Fuzzy Modeling to Forecast an Electric Load Time Series - ScienceDirect

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ScienceDirect Procedia Computer Science 55 (2015) 395 – 404

Information Technology and Quantitative Management (ITQM 2015)

Fuzzy modeling to forecast an electric load time series Cesar Machado Pereira *, Nival Nunes de Almeida, Maria L. F. Velloso Department of Electronics and Telecommunications, UERJ. São Francisco Xavier, 524, Rio de Janeiro, 20550-900, Brasil

Abstract This paper tests and compares two types of modelling to predict the same time series. A time series of electric load was observed and, as a case study, we opted for the metropolitan region of Bahia State. The combination of three exogenous variables were attempted in each model. The exogenous variables are: the number of customers connected to the electricity distribution network, the temperature and the precipitation of rain. The linear model time series forecasting used was a SARIMAX. The modelling of computational intelligence used to predict the time series was a Fuzzy Inference System. According to the evaluation of the attempts, the Fuzzy forecasting system presented the lowest error. But among the smallest errors, the results of the attempts also indicated different exogenous variables for each forecast model. © Published by Elsevier B.V. This © 2015 2015The TheAuthors. Authors. Published by Elsevier B.V.is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Selection and/or peer-review under responsibility of the organizers of ITQM 2015 Peer-review under responsibility of the Organizing Committee of ITQM 2015

Forecast; Time Series; Electric Load; SARIMAX; Fuzzy Inference System

1. Introduction Electricity is essential in society, whether it is for home use or in the various segments of the economy. Routine activities such as watching television or using the air conditioner are only possible through the electric load. The eletric load segment receives a large amount of energy from the transmission system and distributes it to the final consumers, homes, small businesses and industries. The eletric load supply is one of the most challenging services of modern society. To have the consumer get the eletric load at the time that he triggers a switch or connects an electrical appliance into the outlet in his home is all the chain system must be able to operate in a coordinated manner, with load plants, transmission lines, substations, lines and distribution transformers. The eletric sector around the world has been undergoing a process of transformation whether it is politically or economically. In Brazil, the privatization of eletric distribution companies and regulation systems were responsible for the changing landscape of this system [24].

* Corresponding author. Tel.: +55 21 986328752. E-mail address: [email protected].

1877-0509 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ITQM 2015 doi:10.1016/j.procs.2015.07.089

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1.1. The Eletric Load Company Coelba Coelba has been responsible for electric load in the State of Bahia, Brazil since 1960. Coelba is the third largest distributor in Brazil in number of customers and the sixth largest in provided energy volume. The time series object of study of this paper is the electric load in the metropolitan region of Coelba. The time series has daily datas, in megawatt-hour (MWh), and were used only on weekdays, so discarding the weekends (Saturday and Sunday) of the time series. The holidays were kept in the time series, except those on weekends. The time series of electric load starts in January 2010 and ends in October 2013, consisting of 983 points as shown below. 26000

24000 22000 20000 18000 16000 14000 01/01/2010

20/07/2010

05/02/2011

24/08/2011

11/03/2012

27/09/2012

15/04/2013

Fig. 1. Eletric load serie in the metropolitan region without weekends.

1.2. Objective This study aims to compare two modeling for time series forecasting. The modelings are compared: a SARIMAX linear model and a Fuzzy Inference System. Both models are applied to predict the same time series of electric load from Bahia’s state metropolitan area. For comparison purposes, it is therefore proposed attempts in both models using different combinations of three external variables. Exogenous variables listed for both models were: temperature, precipitation of rainfall and the number of customers. The combinations are made as shown below: Table 1. Attempts Attempt 12

Attempt 13

Attempt 23

Attempt 123

Customers e Temperature

Customers e Rainfall

Temperature e Rainfall

Customers, Temperature and Rainfall

With the use of exogenous variables, the models seek to provide a forecasting step forward, using the day before’s. 1.3. Related Works Several models are used to forecast electricity. Mathematical models ARMA, ARIMA and SARIMAX and fuzzy inference systems (FIS) were found in several academic works, cited throughout this section. The most widely used statistical model is the autoregressive integrated moving average (ARIMA). In the energy sector this model is being used since 1978. The author [32] used it to predict the monthly occurrence of electric load through behavioral changes in economic and climatic variables.

