the 24 hours in block of every day.

Each of the previous models was applied both for the forecast of one month as for the forecast of 9 months. In the case of one month data from January 2000 until July 2007 were used to do the network training. Afterwards, the month of August 2007 was forecasted. In the case in which 9 months were forecasted the data from January 2000 until December 2006 was used for the training of the network. Afterwards, the forecast from January to August 2007 was done.

The data from the hour consumption average in Bogotá used in this work were supplied by XM and they are shown in megawatts/hour. For the models execution is was necessary to do each one of the entry matrixes starting from the information supplied. All the designs were developed on MATLAB 7.1 [8], where data normalization was necessary to be done, since in this program the networks function with entries between 1 and -1. After the training and entry data normalization process we proceeded to define each one of the networks, where it was necessary to define the amount of neurons and the filtering function in each layer, not necessarily all the layers have the same function. After defining the network we proceeded to the adjustment of the synaptic weights to find the smallest possible error, and once trained the network the entry data were applied to it in order to reach the desired forecast.

results

As it was mentioned in the conceptual framework the neuronal network’s behavior will be simulated with different models to see which one behaves in a better way. It is reasonable to point out that the success or failures of the forecasts done by means of neuronal networks depend on the model’s construction.

For the evaluation of each of the models the forecast error percentage average defined in the following way was used as a comparative error measure:

[pic] (3)
Where

N=Number of cases of study.

[pic]= Real load in hour i.

[pic]= Load forecasting in hour i.

This measurement factor is used since its signification is easily understood and besides it is the most used measurement factor in the short term energy demand forecasts literature, which gives us a comparison point against other studies carried out about the same subject, to see in comparative terms if the results are adequate or not.

As a complement of this measurement factor a peak percent error for each model will be given in each case, in this way may see how deviated the peak values are from the percent medium error.

Before showing the results and making and comparative analysis we can proceed to show a summary of the models done which were explained with more detail in section 4 of this work.

TABLE I
INPUT VARIABLES IN THE MODELS

|FACTORS |MODEL 1 |MODEL 2 |MODEL 3 |MODEL 4 |
|24 HOURS PREVIOUS DAY |X |X |X | |
|24 HOURS PREVIOUS WEEK |X |X |X | |
|24 HOURS PREVIOUS YEAR | |X |X | |
|TYPE DAY WEEKEND AND BETWEEN |X |X | | |
|WEEK | | | | |
|INDIVIDUAL DAY, HOLLY DAY | | |X |X |
|LIKE SUNDAY | | | | |
|BINARY VARIABLE SPECIAL WEEK | | |X |X |
|(JANUARY, EASTER, DECEMBER) | | | | |
|1 HOUR PREVIOUS DAY | | | |X |
|1 HOUR PREVIOUS WEEK | | | |X |

As it was mentioned before each one of the models was simulated to find the 1 and 9 months forecast.
Results to 9 months:

Next a comparative chart is shown with the results obtained in the four simulations:

TABLE ii
RESULTS TO 9 MONTHS
| |MODEL 1 |MODEL 2 |MODEL 3 |MODEL 4 |
|ERROR OBTAINED IN|0,32% |0,14% |0,064% |0,13% |
|THE LEARNING A | | | | |
|STEP FORWARD | | | | |
|NEURONS IN THE |50 |74 |58 |12 |
|INPUT LAYER | | | | |
|NEURONS IN THE |24 |24 |24 |1 |
|OUTPUT LAYER | | | | |
|NEURONS IN THE |74 |98 |82 |13 |
|HIDDEN LAYER | | | | |
|SIMULATION TIME |65 MINUTES |100 MINUTES |80 MINUTES |11 MINUTES |
|AVERAGE |25,2% |27,3% |3,25% |13,2% |
|PERCENTAGE ERROR | | | | |

Next some of the graphics obtained for the percent error in each of the models can be seen (Fig .9):

[pic]
a. Model 1

[pic]
b. Model 2

[pic]
c. Model 3

[pic]
d. Model 4

Fig. 9. Results to 9 months, average percentage error.