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In 1999 the time series analysis model Box-Jenkins was used to forecast production and energy consumption. The countries were: Austria and Spain [4]. The authors [24] investigated different methodologies to perform the electricity forecast in Lebanon. One of the models was the ARIMA. In price forecast generation and consumption of electricity in Spain the authors [7] and [6] also used the ARIMA model. The study [21] and [25] used mathematical modeling techniques to predict the energy consumption behavior through weather conditions, working hours, days of the week and the time of day. The FIS has been used in time series forecasting models of the electricity sector in short term, half-hour series, hourly and daily. The works [13] and [21] are some examples. The researches [21] and [25] show that the energy consumption behavior in a short-term perspective, is sensitive to weather conditions, the part of the working day, the day of the week and hours of the day . For this reason, having a good load forecasting system, using these climatic variables associated with load is of paramount importance. The authors [26-28] proposed the use of Fuzzy systems in time series in their works. And the authors [17] proposed a fuzzy time series, where characterized as an ordered sequence of linguistic terms of time. The work proposed by the authors [1] uses a Fuzzy system by ANFIS for the annual energy forecast on industrialized countries like USA, UK, Japan, France and Italy. System input variables are the gross domestic product (GDP) and population. 2. Description of Modeling Techniques There are several different approaches to time series modeling. The most common time series model is the Box-Jenkins. The Box-Jenkins article named "Time Series Analysis: Forecasting and Control" was created in 1976 and has been used in numerous academic papers [2]. 2.1. Autoregressive Integrated Moving Average Model (ARIMA) From an operator difference it´s possible construct an ARIMA model, consisting of a variable autoregressive part, explained by the past behavior of the variable itself, a part of moving averages explained by prior period disorders, and another set to integration, where the number of lags (d) turns the series into a stationary series. According [29], the ARIMA (p, d, q) model can be written as follows: (1) ߘܼ௧ ൌ ሺͳ െ ‫ܤ‬ሻܼ௧ ൌ ‫ݓ‬௧ ݁ߦሺ‫ܤ‬ሻ ൌ ߶ሺ‫ܤ‬ሻߘ ௗ ൌ ߶ሺ‫ܤ‬ሻሺͳ െ ‫ܤ‬ሻௗ Where: ߘܼ௧ is the difference operator, ܼ௧ is an integral ‫ݓ‬௧ , ߘ is a type AR polynomial, ‫ݓ‬௧ is the difference ܼ௧ , φ (B) is a non-stationary autoregressive operator of order ‫ ݌‬൅ ݀, B is the delay operator and μ is the expectation ܻ௧ generally assumed to be 0. 2.2. Identification of models ARIMA (p, d e q) Several authors argue that to determine the order of terms p, d and q, the main tools to be used are the ACF, the PACF and their correlogram [11], [18] and [20]. The ACF of order k is a widely used tool in the Box-Jenkins models [8]. An autocorrelation coefficient ‫݌‬ଵ fits the correlation between the variable at time t for the period t-1 and lagged is called lag one autocorrelation coefficient and a correlation coefficient of self ‫݌‬௞ is said to lag autocorrelation coefficient k [22]. The ACF is a measure of the extent to which the value taken at time t depends on that taken at time ‫ݐ‬௞ [29]. So the optimal number of lags in the model can be determined based on the values of ACF [19].

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The ACF also aids in the analysis of time series, it is an important tool for the investigation of the empirical properties of the time series, such as: identification of stationary condition and identifying potential models to be used in the modeling and prediction of time series [11] and [20]. As the ACF, the PACF is also used for the identification of models to be used for the modeling and forecasting time series. The PACF can also help identify the stationary condition of the time series [11], [19] and [20]. According to [14] the partial correlation between two variables ‫ݕ‬௧ and ‫ݕ‬௧ି௞ is the simple correlation between them less linearly explained that part of the intermediate lags. 2.3. Seasonal Autoregressive Integrated Moving Average Model (SARIMA) The SARIMA formula contains a seasonal pattern in the first part of the formula, and a non-seasonal of the parameters in the second part of the formula [9]. As a natural extension, you can set the SARIMA model as [29]: ሺ‫ܤ‬ሻǤ ߶ሺ‫ ܤ‬௦ ሻǤ ߘ௦஽ Ǥ ߘ ௗ ܼ௧ ൌ ߶ሺ‫ܤ‬ሻǤ ߆‫ ܤ‬௦ Ǥ ܽ௧ (2) Where: Ԅሺሻ ൌ  ሺͳ െ Ԅଵ  െ ‫ ڮ‬െ Ԅ୮  ୮ ሻ is a non-seasonal autoregressive operator, ‫׏‬ୢ ൌ ሺͳ െ ሻୢ is an operator not seasonal difference order d, Ԅሺ ୱ ሻ ൌ ൫ͳ െ Ԅଵ  ୱ െ Ԅଶ  ଶୱ െ ‫ ڮ‬െ Ԅ୮  ୮ୱ ൯ is a seasonal ୱ ୈ ଶ autoregressive operator, ‫׏‬ୈ ୱ ൌ ሺͳ െ  ሻ is a seasonal operator of order D, Ʌሺሻ ൌ ሺͳ െ Ʌଵ  െ Ʌଶ  െ ‫ ڮ‬െ ୯ ୱ ୱ ଶୱ ୕ୱ Ʌ୯  ሻ is a non-seasonal operator of moving averages, Ʌሺ ሻ ൌ ሺͳ െ Ʌଵ  െ Ʌଶ  െ ‫ ڮ‬െ Ʌ୕  ሻ is a seasonal ୢ operator of moving averages, ‫׏‬ୈ ୗ Ǥ ‫ ୲ ׏‬ൌ ™୲ is a filter to the original series linear not applied ୲ that produces a steady ‫ݓ‬௧ and ܽ௧ process is a white noise. 2.4. Seasonal Autoregressive Integrated Moving Average with Exogenous Variables Model (SARIMAX) In order to improve the performance prediction of future values, another time serie can be incorporated in time series model [5]. The addition of an external input for a model is called using an exogenous variable. For example, the following is a common ARMA model with a single exogenous variable, ARMAX. ௣ ௤ (3) ܻ௧ ൌ σ௜ୀଵሺ߶௜ ܻ௧ି௜ ሻ ൅ σ௝ୀଵ൫ߠ௝ ܽ௧ି௝ ൯ ൅ σ௕௛ୀଵ ߚ௛ ܺ௧ି௛ ൅ ߱௧ Where ܺ௧ are the exogenous variables. 3. Fuzzy Inference System This chapter contains the Fuzzy Logic along with concepts of fuzzy sets, membership functions, linguistic variables and logical operations. The last subsection is intended for the fuzzy system proposed. 3.1. Fuzzy Lógic Fuzzy logic is a computational intelligence technique that aims to model the approximate way of thinking of human beings [3]. So it´s possible make decisions in an environment of uncertainty and imprecision [33], which does not occur with discrete logic. Fuzzy logic is formed from the set theory fuzzy. Formally, a fuzzy set universe of discourse is defined by a membership function, uA: U → [0,1]. A fuzzy set is a set without a clear boundary. In the classical form the combination of an element relative to a set of bit words is assumed. That is, the element can only belong or not to belong. The Membership Function (MF) is a curve that defines how each point in the input space is mapped to a membership value. The membership function must be between 0 and 1.