Results to 1 month:

Next a comparative chart is shown with the results obtained in the four simulations:

TABLE III
RESULTS TO 1 MONTH
| |MODEL 1 |MODEL 2 |MODEL 3 |MODEL 4 |
|ERROR OBTAINED IN|0,32% |0,14% |0,064% |0,13% |
|THE LEARNING A | | | | |
|STEP FORWARD | | | | |
|NEURONS IN THE |50 |74 |58 |12 |
|INPUT LAYER | | | | |
|NEURONS IN THE |24 |24 |24 |1 |
|OUTPUT LAYER | | | | |
|NEURONS IN THE |74 |98 |82 |13 |
|HIDDEN LAYER | | | | |
|SIMULATION TIME |67 MINUTES |105 MINUTES |83 MINUTES |12 MINUTES |
|AVERAGE |17,12% |21,14% |1,3% |15,23% |
|PERCENTAGE ERROR | | | | |

Next some of the graphics obtained for the percent error in each of the models can be seen (Fig. 10):

[pic]
a. Model 1

[pic]
b. Model 2

[pic]
c. Model 3

[pic]
d. Model 4
Fig. 10. Results to 1 month, average percentage error.

analysis and conclusions

Starting with the results obtained in the previous section of this document it is possible to see that better results are found when doing a one month forecast than a nine months one. This result may be due to several explanations, on one hand it is possible to think that when doing 9 months forecast there may be variables influence which has not been taken into account in the charge curve, these factors may be macro-economical such as inflation, economic growth, among others. Another possible explanation for this phenomenon which may be observed graphically is that when greater the distance between the forecast done greater the error will be, therefore when obtaining the average error of the 9 months it will be greater than obtaining it for one month. For the reason previously mentioned we will focus on the forecast analysis done on one month, since the one for 9 months do not provides us adequate information.

When observing the 4 proposed models in section 4 of this article it is possible to reach different conclusions. If model 1 is compared with model 2 it is easy to observe that not necessarily more entries in a network mean better results. Despite model 2 has 24 more entry hours than the model 1, the result of model 1 is much better. It is probable to think that this result is explained because the previous year data may generate noise in the network, reason why not expected results are found. In some articles investigated in the bibliography the authors recommend the addition of new entry variables which in effect have incidence in the result, since at greater number of entry data greater the training time of the network will be. In this investigation it was observed that not necessarily more data are needed, since these may increase the forecast medium percent error.

The most amazing result is found in model 3 where the percent medium error is only of 1.13%. This value is very low compared to other studies done about this subject with neuronal networks. Next a chart is shown with some studies done previously with neuronal networks and short term energy demand. In the chart the reached medium percent error can be observed.

TABLE Iv
OTHER STUDIES WITH ANN
|STUDY |AVERAGE PERCENTAGE ERROR |
|ACOSTA, 2000 [4] |4% |
|MURTO, 1998 [2] |3,5% |
|TAYLOR, 2006 [14] |3,8% |
|SLAVISA,2000 [15] |3,2% |
|DEMIROREN, 2006 [16] |2,8% |
|CANIZARES, 2000 [1] |3% |

All the mentioned studies previously were done with different networks and variables. In some cases temperature and price variables were added, these studies have been developed in different countries such as: Brazil, Finland, USA, Canada, among others.

It is important to see that in this study a medium percent error of almost one third of what if found in other studies was accomplished and without including temperature and price variable. On the other hand it is the only study that uses 7 binary variables to differentiate each day including holidays as Sundays. Besides it uses 3 additional variables to differentiate special weeks in the year. In other studies something similar is done but it is done with the 4 seasons of the year, because climate conditions the charge curve.

Despite the medium percent error is low, in the Fig. 10c it can be observed that the maximum errors take place the 7 and 20 of August, which are holidays, indicating it is possible to get a better network when adding other variables that allow a better adaptation for the holidays.

The resulting network in the model 3 must not be re-trained since 2000. When September is wanted to be forecasted it is only necessary to train the network with the data from August and it will fix again the synaptic weights to found the September’s forecast, that is, beginning with the network trained the companies interested in finding the forecast just need to re-train the network with new data and they will obtain new forecasts.

Since excellent results were obtained with this model we proceeded to apply the network for the forecast of the month of April 2007 since it includes Holy Week which is one of the most critical weeks of the year. In the case of the April’s forecast a medium percent error of 2,55% was found, which signifies that it is possible to improve the model in the future with other works for the forecast of these special weeks adding some new variables to the model.

Next the Fig, 11 is presented with the excellent results found in the model 3 where the forecast and the real demand of the first week of 2007 can be seen:

[pic] Fig. 11. Forecasting load first week of year. Model 3.

As the results of model 3 were so good, it was come to observe if the forecast can be done hour per hour of individual form or is necessary to do it in blocks of 24 hours, for which model 4 for the forecasting of the hour with smaller variability was simulated. The results show that the average percentage error ascends to 15.23% which is a very high against the obtained in model 3.