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It is very important to choose the type of membership function, because it influences directly in the Fuzzy system results. Therefore your choice depends on the context of the problem studied and user experience. A linguistic variable is one whose values are words instead of numbers [31]. In fuzzy logic, a linguistic variable is a variable whose values are names of fuzzy sets. The main function of the linguistic variables is a systematic way to provide a rough characterization of phenomena through linguistic description. This allows a better understanding of the problem being analyzed, allowing treatment systems that are too complex to be analyzed by conventional mathematical mechanisms [12]. 3.2. The Fuzzy System The Fuzzy systems are one of the most important areas of application of the Fuzzy Set Theory, the most widely used model based on the rules. They are an extension of the classical rule-based systems because they use rules such as "if-then" where the antecedents and consequences are fuzzy propositions [15] and [16]. There are two types of inference systems that vary according to the particular type of output: Mamdani and Sugeno. The Mamdani is used by the theory of fuzzy sets. Sugeno is similar in many respects to the Mamdani. The main difference is that the output membership functions Sugeno are either linear or constant [30]. The fuzzy rules "if-then", "or", "but" and "and" are rules whose output can be number or language. For example, "If x is A then y is B" as a Fuzzy point A x B [33]. The most usual structure of a fuzzy controlled by a process controller comprises a fuzzification interface, an inference unit, a defuzzification interface and a knowledge base, which in turn splits into Base and the Rule Base. 4. The Data Although there is no consensus on the best variables that describe the functioning of the energy distribution service [23], was opted for the use of three exogenous variables: the number of customers, precipitation of rain and temperature were used in both models. The data of climatological normals was provided by the National Institute of Meteorology of Brazil (INMET). And the measured data was provided by the company under study. 4.1. Number of Customers The variable "customers" is the amount of commercial or residential units connected to the eletric load. It was defined as the percentage variation of the input amount of consumers from among two consecutive days. The formula below shows this representation. (4) ‫ݔ‬ଵ ൌ ܿ௧ Τܿ௧ିଵ െ ͳ Where ܿ௧ is the instant in question e ܿ௧ିଵ is in the immediately preceding instant. 4.2. Rainfall For the formation of the variable, we opted for the difference of the daily precipitation in relation to the climatological normal. The formula below shows this representation. (5) ‫ݔ‬ଶ ൌ ‫ܥ‬௠ െ ‫ܥ‬ே஼ Where ‫ܥ‬௠ is the precipitation on the day in question, and ‫ܥ‬ே஼ is the climatological normal month's rainfall in a matter of Salvador.

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4.3. Temperature For the formation of variable, we used the difference in the value of daily temperature in relation to normal climatic. For use in models: made up the difference calculation of the maximum daily temperature measurement and the normal maximum climatological and the difference between the minimum daily temperature measurement and the minimum standard climatological each day. For thus, with the sum enters the differences maximum temperature and minimum temperature afforded a unique value of temperature sensitivity. The formula below shows this representation. (6) ‫ݔ‬ଷ ൌ ሺܶ௠ž௫ െ ܰ‫ܥ‬௠ž௫ ሻ ൅ ሺܶ௠À௡ െ ܰ‫ܥ‬௠À௡ ሻ Where: ܶ௠ž௫ e ܶ௠À௡ are the maximum and minimum temperatures measured on the day and ܰ‫ܥ‬௠ž௫ e ܰ‫ܥ‬௠À௡ are the climatological normal maximum and minimum temperature of the month in question in Salvador. 5. Experiments and Models Obtained Before presenting the proposed models it is important to formalize the form of assessment models. 5.1. Evaluation To evaluate the two algorithms the mean average percentage error (MAPE) and standard deviation (SD) were used. The MAPE measures, in absolute terms, the deviation from the original series with the output of the model series. The formula of MAPE is highlighted below: (7) ‫ ܧܲܣܯ‬ൌ ͳȀ݊ σ௡௧ୀଵȁሺ‫ܣ‬௧ െ ܻ௧ ሻΤ‫ܣ‬௧ ȁ Where ‫ܣ‬௧ is the original value and ܻ௧ is the value provided by the model. The SD is the difference between the original serie and the serie predicted by the model. It can be written using the formula below: (8) ܵ‫ ܦ‬ൌ  ඥσሺ‫ܣ‬௧ െ ܻഥ௧ ሻଶ Τሺ݊ െ ͳሻ From both evaluators in this section it is possible define the most accurate model for predicting the time series. 5.2. SARIMAX forecast model The following sections describe the SARIMAX model, with the OLS method. For SARIMAX time series analysis were used all 983 time-series data. The same 983 data were also used for testing. 5.2.1. Time Series Transformation. As the time series isn´t stationary, it is necessary to differ until it becomes stationary. As in the image below the ACF indicates a seasonality of 5 periods, and was also made the first difference into 5 periods.

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Fig. 2. ACF

At this point it is possible to analyze the series for the search of the SARIMAX parameters model. 5.2.2. Descriptive Analysis. In the ACF and PACF, shown in Figure 3, it was assumed the white noise hypothesis for the time series with the lower limit and the upper limit of 95% was assumed, the two dashed lines shown in the picture follow.

Fig. 3. ACF and PACF.

Following the description of the interpretability in Chapter 2: the ACF and PACF indicate the parameters p, P, q and Q. Therefore, it was assumed the displayed SARIMAX model in the following table: Table 2. The SARIMAX model p

d

q

P

D

Q

S

1

1

1

1

1

1

5

401

402

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Their parameters and their errors in 4 attemps applied are highlighted in the appendix. 5.3. Forecasting Model of Fuzzy Inference System This section describes the forecasting model of Fuzzy Inference System. The proposed model is an Inference System Adaptive Neuro-Fuzzy (ANFIS). The proposed system uses the Sugeno method with constant output. 30 was used as the number of epochs for fitting the model, which fits through a hybrid method. The hybrid method in question is a combination of the method of least squares and the backpropagation gradient descent method. To adjust the ANFIS parameter was used as default to input all the 983 data. As in SARIMAX model, the same 983 data was also used for testing. For the ANFIS it is optional to use the validation of the model. Set up four the Gaussian membership functions for each entry. For comparison purposes, as reported in SARIMAX was defined as input in the Fuzzy Inference System the eletric load serie of the previous day and the previous five days. We also used the same combinations of exogenous variables for modeling the Fuzzy System attempts. The table below highlights the MAPE’s to select the parameter. In bold the test that had the lowest MAPE. Table 3. Parameters of the Fuzzy Inference System MF

Attempt 12

Attempt 13

Attempt 23

Attempt 123

Gbellmf

2,25%

2,40%

2,33%

2,07%

Gaussmf

2,18%

2,31%

2,23%

1,99%

Gauss2mf

2,79%

2,99%

2,86%

2,63%

Pimf

3,04%

3,32%

3,11%

2,94%

dsigmf

2,32%

2,71%

2,51%

2,32%

6. Results Obtained by the Models and Conclusion This chapter presents the results and the differences between the attempts obtained using the different tools described in chapter 5. The results of SARIMAX model and Fuzzy System are presented in the table below with the different combinations of exogenous variables. Table 4. Results Attempts

Customers

Temperature

Attempt 12

X

X

Attempt 13

X

Attempt 23 Attempt 123

X

Rainfall

SARIMAX

SARIMAX

ANFIS

ANFIS

MAPE

SD

MAPE

SD

3,20%

964

2,18%

705

X

2,69%

867

2,31%

762

X

X

3,07%

919

2,23%

727

X

X

3,08%

919

1,99%

662

The result of the attempt 13 indicated in bold in table 4 highlights the best combination of exogenous variables for SARIMAX model. As the result of attempt 123 indicated in bold highlights is the best combination of