Starting with this work the following conclusions could be identified for the forecast of the short term energetic demand with neuronal networks:

- More training data for a network does not mean better results. - It is necessary to differentiate with the entry variables each one of the 7 days of the week and to catalogue the holidays as Sundays. - To obtain better results it is necessary to do the 24 hours ahead forecast in block. By doing the hour by hour forecast in an individual way the results are not satisfactory. It is important to observe that with more variability in the one hour consumption, the fewer its percentage average error forecast will be. - It is possible to improve the current result by finding the factors that could affect the charge curve on the holidays and the special weeks. - The best model is achieved while forecasting 24 hours ahead, using as entry variables the 24 hours of the previous day, the 24 hours of the same day on the previous week and the 10 binary variables that represent the 7 days of the week plus the 3 variables for the special weeks. - In general the networks can have better results than the time series, since the network allows modeling the non-lineal relationship among the variables. - When the network is trained it is not necessary to do a complete training but a data update that allows adjusting the synaptic weights in the network. - It is possible to do an ARIMA model in the future with the variable used in the model 3 of this work to do a comparison between the neuronal networks and the time series methods that are widely used in this context.

.

vi. References
1] C,CANIZARES,”ANN- BASED SHORT-TERM LOAD FORECASTING IN ELECTRICITY MARKETS”, DEPT. ELECT. ENG, UNIVERSITY OF WATERLOO, 2000.
2] P. Murto, “Neural Network Models For Short- Term Load Forecasting”, Dept. Elect. Eng, University of Technology. Helsinki,1998.
3] G.Box, "Time series analysis: forecasting and control",Holden-Day, Prentice Hall. San Francisco, 1994.
4] M. Acosta and C. Zuluaga, “Tutorial sobre redes neuronales aplicadas en Ingeniería Eléctrica y su implementación en un sitio Web”, Dept. Elect. Eng Universidad Tecnológica de Pereira, Pereira, 2000.
5] M. Shahidehpour, H. Yamin and Z. Li, “Market Operations in Electric Power Systems”. Wiley Interscience. the institute of Electrical and Electronics engineers, Inc , new York, 2002, pp 21-55.
6] H. Camacho, “Empleo de Redes Neuronales para la Predicción de Series de Tiempo Univariadas”, Dept. Elect. Eng, Universidad de los Andes, Bogotá, 2001.
7] S. Londoño, C. Lozano and G. Caicedo, “ Pronostico de Precio en el Mercado de Electricidad Colombiano usando Redes Neuronales” , Dept. Elect. Eng, Universidad del Valle, Cali, Colombia, 2004.
8] H. Demuth, “ Neural Network Toolbox for use with MATLAB:. User Guide”, The Math Works, 2004, pp 22-103.
9] G. Gomez, “Introduccion a las Redes Neuronales Artificiales” , Dept. Elect. Eng, Universidad del Valle, Cali, Colombia, 1992.
10] S. Haykin, "Neural networks – a comprehensive foundation", MacMillan College Publ. Co., New York, 1994.
11] G. Zhang, “Forecasting with Artificial Neural Networks: The State of Art” . International journal of forecasting, Vol 1, pp 35-62, 1998.
12] A. Zapranis, “Neural Model Identification, Variable Selection and Model Adequacy”, Journal of Forecasting, London Business School, Vol 18, pp 299-332, 1999.
13] M. Pantoja, “Análisis Comparativo de pronósticos Realizados con Redes Neuronales, Modelos Arima y Procesos Garch para Series de Tiempo no Estacionarias”, Dept. Elect. Eng, Universidad de los Andes, Bogota, 2004.
14] J. Taylor, “A Comparison of Univariate Methods of Forecasting Electricity demand up to a Day ahead”, International Journal of Forecasting, Vol 22, pp 1-16, 2006.
15] M. Slavisa, “An Improved Neural Network Application for Short- Term Load Forecasting in Power Systems”, Banjaluka Faculty of Electrical Engineering, 2000.
16] A. Demiroren, “Middle Anatolian Region Short-Term Load Forecasting Using Artificial Networks”, Dept. Elect. Eng, Istanbul Technical University, 2006.

Joaquin. E. Mejia is student of the University of the Andes, Bogotá, Colombia (e-mail: je.mejia51@egresados.uniandes.edu.co). Maria E. Correal is with the Department of Industrial Engineering, university of the Andes, Bogotá, Colombia (e-mail: mcorreal@uniandes.edu.co).

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0,00E+00

5,00E+02

1,00E+03

1,50E+03

2,00E+03

2,50E+03

1

1001

2001

3001

4001

5001

6001

7001

8001

Carga (MW)

[pic]

wk1

wk2

wk3

wkm

x1

x2

x3

xm

F()

Yk

Synaptic weight

Input

Body cell

Transference function

Output

[pic]

Hidden layer

Input layer

Output layer