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403

exogenous variables of the Fuzzy Inference System. That is, the combinations that showed the lowest MAPE and the lowest SD. The attempt 13 is made from a combination of exogenous variable number of customers and rainfall. In attempt 13 in the SARIMAX model the MAPE is 2.69% and the SD is 867. The attempt 123 is made from a combination of all three exogenous variables: number of customers, temperature and rainfall. In attempt 123 in the Fuzzy Inference System the MAPE is 1.99%, and SD is 662. The table above highlights that the best combination of variables for the model SARIMAX was not the best combination of variables for Fuzzy System. The Fuzzy Inference System had the lowest error in the attempt 123 and the SARIMAX the attempt 13. The attempts results with Fuzzy Inference System showed the best results in all combinations of exogenous variables. It was concluded that, in general, in this work, using the time series of Coelba the metropolitan region in the two proposed models, there is no "best" exogenous variables to predict the daily amount of energy distributed in Bahia’s Metropolitan region. It can be said that the SARIMAX model applied the "best" combination of exogenous variables was the amount of customers and the precipitation of rain. And for the Fuzzy Inference System applied the "best" combination were all three exogenous variables. It is possible to say that the insertion SARIMAX model applied more "information" to the system, or another input variable in the system did not result in a minor error in prediction. Unlike the Fuzzy Inference System. The Fuzzy Inference System lost its interpretability since it generated a very large amount of rules. 256 rules being generated in the attempt 12, attempt 13 and attempt 23 and rules 1.024 in the attempt 123. To recover the interpretability another system is required to reduce the number of rules. Acknowledgements Gratitude to Iberdrola company for providing the data used in this paper. References [1] Azadeh V. N. A., Pazhouheshfar, P.; Saberic, M. An Adaptive-Network-Based Fuzzy Inference System for Long-Term Electric Consumption Forecasting (2008-2015): A Case Study of the Group of Seven (G7) Industrialized Nations: U.S.A, Canada, Germany, United Kingdom, Japan, France and Italy. IEEE. 2010. [2] Box, G. E. P.; Jenkins, G. M. Time Series Analysis: Forecasting and Control. San Francisco. Holden Day, 1970. [3] Campos, R. J.; Jesus, T. A.; Mendes, E. M. A. M. Uma abordagem fuzzy para a previsão de curto-prazo do consumo de energia elétrica. XXX Congresso Nacional de Matemática Aplicada (CNMAC), Brasil, Florianópolis, Setembro, 2007. [4] Chavez, G. S.; Bernat, J. X.; Coalla, H. L. Forecasting of energy production and consumption in Asturias (Northern Spain). Energy, 24, 183–98. 1999. [5] Christo, E. S.; Souza, R. C. Uma abordagem estatística para a previsão de potência reativa em sistemas elétricos. Pesquisa Oper. vol.26. número 2. Rio de Janeiro. Maio-Agosto. 2006 [6] Conejo, A. J.; Contreras, J.; Espínola, R.; Plazas, M. A. Forecasting electricity prices for a day-ahead pool-based electric energy market. International Journal of Forecasting 2005;21:435-65. [7] Cuaresma, J. C.; Hlouskova, J.; Kossmeier, S.; Obersteiner, M. Forecasting electricity spot-prices using linear univariate time series models. Applied Energy 2004;77:87-106. [8] Enders, W. Applied Econometric Time Series. 2 ed. United States of America: Ed. Wiley Series In Probability And Statistics, 2004, 466 p. ISBN 0-471-23065-0. [9] Espinosa, M. M.; Prado, S. M.; Ghellere, M. Uso do modelo SARIMA na previsão do número de focos de calor para os meses de junho a outubro no Estado de Mato Grosso. Departamento de Estatística/ICET. Universidade Federal de Mato Grosso, Cuiaba, MT, 2010 [11] Gujarati, D. N.; Econometria básica. 3 ed. São Paulo: Makron Books, 2000. [12] Gomide, F.; Gudwin, R.; Tanscheit, R. Conceitos fundamentais da teoria de conjuntos fuzzy, lógica fuzzy e aplicações. 6th IFSA Congress-Tutorials. 1995 [13] Gooijer, de; Jan,G.; Hydman, R. J. 25 years of time series forecasting.International Journal of Forecasting 2006;22(3):443-473.

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[14] Greene, Willian H.Econometric analysis. 5 ed. New Jersey: Prentice Hall, 2003. [15] Herrera, F. Genetic fuzzy systems: taxonomy, current research trends and prospects. Evolutionary Intelligence 2008;1(1):27-46. [16] Jang, J. S. R.; Gulley, N. The Fuzzy Logic Toolbox for use with MATLAB. The MathWorks, Inc., Natick, Massachusetts, 1995. [17] Li, S.T.; Cheng, Y. C. A Stochastic HMM-Based Forecasting Model for Fuzzy Time Series. IEE Transactions on Systems, Man, and Cybernetics – Part B: Cybernetic 2010;40(5). [18] Liu, L; Hudak, G. P. Forecastingand Time Series AnalysisUsing the SCA (Statistical System.Scientific Computing Associated), 1994. [19] Matos, O. C. Econometria básica: teoria e aplicações. São Paulo: Atlas, 2000. [20] Morettin, Pedro A.. Econometria Financeira: um curso de séries temporais financeiras. ABE: São Paulo, 2008 [21] Pandian, S. C.; Duraiswamy, K.; Rajan, C. C. A.; Kanagarajn N. Fuzzy approach for short term load forecasting. Electric Power Systems Research 2006;76:541-548. [22] Pellegrini, F.R. Metodologia para implementação de sistemas de previsão de demanda. Master's thesis in Production Engineering. Porto Alegre: UFRGS, 2000. [23] Pinheiro, T. M. M. Distribuidoras de Energia Elétrica no Brasil. Dissertação de mestrado. Universidade de Brasília, 2012, available at: http://www.aneel.gov.br/biblioteca/trabalhos/trabalhos/Dissertacao_Thelma_Pinheiro.pdf [24] Saab, S.; Badr, E.; Nasr, G. Univariate modeling and forecasting of energy consumption: the case of electricity in Lebanon. Energy 2001;26:1-14. [25] Serrão, F. C. C. Modelo de previsão de carga de curto-prazo utilizando redes neurais e lógica fuzzy. Master's thesis. Pontifica Universidade Católica, Brasil, Rio de Janeiro, 2003. [26] Song, Q.; Chissom, B. S. Forecasting enrollments with fuzzy time series - Part I, Fuzzy Sets Syst. 1993;54:1-9. [27] Song, Q.; Chissom, B. S. Fuzzy time series and its models, Fuzzy Sets Syst. 1993;54:269-277. [28] Song, Q.; Chissom, B. S. Forecasting enrollments with fuzzy time series - Part II, Fuzzy Sets Syst. 1994;62:1-8. [29] Souza, R. C.; Camargo, M. E. Análise e previsão de séries temporais: os modelos ARIMA. SEDIGRAF, 1996. [30] Sugeno, M. Industrial applications of fuzzy control. Elsevier Science Pub. Co., 1985 [31] Tanscheit, R. Sistemas fuzzy, In: Inteligência computacional aplicada a administração, economia e engenharia em Matlab, São Paulo, Thomson Learning, 2007,p.229-264,. [32] Uri, N. D. Forecasting peak system load using a combined time series and econometric model. Applied Energy 1978;4:219-227. [33] Zadeh, L. A. Fuzzy sets. Fuzzy sets, Information and Contro 1965;8:338-353.

Appendix A. The SARIMAX parameters and their errors in 4 attempts applied are highlighted below. Since the parameter values the principal and his error in brackets: Table 5.Parameters of the SARIMAX model Parameters

Attempt 12

Attempt 13

Attempt 23

Attempt 123

AR

0,054 (0,032)

0,561 (0,032)

0,491 (0,032)

0,491 (0,034)

SAR

0,022 (0,034)

0,082 (0,034)

0,081 (0,034)

0,081 (0,032)

MA

-0,480 (0,033)

-0,928 (0,033)

-0,937 (0,033)

-0,937 (0,014)

SMA

-0,999 (0,001)

-1,000 (0,001)

-1,000 (0,001)

-1,000 (0,000)

Constant

20.836,10 (-)

21.093,33 (-)

20.840,38 (-)

20.840,01 (-)

Customers

5.598,59 (1,410)

3.278,408 (0,000)

-

5.697,09 (0,00)

Temperature

282.055 (24,294)

-

287,472 (28,513)

287,533 (28,514)

Rainfall

-

28,057 (3,747)

-3,616 (5,822)

-3,631 (5,822)